1 Introduction

Throughout this paper, let \({\mathbb {R}}\) denote the set of all real numbers and \({\mathbb {N}}\) denote the set of all positive numbers. Let H be a real Hilbert space with inner product \(\langle \cdot , \cdot \rangle \) and induced norm \(||\cdot ||,\) and let C be a nonempty closed convex subset of H. Let \(S: C\rightarrow C\) be a nonlinear mapping. A point \({\hat{x}}\in C\) is called a fixed point of S if \(S{\hat{x}} = {\hat{x}}.\) We denote by F(S),  the set of all fixed points of S,  i.e.

$$\begin{aligned} F(S) = \{{\hat{x}}\in C: S{\hat{x}} = {\hat{x}}\}. \end{aligned}$$
(1.1)

A mapping \(S: C\rightarrow C\) is called a nonexpansive mapping if

$$\begin{aligned} ||Sx-Sy||\le ||x-y||\quad \forall ~ x,y\in C. \end{aligned}$$

The study of fixed point theory for nonexpansive mappings has flourished in recent years due to its vast applications in fields like economics, compressed sensing, and other applied sciences. In particular, certain problems such as variational inequalities problems, convex optimization problems, convex feasibility problems, monotone inclusion problems and image restoration problems can be formulated as finding the fixed points of nonexpansive mapping (see Bauschke and Borwein 1996; Chen et al. 2013). Several researchers have put considerable efforts in the study and formulation of iterative methods to approximate the fixed points of nonexpansive mappings (for example, see Halpern 1967; Moudafi 2000 and the reference therein).

In 2007, Agarwal et al. (2007) introduced the following iterative method known as the S-iteration. Let C be a convex subset of a linear space X and S a mapping of C into itself. The sequence \(\{x_n\}\) is generated as follows:

figure a

where \(\{\alpha _n\}\) and \(\{\beta _n\}\) are sequences in (0, 1). The authors showed that the S-iteration process has a better rate of convergence than Mann and Ishikawa iteration processes.

Some authors have used the S-iteration process and its modifications to find common fixed points of two mappings; see for example (Bussaban and Kettapun 2018; Pandey et al. 2019) and the references therein.

Let \( F:C\times C\rightarrow {\mathbb {R}}\) be a bifunction. The equilibrium problem (EP) for the bifunction F on C is formulated as finding a point \({\hat{x}}\in C\) such that

$$\begin{aligned} F({\hat{x}}, y) \ge 0,\quad \forall y\in C. \end{aligned}$$
(1.2)

The solution set of EP (1.2) is denoted by EP(F). The EP covers a wide range of topics that have emerged from the social sciences, economics, finance, image restoration, ecology, transport, networking, elasticity and optimization problems (see Olona et al. 2021a; Patriksson 2015). The problem is a generalized concept that unifies several mathematical problems as special cases, namely minimization problems, variational inequality problems, mathematical programming problems, complementarity problems, saddle point problems, Nash equilibrium problems in noncooperative games, minimax inequality problems, fixed point problems, scalar and vector minimization problems, and others; see (Alakoya et al. 2021; Blum 1994) and the references therein. Recently, the EP and its various generalizations have attracted considerable research efforts and various iterative methods have been proposed for approximating their solutions (see Alakoya et al. 2021; Jolaoso et al. 2020; Ogwo et al. 2021; Olona et al. 2021b; Oyewole et al. 2021; Taiwo et al. 2021a, b and the references therein).

In the recent time, the Split Inverse Problem (SIP) has attracted the attention of several authors (see Alakoya et al. 2021; Oyewole et al. 2021; Taiwo et al. 2021 and the references therein) due to its wide areas of applications, for example, in phase retrieval, image recovery, signal processing, data compression, intensity-modulated radiation therapy, among others (see Censor et al. 2006; Censor and Elfving 1994 and the references therein). The SIP model is formulated as follows: Find a point

$$\begin{aligned} {\hat{x}}\in H_1 \quad \text {that solves IP}_1 \end{aligned}$$
(1.3)

such that

$$\begin{aligned} {\hat{y}} := A{\hat{x}}\in H_2 \quad \text {solves IP}_2, \end{aligned}$$
(1.4)

where \(H_1\) and \(H_2\) are real Hilbert spaces, \(\hbox {IP}_1\) denotes an inverse problem formulated in \(H_1\) and \(\hbox {IP}_2\) denotes an inverse problem formulated in \(H_2,\) and \(A : H_1 \rightarrow H_2\) is a bounded linear operator.

Censor and Elfving (1994) introduced the first instance of the SIP called the split feasibility problem (SFP) in 1994 for modelling inverse problems which arise from medical image reconstruction. Since then, the SFP has been studied intensively by several authors due to its wide areas of application such as in signal processing, control theory, approximation theory, geophysics, biomedical engineering, communications, etc (Byrne 2002; Censor et al. 2006; Godwin et al. 2020). Let C and Q be nonempty closed convex subsets of Hilbert spaces \(H_1\) and \(H_2,\) respectively, and \(A:H_1\rightarrow H_2\) be a bounded linear operator. The SFP is formulated as follows:

$$\begin{aligned} \text {find a point}~~ {\hat{x}}\in C~~ \text {such that}~~ {\hat{y}}= A{\hat{x}}\in Q. \end{aligned}$$
(1.5)

Moudafi (2011) introduced another instance of the SIP known as the split monotone variational inclusion problem (SMVIP). Let \(H_1, H_2\) be real Hilbert spaces, \(f_1:H_1\rightarrow H_1, f_2:H_2\rightarrow H_2,\) are inverse strongly monotone mappings, \(A:H_1\rightarrow H_2\) is a bounded linear operator, \(B_1:H_1\rightarrow 2^{H_1}, B_2:H_2\rightarrow 2^{H_2}\) are multivalued maximal monotone mappings. The SMVIP is formulated as follows:

$$\begin{aligned} \text {find a point}~ {\hat{x}}\in H_1~ \text {such that}~ 0\in f_1({\hat{x}}) + B_1({\hat{x}}) \end{aligned}$$
(1.6)

and

$$\begin{aligned} {\hat{y}}=A{\hat{x}}\in H_2~ \text {such that}~ 0\in f_2({\hat{y}}) + B_2({\hat{y}}). \end{aligned}$$
(1.7)

If \(f_1\equiv 0\equiv f_2,\) then the SMVIP (1.6)-(1.7) reduces to the following split variational inclusion problem (SVIP):

$$\begin{aligned} \text {find a point}~ {\hat{x}}\in H_1~ \text {such that}~ 0\in B_1({\hat{x}}) \end{aligned}$$
(1.8)

and

$$\begin{aligned} {\hat{y}}=A{\hat{x}}\in H_2~ \text {such that}~ 0\in B_2({\hat{y}}). \end{aligned}$$
(1.9)

The SVIP (1.8)-(1.9) constitutes a pair of variational inclusion problems which have to be solved so that the image \({\hat{y}} = A{\hat{x}}\) under a given bounded linear operator A of the solution of the SVIP (1.8) in \(H_1\) is the solution of the other SVIP (1.9) in another Hilbert space \(H_2.\) Moudafi (2011), showed that the SVIP (1.8)-(1.9) includes as a special case the SFP (1.11). The SVIP is at the core of modelling many inverse problems arising from phase retrieval and other real world problems, for instance, in sensor networks in computerized and data compression (Byrne 2002; Combettes 1996). We denote the solution set of SVIP (1.8) by SOLVIP(\(B_1\)) while the solution set of SVIP (1.9) is denoted by SOLVIP(\(B_2\)). Hence, the solution set of the SVIP (1.8)–(1.9) is denoted by

$$\begin{aligned} {\mathcal {F}} = \{x^*\in H_1: x^*\in \text {SOLVIP(}B_1)~~ \text {and}~~ Ax^*\in \text {SOLVIP(}B_2)\}. \end{aligned}$$
(1.10)

To solve the SVIP for two maximal monotone operators \(B_1\) and \(B_2\) in Hilbert spaces, Byrne et al. (2012) proposed the following algorithm:

figure b

for \(\lambda >0\) and \(A^*\) is the adjoint operator of the bounded linear operator \(A, \gamma \in (0, \frac{2}{L}), L = ||A^*A||, J_\lambda ^{B_1}:= (I +\lambda B_1)^{-1}, J_\lambda ^{B_2}:= (I +\lambda B_2)^{-1}\) are the resolvent operators of \(B_1\) and \(B_2\) respectively, and \(\{\alpha _n\}\) is a sequence in [0, 1] satisfying \(\lim _{n\rightarrow \infty }\alpha _n=0\) and \(\sum _{n=1}^{\infty }\alpha _n=\infty .\) The authors obtained a strong convergence result for the proposed algorithm under some mild conditions.

In the recent time, the problems of finding common solutions of the set of fixed points of nonlinear mappings and the set of solutions of optimization problems have been considered by some authors (for instance, see Alakoya et al. 2021; Cholamjiak and Suantai 2013; Khan et al. 2020 and the references therein). The importance and motivation for studying such a common solution problem lies in its potential application to mathematical models whose constraints can be expressed as fixed point problems and optimization problems. In this article we will be studying the problems of finding common solutions of the set of fixed points of nonlinear mappings and the set of solutions of certain optimization problems.

Recently, Wangkeeree et al. (2018) introduced the following general iterative scheme for approximating a common solution of SVIP and FPP for a nonexpansive mapping in the setting of real Hilbert spaces.

figure c

where \(f: H_1\rightarrow H_1\) is a contraction with constant \(k\in (0,1),~ S: H_1\rightarrow H_1\) is a nonexpansive mapping, \(D: H_1\rightarrow H_1\) is a strongly positive bounded linear operator with constant \(\mu \) and \(0<\beta <\frac{\mu }{k}, \lambda >0, \gamma \in (0, \frac{1}{L}),\) where L is the spectral radius of the operator \(A^*A,\) \(\{\alpha _n\}\subset (0,1)\) and \(B_1: H_1\rightarrow 2^{H_1}, B_2: H_2\rightarrow 2^{H_2}\) are two multi-valued maximal monotone operators on \(H_1\) and \(H_2\) respectively. Under certain conditions, the sequence generated by the proposed Algorithm 1.3 was proved to converge strongly to a common solution of split variational inclusion problem and fixed point problem for a nonexpansive mapping.

Remark 1.4

Here, we remark that the step size \(\gamma \) of Algorithms 1.2 and 1.3 above plays an essential role in the convergence properties of the algorithms. Many of the existing iterative methods for solving SVIP involve step size that depends on the norm of the bounded linear operator A. Such algorithms are usually not easy to implement because they require computation of the operator norm which oftentimes is difficult to compute. In addition, the step size defined by these methods are often very small and deteriorates the convergence rate of the algorithm. A larger step size can often be used in practice to yield better numerical results.

Very recently, Tang (2020) proposed the following Halpern-type algorithm with self-adaptive step size for solving SVIP in the framework of Hilbert spaces.

figure d

where \(g(x) = \frac{1}{2}||(I - J_\lambda ^{B_2})Ax||^2, G(x) = A^*(I - J_\lambda ^{B_2})Ax, H(x) = (I - J_\lambda ^{B_1})x,\) and \(\{\alpha _n\}\) is a sequence in (0, 1) satisfying \(\lim _{n\rightarrow \infty }\alpha _n=0\) and \(\sum _{n=1}^{\infty }\alpha _n=\infty .\) Under mild conditions, strong convergence result was obtained for the proposed algorithm.

Based on the heavy ball methods of a two-order time dynamical system, Polyak (1964) first introduced an inertial extrapolation as an acceleration process to solve the smooth convex minimization problem. The inertial algorithm is a two-step iteration where the next iterate is defined by making use of the previous two iterates. In the recent time, there has been an increasing interest in the study of inertial type algorithms and several authors have constructed some fast iterative methods by employing the inertial technique (see, e.g., Alakoya et al. 2021, 2020, 2021; Izuchukwu et al. 2020; Ogwo et al. 2021; Owolabi et al. 2021).

For approximating the zero point of a maximal monotone operator B,  Alvarez and Attouch (2001) introduced the following inertial proximal algorithm:

figure e

The authors obtained a weak convergence result for the algorithm under the following conditions:

  1. (B1)

       There exists \(\lambda >0\) such that for all \(n\in {\mathbb {N}}, ~ \lambda _n\ge \lambda .\)

  2. (B2)

       There exists \(\alpha \in [0,1)\) such that for all \(n\in {\mathbb {N}},~ 0\le \alpha _n\le \alpha .\)

  3. (B3)

       \(\sum _{n=1}^\infty \alpha _n\Vert x_n - x_{n-1}\Vert ^2< \infty .\)

Recently, authors have pointed out one of the drawbacks of the summability condition (B3) of the Algorithm 1.6, that is, to satisfy the summability condition, it is necessary to first calculate \(\alpha _n\) at each step (see Moudafi and Oliny 2003).

Very recently, Shehu et al. (2021), Iyiola et al. (2018), Shehu et al. (2020), and Shehu and Iyiola (2020) proposed some efficient inertial iterative methods with self-adaptive step size for approximating solutions of certain classes of optimization problems and the authors were able to establish convergence of the proposed methods under some mild conditions imposed on the control parameters.

From the above review, the following natural question arises:

Question: Can we construct an inertial iterative method with self-adaptive step size for approximating a common solution of split variational inclusion problem, equilibrium problem and fixed point problem in Hilbert spaces such that condition (B3) of Algorithm  1.6 is dispensed with?

Our interest in this paper is to provide an affirmative answer to the above question.

Inspired by the above results and the ongoing research interest in this direction, in this paper, we introduce a new inertial iterative scheme which employs the viscosity S-iteration technique with self-adaptive step size for approximating a common element of the set of solutions of split variational inclusion problem, equilibrium problem and common fixed point problem for nonexpansive mappings in Hilbert spaces. Our motivation for studying such a common solution problem lies in its potential application to mathematical models whose constraints can be expressed as split variational inclusion problem, equilibrium problem and common fixed point problem. This arises in practical problems such as signal processing, network resource allocation, image recovery. A scenario is in network bandwidth allocation problem for two services in a heterogeneous wireless access networks in which the bandwidth of the services are mathematically related (see, for instance, Iiduka 2012; Luo et al. 2009 and the references therein). Unlike in Algorithms 1.2 and 1.3 and several other algorithms in the literature, our algorithm is designed such that its implementation does not require the knowledge of the norm of the bounded linear operator. Moreover, our work extend the results in Agarwal et al. (2007), Wangkeeree et al. (2018), Tang (2020), Alvarez and Attouch (2001) to the problem of finding a common solution of split variational inclusion problem, equilibrium problem and common fixed point problem for nonexpansive mappings and the inertial technique employed is more efficient than that used in Alvarez and Attouch (2001). Under some mild conditions, we prove strong convergence theorem for the proposed algorithm. Furthermore, we apply our result to study other optimization problems and we provide some numerical experiments with graphical illustrations to demonstrate the efficiency of the proposed algorithm in comparison with some existing state of the art algorithms in the literature.

The outline of the paper is as follows: In Sect. 2, we recall some basic definitions and existing results which are needed for the convergence analysis of the proposed algorithm. In Sect. 3, we present the proposed algorithm and highlight some of its important features while in Sect. 4 we discuss its convergence. In Sect. 5 we apply our results to study split minimization problem and split feasibility problems. In Sect. 6, numerical examples and comparison with some related algorithms are presented to demonstrate the performance of our new algorithm. Finally, we give the concluding remarks in Sect. 7.

2 Preliminaries

In this section, we recall some concepts and results which will be employed in the sequel. Let H be a real Hilbert space, for a nonempty closed and convex subset C of H, the metric projection \(P_C: H\rightarrow C\) is defined, for each \(x\in H,\) as the unique element \(P_Cx\in C\) such that

$$\begin{aligned} ||x - P_Cx|| = \inf \{||x-z||: z\in C\}. \end{aligned}$$

It is known that \(P_C\) is firmly nonexpansive, i.e.,

$$\begin{aligned} ||P_Cx - P_Cy||^2 \le \langle P_Cx - P_Cy, x-y \rangle , \end{aligned}$$
(2.1)

for all \(x,y\in H.\) Moreover, for any \(x\in H\) and \(z\in C, z=P_Cx\) if and only if (see [?])

$$\begin{aligned} \langle x - z, z - y \rangle \ge 0 \quad \forall ~ y\in C. \end{aligned}$$
(2.2)

In what follows, we denote the weak and strong convergence of a sequence \(\{x_n\}\) to a point \(x \in H\) by \(x_n \rightharpoonup x\) and \(x_n \rightarrow x\), respectively and \(w_\omega (x_n)\) denotes set of weak limits of \(\{x_n\},\) that is,

$$\begin{aligned} \omega _w(x_n):= \{x\in H: x_{n_j}\rightharpoonup x~ \text {for some subsequence}~ \{x_{n_j}\}~ \text {of} ~\{x_{n}\}\}. \end{aligned}$$

Definition 2.1

Let H be a real Hilbert space, \(C\subset H\) be a subset of H and \(h:C\rightarrow H\) be an operator from C onto H. The operator h is said to be

  1. (1)

    firmly nonexpansive if

    $$\begin{aligned} \langle h(x)-h(y),x-y \rangle \ge ||h(x)-h(y)||^2,\quad \forall x,y\in C; \end{aligned}$$
  2. (2)

    L-Lipschitz continuous, where \(L>0,\) if

    $$\begin{aligned} ||hx - hy|| \le L||x-y||,\quad \forall ~~x,y\in C; \end{aligned}$$

    if \(L\in [0,1),\) then T is called a contraction mapping;

  3. (3)

    nonexpansive if T is \(1-\)Lipschitz continuous;

  4. (4)

    hemicontinuous if it is continuous along each line segment in C.

Lemma 2.2

(Zhao et al. 2018) (Demiclosedness Principle). Let T be a nonexpansive mapping on a closed convex subset C of a real Hilbert space H. Then \(I-T\) is demiclosed at any point \(y\in H,\) that is, if \(x_n\rightharpoonup x\) and \(x_n-Tx_n\rightarrow y\in H,\) then \(x-Tx=y.\)

Definition 2.3

Let H be a real Hilbert space. A function \(f:H\rightarrow {\mathbb {R}}\cup \{+\infty \}\) is said to be weakly lower semicontinuous (w-lsc) at \(x\in H,\) if

$$\begin{aligned} f(x)\le \liminf _{n\rightarrow \infty } f(x_n) \end{aligned}$$

holds for an arbitrary sequence \(\{x_n\}_{n=0}^\infty \) in H satisfying \(x_n\rightharpoonup x.\)

Definition 2.4

Let H be a real Hilbert space and \(\lambda >0.\) The operator \(B: H\rightarrow 2^{H}\) is said to be

  • monotone if

    $$\begin{aligned} \langle u -v, x-y \rangle \ge 0~ \text {for all}~ u\in B(x), v\in B(y). \end{aligned}$$

  • maximal monotone mapping if the graph G(B) of B, 

    $$\begin{aligned} G(B) := \{(x,u)\in H \times H| u\in B(x)\}, \end{aligned}$$

    is not properly contained in the graph of any other monotone mapping.

  • The resolvent of B with parameter \(\lambda > 0\) denoted by \(J_\lambda ^B\) is given by

    $$\begin{aligned} J_\lambda ^B := (I + \lambda B)^{-1}, \end{aligned}$$

    where I is the identity operator.

Remark 2.5

For \(\lambda > 0,\) the following results hold (Tang 2020):

  1. (1)

    B is maximal monotone if and only if \(J_\lambda ^B\) is single-valued, firmly nonexpansive and \(dom(J_\lambda ^B) = H,\) where \(dom(B): = \{x\in H| B(x) \ne \emptyset \}.\)

  2. (2)

    The point \(x^*\in B^{-1}(0)\) if and only if \(x^* = J_\lambda ^Bx^*.\)

  3. (3)

    The solution set \({\mathcal {F}}\) of the SVIP (1.8)-(1.9) is equivalent to the following:

    $$\begin{aligned} \text {Find}~~ x^*\in H_1~~ \text {with}~~ x^* = J_\lambda ^{B_1}x^*~~ \text {such that}~~ y^* = Ax^*\in H_2 ~\text {and}~ y^* = J_\lambda ^{B_2}y^* \end{aligned}$$
    (2.3)

Assumption 2.6

For solving the EP, we assume that the bifunction \(F: C \times C \rightarrow {\mathbb {R}}\) satisfies the following conditions:

  1. (A1)

    \(F(x,x) = 0\) for all \(x \in C;\)

  2. (A2)

    F is monotone, that is, \(F(x,y) + F(y,x) \le 0\) for all \(x,y \in C;\)

  3. (A3)

    F is upper hemicontinuous, that is, for all \( x,y,z \in C\), \(\lim _{t \downarrow 0} F\big (tz + (1-t)x,y \big )\le F(x,y);\)

  4. (A4)

    for each \(x \in C, y \mapsto F(x,y)\) is convex and lower semicontinuous.

Lemma 2.7

Ma et al. (2015) Let C be a nonempty closed convex subset of a Hilbert space H and \(F:C \times C \rightarrow {\mathbb {R}}\) be a bifunction satisfying Assumption 2.6. For \(r > 0\) and \(x \in H,\) define a mapping \( T^F_r : H \rightarrow C\) as follows:

$$\begin{aligned} T^F_r (x) = \{z \in C : F(z,y) + \frac{1}{r} \langle y - z, z - x \rangle \ge 0, ~~ \forall ~ y \in C \}. \end{aligned}$$
(2.4)

Then \(T^F_r\) is well defined and the following hold:

  1. (1)

    for each \(x \in H, T^F_r(x) \ne \emptyset ;\)

  2. (2)

    \(T^F_r\) is single-valued;

  3. (3)

    \(T^F_r\) is firmly nonexpansive, that is, for any \(x,y \in H,\)

    $$\begin{aligned} \Vert T^F_r x - T^F_r y\Vert ^2 \le \langle T^F_r x - T^F_r y, x - y \rangle ; \end{aligned}$$
  4. (4)

    \(F(T^F_r) = EP(F);\)

  5. (5)

    EP(F) is closed and convex.

Lemma 2.8

Chuang (2013), Ogwo et al. (2021) Let H be a real Hilbert space. Then the following results hold for all \(x,y\in H\) and \(\delta \in {\mathbb {R}}:\)

  1. (i)

    \(||x + y||^2 \le ||x||^2 + 2\langle y, x + y \rangle ;\)

  2. (ii)

    \(||x + y||^2 = ||x||^2 + 2\langle x, y \rangle + ||y||^2;\)

  3. (iii)

    \(||x - y||^2 = ||x||^2 - 2\langle x, y \rangle + ||y||^2.\)

  4. (iv)

    \(||\delta x + (1-\delta ) y||^2 = \delta ||x||^2 + (1-\delta )||y||^2 -\delta (1-\delta )||x-y||^2.\)

Lemma 2.9

Maingé (2007) Let \(\{a_n\}, \{c_n\}\subset \mathbb {R_+}, \{\sigma _n\}\subset (0,1)\) and \(\{b_n\}\subset {\mathbb {R}}\) be sequences such that

$$\begin{aligned} a_{n+1}\le (1-\sigma _n)a_n + b_n + c_n~~ \text {for all}~ n\ge 0. \end{aligned}$$

Assume \(\sum _{n=0}^{\infty }|c_n|<\infty .\) Then the following results hold:

  1. (1)

       If \(b_n\le \beta \sigma _n\) for some \(\beta \ge 0,\) then \(\{a_n\}\) is a bounded sequence.

  2. (2)

       If we have

    $$\begin{aligned} \sum _{n=0}^\infty \sigma _n = \infty ~~ \text {and}~~ \limsup _{n\rightarrow \infty }\frac{b_n}{\sigma _n}\le 0, \end{aligned}$$

    then \(\lim _{n\rightarrow \infty }a_n =0.\)

Lemma 2.10

Saejung and Yotkaew (2012) Let \(\{a_n\}\) be a sequence of non-negative real numbers, \(\{\alpha _n\}\) be a sequence in (0, 1) with \(\sum _{n=1}^\infty \alpha _n = \infty \) and \(\{b_n\}\) be a sequence of real numbers. Assume that

$$\begin{aligned} a_{n+1}\le (1 - \alpha _n)a_n + \alpha _nb_n, ~~~ \text {for all}~~ n\ge 1, \end{aligned}$$

if \(\limsup _{k\rightarrow \infty }b_{n_k}\le 0\) for every subsequence \(\{a_{n_k}\}\) of \(\{a_n\}\) satisfying \(\liminf _{k\rightarrow \infty }(a_{n_{k+1}} - a_{n_k})\ge 0,\) then \(\lim _{n\rightarrow \infty }a_n =0.\)

3 Proposed method

In this section, we present the proposed algorithm. First, we define the following functions:

$$\begin{aligned} g(x) = \frac{1}{2}||(I - J_{\lambda _2}^{B_2})Ax||^2,\quad \quad h(x) = \frac{1}{2}||(I - J_{\lambda _1}^{B_1})x||^2 \end{aligned}$$

and

$$\begin{aligned} G(x) = A^*(I - J_{\lambda _2}^{B_2})Ax, \quad \quad H(x) = (I - J_{\lambda _1}^{B_1})x. \end{aligned}$$

From Aubin (1993), it can easily be verified that g and h are weak lower semi-continuous and convex differentiable. Moreover, G and H are Lipschitz continuous (see Tang 2020).

In what follows, we assume that C and Q are nonempty closed convex subsets of real Hilbert spaces \(H_1\) and \(H_2,\) respectively, \(B_1 : H_1 \rightarrow 2^{H_1}\) and \(B_2 : H_2 \rightarrow 2^{H_2}\) are maximal monotone mappings, \(F:C\times C\rightarrow {\mathbb {R}}\) is a bifunction satisfying Assumption 2.6, and \(A : H_1 \rightarrow H_2\) is a bounded linear operator with \(A^*\) being its adjoint operator (\(A^*=A^T\) in finite dimensional spaces). Let \(S,T:H_1\rightarrow H_1\) be nonexpansive mappings and \(f:H_1\rightarrow H_1\) be a contraction with coefficient \(k\in (0,1).\) We denote the solution set by \(\Omega = F(S)\cap F(T)\cap {\mathcal {F}} \cap EP(F) \ne \emptyset \). We establish the convergence of the algorithm under the following conditions on the control parameters:

  1. (C1)

    Let \( \{\alpha _n\} \subset (0,1)\) such that \(\lim _{n\rightarrow \infty }\alpha _n=0\) and \(\sum _{n=0}^\infty \alpha _n = \infty ;\)

  2. (C2)

    Let \(\{\beta _n\}, \{\sigma _n\}, \{\delta _n\}, \{\xi _n\}\subset [a,b], a,b\in (0,1)\) and such that \(\alpha _n+\delta _n+\xi _n=1;\)

  3. (C3)

    Let \(\theta >0, \{\mu _n\}\) be a positive sequence such that \(\lim _{n\rightarrow \infty }\frac{\mu _n}{\alpha _n}=0;\)

  4. (C4)

    \(0< a \le \rho _n \le b < 4, \{r_n\}\subset (0,\infty )\) such that \(\liminf _{n\rightarrow \infty }r_n>0,\) and \(\lambda _i>0,~~ i=1,2.\)

Now, our main algorithm is presented as follows:

figure f

Remark 3.2

By conditions (C1) and (C3), one can easily verify from (3.1) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\theta _n||x_n - x_{n-1}|| = 0 ~~ \text {and}~~ \lim _{n\rightarrow \infty }\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}|| = 0. \end{aligned}$$

Remark 3.3

Observe that the step size of our proposed Algorithm (3.1) is constructed such that its implementation does not require knowledge of the operator norm. Moreover, the implementation of the inertial term does not require the very stringent summability condition (B3) of Algorithm 1.6 (Alvarez and Attouch 2001). These important features make our proposed method easily implementable.

Remark 3.4

We remark that since the split variational inclusion problem generalizes the split feasibility problem and variational inequality problem, our proposed method could be viewed as an extension of the methods proposed in Shehu and Ogbuisi (2015), Cai et al. (2018), Gibali and Shehu (2019).

Remark 3.5

Observe that according to Lemma 2.7, the operator defined in (2.4) is single-valued. To compute \(u_n\) in (3.2) is equivalent to evaluating \(T^F_{r_n} (w_n).\) In doing this, we find an element \(u_n\in C\) such that inequality (3.2) holds. By applying the given definition of \(F(u_n,y)\) and simplifying the resulting inequality, we will obtain a quadratic function in variable y. Since the operator \(T^F_{r_n}\) is single-valued, then the quadratic function will have at most one solution in \({\mathbb {R}}.\) Hence, the value of \(u_n\) for which the discriminant of the quadratic function is zero is determined and this gives the value of the operator \(T^F_{r_n}\) at \(w_n.\)

4 Convergence analysis

Next, we state the strong convergence theorem for the proposed algorithm as follows.

Theorem 4.1

Let \(H_1\) and \(H_2\) be real Hilbert spaces, and \(A:H_1\rightarrow H_2\) be a bounded linear operator with adjoint \(A^*.\) Suppose \(S,T:H_1\rightarrow H_1\) are nonexpansive mappings, and \(f:H_1\rightarrow H_1\) is a contraction with coefficient \(k\in (0,1).\) Let \(\{x_n\}\) be the sequence generated by Algorithm 3.1 such that conditions (A1)-(A4) and (C1)-(C4) are satisfied. Then \(\{x_n\}\) converges strongly to a point \({\hat{x}}\in \Omega ,\) where \({\hat{x}}=P_\Omega \circ f({\hat{x}}).\)

We divide the proof of the strong convergence theorem into the following lemmas.

Lemma 4.2

Suppose that \(\{x_n\}\) is the sequence generated by Algorithm 3.1. Then \(\{x_n\}\) is bounded.

Proof

Observe that by (3.2), \(u_n=T_{r_n}^Fw_n.\) Also, we note that the mapping \(P_\Omega \circ f\) is a contraction. Then by the Banach Contraction Principle, there exists \(p\in H_1\) such that \(p=P_\Omega \circ f(p)\) and in particular \(p\in \Omega .\) Thus, it follows that \(Sp=p=Tp, T_{r_n}p=p, J_{\lambda _1}^{B_1}p=p\) and \(J_{\lambda _2}^{B_2}(Ap)=Ap.\) Since \(T_{r_n}^F\) is nonexpansive, then we have

$$\begin{aligned} ||u_n-p||= ||T_{r_n}^Fw_n-p||\le ||w_n-p||. \end{aligned}$$
(4.1)

Applying (4.1), we get

$$\begin{aligned} ||v_n-p||&= ||\beta _nw_n + (1-\beta _n)u_n-p||\nonumber \\&\le \beta _n||w_n-p|| + (1-\beta _n)||u_n-p||\nonumber \\&\le \beta _n||w_n-p|| + (1-\beta _n)||w_n-p||\nonumber \\&=||w_n-p||. \end{aligned}$$
(4.2)

Next, by the definition of G(x) and the firmly nonexpansivity of \(I-J_{\lambda _2}^{B_2},\) we have

$$\begin{aligned} \langle G(v_n), v_n-p\rangle&= \langle A^*(I-J_{\lambda _2}^{B_2})Av_n, v_n-p \rangle \nonumber \\&= \langle (I-J_{\lambda _2}^{B_2})Av_n, Av_n-Ap \rangle \nonumber \\&= \langle (I-J_{\lambda _2}^{B_2})Av_n- (I-J_{\lambda _2}^{B_2})Ap, Av_n-Ap \rangle \nonumber \\&\ge ||(I-J_{\lambda _2}^{B_2})Av_n||^2\nonumber \\&=2g(v_n). \end{aligned}$$
(4.3)

Then by Lemma 2.8(iii) and applying (4.3) together with the nonexpansivity of \(J_{\lambda _1}^{B_1},\) it follows that

$$\begin{aligned} ||t_n-p||^2&= ||J^{B_1}_{\lambda _1}(I - \gamma _nA^*(I- J^{B_2}_{\lambda _2})A)v_n -p||^2\nonumber \\&\le ||v_n - \gamma _nA^*(I- J^{B_2}_{\lambda _2})Av_n -p||^2\nonumber \\&=||v_n-p-\gamma _nG(v_n)||^2\nonumber \\&=||v_n-p||^2+\gamma _n^2||G(v_n)||^2-2\gamma _n\langle G(v_n), v_n-p \rangle \nonumber \\&\le ||v_n-p||^2+\gamma _n^2||G(v_n)||^2-4\gamma _ng(v_n)\nonumber \\&=||v_n-p||^2 + \frac{\rho _n^2g^2(v_n)}{(||G(v_n)||^2+||H(v_n)||^2)^2}||G(v_n)||^2 - \frac{4\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\nonumber \\&\le ||v_n-p||^2 - \frac{(4-\rho _n)\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}. \end{aligned}$$
(4.4)

By the condition on \(\rho _n,\) we have that

$$\begin{aligned} ||t_n-p||\le ||v_n-p||. \end{aligned}$$
(4.5)

Applying (4.5), we obtain

$$\begin{aligned} ||z_n-p||&= ||(1-\sigma _n)v_n +\sigma _nSt_n -p||\nonumber \\&\le (1-\sigma _n)||v_n-p|| +\sigma _n||St_n -p||\nonumber \\&\le (1-\sigma _n)||v_n-p|| +\sigma _n||t_n -p||\nonumber \\&\le (1-\sigma _n)||v_n-p|| +\sigma _n||v_n -p||\nonumber \\&=||v_n-p||. \end{aligned}$$
(4.6)

Next, applying the triangle inequality, we get

$$\begin{aligned} ||w_n - p||&= ||x_n + \theta _n(x_n - x_{n-1}) - p||\nonumber \\&\le ||x_n - p|| + \theta _n||x_n - x_{n-1}||\nonumber \\&= ||x_n - p|| + \alpha _n\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||. \end{aligned}$$
(4.7)

By Remark 3.2, \(\lim _{n\rightarrow \infty }\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}|| = 0.\) Then, it follows that there exists a constant \(M_1 > 0\) such that \(\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||\le M_1\) for all \(n\ge 1.\) Hence, from (4.7) we obtain

$$\begin{aligned} ||w_n - p||\le ||x_n - p|| + \alpha _nM_1. \end{aligned}$$
(4.8)

Applying (4.2), (4.5), (4.6) and (4.8), we have

$$\begin{aligned} ||x_{n+1}-p||&= ||\alpha _nf(x_n) + \delta _nSt_n + \xi _nTz_n-p||\\&= ||\alpha _n(f(x_n)-fp) + \alpha _n(f(p)-p) + \delta _n(St_n-p) + \xi _n(Tz_n-p)||\\&\le \alpha _nk||x_n-p|| + \alpha _n||f(p)-p|| + \delta _n||t_n-p|| + \xi _n||z_n-p||\\&\le \alpha _nk||x_n-p|| + \alpha _n||f(p)-p|| + \delta _n(||x_n - p|| + \alpha _nM_1) \\&\quad + \xi _n(||x_n - p|| + \alpha _nM_1)\\&=(\alpha _nk+(1-\alpha _n))||x_n-p|| + \alpha _n||f(p)-p|| + (1-\alpha _n)\alpha _nM_1\\&=(1-\alpha _n(1-k))||x_n-p|| + \alpha _n(1-k)\Big \{\frac{||f(p)-p||}{1-k} + \frac{(1-\alpha _n)M_1}{1-k} \Big \}\\&\le (1-\alpha _n(1-k))||x_n-p|| + \alpha _n(1-k)M^*, \end{aligned}$$

where \(M^* :=\sup _{n\in {\mathbb {N}}}\Big \{\frac{||f(p)-p||}{1-k} + \frac{(1-\alpha _n)M_1}{1-k}\}.\) Setting \(a_n:=||x_n-p||;~ b_n:=\alpha _n(1-k)M^*;~ c_n:=0,\) and \(\sigma _n:=\alpha _n(1-k).\) By invoking Lemma 2.9 together with the assumptions on the control parameters, we have that \(\{||x_n-p||\}\) is bounded and this implies that \(\{x_n\}\) is bounded. Consequently, \(\{w_n\}, \{u_n\}, \{v_n\}, \{t_n\}\) and \(\{z_n\}\) are all bounded. \(\square \)

Lemma 4.3

Let \(\{x_n\}\) be the sequence generated by Algorithm 3.1 and \(p\in \Omega .\) Then, under conditions (C1)-(C4) and for all \(n\in {\mathbb {N}},\) we have

$$\begin{aligned} ||x_{n+1}-p||^2 \le&\Big (1-\frac{2\alpha _n(1-k)}{(1-\alpha _nk)}\Big ) ||x_n - p||^2 + \frac{2\alpha _n(1-k)}{(1-\alpha _nk)}\Big \{\frac{\alpha _nM_3}{2(1-k)}\\&+ \frac{3M_2(1-\alpha _n)^2}{2(1-k)} \frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||\\&+ \frac{1}{(1-k)}\langle f(p)-p, x_{n+1}-p \rangle \Big \}- \frac{\xi _n(1-\alpha _n)}{(1-\alpha _nk)} \Big \{\beta _n(1-\beta _n)||w_n-u_n||^2 \\&+ \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\\&+\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \}. \end{aligned}$$

Proof

Let \(p\in \Omega \). Then, applying Lemma 2.8(iv) and (4.1), we get

$$\begin{aligned} ||v_n-p||^2&= ||\beta _nw_n + (1-\beta _n)u_n-p||^2\nonumber \\&= \beta _n||w_n-p||^2 + (1-\beta _n)||u_n-p||^2-\beta _n(1-\beta _n)||w_n-u_n||^2\nonumber \\&\le \beta _n||w_n-p||^2 + (1-\beta _n)||w_n-p||^2-\beta _n(1-\beta _n)||w_n-u_n||^2\nonumber \\&\le ||w_n-p||^2 -\beta _n(1-\beta _n)||w_n-u_n||^2. \end{aligned}$$
(4.9)

Again, by invoking Lemma 2.8(ii) and applying Cauchy-Schwartz inequality we have

$$\begin{aligned} ||w_n - p||^2&= ||x_n + \theta _n(x_n - x_{n-1}) - p||^2\nonumber \\&= ||x_n - p||^2 + \theta _n^2||x_n - x_{n-1}||^2 + 2\theta _n\langle x_n - p, x_n - x_{n-1} \rangle \nonumber \\&\le ||x_n - p||^2 + \theta _n^2||x_n - x_{n-1}||^2 + 2\theta _n||x_n - x_{n-1}||||x_n - p|| \nonumber \\&= ||x_n - p||^2 + \theta _n||x_n - x_{n-1}||(\theta _n||x_n - x_{n-1}|| + 2||x_n - p||)\nonumber \\&\le ||x_n - p||^2 + 3M_2\theta _n||x_n - x_{n-1}||\nonumber \\&= ||x_n - p||^2 + 3M_2\alpha _n\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||, \end{aligned}$$
(4.10)

where \(M_2:= \sup _{n\in {\mathbb {N}}}\{||x_n - p||, \theta _n||x_n - x_{n-1}||\}>0.\)

Next, applying Lemma 2.8(iv), (4.4), (4.9) and (4.10), we obtain

$$\begin{aligned} ||z_n-p||^2&= ||(1-\sigma _n)v_n + \sigma _nSt_n-p||^2\nonumber \\&= (1-\sigma _n)||v_n-p||^2 + \sigma _n||St_n-p||^2-\sigma _n(1-\sigma _n)||v_n-St_n||^2\nonumber \\&\le (1-\sigma _n)||v_n-p||^2 + \sigma _n||t_n-p||^2-\sigma _n(1-\sigma _n)||v_n-St_n||^2\nonumber \\&\le (1-\sigma _n)||v_n-p||^2 + \sigma _n\Big \{||v_n-p||^2 - \frac{(4-\rho _n)\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\Big \}\nonumber \\&\quad -\sigma _n(1-\sigma _n)||v_n-St_n||^2\nonumber \\&= ||v_n-p||^2 - \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\nonumber \\&\quad -\sigma _n(1-\sigma _n)||v_n-St_n||^2\nonumber \\&\le ||w_n-p||^2 -\beta _n(1-\beta _n)||w_n-u_n||^2 - \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2} \nonumber \\&\quad -\sigma _n(1-\sigma _n)||v_n-St_n||^2. \end{aligned}$$
(4.11)

Now, invoking Lemma 2.8 and applying (4.2), (4.5) and (4.11) we have

$$\begin{aligned} ||x_{n+1}-p||^2&= ||\alpha _nf(x_n) + \delta _nSt_n + \xi _nTz_n -p||^2\\&\le ||\delta _n(St_n-p) + \xi _n(Tz_n -p)||^2 + 2\alpha _n\langle f(x_n)-p, x_{n+1}-p \rangle \\&\le \delta _n^2||St_n-p||^2 + \xi _n^2||Tz_n -p||^2 + 2\delta _n\xi _n||St_n-p||||Tz_n -p|| \\&\quad + 2\alpha _n\langle f(x_n)-p, x_{n+1}-p \rangle \\&\le \delta _n^2||St_n-p||^2 + \xi _n^2||Tz_n -p||^2 + \delta _n\xi _n\big (||St_n-p||^2 + ||Tz_n -p||^2\big ) \\&\quad + 2\alpha _n\langle f(x_n)-p, x_{n+1}-p \rangle \\&=\delta _n(\delta _n+\xi _n)||St_n-p||^2 + \xi _n(\xi _n+\delta _n)||Tz_n -p||^2 \\&\quad + 2\alpha _n\langle f(x_n)-p, x_{n+1}-p \rangle \\&\le \delta _n(1-\alpha _n)||t_n-p||^2 + \xi _n(1-\alpha _n)||z_n -p||^2\\&\quad + 2\alpha _n\langle f(x_n)-f(p), x_{n+1}-p \rangle + 2\alpha _n\langle f(p)-p, x_{n+1}-p \rangle \\&\le \delta _n(1-\alpha _n)||w_n-p||^2 + \xi _n(1-\alpha _n)\Big \{||w_n-p||^2 \\&\quad -\beta _n(1-\beta _n)||w_n-u_n||^2- \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\\&\quad -\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \} + 2\alpha _nk||x_n-p||||x_{n+1}-p|| \\&\quad + 2\alpha _n\langle f(p)-p, x_{n+1}-p \rangle \\&\le (1-\alpha _n)^2||w_n-p||^2 - \xi _n(1-\alpha _n)\Big \{\beta _n(1-\beta _n)||w_n-u_n||^2\\&\quad + \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\\&\quad +\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \} + \alpha _nk\big (||x_n-p||^2 + ||x_{n+1}-p||^2\big ) \\&\quad + 2\alpha _n\langle f(p)-p, x_{n+1}-p \rangle \\&\le (1-\alpha _n)^2\big (||x_n - p||^2 + 3M_2\alpha _n\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||\big ) \\&\quad - \xi _n(1-\alpha _n)\Big \{\beta _n(1-\beta _n)||w_n-u_n||^2\\&\quad + \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2} +\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \} \\&\quad + \alpha _nk\big (||x_n-p||^2 + ||x_{n+1}-p||^2\big )\\&\quad + 2\alpha _n\langle f(p)-p, x_{n+1}-p \rangle \\&=\big ((1-\alpha _n)^2+\alpha _nk\big )||x_n - p||^2 + \alpha _nk||x_{n+1}-p||^2 \\&\quad + 3M_2(1-\alpha _n)^2\alpha _n\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||\\&\quad - \xi _n(1-\alpha _n)\Big \{\beta _n(1-\beta _n)||w_n-u_n||^2 + \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\\&\quad +\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \} \\&\quad + 2\alpha _n\langle f(p)-p, x_{n+1}-p \rangle \\ \end{aligned}$$

From this, we obtain

$$\begin{aligned} ||x_{n+1}-p||^2&\le \frac{(1-2\alpha _n+\alpha _n^2 + \alpha _nk)}{(1-\alpha _nk)}||x_n - p||^2 + \frac{\alpha _n}{(1-\alpha _nk)}\Big \{ 3M_2(1-\alpha _n)^2\frac{\theta _n}{\alpha _n}||x_n \\&\quad - x_{n-1}|| + 2\langle f(p)-p, x_{n+1}-p \rangle \Big \} \\&\quad - \frac{\xi _n(1-\alpha _n)}{(1-\alpha _nk)} \Big \{\beta _n(1-\beta _n)||w_n-u_n||^2 + \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\\&\quad +\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \} \\&= \frac{(1-2\alpha _n+ \alpha _nk)}{(1-\alpha _nk)}||x_n - p||^2 + \frac{\alpha _n^2}{(1-\alpha _nk)}||x_n - p||^2 \\&\quad + \frac{\alpha _n}{(1-\alpha _nk)}\Big \{ 3M_2(1-\alpha _n)^2\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}|| \\&\quad + 2\langle f(p)-p, x_{n+1}-p \rangle \Big \}- \frac{\xi _n(1-\alpha _n)}{(1-\alpha _nk)} \Big \{\beta _n(1-\beta _n)||w_n-u_n||^2 \\&\quad + \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\\&\quad +\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \} \\&\le \Big (1-\frac{2\alpha _n(1-k)}{(1-\alpha _nk)}\Big ) ||x_n - p||^2 + \frac{2\alpha _n(1-k)}{(1-\alpha _nk)} \Big \{\frac{\alpha _nM_3}{2(1-k)}\\&\quad + \frac{3M_2(1-\alpha _n)^2}{2(1-k)} \frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||\\&\quad + \frac{1}{(1-k)}\langle f(p)-p, x_{n+1}-p \rangle \Big \}- \frac{\xi _n(1-\alpha _n)}{(1-\alpha _nk)} \Big \{\beta _n(1-\beta _n)||w_n-u_n||^2 \\&\quad + \frac{(4-\rho _n)\sigma _n\rho _ng^2(v_n)}{||G(v_n)||^2+||H(v_n)||^2}\\&\quad +\sigma _n(1-\sigma _n)||v_n-St_n||^2\Big \}, \end{aligned}$$

where \(M_3:=\sup \{||||x_n - p||^2 : n\in {\mathbb {N}}\}.\) This completes the proof. \(\square \)

Lemma 4.4

Let \(\{x_n\}\) be a sequence generated by Algorithm 3.1 such that conditions (C1)-(C4) hold. Then, the following inequality holds for all \(p\in \Omega \) and \(n\in {\mathbb {N}}:\)

$$\begin{aligned} ||x_{n+1}-p||^2\le&(1-\alpha _n)||x_n - p||^2 + \alpha _n||f(x_n)-p||^2 \\&+ 3M_2(1-\alpha _n)\alpha _n\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}|| -\delta _n||t_n-v_n||^2\\&+ 2M_4\delta _n||A^*(I- J^{B_2}_{\lambda _2})Av_n|| - \delta _n\xi _n||St_n-Tz_n||^2 . \end{aligned}$$

Proof

Let \(p\in \Omega .\) From (4.4), observe that

$$\begin{aligned} ||v_n - \gamma _nA^*(I- J^{B_2}_{\lambda _2})Av_n -p||^2\le ||v_n-p||^2. \end{aligned}$$

Applying Lemma 2.8 and the firmly nonexpansivity of \(J_{\lambda _1}^{B_1},\) we have

$$\begin{aligned} ||t_n-p||^2&= ||J^{B_1}_{\lambda _1}(I - \gamma _nA^*(I- J^{B_2}_{\lambda _2})A)v_n -p||^2\\&\le \langle t_n-p, v_n - \gamma _nA^*(I- J^{B_2}_{\lambda _2})Av_n -p\rangle \\&=\frac{1}{2}\big (||t_n-p||^2 + ||v_n - \gamma _nA^*(I- J^{B_2}_{\lambda _2})Av_n -p||^2 - ||t_n-v_n \\&\quad + \gamma _nA^*(I- J^{B_2}_{\lambda _2})Av_n||^2\big )\\&\le \frac{1}{2}\big (||t_n-p||^2 + ||v_n -p||^2 - ||t_n-v_n + \gamma _nA^*(I- J^{B_2}_{\lambda _2})Av_n||^2\big )\\&=\frac{1}{2}\big (||t_n-p||^2 + ||v_n -p||^2 - \big (||t_n-v_n||^2 + \gamma _n^2||A^*(I- J^{B_2}_{\lambda _2})Av_n||^2 \\&\quad - 2\gamma _n\langle v_n-t_n, A^*(I- J^{B_2}_{\lambda _2})Av_n \rangle \big )\big )\\&\le \frac{1}{2}\big (||t_n-p||^2 + ||v_n -p||^2 -||t_n-v_n||^2 - \gamma _n^2||A^*(I- J^{B_2}_{\lambda _2})Av_n||^2 \\&\quad + 2\gamma _n||v_n-t_n||||A^*(I- J^{B_2}_{\lambda _2})Av_n||\big )\\&\le \frac{1}{2}\big (||t_n-p||^2 + ||v_n -p||^2 -||t_n-v_n||^2 + 2\gamma _n||v_n-t_n||||A^*\\&\quad \times (I- J^{B_2}_{\lambda _2})Av_n||\big ). \end{aligned}$$

Consequently, we have that

$$\begin{aligned} ||t_n-p||^2&\le ||v_n -p||^2 -||t_n-v_n||^2 + 2\gamma _n||v_n-t_n||||A^*(I- J^{B_2}_{\lambda _2})Av_n||\nonumber \\&\le ||v_n -p||^2 -||t_n-v_n||^2 + 2M_4||A^*(I- J^{B_2}_{\lambda _2})Av_n||\nonumber \\&\le ||w_n -p||^2 -||t_n-v_n||^2 + 2M_4||A^*(I- J^{B_2}_{\lambda _2})Av_n||, \end{aligned}$$
(4.12)

where \(M_4:=\sup \{\gamma _n||v_n-t_n||: n\in {\mathbb {N}}\}.\)

Next, by invoking Lemma 2.8(iv) and applying (4.2), (4.6), (4.10) and (4.12) we obtain

$$\begin{aligned} ||x_{n+1}-p||^2&= ||\alpha _nf(x_n) + \delta _nSt_n + \xi _nTz_n -p||^2\\&= \alpha _n||f(x_n)-p||^2 + \delta _n||St_n-p||^2 + \xi _n||Tz_n -p||^2- \delta _n\xi _n||St_n-Tz_n||^2\\&\le \alpha _n||f(x_n)-p||^2 + \delta _n||t_n-p||^2 + \xi _n||z_n -p||^2- \delta _n\xi _n||St_n-Tz_n||^2\\&\le \alpha _n||f(x_n)-p||^2 + \delta _n\big (||w_n -p||^2 -||t_n-v_n||^2 \\&\quad + 2M_4||A^*(I- J^{B_2}_{\lambda _2})Av_n||\big ) + \xi _n||w_n -p||^2\\&\quad - \delta _n\xi _n||St_n-Tz_n||^2\\&= \alpha _n||f(x_n)-p||^2 + (1-\alpha _n)||w_n -p||^2 -\delta _n||t_n-v_n||^2 \\&\quad + 2M_4\delta _n||A^*(I- J^{B_2}_{\lambda _2})Av_n||\\&\quad - \delta _n\xi _n||St_n-Tz_n||^2\\&\le \alpha _n||f(x_n)-p||^2 + (1-\alpha _n)\Big (||x_n - p||^2 + 3M_2\alpha _n\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||\Big )\\&\quad -\delta _n||t_n-v_n||^2\\&\quad + 2M_4\delta _n||A^*(I- J^{B_2}_{\lambda _2})Av_n|| - \delta _n\xi _n||St_n-Tz_n||^2\\&= (1-\alpha _n)||x_n - p||^2 + \alpha _n||f(x_n)-p||^2 \\&\quad + 3M_2(1-\alpha _n)\alpha _n\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}|| -\delta _n||t_n-v_n||^2\\&\quad + 2M_4\delta _n||A^*(I- J^{B_2}_{\lambda _2})Av_n|| - \delta _n\xi _n||St_n-Tz_n||^2, \end{aligned}$$

which is the required inequality. \(\square \)

Lemma 4.5

Let \(\{x_n\}\) be the sequence generated by Algorithm 3.1 such that conditions (A1)-(A4) and (C1)-(C4) are satisfied. Then \(\{x_n\}\) converges strongly to a point \({\hat{x}}\in \Omega ,\) where \({\hat{x}}=P_\Omega \circ f({\hat{x}}).\)

Proof

Let \({\hat{x}}=P_\Omega \circ f({\hat{x}}).\) It then follows from Lemma 4.3 that

$$\begin{aligned} ||x_{n+1}-{\hat{x}}||^2&\le \Big (1-\frac{2\alpha _n(1-k)}{(1-\alpha _nk)}\Big ) ||x_n - {\hat{x}}||^2 + \frac{2\alpha _n(1-k)}{(1-\alpha _nk)}\Big \{\frac{\alpha _nM_3}{2(1-k)}\nonumber \\&\quad + \frac{3M_2(1-\alpha _n)^2}{2(1-k)} \frac{\theta _n}{\alpha _n}||x_n - x_{n-1}||\nonumber \\&\quad + \frac{1}{(1-k)}\langle f({\hat{x}})-{\hat{x}}, x_{n+1}-{\hat{x}} \rangle \Big \}. \end{aligned}$$
(4.13)

Now, we claim that the sequence \(\{||x_n - {\hat{x}}||\}\) converges to zero. To do this, by Lemma 2.10 it suffices to show that \(\limsup _{k\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, x_{n_k+1} -{\hat{x}} \rangle \le 0\) for every subsequence \(\{||x_{n_k} - {\hat{x}}||\}\) of \(\{||x_n - {\hat{x}}||\}\) satisfying

$$\begin{aligned} \liminf _{k\rightarrow \infty }(||x_{n_k+1} - {\hat{x}}|| - ||x_{n_k} - {\hat{x}}||) \ge 0. \end{aligned}$$

Now, suppose that \(\{||x_{n_k} - {\hat{x}}||\}\) is a subsequence of \(\{||x_n - {\hat{x}}||\}\) such that

$$\begin{aligned} \liminf _{k\rightarrow \infty }(||x_{n_k+1} - {\hat{x}}|| - ||x_{n_k} - {\hat{x}}||) \ge 0. \end{aligned}$$
(4.14)

From Lemma 4.3 we have

$$\begin{aligned}&\frac{\xi _{n_k}(1-\alpha _{n_k})}{(1-\alpha _{n_k}k)}\beta _{n_k}(1-\beta _{n_k})||w_{n_k}-u_{n_k}||^2\\&\quad \le \Big (1-\frac{2\alpha _{n_k}(1-k)}{(1-\alpha _{n_k}k)}\Big ) ||x_{n_k} - p||^2 - ||x_{{n_k}+1}-p||^2 + \frac{2\alpha _{n_k}(1-k)}{(1-\alpha _{n_k}k)}\Big \{\frac{\alpha _{n_k}M_3}{2(1-k)}\\&\qquad + \frac{3M_2(1-\alpha _{n_k})^2}{2(1-k)} \frac{\theta _{n_k}}{\alpha _{n_k}}||x_{n_k} - x_{{n_k}-1}||+ \frac{1}{(1-k)}\langle f(p)-p, x_{{n_k}+1}-p \rangle \Big \}. \end{aligned}$$

By (4.14) together with the fact that \(\lim _{k\rightarrow \infty }\alpha _{n_k}=0,\) we obtain

$$\begin{aligned} \frac{\xi _{n_k}(1-\alpha _{n_k})}{(1-\alpha _{n_k}k)}\beta _{n_k}(1-\beta _{n_k})||w_{n_k}-u_{n_k}||^2\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$

Consequently, we have

$$\begin{aligned} ||w_{n_k}-u_{n_k}||\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$
(4.15)

Following similar argument, from Lemma 4.3 we obtain

$$\begin{aligned} ||v_{n_k}-St_{n_k}|\rightarrow 0,\quad k\rightarrow \infty , \end{aligned}$$
(4.16)

and

$$\begin{aligned} \frac{(4-\rho _{n_k})\sigma _{n_k}\rho _{n_k}g^2(v_{n_k})}{||G(v_{n_k})||^2+||H(v_{n_k})||^2}\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$

Since G and H are Lipschitz continuous, then by the condition on \(\rho _n\) it follows that

$$\begin{aligned} g^2(v_{n_k})\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$

From this, we obtain

$$\begin{aligned} \lim _{k\rightarrow \infty }g(v_{n_k})= \lim _{k\rightarrow \infty }\frac{1}{2}||(I - J_{\lambda _2}^{B_2})Av_{n_k}||^2=0. \end{aligned}$$
(4.17)

Consequently, we have

$$\begin{aligned} ||(I - J_{\lambda _2}^{B_2})Av_{n_k}||\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$
(4.18)

From this, we get

$$\begin{aligned} ||A^*(I - J_{\lambda _2}^{B_2})Av_{n_k}||\le ||A^*||||(I - J_{\lambda _2}^{B_2})Av_{n_k}||= ||A||||(I - J_{\lambda _2}^{B_2})Av_{n_k}||\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$
(4.19)

Also, from Lemma 4.4 we have

$$\begin{aligned} \delta _{n_k}||t_{n_k}-v_{n_k}||^2 \le&(1-\alpha _{n_k})||x_{n_k} - p||^2 - ||x_{{n_k}+1}-p||^2 + \alpha _{n_k}||f(x_{n_k})-p||^2 \\&+ 3M_2(1-\alpha _{n_k})\alpha _{n_k}\frac{\theta _{n_k}}{\alpha _{n_k}}||x_{n_k} - x_{{n_k}-1}|| + 2M_4\delta _{n_k}||A^*(I- J^{B_2}_{\lambda _2})Av_{n_k}||. \end{aligned}$$

By (4.14), and by applying (4.19) together with Remark 3.2 and the fact that \(\lim _{k\rightarrow \infty }\alpha _{n_k}=0,\) we obtain

$$\begin{aligned} ||t_{n_k}-v_{n_k}||\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$
(4.20)

Similarly, from Lemma 4.4 we get

$$\begin{aligned} ||St_{n_k}-Tz_{n_k}||\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$
(4.21)

By Remark 3.2, we obtain

$$\begin{aligned} ||w_{n_k}-x_{n_k}||= \theta _{n_k}||x_{n_k}-x_{n_k-1}||\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$
(4.22)

Applying (4.15) and (4.22), we get

$$\begin{aligned} ||x_{n_k}-u_{n_k}||\rightarrow 0,\quad k\rightarrow \infty ;\quad \quad ||v_{n_k}-x_{n_k}||\rightarrow 0,\quad k\rightarrow \infty . \end{aligned}$$
(4.23)

On the other hand, by applying (4.16), (4.20), (4.21) and (4.23) we obtain

$$\begin{aligned}&||x_{n_k}-t_{n_k}||\rightarrow 0,\quad k\rightarrow \infty ;\quad ||x_{n_k}-St_{n_k}||\rightarrow 0,\quad k\rightarrow \infty ;\quad ||x_{n_k}-Tz_{n_k}||\rightarrow 0,\quad \nonumber \\&k\rightarrow \infty . \end{aligned}$$
(4.24)

Also, by applying (4.23)–(4.25) we get

$$\begin{aligned}&||z_{n_k}-x_{n_k}||\rightarrow 0,\quad k\rightarrow \infty ;\quad ||t_{n_k}-St_{n_k}||\rightarrow 0,\quad k\rightarrow \infty ;\quad ||z_{n_k}-Tz_{n_k}||\rightarrow 0,\quad \nonumber \\&k\rightarrow \infty . \end{aligned}$$
(4.25)

Now, by using (4.24) together with the fact that \(\lim _{k\rightarrow \infty }\alpha _{n_k}=0,\) we have

$$\begin{aligned}&||x_{n_k+1}-x_{n_k}||\le \alpha _{n_k}||f(x_{n_k})-x_{n_k}|| + \delta _{n_k} ||St_{n_k}-x_{n_k}||+ \xi _{n_k} ||Tz_{n_k}-x_{n_k}||\rightarrow 0,\quad \nonumber \\&k\rightarrow \infty . \end{aligned}$$
(4.26)

To complete the proof, we need to show that \(w_\omega (x_n)\subset \Omega .\) First, we claim that \(w_\omega (x_n)\subset EP(F).\) Since \(\{x_n\}\) is bounded, then \(w_\omega (x_n)\) is nonempty. Let \(x^*\in w_\omega (x_n)\) be an arbitrary element. Then there exists a subsequence \(\{x_{n_k}\}\) of \(\{x_n\}\) such that \(x_{n_k}\rightharpoonup x^*\) as \(k\rightarrow \infty .\) By (4.23), it follows that \(u_{n_k}\rightharpoonup x^*\) as \(k\rightarrow \infty .\) By the definition of \(T_{r_{n_k}}^Fw_{n_k},\) we have that

$$\begin{aligned} F(u_{n_k}, y) + \frac{1}{r_{n_k}}\langle y - u_{n_k}, u_{n_k} - w_{n_k}\rangle \ge 0,~~\forall y\in C. \end{aligned}$$

It follows from the monotonicity of F that

$$\begin{aligned} \frac{1}{r_{n_k}}\langle y - u_{n_k}, u_{n_k} - w_{n_k} \rangle \ge F(y, u_{n_k}), \quad \forall ~ y\in C. \end{aligned}$$

By (4.15), \(\liminf _{k \rightarrow \infty }r_{n_k}>0,\) and condition (A4), we have that

$$\begin{aligned} F(y, x^*)\le 0, \quad \forall ~ y\in C. \end{aligned}$$
(4.27)

Let \(y_t = ty + (1 - t)x^*,~~\forall ~ t \in (0, 1]\) and \(y \in C.\) This implies that \(y_t\in C,\) and it follows from (4.27) that \(F(y_t, x^*)\le 0.\) So, by applying conditions (A1)-(A4), we have

$$\begin{aligned} 0&= F(y_t,y_t)\\&\le tF(y_t,y) +(1-t)F(y_t,x^*)\\&\le tF(y_t,y). \end{aligned}$$

Hence, we have

$$\begin{aligned} F(y_t,y)\ge 0,\quad \forall ~y\in C. \end{aligned}$$

Letting \(t\rightarrow 0,\) by condition (A3), we get

$$\begin{aligned} F(x^*,y)\ge 0,\quad \forall ~y\in C. \end{aligned}$$

This implies that \(x^*\in EP(F).\)

Next, we show that \(x^* \in {\mathcal {F}}.\) By the lower semicontinuity of g,  it follows from (4.17) that

$$\begin{aligned} 0\le g(x^*)\le \lim _{k\rightarrow \infty }g(v_{n_k})= \lim _{n\rightarrow \infty }g(v_n)=0, \end{aligned}$$

which implies that

$$\begin{aligned} g(x^*)= \frac{1}{2}||(I - J_{\lambda _2}^{B_2})Ax^*||^2=0. \end{aligned}$$

Thus, by Remark 2.5 we have that

$$\begin{aligned} Ax^*\in B_2^{-1}(0)\quad \text {or}\quad 0\in B_2(Ax^*). \end{aligned}$$
(4.28)

Since \(t_{n_k} = J_{\lambda _1}^{B_1}(v_{n_k} - \gamma _{n_k}A^*(I - J_{\lambda _2}^{B_2})Av_{n_k})\) can be rewritten as

$$\begin{aligned} v_{n_k} - \gamma _{n_k}A^*(I - J_{\lambda _2}^{B_2})Av_{n_k}\in t_{n_k} + \lambda _1B_1(t_{n_k}) \end{aligned}$$

or, equivalently

$$\begin{aligned} \frac{(v_{n_k} - t_{n_k}) - \gamma _{n_k}A^*(I - J_\lambda ^{B_2})Av_{n_k}}{\lambda _1}\in B_1(t_{n_k}). \end{aligned}$$
(4.29)

By passing to limit as \(k\rightarrow \infty \) in (4.29), applying (4.19), (4.20) and (4.24), and taking into consideration the fact that the graph of a maximal monotone operator is weakly-strongly closed, we obtain \(0\in B_1(x^*).\) This together with (4.28) implies that \(x^*\in \Gamma .\)

Next, we show that \(x^*\in F(S)\cap F(T).\) By (4.24) and (4.25), we have \(t_{n_k}\rightharpoonup x^* \) and \(z_{n_k}\rightharpoonup x^* \) as \(k\rightarrow \infty .\) Since S and T are nonexpansive, by the demiclosedness principle, it follows from (4.25) that \(x^*\in F(S)\cap F(T).\) Consequently, we have that \(w_\omega (x_n)\subset \Omega .\)

Moreover, from (4.24) and (4.25) it follows that \(w_\omega \{t_n\} = w_\omega \{x_n\} =w_\omega \{z_n\}.\) By the boundedness of \(\{x_{n_k}\},\) there exists a subsequence \(\{x_{n_{k_j}}\}\) of \(\{x_{n_k}\}\) such that \(x_{n_{k_j}}\rightharpoonup x^\dagger \) and

$$\begin{aligned} \lim _{j\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, x_{n_{k_j}} -{\hat{x}} \rangle = \limsup _{k\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, x_{n_k} -{\hat{x}} \rangle = \limsup _{k\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, t_{n_k} -{\hat{x}} \rangle . \end{aligned}$$

Since \({\hat{x}}=P_\Omega \circ f({\hat{x}}),\) then it follows that

$$\begin{aligned} \limsup _{k\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, x_{n_k} -{\hat{x}} \rangle = \lim _{j\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, x_{n_{k_j}} -{\hat{x}} \rangle = \langle f({\hat{x}}) - {\hat{x}}, x^\dagger -{\hat{x}} \rangle \le 0. \end{aligned}$$
(4.30)

Now, from (4.26) and (4.30), we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, x_{n_k+1} -{\hat{x}} \rangle = \limsup _{k\rightarrow \infty }\langle f({\hat{x}}) - {\hat{x}}, x_{n_k} -{\hat{x}} \rangle = \langle f({\hat{x}}) - {\hat{x}}, x^\dagger -{\hat{x}} \rangle \le 0. \end{aligned}$$
(4.31)

Applying Lemma 2.10 to (4.13), and using (4.31) together with the fact that \(\lim _{n\rightarrow \infty }\frac{\theta _n}{\alpha _n}||x_n - x_{n-1}|| =0\) and \(\lim _{n\rightarrow \infty }\alpha _n = 0,\) we deduce that \(\lim _{n\rightarrow \infty }||x_n - {\hat{x}}||=0\) as desired. \(\square \)

5 Applications

In this section we apply our results to study some related optimization problems.

5.1 Split minimization problem

Let \(H_1\) and \(H_2\) be real Hilbert spaces, \(A:H_1 \rightarrow H_2\) be a bounded linear operator. Given some proper, lower semicontinuous and convex functions \(f_1:H_1 \rightarrow {\mathbb {R}}\cup \{+\infty \}\) and \(f_2:H_2 \rightarrow {\mathbb {R}} \cup \{+\infty \},\) the Split Minimization Problem (SMP) is defined as

$$\begin{aligned} \text {Find} \quad {\bar{x}} \in H_1 \quad \text {such that}\quad {\bar{x}} \in \arg \min _{x\in H_1}~f_1(x) \quad \text {and}\quad A{\bar{x}} \in \arg \min _{y\in H_2}~f_2(y). \end{aligned}$$
(5.1)

We denote the set of solution of the SMP (5.1) by \(\Gamma _{SMP}.\) The SMP was first introduced by Moudafi and Thakur (2014). It has attracted lots of attention in recent years and has been applied in the study of many applied science problems such as multi-resolution sparse regularization, Fourier regularization, hard-constrained inconsistent feasibility and alternating projection signal synthesis problems (see (Abbas et al. 2018) and the references therein).

In a real Hilbert space H,  the proximal operator of f is defined by

$$\begin{aligned} prox_{\lambda ,f}(x) := \arg \min _{z \in H}\left\{ f(z) + \frac{1}{2\lambda }\Vert x-z\Vert ^2\right\} \quad \forall ~ x\in H, ~~ \lambda >0. \end{aligned}$$

It is well known that

$$\begin{aligned} prox_{\lambda , f}(x) = (I+\lambda \partial f)^{-1}(x) = J_\lambda ^{\partial f}(x), \end{aligned}$$
(5.2)

where \(\partial f\) is the subdifferential of f defined by

$$\begin{aligned} \partial f(x) = \{z \in H : f(x) - f(y) \le \langle z, x - y\rangle , \forall y \in H\}, \end{aligned}$$

for each \(x\in H.\) From [?], \(\partial f\) is a maximal monotone operator and \(prox_{\lambda ,f}\) is firmly nonexpansive.

By setting \(B_1 = \partial f_1\) and \(B_2 = \partial f_2\) in Theorem 4.1, we obtain the following result for approximating a common solution of split minimization problem, equilibrium problem and and common fixed point of nonexpansive mappings in Hilbert spaces.

Theorem 5.1

Let \(H_1\) and \(H_2\) be real Hilbert spaces, and \(A:H_1\rightarrow H_2\) be a bounded linear operator with adjoint \(A^*.\) Let \(f_1:H_1 \rightarrow {\mathbb {R}}\cup \{+\infty \}\) and \(f_2:H_2 \rightarrow {\mathbb {R}} \cup \{+\infty \}\) be proper, lower semicontinuous and convex functions, \(S,T:H_1\rightarrow H_1\) be nonexpansive mappings, and \(f:H_1\rightarrow H_1\) be a contraction with coefficient \(k\in (0,1).\) Suppose that \(\Omega = F(S)\cap F(T)\cap \Gamma _{SMP}\cap EP(F) \ne \emptyset ,\) and conditions (A1)-(A4) and (C1)-(C4) are satisfied. Then the sequence \(\{x_n\}\) generated by the following algorithm converges strongly to a point \({\hat{x}}\in \Omega ,\) where \({\hat{x}}=P_\Omega \circ f({\hat{x}}).\)

figure g

where

$$\begin{aligned} g(x) = \frac{1}{2}\Vert (I - prox_{\lambda _2,f_2})Ax\Vert ^2, \quad \quad h(x) = \frac{1}{2}\Vert (I - prox_{\lambda _1,f_1})x\Vert ^2 \end{aligned}$$

and

$$\begin{aligned} G(x) = A^*(I - prox_{\lambda _2,f_2})Ax,\quad \quad H(x) = (I - prox_{\lambda _1,f_1})x. \end{aligned}$$

5.2 Split feasibility problem

Let \(H_1\) and \(H_2\) be two real Hilbert spaces and let C and Q be nonempty closed convex subsets of \(H_1\) and \(H_2,\) respectively. The Split Feasibility Problem (SFP) is defined as follows:

$$\begin{aligned} \text {Find}~~x^*\in C~~\text {such that}~~Ax^*\in Q, \end{aligned}$$
(5.5)

where \(A : H_1 \rightarrow H_2\) is a bounded linear operator. Let the solution set of SFP (5.5) be denoted by \(\Gamma _{SFP}\).

In 1994, the SFP was introduced by Censor and Elfving (1994) in finite dimensional Hilbert spaces for modelling inverse problems which arise from phase retrievals and in medical image reconstruction (Byrne 2004). Furthermore, the problem (5.5) is also useful in various disciplines such as computer tomography, image restoration, and radiation therapy treatment planning (Censor et al. 2006, 2005). The problem has been studied by numerous researchers, (see Byrne 2004; Censor et al. 2006, 2005 and the references therein).

Let C be a nonempty closed convex subset of a real Hilbert space H and \(i_c\) be the indicator function on C, that is

$$\begin{aligned} i_c(x) = {\left\{ \begin{array}{ll} 0 &{} \text {if}~~x \in C;\\ \infty &{} \text {if}~~x \notin C. \end{array}\right. } \end{aligned}$$

Moreover, we define the normal cone \(N_Cu\) of C at \(u \in C\) as follows:

$$\begin{aligned} N_Cu = \{z\in H : \langle z, v - u\rangle \le 0, \forall v \in C\}. \end{aligned}$$

It is known that \(i_C\) is a proper, lower semicontinuous and convex function on H. Hence, the subdifferential \(\partial i_C\) of \(i_C\) is a maximal monotone operator. Therefore, we define the resolvent \(J^{\partial i_C}_r\) of \(\partial i_C\), \(\forall r > 0\) as follows:

$$\begin{aligned} J^{\partial i_C}_rx = (I + r\partial i_C)^{-1}x, \forall x \in H. \end{aligned}$$

Moreover, for each \(x \in C\), we have

$$\begin{aligned} \partial i_Cx&= \{z \in H : i_Cx + \langle z, u - x\rangle \le i_Cu, \forall u \in H\}\\&= \{z \in H : \langle z, u - x\rangle \le 0, \forall u \in C\}\\&= N_Cx. \end{aligned}$$

Hence, for all \(\alpha > 0,\) we derive

$$\begin{aligned} u = J^{\partial i_C}_rx&\Leftrightarrow x \in u + r\partial i_Cu\\&\Leftrightarrow x - u \in r\partial i_Cu\\&\Leftrightarrow \langle x-u, z-u \rangle \le 0~~ \forall z\in C\\&\Leftrightarrow u = P_Cx. \end{aligned}$$

Now, by applying Theorem 4.1 we obtain the following result for approximating a common solution of split feasibility problem, equilibrium problem and common fixed point of nonexpansive mappings in Hilbert spaces.

Theorem 5.3

Let C and Q be nonempty closed convex subsets of real Hilbert spaces \(H_1\) and \(H_2,\) respectively, \(A:H_1\rightarrow H_2\) be a bounded linear operator with adjoint \(A^*.\) Let \(S,T:H_1\rightarrow H_1\) be nonexpansive mappings, and \(f:H_1\rightarrow H_1\) be a contraction with coefficient \(k\in (0,1).\) Suppose that \(\Omega = F(S)\cap F(T)\cap \Gamma _{SFP}\cap EP(F) \ne \emptyset ,\) and conditions (A1)-(A4) and (C1)-(C4) are satisfied. Then the sequence \(\{x_n\}\) generated by the following algorithm converges strongly to a point \({\hat{x}}\in \Omega ,\) where \({\hat{x}}=P_\Omega \circ f({\hat{x}}).\)

figure h

where

$$\begin{aligned} g(x) = \frac{1}{2}\Vert (I - P_Q)Ax\Vert ^2, \quad \quad h(x) = \frac{1}{2}\Vert (I - P_C)x\Vert ^2 \end{aligned}$$

and

$$\begin{aligned} G(x) = A^*(I - P_Q)Ax,\quad \quad H(x) = (I - P_C)x. \end{aligned}$$

5.3 Relaxed split feasibility problem

Next, we study the Relaxed Split Feasibility Problem (RSFP) which is a special case of the split feasibility problem when the sets C and Q are defined as follows:

$$\begin{aligned} C:= \{u\in H_1: c(u)\le 0 \}\quad \text {and}\quad Q:= \{v\in H_2: q(v)\le 0 \}, \end{aligned}$$
(5.8)

where \(c:H_1\rightarrow {\mathbb {R}}\) and \(q:H_2\rightarrow {\mathbb {R}}\) are convex and lower semicontinuous functions such that \(\partial c\) and \(\partial q\) are bounded on bounded sets. We denote the solution set of the RSFP by \(\Gamma _{RSFP}.\) Now, by applying Theorem 4.1 we obtain the following result for approximating a common solution of relaxed split feasibility problem, equilibrium problem and common fixed point of nonexpansive mappings in Hilbert spaces.

Theorem 5.5

Let \(H_1\) and \(H_2\) be real Hilbert spaces \(H_1\) and \(H_2,\) and let \(A:H_1\rightarrow H_2\) be a bounded linear operator with adjoint \(A^*.\) Let \(S,T:H_1\rightarrow H_1\) be nonexpansive mappings, and \(f:H_1\rightarrow H_1\) be a contraction with coefficient \(k\in (0,1).\) Suppose that \(\Omega = F(S)\cap F(T)\cap \Gamma _{RSFP}\cap EP(F) \ne \emptyset ,\) and conditions (A1)-(A4) and (C1)-(C4) are satisfied. Then the sequence \(\{x_n\}\) generated by the following algorithm converges strongly to a point \({\hat{x}}\in \Omega ,\) where \({\hat{x}}=P_\Omega \circ f({\hat{x}}).\)

figure i

where

$$\begin{aligned} g(x) = \frac{1}{2}\Vert (I - P_{Q_n})Ax\Vert ^2, \quad \quad h(x) = \frac{1}{2}\Vert (I - P_{C_n})x\Vert ^2 \end{aligned}$$

and

$$\begin{aligned} G(x) = A^*(I - P_{Q_n})Ax,\quad \quad H(x) = (I - P_{C_n})x. \end{aligned}$$

6 Numerical examples

In this section, we present some numerical experiments to illustrate the performance of our method, Algorithm 3.1 in comparison with Algorithms 1.2, Algorithm 1.3, Algorithm 1.5 and Algorithm 7.2 in the literature. All numerical computations were carried out using Matlab version R2019(b).

In our computations, we choose \(\rho _n = 3-\frac{1}{2n+1}, \beta _n=\frac{n}{n+3}, \sigma _n=\frac{n}{2n+1}, \alpha _n = \frac{1}{2n+3}, \delta _n=\xi _n = \frac{n+1}{2n+3}, \mu _n=\frac{1}{(2n+3)^3}, r_n=\frac{n}{n+3}, \lambda = \lambda _1=\lambda _2=0.5, \theta =0.8\) for each \(n\in {\mathbb {N}}\). It can easily be checked that all the conditions of Theorem 4.1 are satisfied. We take \(\gamma =0.0001\) in Algorithms 1.2, 1.3 and 7.2, and \(Dx=\frac{1}{3}x, \beta =0.5\) in Algorithm 1.3.

Example 6.1

Let \(H_1={\mathbb {R}}^3=H_2\) and \(C=\{x = (x_1,x_2,x_3)\in {\mathbb {R}}^3| \langle a,x\rangle \ge b\}.\) For \(r>0,\) \(T_r^Fx = P_Cx = \frac{b - \langle a,x\rangle }{\Vert a\Vert _{2}^{2}}a + x.\) Here we take \(a = (8,-3,1), b = -1\) and define \(S,T:H_1\rightarrow H_1\) by \(Sx=\frac{1}{2}x, Tx=\frac{1}{3}x\) for all \(x\in H_1.\) Define the operators \(A, B_1\) and \(B_2\) as follows:

$$\begin{aligned} Ax= \begin{pmatrix} 6&{}\quad 3&{}\quad 1\\ 8&{}\quad 7&{}\quad 5\\ 3&{}\quad 6&{}\quad 2 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}, \quad B_1x= \begin{pmatrix} 6&{}\quad 0&{}\quad 0\\ 0&{}\quad 4&{}\quad 0\\ 0&{}\quad 0&{}\quad 3 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}, \quad B_2x= \begin{pmatrix} 7&{}\quad 0&{}\quad 0\\ 0&{}\quad 5&{}\quad 0\\ 0&{}\quad 0&{}\quad 2 \end{pmatrix} \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix} \end{aligned}$$

where \(x=(x_1,x_2,x_3)\in {\mathbb {R}}^3\) and set \(f(x) = \frac{1}{5}x.\)

We choose different initial values as follows:

Case I: \(x_0 = (-3,2,5)^T, x_1 = (2,1,-1)^T;\)

Case II: \(x_0 = (7,2.1,3.5)^T, x_1 = (5,1,2)^T;\)

Case III: \(x_0 = (2.3,4.7,-3.5)^T, x_1 = (3,1,0)^T;\)

Case IV: \(x_0 = (8,2,5)^T, x_1 = (-5,1,-1)^T.\)

Using MATLAB 2019(b), we compare the performance of Algorithm 3.1 with Algorithm 1.2, Algorithm 1.3, Algorithm 1.5 and Algorithm 7.2. The stopping criterion used for our computation is \(||x_{n+1}-x_{n}||< 10^{-3}\). We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Fig. 1 and Table 1.

Table 1 Numerical results for Example 6.1
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Fig. 1
figure 1

Top left: Case I; Top right: Case II; Bottom left: Case III; Bottom right: Case IV

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Example 6.2

Let \(H_1=( l _2({\mathbb {R}}), \Vert \cdot \Vert _2)=H_2,\) where \( l _2({\mathbb {R}}):=\{x=(x_1,x_2,\ldots ,x_n,\ldots ), x_j\in {\mathbb {R}}:\sum _{j=1}^{\infty }|x_j|^2<\infty \}, ||x||_2=(\sum _{j=1}^{\infty }|x_j|^2)^{\frac{1}{2}}\) for all \(x\in l _2({\mathbb {R}}).\) We set \(f(x) = \frac{1}{2}x\) and define \(S,T:H_1\rightarrow H_1\) by \(Sx=\frac{1}{3}x, Tx=\frac{1}{5}x,\) \(A:H_1\rightarrow H_2\) be defined by \(Ax = \frac{x}{3}~~ \text {for all}~~x\in H_1,\) then \(A^*y = \frac{y}{3} ~~ \text {for all}~~ y\in H_2.\) Define \(B_1:H_1\rightarrow H_1\) by \(B_1x = \frac{5}{2}x,\) and \(B_2:H_2\rightarrow H_2\) by \(B_2x = \frac{3}{2}x.\) Then \(B_1\) and \(B_2\) are maximal monotone operators. Define the bifunction F by \(F(x,y)=x(y-x).\) It can be verified that

$$\begin{aligned} T_r^Fx= \frac{x}{1 + r} \quad \text {for all}~~x\in H_1. \end{aligned}$$

We choose different initial values as follows:

Case I: \(x_0 = (0, -3, 7, \dots ),\) \(x_1 = (-1, 2, 3,\dots ),\)

Case II: \(x_0 = (5, -1, \frac{1}{5}, \dots ),\) \(x_1 = (3, 0.3, 0.03, \dots ),\)

Case III: \(x_0 = (0, 3, -7, \dots ),\) \(x_1 = (1, -2, 3, \dots ),\)

Case IV: \(x_0 = (7, -3, -\frac{1}{7},\dots ),\) \(x_1 = (4, 0.4, 0.04, \dots ).\)

Using MATLAB 2019(b), we compare the performance of Algorithm 3.1 with Algorithm 1.2, Algorithm 1.3, Algorithm 1.5 and Algorithm 7.2. The stopping criterion used for our computation is \(||x_{n+1}-x_{n}||< 10^{-3}\). We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Fig. 2 and Table 2.

Table 2 Numerical results for Example 6.2
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Fig. 2
figure 2

Top left: Case I; Top right: Case II; Bottom left: Case III; Bottom right: Case IV

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Remark 6.3

By using different starting points and plotting the graphs of errors against the number of iterations in each example (Examples 6.1–6.2), we obtain the numerical results displayed in Tables 1 and 2 and Figs. 1 and 2. We compared our proposed Algorithm 3.1 with Algorithm 1.2, Algorithm 1.3, Algorithm 1.5 and Algorithm 7.2. Furthermore, we note the following from our numerical experiments:

  • We observe that the different choices of the starting point and key parameters does not have a significant effects on the output of our method with respect to the performance of the proposed algorithm.

  • In all the examples, we can see from the tables and figures that the number of iterations for our proposed method remain consistent (well-behaved).

  • From the Table 2 and Fig. 2, we can see clearly that in terms of number of iterations, our proposed Algorithm 3.1 outperforms the other four existing methods. Table 1 and Fig. 1 also show that our method performs favourable well compared with the four existing methods.

7 Conclusion

In this paper, we studied the problem of finding the common solution of split variational inclusion problem, equilibrium problem and common fixed point of nonexpansive mappings. We introduced a new inertial viscosity S-iteration method for approximating the solution of the problem and we proved strong convergence theorem for the proposed algorithm without the knowledge of the operator norm. Finally, we applied our results to study other optimization problems and provided some numerical experiments with graphical illustrations to demonstrate the efficiency of our method in comparison with some existing methods in the current literature.

figure j

where \(f: H_1\rightarrow H_1\) is a contraction with constant \(k\in (0,1),~ S: H_1\rightarrow H_1\) is a nonexpansive mapping, \(\gamma \in (0, \frac{1}{L}),\) where L is the spectral radius of the operator \(A^*A,\) and \(A^*\) is the adjoint of \(A, \{\alpha _n\}\subset (0,1)\) and \(B_1: H_1\rightarrow 2^{H_1}, B_2: H_2\rightarrow 2^{H_2}\) are two multi-valued maximal monotone operators on \(H_1\) and \(H_2,\) respectively.