An iterative method for split inclusion problems without prior knowledge of operator norms

Abstract

In this paper, we study the approximation of solution (assuming existence) for the split inclusion problem in uniformly convex Banach spaces which are also uniformly smooth. We introduce an iterative algorithm in which the stepsizes are selected without the need for any prior information about the bounded linear operator norm and strong convergence obtained. The novelty of our algorithm is that the bounded linear operator norm is not given a priori and stepsizes are constructed step by step in a natural way. Our results extend and improve many recent and important results obtained in the literature on the split inclusion problem and its variations.

1 Introduction

Let $\mathcal {H}_1$ and $\mathcal {H}_2$ be two real Hilbert spaces, $T_1:\mathcal {H}_1\rightrightarrows \mathcal {H}_1$ and $T_2:\mathcal {H}_2\rightrightarrows \mathcal {H}_2$ be two set-valued maximal monotone mappings, $A:\mathcal {H}_1\rightarrow \mathcal {H}_2$ be a bounded linear operator, and $A^*$ be the adjoint of A. Many authors [5, 10, 15, 17, 37, 38] have studied the following split inclusion problem (SIP) in Hilbert spaces:

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\mathrm{Find}~~x^*\in \mathcal {H}_{1}\,\mathrm{such} \, \mathrm{that }\quad 0\in T_1(x^*),~~\mathrm{and}\\ &{}0\in T_2(Ax^*).\end{array}\right. \end{aligned}$$

(SIP)

The split inclusion problem has studied extensively by many authors and applied to solving many real life problems, such as in modelling intensity-modulated radiation therapy treatment planning [7, 8], modelling of inverse problems arising from phase retrieval, and in sensor networks in computerized tomography and data compression [4, 12].

Next, we give some examples which serve as motivations for studying the split inclusion problem and to understand its relationship with other problems in the literature. Byrne et al. [6] studied a split inverse problem (SInvP) formulated in Sect. 2 of [9] which concerns a model in which there are two given vector spaces $\mathcal {X}_1$ and $\mathcal {X}_2$ and a linear operator $A:\mathcal {X}_1\rightarrow \mathcal {X}_2$. In this model, two Inverse Problems (IP) are involved. The first one, denoted by (IP1), is formulated in the space $\mathcal {X}_1$, and the second one, denoted by (IP2), is in $\mathcal {X}_2$. Given these data, the Split Inverse Problem (SInvP) is formulated as follows:

$$\begin{aligned} \left\{ \begin{array}{ll}&{}\mathrm{Find}\quad x^* \in \mathcal {X}_1\;\; \text {solving (IP1)},~\text {and}\\ &{}Ax^*\in \mathcal {X}_2 \;\; \text {solves (IP2).}\end{array}\right. \end{aligned}$$

(SInvP)

Now, recall the split variational inequality problem (SVIP) introduced in [9], which is an SIP with a variational inequality problem (VIP) in each one of the two spaces. Let $\mathcal {H}_1$ and $\mathcal {H}_2$ be two real Hilbert spaces, and assume that there are given two point-to-point operators $F_1:\mathcal {H}_1\rightarrow \mathcal {H}_1$ and $F_2:\mathcal {H}_2\rightarrow \mathcal {H}_2$, a bounded linear operator $A:\mathcal {H}_1\rightarrow \mathcal {H}_2$, and nonempty, closed, and convex subsets $\Omega _1\subseteq \mathcal {H}_1$ and $\Omega _2\subseteq \mathcal {H}_2$. The SVIP is then formulated as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\mathrm{Find}~~ x^* \in \Omega _1 \;~~\mathrm{such~~ that}~~\; \langle F_1(x^*),x-x^*\rangle \ge 0, \quad \forall x \in \Omega _1, \;\; \text{ and }\\ &{}y^* = Ax^* \in \Omega _2 ~~\mathrm{~~solves}\;\;\, \langle F_2(y^*), y-y^*\rangle \ge 0, \quad \forall y \in \Omega _2. \end{array}\right. \end{aligned}$$

(SVIP)

The above problem (SVIP) can be structurally considered a special case of the split monotonic variational inclusion problem (SMVIP) when the operators $F_1$ and $F_2$ are monotones. It is formulated as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\mathrm{Find} ~~x^*\in ~~\mathrm{SOL} (F_1,\Omega _1) ~~\mathrm{~~such~~ that}\\ &{} Ax^* \in ~~\mathrm{SOL} (F_2,\Omega _2),\end{array}\right. \end{aligned}$$

(SMVIP)

where we denote by SOL$(F_1,\Omega _1)$ and SOL$(F_2,\Omega _2)$ the solution sets of the VIPs in (SVIP). Taking in (SMVIP), $\Omega _1 = \mathcal {H}_1, \Omega _2 = \mathcal {H}_2$ and choosing $x:= x^*-F_1(x^*) \in \mathcal {H}_1$ and $y = Ax^* -F_2(Ax^*) \in \mathcal {H}_2$, we obtain the Split Zeros Problem (SZP) for two operators $F_1:\mathcal {H}_1\rightarrow \mathcal {H}_1$ and $F_2:\mathcal {H}_2\rightarrow \mathcal {H}_2$, which were introduced in Sect. 7.3 of [9]. It is formulated as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\mathrm{Find} ~~x^*\in \mathcal {H}_1 ~~\mathrm{~~such~~ that~~} F_1(x^*)=0~~\mathrm{and}\\ &{}F_2(Ax^*)=0. \end{array}\right. \end{aligned}$$

(SZP)

Furthermore, we can define the set-valued mapping ${T}_1:\Omega _1\subset \mathcal {H}_1\rightrightarrows \mathcal {H}_1$ by

$$\begin{aligned} T_1(x):= \left\{ \begin{array}{lll} F_1(x)+\mathcal {N}_{\Omega _1}(x),&{}&{}x \in \Omega _1\\ \emptyset ,&{}&{}\mathrm{Otherwise}, \end{array} \right. \end{aligned}$$

and ${T}_2:\Omega _2\subset \mathcal {H}_2\rightrightarrows \mathcal {H}_2$ by

$$\begin{aligned} T_2(y):= \left\{ \begin{array}{lll} F_2(y)+\mathcal {N}_{\Omega _2}(y),&{}&{}y \in \Omega _2\\ \emptyset ,&{}&{} \mathrm{Otherwise}, \end{array} \right. \end{aligned}$$

where $\mathcal {N}_\Omega :\mathcal {H}\rightrightarrows \mathcal {H}$ is the normal cone of some nonempty, closed, and convex set $\Omega \subset \mathcal {H}$, which is defined at a point $u\in \Omega$ as

$$\begin{aligned} \mathcal {N}_\Omega (u):=\{d \in \mathcal {H}:\langle d,u-v\rangle \le 0,\quad \forall u \in \Omega \} , \end{aligned}$$

and the empty set otherwise. Then, under maximal monotonicity assumption on $F_1$ and $F_2$, Theorem 3 of [19] implies that $T_1$ and $T_2$ are maximal monotone mappings and so $x^*\in \mathrm{SOL}(F_1,\Omega _1)$ and $Ax^*\in \mathrm{SOL}(F_2,\Omega _2)$ if and only if $0\in T_1(x^*)$ and $0\in T_2(Ax^*)$, respectively. Hence, we observe that problem (SMVIP) is a special case of Split Inclusion Problem defined in (SIP). Thus, SIP is a remalkable problem, which contains a large class of applications; for further discussions of this problem, the reader can see [10, 17] and the references therein.

2 Some notations and definitions on uniformly convex Banach spaces

We start this part by recalling some definitions and notation used in this paper, which are standard and follows from [3, 39]. Throughout this paper, we write $\mathcal {S}:=\mathcal {R}$ to indicate that $\mathcal {S}$ is defined to be equal to $\mathcal {R}$. Let $\mathcal {B}$ be a real Banach space. The modulus of smoothness of $\mathcal {B}$ is the function $\rho _\mathcal {B}:[0,\infty ) \rightarrow [0,\infty )$ defined by

$$\begin{aligned} \rho _\mathcal {B}(t) := \sup \left\{ {1\over 2}(\Vert x+y\Vert + \Vert x-y\Vert )-1:\Vert x\Vert \le 1,\Vert y\Vert \le t\right\} . \end{aligned}$$

We assume that $1<p,\,q<\infty$ such that $\frac{1}{p}+\frac{1}{q}=1$. $\mathcal {B}$ is uniformly smooth if and only if

$$\begin{aligned} \lim \limits _{t\rightarrow 0} {\rho _\mathcal {B}(t)\over t} = 0, \end{aligned}$$

and q-uniformly smooth if there exists a $C_{q}>0$, such that $\rho _{\mathcal {B}}(t)\le C_{q}t^{q}$ for any $t >0$.

Let dim $\mathcal {B}\ge 2$. The modulus of convexity of $\mathcal {B}$ is the function $\delta _\mathcal {B}:(0,2]\rightarrow [0,1]$ defined by

$$\begin{aligned} \delta _\mathcal {B}(\epsilon ):=\inf \left\{ 1-\Big |\Big | \frac{x+y}{2}\Big |\Big |:\Vert x\Vert =\Vert y\Vert =1; \epsilon =\Vert x-y\Vert \right\} . \end{aligned}$$

$\mathcal {B}$ is uniformly convex if and only if $\delta _\mathcal {B}(\epsilon ) >0$ for all $\epsilon \in (0,2]$ and p-uniformly convex if there is a $C_{p}>0$, so that $\delta _\mathcal {B}(\epsilon )\ge C_{p}\epsilon ^{p}$ for any $\epsilon \in (0,2]$. The uniform convexity is a geometric property of the unit ball. In particular, the unit sphere must be round and cannot include any line segment. Surprisingly, this geometric property ensures a topological one. That is, it follows from Milman–Pettis’ Theorem that every uniformly convex Banach space is reflexive; see Theorem 3.31 of [3].

It is well known that $\mathcal {B}$ is p-uniformly convex and uniformly smooth if and only if its dual $\mathcal {B}^*$ is q-uniformly smooth and uniformly convex. Then, from now on, mainly in Sect. 3, we assume that $\mathcal {B}$ is p-uniformly convex and uniformly smooth with constant $C_p$ and we denote $C_q$ with $1/p+1/q=1$ the associated constant to $\mathcal {B}^*$. It is also a common knowledge that the duality mapping $\mathcal {J}^\mathcal {B}_p$ is one-to-one, single valued and satisfies $\mathcal {J}^\mathcal {B}_p=(\mathcal {J}^{\mathcal {B}^*}_q)^{-1}$, where $\mathcal {J}^{\mathcal {B}^*}_q$ is the duality mapping of $\mathcal {B}^*$ (see [1, 11]).

Definition 2.1

The duality mapping $\mathcal {J}^\mathcal {B}_p:\mathcal {B}\rightrightarrows \mathcal {B}^{*}$ is defined by

$$\begin{aligned} \mathcal {J}^\mathcal {B}_p(x):=\left\{ x^{*}\in \mathcal {B}^*:\langle x,x^*\rangle =\Vert x\Vert ^{p},\Vert x^*\Vert =\Vert x\Vert ^{p-1}\right\} . \end{aligned}$$

We denote it as $\mathcal {J}$ when $p=1$ and the domain space $\mathcal {B}$ is clear. The duality mapping $\mathcal {J}^\mathcal {B}_p$ is said to be weak-to-weak star continuous if

$$\begin{aligned} x_{n}\rightharpoonup x\Rightarrow \langle \mathcal {J}^\mathcal {B}_p(x_{n}),y\rangle \rightarrow \langle \mathcal {J}^\mathcal {B}_p(x),y\rangle \end{aligned}$$

holds true for any $y\in \mathcal {B}$. We note here that $l_{p}(p>1)$ spaces have such a property, but $L_{p}(p>2)$ does not share this property.

Let $\mathcal {B}$ be a reflexive, strictly convex, and smooth Banach space and let $T:\mathcal {B}\rightrightarrows \mathcal {B}^*$ be a maximal monotone operator. Then, for $\lambda >0$ and $x\in \mathcal {B}$, consider the metric resolvent of T, for $\lambda >0$, as

$$\begin{aligned} J_{\lambda T}(x):=\{z\in \mathcal {B}:0\in \mathcal {J}(z-x)+\lambda T(z)\} \end{aligned}$$

or equivalently $J_{\lambda T}=(I+\lambda \mathcal {J}^{-1}T)^{-1}: \mathcal {B}\rightarrow \mathcal {B}^*$, which is point-to-point, full domain, and nonexpansive operator. Then

$$\begin{aligned} 0\le \left\langle J_{\lambda T}(x)-J_{\lambda T}(y),\mathcal {J}(x-J_{\lambda T}(x))-\mathcal {J}(y-J_{\lambda T}(y))\right\rangle \end{aligned}$$

holds, for all x,  $y\in \mathcal {B}$; see, for instance, Proposition 57.5(b) of [40].

In addition, we consider the relative resolvent of T, for $\lambda >0$, as

$$\begin{aligned} Q_{\lambda T}(x):=\{z\in \mathcal {B}:Jx\in Jz+\lambda T(z)\}, \end{aligned}$$

or equivalently $Q_{\lambda T}=(J+\lambda T)^{-1}J:\mathcal {B}\rightarrow \mathcal {B}^*$, which is point-to-point and relatively nonexpansive mapping. That is, for $\lambda >0$,

$$\begin{aligned} 0 \le \langle Q_{\lambda T}(x) - Q_{\lambda T}(y), \mathcal {J}(x) - \mathcal {J}(Q_{\lambda T}(x)) - (\mathcal {J}y - Q_{\lambda T}(y))\rangle \end{aligned}$$

for all x,  $y\in \mathcal {B}$; see, Theorem 5.2. of [27].

3 Previous related schemes and our proposal

In this part, we describe some previous schemes, which use the norm of the operator A to find the stepsizes. In 2011, Byrne et al. [5] gave the following convergence theorem for the split inclusion problem (SIP) working in Hilbert spaces.

Theorem 1.1

(Theorem 3.2 of [5]). Let $\mathcal {H}_1$ and $\mathcal {H}_2$ be real Hilbert spaces, $A:\mathcal {H}_1\rightarrow \mathcal {H}_2$ be a linear and bounded operator, and $A^*$ denote the adjoint of A. Let $T_1:\mathcal {H}_1\rightrightarrows \mathcal {H}_1$ and $T_2:\mathcal {H}_2\rightrightarrows \mathcal {H}_2$ be set-valued maximal monotone mappings, $\lambda > 0$ and $\gamma \in \Big (0,\frac{2}{\Vert A^*A\Vert }\Big )$. Suppose that $\Gamma \ne \emptyset$, the solution set of (SIP). The sequence $(x_n)_{n\in \mathbb {N}}$ defined as follows:

$$\begin{aligned} x_{n+1}=J_{\lambda T_1}\Big [x_n-\gamma A^*\Big (I-J_{\lambda T_2} \Big )Ax_n\Big ] \end{aligned}$$

converges weakly to an element $x^* \in \Gamma$.

Furthermore, in 2013, Chuang [10] gave the following strong convergence theorem for problem (SIP).

Theorem 1.2

(Theorem 4.1 of [10]). Let $\mathcal {H}_1$ and $\mathcal {H}_2$ be real Hilbert spaces, $A:\mathcal {H}_1\rightarrow \mathcal {H}_2$ be a linear and bounded operator, and $A^*$ denote the adjoint of A. Let $T_1:\mathcal {H}_1\rightrightarrows \mathcal {H}_1$ and $T_2:\mathcal {H}_2\rightrightarrows \mathcal {H}_2$ be set-valued maximal monotone mappings. Let $(\alpha _n)_{n\in \mathbb {N}}\subset (0,1)$, $(\lambda _n)_{n\in \mathbb {N}}$ be a sequence in $(0,\infty )$ and $(\gamma _n)_{n\in \mathbb {N}}\subset \left( 0,2/(\Vert A\Vert ^2+2)\right)$. Let $\Gamma$ be the solution set of (SIP) and suppose that $\Gamma \ne \emptyset$. Let $(x_n)_{n\in \mathbb {N}}$ be defined by

$$\begin{aligned} x_{n+1}=J_{\lambda _nT_1}\Big [(1-\alpha _n\gamma _n)x_n-\gamma _n A^*\Big (I-J_{\lambda _nT_2} \Big )Ax_n\Big ],~~n \ge 1. \end{aligned}$$

Assume that

$$\begin{aligned} \lim _{n\rightarrow \infty } \alpha _n =0,~~\sum \limits _{n=1}^{\infty } \alpha _n\gamma _n =\infty ,~~ \underset{n\rightarrow \infty }{\liminf }\gamma _n>0,~~ \underset{n\rightarrow \infty }{\liminf }\lambda _n >0. \end{aligned}$$

Then, $\lim _{n\rightarrow \infty } x_n=\bar{x}:= P_\Gamma 0$, that is, $\bar{x}$ is the minimal norm solution of (SIP).

Recently, Takahashi and Takahashi [29] considered the problem (SIP) in uniformly convex and smooth Banach spaces, which are higher spaces other than Hilbert spaces. In other words, let $\mathcal {B}_1$ and $\mathcal {B}_2$ be two real uniformly convex and smooth Banach spaces, $T_1:\mathcal {B}_1\rightrightarrows \mathcal {B}^*_1$ and $T_2:\mathcal {B}_2\rightrightarrows \mathcal {B}^*_2$ be two set-valued maximal monotone mappings, $A:\mathcal {B}_1\rightarrow \mathcal {B}_2$ be a bounded linear operator, and $A^*:\mathcal {B}_2^*\rightarrow \mathcal {B}_1^*$ be the adjoint of A. Takahashi and Takahashi [29] studied the Split Inclusion Problem given in (SIP) in Banach spaces, that is

$$\begin{aligned} \left\{ \begin{array}{ll} &{} \mathrm{Find}~~x^*\in \mathcal {B}_1~~\mathrm{~~ such~~ that~~} 0\in T_1(x^*),~~\mathrm{and}\\ &{}0\in T_2(Ax^*).\end{array}\right. \qquad (\mathrm{SIP}_\mathrm{\mathcal {B}}) \end{aligned}$$

Let the set of solutions of the split inclusion problem (SIP$_\mathrm{\mathcal {B}}$) be denoted by $\Gamma$. Using the shrinking projection method, Takahashi and Takahashi [29] proved the following strong convergence theorem for finding a solution of (SIP$_\mathrm{\mathcal {B}}$) in uniformly convex and smooth Banach spaces.

Theorem 1.3

(Theorem 6 of [29]). Let $\mathcal {B}_1$ and $\mathcal {B}_2$ be uniformly convex and smooth Banach spaces and let $\mathcal {J}^{\mathcal {B}_1}$ and $\mathcal {J}^{\mathcal {B}_2}$ be the duality mappings on $\mathcal {B}_1$ and $\mathcal {B}_2$, respectively. Let $T_1:\mathcal {B}_1\rightrightarrows \mathcal {B}^*_1$ and $T_2:\mathcal {B}_2\rightrightarrows \mathcal {B}^*_2$ be maximal monotone operators, such that $T^{-1}_1(0)\ne \emptyset$ and $T^{-1}_2(0)\ne \emptyset ,$ respectively. Let $Q_{\lambda T_2}$ be the relative resolvent of $T_2$ for $\lambda > 0$. Let $A:\mathcal {B}_1\rightarrow \mathcal {B}_2$ be a bounded linear operator, such that $A\ne 0$ and let $A^*$ be the adjoint operator of A. Suppose that $\Gamma \ne \emptyset$. Let $x_1 \in \mathcal {B}_1$ and let $C_1 = T^{-1}_1(0).$ Let $(x_n)_{n\in \mathbb {N}}$ be a sequence generated by

$$\begin{aligned} \left\{ \begin{array}{lll} z_n &{}=&{} x_n-t_n (\mathcal {J}^{\mathcal {B}_1})^{-1}A^*\mathcal {J}^{\mathcal {B}_2}(Ax_n-Q_{\lambda _n T_2}(Ax_n)), \\ C_{n+1}&{}=&{}\left\{ z \in C_n:\langle z_n-z, \mathcal {J}^{\mathcal {B}_1}(x_n-z_n)\rangle \ge 0\right\} \\ x_{n+1} &{}=&{} P_{C_{n+1}}x_1,~~\forall n \ge 1, \end{array} \right. \end{aligned}$$

(1.1)

where $\{t_n\}, (\lambda _n)_{n\in \mathbb {N}}\subset (0,\infty )$ satisfy the conditions, such that for some $a, b, c\in \mathbb {R}$

$$\begin{aligned} 0<a\le t_n\Vert A\Vert ^2\le b<1~~\mathrm{and}~~0<c\le \lambda _n,~~\forall n \ge 1. \end{aligned}$$

Then, the sequence $(x_n)_{n\in \mathbb {N}}$ converges strongly to a point $z_0\in \Gamma$, where $z_0 = P_{\Gamma }x_1$.

Closely studying the result of Takahashi and Takahashi [29], we see that the choice of step-size $t_n$ is operator norm dependent. This is a drawback in terms of efficiency and implementation of the iteration process (1.1). Furthermore, the set $C_n$ has to be computed in each iteration step in the iteration process (1.1). This makes the iteration process (1.1), above, computationally expensive, because the complexity of the proposed iteration grows dramatically when n is large. Note that the set where is doing the projection step can be written as

$$\begin{aligned} C_{n+1}= & {} C_n\cap \{z\in \mathcal {B}_1: \langle z_n-z, \mathcal {J}^{\mathcal {B}_1}(x_n-z_n)\rangle \ge 0\}\\= & {} \displaystyle \cap _{\ell =2}^n\{z\in C_1: \langle z_\ell -z, \mathcal {J}^{\mathcal {B}_1}(x_\ell -z_\ell )\rangle \ge 0\}. \end{aligned}$$

Moreover, it is worth mentioning that in most of the results on the split inclusion problem studied so far, the choice of the stepsize depends on the operator norm $\Vert A\Vert$. This involves knowing a priori the norm (or at least an estimate of the norm) of the bounded linear operator, which is in general not an easy work in practice. Hence, it makes the implementation of the iteration process inefficient when the computation of the operator norm $\Vert A\Vert$ is not explicit (see [13, 16]). Even in finite dimensions, computing the norm of bounded linear operator is a difficult task as shown by the following Theorem of Hendrickx and Olshevsky [14]:

Theorem 1.4

(Theorem 2.3 of [14]). For any rational $p\in [1,\infty )$ except $p=1, 2$, unless $P = NP$, there is no algorithm which computes the p-norm of a matrix with entries in $\{-1, 0, 1\}$ to relative error $\epsilon$ with running time polynomial in the dimensions.

These observations lead to the following natural question.

Question: Can we obtain strong convergence result for the split inclusion problem in higher spaces other than Hilbert spaces in which the iteration process is operator norm independent for solving the Split Inclusion Problem (SIP)? Moreover, could the proposed iteration process be more efficient and implementable than the iteration processes already obtained for the split inclusion problem in the literature?

Our main purpose, in this paper, is to give an affirmative answer to the above question. Thus, we propose an iterative scheme, which generated a sequence strongly convergent to some solution of the split inclusion problem in p-uniformly convex real Banach spaces which are also uniformly smooth. The proposed iteration process is constructed in such a way that we do not need to know a priori the norm or an estimate of the norm of the bounded linear operator for approximating solution of the split inclusion problems and so it can be more efficiently implemented. Our results complement and extend many recent and important results on the split inclusion problems both in Hilbert spaces and Banach spaces.

Organization of the paper: The next subsection provides some preliminaries results that will be used in the remainder of this paper. In Sect. 3, the proposed algorithm and its strong convergence are analyzed by choosing stepsizes without norm dependence, and moreover, we remark some consequences. Finally, Sect. 4 gives some concluding remarks.

4 Preliminaries

This section discusses some preliminary results which will be used throughout the paper.

Lemma 2.1

(Corollary $1'$ of [34]). Let $x,y \in \mathcal {B}$. If $\mathcal {B}$ is q-uniformly smooth Banach space, then there exists $C_{q}>0$, such that

$$\begin{aligned} \Vert x-y\Vert ^{q}\le \Vert x\Vert ^{q}-q\langle \mathcal {J}_{q}(x),y\rangle +C_{q}\Vert y\Vert ^{q}. \end{aligned}$$

(2.1)

Definition 2.2

Let $\mathcal {B}$ be a p-uniformly convex real Banach spaces which is also uniformly smooth. Given a Gâteaux differentiable convex function $f:\mathcal {B}\rightarrow \mathbb {R}$, the Bregman distance with respect to f is defined as

$$\begin{aligned} \Delta _f(x,y)=f(y)-f(x)-\langle f'(x),y-x\rangle , \quad x,y \in \mathcal {B}. \end{aligned}$$

We note here also that the duality mapping $\mathcal {J}^\mathcal {B}_p$ is in fact the derivative of the function $f_p(x):= \frac{1}{p}\Vert x\Vert ^p$, for $2\le p<\infty$. Hence, the Bregman distance with respect to $f_p$ (see [24]) is given by

$$\begin{aligned} \Delta _p(x,y)=&\,\frac{1}{q}\Vert x\Vert ^p-\langle \mathcal {J}^\mathcal {B}_p(x),y\rangle +\frac{1}{p}\Vert y\Vert ^p. \end{aligned}$$

(2.2)

In addition, the Bregman distance possesses the following important properties:

$$\begin{aligned} \Delta _{p}(x,y)=\Delta _{p}(x,z)+\Delta _{p}(z,y)+\left\langle z-y,\mathcal {J}^\mathcal {B}_p(x)-\mathcal {J}^\mathcal {B}_p(y)\right\rangle ,\quad \forall x,y,z\in \mathcal {B}.\nonumber \\ \end{aligned}$$

(2.3)

$$\begin{aligned} \Delta _{p}(x,y)+\Delta _{p}(y,x)=\left\langle x-y,\mathcal {J}^\mathcal {B}_p(x)-\mathcal {J}^\mathcal {B}_p(y)\right\rangle ,\quad \forall x,y\in \mathcal {B}. \end{aligned}$$

(2.4)

When considering the p-uniformly convex space, the Bregman distance and the metric distance have the following relation (see (7) of [21]):

$$\begin{aligned} \tau \Vert x-y\Vert ^{p}\le \Delta _{p}(x,y)\le \left\langle x-y,\mathcal {J}^\mathcal {B}_p(x)-\mathcal {J}^\mathcal {B}_p(y)\right\rangle , \end{aligned}$$

(2.5)

where $\tau >0$ is some fixed number.

Let $\Omega$ be a nonempty, closed, and convex subset of $\mathcal {B}$. The metric projection

$$\begin{aligned} P_{\Omega }x:=\arg \min _{y\in \Omega } \Vert x-y\Vert ,\quad x\in \mathcal {B}, \end{aligned}$$

is the unique minimizer of the norm distance, which can be characterized by a variational inequality:

$$\begin{aligned} \langle \mathcal {J}^\mathcal {B}_{p}(x-P_{\Omega }x),z-P_{\Omega }x\rangle \le 0,\quad \forall z\in \Omega . \end{aligned}$$

(2.6)

Similar to the metric projection, the Bregman projection is defined as

$$\begin{aligned} \Pi _{\Omega }x=\arg \min _{y\in \Omega }\Delta _{p}(x,y),\quad x\in \mathcal {B}, \end{aligned}$$

the unique minimizer of the Bregman distance (see Proposition 1.25 of [20]). The Bregman projection can also be characterized by a variational inequality:

$$\begin{aligned} \langle \mathcal {J}^\mathcal {B}_{p}(x)-\mathcal {J}^\mathcal {B}_p(\Pi _{\Omega }x),z-\Pi _{\Omega }x\rangle \le 0,\quad \forall z\in \Omega , \end{aligned}$$

(2.7)

from which one has

$$\begin{aligned} \Delta _{p}(\Pi _{\Omega }x,z)\le \Delta _{p}(x,z)-\Delta _{p}(x,\Pi _{\Omega }x),~\forall z\in \Omega . \end{aligned}$$

(2.8)

Following [1] [see equation (6.3), page 29], we make use of the function $V_{p}:\mathcal {B}^*\times \mathcal {B}\rightarrow [0,+\infty )$, which is defined by

$$\begin{aligned} V_{p}(x,y):=\frac{1}{q}\Vert x\Vert ^{q}-\langle x,y\rangle + \frac{1}{p}\Vert y\Vert ^{p},~\forall x\in \mathcal {B}^*,y\in \mathcal {B}. \end{aligned}$$

Then, $V_{p}$ is nonnegative and $V_{p}(x,y)=\Delta _{p}(\mathcal {J}^{\mathcal {B}^*}_{q}(x),y)$ for all $x\in \mathcal {B}^*$ and $y\in \mathcal {B}$. Moreover, by the subdifferential inequality

$$\begin{aligned} \langle f^{\prime }(x),y-x\rangle \le f(y)-f(x), \end{aligned}$$

with $f(x)=\frac{1}{q}\Vert x\Vert ^{q},x\in \mathcal {B}^*,~ \mathrm{then}~f^{\prime }(x)=\mathcal {J}_{q}^{\mathcal {B}^*}$. Then, we have

$$\begin{aligned} \langle \mathcal {J}_{q}^{\mathcal {B}^*}(x),y\rangle \le \frac{1}{q}\Vert x+y\Vert ^{q}-\frac{1}{q}\Vert x\Vert ^{q} \end{aligned}$$

(2.9)

and from (2.9), we obtain (see [24])

$$\begin{aligned} V_{p}(x^*+y^*,x) \ge&\,V_{p}(x^*,x)+\langle y^*,\mathcal {J}_{q}^{\mathcal {B}^*}(x^*)-x\rangle , \end{aligned}$$

(2.10)

for all $x\in \mathcal {B}$ and $x^*,y^*\in \mathcal {B}^*$. In addition, since $f=f_p$ is a proper lower semi-continuous and convex function, we have $f^*=f_{p}^*$ is a proper weak$^*$ lower semi-continuous and convex function; see [18]. Hence, $V_{p}$ is convex in the second variable. Thus, for all $z\in \mathcal {B}$, we can show that (see [24])

$$\begin{aligned} \Delta _{p}\big (\mathcal {J}_{q}^{\mathcal {B}^*}\big (\sum _{i=1}^{N}t_{i}\mathcal {J}_{p}^B(x_{i})\big ),z\big ) \le V_{p}\Big (\sum _{i=1}^{N}t_{i}\mathcal {J}_{p}^{B}(x_{i}),z\Big ) =\, \sum _{i=1}^{N}t_{i}\Delta _{p}(x_{i},z)\nonumber \\ \end{aligned}$$

(2.11)

where $(x_i)_{i=1,\ldots ,N}\subset \mathcal {B}$ and $(t_i)_{i=1,\ldots ,N}\subset (0,1)$ with $\sum _{i=1}^Nt_{i}=1.$ For more details, see [22, 23, 25, 26, 32].

We need the following useful lemma in the sequel.

Lemma 2.2

(Lemma 2.5 of [35]). Let $(a_{n})_{n\in \mathbb {N}}$ be a sequence of nonnegative real numbers satisfying the following relation:

$$\begin{aligned} a_{n+1}\le (1-\alpha _{n})a_{n}+\alpha _{n}\sigma _{n}+\gamma _{n},~n\ge 0, \end{aligned}$$

where $(\alpha _n)_{n\in \mathbb {N}},~~(\gamma _n)_{n\in \mathbb {N}}$ and $(\sigma _n)_{n\in \mathbb {N}}$ are sequences of real numbers satisfying

• (i) $(\alpha _{n})_{n\in \mathbb {N}}\subset [0,1],~\sum \alpha _{n}=\infty ;$

• (ii) $\limsup _{n\rightarrow \infty } \sigma _{n}\le 0;$

• (iii) $\gamma _{n}\ge 0,~\sum \gamma _{n}<\infty$.

Then, $a_{n}\rightarrow 0$ as $n\rightarrow \infty .$

5 Main results

Using the ideas of [41], we introduce an iterative algorithm in which the stepsize does not depend on the norm $\Vert A\Vert$ and then prove the strong convergence of the sequence generated by the algorithm in p-uniformly convex real Banach spaces which are also uniformly smooth.

Algorithm Let $\mathcal {B}_1$ and $\mathcal {B}_2$ be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let $T_1:\mathcal {B}_1\rightrightarrows \mathcal {B}_1$ and $T_2:\mathcal {B}_2\rightrightarrows \mathcal {B}_2$ be maximal monotone operators, such that $T^{-1}_1(0)\ne \emptyset$ and $T^{-1}_2(0)\ne \emptyset$, respectively. Let $J_{\lambda T_1}$ and $Q_{\lambda T_2}$, for $\lambda > 0$, be the metric and relative resolvents of $T_1$ and $T_2$, respectively. Let $A:\mathcal {B}_1 \rightarrow \mathcal {B}_2$ be a bounded linear operator, such that $A\ne 0$ and $A^*:\mathcal {B}_2^* \rightarrow \mathcal {B}_1^*$ be the adjoint of A. For a fixed $u \in \mathcal {B}_1$, choose an initial guess $x_1 \in \mathcal {B}_1$ arbitrarily. Let $(\alpha _n)_{n\in \mathbb {N}} \subset [0,1]$. Assume that the nth iterate $x_n \in \mathcal {B}_1$ has been constructed; then, we calculate the $(n+1)$th iterate $x_{n+1}$ via the formula:

$$\begin{aligned} \left\{ \begin{array}{rcl} u_{n}&{}=&{}\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\left[ \mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n)-t_nA^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\right] ,\\ x_{n+1} &{}=&{}\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\Big [\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}\left( J_{\lambda T_1}(u_{n})\right) \Big ]. \end{array} \right. \end{aligned}$$

(3.1)

Let the stepsize $t_n$ be chosen in such a way that

$$\begin{aligned} t_n^{q-1}\in \left( 0,\Big (\frac{q\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p}{C_q\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q}\Big )\right) ,~~ n\in \Omega , \end{aligned}$$

(3.2)

where the index set $\Omega := \{n \in \mathbb {N}:(I-Q_{\lambda T_2})Ax_n\ne 0\}$ otherwise $t_n=t$ (t being any nonnegative value).

Before the formal analysis of the convergence properties of the algorithm (3.1)–(3.2), we note that the step-size $t_n$ does need compute the norm of any operator, and also, it is possible to use a backtracking procedure to find it without multiple evaluations of the vector norm.

In the following, we present two lemmas establishing that the proposed algorithm is well-defined and the generated sequences are bounded.

Lemma 3.1

Suppose that the split inclusion problem (SIP$_\mathrm{\mathcal {B}}$) has a nonempty solution set $\Gamma$. Then, $t_n$ defined by (3.2) is well defined.

Proof

First, we observe that in algorithm (3.1)–(3.2) the choice of the step-size $t_n$ is independent of the norm $\Vert A\Vert$. In addition, the value of t does not influence the considered algorithm, but it was introduced just for the sake of clarity. Furthermore, we show that $t_n$ is well defined. Now, let $x \in \Gamma$. Then, $Ax=Q_{\lambda T_2} Ax$. Hence

$$\begin{aligned}&\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p\\&\quad = \langle \mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n, (I-Q_{\lambda T_2})Ax_n\rangle \\&\quad =\langle \mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n, Ax_n-Ax+Q_{\lambda T_2} Ax-Q_{\lambda T_2} Ax_n\rangle \\&\quad =\langle \mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n, Ax_n-Ax\rangle \\&\qquad +\langle \mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n, Q_{\lambda T_2} Ax-Q_{\lambda T_2} Ax_n\rangle \\&\quad =\langle A^*\mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n, x_n-x\rangle \\&\qquad +\langle \mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n, Q_{\lambda T_2} Ax-Q_{\lambda T_2} Ax_n\rangle \\&\quad \le \Vert A^*\mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n\Vert \Vert x_n-x\Vert \\&\qquad + \Vert \mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n\Vert \Vert Q_{\lambda T_2} Ax-Q_{\lambda T_2} Ax_n\Vert \\&\quad = \Vert A^*\mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n\Vert \Vert x_n-x\Vert \\&\qquad + \Vert (I-Q_{\lambda T_2})Ax_n\Vert ^{p-1}\Vert Q_{\lambda T_2} Ax-Q_{\lambda T_2} Ax_n\Vert . \end{aligned}$$

Consequently, for $n \in \Omega$, that is, $\Vert (I-Q_{\lambda T_2})Ax_n\Vert > 0$, we get $\Vert A^*\mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n\Vert \Vert x_n-x\Vert >0$ and $\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^{p-1}\Vert Q_{\lambda T_2} Ax-Q_{\lambda T_2} Ax_n\Vert >0$. Since $\Vert A^*\mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n\Vert \Vert x_n-x\Vert >0$, then we obtain that $\Vert A^*\mathcal {J}_{p}^{\mathcal {B}_2}(I-Q_{\lambda T_2})Ax_n\Vert \ne 0$. This implies that $t_n$ is well defined. $\square$

Lemma 3.2

Let $\mathcal {B}_1$ and $\mathcal {B}_2$ be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let $T_1:\mathcal {B}_1\rightrightarrows \mathcal {B}_1$ and $T_2:\mathcal {B}_2\rightrightarrows \mathcal {B}_2$ be maximal monotone operators, such that $T^{-1}_1(0)\ne \emptyset$ and $T^{-1}_2(0)\ne \emptyset$, respectively. Let $Q_{\lambda T_2}$ be the metric resolvent of $T_2$ for $\lambda > 0.$ Let $A:\mathcal {B}_1 \rightarrow \mathcal {B}_2$ be a bounded linear operator, such that $A\ne 0$ and $A^*:\mathcal {B}_2^* \rightarrow \mathcal {B}_1^*$ be the adjoint of A. Suppose that $\Gamma \ne \emptyset$ and $(\alpha _n)_{n\in \mathbb {N}}\subset (0,1)$. Let the sequence $(x_n)_{n\in \mathbb {N}}$ be generated by (3.1)–(3.2). Assume that for small enough $\epsilon >0,$

$$\begin{aligned} t_n\in \left( \epsilon ,\Big (\frac{q\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p}{C_q\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q}-\epsilon \Big )^{\frac{1}{q-1}}\right) ,~~ n\in \Omega . \end{aligned}$$

Then, the sequences $(x_n)_{n\in \mathbb {N}}$ and $(u_n)_{n\in \mathbb {N}}$ are bounded.

Proof

It follows from (2.6) that for any $z \in \Gamma$, where $\Gamma$ is the solution set of problem (SIP$_\mathrm{\mathcal {B}}$), we have

$$\begin{aligned}&\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n), Ax_n-Az \rangle \nonumber \\&\quad = \Vert Ax_n-Q_{\lambda T_2}(Ax_n)\Vert ^{p} +\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n), Q_{\lambda T_2}(Ax_n)-Az \rangle \nonumber \\&\quad \ge \Vert Ax_n-Q_{\lambda T_2}(Ax_n)\Vert ^{p}. \end{aligned}$$

(3.3)

In addition, from (3.3) and Lemma 2.1, we obtain that

$$\begin{aligned} \Delta _p(u_n,z)\le&\, \Delta _p(\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}[\mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n)-t_nA^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n],z)\nonumber \\ =&\, \frac{1}{q}\Vert \mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n)-t_nA^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n)\Vert ^{q}-\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n),z \rangle \nonumber \\&\,+t_n\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n),Az\rangle +\frac{1}{p}\Vert z\Vert ^{p}\nonumber \\ \le&\, \frac{1}{q}\Vert \mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n)\Vert ^q-t_n\langle Ax_n,\mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n)\rangle \nonumber \\&\,+\frac{C_qt_n^q}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\nonumber \\&\, -\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n),z\rangle +t_n\langle Az,\mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n)\rangle + \frac{1}{p}\Vert z\Vert ^p\nonumber \\ =&\, \frac{1}{q}\Vert x_n\Vert ^{p}-\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n),z\rangle +\frac{1}{p}\Vert z\Vert ^p\nonumber \\&\, + t_n\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n), Az-Ax_n\rangle \nonumber \\&\, + \frac{C_qt_n^q}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\nonumber \\ =&\, \Delta _p(x_n,z)+t_n\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n, Az-Ax_n\rangle \nonumber \\&\,+ \frac{C_qt_n^q}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q. \end{aligned}$$

(3.4)

Now, using (3.3) in (3.4), we obtain

$$\begin{aligned}&\Delta _p(u_n,z)\le \Delta (x_n,z)-t_n\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p+\frac{C_qt_n^q}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\nonumber \\&\quad =\Delta (x_n,z)-t_n\Big [\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p-\frac{C_qt_n^{q-1}}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\Big ]. \end{aligned}$$

(3.5)

By the condition on $t_n$, it follows from (3.5) that

$$\begin{aligned} \Delta _p(u_n,z)\le \Delta _p(x_n,z). \end{aligned}$$

Furthermore, using (2.8) in (3.1), we get that

$$\begin{aligned} \Delta _p(x_{n+1},z)\le&\, \Delta _p\Big (\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\Big [\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n)\Big ] ,z\Big )\nonumber \\ =&\, \alpha _n\Delta _p(u,z)+(1-\alpha _n)\Delta _p(u_n,z)\nonumber \\ \le&\, \alpha _n\Delta _p(u,z)+(1-\alpha _n)\Delta _p(x_n,z)\nonumber \\ \le&\, \max \{\Delta _p(u,z),\Delta _p(x_n,z)\}\nonumber \\ \vdots \nonumber \\ \le&\, \max \{\Delta _p(u,z),\Delta _p(x_1,z)\}. \end{aligned}$$

(3.6)

Therefore, $(\Delta _p(x_n,z))_{n\in \mathbb {N}}$ is bounded and consequently, the sequence $(\Delta _p(u_n,z))_{n\in \mathbb {N}}$ is bounded. Thus, the sequences $(x_n)_{n\in \mathbb {N}}$ and $(u_n)_{n\in \mathbb {N}}$ are bounded. $\square$

Now, we proceed by proving strong convergence of the sequence generated by Algorithm (3.1)–(3.2).

Theorem 3.3

Let $\mathcal {B}_1$ and $\mathcal {B}_2$ be two p-uniformly convex real Banach spaces which are also uniformly smooth. Let $T_1:\mathcal {B}_1\rightrightarrows \mathcal {B}_1$ and $T_2:\mathcal {B}_2\rightrightarrows \mathcal {B}_2$ be maximal monotone operators, such that $T^{-1}_1(0)\ne \emptyset$ and $T^{-1}_2(0)\ne \emptyset$, respectively. Let $Q_{\lambda T_2}$ be the metric resolvent of $T_2$ for $\lambda > 0.$ Let $A:\mathcal {B}_1 \rightarrow \mathcal {B}_2$ be a bounded linear operator such that $A\ne 0$ and $A^*:\mathcal {B}_2^* \rightarrow \mathcal {B}_1^*$ be the adjoint of A. Suppose that $\Gamma \ne \emptyset$ and $(\alpha _n)_{n\in \mathbb {N}}\subset (0,1)$. Let the sequence $(x_n)_{n\in \mathbb {N}}$ be generated by (3.1)–(3.2). Assume that for small enough $\epsilon >0$:

$$\begin{aligned} t_n\in \left( \epsilon ,\Big (\frac{q\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p}{C_q\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q}-\epsilon \Big )^{\frac{1}{q-1}}\right) ,~~ n\in \Omega . \end{aligned}$$

Suppose the following conditions are satisfied:

• (i) $\lim _{n\rightarrow \infty }\alpha _n= 0$;

• (ii) $\sum _{n=1}^{\infty }\alpha _{n}=\infty$.

Then, $(u_n)_{n\in \mathbb {N}}$ and $(x_n)_{n\in \mathbb {N}}$ both converge strongly to $\bar{x}$, where $\bar{x}=\Pi _{\Gamma }u$.

Proof

Let $\bar{x}=\Pi _{\Gamma }u$ and let us set $v_n:=\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\Big [\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n)\Big ], n \ge 1$. Then, using (2.10) and (2.11), we obtain

$$\begin{aligned}&\Delta _p(x_{n+1},\bar{x})\le \Delta _p(\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\Big [\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n)\Big ],\bar{x})\nonumber \\&\quad = V_{p}(\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n),\bar{x})\nonumber \\&\quad \le V_{p}(\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n)-\alpha _{n}(\mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x})),\bar{x})\nonumber \\&\qquad -\langle -\alpha _{n}(\mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x})),\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\Big [\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n)\Big ]-\bar{x}\rangle \nonumber \\&\quad = V_{p}(\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n),\bar{x})+\alpha _{n}\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_{n}-\bar{x}\rangle \nonumber \\&\quad = \Delta _p(\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\Big [\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n)\Big ],\bar{x})+\alpha _{n}\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_{n}-\bar{x}\rangle \nonumber \\&\quad \le \alpha _{n}\Delta _p(\bar{x},\bar{x})+ (1-\alpha _{n})\Delta _p(u_n,\bar{x})+\alpha _{n}\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_{n}-\bar{x}\rangle \nonumber \\&\quad = (1-\alpha _{n})\Delta _p(u_n,\bar{x})+\alpha _{n}\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_{n}-\bar{x}\rangle \nonumber \\&\quad \le (1-\alpha _{n})\Delta _p(x_n,\bar{x})+\alpha _{n}\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_{n}-\bar{x}\rangle . \end{aligned}$$

(3.7)

We now consider two cases to prove the strong convergence.

Case 1: Suppose that there exists $n_{0}\in \mathbb {N}$, such that $(\Delta _p(x_n,\bar{x}))_{n\in \mathbb {N}}$ is monotonically non-increasing. Then, obviously $(\Delta _p(x_n,\bar{x}))_{n\in \mathbb {N}}$ converges and

$$\begin{aligned} \Delta _p(x_n,\bar{x})-\Delta _p(x_{n+1},\bar{x})\rightarrow 0,~~ n \rightarrow \infty . \end{aligned}$$

(3.8)

Since $(x_n)_{n\in \mathbb {N}}$ is bounded and $\mathcal {B}_1$ is reflexive, there exists a subsequence $(x_{\ell _n})_{n\in \mathbb {N}}$ of $(x_n)_{n\in \mathbb {N}}$ that converges weakly to $x^{*}\in \mathcal {B}_1$.

Next, we show that $Ax^{*}\in \mathcal {B}_2$, that is $x^{*}\in \Gamma$. Set $y_{n}=\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}[\mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n)-t_nA^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}((I-Q_{\lambda T_2})Ax_n)],~~ n \ge 1$. Following the same line of arguments in (3.4) and (3.5), we can show that

$$\begin{aligned} \Delta _p(y_{n},\bar{x}) \le \Delta _p(x_n,\bar{x}). \end{aligned}$$

Now, it follows from (2.8), (3.5), and (3.7) that

$$\begin{aligned} \Delta _p(y_{n},u_n)=&\,\Delta _p(y_{n},\Pi _{\Gamma }y_{n})\nonumber \\ \le&\, \Delta _p(y_{n},\bar{x})-\Delta _p(u_n,\bar{x})\nonumber \\ \le&\, \Delta _p(x_n,\bar{x})-\Delta _p(u_n,\bar{x})\nonumber \\ \le&\, \alpha _nM+\Delta _p(x_n,\bar{x})-\Delta _p(x_{n+1},\bar{x})\rightarrow 0,~~n\rightarrow \infty , \end{aligned}$$

(3.9)

where $M>0$ is such that $\Delta _p(x_{n+1},\bar{x})+\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}u-\mathcal {J}_{p}^{\mathcal {B}_{1}}\bar{x},v_n-\bar{x}\rangle \le M.$ Thus, we have from (3.9) that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert u_n-y_n\Vert =0. \end{aligned}$$

(3.10)

Moreover, it follows from (3.5) that, for some $M_1>0$

$$\begin{aligned}&t_n\Big (\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p- \frac{C_qt_n^{q-1}}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\Big )\\&\quad \le \Delta _p(x_n,\bar{x})-\Delta _p(u_n,\bar{x})\\&\quad \le \Delta _p(x_n,\bar{x})-\Delta _p(x_{n+1},\bar{x}) +\alpha _n[\Delta _p(u,\bar{x})-\Delta _p(u_n,\bar{x})]. \end{aligned}$$

Therefore, since $\Delta _p(x_n,\bar{x})-\Delta _p(x_{n+1},\bar{x}) +\alpha _n[\Delta _p(u,\bar{x})-\Delta _p(u_n,\bar{x})]\rightarrow 0$ when $n\rightarrow \infty$, the above inequality implies

$$\begin{aligned} t_n\Big (\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p- \frac{C_qt_n^{q-1}}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\Big )\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

Using the condition on $t_n$, that is

$$\begin{aligned} t_n^{q-1}<\frac{q\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p}{C_q\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q}-\epsilon , \end{aligned}$$

which implies

$$\begin{aligned}&C_qt_n^{q-1}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q\\&\quad <q\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p-\epsilon C_q\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q, \end{aligned}$$

and then

$$\begin{aligned}&\frac{\epsilon C_q}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q\\&\quad <\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p- \frac{C_qt_n^{q-1}}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

Hence

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q=0. \end{aligned}$$

Furthermore, we obtain from (3.5) that

$$\begin{aligned} 0&<\epsilon \Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p\le t_n\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p \le \Delta _p(x_n,\bar{x})-\Delta _p(u_n,\bar{x})\\&\quad +\frac{C_qt_n^q}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\\&\le \Delta _p(x_n,\bar{x})-\Delta _p(x_{n+1},\bar{x}) +\alpha _n[\Delta _p(u,\bar{x})-\Delta _p(u_n,\bar{x})]\\&\quad +\frac{C_qt_n^q}{q}\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_n\Vert ^q\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

Thus

$$\begin{aligned} \Vert Ax_n-Q_{\lambda T_2}(Ax_n)\Vert \rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

(3.11)

Then, it follows from the definition of $y_{n}$ that

$$\begin{aligned} 0\le&\, \Vert \mathcal {J}_{p}^{\mathcal {B}_{1}}(y_{n})-\mathcal {J}_{p}^{\mathcal {B}_{1}}(x_n)\Vert \\ \le&\, t_n\Vert A^{*}\Vert \Vert \mathcal {J}_{p}^{\mathcal {B}_{2}}(Ax_n-Q_{\lambda T_2}(Ax_n))\Vert \\ =&\, t_n\Vert A^{*}\Vert \Vert Ax_n-Q_{\lambda T_2}(Ax_n)\Vert ^{p-1}\rightarrow 0,\;n\rightarrow \infty . \end{aligned}$$

Since $\mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}$ is norm-to-norm uniformly continuous on bounded subsets of $\mathcal {B}_{1}^{*}$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert y_{n}-x_n\Vert \rightarrow 0,~~ n\rightarrow \infty . \end{aligned}$$

(3.12)

It follows from (3.10) and (3.12) that

$$\begin{aligned} \Vert u_n-x_n\Vert \le \Vert u_n-y_{n}\Vert +\Vert y_{n}-x_n\Vert \rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

(3.13)

Furthermore, we get from (3.1) that

$$\begin{aligned} \Delta _p(x_{n+1},u_n)\le&\, \alpha _n\Delta _p(u,u_n)+(1-\alpha _n)\Delta _p(u_n,u_n)\\ =&\,\alpha _n\Delta _p(u,u_n)\rightarrow 0, n\rightarrow \infty . \end{aligned}$$

Then

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert x_{n+1}-u_n\Vert =0, \end{aligned}$$

and this together with (3.13) implies that

$$\begin{aligned} \Vert x_{n+1}-x_n\Vert \le \Vert u_n-x_n\Vert +\Vert x_{n+1}-u_n\Vert \rightarrow 0, n\rightarrow \infty . \end{aligned}$$

Similarly, using (2.6), we get

$$\begin{aligned}&\Vert (I-Q_{\lambda T_2})Ax^{*}\Vert ^{p}\nonumber \\&\quad =\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}(Ax^{*}-Q_{\lambda T_2} Ax^{*}),Ax^{*}-Q_{\lambda T_2} Ax^{*}\rangle \nonumber \\&\quad =\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}(Ax^{*}-Q_{\lambda T_2} Ax^{*}),Ax^{*}-Au_{\ell _n}\rangle \nonumber \\&\qquad +\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}(Ax^{*}-Q_{\lambda T_2} Ax^{*}),Au_{\ell _n}-Q_{\lambda T_2} Au_{\ell _n}\rangle \nonumber \\&\qquad +\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}(Ax^{*}-Q_{\lambda T_2} Ax^{*}),Q_{\lambda T_2} Au_{\ell _n}-Q_{\lambda T_2} Ax^{*}\rangle \nonumber \\&\quad \le \langle \mathcal {J}_{p}^{\mathcal {B}_{2}}(Ax^{*}-Q_{\lambda T_2} Ax^{*}),Ax^{*}-Au_{\ell _n}\rangle \nonumber \\&\qquad +\langle \mathcal {J}_{p}^{\mathcal {B}_{2}}(Ax^{*}-Q_{\lambda T_2} Ax^{*}),Au_{\ell _n}-Q_{\lambda T_2} Au_{\ell _n}\rangle . \end{aligned}$$

(3.14)

By the linearity and continuity of $A,~~Ax_{\ell _n}\rightharpoonup Ax^{*}, n\rightarrow \infty$ and $\Vert u_n-x_n\Vert \rightarrow 0, n\rightarrow \infty$, which imply that $Au_{\ell _n}\rightharpoonup Ax^{*}, n\rightarrow \infty$. Hence, letting $n\rightarrow \infty$ in (3.14), we have

$$\begin{aligned} \Vert Ax^{*}-Q_{\lambda T_2} Ax^{*}\Vert =0. \end{aligned}$$

Therefore, $Ax^{*}=Q_{\lambda T_2} Ax^{*}$, that is, $Ax^{*}\in \mathcal {B}_2$. Thus, $x^{*}\in \Gamma .$

Next, we show that $(x_n)_{n\in \mathbb {N}}$ converges strongly to $\bar{x}=\Pi _{\Gamma }u$. Now, note that

$$\begin{aligned} \Delta _p(v_{n},u_n)=&\, \Delta _p\left( \mathcal {J}_{q}^{\mathcal {B}_{1}^{*}}\Big [\alpha _{n}\mathcal {J}_{p}^{\mathcal {B}_{1}}(u) +(1-\alpha _{n})\mathcal {J}_{p}^{\mathcal {B}_{1}}(u_n)\Big )\Big ],u_n\right) \nonumber \\ \le&\,\alpha _{n}\Delta _p(u,u_n)+(1-\alpha _{n})\Delta _p(u_n,u_n)\rightarrow 0,~~ n\rightarrow \infty . \end{aligned}$$

Hence

$$\begin{aligned} \Vert v_{n}-u_n\Vert \rightarrow 0,n\rightarrow \infty , \end{aligned}$$

and so

$$\begin{aligned} \Vert v_n-x_n\Vert \le \Vert v_{n}-u_n\Vert +\Vert u_n-x_n\Vert \rightarrow 0,n\rightarrow \infty . \end{aligned}$$

We now can chose a subsequence $\{v_{\ell _n}\}$ of $\{v_n\}$, such that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_n-\bar{x}\rangle = \lim _{n\rightarrow \infty }\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_{\ell _n}-\bar{x}\rangle . \end{aligned}$$

Since $\Vert x_n-v_{n}\Vert \rightarrow 0,~~ n\rightarrow \infty$ and $x_{\ell _n}\rightharpoonup x^{*}$, we have $v_{\ell _n}\rightharpoonup x^{*}$. Then, it follows from (2.7) that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_n-\bar{x}\rangle= & {} \lim _{n\rightarrow \infty }\langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),v_{\ell _n}-\bar{x}\rangle \\= & {} \langle \mathcal {J}_{p}^{\mathcal {B}_{1}}(u)-\mathcal {J}_{p}^{\mathcal {B}_{1}}(\bar{x}),x^{*}-\bar{x}\rangle \le 0. \end{aligned}$$

Using Lemma 2.2 in (3.7), we conclude that $\Delta _p(x_n,\bar{x})\rightarrow 0,~~n\rightarrow \infty .$ Thus, $x_n\rightarrow \bar{x},~~n\rightarrow \infty$. By $\Vert x_n-u_n\Vert \rightarrow 0,~~n\rightarrow \infty$, we conclude $u_n\rightarrow \bar{x},~~n\rightarrow \infty$.

Case 2: Now, we assume that $(\Delta _p(x_n,\bar{x}))_{n\in \mathbb {N}}$ is not monotonically decreasing sequence. Set $\Gamma _{n}=\Delta _p(x_n,\bar{x}),\quad \forall n \ge 1$ and let $\tau :\mathbb {N}\rightarrow \mathbb {N}$ be a mapping for all $n\ge n_{0}$ (for some $n_{0}$ large enough) by

$$\begin{aligned} \tau (n):=\max \{k\in \mathbb {N}:k\le n,\Gamma _{k}\le \Gamma _{k+1}\}. \end{aligned}$$

Obviously, $\tau$ is a non decreasing sequence, such that $\tau (n)\rightarrow \infty$ as $n \rightarrow \infty$ and

$$\begin{aligned} 0\le \Gamma _{\tau (n)}\le \Gamma _{\tau (n)+1}, \forall n\ge n_{0}. \end{aligned}$$

After a similar conclusion from (3.11), it is easy to see that

$$\begin{aligned} \Vert Ax_{\tau (n)} - Q_{\lambda T_2} x_{\tau (n)} \Vert \rightarrow 0, n\rightarrow \infty . \end{aligned}$$

By the similar argument as above in Case 1, we immediately conclude that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert A^*\mathcal {J}_{p}^{\mathcal {B}_2}(Ax_{\tau (n)}-Q_{\lambda T_2}(Ax_{\tau (n)}))\Vert =0, \end{aligned}$$

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert x_{\tau (n)+1}-x_{\tau (n)}\Vert =0 \end{aligned}$$

and

$$\begin{aligned} \limsup _{n\rightarrow \infty } \langle \mathcal {J}_{p}^{\mathcal {B}_1}(u)-\mathcal {J}_{p}^{\mathcal {B}_1}(\bar{x}),v_{\tau (n)}-\bar{x}\rangle \le 0. \end{aligned}$$

Since $\{x_{\tau (n)}\}$ is bounded, there exists a subsequence of $\{x_{\tau (n)}\}$, still denoted by $\{x_{\tau (n)}\}$ which converges weakly to $x^{*}\in \mathcal {B}_1$ and $Ax^{*} \in \mathcal {B}_2$. It follows from (3.7) that

$$\begin{aligned} \Delta _p(x_{\tau (n)+1},\bar{x})\le (1-\alpha _{\tau (n)})\Delta _p(x_{\tau (n)},\bar{x})+\alpha _{\tau (n)} \langle w_{\tau (n)}-\bar{x}, \mathcal {J}_{p}^{\mathcal {B}_1}(u)-\mathcal {J}_{p}^{\mathcal {B}_1}(\bar{x})\rangle , \end{aligned}$$

which leads that (noting that $\Gamma _{\tau (n)}\le \Gamma _{\tau (n)+1}$ and $\alpha _{\tau (n)} >0$)

$$\begin{aligned} \Delta _p(x_{\tau (n)},\bar{x})\le \langle \mathcal {J}_{p}^{\mathcal {B}_1}(u)-\mathcal {J}_{p}^{\mathcal {B}_1}(\bar{x}),v_{\tau (n)}-\bar{x}\rangle . \end{aligned}$$

This implies that

$$\begin{aligned} \limsup _{n\rightarrow \infty } \Delta _p(x_{\tau (n)},\bar{x})\le 0. \end{aligned}$$

Thus, $\lim _{n\rightarrow \infty } \Delta _p(x_{\tau (n)},\bar{x})=0.$ Therefore

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert x_{\tau (n)}-\bar{x}\Vert =0. \end{aligned}$$

(3.15)

Since $\lim _{n\rightarrow \infty }\Vert x_{\tau (n)+1}-x_{\tau (n)}\Vert =0$, we get that $\lim _{n\rightarrow \infty } \Vert x_{\tau (n)+1}-\bar{x}\Vert =0$. Now, it follows from (2.5) that

$$\begin{aligned} 0\le \Delta _p(x_{\tau (n)+1},\bar{x})\le&\, \langle x_{\tau (n)+1}-\bar{x}, \mathcal {J}_{p}^{\mathcal {B}_1}(x_{\tau (n)+1})-\mathcal {J}_{p}^{\mathcal {B}_1}(\bar{x})\rangle \\ \le&\, \Vert x_{\tau (n)+1}-\bar{x}\Vert \Vert \mathcal {J}_{p}^{\mathcal {B}_1}(x_{\tau (n)+1})-\mathcal {J}_{p}^{\mathcal {B}_1}(\bar{x})\Vert \rightarrow 0, n\rightarrow \infty . \end{aligned}$$

Furthermore, for $n\ge n_{0}$, it is easy to see that $\Gamma _{\tau (n)}\le \Gamma _{\tau (n)+1}$ if $n\ne \tau (n)$ (that is, $\tau (n)<n$), because $\Gamma _{j}\ge \Gamma _{j+1}$ for $\tau (n)+1\le j\le n.$ As a consequence, we obtain for all $n\ge n_{0}$

$$\begin{aligned} 0\le \Gamma _n\le \max \{\Gamma _{\tau (n)},\Gamma _{\tau (n)+1}\}=\Gamma _{\tau (n)+1}. \end{aligned}$$

Hence, $\lim _{n\rightarrow \infty }\Gamma _n=0$, that is, $(x_n)_{n\in \mathbb {N}}$ converges strongly to $\bar{x}$. This completes the proof. $\square$

Next, we enunciate some direct consequences of the above theorem.

Corollary 3.4

Let $\mathcal {B}_1$ and $\mathcal {B}_2$ be two $L_p$ spaces with $2\le p<\infty$. Let $T_1:\mathcal {B}_1\rightrightarrows \mathcal {B}_1$ and $T_2:\mathcal {B}_2\rightrightarrows \mathcal {B}_2$ be maximal monotone operators such that $T^{-1}_1(0)\ne \emptyset$ and $T^{-1}_2(0)\ne \emptyset$, respectively. Let $Q_{\lambda T_2}$ be the metric resolvent of $T_2$ for $\lambda > 0.$ Let $A:\mathcal {B}_1 \rightarrow \mathcal {B}_2$ be a bounded linear operator, such that $A\ne 0$ and $A^*:\mathcal {B}_2^* \rightarrow \mathcal {B}_1^*$ be the adjoint of A. Suppose that $\Gamma \ne \emptyset$ and $(\alpha _n)_{n\in \mathbb {N}}\subset (0,1)$. Let the sequence $(x_n)_{n\in \mathbb {N}}$ be generated by (3.1)–(3.2). Assume that for small enough $\epsilon >0$

$$\begin{aligned} t_n\in \left( \epsilon ,\Big (\frac{q\Vert (I-Q_{\lambda T_2})Ax_n\Vert ^p}{C_q\Vert A^{*}\mathcal {J}_{p}^{\mathcal {B}_{2}}(I-Q_{\lambda T_2})Ax_{n}\Vert ^q}-\epsilon \Big )^{\frac{1}{q-1}}\right) ,~~ n\in \Omega . \end{aligned}$$

Suppose the following conditions are satisfied:

• (i) $\lim _{n\rightarrow \infty }\alpha _n= 0$;

• (ii) $\sum _{n=1}^{\infty }\alpha _{n}=\infty$.

Then, $(u_n)_{n\in \mathbb {N}}$ and $(x_n)_{n\in \mathbb {N}}$ both converge strongly to $\bar{x}$, where $\bar{x}=\Pi _{\Gamma }u$.

Corollary 3.5

Let $\mathcal {H}_1$ and $\mathcal {H}_2$ be two real Hilbert spaces. Let $T_1:\mathcal {H}_1\rightrightarrows \mathcal {H}_1$ and $T_2:\mathcal {H}_2\rightrightarrows \mathcal {H}_2$ be maximal monotone operators, such that $T^{-1}_1(0)\ne \emptyset$ and $T^{-1}_2(0)\ne \emptyset$, respectively. Let $J_{\lambda T_k}$ be the metric resolvent of $T_k$ for $\lambda > 0$ and $k=1, 2$. Let $A:\mathcal {H}_1 \rightarrow \mathcal {H}_2$ be a bounded linear operator, such that $A\ne 0$ and $A^*:\mathcal {H}_2 \rightarrow \mathcal {H}_1$ be the adjoint of A. Suppose that $\Gamma \ne \emptyset$ and $(\alpha _n)_{n\in \mathbb {N}}\subset (0,1)$. For a fixed $u \in C$, let sequences $(u_n)_{n\in \mathbb {N}}$ and $(x_n)_{n\in \mathbb {N}}$ be iteratively generated by $x_1 \in C$

$$\begin{aligned} \left\{ \begin{array}{rcl} u_n&{}\,=&{}\,x_n-t_nA^*(I-J_{\lambda T_2})Ax_n,\\ x_{n+1} &{}\,=&{}\,\alpha _nu +(1-\alpha _n)J_{\lambda T_1}(u_n). \end{array} \right. \end{aligned}$$

(3.16)

Let the step-size $t_n$ be chosen in such a way that for some $\epsilon >0$

$$\begin{aligned} t_n\in \left( \epsilon ,\frac{2\Vert (I-J_{\lambda T_2})Ax_n\Vert ^2}{\Vert A^*(I-J_{\lambda T_2})Ax_n\Vert ^2}-\epsilon \right) ,~~ n\in \Omega , \end{aligned}$$

otherwise $t_n=t$ (t being any nonnegative value), where the index set $\Omega = \{n \in \mathbb {N}:(I-J_{\lambda T_2})Ax_n\ne 0\}.$ Suppose the following conditions are satisfied:

• (i) $\lim _{n\rightarrow \infty }\alpha _n= 0$;

• (ii) $\sum _{n=1}^{\infty }\alpha _{n}=\infty$.

Then, $(u_n)_{n\in \mathbb {N}}$ and $(x_n)_{n\in \mathbb {N}}$ both converge strongly to $\bar{x}$, where $\bar{x}=\Pi _{\Gamma }u$.

To finish this section, we make some remarks and comments which highlight our contributions in this paper.

Remark 3.6

It is worth pointing out here that the implementation of the recent iterative schemes of Byrne et al. [5], Chuang [10], Moudafi [17], Alsulami and Takahashi [2] and Takahashi and Takahashi [29] depends largely on knowing the norm of the bounded linear operator A. Here, in our results, we do not need to know a priori the norm or an estimate of the norm of the bounded linear operator A to implement the proposed iterative method. Moreover, we only use just one vector norm evaluation per iteration to find the stepsize, which could be found via any backtracking procedure. Therefore, our results improve, extend and more efficient and implementable than the corresponding results of Byrne et al. [5], Chuang [10], Moudafi [17], Wen and Chen [33] in real Hilbert spaces and improve the result of Takahashi and Takahashi [29] and Takahashi and Yao [30] and Takahashi [28] in p-uniformly convex Banach spaces which are also uniformly smooth.

Remark 3.7

Suppose that in our Corollary 3.5, we take $J_{\lambda T_2}=T$, where $T:\mathcal {H}_2\rightarrow \mathcal {H}_2$ is a nonexpansive mapping. For this case, Takahashi et al. [31] introduced an iterative scheme which is dependent on the norm of the bounded linear operator and obtained weak convergence result. In our Corollary 3.5 of this paper, we obtain strong convergence result using an iterative method independent of the bounded linear operator norm for the same problem considered by Takahashi et al. [31].

Remark 3.8

It is worth mentioning that following the same lines of arguments in this paper, similar algorithm for solving Split Minimization Problem (SMP) when $T_1=\partial (f+\delta _{\Omega _1}),~~T_2=\partial (g +\delta _{\Omega _2})$ with $f:\mathcal {H}_1\rightarrow \mathbb {R},~~g:\mathcal {H}_2\rightarrow \mathbb {R}$ and $\Omega _1$ and $\Omega _2$ are nonempty, closed, and convex subsets of $\mathcal {H}_1$ and $\mathcal {H}_2$, respectively. That is

$$\begin{aligned} \left\{ \begin{array}{ll} &{}\mathrm{Find}~~ x^*=\arg \min _{x\in \Omega _1} f(x) \;~~\mathrm{such~~ that}\\ &{}Ax^*=\arg \min _{y\in \Omega _2} g(y). \end{array}\right. \end{aligned}$$

(SMP)

Xu [36] considered Problem (SMP) when $\mathcal {H}_1=\mathcal {H}_2,~~f=g$ and $A=I$, where I is an identity map and obtained strong convergence result in real Hilbert spaces. Here, in this paper, our results extend the corresponding results of Xu [36].

6 Conclusions

In this paper, we have presented strong convergence results for solving the split inclusion problem with a way of selecting the stepsizes, such that its implementation does not need any prior information about the operator norm $\Vert A\Vert$ in uniformly convex Banach spaces which are also uniformly smooth. Comparing our scheme with the iterative methods proposed in [5, 10, 15, 17, 37, 38] to solve the split inclusion problem in Banach spaces, we see that all of them need to know a priori the norm (or at least an estimate of the norm) of the bounded linear operator $\Vert A\Vert$, which is in general a not easy task. Here, we develop an algorithm which is designed to address a way of selecting the stepsizes, such that its implementation does not need any prior information about the operator norm, and moreover, we prove its related strong convergence results in uniformly convex and smooth Banach spaces. Furthermore, note that our results in this paper are not applicable in $L_p$ spaces with $1<p<2$. It is, therefore, worth of further studying some amendment to this our algorithm (3.1)–(3.2), such that it will be applicable in $L_p$ spaces with $1<p<2$ and in a more general uniformly convex and smooth Banach spaces. This is our future work.