Averaged Iterative Algorithms for Convex Optimization Problems over a Common Fixed-Points Set of Demicontractive Mappings

Abstract

In this article, we introduce a novel averaged-type iterative scheme designed for solving convex minimization problems over the set of common fixed points of a pair of demicontractive mappings. Under suitable assumptions, we prove that the proposed algorithm converges strongly to the solution of the considered problem in a Hilbert space setting. We further demonstrate the applicability of our method to quadratic optimization problems with a bounded linear operator. In addition, we also report the numerical experiments that were performed in order to demonstrate the convergence behavior of the algorithm and to highlight its superiority over related existing methods.

1. Introduction

Throughout this paper, we are working in a real Hilbert space $\mathcal{H}$ whose norm and inner product are denoted by $\parallel ·\parallel$ and $\langle ·,·\rangle ,$ respectively. Let $\mathcal{C}$ be a closed and convex subset of $\mathcal{H}$ and $F:\mathcal{C}\to \mathcal{C}$. The notation

$$Fix\left(F\right):=\{u\in \mathcal{C}:Fu=u\}.$$

will designate the set of all fixed points of F.

Let $f:\mathcal{C}\to \mathbb{R}$ be a convex objective function and $F:\mathcal{C}\to \mathcal{C}$ a nonexpansive-type mapping with $Fix\left(F\right)\ne \varnothing$. In this paper, we are interested to solve the following constrained optimization problem:

$$Minimize f\left(x\right)$$

(1)

subject to

$$x\in Fix\left(F\right)\cap Fix\left(G\right),$$

where F and G are two demicontractive mappings.

Under its various versions, like the one when the constraint consists of $x\in Fix\left(F\right)$, with F as a nonexpansive-type mapping, this problem is one of the central problems in nonlinear analysis with various important applications in communication networks; see, for example, [1].

Various algorithms were proposed for solving the problem (1); see, for example, [1,2,3,4], and references therein. In Iiduka [2], a fixed-point optimization algorithm has been proposed for the case in which F is nonexpansive, while in Iiduka [3], two methods were proposed for solving a networked optimization system with a finite number of users under the assumption that each user tries to minimize its own private objective function over its own private constraint set. In fact, each user’s constrained set was considered of the fixed-point set of a certain quasi-nonexpansive mapping.

In the case of two users, Sow [1] presented an explicit iterative method for solving the convex optimization problem (1) over the set of common fixed points of two mappings $T_1$ and $T_2$, i.e., 

$$Fix\left(T_1\right)\cap Fix\left(T_2\right),$$

where $T_1$ is a demicontractive mapping and $T_2$ is a quasi-nonexpansive mapping.

Motivated by the above results, on one hand, and by the fact that the class of demicontractive mappings strictly includes the quasi-nonexpansive mappings as well as the nonexpansive mappings, our aim in this work is to consider the convex optimization problem over the set of common fixed points of two demicontractive mappings, thus extending all previous results that referred to fixed points or common fixed points of nonexpansive or quasi-nonexpansive mappings.

To this end, we introduce an averaged algorithm intended to solve the convex optimization problem (1) over the set of common fixed points of two demicontractive mappings. We establish convergence results and illustrate the approach with applications to quadratic optimization problems involving bounded linear operators. A report on the numerical experiments we performed is also presented to illustrate the convergence behavior of the iterative algorithms.

Our results extend, generalize, or are related to several others from the previous literature, e.g.,  [1,2,3,4,5,6,7].

2. Preliminaries

We call F Lipschitz continuous (or Lipschitzian) if there exists $M\ge 0$ satisfying

$$\parallel Fu-Fv\parallel \le M\parallel u-v\parallel ,\forall u,v\in \mathcal{C}.$$

If $M<1,$ then F is called a Banach contraction or strict contraction and if $M=1$, then F is said to be nonexpansive. Furthermore, F is said to be

(a)

strictly pseudocontractive if there exists $\beta \in (0,1)$ such that

$${\parallel Fu-Fv\parallel }^2\le {\parallel u-v\parallel }^2+\beta {\parallel (I-F)u-(I-F)v\parallel }^2,\forall u,v\in \mathcal{C}.$$

(2)

(b)

quasi-nonexpansive if $Fix\left(F\right)\ne \varnothing$ and

$$\parallel Fu-v\parallel \le \parallel u-v\parallel ,\mathrm{for}\mathrm{all}u\in \mathcal{C}\mathrm{and}v\in Fix\left(F\right);$$

(3)

(c)

demicontractive if $Fix\left(F\right)\ne \varnothing$ and there exists $\varrho \in (0,1)$ satisfying

$${\parallel Fu-v\parallel }^2\le {\parallel u-v\parallel }^2+\varrho {\parallel u-Fu\parallel }^2,\mathrm{for}\mathrm{all}u\in \mathcal{C}\mathrm{and}v\in Fix\left(F\right).$$

(4)

From the above definitions, it follows that every nonexpansive mapping is strictly pseudocontractive, and every quasi-nonexpansive mapping is also demicontractive. Moreover, if F is nonexpansive and $Fix\left(T\right)\ne \varnothing$, then F is quasi-nonexpansive.

However, the converses of all these implications fail in general, as illustrated in the next two examples. For a complete diagram which depicts the relationships among the above class mappings, we refer the reader to [8].

Example 1.

Consider $\mathcal{H}=\mathbb{R}$, $\mathcal{C}=[0,1]$ and define the mapping F on $\mathcal{C}$ as

$$F\left(u\right)=\left\{\begin{array}{cc}{\displaystyle \frac{2}{3}},\hfill & if 0\le u\le {\displaystyle \frac{2}{3}}\hfill \\ u,\hfill & if {\displaystyle \frac{2}{3}}<u<{\displaystyle \frac{3}{4}}\hfill \\ {\displaystyle \frac{3}{4}},\hfill & if {\displaystyle \frac{3}{4}}\le u<{\displaystyle \frac{9}{10}}\hfill \\ {\displaystyle \frac{1}{3}},\hfill & if {\displaystyle \frac{9}{10}}\le u\le 1.\hfill \end{array}\right.$$

(5)

Then

(a)

F is β-demicontractive with $\beta ={\displaystyle \frac{7}{17}};$

(b)

F is neither quasi-nonexpasive nor nonexpansive;

(c)

F is not strictly pseudocontractive.

Proof. 

(a)

$Fix\left(F\right)=\left[{\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}\right].$ Let $v_1={\displaystyle \frac{2}{3}}\in Fix\left(F\right).$ It is straightforward to prove that inequality (4) holds for any $u\in \left[0,{\displaystyle \frac{9}{10}}\right).$ If $u\in \left[{\displaystyle \frac{9}{10}},1\right]$, then inequality (4) becomes

$${\left|{\displaystyle \frac{1}{3}}-{\displaystyle \frac{2}{3}}\right|}^2\le {\left|u-{\displaystyle \frac{2}{3}}\right|}^2+\beta {\left|u-{\displaystyle \frac{1}{3}}\right|}^2,$$

which holds for $\beta \ge {\displaystyle \frac{7}{17}}.$ Let $v_2={\displaystyle \frac{3}{4}}\in Fix\left(F\right).$ It is straightforward to prove that inequality (4) holds for any $u\in \left[0,{\displaystyle \frac{9}{10}}\right).$ If $u\in \left[{\displaystyle \frac{9}{10}},1\right],$ then inequality (4) becomes

$${\left|{\displaystyle \frac{1}{3}}-{\displaystyle \frac{3}{4}}\right|}^2\le {\left|u-{\displaystyle \frac{3}{4}}\right|}^2+\beta {\left|u-{\displaystyle \frac{1}{3}}\right|}^2$$

that holds for $\beta \ge {\displaystyle \frac{7}{17}}.$ Now take $v_3\in \left({\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}\right)\subset Fix\left(F\right).$ Obviously, inequality (4) is valid for every $u\in \left({\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}\right).$ If $u\in \left[0,{\displaystyle \frac{2}{3}}\right],$ then inequality (4) becomes

$${\left|{\displaystyle \frac{2}{3}}-v_3\right|}^2\le {\left|u-v_3\right|}^2+\beta {\left|u-{\displaystyle \frac{2}{3}}\right|}^2$$

Since $u\le {\displaystyle \frac{2}{3}}<v_3,$ this inequality holds for any positive $\beta .$ If $u\in \left[{\displaystyle \frac{3}{4}},{\displaystyle \frac{9}{10}}\right)$ then inequality (4) becomes

$${\left|{\displaystyle \frac{3}{4}}-v_3\right|}^2\le {\left|u-v_3\right|}^2+\beta {\left|u-{\displaystyle \frac{3}{4}}\right|}^2.$$

Since $u\ge {\displaystyle \frac{3}{4}}>v_3,$ this inequality holds for any positive $\beta .$ If $u\in \left[{\displaystyle \frac{9}{10}},1\right]$ then inequality (4) becomes

$${\left|{\displaystyle \frac{1}{3}}-v_3\right|}^2\le {\left|u-v_3\right|}^2+\beta {\left|u-{\displaystyle \frac{1}{3}}\right|}^2.$$

Since $\left|{\displaystyle \frac{1}{3}}-v_3\right|<\left|u-{\displaystyle \frac{1}{3}}\right|,$ we can always find a positive β with ${\displaystyle \frac{7}{17}}<\beta <1,$ such that this inequality holds. By summarizing all cases, we can conclude that F is a ${\displaystyle \frac{7}{17}}$-demicontractive mapping.

(b)

To show that F is not quasi-nonexpansive, check inequality (3) for $u=1$ and $v={\displaystyle \frac{3}{4}}\in Fix\left(F\right)$ to get the contradiction ${\displaystyle \frac{5}{12}}\le {\displaystyle \frac{1}{4}}.$

The dicontinuity of F implies that it is not nonexpansive, too.

(c)

Assume that F is β-strictly pseudocontractive; that is, there is a positive constant $\beta <1$ such that the inequality (2) applies for all $u,v\in [0,1].$ By taking $u\in \left[{\displaystyle \frac{3}{4}},{\displaystyle \frac{9}{10}}\right)$ and $v={\displaystyle \frac{9}{10}},$ in (2) we get

$${\left({\displaystyle \frac{5}{12}}\right)}^2\le {\left|u-{\displaystyle \frac{9}{10}}\right|}^2+\beta {\left|u-{\displaystyle \frac{9}{10}}-{\displaystyle \frac{5}{12}}\right|}^2,$$

from which, by letting $u\to {\displaystyle \frac{9}{10}},$ we obtain the contradiction $1\le \beta <1.$

Therefore, F is not strictly pseudocontractive.

     □

Example 2.

Define G on the interval $[0,1]$ as

$$G\left(u\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}u+{\displaystyle \frac{1}{3}},\hfill & if 0\le u\le {\displaystyle \frac{2}{3}}\hfill \\ u,\hfill & if {\displaystyle \frac{2}{3}}<u<{\displaystyle \frac{3}{4}}\hfill \\ {\displaystyle \frac{3}{4}},\hfill & if {\displaystyle \frac{3}{4}}\le u<1\hfill \\ {\displaystyle \frac{2}{9}},\hfill & if u=1.\hfill \end{array}\right.$$

(6)

Arguing similarly to the proof in Example 1, one can verify that:

(a)

G is β-demicontractive with $\beta ={\displaystyle \frac{5}{14}}$;

(b)

G is neither quasi-nonexpasive (indeed, checking (3) for $u=1$ and $v={\displaystyle \frac{2}{3}}\in Fix\left(G\right)$ to get the contradiction ${\displaystyle \frac{4}{9}}\le {\displaystyle \frac{1}{3}}$) nor nonexpansive (G is not continuous);

(c)

G is not strictly pseudocontractive.

Remark 1.

We note that all points on the interval $\left[{\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}\right]$ are common fixed points of F in Example 1 and G in Example 2, i.e., $Fix\left(F\right)=Fix\left(G\right)=\left[{\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}\right].$

The interest in the study of fixed points of nonexpansive-type mappings has been and continues to be an active area of investigation (see [8,9,10]). Its importance is underscored by numerous relevant real-world applications, particularly in convex optimization, game theory, market economics, and many other branches of applied mathematics.

A function $g:\mathcal{H}\to \mathbb{R}$ is said to be k-strongly convex if $\exists k>0$ such that

$$g(\alpha u+(1-\alpha )v)\le {\alpha g\left(u\right)+(1-\alpha )g\left(v\right)-k\parallel u-v\parallel }^2,$$

(7)

for every $u,v\in \mathcal{H}$ with $u\ne v$ and $\alpha \in (0,1)$.

In order to prove the main results of this paper, we shall need the following lemmas.

Lemma 1

([11]). If $\mathcal{H}$ is a real Hilbert space, $\mathcal{D}$ is a closed convex subset of $\mathcal{H}$ and $F:\mathcal{D}\to \mathcal{D}$ is nonexpansive, with $Fix\left(F\right)\ne emptyset$, then $I-F$ is demiclosed at 0.

Lemma 2

([12]). For any $u,v\in \mathcal{H}$ (a real Hilbert space), we have

(a)

${\parallel u+v\parallel }^2\le {\parallel u\parallel }^2+\langle v,u+v\rangle$;

(b)

${\parallel \mu u+(1-\mu )v\parallel }^2={\mu \parallel u\parallel }^2+{(1-\mu )\parallel v\parallel }^2-\mu (1-\mu ){\parallel u-v\parallel }^2,\mu \in (0,1)$.

Lemma 3

([13]). Let $\left\{a_p\right\}\subset (0,1)$ and $\left\{b_p\right\}\subset \mathbb{R}$ be two sequences satisfying

(a)

${\sum }_{p=0}^{\infty }{a}_p=\infty ;$

(b)

$\underset{p\to \infty }{lim\; sup}{b}_p\le 0$ or    ${\sum }_{p=0}^{\infty }\left|a_p{b}_p\right|<\infty$.

If $\left\{{\alpha }_p\right\}\subset [0,\infty )$ is such that

$${\alpha }_{p+1}\le (1-a_p){\alpha }_p+a_p{b}_p$$

for all $p\ge 0$, then $\underset{p\to \infty }{lim}{\alpha }_p=0.$

Lemma 4

([14]). Let $\mathcal{D}$ be a nonempty closed convex subset of a real Hilbert space $\mathcal{H}$ and let $\phi :\mathcal{D}\to \mathcal{H}$ be c-strongly monotone and M-Lipschitzian, with $c>0, M>0.$ Assume that $0<\nu <{\displaystyle \frac{2m}{M^2}}$ and $\delta =\alpha c-{\displaystyle \frac{M^2{\alpha }^2}{2}}.$ Then, for each $s\in \left(0,min\left\{1,{\displaystyle \frac{1}{r}}\right\}\right),$ we have

$$\parallel (I-s\alpha \phi )u-(I-s\alpha \phi )v\parallel \le (1-s\delta )\parallel u-v\parallel ,\forall u,v\in \mathcal{D},$$

(that is, $I-s\alpha A$ is $(1-s\delta )$-Lipschitzian on $\mathcal{D}$).

Let $\mathcal{D}\subset \mathcal{H}$ be closed convex and $T:\mathcal{D}\to \mathcal{D}$ a mapping. Then the map $T_{\lambda }:\mathcal{D}\to \mathcal{D}$ defined by $T_{\lambda }\left(x\right)=(1-\lambda )x+\lambda Tx,x\in C$, will be called the averaged mapping associated to T.

Lemma 5

([15]). Let $\mathcal{D}\subset \mathcal{H}$ be closed convex and $G:\mathcal{D}\to \mathcal{D}$ be μ-demicontractive. Then,

(a)

$Fix\left(G\right)$ is closed and convex;

(b)

For any $\lambda \in (0,1-\mu )$, $T_{\lambda }$ is quasi-nonexpansive.

Lemma 6

([16]). Let $\mathcal{D}$ be a nonempty closed convex subset of the normed linear space $\mathcal{X}$. If $f:\mathcal{D}\to \mathbb{R}$ is a differentiable and convex function, then $u^{\ast }$ is a minimizer of f over $\mathcal{D}$⟺$u^{\ast }$ solves the variational inequality $\langle \nabla f\left(u^{\ast }\right),v-u^{\ast }\rangle \ge 0$ for all $v\in \mathcal{D}.$

Remark 2.

By Lemma 6 , $u^{\ast }\in \Omega$⟺$u^{\ast }$ solves the following variational inequality problem:

$$\langle \nabla f\left(u^{\ast }\right),u^{\ast }-z\rangle \le 0,\forall z\in \mathsf{\Gamma }.$$

(8)

We denote by $\mathcal{VI}(\nabla f,\mathsf{\Gamma })$ the set of solutions of the variational inequality problem (8).

The projection (nearest point) from $\mathcal{H}$ to $\mathcal{D},$ denoted by $P_{\mathcal{D}}$ assigns to each $u\in \mathcal{H}$ the unique point $P_{\mathcal{D}}u$ with the property that

$$\parallel u-P_{\mathcal{D}}u\parallel \le \parallel u-v\parallel ,\forall v\in \mathcal{D}.$$

Recall that $P_{\mathcal{D}}$ satisfies

$$\langle u-v,P_{\mathcal{D}}u-P_{\mathcal{D}}v\rangle \ge {\parallel P_{\mathcal{D}}u-P_{\mathcal{D}}v\parallel }^2,\forall v\in \mathcal{H}$$

(9)

and

$$\langle P_{\mathcal{D}}w-v,w-P_{\mathcal{D}}w\rangle \ge 0,\forall w\in \mathcal{D}\mathrm{and}v\in \mathcal{H}.$$

(10)

An operator $\phi :\mathcal{D}\to \mathcal{H}$ is called monotone if

$$\langle \phi u-\phi v,u-v\rangle \ge 0,\forall u,v\in \mathcal{D};$$

$\phi$ is called c-strongly monotone if there exists $c\in (0,1)$ such that

$$\langle \phi u-\phi v,u-v\rangle \ge {k\parallel u-v\parallel }^2,\forall u,v\in \mathcal{D}.$$

An operator $\phi :\mathcal{H}\to \mathcal{H}$ is said to be strongly positive bounded if there exists a constant $b>0$ such that

$$\langle \phi u,u\rangle \ge {b\parallel u\parallel }^2,\forall u\in \mathcal{H}.$$

Remark 3.

Observe that a bounded linear operator φ which is strongly positive is both $\parallel \phi \parallel$-Lipchitzian and c-strongly monotone.

3. Main Results

We now introduce our averaged iterative algorithms for solving the convex minimization problem and establish the corresponding strong convergence results.

Let $f:\mathcal{D}\to \mathbb{R}$ be a differentiable and k-strongly convex objective function. Assume that the gradient $\nabla f:\mathcal{H}\to \mathcal{H}$ is L-Lipschitzian and consider two demicontractive mappings $F,G:\mathcal{D}\to \mathcal{D}$ such that $\mathsf{\Gamma }=Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing$. We are interested in studying the minimization problem:

$$\mathrm{find}{u}^{\ast }\in \mathsf{\Gamma }\mathrm{such}\mathrm{that}f\left(u^{\ast }\right)=\underset{u\in \mathsf{\Gamma }}{min}f\left(u\right),$$

(11)

whose set of solutions is denoted by $\Omega .$

Lemma 7.

Let $\mathcal{D}$ be a nonempty closed convex subset of a Hilbert space $\mathcal{H},$ and suppose that $f:\mathcal{H}\to \mathbb{R}$ is differentiable and k-strongly convex. Assume that $\nabla f:\mathcal{D}\to \mathcal{H}$ is K-Lipschitzian. Let $F,G:\mathcal{D}\to \mathcal{D}$ be two μ-demicontractive mappings such that

$$\mathsf{\Gamma }:=Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$$

Then, the solution set of (8) is nonempty.

Proof. 

Let $\alpha$ and $\delta$ be two constants such that $0<\alpha <{\displaystyle \frac{2k}{K^2}}$ and $\delta =\alpha \left(k-{\displaystyle \frac{K^2\alpha }{2}}\right),$ and let ${\tau }_0\in \left(0,min\{1,{\displaystyle \frac{1}{r}}\}\right).$

Notice that $P_{\mathsf{\Gamma }}(I-{\tau }_0\alpha \nabla f)$ is a strict contraction, a fact which follows Lemma 4:

$$\parallel P_{\mathsf{\Gamma }}(I-{\tau }_0\alpha \nabla f)u-P_{\mathsf{\Gamma }}(I-{\tau }_0\alpha \nabla f)v\parallel \le \parallel (I-{\tau }_0\alpha \nabla f)u-(I-{\tau }_0\alpha \nabla f)v\parallel \le (1-{\tau }_0\delta )\parallel u-v\parallel ,$$

for all $u,v\in \mathcal{D}.$ Hence, $P_{\mathsf{\Gamma }}(I-{\tau }_0\alpha \nabla f)$ has a unique fixed point $u_1\in \mathcal{H}.$ We have $u_1=P_{\mathsf{\Gamma }}(I-{\tau }_0\alpha \nabla f)u_1.$ Thus, by inequality (9), the problem can be restated under the form of a variational inequality:

$$\langle \nabla f\left(u_1\right),u_1-z\rangle \le 0,\forall z\in \mathsf{\Gamma }.$$

(12)

From Lemma 6, we deduce that $u_1\in \Omega .$    □

Theorem 1.

Let $\mathcal{D}$ be a nonempty closed convex subset of the Hilbert space $\mathcal{H}.$ Let $f:\mathcal{D}\to \mathbb{R}$ be a differentiable and k-strongly convex function. Let $F,G:\mathcal{D}\to \mathcal{D}$ be two demicontractive mappings such that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$ Assume that the differential map $\nabla f:\mathcal{D}\to \mathcal{H}$ is L-Lipschitz and that $I-F$ and $I-G$ are demiclosed at the origin. Suppose that $\left\{a_p\right\},\left\{b_p\right\}$ and $\left\{{\phi }_p\right\}$ are sequences satisfying:

(a)

$\underset{p\to \infty }{lim}{a}_p=0;{\sum }_{p=0}^{\infty }{a}_p=\infty ;$

(b)

${\phi }_p\in (\mu ,1),\underset{p\to \infty }{lim\; inf}(1-{\phi }_p)({\phi }_p-\mu )>0$ and ${lim\; inf}_{p\to \infty }{b}_p(1-b_p)>0.$

Assume also that $0<\alpha <{\displaystyle \frac{2k}{L^2}}.$ Then, the sequence $\left\{u_p\right\}$ generated by Algorithm 1 strongly converges to the unique solution of Problem (11).

Algorithm 1 The sequence $\left\{u_p\right\}$

Step 1. Take $\left\{a_p\right\}\subset (0,1),\left\{{\phi }_p\right\}\subset (0,1),\left\{b_p\right\}\subset (0,1),$ and $\alpha >0.$ Choose $u_0\in \mathcal{D};$ and let $p=0.$ Step 2. Given $u_p\in \mathcal{D},$ compute $u_{p+1}\in \mathcal{D}$ as $$\left\{\begin{array}{cc}{w}_p={\phi }_p{u}_p+(1-{\phi }_p)F{u}_p,\hfill & \\ v_p=b_p{w}_p+(1-b_p)\left[(1-\mu )w_p+\mu G{w}_p\right],\hfill & \\ u_{p+1}=P_{\mathcal{D}}(I-\alpha a_p\nabla f)v_p,\hfill \end{array}\right.$$ (13) Update $p:=p+1$ and go to Step 2.

Proof. 

Lemma 7 implies that $\mathcal{VI}(\nabla f,\mathsf{\Gamma })\ne \varnothing .$ Let $u^{\ast }\in \mathcal{VI}(\nabla f,\mathsf{\Gamma }).$ Assume that

$$a_p\in (0,min\{1,1/\delta \}),\delta =\alpha \left(k-{\displaystyle \frac{L^2\alpha }{2}}\right).$$

Let $z\in \mathsf{\Gamma }.$ From (13), Lemma 2.2 and demicontrativeness of $F,$

$$\begin{array}{cc}\hfill \parallel w_p{-z\parallel }^2& ={∥{\phi }_p(u_p-z)+(1-{\phi }_p)(F{u}_p-z)∥}^2\hfill \\ \hfill & ={\phi }_p\parallel u_p{-z\parallel }^2+(1-{\phi }_p)\parallel F{u}_p{-z\parallel }^2-{\phi }_p(1-{\phi }_p){\parallel F{u}_p-u_p\parallel }^2.\hfill \\ \hfill & \le {\phi }_p\parallel u_p{-z\parallel }^2+(1-{\phi }_p)(\parallel u_p{-z\parallel }^2+\mu \parallel F{u}_p-u_p{\parallel }^2)\hfill \\ \hfill & \phantom{=.}-{\phi }_p(1-{\phi }_p){\parallel F{u}_p-u_p\parallel }^2\hfill \\ \hfill & \le \parallel u_p{-z\parallel }^2-(1-{\phi }_p)({\phi }_p-\mu ){\parallel F{u}_p-u_p\parallel }^2.\hfill \end{array}$$

Since ${\phi }_p\in (\mu ,1),$ it follows that

$$\parallel w_p-z\parallel \le \parallel u_p-z\parallel .$$

(14)

Thus,

$$\begin{array}{cc}\hfill \parallel v_p-z\parallel & =\parallel b_p{w}_p+(1-b_p)\widehat{G}{w}_p-z\parallel \hfill \\ \hfill & \le b_p\parallel w_p-z\parallel +(1-b_p)\parallel \widehat{G}{w}_p-z\parallel \hfill \\ \hfill & \le \parallel w_p-z\parallel ,\hfill \end{array}$$

where $\widehat{G}=(1-\mu )I+\mu G.$ Thus,

$$\parallel v_p-z\parallel \le \parallel w_p-z\parallel \le \parallel u_p-z\parallel .$$

(15)

By Lemma 4, the nonexpansiveness of $P_{\mathcal{D}},$ and inequality (15),

$$\begin{array}{cc}\hfill \parallel u_{p+1}-z\parallel & = \parallel P_{\mathcal{D}}(I-a_p\alpha \nabla f)v_p-z\parallel \hfill \\ \hfill & = \parallel P_{\mathcal{D}}{v}_p-a_p\alpha P_{\mathcal{D}}\nabla f{v}_p-z+a_p\alpha P_{\mathcal{D}}\nabla f\left(z\right)-a_p\alpha P_{\mathcal{D}}\nabla f\left(z\right)\parallel \hfill \\ \hfill & \le \parallel P_{\mathcal{D}}{v}_p-a_p\alpha P_{\mathcal{D}}\nabla f{v}_p-P_{\mathcal{D}}z+a_p\alpha P_{\mathcal{D}}\nabla f\left(z\right)-a_p\alpha P_{\mathcal{D}}\nabla f\left(z\right)\parallel \hfill \\ \hfill & \le \parallel v_p-a_p\alpha \nabla f{v}_p-z+a_p\alpha \nabla f\left(z\right)-a_p\alpha \nabla f\left(z\right)\parallel \hfill \\ \hfill & \le \parallel (I-a_p\alpha \nabla f)v_p-(I-a_p\alpha \nabla f)z-a_p\alpha \nabla f\left(z\right)\parallel \hfill \\ \hfill & \le (1-\delta a_p)\parallel v_p-z\parallel +a_p\parallel \alpha \nabla f\left(z\right)\parallel \hfill \\ \hfill & \le (1-\delta a_p)\parallel u_p-z\parallel +a_p\parallel \alpha \nabla f\left(z\right)\parallel \hfill \\ \hfill & \le max\left\{\parallel u_p-z\parallel ,{\displaystyle \frac{\parallel \alpha \nabla f\left(z\right)\parallel }{\delta }}\right\}.\hfill \end{array}$$

By induction, we deduce that

$$\parallel u_p-z\parallel \le max\left\{\parallel u_0-z\parallel ,{\displaystyle \frac{\parallel \alpha \nabla f\left(z\right)\parallel }{\delta }}\right\},p\ge 1.$$

Hence, the sequence $\left\{u_p\right\}$ is bounded, as are $\left\{v_p\right\}$ and $\{\nabla f\left(u_p\right)\}.$

Consequently,

$$\begin{array}{cc}\hfill \parallel u_{p+1}{-z\parallel }^2& \le \parallel (I-\alpha a_p\nabla f)(v_p-z)-a_p\alpha \nabla f\left(z\right){\parallel }^2\hfill \\ \hfill & \le a_p^2{\parallel \alpha \nabla f\left(z\right)\parallel }^2+{(1-\delta a_p)}^2{\parallel v_p-z\parallel }^2\hfill \\ \hfill & \phantom{=.}2a_p(1-\delta a_p)\parallel \alpha \nabla f\left(z\right)\parallel \parallel v_p-z\parallel \hfill \\ \hfill & \le a_p^2{\parallel \alpha \nabla f\left(z\right)\parallel }^2+{(1-\delta a_p)}^2{\parallel v_p-z\parallel }^2\hfill \\ \hfill & \phantom{=.}-{(1-\delta a_p)}^2(1-{\phi }_p){({\phi }_p-\mu )}^2{\parallel F{u}_p-u_p\parallel }^2\hfill \\ \hfill & \phantom{=.}2a_p(1-\delta a_p)\parallel \alpha \nabla f\left(z\right)\parallel \parallel v_p-z\parallel .\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill {(1-\delta a_p)}^2(1-{\phi }_p)({\phi }_p-\mu ){\parallel F{u}_p-u_p\parallel }^2& \le \parallel u_p{-z\parallel }^2-\parallel u_{p+1}{-z\parallel }^2+a_p^2{\parallel \alpha \nabla f\left(z\right)\parallel }^2\hfill \\ \hfill & \phantom{=.}+2a_p(1-\delta a_p)\parallel \alpha \nabla f\left(z\right)\parallel \parallel u_p-z\parallel .\hfill \end{array}$$

Since $\left\{u_p\right\}$ is bounded, it follows that there exists a real number $m>0$ such that:

$${(1-\delta a_p)}^2(1-{\phi }_p)({\phi }_p-\mu )\parallel F{u}_p-u_p{\parallel }^2\le \parallel u_p{-z\parallel }^2-{\parallel u_{p+1}-z\parallel }^2+a_{p}m.$$

(16)

We proceed by considering two separate cases for the remainder of the proof.

Case 1.

Suppose that $\{\parallel u_p-z\parallel \}$ is monotonically decreasing, which implies that it is convergent. Clearly,

$$\underset{p\to \infty }{lim}(\parallel u_p{-z\parallel }^2-\parallel u_{p+1}-z{\parallel }^2)=0.$$

(17)

It follows from (16) that

$$\underset{p\to \infty }{lim}(1-{\phi }_p)({\phi }_p-\mu ){\parallel F{x}_n-u_p\parallel }^2=0.$$

(18)

Since $\underset{p\to \infty }{lim\; inf}(1-{\phi }_p)({\phi }_p-\mu )>0,$ we deduce that

$$\underset{p\to \infty }{lim}\parallel u_p-F{u}_p\parallel =0.$$

(19)

Observe that

$$\begin{array}{cc}\hfill \parallel w_p-u_p\parallel & = \parallel {\phi }_p{u}_p+(1-{\phi }_p)F{u}_p-u_p\parallel \hfill \\ \hfill & = \parallel {\phi }_p{u}_p+(1-{\phi }_p)F{u}_p-{\phi }_p{u}_p-(1-{\phi }_p)u_p\parallel \hfill \\ \hfill & =(1-{\phi }_p)\parallel F{u}_p-u_p\parallel \hfill \\ \hfill & \le \parallel F{u}_p-u_p\parallel .\hfill \end{array}$$

It follows from (19) that

$$\underset{p\to \infty }{lim}\parallel w_p-u_p\parallel =0.$$

(20)

Since $\left\{u_p\right\}$ is bounded, there exists a subsequence $\left\{u_{n_j}\right\}$ of $\left\{u_p\right\}$ such that $\left\{u_{n_j}\right\}$ converges weakly to $y\in \mathcal{D}$ and

$$\underset{p\to \infty }{lim\; sup}\langle \nabla f\left(u^{\ast }\right),u^{\ast }-u_p\rangle =\underset{j\to \infty }{lim}\langle g\left(u^{\ast }\right),u^{\ast }-u_{p_j}\rangle .$$

From (19) and the demiclosedness of $I-F,$ we have that $y\in Fix\left(F\right).$ Additionally, by Lemma 2 and the fact that $\widehat{G}$ is quasinonexpansive, we obtain

$$\begin{array}{cc}\hfill \parallel v_p{-z\parallel }^2& = \parallel b_p{w}_p+(1-b_p)\widehat{G}{w}_p{-z\parallel }^2\hfill \\ \hfill & =b_p\parallel w_p{-z\parallel }^2+(1-b_p)\parallel \widehat{G}{w}_p{-z\parallel }^2-(1-b_p)b_p{\parallel \widehat{G}{w}_p-w_p\parallel }^2\hfill \\ \hfill & \le \parallel u_p{-z\parallel }^2-(1-b_p)b_p{\parallel \widehat{G}{w}_p-w_p\parallel }^2.\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \parallel u_{p+1}{\parallel }^2& \le \parallel (I-a_p\alpha \nabla f)v_p{-z\parallel }^2\hfill \\ \hfill & \le \parallel (I-a_p\alpha \nabla f)(v_p-z)-a_p\alpha \nabla f\left(z\right){\parallel }^2\hfill \\ \hfill & \le a_p^2{\parallel \alpha \nabla f\left(z\right)\parallel }^2+{(1-a_p\delta )}^2{\parallel v_p-z\parallel }^2\hfill \\ \hfill & \phantom{=.}+2a_p(1-a_p\delta )\parallel \alpha \nabla f\left(z\right)\parallel \parallel v_p-z\parallel \hfill \\ \hfill & \le a_p^2{\parallel \alpha \nabla f\left(z\right)\parallel }^2+{(1-a_p\delta )}^2{\parallel v_p-z\parallel }^2\hfill \\ \hfill & \phantom{=.}-{(1-a_p\delta )}^2(1-b_p)b_p{\parallel \widehat{G}{w}_p-w_p\parallel }^2\hfill \\ \hfill & \phantom{=.}+2a_p(1-a_p\delta )\parallel \alpha \nabla f\left(z\right)\parallel \parallel v_p-z\parallel .\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill {(1-\delta a_p)}^2{b}_p(1-b_p){\parallel \widehat{G}{w}_p-w_p\parallel }^2& \le \parallel u_p{-z\parallel }^2-\parallel u_{p+1}{-z\parallel }^2+a_p^2{\parallel \alpha \nabla f\left(z\right)\parallel }^2\hfill \\ \hfill & \phantom{=.}+2a_p(1-a_p\delta )\parallel \alpha \nabla f\left(z\right)\parallel \parallel u_p-z\parallel .\hfill \end{array}$$

Since $\left\{u_p\right\}$ is bounded, there exists a real number $K>0$ such that

$${(1-\delta a_p)}^2{b}_p(1-b_p)\parallel \widehat{G}{w}_p-w_p{\parallel }^2\le \parallel u_p{-z\parallel }^2-{\parallel u_{p+1}-z\parallel }^2+a_{p}K.$$

(21)

It follows from (21) and (17) that

$$\underset{p\to \infty }{lim}{b}_p(1-b_p)\parallel \widehat{G}{w}_p-w_p\parallel =0.$$

(22)

Since $\underset{p\to \infty }{lim\; inf}{b}_p(1-b_p)>0,$ we conclude that

$$\underset{p\to \infty }{lim}\parallel \widehat{G}{w}_p-w_p\parallel =0.$$

(23)

Since $\left\{w_{p_j}\right\}$ converges weakly to $y,$ it follows from (23) and Lemma 1 that $y\in Fix\left(\widehat{G}\right)=Fix\left(G\right).$ Hence, $y\in \mathsf{\Gamma }.$ Given that $u^{\ast }\in \mathcal{VI}(\nabla f,\mathsf{\Gamma }),$ we obtain

$$\begin{array}{cc}\hfill \underset{p\to \infty }{lim\; sup}\langle \nabla f\left(u^{\ast }\right),u^{\ast }-u_p\rangle & =\underset{j\to \infty }{lim}\langle g\left(u^{\ast }\right),u^{\ast }-u_{p_j}\rangle \hfill \\ \hfill & =\langle \nabla f\left(u^{\ast }\right),u^{\ast }-y\rangle \le 0.\hfill \end{array}$$

Observe that

$$\begin{array}{cc}\hfill \parallel u_{p+1}-u^{\ast }\parallel & \le \langle (I-\alpha a_p\nabla f)v_p-u^{\ast },u_{p+1}-u^{\ast }\rangle \hfill \\ \hfill & \le \langle (I-\alpha a_p\nabla f)v_p-u^{\ast }+\alpha a_p\nabla f\left(u^{\ast }\right)-\alpha a_p\nabla f\left(u^{\ast }\right),u_{p+1}-u^{\ast }\rangle \hfill \\ \hfill & \le \parallel (I-\alpha a_p\nabla f)v_p-u^{\ast }\parallel \parallel u_{p+1}-u^{\ast }\parallel +a_p\langle \alpha \nabla f\left(u^{\ast }\right),u_{p+1}-u^{\ast }\rangle \hfill \\ \hfill & \le (1-a_p\delta ){\parallel u_p-u^{\ast }\parallel }^2+2a_p\alpha \langle \nabla f\left(u^{\ast }\right),u_{p+1}-u^{\ast }\rangle .\hfill \end{array}$$

Lemma 3 implies that $u_p\to u^{\ast }.$

Case 2.

Assume that the sequence $\{\parallel u_p-u^{\ast }\parallel \}$ is not a monotonically decreasing sequence. Let $B_p={\parallel u_p-u^{\ast }\parallel }^2.$ Define a mapping $\kappa :\mathbb{N}\to \mathbb{N}$ for all $p\ge p_0$ (for some $p_0\in \mathbb{N}$ large enough) by

$$\kappa \left(p\right)=max\{n\in \mathbb{N}:n\le p,B_n\le B_{n+1}\}.$$

The sequence $\left\{\kappa \right(p\left)\right\}$ is non-decreasing, with $\kappa \left(p\right)\to \infty$ as $p\to \infty ,$ and $B_{\kappa \left(p\right)}\le B_{\kappa \left(p\right)+1}$ for all $p\ge p_0.$ It follows from (16) that

$$(1-{\phi }_{\kappa \left(p\right)})({\phi }_{\kappa \left(p\right)}-\mu ){\parallel u_{\kappa \left(p\right)}-F{u}_{\kappa \left(p\right)}\parallel }^2\le a_{\kappa \left(p\right)}C.$$

Since ${\phi }_{\kappa \left(p\right)}\in (\mu ,1),$ we have

$$\underset{p\to \infty }{lim}\parallel u_{\kappa \left(p\right)}-F{u}_{\kappa \left(p\right)}\parallel =0$$

(24)

By the same argument as in Case 1, it follows that the sequences $\left\{u_{\kappa \left(p\right)}\right\}$ and $\left\{v_{\kappa \left(p\right)}\right\}$ are bounded and $\underset{\kappa \left(p\right)\to \infty }{lim\; sup}\langle \nabla f\left(u^{\ast }\right),u^{\ast }-u_{\kappa \left(p\right)}\rangle \le 0.$ For all $p\ge p_0,$ we have

$$\begin{array}{cc}\hfill 0& \le \parallel u_{\kappa \left(p\right)+1}-u^{\ast }{\parallel }^2-{\parallel u_{\kappa \left(p\right)}-u^{\ast }\parallel }^2\hfill \\ \hfill & \le a_{\kappa \left(p\right)}\left(-\kappa \left(p\right)\parallel u_{\kappa \left(p\right)}-u^{\ast }{\parallel }^2+2\alpha \langle \nabla f\left(u^{\ast }\right),u^{\ast }-x_{\kappa \left(p\right)+1}\rangle \right).\hfill \end{array}$$

This implies that

$$\parallel u_{\kappa \left(p\right)}-u^{\ast }{\parallel }^2\le {\displaystyle \frac{2\alpha }{\kappa \left(p\right)}}\langle \nabla f\left(u^{\ast }\right),u^{\ast }-u_{\kappa \left(p\right)+1}\rangle .$$

Now we can write

$$\underset{p\to \infty }{lim}\parallel u_{\kappa \left(p\right)}-u^{\ast }\parallel =0.$$

Hence

$$\underset{p\to \infty }{lim}{B}_{\kappa \left(p\right)}=\underset{p\to \infty }{lim}{B}_{\kappa \left(p\right)+1}=0.$$

Moreover, for all $p\ge p_0,$ if $p\ne \kappa \left(p\right)$ (that is, $p>\kappa \left(p\right)$), then $B_{\kappa \left(p\right)}\le B_{\kappa \left(p\right)+1},$ due to the fact that $B_j>B_{j+1}$ for $\kappa \left(p\right)+1\le j\le p.$ Consequently, for $p\ge p_0$

$$0\le B_p\le max\{B_{\kappa \left(p\right)},B_{\kappa \left(p\right)+1}\}=B_{\kappa \left(p\right)+1}.$$

Therefore, $\underset{p\to \infty }{lim}{B}_p=0,$ that is, $\left\{u_p\right\}$ converges strongly to $u^{\ast }.$ □

Theorem 2.

Let $\mathcal{D}$ be a nonempty closed convex subset of the Hilbert space $\mathcal{H}.$ Let $f:\mathcal{D}\to \mathbb{R}$ be a differentiable and k-strongly convex function. Let $F,G:\mathcal{D}\to \mathcal{D}$ be two strictly pseudocontractive mappings such that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$ Assume that the differential map $\nabla f:\mathcal{D}\to \mathcal{H}$ is L-Lipschitz and $0<\alpha <{\displaystyle \frac{2k}{L^2}}.$ Let $u_0\in \mathcal{D}$ and define the sequence $\left\{u_p\right\}$ as follows:

$$\left\{\begin{array}{cc}{w}_p={\phi }_p{u}_p+(1-{\phi }_p)F{u}_p,\hfill & \\ v_p=b_p{u}_p+(1-b_p)\left[(1-\mu )u_p+\mu G{u}_p\right],\hfill & \\ u_{p+1}=P_{\mathcal{D}}(I-\alpha a_p\nabla f)v_p.\hfill \end{array}\right.$$

(25)

Suppose that $\left\{a_p\right\},\left\{b_p\right\}$ and $\left\{{\phi }_p\right\}$ are the sequences such that:

(i)

$\underset{p\to \infty }{lim}{a}_p=0,{\displaystyle \sum _{p=0}^{\infty }}{a}_p=\infty ,$

(ii)

${\phi }_p\in (0,1),$ and $\underset{p\to \infty }{lim\; inf}{b}_p(1-b_p)>0.$

Then $\left\{u_p\right\}$ in (25) converges strongly to a minimizer of f over $Fix\left(F\right)\cap Fix\left(G\right).$

Proof. 

Since F is $\beta$-strictly pseudocontractive, and G is $\gamma$-strictly pseudocontractive, with $\alpha ,\beta \in (0,1)$; from Lemma 5, we observe that $\widehat{F}=(1-\mu )I+\mu F$ and $\widehat{G}=(1-\mu )I+\mu G$ are both nonexpansive mappings, for $\mu \in (0,min\{\beta ,\gamma \left\}\right)$. Thus, F and G are 0-demicontractive. The rest of the proof follows from Theorem 1. □

Next, Theorem 1 is employed to solve the quadratic optimization problem given by:

$$\mathrm{find}{u}^{\ast }\in \mathsf{\Gamma }\mathrm{such}\mathrm{that}f\left(u^{\ast }\right)=\underset{u\in \mathsf{\Gamma }}{min}f\left(u\right),\mathrm{where}f\left(u\right)={\displaystyle \frac{1}{2}}\langle Au,u\rangle .$$

(26)

Theorem 3.

Let $\mathcal{D}$ be a nonempty, closed convex subset of the real Hilbert space $\mathcal{H}.$ Let $F,G:\mathcal{D}\to \mathcal{D}$ be two μ-demicontractive mappings such that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing ,$ and let $A:\mathcal{D}\to \mathcal{H}$ be a strongly bounded linear operator with coefficient $c>0.$ Assume that $I-F$ and $I-G$ are demiclosed at the origin and $0<\alpha <{\displaystyle \frac{2k}{L^2}}.$ Let $u_0\in \mathcal{D}$ and define the sequence $\left\{u_p\right\}$ as follows:

$$\left\{\begin{array}{cc}{w}_p={\phi }_p{u}_p+(1-{\phi }_p)F{u}_p,\hfill & \\ v_p=b_p{u}_p+(1-b_p)\left[(1-\mu )u_p+\mu G{u}_p\right],\hfill & \\ u_{p+1}=P_{\mathcal{D}}(I-\alpha a_{p}A)v_p.\hfill \end{array}\right.$$

(27)

Let $\left\{a_p\right\},\left\{b_p\right\}$ and $\left\{{\phi }_p\right\}$ be sequences such that:

(i)

$\underset{p\to \infty }{lim}{a}_p=0,{\displaystyle \sum _{p=0}^{\infty }}{a}_p=\infty ,$

(ii)

${\phi }_p\in (0,1),$ and $\underset{p\to \infty }{lim\; inf}{b}_p(1-b_p)>0.$

Then $\left\{u_p\right\}$ in (27) converges strongly to a solution of (26).

Proof. 

Notice that A is a $\parallel A\parallel$-Lipchitzian and a c-strongly monotone operator. Setting $\nabla fu=Au,$ the conclusion of this theorem follows from Theorem 1. □

The main results in [1], i.e., Theorems 3.1, 3.2 and 3.3, are direct consequences of our Theorems 1, 2 and 3, respectively.

Corollary 1.

Let $\mathcal{D}$ be a nonempty closed convex subset of a real Hilbert space $\mathcal{H}.$ Let $f:\mathcal{D}\to \mathbb{R}$ be a differentiable, k-strongly convex real-valued function. Let $F:\mathcal{D}\to \mathcal{D}$ be a β-demicontractive mapping and $G:\mathcal{D}\to \mathcal{D}$ be a quasi-nonexpansive mapping such that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$ Assume that the differential map $\nabla f:\mathcal{D}\to \mathcal{H}$ is L-Lipschitz and that $0<\alpha <{\displaystyle \frac{2k}{L^2}}.$ Let $u_0\in \mathcal{D}$ and define the sequence $\left\{u_p\right\}$ as follows:

$$\left\{\begin{array}{cc}{w}_p={\phi }_p{u}_p+(1-{\phi }_p)F{u}_p,\hfill & \\ v_p=b_p{u}_p+(1-b_p)\left[(1-\mu )u_p+\mu G{u}_p\right],\hfill & \\ u_{p+1}=P_{\mathcal{D}}(I-\alpha a_p\nabla f)v_p.\hfill \end{array}\right.$$

(28)

Let $\left\{a_p\right\},\left\{b_p\right\}$ and $\left\{{\phi }_p\right\}$ be sequences such that:

(i)

$\underset{p\to \infty }{lim}{a}_p=0,{\displaystyle \sum _{p=0}^{\infty }}{a}_p=\infty ,$

(ii)

${\phi }_p\in (0,1),$ and $\underset{p\to \infty }{lim\; inf}{b}_p(1-b_p)>0.$

Then $\left\{u_p\right\}$ in (28) converges strongly to a minimizer of f over $Fix\left(F\right)\cap Fix\left(G\right).$

Corollary 2.

Let $\mathcal{D}$ be a nonempty closed convex subset of a real Hilbert space $\mathcal{H}.$ Let $f:\mathcal{D}\to \mathbb{R}$ be a differentiable, k-strongly convex real-valued function. Let $F,G:\mathcal{D}\to \mathcal{D}$ be two nonexpansive mappings such that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$ Assume that the differential map $\nabla f:\mathcal{D}\to \mathcal{H}$ is L-Lipschitz and $0<\alpha <{\displaystyle \frac{2k}{L^2}}.$ Let $u_0\in \mathcal{D}$ and define $\left\{u_p\right\}$ as follows:

$$\left\{\begin{array}{cc}{w}_p={\phi }_p{u}_p+(1-{\phi }_p)F{u}_p,\hfill & \\ v_p=b_p{u}_p+(1-b_p)\left[(1-\mu )u_p+\mu G{u}_p\right],\hfill & \\ u_{p+1}=P_{\mathcal{D}}(I-\alpha a_p\nabla f)v_p.\hfill \end{array}\right.$$

(29)

Suppose that $\left\{a_p\right\},\left\{b_p\right\}$ and $\left\{{\phi }_p\right\}$ are sequences such that:

(i)

$\underset{p\to \infty }{lim}{a}_p=0,{\displaystyle \sum _{p=0}^{\infty }}{a}_p=\infty ,$

(ii)

${\phi }_p\in (0,1),$ and $\underset{p\to \infty }{lim\; inf}{b}_p(1-b_p)>0.$

Then $\left\{u_p\right\}$ defined in (29) converges strongly to a minimizer of f over $Fix\left(F\right)\cap Fix\left(G\right).$

Corollary 3.

Let $\mathcal{D}$ be a nonempty closed convex subset of a real Hilbert space $\mathcal{H}.$ Let $A:\mathcal{D}\to \mathcal{H}$ be a strongly bounded linear operator with coefficient $k>0.$ Let $F:\mathcal{D}\to \mathcal{D}$ be a β-demicontractive mapping and $G:\mathcal{D}\to \mathcal{D}$ be a quasi-nonexpansive mapping such that $Fix\left(F\right)\cap Fix\left(G\right)\ne \varnothing .$ Assume that $I-F$ and $I-G$ are demiclosed at the origin and $0<\alpha <{\displaystyle \frac{2k}{L^2}}.$ Let $u_0\in \mathcal{D}$ and define $\left\{u_p\right\}$ as follows:

$$\left\{\begin{array}{cc}{w}_p={\phi }_p{u}_p+(1-{\phi }_p)F{u}_p,\hfill & \\ v_p=b_p{u}_p+(1-b_p)\left[(1-\mu )u_p+\mu G{u}_p\right],\hfill & \\ u_{p+1}=P_{\mathcal{D}}(I-\alpha a_{p}A)v_p.\hfill \end{array}\right.$$

(30)

Suppose that $\left\{a_p\right\},\left\{b_p\right\}$ and $\left\{{\phi }_p\right\}$ are sequences such that:

(i)

$\underset{p\to \infty }{lim}{a}_p=0,{\displaystyle \sum _{p=0}^{\infty }}{a}_p=\infty ,$

(ii)

${\phi }_p\in (0,1),$ and $\underset{p\to \infty }{lim\; inf}{b}_p(1-b_p)>0.$

Then $\left\{u_p\right\}$ defined in (30) converges strongly to a solution of (26).

4. Numerical Illustrations

In order to cover the diversity of our theoretical results in the previous section, we consider three different cases: (1) a pair of demicontractive mappings (Example 3); (2) a pair consisting of a demicontractive mapping and a quasi-nonexpanive mapping (Example 4); and (3) a pair consisting of a demicontractive mapping and a strictly pseudocontactive mapping (Example 5). These three pairs of mappings are intended to illustrate the fact that the rate of convergence of our averaged Algorithm 1 slightly decreases its rate of convergence when one passes from a certain class of nonexpansive-type mappings to a larger class of nonexpanisve type mappings.

Example 3.

Consider the two demicontractive mappings given in Example 1 and Example 2. The set of common fixed points of F and G is $\mathsf{\Gamma }=Fix\left(F\right)\cap Fix\left(G\right)=\left[{\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}\right].$

We use the ${\displaystyle \frac{2}{3}}$-strongly convex function $f\left(u\right)={\displaystyle \frac{u^2}{3}}.$ Note that f is also ${\displaystyle \frac{2}{3}}$-Lipschitz function. We set the following sequences and constant:

$$\begin{array}{cc}\hfill a_p& ={\displaystyle \frac{1}{100p}},b_p={\displaystyle \frac{p}{100p+1}},{\phi }_p={\displaystyle \frac{9}{25}},\forall p\in \mathbb{N}\mathit{and}\alpha =0.45.\hfill \end{array}$$

It is easy to verify that all assumptions in Theorem 1 are satisfied and hence, the sequence $\left\{u_p\right\}$ from Algorithm 1 converges to the solution of Problem (11), which is $u={\displaystyle \frac{2}{3}}\in \mathsf{\Gamma }$ giving the minimum value of $f\left(u\right)$ in $\mathsf{\Gamma }.$

We perform several numerical experiments in Python using Algorithm 1 with various choices of parameters values to solve the minimization problem. We select four different initial values: one from each of the intervals $[0,{\displaystyle \frac{2}{3}}],({\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}),[{\displaystyle \frac{3}{4}},{\displaystyle \frac{9}{10}})$ and $[{\displaystyle \frac{9}{10}},1].$ See the numerical results in

Table 1. Numerical results for Example 3 with initial values $u_0=0.1,0.7,0.8$ and $u_0=1$.

Iteration (p)	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$
0	0.100000	0.700000	0.800000	1.000000
1	0.516982	0.691600	0.748219	0.544454
2	0.625012	0.687450	0.743730	0.631923
3	0.653500	0.684701	0.740755	0.655242
4	0.661345	0.682646	0.738533	0.661784
5	0.663723	0.681008	0.736760	0.663834
…	…	…	…	…
10	0.665551	0.675748	0.731070	0.665551
…	…	…	…	…
500	0.666645	0.666645	0.697942	0.666645
…	…	…	…	…
9180	0.666665	0.666665	0.673996	0.666665
9181	0.666666	0.666666	0.673996	0.666666
…	…	…	…	…
64,264	0.666666	0.666666	0.666666	0.666666
64,265	0.666667	0.666667	0.666667	0.666667
64,266	0.666667	0.666667	0.666667	0.666667

.

Example 4.

We consider the β-demicontractive mapping F in Example 1 and the following quasi-nonexansive mapping:

$$T\left(u\right)=\left\{\begin{array}{cc}{\displaystyle \frac{3u^2+2}{3u+3}},\hfill & if 0\le u\le {\displaystyle \frac{2}{3}}\hfill \\ u,\hfill & if {\displaystyle \frac{2}{3}}<u<{\displaystyle \frac{3}{4}}\hfill \\ {\displaystyle \frac{3}{4}},\hfill & if {\displaystyle \frac{3}{4}}\le u\le 1.\hfill \end{array}\right.$$

(31)

We have $\mathsf{\Gamma }=Fix\left(F\right)\cap Fix\left(T\right)=\left[{\displaystyle \frac{2}{3}},{\displaystyle \frac{3}{4}}\right].$ All other values are the same as in Example 3. Observe that the assumptions in Theorem 1 are satisfied and thus, the sequence $\left\{u_p\right\}$ in Algorithm 1 converges to the solution of Problem (11), which is $u={\displaystyle \frac{2}{3}}\in \mathsf{\Gamma }$, giving the minimum value of $f\left(u\right)$ in $\mathsf{\Gamma }.$ See

Table 2. Numerical results for Example 4 with initial values $u_0=0.1,0.7,0.8$ and $u_0=1$.

Iteration (p)	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$
0	0.100000	0.700000	0.800000	1.000000
1	0.563625	0.691600	0.748219	0.777096
2	0.638105	0.687450	0.743730	0.749436
3	0.656963	0.684701	0.740755	0.746439
4	0.662256	0.682646	0.738533	0.744199
5	0.663968	0.681008	0.736760	0.742413
…	…	…	…	…
10	0.665557	0.675748	0.731070	0.736679
…	…	…	…	…
500	0.666645	0.666645	0.697942	0.703297
…	…	…	…	…
9144	0.666665	0.666665	0.674028	0.679200
9145	0.666666	0.666666	0.674027	0.679199
…	…	…	…	…
64,008	0.666666	0.666666	0.666666	0.666666
64,009	0.666667	0.666667	0.666667	0.666667
64,010	0.666667	0.666667	0.666667	0.666667

for the numerical results.

Example 5.

We consider the β-demicontractive mapping F in Example 1 and the following ${\displaystyle \frac{3}{5}}$-strictly pseudocontractive mapping:

$$T_2\left(u\right)={\displaystyle \frac{4}{9u}},0\le u\le 1.$$

(32)

Here, $\mathsf{\Gamma }=Fix\left(F\right)\cap Fix\left(T_2\right)=2/3.$ All other values are the same as in Example 3. Hence, the assumptions in Theorem 1 are satisfied and hence, the sequence $\left\{u_p\right\}$ in Algorithm 1 converges to the solution of Problem (11), which is $u={\displaystyle \frac{2}{3}}\in \mathsf{\Gamma }.$ See the numerical results in

Table 3. Numerical results for Example 5 with initial values $u_0=0.1,0.7,0.8$ and $u_0=1$.

Iteration (p)	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$	${\mathit{u}}_{\mathit{p}}$
0	0.100000	0.700000	0.800000	1.000000
1	1.000000	0.655331	0.649398	0.649754
2	0.653699	0.665946	0.667765	0.667654
3	0.667780	0.664201	0.663876	0.663889
4	0.664541	0.665361	0.665453	0.665450
5	0.665665	0.665433	0.665407	0.665408
…	…	…	…	…
10	0.666056	0.666056	0.666056	0.666056
…	…	…	…	…
500	0.666654	0.666645	0.666654	0.666654
…	…	…	…	…
5356	0.666665	0.666665	0.666665	0.666665
5357	0.666666	0.666666	0.666666	0.666666
…	…	…	…	…
37,499	0.666666	0.666666	0.666666	0.666666
37,500	0.666667	0.666667	0.666667	0.666667
37,501	0.666667	0.666667	0.666667	0.666667

.

5. Conclusions

In this paper we introduce an averaged three-step iterative scheme (Algorithm 1) designed to solve the convex optimization problem (11) over the set of common fixed points of two demicontractive mappings F and G. Our first main result (Theorem 1) shows that, if f is a differentiable and k-strongly convex function, $\nabla f$ is Lipschitzian, and $I-F$ and $I-G$ are demiclosed at zero, then the sequence generated by Algorithm 1 converges strongly to the solution of problem (11). Similarly, Theorem 2 establishes a convergence result for the averaged iterative scheme (25).

The main argument used in our investigation relies on an embedding technique using averaged mappings: if G is $\beta$-demicontractive, then for any $\mu \in (0,1-\beta ),$ $G_{\mu }=(1-\mu )I+\mu G$ is quasi-nonexpansive.

The main results are then used to solve the minimization problem in the case when F and G are strictly pseudocontractive Theorem 2 and to solve a quadratic optimization problem involving bounded linear operators Theorem 3.

We also prove that the main results in [1] (Theorems 3.1. 3.2 and 3.3) are particular cases that can be seen in our results; see Corollaries 1, 2 and 3, respectively.

As demonstrated by the numerical experiments in Section 4 (Examples 3–5), our theoretical results can be applied successfully to minimization problems in the broader class of demicontractive mappings.

We thus extend several existing results in the literature—originally established for nonexpansive or quasi-nonexpansive mappings—to the more general settings of strictly pseudocontractive and demicontractive mappings; see, for example [17,18,19,20,21,22].

A report on the numerical experiments we performed is also presented to illustrate the convergence behavior of the iterative algorithms.