Iterative methods for convex proximal split feasibility problems and fixed point problems

Abstract

In this paper we prove strong convergence result for a problem of finding a point which minimizes a proper convex lower-semicontinuous function f which is also a fixed point of a total asymptotically strict pseudocontractive mapping such that its image under a bounded linear operator A minimizes another proper convex lower-semicontinuous function g in real Hilbert spaces. In our result in this work, our iterative scheme is proposed with a way of selecting the step-size such that its implementation does not need any prior information about the operator norm ||A|| because the calculation or at least an estimate of the operator norm ||A|| is very difficult, if it is not an impossible task. Our result complements many recent and important results in this direction.

1 Introduction

In this paper, we shall assume that H is a real Hilbert space with inner product $\langle .,.\rangle$ and norm ||.||. Let I denote the identity operator on H. Now, let us recall the definitions of some operators that will be used in this paper.

Let $T:H\rightarrow H$ be a mapping. A point $x \in H$ is called a fixed point of T if $Tx=x$. The set of fixed points of T is denoted by F(T). A point $x^*\in H$ is called a minimum norm fixed point of T if and only if $x^* \in F(T)$ and $||x^*|| =\min \{ ||x||: x \in F(T)\}$.

Definition 1.1

The mapping $T:H\rightarrow H$ is said to be

• (a) nonexpansive if

  $$\begin{aligned} ||Tx-Ty||\le ||x-y||, \forall x, y \in H. \end{aligned}$$

• (b) asymptotically nonexpansive mapping if there exists a sequence $\{\mu _n\}$ of real positive numbers such that $\lim \mu _n=0$ and

  $$\begin{aligned} ||T^nx-T^ny||\le (1+\mu _n)||x-y||, \forall x,y \in H, ~~\forall n\ge 1. \end{aligned}$$

• (c) k-strictly pseudocontractive (see, [2]) if for $0\le k <1$,

  $$\begin{aligned} || Tx-Ty||^{2} \le || x-y||^{2}+k||(I-T)x-(I-T)y||^{2}, ~~ \forall ~~x,y\in H. \end{aligned}$$

  (1.1)

• (d) asymptotically k-strict pseudo-contraction mapping in the intermediate sense if there exist a constant $k\in [0,1)$ and sequences $\{\mu _n\}\subset [0, \infty ), \{\xi _n\}\subset [0, \infty )$ with $\mu _n\rightarrow 0$ and $\xi _n\rightarrow 0$ as $n\rightarrow \infty$ such that for all $n\ge 1,$

  $$\begin{aligned} ||T^nx-T^ny||^2\le (1+\mu _n)||x-y||^2+k||(x-y)-(Tx-Ty)||^2+\xi _n, \forall x,y \in H. \end{aligned}$$

• (e) $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping [6], if there exist a constant $k\in [0,1)$ and sequences $\{\mu _n\}\subset [0, \infty ), \{\xi _n\}\subset [0, \infty )$ with $\mu _n\rightarrow 0$ and $\xi _n\rightarrow 0$ as $n\rightarrow \infty$, and a continuous and strictly increasing function $\phi :[0,\infty )\rightarrow [0,\infty )$ with $\phi (0)=0$ such that for all $n\ge 1,$

  $$\begin{aligned} ||T^nx-T^ny||^2\le & {} ||x-y||^2+k||(x-y)-(Tx-Ty)||^2\\&+\mu _n\phi (||x-y||)+\xi _n, \forall x,y \in H. \end{aligned}$$

For an example of a total asymptotically strict pseudocontractive mapping, we refer the reader to Chang et al. [6].

In this paper, we shall assume that H is a real Hilbert space with inner product $\langle .,.\rangle$ and norm ||.||. Let I denote the identity operator on H. Let C and Q be nonempty, closed and convex subsets of real Hilbert spaces $H_1$ and $H_2$, respectively. The split feasibility problem (SFP) is to find a point

$$\begin{aligned} x \in C~~ \mathrm{such ~~that~~} Ax\in Q, \end{aligned}$$

(1.2)

where $A:H_1\rightarrow H_2$ is a bounded linear operator. The SFP in finite-dimensional Hilbert spaces was first introduced by Censor and Elfving [5] for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [3]. The SFP attracts the attention of many authors due to its application in signal processing. Various algorithms have been invented to solve it (see, for example, [4, 12, 15, 17, 19, 21, 23, 26] and references therein).

Note that the split feasibility problem (1.2) can be formulated as a fixed-point equation by using the fact

$$\begin{aligned} P_C(I-\gamma A^*(I-P_Q)A)x^*=x^*, \end{aligned}$$

(1.3)

where $P_C$ is the metric projection from $H_1$ onto C [defined as $||x-P_Cx||=\min _{y\in C}||x-y||$]. Hence, $x^*$ solves the SFP (1.2) if and only if $x^*$ solves the fixed point equation (1.3) (see [18] for the details). This implies that we can use fixed-point algorithms (see [24, 25, 27]) to solve SFP. A popular algorithm that solves the SFP (1.2) is due to Byrne’s CQ algorithm [3] which is found to be a gradient-projection method (GPM) in convex minimization. Subsequently, Byrne [4] applied Krasnoselskii-Mann iteration to the CQ algorithm, and Zhao and Yang [28] applied Krasnoselskii-Mann iteration to the perturbed CQ algorithm to solve the SFP. It is well known that the CQ algorithm and the Krasnoselskii-Mann algorithm for a split feasibility problem do not necessarily converge strongly in the infinite-dimensional Hilbert spaces.

Let $h: H\rightarrow \mathbb {R}\cup \{+\infty \}$ be a functional on a real Hilbert space H.

Definition 1.2

• 1. The Moreau-Yosida approximate of the function h of parameter $\lambda >0$ is defined as $h_\lambda (y):=\min _{u\in H} \{h(u) + \frac{1}{2 \lambda }||u-y||^2\}$.

• 2. $\mathrm{argmin} f:= \{ \bar{x} \in H: f(\bar{x})\le f(x), \forall x \in H\}$.

• 3. The proximal mapping of f is defined as $\mathrm{prox}_{\lambda f}(y) = \mathrm{argmin}_{u\in H} \{f(u) + \frac{1}{2 \lambda }||u-y||^2\}$.

• 4. The subdifferential of f at x is the set

  $$\begin{aligned} \partial f(x):=\{u \in H:f(y)\ge f(x)+\langle u, y-x\rangle , \forall y \in H \}. \end{aligned}$$

Let us consider the following problem: find a solution $x^*\in H_1$ such that

$$\begin{aligned} \underset{x \in H_1}{\min }\{f(x)+g_\lambda (Ax)\}, \end{aligned}$$

(1.4)

where $H_1, H_2$ are two real Hilbert spaces, $f:H_1\rightarrow \mathbb {R}\cup \{+\infty \}, g:H_2 \rightarrow \mathbb {R}\cup \{+\infty \}$ two proper, convex, lower semicontinuous functions and $A:H_1\rightarrow H_2$ a bounded linear operator, $g_\lambda (y) = \min _{u\in H_2} \{g(u) + \frac{1}{2 \lambda }||u-y||^2\}$ stands for the Moreau-Yosida approximate of the function g of parameter $\lambda$.

Observe that by taking $f =\delta _C$ [defined as $\delta _C(x) = 0$ if $x\in C$ and $+\infty$ otherwise], $g =\delta _Q$ the indicator functions of two nonempty, closed and convex sets C, Q of $H_1$ and $H_2$ respectively, Problem (1.4) reduces to

$$\begin{aligned} \underset{x \in H_1}{\min }\{\delta _C(x)+(\delta _{Q})_\lambda (Ax)\}\Leftrightarrow \underset{x \in C}{\min }\left\{ \frac{1}{2\lambda }||(I-P_Q)(Ax)||^2\right\} \end{aligned}$$

(1.5)

which, when $C\cap A^{-1}(Q)\ne \emptyset$, is equivalent to (1.2).

By the differentiability of the Moreau-Yosida approximate $g_\lambda$, see for instance [16], we have the additivity of the subdifferentials and thus we can write

$$\begin{aligned} \partial (f(x)+g_\lambda (Ax))=\partial f(x)+A^*\nabla g_\lambda (Ax)=\partial f(x)+A^*\Big (\frac{I-prox_{\lambda g}}{\lambda } \Big )(Ax). \end{aligned}$$

This implies that the optimality condition of (1.4) can be then written as

$$\begin{aligned} 0 \in \lambda \partial f(x)+A^*(I-prox_{\lambda g})(Ax). \end{aligned}$$

(1.6)

Inclusion (1.6) in turn yields to the following equivalent fixed point formulation

$$\begin{aligned} \mathrm{prox}_{\mu \lambda f}\left( x^*-\mu A^*(I-\mathrm{prox}_{\lambda g})Ax^*\right) =x^*. \end{aligned}$$

(1.7)

To solve (1.4), relation (1.7) suggests to consider the following split proximal algorithm

$$\begin{aligned} x_{n+1}=\mathrm{prox}_{\mu _n \lambda f}\left( x_n-\mu _n A^*(I-\mathrm{prox}_{\lambda g})Ax_n\right) . \end{aligned}$$

(1.8)

Based on an idea introduced in Lopez et al. [11], Moudafi and Thakur [13] recently proved weak convergence results for solving (1.4) in the case $\mathrm{argmin} f\cap A^{-1}(\mathrm{argmin} g)\ne \emptyset$, or in other words: in finding a minimizer $x^*$ of f such that $Ax^*$ minimizes g, namely

$$\begin{aligned} x^* \in \mathrm{argmin} f~~\mathrm{such~~ that}~~ Ax^*\in \mathrm{argmin}~~ g, \end{aligned}$$

(1.9)

f, g being two proper, lower semicontinuous convex functions. We will denote the solution set of (1.9) by $\Gamma$. Concerning problem (1.9), Moudafi and Thakur [13] introduced a new way of selecting the step-sizes: Set $\theta (x_n):=\sqrt{||\nabla h(x_n)||^2+||\nabla l(x_n)||^2}$ with $h(x_n)=\frac{1}{2}||(I-\mathrm{prox}_{\lambda g})Ax_n||^2, l(x_n)=\frac{1}{2}||(I-\mathrm{prox}_{\lambda \mu _n f})x_n||^2$ and introduced the following split proximal algorithm:

Split Proximal Algorithm: Given an initial point $x_1\in H_1$. Assume that $x_n$ has been constructed and $\theta (x_n) \ne 0$, then compute $x_{n+1}$ via the rule

$$\begin{aligned} x_{n+1} =\mathrm{prox}_{\lambda \mu _n f}( x_n-\mu _n A^*(I-\mathrm{prox}_{\lambda g})Ax_n),~~n \ge 1, \end{aligned}$$

(1.10)

where stepsize $\mu _n:=\rho _n\frac{h(x_n)+l(x_n)}{\theta ^2(x_n)}$ with $0<\rho _n<4$. If $\theta (x_n)=0$, then $x_{n+1}=x_n$ is a solution of (1.4) and the iterative process stops, otherwise, we set $n:= n + 1$ and go to (1.10).

Using the split proximal algorithm (1.10), Moudafi and Thakur [13] proved the following weak convergence theorem for approximating a solution of (1.9).

Theorem 1.3

Assume that f and g are two proper convex lower-semicontinuous functions and that (1.9) is consistent (i.e., $\Gamma \ne \emptyset$). If the parameters satisfy the following conditions $\epsilon \le \rho _n \le \frac{4 h(x_n)}{h(x_n)+l(x_n)}-\epsilon$ (for some $\epsilon >0$ small enough), then the sequence $\{x_n\}$ generated by (1.10) weakly converges to a solution of (1.9).

We remark here that it is quite usual to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a nonempty, closed and convex subset C of a Hilbert space $H_1$ and a bounded linear operator $A:H_1\rightarrow H_2$, where $H_2$ is another Hilbert space. The C-constrained pseudoinverse of A, $A^{\dag }_C$ is then defined as the minimum-norm solution of the constrained minimization problem

$$\begin{aligned} A^{\dag }_C(b):=\mathrm{argmin}_{x \in C}||Ax-b|| \end{aligned}$$

which is equivalent to the fixed point problem

$$\begin{aligned} x=P_C(x-\lambda A^*(Ax-b)), \end{aligned}$$

where $P_C$ is the metric projection from $H_1$ onto C, $A^*$ is the adjoint of A, $\lambda >0$ is a constant, and $b\in H_2$ is such that $P_{\overline{A(C)}}(b)\in A(C)$. It is therefore our aim in this paper to introduce an iterative algorithm that can generate sequence which converges strongly to the minimum-norm solution of a given convex proximal split feasibility problem and fixed point problems for total asymptotically strict pseudocontractive mapping in real Hilbert spaces.

2 Preliminaries

Definition 2.1

A mapping $T:H\rightarrow H$ is said to be uniformly L-Lipschitzian continuous if there exists a constant $L > 0$ such that $||T^nx - T^ny||\le L||x-y||, \forall x, y \in H, n\ge 1.$

Lemma 2.2

Let K be a nonempty, closed and convex subset of a real Hilbert space H and let $T:K\rightarrow K$ be a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping such that $F(T)\ne \emptyset$. Suppose there exist constants $M_0>0, K_1 > 0$ such that $\phi (\lambda )\le M_0 \lambda ^2, \forall \lambda >K_1$. Then F(T) is closed and convex.

Proof

Since T is a uniformly L-Lipschitzian continuous, F(T) is closed. Next, we show that F(T) is convex. Next, we show that F(T) is convex. For $t\in [0,1]$ and $x,y \in F(T)$, put $z:= tx + (1- t)y$, we show that $z=Tz$. Since $\phi$ is continuous, it follows that $\phi$ attains maximum (say M) in $[0,K_1]$ and by our assumption, $\phi (t)\le M_0 t^2, \forall t >K_1$. In either case, we have that

$$\begin{aligned} \phi (t)\le M+M_0 t^2, \forall t \in [0,\infty ). \end{aligned}$$

Using Lemma 2.3 (iii), we obtain

$$\begin{aligned} ||T^nz-z||^2= & {} ||t(T^nz-x)+(1-t)(T^nz-y)||^2\nonumber \\= & {} t||T^nz-x||^2+(1-t)||T^nz-y||^2-t(1-t)||x-y||^2\nonumber \\\le & {} t\Big (||z-x||^2+k||T^nz-z||^2+\mu _n\phi (||z-x||)+\xi _n\Big ) \nonumber \\&+\,(1-t)\Big (||z-y||^2+k||T^nz-z||^2+\mu _n\phi (||z-y||)+\xi _n\Big )\nonumber \\&-\,t(1-t)||x-y||^2 \nonumber \\\le & {} t\Big (||z-x||^2+k||T^nz-z||^2+\mu _n(M+M_0||z-x||^2)+\xi _n\Big ) \nonumber \\&+\,(1-t)\Big (||z-y||^2+k||T^nz-z||^2+\mu _n(M+M_0||z-y||^2)+\xi _n\Big )\nonumber \\&-\,t(1-t)||x-y||^2 \nonumber \\= & {} t\Big ((1\,{-}\,t)^2||x\,{-}\,y||^2+k||T^nz\,{-}\,z||^2+\mu _n(M+M_0||z\,{-}\,x||^2)+\xi _n\Big )(1\,{-}\,t) \nonumber \\&\times \,\Big (t^2||x-y||^2+k||T^nz-z||^2+\mu _n(M+M_0||z-y||^2)+\xi _n\Big )\nonumber \\&-\,t(1-t)||x-y||^2 \nonumber \\= & {} k||T^nz-z||^2+\mu _n(M+M_0||z-y||^2)+\xi _n. \end{aligned}$$

(2.1)

This implies from (2.1) that

$$\begin{aligned} (1-k)||T^nz-z||^2\le \mu _n(M+M_0||z-y||^2)+\xi _n. \end{aligned}$$

Thus, $\lim \nolimits _{n\rightarrow \infty } ||T^nz-z||=0$, which implies that $\lim \nolimits _{n\rightarrow \infty } T^nz=z$. By continuity of T, we obtain that

$$\begin{aligned} z=\underset{n\rightarrow \infty }{\lim }T^nz=\underset{n\rightarrow \infty }{\lim }T(T^{n-1}z)= T(\underset{n\rightarrow \infty }{\lim }T^{n-1}z)=Tz. \end{aligned}$$

Hence, $z \in F(T)$, that F(T) is convex. $\square$

We state the following well-known lemmas which will be used in the sequel.

Lemma 2.3

Let H be a real Hilbert space. Then there holds the following well-known results:

$$\begin{aligned} (i)~~||x+y||^2=||x||^2+2\langle x,y\rangle +||y||^2, \forall x, y \in H. \end{aligned}$$

$$\begin{aligned} (ii)~~||x+y||^2 \le ||x||^2+ 2\langle y, x+y \rangle , \forall x, y \in H. \end{aligned}$$

$$\begin{aligned} (iii)~~||tx+(1\!-\!t)y||^2=t||x||^2+(1\!-\!t)||y||^2\!-\!t(1\!-\!t)||x\!-\!y||^2, \forall x, y \in H, \forall t \in [0,1]. \end{aligned}$$

Lemma 2.4

(Chang et al. [6]) Let $T:H\rightarrow H$ be a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping, then $I-T$ is demiclosed at 0, i.e., if $x_n\rightharpoonup x\in H$ and $x_{n}-Tx_{n}\rightarrow 0$, then $x=Tx.$

Lemma 2.5

(Alber et al. [1]) Let $\{\lambda _n\}$ and $\{\gamma _n\}$ be nonnegative, $\{\alpha _n\}$ be positive real numbers such that

$$\begin{aligned} \lambda _{n+1} \le \lambda _n-\alpha _n\lambda _n +\gamma _n, n\ge 1. \end{aligned}$$

Let for all $n > 1$,

$$\begin{aligned} \frac{\gamma _n}{\alpha _n}\le c_1~~\mathrm{and}~~\alpha _n \le \alpha . \end{aligned}$$

Then $\lambda _n \le \max \{\lambda _1,K_{*}\},$ where $K_{*}=(1 + \alpha )c_1$.

Lemma 2.6

(Xu, [20]) Let $\{a_n\}$ be a sequence of nonnegative real numbers satisfying the following relation:

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

where

• (i) $\{a_n\}\subset [0,1],~~\sum \alpha _n=\infty ;$

• (ii) $\limsup \sigma _n \le 0$;

• (iii) $\gamma _n \ge 0;~~(n \ge 1),~~\Sigma \gamma _n <\infty .$

Then, $a_n\rightarrow 0$ as $n\rightarrow \infty$.

Now, our interest is in studying the convergence properties of the following algorithm: Given an initial point $x_1\in H_1$, then compute $u_n$ using $u_n=(1-\alpha _n)x_n$. Set $\theta (u_n):=\sqrt{||\nabla h(u_n)||^2+||\nabla l(u_n)||^2}$ with $h(u_n)=\frac{1}{2}||(I-\mathrm{prox}_{\lambda g})Au_n||^2, l(u_n)=\frac{1}{2}||(I-\mathrm{prox}_{\lambda \tau _n f})u_n||^2$ and introduce the following algorithm:

Algorithm: Let $T:H_1\rightarrow H_1$ is a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping such that $F(T)\ne \emptyset$. Given an initial point $x_1\in H_1$, the compute $u_n$ using $u_n=(1-\alpha _n)x_n$ and $\theta (u_n) \ne 0$, then compute $x_{n+1}$ via the rule

$$\begin{aligned} \left\{ \begin{array}{l} u_n=(1-\alpha _n)x_n,\\ y_n =\mathrm{prox}_{\lambda \tau _n f}(u_n-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Au_n),\\ x_{n+1} =(1-\beta _n)y_n+ \beta _nT^ny_n,~~n \ge 1, \end{array} \right. \end{aligned}$$

(2.2)

where stepsize $\tau _n:=\rho _n\frac{h(u_n)+l(u_n)}{\theta ^2(u_n)}$ with $0<\rho _n<4$. If $\theta (u_n)=0$, then $x_{n+1}=x_n$ is a solution of (1.9) which is also a fixed point of a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping T and the iterative process stops, otherwise, we set $n:= n + 1$ and go to (2.2).

3 Main results

Theorem 3.1

Let $T:H_1\rightarrow H_1$ is a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping such that $F(T)\ne \emptyset$. Assume that $f:H_1\rightarrow \mathbb {R}\cup \{+\infty \}$ is a proper convex lower-semicontinuous function, $g:H_2 \rightarrow \mathbb {R}\cup \{+\infty \}$ is a proper convex lower-semicontinuous function such that (1.9) is consistent (i.e., $\Gamma \ne \emptyset$) and $\Gamma \cap F(T)\ne \emptyset$. Let $\{\alpha _n\}$ and $\{\beta _n\}$ be sequences in (0, 1). If the parameters satisfy the following conditions

• (a) $\sum \limits _{n=1}^{\infty }\mu _n<\infty ; \sum \limits _{n=1}^{\infty } \xi _n<\infty ;$

• (b) $\mu _n=o(\alpha _n); \xi _n=o(\alpha _n); \underset{n\rightarrow \infty }{\lim }\alpha _n =0; \sum \limits _{n=1}^{\infty } \alpha _n =\infty ;$

• (c) $\epsilon \le \rho _n \le \frac{4 h(u_n)}{h(u_n)+l(u_n)}-\epsilon$ for some $\epsilon >0$;

• (d) $0<\underset{n\rightarrow \infty }{\liminf }\beta _n \le \underset{n\rightarrow \infty }{\limsup }\beta _n<1-k$;

• (e) there exist constants $M_0>0, K_1 > 0$ such that $\phi (t)\le M_0 t^2, \forall t >K_1$;

then sequence $\{x_n\}$ generated by (2.2) converges strongly to $x^*\in \Gamma \cap F(T)$ which is also the minimum-norm solution (i.e., $x^*\in \Gamma \cap F(T)$ and $||x^*|| =\min \{ ||x||: x \in \Gamma \cap F(T)\}$).

Proof

Let $x^{*}\in \Gamma$. Observe that $\nabla h(u_n)= A^*(I-\mathrm{prox}_{\lambda g})Au_n, \nabla l(u_n) = (I-\mathrm{prox}_{\tau _n \lambda f})x$. Using the fact that $\mathrm{prox}_{\tau _n \lambda f}$ is nonexpansive, $x^{*}$ verifies (1.9) (since minimizers of any function are exactly fixed-points of its proximal mapping) and having in hand

$$\begin{aligned} \langle \nabla h(u_n), u_n-x^{*} \rangle =\langle (I-\mathrm{prox}_{\lambda g})Au_n, Au_n-Ax^{*}\rangle \ge ||(I-\mathrm{prox}_{\lambda g})Au_n||^2=2h(u_n), \end{aligned}$$

thanks to the fact that $I-\mathrm{prox}_{\lambda g}$ is firmly nonexpansive, we can write

$$\begin{aligned} ||y_n-x^{*}||^2= & {} ||u_n-x^{*}||^2+\tau _n^2||\nabla h(u_n)||^2-2\tau _n\langle \nabla h(u_n), u_n-x^{*}\rangle \nonumber \\\le & {} ||u_n-x^{*}||^2+\tau _n^2||\nabla h(u_n)||^2-4\tau _nh(u_n) \nonumber \\= & {} ||u_n-x^{*}||^2+\rho ^2_n\frac{(h(u_n)+l(u_n))^2}{(\theta ^2(u_n))^2}||\nabla h(u_n)||^2-4\rho _n\frac{h(u_n)+l(u_n)}{\theta ^2(u_n)}h(u_n) \nonumber \\\le & {} ||u_n-x^{*}||^2+\rho ^2_n\frac{(h(u_n)+l(u_n))^2}{\theta ^2(u_n)}-4\rho _n\frac{(h(u_n)+l(u_n))^2}{\theta ^2(u_n)}\frac{h(u_n)}{h(u_n)+l(u_n)} \nonumber \\= & {} ||u_n-x^{*}||^2-\rho _n\Big (\frac{4h(u_n)}{h(u_n)+l(u_n)}-\rho _n\Big )\frac{(h(u_n)+l(u_n))^2}{\theta ^2(u_n)}. \end{aligned}$$

(3.1)

Using (3.1) in (2.2), we obtain that

$$\begin{aligned} || x_{n+1}-x^{*}||^2= & {} ||(1-\beta _n)y_n+\beta _n T^ny_n-x^{*}||^2 \nonumber \\= & {} || (1-\beta _n)(y_n-x^{*})+\beta _n(T^ny_n-x^{*})||^2 \nonumber \\= & {} (1-\beta _n)|| y_n-x^{*}||^2 +\beta _n|| T^ny_n-x^{*}||^2-\beta _n(1-\beta _n)||T^ny_n-y_n||^2 \nonumber \\\le & {} (1-\beta _n)|| y_n-x^{*}||^2 +\beta _n\Big [|| y_n-x^{*}||^2 +k||T^ny_n-y_n||^2 \nonumber \\&+\,\mu _n \phi (||y_n-x^{*}||)+\xi _n\Big ]-\beta _n(1-\beta _n)||T^ny_n-y_n||^2 \nonumber \\= & {} || y_n-x^{*}||^2+ \beta _n(\beta _n-(1-k))||T^ny_n-y_n||^2+\beta _n\mu _n\phi (||y_n-x^{*}||) \nonumber \\&+\,\beta _n \xi _n. \end{aligned}$$

(3.2)

Since $\phi$ is continuous, it follows that $\phi$ attains maximum (say M) in $[0,K_1]$ and by our assumption, $\phi (t)\le M_0 t^2, \forall t >K_1$. In either case, we have that

$$\begin{aligned} \phi (t)\le M+M_0 t^2, \forall t \in [0,\infty ). \end{aligned}$$

(3.3)

Using (3.2) and (3.3), we have that

$$\begin{aligned} || x_{n+1}-x^{*}||^2\le & {} || y_n-x^{*}||^2+ \beta _n(\beta _n-(1-k))||T^ny_n-y_n||^2+\beta _n\mu _n M_0||y_n-x^{*}||^2 \nonumber \\&+\,\beta _n \xi _n+\beta _n\mu _n M_1\nonumber \\\le & {} || y_n-x^{*}||^2+\beta _n\mu _n M_0||y_n-x^{*}||^2 +\beta _n \xi _n+\beta _n\mu _n M_1\nonumber \\\le & {} ||u_n-x^{*}||^2+\beta _n\mu _n M_0 ||u_n-x^{*}||^2 +\beta _n \xi _n+\beta _n\mu _n M_1\nonumber \\\le & {} \Big [(1-\alpha _n)||x_n-x^{*}||^2+\alpha _n||x^{*}||^2\Big ]\nonumber \\&+\,\beta _n\mu _n M_0\Big [(1-\alpha _n)||x_n-x^{*}||^2+\alpha _n||x^{*}||^2\Big ] +\beta _n \xi _n+\beta _n\mu _n M_1\nonumber \\\le & {} \Big [(1-\alpha _n)||x_n-x^{*}||^2+\alpha _n||x^{*}||^2\Big ]+\beta _n\mu _n M_0||x_n-x^{*}||^2\nonumber \\&+\,\beta _n\mu _n M_0||x^{*}||^2+\beta _n \xi _n+\beta _n\mu _n M_1\nonumber \\= & {} ||x_n-x^{*}||^2-\Big [\alpha _n-\beta _n\mu _n M_0\Big ]||x_n-x^{*}||^2+\sigma _n, \end{aligned}$$

(3.4)

where $\sigma _n= \alpha _n||x^{*}||^2+\beta _n\mu _n M_0||x^{*}||^2+\beta _n \xi _n+\beta _n\mu _n M_1,~~\forall n\ge 1.$ From (3.4), we have

$$\begin{aligned} || x_{n+1}-x^{*}||^2\le & {} ||x_n-x^{*}||^2-\Big [\alpha _n-\mu _n M_0\Big ]||x_n-x^{*}||^2+\sigma _n. \end{aligned}$$

(3.5)

Since $\mu _n=o(\alpha _n),~~\lambda _n=o(\alpha _n)$ and $\xi _n=o(\alpha _n)$, we may assume without loss of generality that there exist constants $k_0\in (0, 1)$ and $M_2 > 0$ such that for all $n\ge 1$,

$$\begin{aligned} \frac{\mu _n}{\alpha _n}M_0\le 1-k_0,~~\mathrm{and}~~\frac{\sigma _n}{\alpha _n}\le M_2. \end{aligned}$$

Thus, we obtain from (3.5) that

$$\begin{aligned} ||x_{n+1}-x^{*}||^2 \le ||x_n-x^{*}||^2-\alpha _nk_0 ||x_n-x^{*}||^2 + \sigma _n. \end{aligned}$$

By Lemma 2.5, we have that

$$\begin{aligned} ||x_n-x^{*}||^2 \le \max \{|| x_1-x^{*}||^2,(1+k_0)M_2\}. \end{aligned}$$

Therefore, $\{x_n\}$ is bounded. Furthermore, the sequences $\{y_n\}$ and $\{u_n\}$ are bounded.

Observe that since T is a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive, then

$$\begin{aligned}&\!\!\!|| T^nx-x^{*}||^{2} \le || x-x^{*}||^{2}+k ||x-T^nx||^{2}+\mu _n\phi (||x-x^{*}||)+\xi _n\nonumber \\&\!\!\!\quad \Rightarrow \langle T^nx-x^{*},T^nx-x^{*}\rangle \le \langle x-x^{*},x-T^nx\rangle +\langle x-x^{*},T^nx-x^{*}\rangle + k || x-T^nx||^{2}\nonumber \\&\!\!\!\qquad +\mu _n\phi (||x-x^{*}||)+\xi _n \nonumber \\&\!\!\!\quad \Rightarrow \langle T^nx-x^{*},T^nx-x\rangle \le \langle x-x^{*},x-T^nx\rangle + k || x-T^nx||^{2} +\mu _n\phi (||x-x^{*}||)+\xi _n \nonumber \\&\!\!\!\quad \Rightarrow \langle T^nx-x,T^nx-x\rangle +\langle x-x^{*},T^nx-x\rangle \le \langle x-x^{*},x-T^nx\rangle + k || x-T^nx||^{2}\nonumber \\&\!\!\!\qquad +\mu _n\phi (||x-x^{*}||)+\xi _n \nonumber \\&\!\!\!\quad \Rightarrow (1-k)|| x-T^nx||^{2}\le 2\langle x-x^{*},x-T^nx\rangle +\mu _n\phi (||x-x^{*}||)+\xi _n. \end{aligned}$$

(3.6)

It follows from (2.2) and (3.6) that

$$\begin{aligned} || x_{n+1}-x^{*}||^{2}= & {} || (1-\beta _n)y_n+\beta _nT^ny_n-x^{*}||^{2}\nonumber \\= & {} || (y_n-x^{*})+\beta _n(Ty_n-y_n)||^{2}\nonumber \\= & {} || y_n-x^{*}||^{2}+\beta _n^{2}|| y_n-T^ny_n||^{2}-2\beta _n\langle y_n-x^{*},y_n-Ty_n\rangle \nonumber \\\le & {} || y_n\,{-}\,x^{*}||^{2}+\beta _n\Big [\beta _n\,{-}\,(1\,{-}\,k))|| y_n\,{-}\,T^ny_n||^{2}+\mu _n\phi (||y_n\,{-}\,x^{*}||)+\xi _n\Big ]\nonumber \\\le & {} || u_n\!-\!x^{*}||^{2}\,{+}\,\beta _n(\beta _n\!-\!(1\!-\!k))|| y_n\!-\!T^ny_n||^{2}\,{+}\,\beta _n\Big [\mu _n\phi (||y_n\!-\!x^{*}||)\,{+}\,\xi _n\Big ]\nonumber \\= & {} || (1-\alpha _n)x_n-x^{*}||^{2}+\beta _n(\beta _n-(1-k))|| y_n-T^ny_n||^{2}\nonumber \\&+\,\beta _n\Big [\mu _n\phi (||y_n-x^{*}||)+\xi _n\Big ]\nonumber \\= & {} (1-\alpha _n)^{2}|| x_n-x^{*}||^{2}+\alpha _n^{2}|| x^{*}||^{2}-\alpha _n(1-\alpha _n)\langle x_n-x^{*},x^{*}\rangle \nonumber \\&+\,\beta _n(\beta _n-(1-k))|| y_n-T^ny_n||^{2}+\beta _n\Big [\mu _n\phi (||y_n-x^{*}||)+\xi _n\Big ]\nonumber \\\le & {} || x_n-x^{*}||^{2}+\alpha _n^{2}|| x^{*}||^{2}-\alpha _n(1-\alpha _n)\langle x_n-x^{*},x^{*}\rangle \nonumber \\&+\,\beta _n(\beta _n-(1-k))|| y_n-T^ny_n||^{2}+\beta _n\Big [\mu _n\phi (||y_n-x^{*}||)+\xi _n\Big ]\nonumber \\\le & {} || x_n-x^{*}||^{2}+\alpha _n^{2}|| x^{*}||^{2}-\alpha _n(1-\alpha _n)\langle x_n-x^{*},x^{*}\rangle \nonumber \\&+\,\beta _n(\beta _n\!-\!(1-k))|| y_n-T^ny_n||^{2}+\beta _n\Big [\mu _n(M_1+M_0||y_n\!-\!x^{*}||^2)+\xi _n\Big ].\nonumber \\ \end{aligned}$$

(3.7)

Since $\{x_n\}$ and $\{y_n\}$ are bounded, $\exists M>0$ such that

$$\begin{aligned} -(1-\alpha _n)\langle x_n-x^{*},x^{*}\rangle \le M,~~ \forall n\ge 1. \end{aligned}$$

Therefore,

$$\begin{aligned} \beta _n((1-k)-\beta _n)|| y_n-T^ny_n||^{2}\le & {} \alpha _n^{2}|| x^{*}||^{2}+\alpha _nM-|| x_{n+1}-x^{*}||^{2}+|| x_n-x^{*}||^{2}\nonumber \\&+\,\mu _n(M_1+M_0||y_n-x^{*}||^2)+\xi _n. \end{aligned}$$

(3.8)

Now we divide the rest of the proof into two cases.

$\mathbf{Case1}$

Assume that $\{||x_n-x^{*}||\}$ is monotonically decreasing sequence. Then $\{||x_n-x^{*}||\}$ is convergent, obviously

$$\begin{aligned} || x_{n+1}-x^{*}||^2-|| x_n-x^{*}||^2\rightarrow 0, n\rightarrow \infty . \end{aligned}$$

(3.9)

This together with (3.8) and the conditions that $\alpha _n\rightarrow 0, \mu _n\rightarrow 0$ and $\xi _n\rightarrow 0$ imply that

$$\begin{aligned} || y_n-T^ny_n||\rightarrow 0, n\rightarrow \infty . \end{aligned}$$

From (3.1) and (2.2), we have that

$$\begin{aligned}&\rho _n\Big (\frac{4h(u_n)}{h(u_n)+l(u_n)}-\rho _n\Big )\frac{(h(u_n)+l(u_n))^2}{\theta ^2(u_n)} \le (1+M^*\lambda _n)||u_n-x^{*}||^2-||y_n-x^{*}||^2\\&\quad \le (1-\alpha _n)||x_n-x^{*}||^2+\alpha _n||x^{*}||^2-||x_{n+1}-x^{*}||^2+M^*\lambda _n||u_n-x^{*}||^2\\&\quad \le ||x_n-x^{*}||^2-||x_{n+1}-x^{*}||^2 +\alpha _n||x^{*}||^2+M^*\lambda _n||u_n-x^{*}||^2. \end{aligned}$$

Using conditions that $\alpha _n\rightarrow 0$ and $\lambda _n\rightarrow 0$, we have that

$$\begin{aligned} \rho _n\Big (\frac{4h(u_n)}{h(u_n)+l(u_n)}-\rho _n\Big )\frac{(h(u_n)+l(u_n))^2}{\theta ^2(u_n)} \rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

Hence, we obtain

$$\begin{aligned} \frac{(h(u_n)+l(u_n))^2}{\theta ^2(u_n)} \rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

(3.10)

Consequently, we have

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }(h(u_n)+l(u_n))=0\Leftrightarrow \underset{n\rightarrow \infty }{\lim }h(u_n)=0~~\mathrm{and}~~\underset{n\rightarrow \infty }{\lim }l(u_n)=0, \end{aligned}$$

because $\theta ^2(u_n)=||\nabla h(u_n)||^2+||\nabla l(u_n)||^2$ is bounded. This follows from the fact that $\nabla h$ is Lipschitz continuous with constant $||A||^2$, $\nabla l$ is nonexpansive and $\{u_n\}$ is bounded. More precisely, for any $x^{*}$ which solves (1.9), we have

$$\begin{aligned} ||\nabla h(u_n)||=||\nabla h(u_n)-\nabla x^{*}|| \le ||A||^2||u_n-x^{*}||~~\mathrm{and} \end{aligned}$$

$$\begin{aligned} ||\nabla l(u_n)||=||\nabla l(u_n)-\nabla x^{*}||\le ||u_n-x^{*}||. \end{aligned}$$

Now, let z be a weak cluster point of $\{u_n\}$, there exists a subsequence $\{u_{n_j}\}$ which weakly converges to z. The lower-semicontinuity of h then implies that

$$\begin{aligned} 0\le h(z)\le \underset{j\rightarrow \infty }{\liminf }h(u_{n_j})=\underset{n\rightarrow \infty }{\lim }h(u_n)=0. \end{aligned}$$

That is, $h(z)=\frac{1}{2}||(I- \mathrm{prox}_{\lambda g})Az||= 0$, i.e., Az is a fixed point of the proximal mapping of g or equivalently, $0\in \partial _{pg}(Az)$. In other words, Az is a minimizer of g.

Likewise, the lower-semicontinuity of l implies that

$$\begin{aligned} 0\le l(z)\le \underset{j\rightarrow \infty }{\liminf }l(u_{n_j})=\underset{n\rightarrow \infty }{\lim }l(u_n)=0. \end{aligned}$$

That is, $l(z)=\frac{1}{2}||(I- \mathrm{prox}_{\tau \lambda f})z||= 0$, i.e., z is a fixed point of the proximal mapping of f or equivalently, $0\in \partial f(z)$. In other words, z is a minimizer of f. Hence, $z \in \Gamma$.

Next, we show that $z \in F(T)$. Since $x^{*}=\mathrm{prox}_{\lambda \tau _n f}( x^{*}-\mu _n A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})$ and $A^*(I-\mathrm{prox}_{\lambda g})A$ is Lipschitz with constant $||A||^2$, we have from (2.2) that

$$\begin{aligned}&\!\!\!|| y_n-x^{*}||^2\\&\!\!\!\quad = ||\mathrm{prox}_{\lambda \tau _n f}( u_n-\mu _n A^*(I-\mathrm{prox}_{\lambda g})Au_n)-\mathrm{prox}_{\lambda \tau _n f}\left( x^{*}-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Ax^{*}\right) ||^2\nonumber \\&\!\!\!\quad \le \langle ( u_n-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Au_n)-\left( x^{*}-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Ax^{*}\right) , y_n-x^{*}\rangle \nonumber \\&\!\!\!\quad =\frac{1}{2}\Big [ ||( u_n-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Au_n)-\left( x^{*}-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Ax^{*}\right) ||^2 +||y_n-x^{*}||^2\\&\!\!\!\qquad -\,||( u_n-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Au_n)-\left( x^{*}-\tau _n A^*(I-\mathrm{prox}_{\lambda g})Ax^{*}\right) -(y_n-x^{*})||^2\Big ]\\&\!\!\!\quad \le \frac{1}{2}\Big [ (1+\tau _n||A||^2)^2||u_n-x^{*}||^2 +||y_n-x^{*}||^2-||u_n-y_n -\tau _n(A^*(I-\mathrm{prox}_{\lambda g})Au_n\\&\!\!\!\qquad -\, A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})||^2\Big ]\\&\!\!\!\quad =\frac{1}{2}\Big [ (1+\tau _n||A||^2)^2||u_n-x^{*}||^2 +||y_n-x^{*}||^2\\&\!\!\!\qquad -\,||u_n-y_n||^2+2\tau _n \langle u_n-y_n, A^*(I-\mathrm{prox}_{\lambda g})Au_n- A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})\rangle \\&\!\!\!\qquad -\,\tau _n^2||A^*(I-\mathrm{prox}_{\lambda g})Au_n- A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})||^2\Big ]. \end{aligned}$$

Thus,

$$\begin{aligned} || y_n-x^{*}||^2\le & {} (1+\tau _n||A||^2)^2||x_n-x^{*}||^2-||u_n-y_n||^2\nonumber \\&+\,2\tau _n \langle u_n-y_n, A^*(I-\mathrm{prox}_{\lambda g})Au_n- A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})\rangle \nonumber \\&-\,\tau _n^2||A^*(I-\mathrm{prox}_{\lambda g})Au_n- A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})||^2. \end{aligned}$$

(3.11)

We observe that

$$\begin{aligned} 0<\tau _n<4\frac{h(u_n)+l(u_n)}{\theta ^2(u_n)} \rightarrow 0,~~n\rightarrow \infty \end{aligned}$$

implies that $\tau _n \rightarrow 0,~~n\rightarrow \infty$. Furthermore, we obtain from (3.11) and (2.2) that

$$\begin{aligned} ||u_n-y_n||^2\le & {} (1+\tau _n||A||^2)^2||u_n-x^{*}||^2-|| y_n-x^{*}||^2\nonumber \\&+\,2\tau _n \langle u_n-y_n, A^*(I-\mathrm{prox}_{\lambda g})Au_n- A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})\rangle \nonumber \\= & {} ||u_n-x^{*}||^2+\tau _n||A||^2(2+\tau _n||A||^2)||u_n-x^{*}||^2-|| y_n-x^{*}||^2\nonumber \\&+\,2\tau _n \langle u_n-y_n, A^*(I-\mathrm{prox}_{\lambda g})Au_n- A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})\rangle \nonumber \\\le & {} (1-\alpha _n)||x_n-x^{*}||^2+\alpha _n||x^{*}||^2-||x_{n+1}-x^{*}||^2\nonumber \\&+\,\tau _n||A||^2(2+\tau _n||A||^2)||u_n-x^{*}||^2+2\tau _n \langle u_n-y_n, A^*(I-\mathrm{prox}_{\lambda g})Au_n\nonumber \\&-\, A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})\rangle \nonumber \\\le & {} ||x_n\!-\!x^{*}||^2\!-\!||x_{n+1}-x^{*}||^2+\alpha _n||x^{*}||^2\!+\!\tau _n||A||^2(2+\tau _n||A||^2)||u_n\!-\!x^{*}||^2\nonumber \\&+\,2\tau _n \langle u_n-y_n, A^*(I-\mathrm{prox}_{\lambda g})Au_n- A^*(I-\mathrm{prox}_{\lambda g})Ax^{*})\rangle . \end{aligned}$$

(3.12)

Since $\tau _n \rightarrow 0,~~n\rightarrow \infty$ and $\alpha _n \rightarrow 0,~~n\rightarrow \infty$, we obtain that

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

We observe that $||u_n-x_n||\le \alpha _n||x_n||\rightarrow 0,~~n\rightarrow \infty$ and

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

Using Lemma 2.3 (i), we have that

$$\begin{aligned} ||x_{n+1}-x_n||^2= & {} ||x_n-y_n+y_n-x_{n+1}||^2 \nonumber \\\le & {} ||y_n-x_{n+1}||^2+2\langle x_n-y_n,x_n-x_{n+1}\rangle \nonumber \\\le & {} ||y_n-x_{n+1}||^2+2||x_n-y_n||||x_n-x_{n+1}||. \end{aligned}$$

(3.13)

Using (2.2) and (3.13), we have

$$\begin{aligned} ||y_n-T^ny_n||^2= & {} \frac{1}{\beta _n^2}||y_n-x_{n+1}||^2 \nonumber \\\ge & {} \frac{1}{\beta _n^2}\Big [||y_n-x_{n+1}||^2+2||x_n-y_n||||x_n-x_{n+1}||\Big ]. \end{aligned}$$

(3.14)

Since $\lim \nolimits _{n\rightarrow \infty } ||y_n-T^ny_n||=0$ and $0<\liminf \nolimits _{n\rightarrow \infty } \beta _n \le \limsup \nolimits _{n\rightarrow \infty }\beta _n<1-k$, then we have that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\Big [||y_n-x_{n+1}||^2+2||x_n-y_n||||x_n-x_{n+1}||\Big ]=0, \end{aligned}$$

from which we have

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||x_n-x_{n+1}||=0. \end{aligned}$$

Consequently,

$$\begin{aligned} ||u_{n+1}-u_n||= & {} ||(1-\alpha _{n+1})x_{n+1}-(1-\alpha _n)x_n||\nonumber \\\le & {} |\alpha _{n+1}-\alpha _n|||x_{n+1}||+(1-\alpha _n)||x_{n+1}-x_n||\rightarrow 0,~~n\rightarrow \infty .\qquad \quad \end{aligned}$$

(3.15)

Now,

$$\begin{aligned} ||y_{n+1}-y_n||\le & {} ||y_{n+1}-u_{n+1}||+||u_n-y_n||+||u_{n+1}-u_n||. \end{aligned}$$

(3.16)

Using (3.15) and the fact that $\lim \nolimits _{n\rightarrow \infty } ||u_n-y_n||=0$ in (3.16), we obtain

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||y_{n+1}-y_n||=0. \end{aligned}$$

Using the fact that T is uniformly L-Lipschitzian, we have

$$\begin{aligned} ||Ty_n-y_n||\le & {} ||Ty_n-T^{n+1}y_n||+||T^{n+1}y_n-T^{n+1}y_{n+1}||\nonumber \\&+\,||T^{n+1}y_{n+1}-y_{n+1}||+||y_{n+1}-y_n||\nonumber \\\le & {} L||y_n-T^ny_n||+(L+1)||y_{n+1}-y_n||+||T^{n+1}y_{n+1}-y_{n+1}||\nonumber \\\le & {} L||y_n-T^ny_n||+(L+1)||A|| ||y_{n+1}-y_n||+||T^{n+1}y_{n+1}-y_{n+1}||.\nonumber \\ \end{aligned}$$

(3.17)

By using $\lim \nolimits _{n\rightarrow \infty } ||y_{n+1}-y_n||=0$ and $\lim \nolimits _{n\rightarrow \infty } ||y_n-T^ny_n||=0$ in (3.17), we obtain

$$\begin{aligned} || y_n-Ty_n||\rightarrow 0, n\rightarrow \infty . \end{aligned}$$

(3.18)

Using the fact that $u_{n_j}\rightharpoonup z \in H_1$ and $||u_n-y_n|| \rightarrow 0,~~n\rightarrow \infty$, we have that $y_{n_j}\rightharpoonup z \in H_1$. Similarly, $u_{n_j}\rightharpoonup z \in H_1$ since $||u_n-x_n||\rightarrow 0,~~n\rightarrow \infty$. Using Lemma 2.4 and (3.18), we have that $z \in F(T)$. Therefore, $z \in \Gamma \cap F(T).$

From (2.2), we have

$$\begin{aligned} || x_{n+1}-x^{*}||^2= & {} ||(1-\beta _n)y_n+\beta _n T^ny_n-x^{*}||^2 \nonumber \\= & {} || (1-\beta _n)(y_n-x^{*})+\beta _n(T^ny_n-x^{*})||^2 \nonumber \\= & {} (1-\beta _n)^2|| y_n-x^{*}||^2 +\beta _n^2|| T^ny_n-x^{*}||^2\nonumber \\&+\, 2\beta _n(1-\beta _n)\langle y_n-x^{*},T^ny_n-x^{*}\rangle \nonumber \\\le & {} (1-\beta _n)^2|| y_n-x^{*}||^2 +\beta _n^2[|| y_n-x^{*}||^2 +k|| y_n-T^ny_n||^2] \nonumber \\&+\,\beta _n^2\mu _n\phi (||y_n-x^{*}||)+\beta _n^2\xi _n+2\beta _n(1-\beta _n)[|| y_n-x^{*}||^2\nonumber \\&-\,\frac{1-k}{2}|| y_n-T^ny_n||^2]\nonumber \\= & {} (1-2\beta _n+\beta _n^2)|| y_n-x^{*}||^2+ \beta _n^2[|| y_n-x^{*}||^2 +k|| y_n-T^ny_n||^2] \nonumber \\&+\,2\beta _n|| y_n-x^{*}||^2-2\beta _n^2|| y_n-x^{*}||^2 -\beta _n(1-\beta _n)(1-k)|| y_n-T^ny_n||^2 \nonumber \\&+\,\beta _n^2\mu _n(M+M_0||y_n-x^{*}||^2)+\beta _n^2\xi _n \nonumber \\= & {} || y_n-x^{*}||^2+[\beta _n^2k -\beta _n(1-\beta _n)(1-k)]|| y_n-T^ny_n||^2\nonumber \\&+\,\beta _n^2\mu _n(M+M_0||y_n-x^{*}||^2)+\beta _n^2\xi _n \nonumber \\= & {} || y_n-x^{*}||^2 + \beta _n[k + \beta _n -1]|| y_n-T^ny_n||^2 \nonumber \\\le & {} || y_n-x^{*}||^2+\beta _n^2\mu _n(M+M_0||y_n-x^{*}||^2)+\beta _n^2\xi _n \nonumber \\= & {} || y_n-x^{*}||^2+a_n, \end{aligned}$$

(3.19)

where $a_n=\beta _n^2\mu _n(M+M_0||y_n-x^{*}||^2)+\beta _n^2\xi _n.$

Now, from (3.1), (3.19) and Lemma 2.3 (ii), we have

$$\begin{aligned} ||x_{n+1}-x^{*}||^2\le & {} ||y_n-x^{*}||^2 +a_n\le ||u_n-x^{*}||^2+a_n\nonumber \\= & {} ||(1-\alpha _n)(x_n-x^{*})-\alpha _nx^{*}||^2+a_n \nonumber \\\le & {} (1-\alpha _n)||x_n-x^{*}||^2-2\alpha _n \langle u_n-x^{*},x^{*}\rangle +a_n. \end{aligned}$$

(3.20)

It is clear that

$$\begin{aligned} \underset{n\rightarrow \infty }{\limsup }-2 \langle u_n-x^{*},x^{*}\rangle= & {} \underset{j\rightarrow \infty }{\lim }-2 \langle u_{n_j}-x^{*},x^{*}\rangle -2 \langle z-x^{*},x^{*}\rangle \le 0, \end{aligned}$$

since $\{u_n\}$ converges weakly to z and $x^{*}$ is the minimum-norm solution (i.e., $x^*=P_{\Gamma \cap F(T)}0$). Also, we observe that $\sum \nolimits _{n=1}^{\infty } a_n <\infty .$ Now, using Lemma 2.6 in (3.20), we have $||x_n-x^{*}||\rightarrow 0$. That is, $x_n\rightarrow x^{*}, n\rightarrow \infty$.

Case 2

Assume that $\{|| x_n-x^{*}||\}$ is not monotonically decreasing sequence. Set $\Gamma _{n}=||x_{n}-x^{*}||^{2}$ and let $\tau :\mathbb {N}\rightarrow \mathbb {N}$ be a mapping for all $n\ge n_{0}$ (for some $n_{0}$ large enough)by

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

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

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

From (3.18), it is easy to see that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }||y_{\tau (n)}-Ty_{\tau (n)}||=0. \end{aligned}$$

Furthermore, we can show that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }h(x_{\tau (n)})=0~~\mathrm{and}~~\underset{n\rightarrow \infty }{\lim }l(x_{\tau (n)})=0. \end{aligned}$$

By similar argument as above in Case 1, we conclude immediately that $x_{\tau (n)}, y_{\tau (n)}$ and $u_{\tau (n)}$ weakly converge to z as $\tau (n)\rightarrow \infty .$ At the same time, from (3.20), we note that, for all $n\ge n_{0},$

$$\begin{aligned} 0\le & {} ||x_{\tau (n)+1}-x^{*}||^{2}-||x_{\tau (n)}-x^{*}||^{2}\\\le & {} \alpha _{\tau (n)}[-2\langle u_{\tau (n)}-x^{*},x^{*}\rangle - ||x_{\tau (n)}-x^{*}||^{2}]+a_{\tau (n)}, \end{aligned}$$

which implies

$$\begin{aligned} ||x_{\tau (n)}-x^{*}||^{2}\le -2\langle u_{\tau (n)}-x^{*},x^{*}\rangle +a_{\tau (n)}. \end{aligned}$$

Hence, we deduce that

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

Therefore,

$$\begin{aligned} \lim _{n\rightarrow \infty }\Gamma _{\tau (n)}=\lim _{n\rightarrow \infty }\Gamma _{\tau (n)+1}= 0. \end{aligned}$$

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, $\lim \Gamma _{n}=0$, that is, $\{x_n\}$ converges strongly to $x^{*}$. This completes the proof. $\square$

4 Applications

Applying Theorem 3.1 to the case where $f =\delta _C$, $g =\delta _Q$ the indicator functions of two nonempty, closed and convex sets C, Q of $H_1$ and $H_2$ respectively, we have the following result.

Theorem 4.1

Let C, Q be nonempty, closed and convex subset of real Hilbert spaces $H_1$ and $H_2$ respectively. Let $T:C\rightarrow C$ is a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping such that $F(T)\ne \emptyset$. Assume that problem (1.2) is consistent (i.e., $\Gamma \ne \emptyset$) and $\Gamma \cap F(T)\ne \emptyset$. Let $\{\alpha _n\}$ and $\{\beta _n\}$ be sequences in (0, 1). If the parameters satisfy the following conditions

• (a) $\sum \limits _{n=1}^{\infty }\mu _n<\infty ; \sum \limits _{n=1}^{\infty } \xi _n<\infty ;$

• (b) $\mu _n=o(\alpha _n); \xi _n=o(\alpha _n); \underset{n\rightarrow \infty }{\lim }\alpha _n =0; \sum \limits _{n=1}^{\infty } \alpha _n =\infty ;$

• (c) $\epsilon \le \rho _n \le \frac{4 h(u_n)}{h(u_n)+l(u_n)}-\epsilon$ for some $\epsilon >0$;

• (d) $0<\underset{n\rightarrow \infty }{\liminf }\beta _n \le \underset{n\rightarrow \infty }{\limsup }\beta _n<1-k$;

• (e) there exist constants $M_0>0, K_1 > 0$ such that $\phi (t)\le M_0 t^2, \forall t >K_1$;

then sequence $\{x_n\}$ generated by

$$\begin{aligned} \left\{ \begin{array}{l} u_n=(1-\alpha _n)x_n,\\ y_n =P_C(u_n-\tau _n A^*(I-P_Q)Au_n),\\ x_{n+1} =(1-\beta _n)y_n+ \beta _nT^ny_n,~~n \ge 1, \end{array} \right. \end{aligned}$$

(4.1)

converges strongly to $x^*\in \Gamma \cap F(T)$ which is also the minimum-norm solution (i.e., $x^*\in \Gamma \cap F(T)$ and $||x^*|| =\min \{ ||x||: x \in \Gamma \cap F(T)\}$).

Proof

Take $f =\delta _C$ and $g =\delta _Q$ in Theorem 3.1. Then we have $\mathrm{prox}_{\lambda f}=P_C$ and $\mathrm{prox}_{\lambda g}=P_Q$. Furthermore, we see that algorithm reduces to (2.2) reduces to (4.1) and the conclusion of Theorem 4.1. $\square$

We next apply our results to split equilibrium problem and fixed point problem. Let C, Q be nonempty, closed and convex subsets of $H_1$ and $H_2$ respectively. Let f be a bifunction of $C \times C$ into $\mathbb {R}$ and g a bifunction of $Q \times Q$ into $\mathbb {R}$. Suppose $A:H_1\rightarrow H_2$ is a bounded linear operator. Let us consider the following split equilibrium problem. The split equilibrium problem is to find $x^{*} \in C$ such that

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

(4.2)

and

$$\begin{aligned} g(Ax^{*}, y)\ge 0,~~\forall y \in Q. \end{aligned}$$

(4.3)

We shall denote the solutions set of (4.2)-(4.3) by $\Gamma$. In order to solve the split equilibrium problem for a bifunctions $f:C \times C\rightarrow \mathbb {R}$ and $g:Q \times Q\rightarrow \mathbb {R}$, let us assume that f and g satisfy the following conditions:

• (A1) $f(x,x)=0$ for all $x \in C$ and $g(x,x)=0$ for all $x \in Q$;

• (A2) f and g are monotone, i.e., $f(x,y)+f(y,x) \le 0$ for all $x, y, \in C$ and $g(x,y)+g(y,x) \le 0$ for all $x, y, \in Q$;

• (A3) for each $x, y \in C,~~\lim \nolimits _{t\rightarrow 0} f(tz+(1-t)x, y) \le f(x, y)$ and for each $x, y \in Q,~~\lim \nolimits _{t\rightarrow 0} g(tz+(1-t)x, y) \le g(x, y)$;

• (A4) for each $x \in C,~~y\mapsto f(x,y)$ is convex and lower semicontinuous and for each $x \in Q,~~y\mapsto g(x,y)$ is convex and lower semicontinuous.

The following lemma follows from [8].

Lemma 4.2

(Combettes and Hirstoaga, [8]) Let C, Q be nonempty, closed and convex subsets of $H_1$ and $H_2$ respectively. Assume that f, g satisfy (A1)-(A4). For $r>0$, define mappings $T^f_r:H_1\rightarrow C$ and $T^g_r:H_2\rightarrow Q$ as follows:

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

and

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

Then, the following assertions:

• 1. $T^f_r$ and $T^g_r$ are single-valued;

• 2. $T^f_r$ and $T^g_r$ are firmly nonexpansive-type mapping, i.e., for any $x, y \in H_1$,

  $$\begin{aligned} ||T^f_r x-T^f_r y||^2 \le \langle T^f_r x- T^f_r y, x-y \rangle ; \end{aligned}$$

• 3. $F(T^f_r)=EP(f)$ and $F(T^g_r)=EP(g)$;

• 4. EP(f) and EP(g) are closed and convex.

Now, we prove the following theorem.

Theorem 4.3

Let $T:C\rightarrow C$ is a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping such that $F(T)\ne \emptyset$. Let f be a bifunction from $C \times C$ and g a bifunction from $Q \times Q$ both satisfying ($A1) - (A4$) such that $\Gamma \ne \emptyset$) and $\Gamma \cap F(T)\ne \emptyset$. Let $\{\alpha _n\}$ and $\{\beta _n\}$ be sequences in (0, 1). If the parameters satisfy the following conditions

• (a) $\sum \limits _{n=1}^{\infty }\mu _n<\infty ; \sum \limits _{n=1}^{\infty } \xi _n<\infty ;$

• (b) $\mu _n=o(\alpha _n); \xi _n=o(\alpha _n); \underset{n\rightarrow \infty }{\lim }\alpha _n =0; \sum \limits _{n=1}^{\infty } \alpha _n =\infty ;$

• (c) $\epsilon \le \rho _n \le \frac{4 h(u_n)}{h(u_n)+l(u_n)}-\epsilon$ for some $\epsilon >0$;

• (d) $0<\underset{n\rightarrow \infty }{\liminf }\beta _n \le \underset{n\rightarrow \infty }{\limsup }\beta _n<1-k$;

• (e) there exist constants $M_0>0, K_1 > 0$ such that $\phi (t)\le M_0 t^2, \forall t >K_1$;

then sequence $\{x_n\}$ generated by $x_1 \in C$;

$$\begin{aligned} \left\{ \begin{array}{l} u_n=(1-\alpha _n)x_n,\\ y_n =T^f_r(u_n-\tau _n A^*(I-T^g_r)Au_n),\\ x_{n+1} =(1-\beta _n)y_n+ \beta _nT^ny_n,~~n \ge 1, \end{array} \right. \end{aligned}$$

(4.4)

converges strongly to $x^*\in \Gamma \cap F(T)$ which is also the minimum-norm solution (i.e., $x^*\in \Gamma \cap F(T)$ and $||x^*|| =\min \{ ||x||: x \in \Gamma \cap F(T)\}$).

Proof

Replace the proximal mappings of the convex functions f and g in Theorem 3.1 by the resolvent operators associated to the two monotone equilibrium bifunctions $T^f_r$ and $T^g_r$. Hence, we have the desired result. $\square$

Remark 1

The following are our contributions in this paper.

• 1. We obtain strong convergence result concerning convex split feasibility problem and fixed point problem in real Hilbert spaces. We recall that Moudafi and Thakur [13] obtained weak convergence result for split feasibility problem alone and thus our result improves on and extends the results of Moudafi and Thakur [13].

• 2. It is worth mentioning here that our result in this paper is more applicable than the result of Moudafi and Thakur [13] in the sense that our result can be applied to finding an approximate common solution to proximal split feasibility problem and fixed point problem for a uniformly L-Lipschitzian continuous and $(k, \{\mu _n\}, \{\xi _n\}, \phi )$-total asymptotically strict pseudocontractive mapping.

• 3. In all our results in this paper, our iterative scheme is proposed with a way of selecting the step-size such that the implementation of our algorithm does not need any prior information about the operator norm ||A|| because the calculation or at least an estimate of the operator norm ||A|| is very difficult, if not an impossible task. Therefore, improved on the results of Chang et al. [6], Yang et al. [22], Cholamjiak and Shehu [7] and other related works.

• 4. Our iterative algorithm in this paper appears more efficient and implementable. Our algorithm appears simpler than the “CQ” algorithm used [10] and other related papers for similar problems. Furthermore, our iterative scheme gives strong convergence without imposing any extra compactness type condition ( like semi-compactness) on the mapping T. This compactness condition appears strong as only few mappings are semi-compact. Therefore, we improved on the results of Chang et al. [6], Qin et al. [14], Ding and Quan [9] and other related results.