A Projection-Type Implicit Algorithm for Finding a Common Solution for Fixed Point Problems and Variational Inequality Problems

Abstract

This paper deals with the problem of finding a common solution for a fixed point problem for strictly pseudocontractive mappings and for a certain variational inequality problem. We propose a projection-type implicit averaged algorithm and establish the strong convergence of the sequences generated by this method to the common solution for the fixed point problem and the variational inequality problem. In order to illustrate the feasibility of the hypotheses and the superiority of our theoretical results over the existing literature, an example is also presented.

1. Introduction

Several problems in nonlinear analysis can be solved by transforming them into equivalent fixed point problems of the form $x=Tx$, with T as an appropriate operator. This means that any fixed point of the operator T, that is, any element of the set

$$Fix\left(T\right)=\{x:Tx=x\}$$

is a solution of the original problem and vice versa. The main benefit of such a fixed point formulation of the initial problem is that in order to approximate its solution we can naturally construct an iterative scheme of the form

$$x_{n+1}=F{x}_n,n\ge 0,$$

with F depending on T, and starting from an initial value $x_0$.

For example, in the case $F=T$, the above iterative scheme reduces to the well-known Picard iteration (or sequence of successive approximations).

As a simple illustration of the fixed point approach, let us consider the problem of constrained optimization, which seeks a vector x in a given convex and closed subset C of a real Hilbert space H that minimizes a certain function $f:C\to \mathbb{R}$.

It is well known that, under suitable conditions (see for example Byrne [1]), this minimization problem is equivalent to a fixed point problem $x=Tx$ with

$$T=P_C\left(I-\gamma \nabla f\right),$$

where $P_C$ is the metric projection of H onto C, I is the identity operator and $\gamma >0$ is a constant.

Since $P_C$ is nonexpansive and, under certain conditions, $I-\gamma \nabla f$ is a contraction, it follows that T is a contraction and so its unique fixed point is the solution of the considered constrained optimization problem that can be approximated by the iterative scheme

$$x_{n+1}=P_C\left(I-\gamma \nabla f\right)x_n,n\ge 0,$$

which is the Picard scheme corresponding to the iteration mapping $T=P_C\left(I-\gamma \nabla f\right)$.

In many instances that occur, for example, in the signal processing and image reconstruction problems, T is in general not a contraction but a nonexpansive mapping or a nonexpansive-type mapping. In such a situation, Picard iteration does not converge or, even though it converges, its limit is not a fixed point of T and it is needed to use more elaborated iterative schemes, like the Krasnoselskij–Mann iteration, defined by

$$x_{n+1}=(1-\lambda )x_n+\lambda T{x}_n,n\ge 0,$$

(1)

with $\lambda \in (0,1)$. Marino and Xu [2] used the Krasnoselskij–Mann iteration scheme (1) to approximate the fixed points of k-strictly pseudocontractive mappings, a class that includes nonexpansive mappings, and proved weak and strong convergence results for this method, thus extending from nonexpansive to k-strictly pseudocontractive mappings the convergence results obtained by Nakajo and Takahashi [3].

An important extension of algorithm defined by (1), that was called "viscosity”-type, was proposed in 2000 by Moudafi [4], and was obtained by replacing the term $(1-\lambda )x_n$ by $(1-\lambda )f\left(x_n\right)$, that is,

$$x_{n+1}=(1-{\alpha }_n)f\left(x_n\right)+{\alpha }_{n}T{x}_n,n\ge 0,$$

(2)

with $f:C\to C$ as a given contraction. The particularity of a viscosity approximation method is the fact that it selects a particular fixed point of a given nonexpansive mapping T.

Amongst the most important extensions of the viscosity iteration method we mention those of Xu [5], who showed that if C is a closed convex subset of the real Hilbert space H, $T:C\to C$ is a nonexpansive mapping with a nonempty fixed point set and $f:C\to C$ is a contraction. Then, the sequence $\left\{x_n\right\}$ generated by (2) converges strongly to a fixed point $\overline{x}$ of T and also $\overline{x}$ solves the following variational inequality:

$$⟨(I-f)\overline{x},x-\overline{x}⟩\ge 0, \mathrm{for} \mathrm{all} x\in Fix\left(T\right).$$

(3)

Moudafi [6] established strong convergence theorems for both explicit and implicit viscosity iteration schemes to a specific fixed point of a nonexpansive mapping.

Another type of iterative algorithm for approximating the fixed point of nonexpansive mappings was proposed in 2014 by Alghamdi et al. [7]. They considered the following implicit midpoint scheme:

$$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T\left({\displaystyle \frac{x_n+x_{n+1}}{2}}\right),n\ge 0,$$

(4)

and proved that if T is nonexpansive, then the sequence $\left\{x_n\right\}$ generated by (4) converges weakly to a fixed point of T.

To obtain a strongly convergent implicit scheme, Xu et al. [8] regularized the implicit midpoint rule (4) by means of a contraction f and proposed the following viscosity type midpoint scheme:

$$x_{n+1}=(1-{\alpha }_n)f\left(x_n\right)+{\alpha }_{n}T\left({\displaystyle \frac{x_n+x_{n+1}}{2}}\right),n\ge 0,$$

(5)

and proved that the sequence $\left\{x_n\right\}$ generated by (5) converges strongly to a fixed point $\overline{x}$ of T which also solves the variational inequality (3).

In the same spirit, Ke and Ma [9] extended the implicit scheme (5) by replacing the arithmetic mean of $x_n$ and $x_{n+1}$ by a convex combination of weight ${\beta }_n$, that is, by ${\beta }_n{x}_n+(1-{\beta }_n)x_{n+1}$, thus obtaining a more general viscosity implicit method for which they have proved similar convergence theorems.

Another important class of iterative schemes that was developed in the last few decades with the aim of approximating fixed points of nonexpansive mappings is based on involving the projection mapping in its body.

In this respect, we mention the paper by Ceng at al. [10] who considered the following implicit continuous type scheme:

$$x_t=P_C\left[t\gamma V{x}_t+(I-t\mu U)T{x}_t\right],t\in (0,1)$$

(6)

and the explicit one:

$$x_{n+1}=P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T{x}_n\right],n\ge 0,$$

(7)

where C is a nonempty closed and convex subset of a real Hilbert space H, T is a nonexpansive self mapping of C with $Fix\left(T\right)\ne \varnothing$, $U:C\to H$ is a Lipschitzian and strongly monotone operator, $V:C\to H$ is a Lipshitzian operator and, as usual, $\left\{{\alpha }_n\right\}\subset (0,1)$ is the parameter sequence.

The main results in Ceng at al. [10] establish the strong convergence of the net ${\left\{x_t\right\}}_{t\in (0,1)}$ and of the sequence $\left\{x_n\right\}$ to a fixed point $\overline{x}$ of T, which also solves the variational inequality

$$⟨(\mu U-\gamma V)\overline{x},x-\overline{x}⟩\ge 0, \mathrm{for} \mathrm{all} x\in Fix\left(T\right).$$

(8)

By considering a generalized projection-type version of the implicit viscosity iterative scheme (5), Sahu et al. [11] proposed a more general implicit nonviscosity algorithm to approximate the fixed points of nonexpansive mappings for which they have established a strong convergence result.

Starting from the above summarized facts, our aim in this paper is to modify the above algorithm in such a way so as to make it able to simultaneously solve a fixed point problem over the solutions of a variational inequality problem, for the class k-strictly pseudocontractive mappings, which is a larger class than the class of nonexpansive mappings.

This is motivated by the fact that there is a significant current interest in finding a common solution for a fixed point problem and for a variational inequality problem (see Alakoya and Mewomo [12], Ansari et al. [13], Batra et al. [14], Eslamian and Kamandi [15], Filali et al. [16], Jolalaoso et al. [17], Mouktonglang et al. [18], Oyewole et al. [19], Peeyada et al. [20], Uba et al. [21], Yin et al. [22], Yu at al. [23]…).

Therefore, our main aim in this paper is to obtain a flexible projection-type implicit iterative scheme and to establish its strong convergence to a fixed point $\overline{x}$ of a k-strictly pseudocontractive mapping T, which also solves the variational inequality (8).

Thus, by inserting an averaged component within the algorithm used in Sahu et al. [11], we obtain (see Theorem 1 in Section 3 and the supporting Examples 1 and 2) more general results that extend and generalize the results previously established by Xu et al. [8], Ke and Ma [9], and Sahu et al. [11], and many other related results.

2. Preliminaries

Let C be a nonempty subset of a normed space X. A mapping $T:C\to C$ is called

(i)

nonexpansive if

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel ,\forall x,y\in C.$$

(9)

(ii)

strictly pseudocontractive or k-strictly pseudocontractive if there exists $k<1$ such that

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k\parallel x-y-(Tx-Ty)\parallel ,\forall x,y\in C.$$

(10)

The notion of a strictly pseudocontractive operator in Hilbert spaces was introduced and studied by Browder and Petryshyn [24] and attracted the interest of many researchers (see Bethke [25], Che et al. [26], Deng [27], Göhde [28], Johnson [29], Liou [30], Marino and Xu [2], Moloney [31], Mukherjee and Som [32], Osilike and Udomene [33], Schu [34,35], Weng [36], Zhou [37], etc.).

It is easy to see that any nonexpansive mapping is k-strictly pseudocontractive, for any $k\in [0,1)$; however, the reverse is not true, as illustrated by the following example.

Example 1 

(Berinde [38]). Let H be the real line with the usual norm, $C=\left[{\displaystyle \frac{1}{2}},2\right]$, and $T:C\to C$ be defined by $T\left(x\right)={\displaystyle \frac{1}{x}},\forall x\in C$.

Then, (1) $Fix\left(T\right)=\left\{1\right\}$;    (2) T is ${\displaystyle \frac{3}{5}}$-strictly pseudocontractive;    (3) T is not nonexpansive.

The following property of strictly pseudocontractive operators in Hilbert spaces will be essential in establishing our new results in this paper.

Lemma 1 

(Zhou [37]). Let C be a nonempty subset of a real Hilbert space and let $T:C\to C$ be a k-strictly pseudocontractive mapping. Then, the averaged mapping $T_{\lambda }=(1-\lambda )I+\lambda T$ is nonexpansive for any $\lambda \in (0,1-k)$.

The following lemmas are very important in the proof of our main results.

Lemma 2 

(Marino and Xu [2]). Assume C is a closed convex subset of a real Hilbert space H and let $T:C\to C$ be a k-strictly pseudocontractive mapping, with $k\in (0,1)$. Then,

(i) 

The mapping $I-T$ is demiclosed (at 0); that is, if $\left\{x_n\right\}$ is a sequence in C such that $x_n\rightharpoonup \overline{x}$ and $(I-T)x_n\to 0$, then $(I-T)\overline{x}=0$.

(ii) 

The fixed point set $Fix\left(T\right)$ of T is closed and convex.

(iii) 

T is Lipschitzian, i.e.,

$$\parallel Tx-Ty\parallel \le {\displaystyle \frac{1+k}{1-k}}·\parallel x-y\parallel ,\forall x,y\in C.$$

Let C be a nonempty closed convex subset of a real Hilbert space H. For any $x\in H$, there exists a unique element in C, denoted by $P_C\left(x\right)$ and called the nearest point projection of x on C, such that

$$\parallel x-P_C\left(x\right)\parallel \le \parallel x-y\parallel , \mathrm{for} \mathrm{all} y\in C.$$

The mapping $P_C:H\to C$ is called the metric projection from H onto C.

The following lemma includes some properties of the metric projection that will be used later. For the proofs and other properties, see Goebel and Kirk [39].

Lemma 3 

(Goebel and Kirk [39]).

(i) 

$P_C\left(x\right)\in C$, for all $x\in H$;

(ii) 

$⟨x-P_C\left(x\right),P_C\left(x\right)-y⟩\ge 0$, for all $x\in H$ and $y\in C$.

Remark 1. 

Based on Lemma 2, (ii), it follows that the projection $P_{Fix\left(T\right)}$ is well defined for any k-strictly pseudocontractive mapping T.

Lemma 4 

(Xu [40], Lemma 2.5). Let $\left\{u_n\right\}\subset [0,\infty )$ be a sequence satisfying the recurrence inequality

$$u_{n+1}\le (1-a_n)u_n+b_n,n\ge 1,$$

where $\left\{a_n\right\}\subset (0,1)$ and $\left\{b_n\right\}\subset \mathbb{R}$ are such that

(1) 

${\displaystyle \sum _{n=1}^{\infty }}{a}_n=\infty$, and

(2) 

either $\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{b_n}{a_n}}\le 0$ or ${\displaystyle \sum _{n=1}^{\infty }}\left|b_n\right|<\infty$.

Then, $\underset{n\to \infty }{lim}{u}_n=0.$

Lemma 5 

(Dincă [41], Lemma 12.10). Let H be a real Hilbert space, $G:H\to H$ and for $\gamma >0$, let us consider the operator $T_{\gamma }:H\to H$, $T_{\gamma }=I-\gamma G$. Then, the following statements are equivalent:

1. 

There exists $\gamma >0$ such that $T_{\gamma }$ is a contraction;

2. 

There exist the constants $L>0$, $m>0$ such that

$$⟨Gu-Gv,u-v⟩\ge {m\parallel u-v\parallel }^2,\forall u,v\in H;$$

(11)

$$\parallel Gu-Gv\parallel \le L\parallel u-v\parallel ,\forall u,v\in H.$$

(12)

Remark 2. 

An operator satisfying property (11) is usually called m-strongly monotone, while the operators satisfying (12) are called Lipschitzian (continuous).

Lemma 5 represents an improved auxiliary result apparently due to Céa [42] (see also Blebea and Dincă [43]).

3. Preparatory Lemmas

Let $C\subset H$ be a nonempty closed and convex subset, $U,V:C\to H$ be two nonself mappings and $T:C\to C$ be a self mapping with $Fix\left(T\right)\ne \varnothing$.

Consider the following implicit iterative algorithm that was introduced in Sahu et al. [11]:

$$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ u_n={\gamma }_n{x}_n+(1-{\gamma }_n)T{x}_n,\hfill \\ x_{n+1}=P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T({\beta }_n{u}_n+(1-{\beta }_n)x_{n+1})\right],n\ge 1,\hfill \end{array}\right.$$

(13)

where $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$ and $\left\{{\gamma }_n\right\}$ are sequences of real numbers, $\gamma \ge 0$, $\mu >0$.

The next lemmas will be crucial in proving our main result in the next section.

Lemma 6. 

Assume that

(1) 

U is $L_1$-Lipschitzian and m-strongly monotone ($L_1>0$, $m>0$);

(2) 

V is L-Lipschitzian ($L>0$)

(3) 

T is nonexpansive with $Fix\left(T\right)\ne \varnothing$;

(4) 

μ and γ are two numbers satisfying the conditions

$$0<\mu <{\displaystyle \frac{2m}{L_1^2}};0<\gamma <{\displaystyle \frac{\tau }{L}},$$

where

$$\tau =1-\sqrt{1-m(2m-L_1^2)}.$$

(14)

Suppose also that ε and the sequences $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}\in (0,1)$ and $\left\{{\gamma }_n\right\}\in [0,1]$ satisfy the following conditions:

(i) 

$\underset{n\to \infty }{lim}{\alpha }_n=0$, ${\displaystyle \sum _{n=1}^{\infty }}{\alpha }_n=\infty$ and ${\displaystyle \sum _{n=1}^{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\infty$;

(ii) 

$0<\epsilon \le {\beta }_n\le {\beta }_{n+1}<1$, for all $n\in \mathbb{N}$, and ${\displaystyle \sum _{n=1}^{\infty }}|{\beta }_{n+1}-{\beta }_n|<\infty$;

(iii) 

${\displaystyle \sum _{n=1}^{\infty }}|{\gamma }_{n+1}-{\gamma }_n|<\infty$.

Then, the sequence $\left\{x_n\right\}$ generated by algorithm (13) is bounded.

Proof. 

Take $p\in Fix\left(T\right)$ arbitrarily. Since T is nonexpansive, we have

$$\parallel u_n-p\parallel = \parallel {\gamma }_n{x}_n+(1-{\gamma }_n)T{x}_n-p\parallel = \parallel {\gamma }_n(x_n-p)+(1-{\gamma }_n)(T{x}_n-Tp)\parallel$$

$$\le {\gamma }_n\parallel x_n-p\parallel +(1-{\gamma }_n)\parallel T{x}_n-Tp\parallel \le {\gamma }_n\parallel x_n-p\parallel +(1-{\gamma }_n)\parallel x_n-p\parallel$$

from which we obtain

$$\parallel u_n-p\parallel \le \parallel x_n-p\parallel .$$

(15)

Now, let us denote

$$y_n={\beta }_n{u}_n+(1-{\beta }_n)x_{n+1}\mathrm{and}{\theta }_n=(1-{\beta }_n)(1-\tau {\alpha }_n),n\ge 1,$$

with $\tau$ given by (14). We have

$$\parallel y_n-p\parallel = \parallel {\beta }_n{u}_n+(1-{\beta }_n)x_{n+1}-p\parallel = \parallel {\beta }_n(u_n-p)+(1-{\beta }_n)(x_{n+1}-p)\parallel$$

$$\le {\beta }_n\parallel u_n-p\parallel +(1-{\beta }_n)\parallel x_{n+1}-p\parallel$$

which, by using (15), yields

$$\parallel y_n-p\parallel \le {\beta }_n\parallel x_n-p\parallel +(1-{\beta }_n)\parallel x_{n+1}-p\parallel .$$

(16)

Now, using the nonexpansiveness of $P_C$ we have

$$\parallel x_{n+1}-p\parallel = \parallel P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T{y}_n\right]-P_{C}p\parallel$$

$$\le \parallel \gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T{y}_n-p\parallel$$

$$= \parallel {\alpha }_n(\gamma V{x}_n-\mu Up)+(I-\mu {\alpha }_{n}U)T{y}_n-(I-\mu {\alpha }_{n}U)p\parallel$$

$$\le {\alpha }_n\parallel \gamma V{x}_n-\mu Up\parallel +(1-\tau {\alpha }_n)\parallel T{y}_n-p\parallel$$

$$\le {\alpha }_n\left(\parallel \gamma V{x}_n-\gamma Vp\parallel +\parallel \gamma Vp-\mu Up\parallel \right)+(1-\tau {\alpha }_n)\parallel y_n-p\parallel .$$

By (16), we further have

$$\parallel x_{n+1}-p\parallel \le \gamma L{\alpha }_n\parallel x_n-p\parallel + {\alpha }_n\parallel \gamma Vp-\mu Up\parallel$$

$$+(1-\tau {\alpha }_n)\left({\beta }_n\parallel x_n-p\parallel +(1-{\beta }_n)\parallel x_{n+1}-p\parallel \right)$$

$$=\gamma L{\alpha }_n\parallel x_n-p\parallel + {\alpha }_n\parallel \gamma Vp-\mu Up\parallel + {\beta }_n(1-\tau {\alpha }_n)\parallel x_n-p\parallel$$

$$+(1-\tau {\alpha }_n)(1-{\beta }_n)\parallel x_{n+1}-p\parallel$$

$$=\left(\gamma L{\alpha }_n+{\beta }_n(1-\tau {\alpha }_n)\right) · \parallel x_n-p\parallel + {\alpha }_n\parallel \gamma Vp-\mu Up\parallel$$

$$+{\theta }_n\parallel x_{n+1}-p\parallel ,$$

which shows that, for all $n\ge 1$,

$$(1-{\theta }_n)· \parallel x_{n+1}-p\parallel \le \left(\gamma L{\alpha }_n+{\beta }_n(1-\tau {\alpha }_n)\right) · \parallel x_n-p\parallel + {\alpha }_n\parallel \gamma Vp-\mu Up\parallel .$$

(17)

Having in view the fact that ${\alpha }_n,{\beta }_n\in (0,1)$, we have

$$1-{\theta }_n=1-(1-{\beta }_n)(1-\tau {\alpha }_n)={\beta }_n+\tau {\alpha }_n(1-{\beta }_n)>0,$$

and so, by (17), we obtain

$$\parallel x_{n+1}-p\parallel \le (1-a_n)·\parallel x_n-p\parallel +a_n·B,$$

(18)

where

$${\displaystyle \frac{\gamma L{\alpha }_n+{\beta }_n(1-\tau {\alpha }_n)}{1-{\theta }_n}}=1-{\displaystyle \frac{(\tau -\gamma L){\alpha }_n}{1-{\theta }_n}}=1-a_n$$

and

$$B={\displaystyle \frac{\parallel \gamma Vp-\mu Up\parallel }{\tau -\gamma L}}.$$

Now, by inequality (18), we infer that

$$\parallel x_{n+1}-p\parallel \le max\left\{\parallel x_n-p\parallel ,{\displaystyle \frac{\parallel \gamma Vp-\mu Up\parallel }{\tau -\gamma L}}\right\}$$

from which we deduce that, indeed, $\left\{x_n\right\}$ is bounded. □

Remark 3. 

From the proof of Lemma 6, it follows that the sequences $\left\{V{x}_n\right\}$, $\left\{u_n\right\}$, $\left\{y_n\right\}$, $\left\{T{x}_n\right\}$, $\left\{T{y}_n\right\}$ and $\left\{UT{x}_n\right\}$ are also bounded.

Lemma 7. 

Under the assumptions of Lemma 5, the sequence $\left\{x_n\right\}$ generated by algorithm (13) is asymptotically regular, that is,

$$\parallel x_{n+1}-x_n\parallel \to 0 \mathrm{as} n\to \infty .$$

Proof. 

From the previous Remark, there exists the constants $A_1$, $A_2$, and $A_3$ given by

$$A_1=\underset{n\to \infty }{lim\; sup}\parallel \gamma V{x}_n-\mu UT{y}_n\parallel ,$$

$$A_2=\underset{n\to \infty }{lim\; sup}\parallel x_n-T{x}_n\parallel ,A_3=\underset{n\to \infty }{lim\; sup}\parallel u_n-x_n\parallel .$$

By (13), we have

$$\parallel u_{n+1}-u_n\parallel = \parallel {\gamma }_{n+1}{x}_{n+1}+(1-{\gamma }_{n+1})T{x}_{n+1}-{\gamma }_n{x}_n+(1-{\gamma }_n)T{x}_n\parallel$$

$$= \parallel {\gamma }_{n+1}(x_{n+1}-x_n)+(1-{\gamma }_{n+1})(T{x}_{n+1}-T{x}_n)+({\gamma }_{n+1}-{\gamma }_n)(x_n-T{x}_n)\parallel$$

$$\le {\gamma }_{n+1}\parallel x_{n+1}-x_n\parallel +(1-{\gamma }_{n+1})\parallel T{x}_{n+1}-T{x}_n\parallel + |{\gamma }_{n+1}-{\gamma }_n|\parallel x_n-T{x}_n\parallel$$

which, by the nonexpansiveness of T, yields

$$\parallel u_{n+1}-u_n\parallel \le \parallel x_{n+1}-x_n\parallel + |{\gamma }_{n+1}-{\gamma }_n|\parallel x_n-T{x}_n\parallel ,n\ge 1.$$

(19)

On the other hand,

$$\parallel y_{n+1}-y_n\parallel = \parallel {\beta }_{n+1}{u}_{n+1}+(1-{\beta }_{n+1})x_{n+2}-{\beta }_n{u}_n+(1-{\beta }_n)x_{n+1}\parallel$$

$$= \parallel (1-{\beta }_{n+1})(x_{n+2}-x_{n+1})+{\beta }_n(u_{n+1}-u_n)+({\beta }_{n+1}-{\beta }_n)(u_{n+1}-x_{n+1}\parallel$$

$$\le (1-{\beta }_{n+1})\parallel x_{n+2}-x_{n+1}\parallel + {\beta }_n\parallel u_{n+1}-u_n\parallel + |{\beta }_{n+1}-{\beta }_n|\parallel u_{n+1}-x_{n+1}\parallel$$

which, by using (19), gives

$$\begin{gathered} \parallel y_{n+1}-y_n\parallel \le (1-{\beta }_{n+1})\parallel x_{n+2}-x_{n+1}\parallel + {\beta }_n\parallel u_{n+1}-u_n\parallel \\ + {\beta }_n|{\gamma }_{n+1}-{\gamma }_n|\parallel x_n-T{x}_n\parallel + |{\beta }_{n+1}-{\beta }_n|\parallel u_{n+1}-x_{n+1}\parallel . \end{gathered}$$

(20)

Now, we evaluate the quantity $\parallel x_{n+2}-x_{n+1}\parallel$ appearing in the right hand side of (20). By (13) and the nonexpansiveness of $P_C$, we have

$$\parallel x_{n+2}-x_{n+1}\parallel$$

$$\le \parallel \gamma {\alpha }_{n+1}V{x}_{n+1}+(I-\mu {\alpha }_{n+1}U)T{y}_{n+1}-\gamma {\alpha }_{n}V{x}_n-(I-\mu {\alpha }_{n}U)T{y}_n\parallel$$

$$\le \parallel (I-\mu {\alpha }_{n+1}U)T{y}_{n+1}-(I-\mu {\alpha }_{n}U)T{y}_n\parallel + \gamma {\alpha }_{n+1}\parallel V{x}_{n+1}-V{x}_n\parallel$$

$$+ |{\alpha }_{n+1}-{\alpha }_n|\parallel \gamma V{x}_n-\mu UT{x}_n\parallel$$

and using the fact that U is $L_1$-Lipschitzian, V is L-Lipschitzian and T is nonexpansive, after some straightforward calculations, we obtain

$$\parallel x_{n+2}-x_{n+1}\parallel \le (1-\tau {\alpha }_{n+1})\parallel y_{n+1-y_n}\parallel + \gamma L{\alpha }_{n+1}\parallel x_{n+1}-x_n\parallel$$

$$+ |{\alpha }_{n+1}-{\alpha }_n|\parallel \gamma V{x}_n-\mu UT{y}_n\parallel$$

$$\le (1-\tau {\alpha }_{n+1})\parallel y_{n+1-y_n}\parallel + \gamma L{\alpha }_{n+1}\parallel x_{n+1}-x_n\parallel + A_1|{\alpha }_{n+1}-{\alpha }_n|.$$

We insert (20) in the previous inequality to obtain

$$\parallel x_{n+2}-x_{n+1}\parallel \le (1-\tau {\alpha }_{n+1})\left[(1-{\beta }_{n+1})\parallel x_{n+2}-x_{n+1}\parallel + {\beta }_n\parallel u_{n+1}-u_n\parallel \right.$$

$$\left.+ {\beta }_n|{\gamma }_{n+1}-{\gamma }_n|\parallel x_n-T{x}_n\parallel + |{\beta }_{n+1}-{\beta }_n|\parallel u_{n+1}-x_{n+1}\parallel \right]$$

$$+ \gamma L{\alpha }_{n+1}\parallel x_{n+1}-x_n\parallel + A_1|{\alpha }_{n+1}-{\alpha }_n|$$

$$=(1-\tau {\alpha }_{n+1})(1-{\beta }_{n+1})\parallel x_{n+2}-x_{n+1}\parallel$$

$$+\left(\gamma L{\alpha }_{n+1}+(1-\tau {\alpha }_{n+1}){\beta }_n\right)\parallel x_{n+1}-x_n\parallel$$

$$+(1-\tau {\alpha }_{n+1})|{\beta }_{n+1}-{\beta }_n|\parallel u_{n+1}-x_{n+1}\parallel$$

$$+(1-\tau {\alpha }_{n+1}){\beta }_n|{\gamma }_{n+1}-{\gamma }_n| · \parallel x_n-T{x}_n\parallel + A_1|{\alpha }_{n+1}-{\alpha }_n|$$

$$\le {\theta }_{n+1}\parallel x_{n+2}-x_{n+1}\parallel +\left(\gamma L{\alpha }_{n+1}+(1-\tau {\alpha }_{n+1}){\beta }_n\right)\parallel x_{n+1}-x_n\parallel +B_n,$$

where

$$B_n=A_1|{\alpha }_{n+1}-{\alpha }_n| +(1-\tau {\alpha }_{n+1}){\beta }_n|{\gamma }_{n+1}-{\gamma }_n|·A_2+(1-\tau {\alpha }_{n+1}){\beta }_n{A}_3.$$

Using the fact that $1-\tau {\alpha }_{n+1}\le 1$ and ${\beta }_n\le 1$, we obtain that

$$(1-{\theta }_{n+1})\parallel x_{n+2}-x_{n+1}\parallel \le \left(\gamma L{\alpha }_{n+1}+(1-\tau {\alpha }_{n+1}){\beta }_n\right)\parallel x_{n+1}-x_n\parallel$$

$$+ A_1|{\alpha }_{n+1}-{\alpha }_n| + A_2|{\beta }_{n+1}-{\beta }_n| + A_3|{\gamma }_{n+1}-{\gamma }_n|$$

which, in view of the fact that $1-{\theta }_{n+1}>0$, can be written as

$$\parallel x_{n+2}-x_{n+1}\parallel \le a_n\parallel x_{n+1}-x_n\parallel +b_n,n\ge 1,$$

(21)

where

$$a_n={\displaystyle \frac{\gamma L{\alpha }_{n+1}+(1-\tau {\alpha }_{n+1}){\beta }_n}{1-{\theta }_{n+1}}}$$

and

$$b_n={\displaystyle \frac{A_1|{\alpha }_{n+1}-{\alpha }_n| + A_2|{\beta }_{n+1}-{\beta }_n| + A_3|{\gamma }_{n+1}-{\gamma }_n|}{1-{\theta }_{n+1}}}.$$

Now, by assumption (ii) in Lemma 6, we have

$${\beta }_{n+1}+{\theta }_{n+1}={\beta }_{n+1}+(1-\tau {\alpha }_{n+1})(1-{\beta }_{n+1})=1-\tau {\alpha }_{n+1}\left((1-{\beta }_{n+1})\right)<1$$

which means that

$$0<\epsilon \le {\beta }_{n+1}<1-{\theta }_{n+1}<1$$

and this implies

$$1<{\displaystyle \frac{1}{1-{\theta }_{n+1}}}.$$

Thus,

$$a_n\le 1-(\tau -\gamma L){\alpha }_{n+1}$$

and so, by denoting $A=max\{A_1,A_2,A_3\}$, from (21) we obtain

$$\parallel x_{n+2}-x_{n+1}\parallel \le (1-(\tau -\gamma L){\alpha }_{n+1})\parallel x_{n+1}-x_n\parallel$$

$$+{\displaystyle \frac{A}{\epsilon }}\left(|{\alpha }_{n+1}-{\alpha }_n| + |{\beta }_{n+1}-{\beta }_n| + |{\gamma }_{n+1}-{\gamma }_n|\right).$$

Now, having in view the assumptions (i), (ii) and (iii) in Lemma 6, the conclusion follows Lemma 4. □

Lemma 8. 

Under the assumptions of Lemma 5, the sequence $\left\{x_n\right\}$ generated by the algorithm (13) is an approximate fixed point sequence, that is,

$$\parallel T{x}_n-x_n\parallel \to 0 \mathrm{as} n\to \infty .$$

Proof. 

By the definition of $\left\{x_n\right\}$, given by (13), we have

$$\begin{gathered} \parallel T{x}_n-x_n\parallel = \parallel x_n-x_{n+1}+x_{n+1}-T{y}_n+T{y}_n-T{x}_n\parallel \\ \le \parallel x_n-x_{n+1}\parallel + \parallel x_{n+1}-T{y}_n\parallel +\parallel T{y}_n-T{x}_n\parallel . \end{gathered}$$

(22)

To evaluate $\parallel x_{n+1}-T{y}_n\parallel$, we make use of the nonexpansiveness of $P_C$ to obtain

$$\parallel x_{n+1}-T{y}_n\parallel = \parallel P_C\left[\gamma {\alpha }_{n}V{x}_n-(I-\mu {\alpha }_{n}U)T{y}_n\right]-P_C\left[T{y}_n\right]\parallel$$

$$\le \parallel \gamma {\alpha }_{n}V{x}_n-(I-\mu {\alpha }_{n}U)T{y}_n-T{y}_n\parallel ={\alpha }_n\parallel \gamma V{x}_n-\mu UT{y}_n)\parallel .$$

By using (22) and the fact that T is nonexpansive, we have

$$\parallel T{x}_n-x_n\parallel \le \parallel x_{n+1}-x_n\parallel + {\alpha }_n\parallel \gamma V{x}_n-\mu UT{y}_n)\parallel + \parallel y_n-x_n\parallel$$

$$\le \parallel x_{n+1}-x_n\parallel + A_1{\alpha }_n+\parallel {\beta }_n{u}_n+(1-{\beta }_n)x_{n+1}-x_n\parallel$$

$$=\parallel x_{n+1}-x_n\parallel +A_1{\alpha }_n + \parallel {\beta }_n(u_n-x_n)+(1-{\beta }_n)(x_{n+1}-x_n)\parallel$$

$$\le (2-{\beta }_n)\parallel x_{n+1}-x_n\parallel + {\beta }_n\parallel u_n-x_n\parallel +A_1{\alpha }_n$$

$$=(2-{\beta }_n)\parallel x_{n+1}-x_n\parallel + {\beta }_n\parallel {\gamma }_n{x}_n+(1-{\gamma }_n)T{x}_n-x_n\parallel +A_1{\alpha }_n$$

$$\le (2-{\beta }_n)\parallel x_{n+1}-x_n\parallel + {\beta }_n(1-{\gamma }_n)\parallel T{x}_n-x_n\parallel +A_1{\alpha }_n$$

which, in view of the fact that $1-{\beta }_n(1-{\gamma }_n)>0(\iff {\beta }_n<1+{\beta }_n{\gamma }_n)$, yields

$$\parallel T{x}_n-x_n\parallel \le {\displaystyle \frac{2-{\beta }_n}{1-{\beta }_n(1-{\gamma }_n)}}·\parallel x_{n+1}-x_n\parallel + M_1{\alpha }_n$$

$$<2\parallel x_{n+1}-x_n\parallel +M_1{\alpha }_n.$$

Now, using (i) in Lemma 6 and letting $n\to \infty$ in the previous inequality, we obtain the desired conclusion. □

4. Main Results

Let $C\subset H$ be a nonempty closed and convex subset, $U,V:C\to H$ be two nonself mappings and $T:C\to C$ be a self mapping with $Fix\left(T\right)\ne \varnothing$.

For $x_1\in C$, arbitrarily chosen, we construct the sequence $\left\{x_n\right\}$ as follows:

$$\left\{\begin{array}{c}{u}_n={\gamma }_n{x}_n+(1-{\gamma }_n)T_{\lambda }{x}_n,\hfill \\ x_{n+1}=P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T_{\lambda }({\beta }_n{u}_n+(1-{\beta }_n)x_{n+1})\right],n\ge 1,\hfill \end{array}\right.$$

(23)

where $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}$ and $\left\{{\gamma }_n\right\}$ are sequences of real numbers, $\gamma \ge 0$, $\mu >0$ and $T_{\lambda }$ is the averaged operator defined by

$$T_{\lambda }x=(1-\lambda )x+\lambda Tx,\lambda \in (0,1).$$

(24)

The main result of this paper is the following strong convergence theorem.

Theorem 1. 

Let $C\subset H$ be a nonempty closed and convex subset, $U,V:C\to H$ be two nonself operators and $T:C\to C$ be a k-strictly pseudocontractive selfmapping with $Fix\left(T\right)\ne \varnothing$ such that

(O1) U is $L_1$-Lipschitzian and m-strongly monotone;

(O2) V is L-Lipschitzian;

(O3) μ and γ are two numbers satisfying the following conditions:

$$0<\mu <{\displaystyle \frac{2m}{L_1^2}};0<\gamma <{\displaystyle \frac{\tau }{L}},$$

where

$$\tau =1-\sqrt{1-\mu (2m-\mu L_1^2)}.$$

(25)

Suppose also that ε and $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}\in (0,1)$ and $\left\{{\gamma }_n\right\}\in [0,1]$ satisfy the following conditions

(i) 

$\underset{n\to \infty }{lim}{\alpha }_n=0$, ${\displaystyle \sum _{n=1}^{\infty }}{\alpha }_n=\infty$ and ${\displaystyle \sum _{n=1}^{\infty }}|{\alpha }_{n+1}-{\alpha }_n|<\infty$;

(ii) 

$0<\epsilon \le {\beta }_n\le {\beta }_{n+1}<1$, for all $n\in \mathbb{N}$, and ${\displaystyle \sum _{n=1}^{\infty }}|{\beta }_{n+1}-{\beta }_n|<\infty$;

(iii) 

${\displaystyle \sum _{n=1}^{\infty }}|{\gamma }_{n+1}-{\gamma }_n|<\infty$.

Then, for any $\lambda \in (0,1-k)$, the sequence $\left\{x_n\right\}$ generated by the algorithm (23) converges strongly to $\overline{x}\in Fix\left(T\right)$, which is the unique solution of the variational inequality

$$⟨(\mu U-\gamma V)\overline{x},\overline{x}-x⟩\le 0,\forall x\in Fix\left(T\right).$$

(26)

Proof. 

Let $\left\{x_n\right\}$ and $\left\{u_n\right\}$ be the sequences generated by algorithm (23). To simplify the writing of formulas, let us denote

$$y_n={\beta }_n{u}_n+(1-{\beta }_n)x_{n+1},{\theta }_n=(1-{\beta }_n)(1-\tau {\alpha }_n),n\ge 1,$$

where $\tau$ is given by (25).

In view of Lemma 1, since T is k-strictly pseudocontractive with $Fix\left(T\right)\ne \varnothing$, it follows that the averaged mapping $T_{\lambda }$ given by (24) is nonexpansive, for any $\lambda \in (0,1-k)$ and $Fix\left(T_{\lambda }\right)=Fix\left(T\right)\ne \varnothing$.

So, the sequence $\left\{x_n\right\}$ is bounded (by Lemma 5), asymptotically regular (by Lemma 6) and an approximate fixed point sequence (by Lemma 7).

Now, by (O1)–(O3), it follows that the operator $\mu U-\gamma V:C\to H$ is $(\mu m-\gamma L)$-strongly monotone. It is also $(\mu L_1+\gamma L)$-Lipschitzian.

Therefore, by Lemma 4, the operator $I-(\mu U-\gamma V)$ is a contraction, which implies that $S=P_{Fix\left(T\right)}(I-(\mu U-\gamma V))$ is a contraction, too. Let $\overline{x}$ be the unique fixed point of S.

Let $\left\{x_{n_k}\right\}$ be a subsequence of $\left\{x_n\right\}$ for which one has

$$\underset{n\to \infty }{lim\; sup}⟨(\mu U-\gamma V)\overline{x},\overline{x}-x⟩=\underset{k\to \infty }{lim}⟨(\mu U-\gamma V)\overline{x},\overline{x}-x_{n_k}⟩.$$

By Lemma 6, there exists a subsequence $\left\{x_{n_{k_l}}\right\}$ of $\left\{x_{n_k}\right\}$ which converges weakly to p. To simplify the notation, denote this subsequence by $\left\{x_{n_k}\right\}$, too. Thus,

$$x_{n_k}\rightharpoonup p, \mathrm{as} k\to \infty .$$

As $T_{\lambda }$ is nonexpansive, by Lemmas 1 and 2, it follows that $p\in Fix\left(T_{\lambda }\right)$.

Then, by Lemma 2, we have

$$\begin{gathered} \underset{n\to \infty }{lim\; sup}⟨(\mu U-\gamma V)\overline{x},\overline{x}-x⟩=\underset{k\to \infty }{lim}⟨(\mu U-\gamma V)\overline{x},\overline{x}-x_{n_k}⟩ \\ =⟨(\mu U-\gamma V)\overline{x},\overline{x}-p⟩=⟨\overline{x}-(I-(\mu U-\gamma V))\overline{x},\overline{x}-p⟩\le 0. \end{gathered}$$

(27)

Now, we shall prove that $\left\{x_n\right\}$ converges strongly to p. To this end, let us denote

$$X_n= \parallel x_n-\overline{x}\parallel ,Y_n=\parallel y_n-\overline{x}\parallel ;{\delta }_n=(1-{\beta }_n)(1-\tau {\alpha }_n)(1-(\tau -\gamma L){\alpha }_n),$$

and

$$A_n={\alpha }_n^2{\parallel \gamma V{x}_n-\mu U\overline{x}\parallel }^2$$

$$+ 2{\alpha }_n⟨(\gamma V-\mu U)\overline{x},(I-\mu {\alpha }_{n}U)T{y}_n-(I-\mu {\alpha }_{n}U)\overline{x}⟩,\forall n\in \mathbb{N}.$$

Using the nonexpansiveness of $P_C$ and (23), we have

$$s_{n+1}^2= \parallel x_{n+1}-\overline{x}{\parallel }^2 ={\parallel P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T_{\lambda }{y}_n\right]-P_C\overline{x}\parallel }^2$$

$$\le \parallel \gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T_{\lambda }{y}_n-\overline{x}{\parallel }^2$$

$$= \parallel {\alpha }_n(\gamma V{x}_n-\mu U\overline{x})+(I-\mu {\alpha }_{n}U)T_{\lambda }{y}_n-(I-\mu {\alpha }_{n}U)\overline{x}{\parallel }^2$$

$$\le {\alpha }_n^2\parallel \gamma V{x}_n-\mu U\overline{x}{\parallel }^2+{(1-\tau {\alpha }_n)}^2·{\parallel T_{\lambda }{y}_n-\overline{x}\parallel }^2$$

$$+ 2{\alpha }_n⟨\gamma V{x}_n-\mu U\overline{x},(I-\mu {\alpha }_{n}U)T_{\lambda }{y}_n-(I-\mu {\alpha }_{n}U)\overline{x}⟩.$$

By denoting

$${\omega }_n=(I-\mu {\alpha }_{n}U)T_{\lambda }{y}_n-(I-\mu {\alpha }_{n}U)\overline{x}$$

we have

$$⟨\gamma V{x}_n-\mu U\overline{x},{\omega }_n⟩=⟨\gamma V{x}_n-\gamma V\overline{x},{\omega }_n⟩+⟨\gamma V\overline{x}-\mu U\overline{x},{\omega }_n⟩$$

and hence, by the inequalities above, we obtain

$$\begin{gathered} s_{n+1}^2\le {(1-\tau {\alpha }_n)}^2{\parallel T_{\lambda }{y}_n-\overline{x}\parallel }^2+2{\alpha }_n⟨\gamma V{x}_n-\gamma V\overline{x},{\omega }_n⟩+A_n \\ \le {(1-\tau {\alpha }_n)}^2\parallel y_n-\overline{x}{\parallel }^2+ 2\gamma (1-\tau {\alpha }_n)\parallel V{x}_n-V\overline{x}\parallel · \parallel T_{\lambda }{y}_n-\overline{x}\parallel + A_n \\ \le {(1-\tau {\alpha }_n)}^2\parallel y_n-\overline{x}{\parallel }^2+ 2\gamma L(1-\tau {\alpha }_n)\parallel x_n-\overline{x}\parallel · \parallel y_n-\overline{x}\parallel + A_n \\ ={(1-\tau {\alpha }_n)}^2{t}_n^2+2\gamma L(1-\tau {\alpha }_n)s_n{t}_n + A_n. \end{gathered}$$

(28)

We can rewrite (28) as a quadratic inequality with respect to $t_n$:

$${(1-\tau {\alpha }_n)}^2{t}_n^2+2\gamma L(1-\tau {\alpha }_n)s_n{t}_n+A_n-s_{n+1}^2\ge 0$$

and solve it for $t_n$ to obtain

$$t_n\ge {\displaystyle \frac{-\gamma L{\alpha }_n{s}_n+\sqrt{w_n}}{1-\tau {\alpha }_n}},$$

(29)

where

$$w_n={\gamma }^2{L}^2{s}_n^2-(A_n-s_{n+1}^2).$$

On the other hand,

$$t_n= \parallel y_n-\overline{x} \parallel \le {\beta }_n · \parallel u_n-\overline{x}\parallel +(1-{\beta }_n)\parallel x_{n+1}-\overline{x}\parallel$$

$$={\beta }_n·\parallel {\gamma }_n{x}_n+(1-{\gamma }_n)T_{\lambda }{x}_n-\overline{x}\parallel +(1-{\beta }_n)s_{n+1}$$

$$\le {\beta }_n\left[{\gamma }_n\parallel x_n-\overline{x}\parallel +(1-{\gamma }_n)\parallel x_n-\overline{x}\parallel \right]+(1-{\beta }_n)s_{n+1}$$

$$={\beta }_n\parallel x_n-\overline{x}\parallel +(1-{\beta }_n)s_{n+1}={\beta }_n{s}_n+(1-{\beta }_n)s_{n+1}.$$

By using (29), we have

$${\beta }_n{s}_n+(1-{\beta }_n)s_{n+1}\ge {\displaystyle \frac{-\gamma L{\alpha }_n{s}_n+\sqrt{w_n}}{1-\tau {\alpha }_n}}$$

which can be written as

$$\begin{gathered} (\gamma L{\alpha }_n+{\beta }_n-\tau {\alpha }_n{\beta }_n)s_n+(1-{\beta }_n)(1-\tau {\alpha }_n)s_{n+1}\ge \sqrt{w_n} \\ \iff (\gamma L{\alpha }_n+{\beta }_n-\tau {\alpha }_n{\beta }_n)s_n+{\theta }_n{s}_{n+1}\ge \sqrt{w_n} \end{gathered}$$

(30)

Since ${\beta }_n-\tau {\alpha }_n{\beta }_n={\beta }_n(1-\tau {\alpha }_n)>0$, by squaring (30), we obtain the recurrence inequality

$$c_n{s}_{n+1}^2\le d_n{s}_n^2+A_n,$$

where

$$c_n=1-{\theta }_n^2-{\theta }_n({\beta }_n-\tau {\alpha }_n{\beta }_n+\gamma L{\alpha }_n)=1-{\delta }_n$$

and

$$d_n={({\beta }_n-\tau {\alpha }_n{\beta }_n+\gamma L{\alpha }_n)}^2+({\beta }_n-\tau {\alpha }_n{\beta }_n+\gamma L{\alpha }_n){\theta }_n-{\gamma }^2{L}^2{\alpha }_n^2.$$

This implies that

$$s_{n+1}^2\le {\displaystyle \frac{d_n}{1-{\delta }_n}}{s}_n^2+{\displaystyle \frac{A_n}{1-{\delta }_n}}.$$

(31)

If we denote

$$h_n={\displaystyle \frac{1}{{\alpha }_n}}\left(1-{\displaystyle \frac{d_n}{1-{\delta }_n}}\right)$$

and simplify the expression of $h_n$, one finally obtains

$$h_n={\displaystyle \frac{(\tau -\gamma L)(2-(\tau -\gamma L){\alpha }_n)-{\gamma }^2{L}^2{\alpha }_n^2}{1-{\delta }_n}}.$$

In view of assumption (ii), we infer that $\underset{n\to \infty }{lim}{\beta }_n$ exists; let it be denoted by $\overline{\beta }$.

Therefore,

$$\underset{n\to \infty }{lim}{h}_n={\displaystyle \frac{2(\tau -\gamma L)}{\overline{\beta }}}.$$

If we let $\sigma$ such that $0<\sigma <{\displaystyle \frac{2(\tau -\gamma L)}{\overline{\beta }}}$, then we can find $n_0$ such that $h_n>\sigma$, for all $n\ge n_0$ and, therefore,

$${\displaystyle \frac{d_n}{1-{\delta }_n}}\le 1-\sigma {\alpha }_n,\mathrm{for} \mathrm{all} n\ge n_0.$$

Then, by (31), it follows that

$$s_{n+1}^2\le (1-\sigma {\alpha }_n)s_n^2+{\displaystyle \frac{A_n}{1-{\delta }_n}},n\ge n_0.$$

(32)

In order to prepare (32) to fit Lemma 4, we shall evaluate $A_n.$ We have

$$A_n={\alpha }_n^2{\parallel \gamma V{x}_n-\mu U\overline{x}\parallel }^2$$

$$+ 2{\alpha }_n⟨(\gamma V-\mu U)\overline{x},(I-\mu {\alpha }_{n}U)T{y}_n-(I-\mu {\alpha }_{n}U)\overline{x}⟩,\forall n\in \mathbb{N}.$$

$$={\alpha }_n\left[\parallel \gamma V{x}_n-\mu U\overline{x}{\parallel }^2\right.$$

$$+ 2⟨(\gamma V-\mu U)\overline{x},(I-\mu {\alpha }_{n}U)T{y}_n-(I-\mu {\alpha }_{n}U)x_n⟩$$

$$\left.+2⟨\gamma V-\mu U)\overline{x},(I-\mu {\alpha }_{n}U)x_n-(I-\mu {\alpha }_{n}U)\overline{x}⟩\right]$$

$$\le {\alpha }_n\left[{\alpha }_n\parallel \gamma V{x}_n-\mu U\overline{x}{\parallel }^2+ 2(1-\tau {\alpha }_n)\parallel (\gamma V-\mu U)\overline{x}\parallel ·\parallel T{y}_n-x_n\parallel \right.$$

$$\left.+ 2\mu L_1{\alpha }_n\parallel (\gamma V-\mu U)\overline{x}\parallel · \parallel x_n-\overline{x}\parallel \right]$$

By using assumption (i) and Lemma 7, we obtain

$$\parallel T{y}_n-x_n\parallel \le \parallel x_n-x_{n+1}\parallel + \parallel x_{n+1}-T{y}_n\parallel$$

$$\le \parallel x_n-x_{n+1}\parallel + {\alpha }_n\parallel \gamma V{x}_n-\mu UT{y}_n\parallel$$

$$\parallel x_n-x_{n+1}\parallel + M_1{\alpha }_n\to 0 \mathrm{as} n\to \infty .$$

By (27) and using, again, assumption (i), we have

$$\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{A_n}{\sigma {\alpha }_n(1-{\delta }_n)}}\le \underset{n\to \infty }{lim\; sup}{\displaystyle \frac{1}{\sigma (1-{\delta }_n)}}·$$

$$\left[{\alpha }_n\parallel \gamma V{x}_n-\mu U\overline{x}{\parallel }^2+ 2(1-\tau {\alpha }_n)\parallel (\gamma V-\mu U)\overline{x}\parallel ·\parallel T{y}_n-x_n\parallel \right.$$

$$\left.+ 2\mu L_1{\alpha }_n ·\parallel (\gamma V-\mu U)\overline{x}\parallel · \parallel x_n-\overline{x}\parallel \right]\le 0.$$

Now, we can apply Lemma 4 to the recurrence inequality (31) to deduce that $s_n\to 0$ as $n\to \infty$, which implies that

$$x_n\to 0 \mathrm{as} n\to \infty .$$

□

Remark 4. 

The next Example illustrates the superiority of Theorem 1 and also of the averaged implicit algorithm (23) over the main results in Sahu et al. [11], that is, Theorem 3.1, Theorem 3.4 and Corollary 3.5.

Example 2. 

Let H, C and T be as in Example 1. Then, T is k-strictly pseudocontractive with $k={\displaystyle \frac{3}{5}}$ and $Fix\left(T\right)=\left\{1\right\}$.

We also consider the following two nonself mappings $U,V:C\to H$, defined by $Ux=2x+2$ and $Vx=x^2+1$, for $x\in C$, respectively.

It is easy to see that U is $L_1$-Lipschitzian, with $L_1=2$, that U is also m-strongly monotone, with $m=2$, and that V is L-Lipschitzian with $L=4$.

We have to choose the parameter μ such that

$$0<\mu <{\displaystyle \frac{2m}{L_1}}\iff 0<\mu <1.$$

Set $\mu ={\displaystyle \frac{2}{3}}$. Then, we have to choose γ such that

$$0<\gamma <{\displaystyle \frac{\tau }{L}}.$$

According to (25), we have

$$\tau =1-\sqrt{1-{\displaystyle \frac{2}{3}}\left(2·2-{\displaystyle \frac{2}{3}}·4\right)}={\displaystyle \frac{2}{3}}$$

and so γ should be taken from the interval $\left(0,{\displaystyle \frac{1}{6}}\right)$. We set $\gamma ={\displaystyle \frac{1}{7}}$.

We also take the parameter sequences as follows: ${\alpha }_n={\displaystyle \frac{1}{n+1}}$, ${\beta }_n={\displaystyle \frac{n-1}{4n+1}}$, and ${\gamma }_n={\displaystyle \frac{2}{n+1}}$.

Then, all assumptions of Theorem 1 are satisfied and, therefore, by applying Theorem 1, it follows that, for $x_1\in \left[{\displaystyle \frac{1}{2}},2\right]$, arbitrarily chosen, and $\lambda =0.3\in (0,1-k)$, the sequence $\left\{x_n\right\}$ generated by algorithm (23) converges strongly to $\overline{x}=1$, the unique fixed point of T.

We note that, as T is not nonexpansive, one can apply neither Theorem 3.1 nor Theorem 3.4 and Corollary 3.5 in Sahu et al. [11] to solve the fixed point problem for T over the variational inequality (8).

Remark 5. 

From our implicit iterative scheme (23) involved in Theorem 1, one can derive various particular implicit schemes that could be useful for solving fixed point problems for nonexpansive or strictly pseudocontractive mappings over the variational inequality (8):

• For ${\gamma }_n=1$, for all $n\in \mathbb{N}$, our algorithm (23) yields the simpler implicit scheme

  $$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}=P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T_{\lambda }({\beta }_n{x}_n+(1-{\beta }_n)x_{n+1})\right],n\ge 1.\hfill \end{array}\right.$$

  (33)
• For ${\gamma }_n=0$, for all $n\in \mathbb{N}$, our algorithm (23) produces the implicit scheme

  $$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}=P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T_{\lambda }({\beta }_n{T}_{\lambda }{x}_n+(1-{\beta }_n)x_{n+1})\right],n\ge 1.\hfill \end{array}\right.$$

  (34)
• For ${\gamma }_n=1$ and ${\beta }_n={\displaystyle \frac{1}{2}}$, for all $n\in \mathbb{N}$, our algorithm (23) reduces to the following midpoint implicit scheme

  $$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}=P_C\left[\gamma {\alpha }_{n}V{x}_n+(I-\mu {\alpha }_{n}U)T_{\lambda }\left({\displaystyle \frac{x_n+x_{n+1}}{2}}\right)\right],n\ge 1.\hfill \end{array}\right.$$

  (35)
• For ${\gamma }_n=1$, for all $n\in \mathbb{N}$, $\gamma =1$, $C=H$, $U={\displaystyle \frac{1}{\mu }}·I$, and $V=f$, where f is a contraction mapping with the coefficient $L\in (0,1)$ satisfying the condition

  $$0<L<\tau =\mu \left(m-{\displaystyle \frac{1}{2}}\mu L_1^2\right),$$

  our algorithm (23) reduces to the following averaged version of the algorithm (3.13) considered by Ke and Ma [9]:

  $$\left\{\begin{array}{c}{x}_1\in C,\hfill \\ x_{n+1}={\alpha }_{n}f\left(x_n\right)+(1-{\alpha }_n)T_{\lambda }({\beta }_n{x}_n+(1-{\beta }_n)x_{n+1}),n\ge 1.\hfill \end{array}\right.$$

  (36)
• For ${\beta }_n={\displaystyle \frac{1}{2}}$, ${\gamma }_n=1$, for all $n\in \mathbb{N}$, $\gamma =1$, $C=H$, $U={\displaystyle \frac{1}{\mu }}·I$, and $V=f$, where f is a contraction mapping with the coefficient $L\in (0,1)$ satisfying the condition

  $$0<L<\tau =\mu \left(m-{\displaystyle \frac{1}{2}}\mu L_1^2\right),$$

  our algorithm (23) reduces to the viscosity algorithm (5) considered by Xu et al. [8].

We end this section by stating the following interesting convergence theorem for the above implicit algorithm (34).

Theorem 2. 

Let $C\subset H$ be a nonempty closed and convex subset. Assume that $U:C\to H$ is a $L_1$-Lipschitzian and m-strongly monotone operator, $V:C\to H$ is L-Lipschitzian and $T:C\to C$ is k-strictly pseudocontractive with $Fix\left(T\right)\ne \varnothing$.

Suppose that μ and γ are two numbers satisfying the conditions

$$0<\mu <{\displaystyle \frac{2m}{L_1^2}};0<\gamma <{\displaystyle \frac{\tau }{L}},$$

where $\tau =1-\sqrt{1-\mu (2m-\mu L_1^2)}.$

Then, for any $\lambda \in (0,1-k)$, the sequence $\left\{x_n\right\}$ generated by the algorithm (34), with $\left\{{\alpha }_n\right\}$, $\left\{{\beta }_n\right\}\in (0,1)$ satisfying the conditions (i) and (ii) in Theorem 1, converges strongly to $\overline{x}\in Fix\left(T\right)$, which is the unique solution of the variational inequality (26).

Proof. 

We apply Theorem 1 by setting ${\gamma }_n=0$ for all $n\in \mathbb{N}$. □

5. Conclusions

• We introduced an averaged implicit iterative algorithm, that is, algorithm (23), for finding a common solution for a fixed point problem and for a variational inequality problem in the class of k-strictly pseudocontractive mappings.
• As illustrated by Example 2, our main result (Theorem 1) extends several related results existing in the literature—mainly the ones established in Xu et al. [8], Ke and Ma [9], and Sahu et al. [11], and many other related results—from the class of nonexpansive mappings to the larger class of k-strictly pseudocontractive mappings.
• From the general algorithm (23), one can obtain some particularly useful implicit iterative algorithms, e.g., (33), (34), (35), (36),…, which could be utilized for solving fixed point problems either in the class of nonexpansive mappings or in the class of k-strictly pseudocontractive mappings.
• For other very recent related works that allow similar developments to the ones in the current paper, we refer the reader to Alakoya and Mewomo [12], Batra et al. [14], Eslamian and Kamandi [15], Filali et al. [16], Jolalaoso et al. [17], Mouktonglang et al. [18], Onifade et al. [44], Oyewole et al. [19], Peeyada et al. [20], Singh and Chandok [45], Tan et al. [46], Uba et al. [21], Yin et al. [22], Yu et al. [23],…