A New Iterative Algorithm for General Variational Inequality Problem with Applications

Abstract

This study aims at investigation of a generalized variational inequality problem. We initiate a new iterative algorithm and examine its convergence analysis. Using this newly proposed iterative method, we estimate the common solution of generalized variational inequality problem and fixed points of a nonexpansive mapping. A numerical example is illustrated to verify our existence result. Further, we demonstrate that the considered iterative algorithm converges with faster rate than normal $S$-iterative scheme. Furthermore, we apply our proposed iterative algorithm to estimate the solution of a convex minimization problem and a split feasibility problem.

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1. Introduction

All through this study, we presume that $\mathrm{\mathscr{H}}$ is a real Hilbert space equipped with norm $\cdot$ induced by inner product $\cdot ,\cdot$. Let $\mathcal{C}$ be a nonempty closed convex subset of $\mathrm{\mathscr{H}}$ and $f,g:\mathcal{C}⟶\mathrm{\mathscr{H}}$ be nonlinear mappings. The generalized nonlinear variational inequality is to locate a point $x^{\ast }\in \mathrm{\mathscr{H}}$ such that$$\begin{array}{}\text{(1)}& f{x}^{\ast },g{x}^{\ast }-g{y}^{\ast }\ge 0,\hspace{1em}\forall y^{\ast }\in \mathcal{C},g{x}^{\ast },g{y}^{\ast }\in \mathrm{\mathscr{H}},\end{array}$$which was introduced by Noor [18]. We denote the set of solutions of (1) by $\text{Sol}\mathcal{C},f,g$.

If $g=I$, then generalized nonlinear variational inequality (1) reduces to the classical variational inequality studied by Stampacchia [23], which is to allocate a point $x^{\ast }\in \mathrm{\mathscr{H}}$, such that$$\begin{array}{}\text{(2)}& f{x}^{\ast },y^{\ast }-x^{\ast }\ge 0,\hspace{1em}\forall y^{\ast }\in \mathrm{\mathscr{H}}.\end{array}$$

If ${\mathcal{C}}^{\ast }=x^{\ast }\in \mathrm{\mathscr{H}}:x^{\ast },y^{\ast }\ge 0,\forall y^{\ast }\in \mathcal{C}$ is a dual cone of a convex cone $\mathcal{C}$, then generalized nonlinear variational inequality (1) coincides to generalized nonlinear complementarity problem which is to locate a point $x^{\ast }\in \mathrm{\mathscr{H}}$ such that$$\begin{array}{c}\text{(3)}& f{x}^{\ast },g{x}^{\ast }=0,g{x}^{\ast }\in \mathcal{C},f{x}^{\ast }\in {\mathcal{C}}^{\ast }.\end{array}$$

It is worthy to adduce that variational inequalities which are unconventional and remarkable augmentation of variational principles provide well organized unified framework for figuring out a wide range of nonlinear problems arising in optimization, economics, physics, engineering science, operations research, and control theory, for example, [2, 8, 15, 20, 21, 24, 26, 33] and references cited therein.

Next, we recall the following definitions of a nonlinear mapping $f:\mathcal{C}\subset \mathrm{\mathscr{H}}⟶\mathrm{\mathscr{H}}$.

Definition 1.

The mapping $f:\mathcal{C}⟶\mathrm{\mathscr{H}}$ is said to be

(i) $a$-inverse strongly monotone or cocoercive if there exists a constant $a>0$, such that$$\begin{array}{}\text{(4)}& fx-fy,x-y\ge a{fx-fy}^2,\hspace{1em}\forall x,y\in \mathcal{C},\end{array}$$

(ii) $L$-Lipschitz continuous if there exists a constant $L>0$, such that

$$\begin{array}{}\text{(5)}& fx-fy\le Lx-y,\hspace{1em}\forall x,y\in \mathcal{C}.\end{array}$$

For $L=1$, $f$ is nonexpansive, and if $0<L<1$, then $f$ is a contraction. Note that $a$-inverse strongly monotone mapping is $1/a$-Lipschitz continuous.

It is customary to mention that variational inequalities, variational inclusions, and related optimization problems can be posed as fixed-point problems. This unusual formulation plays a dominant role in studying variational inequalities and nonlinear problems by employing fixed-point iterative methods.

Lemma 1.

Let ${\mathrm{p}}_{\mathcal{C}}:\mathrm{\mathscr{H}}⟶\mathcal{C}$ be a projection mapping of $\mathrm{\mathscr{H}}$ onto $\mathcal{C}$. For a given $q\in \mathrm{\mathscr{H}},p\in \mathcal{C}$ satisfies the inequality$$\begin{array}{}\text{(6)}& p-q,r-p\ge 0,\hspace{1em}\forall r\in \mathcal{C}\text{if\hspace{0.17em}and\hspace{0.17em}only\hspace{0.17em}if\hspace{0.17em}}p={\mathrm{p}}_{\mathcal{C}}q.\end{array}$$

Note that the projection mapping ${\mathrm{p}}_{\mathcal{C}}$ is nonexpansive [16]. For more details on projection mapping ${\mathrm{p}}_{\mathcal{C}}$, we refer to [12]. By utilizing Lemma 1, the generalized nonlinear variational inequality (1) can be designed as a fixed-point problem as follows:

Lemma 2 (see [17]).

Let ${\mathrm{p}}_{\mathcal{C}}:\mathrm{\mathscr{H}}⟶\mathcal{C}$ be a projection mapping. For any $\rho >0$, $x^{\ast }\in \mathrm{\mathscr{H}},g{x}^{\ast }\in \mathcal{C}$ solves the generalized nonlinear variational inequality (1) if and only if$$\begin{array}{}\text{(7)}& g{x}^{\ast }={\mathrm{p}}_{\mathcal{C}}g{x}^{\ast }-\rho f{x}^{\ast }.\end{array}$$

Relation (7) can be rescripted as$$\begin{array}{}\text{(8)}& x^{\ast }=x^{\ast }-g{x}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g-\rho f{x}^{\ast }.\end{array}$$

Let $T$ be a nonexpansive mapping and $FT$ denotes the set of fixed points of $T$. If $x^{\ast }\in FT\cap \text{Sol}\mathrm{\mathscr{H}},f,g$, then$$\begin{array}{c}\text{(9)}& x^{\ast }=T{x}^{\ast }=x^{\ast }-g{x}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g-\rho f{x}^{\ast }=T{x}^{\ast }-g{x}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g{x}^{\ast }-\rho f{x}^{\ast },\hspace{1em}\rho >0.\end{array}$$

It is significant to achieve better rate of convergence if two or more iterative algorithms converge to the same point for a given problem. We recall the following concepts which are versatile tools to find finer convergence rate for different iterative methods.

Definition 2.

(see [3]). Let $p_n$ and $q_n$ be two real sequences converging to $p$ and $q$, respectively. Suppose that ${\lim }_{n⟶\infty }{p}_n-p/q_n-q=l$ exists. Then,

(i) $p_n$ converges faster than $q_n$ if $l=0$

(ii) $p_n$ and $q_n$ converges with identical rates if $0<l<\infty$

Definition 3 (See [3]).

Let $p_n$ and $q_n$ be two real sequences converging to the same fixed point $t$. If $u_n$ and $v_n$ are two sequences of positive real numbers converging to 0 such that $p_n-t\le u_n$ and $q_n-t\le v_n$ for all $n\in \mathbb{N}$. Then, $p_n$ converges to $t$ faster than $q_n$ if $u_n$ converges faster than $v_n$.

Lemma 3 (see [4]).

Let ${\phi }_n$ and ${\psi }_n$ be nonnegative sequences of real numbers satisfying$$\begin{array}{}\text{(10)}& {\phi }_{n+1}\le \sigma {\phi }_n+{\psi }_n,\hspace{1em}\forall n\in \mathbb{N},\end{array}$$where $\sigma \in \mathrm{0,1}$ and ${\lim }_{n⟶\infty }{\psi }_n=0$. Then ${\lim }_{n⟶\infty }{\phi }_n=0$.

Lemma 4 (see [31]).

Let ${\varphi }_n,{\phi }_n$, and ${\psi }_n$ be nonnegative sequences of real numbers satisfying$$\begin{array}{}\text{(11)}& {\varphi }_{n+1}\le 1-{\phi }_n{\varphi }_n+{\psi }_n,\hspace{1em}\forall n\in \mathbb{N},\end{array}$$where ${\phi }_n\in \mathrm{0,1},{\displaystyle {\sum }_{n=1}^{\infty }{\phi }_n}=\infty$, and ${\psi }_n=o{\phi }_n$. Then, ${\lim }_{n⟶\infty }{\varphi }_n=0$.

Mann, Ishikawa, and Halpern iterative methods are fundamental tools for solving fixed-point problems of nonexpansive mappings. In recent past, a number of fixed point iterative methods have been constructed and implemented to solve various classes of nonlinear mappings [2, 9, 10, 19, 22, 25, 28–30, 34]. Agarwal and others [1] introduced the $S$-iteration method which converges faster than some well-known iterative algorithms such as Mann, Ishikawa, and Picard for contraction as well as nonexpansive mappings. Due to the super convergence rate, it attracted number of researchers to study fixed-point problems, minimization problems, variational inclusions, variational inequalities, and alternate points problems in different settings. In [18], Noor utilized formulation (9) to propose following iterative algorithm:$$\begin{array}{}\text{(12)}& \begin{array}{c}{u}_0\in \mathcal{C},\hfill \\ u_{n+1}=1-a_n{u}_n+a_{n}T{u}_n-g{u}_n+{\mathrm{p}}_{\mathcal{C}}g{u}_n-\rho f{u}_n,\hfill \end{array}\end{array}$$where $a_n$ is a sequence in $\mathrm{0,1}$. The author proved strong convergence of the proposed iterative algorithm. Furthermore, it is customary that the normal $S$-iterative algorithm converges faster than the Mann and Picard iterative algorithm. Owing to its uncomplicated nature and faster convergence rate, Gursoy and others [14] investigated the following normal $S$-iterative algorithm to examine (1) as follows:$$\begin{array}{}\text{(13)}& \begin{array}{c}{p}_0\in \mathcal{C},\hfill \\ p_{n+1}=T{q}_n-g{q}_n+{\mathrm{p}}_{\mathcal{C}}g{q}_n-\rho f{q}_n,\hfill \\ q_n=1-{\xi }_n{p}_n+{\xi }_{n}T{p}_n-g{p}_n+{\mathrm{p}}_{\mathcal{C}}g{p}_n-\rho f{p}_n,{\xi }_n\in \mathrm{0,1}.\hfill \end{array}\end{array}$$

Recently, Ullah and Arshad [27] introduced a more efficient iterative algorithm called the $M$-iterative method for Suzuki’s generalized nonexpansive mappings as follows:$$\begin{array}{}\text{(14)}& \begin{array}{c}{u}_1\in \mathcal{C},\hfill \\ w_n=1-a_n{u}_n+a_{n}T{u}_n,\hfill \\ v_n=T{w}_n,\hfill \\ u_{n+1}=T{v}_n,\hfill \end{array}\end{array}$$where $a_n$ is a sequence in $\mathrm{0,1}$. They analyzed convergence and showed that their iterative procedure converges faster than the Picard $S$ [13] and $S$-iteration process [1]. In recent work, Garodia and Uddin [11] developed a new iterative algorithm for Suzuki’s generalized nonexpansive mappings as follows:$$\begin{array}{}\text{(15)}& \begin{array}{c}{u}_1\in \mathcal{C},\hfill \\ w_n=T{u}_n,\hfill \\ v_n=T1-a_n{w}_n+a_{n}T{w}_n,\hfill \\ u_{n+1}=T{v}_n,\hfill \end{array}\end{array}$$where $a_n$ is a sequence in $\mathrm{0,1}$. The authors approximated fixed-points and inspected the convergence. Also, they proved that the posed iterative method converges with faster rate than that of the $M$-iterative method.

Stimulated by the work discussed in above-mentioned references, in this study, we investigate algorithm (15) to estimate the common solution of fixed points of a nonexpansive mapping $T$ and the generalized nonlinear variational inequality (1) as follows:$$\begin{array}{}\text{(16)}& \begin{array}{c}{u}_1\in \mathcal{C},\hfill \\ w_n=T{u}_n-g{u}_n+{\mathrm{p}}_{\mathcal{C}}g{u}_n-\rho f{u}_n,\hfill \\ r_n=1-a_n{w}_n+a_{n}T{w}_n-g{w}_n+{\mathrm{p}}_{\mathcal{C}}g{w}_n-\rho f{w}_n,\hfill \\ v_n=T{r}_n-g{r}_n+{\mathrm{p}}_{\mathcal{C}}g{r}_n-\rho f{r}_n,\hfill \\ u_{n+1}=T{v}_n-g{v}_n+{\mathrm{p}}_{\mathcal{C}}g{v}_n-\rho f{v}_n,\hfill \end{array}\end{array}$$where $a_n$ is a sequence in $\mathrm{0,1}$ satisfying certain assumptions. We analyze strong convergence of our proposed iterative algorithm (16) under some mild assumptions. We also pose a modified form of our iterative algorithm (16) to investigate convex optimization and split feasibility problems. Theoretical findings are validated by an illustrative numerical example. Our existence and convergence results can be seen as generalizations and prevalent of some known results.

2. Convergence Results

Theorem 1.

Let $f,g:\mathcal{C}⟶\mathrm{\mathscr{H}}$ be $a_1$, $a_2$-inverse strongly monotone mappings, respectively, and $T:\mathrm{\mathscr{H}}⟶\mathcal{C}$ be a nonexpansive mapping such that $FT\cap \text{Sol}\mathcal{C},f,g\ne \varphi$. Suppose that the assumption$$\begin{array}{}\text{(17)}& \rho -a_1<a_{1}1-{\rm Y},\end{array}$$holds, where $\mathrm{{\rm Y}}=2a_2-1/a_2$. Then, the iterative sequence $u_n$ defined by (16) converges strongly to $u^{\ast }\in FT\cap \text{Sol}\mathcal{C},f,g$ with the following error estimates:$$\begin{array}{}\text{(18)}& u_{n+1}-u^{\ast }\le {\zeta }^{3n+1}{u}_0-u^{\ast }{\displaystyle \prod _{k=0}^{n}1-a_{k}1-\zeta },\hspace{1em}\forall n\in \mathbb{N},\end{array}$$where$$\begin{array}{}\text{(19)}& \zeta =2\frac{a_2-1}{a_2}+\frac{a_1-\rho }{a_1}.\end{array}$$

Proof.

Note that$$\begin{array}{}\text{(20)}& u^{\ast }=T{u}^{\ast }-g{u}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g{u}^{\ast }-\rho f{u}^{\ast }.\end{array}$$

Since $f$ being $a_1$-inverse strongly monotone is $1/a_1$-Lipschitz continuous mapping, $T$ and ${\mathrm{p}}_{\mathcal{C}}$ are the nonexpansive mappings. Then, from (16 and 20), we obtain$$\begin{array}{c}\text{(21)}& w_n-u^{\ast }=T{u}_n-g{u}_n+{\mathrm{p}}_{\mathcal{C}}g{u}_n-\rho f{u}_n-T{u}^{\ast }-g{u}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g{u}^{\ast }-\rho f{u}^{\ast }\\ \le 2u_n-u^{\ast }-g{u}_n-g{u}^{\ast }+u_n-u^{\ast }-\rho f{u}_n-f{u}^{\ast }.\end{array}$$

Since $f$ is $a_1$-inverse strongly monotone mapping, then we have$$\begin{array}{c}\text{(22)}& {u_n-u^{\ast }-\rho f{u}_n-f{u}^{\ast }}^2={u_n-u^{\ast }}^2+{\rho }^2{f{u}_n-f{u}^{\ast }}^2-2\rho u_n-u^{\ast },f{u}_n-f{u}^{\ast }\\ \le {u_n-u^{\ast }}^2+\frac{{\rho }^2}{a_1^2}{u_n-u^{\ast }}^2-2\rho a_1{f{u}_n-f{u}^{\ast }}^2\\ \le {\frac{a_1-\rho }{a_1}}^2{u_n-u^{\ast }}^2.\end{array}$$

Also, $g$ is $a_2$-inverse strongly monotone mapping; then we have$$\begin{array}{c}\text{(23)}& {u_n-u^{\ast }-g{u}_n-g{u}^{\ast }}^2={u_n-u^{\ast }}^2+{g{u}_n-g{u}^{\ast }}^2-2u_n-u^{\ast },g{u}_n-g{u}^{\ast }\\ \le {u_n-u^{\ast }}^2+\frac{1}{a_2^2}{u_n-u^{\ast }}^2-2a_2{g{u}_n-g{u}^{\ast }}^2\\ \le {\frac{a_2-1}{a_2}}^2{u_n-u^{\ast }}^2.\end{array}$$

Thus, from (21) to (23), we have$$\begin{array}{}\text{(24)}& w_n-u^{\ast }\le 2\frac{a_2-1}{a_2}+\frac{a_1-\rho }{a_1}{u}_n-u^{\ast }=\zeta u_n-u^{\ast },\end{array}$$where $\zeta$ is defined by (19), and from (17), we have $\zeta <1$. Again, following the same steps (21)–(24) and from (16), we obtain$$\begin{array}{}\text{(25)}& v_n-u^{\ast }\le \zeta r_n-u^{\ast }.\end{array}$$

Next, we estimate$$\begin{array}{c}\text{(26)}& r_n-u^{\ast }=1-a_n{w}_n+a_{n}T{w}_n-g{w}_n+{\mathrm{p}}_{\mathcal{C}}g{w}_n-\rho f{w}_n-u^{\ast }\le 1-a_n{w}_n-u^{\ast }+a_n\zeta w_n-u^{\ast }=1-a_{n}1-\zeta w_n-u^{\ast },\end{array}$$which amounts to say$$\begin{array}{c}\text{(27)}& v_n-u^{\ast }\le \zeta 1-a_{n}1-\zeta w_n-u^{\ast },u_{n+1}-u^{\ast }=T{v}_n-g{v}_n+{\mathrm{p}}_{\mathcal{C}}g{v}_n-\rho f{v}_n-T{u}^{\ast }-g{u}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g{u}^{\ast }-\rho f{u}^{\ast }\\ \le \zeta v_n-u^{\ast }\le {\zeta }^{3}1-a_{n}1-\zeta u_n-u^{\ast }.\end{array}$$

Since, $1-a_{n}1-\zeta <1$. Therefore, we get $u_{n+1}-u^{\ast }\le {\zeta }^3{u}_n-u^{\ast },\forall n\in \mathbb{N}$. By repeating the process in this fashion, we obtain$$\begin{array}{}\text{(28)}& u_{n+1}-u^{\ast }\le {\zeta }^{3n+1}{u}_0-u^{\ast },\hspace{1em}\forall n\in \mathbb{N},\end{array}$$which gives that ${\lim }_{n⟶\infty }{u}_n-u^{\ast }=0$.

Now, we exemplify the existence of solution.

Example 1.

Let $\mathrm{\mathscr{H}}=ℝ,\mathcal{C}=\mathrm{1,2}$ be equipped with norm $u=u$ and inner product $u,v=u.v$. Let $f,g,T:\mathrm{1,2}⟶ℝ$ be defined by$$\begin{array}{}\text{(29)}& fu=u^2,gu=\frac{u^3}{4}+\frac{3}{4},Tu=\frac{u^2+u^3}{16}+\frac{7}{8}.\end{array}$$

Then, for all $u,v\in \mathcal{C}$, observe that$$\begin{array}{c}\text{(30)}& fu-fv,u-v={u-v}^{2}u+v\ge 2{u-v}^2,gu-gv,u-v=\frac{1}{4}{u-v}^2{u}^2+uv+v^2\ge \frac{3}{4}{u-v}^2,\\ Tu-Tv=\frac{1}{16}u-v{u}^2+uv+v^2+u+v\le u-v.\end{array}$$

Then, $f$ and $g$ are 2 and $3/4$-inverse strongly monotone mapping, respectively, and $T$ is nonexpansive mapping. One can easily verify that $u^{\ast }=1\in \mathcal{C}$ is the unique fixed point of $T$. Also,$$\begin{array}{}\text{(31)}& f{u}^{\ast },gv-g{u}^{\ast }=\frac{v^3-1}{4}\ge 0,\hspace{1em}\text{for\hspace{0.17em}all\hspace{0.17em}}v\in \mathcal{C}.\end{array}$$

Thus, we have $u^{\ast }=1\in FT\cap \text{Sol}\mathcal{C},f,g$.

Theorem 2.

Let $\mathrm{\mathscr{H}}$ be a real Hilbert space and $\mathcal{C}$ be a nonempty closed convex subset of $\mathrm{\mathscr{H}}$. Let $f,g,T$, and $\zeta$ be same as defined in Theorem 1. Let $p_n$ and $u_n$ be the sequences defined by (13) and (16), respectively. Suppose that (17) holds and $FT\cap \mathcal{C},f,g\ne \varphi$. Then, the following statements hold:

(i) If $1+{\zeta }^3/{\xi }_n$ is bounded and ${\displaystyle {\sum }_{n=0}^{\infty }{a}_n}=\infty$, then the sequence $u_n-p_n$ converges strongly to 0 with following error estimates:$$\begin{array}{}\text{(32)}& u_{n+1}-p_{n+1}\le 1-{\xi }_{n}1-\zeta u_n-p_n+1+{\zeta }^3{u}_n-u^{\ast },\hspace{1em}\forall n\in \mathbb{N}.\end{array}$$

  $p_n$ converges strongly to $u^{\ast }\in FT\cap \text{Sol}\mathcal{C},f,g$.

(ii) If $p_n$ converges strongly to $u^{\ast }\in FT\cap \text{Sol}\mathcal{C},f,g$, then $p_n-u_n$ converges strongly to 0 with following error estimates:

$$\begin{array}{}\text{(33)}& p_{n+1}-u_{n+1}\le {\zeta }^3{p}_n-u_n+1+{\zeta }^3{p}_n-u^{\ast },\hspace{1em}\forall n\in \mathbb{N}.\end{array}$$

Proof.

(i) It follows from Theorem 1 that ${\lim }_{n⟶\infty }{u}_n-u^{\ast }=0$. Next, we prove that ${\text{lim}}_{n⟶\infty }{p}_n-u^{\ast }=0$. Following (13) and (16) and the steps as in (21)–(24), we obtain$$\begin{array}{c}\text{(34)}& u_{n+1}-p_{n+1}=T{v}_n-g{v}_n+{\mathrm{p}}_{\mathcal{C}}g{v}_n-\rho f{v}_n-T{q}_n-g{q}_n+{\mathrm{p}}_{\mathcal{C}}g{q}_n-\rho f{q}_n\\ \le \zeta v_n-q_n,\end{array}$$

  where $\zeta$ is same as in (19). Again, utilizing (13), (16), and (34), we have$$\begin{array}{c}\text{(35)}& u_{n+1}-p_{n+1}\le \zeta v_n-1-{\xi }_n{p}_n-{\xi }_{n}T{p}_n-g{p}_n+{\mathrm{p}}_{\mathcal{C}}g{p}_n-\rho f{p}_n\le \zeta v_n-u^{\ast }+1-{\xi }_n\zeta p_n-u^{\ast }+{\xi }_n\zeta T{p}_n-g{p}_n\\ +{\mathrm{p}}_{\mathcal{C}}g{p}_n-\rho f{p}_n-T{u}^{\ast }-g{u}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g{u}^{\ast }-\rho f{u}^{\ast }\\ \le \zeta v_n-u^{\ast }+1-{\xi }_n\zeta p_n-u^{\ast }+{\xi }_n{\zeta }^2{p}_n-u^{\ast }\\ =\zeta v_n-u^{\ast }+1-{\xi }_{n}1-\zeta p_n-u^{\ast }\\ \le \zeta {\zeta }^{2}1-a_{n}1-\zeta u_n-u^{\ast }+1-{\xi }_{n}1-\zeta p_n-u^{\ast }\\ =\zeta {\zeta }^{2}1-a_{n}1-\zeta u_n-u^{\ast }+1-{\xi }_{n}1-\zeta u_n-u^{\ast }\\ +1-{\xi }_{n}1-\zeta u_n-p_n\\ \le \zeta 1-{\xi }_{n}1-\zeta u_n-p_n+1+{\zeta }^3\max 1-a_{n}1-\zeta ,\\ 1-{\xi }_{n}1-\zeta u_n-u^{\ast }\\ \le 1-{\xi }_{n}1-\zeta u_n-p_n+1+{\zeta }^3{u}_n-u^{\ast }.\end{array}$$

  Let ${\varphi }_n=u_n-p_n,{\phi }_n={\xi }_{n}1-\zeta ,{\psi }_n=1+{\zeta }^3{u}_n-u^{\ast }$, and ${\delta }_n=u_n-u^{\ast },\forall n\in \mathbb{N}$. It follows from assumption of the theorem that $1+{\zeta }^3/{\xi }_n$ is bounded; therefore, $1+{\zeta }^3/{\xi }_{n}1-\zeta$ is also bounded. Then, there exists a constant $M>0$, such that $1+{\zeta }^3/{\xi }_{n}1-\zeta <M,\forall n\in \mathbb{N}$. Since ${\lim }_{n⟶\infty }{\delta }_n=0$ and $1+{\zeta }^3/{\xi }_{n}1-\zeta$ is bounded, therefore, $1+{\zeta }^3/{\xi }_{n}1-\zeta {\delta }_n⟶0$ as $n⟶\infty$, i.e., ${\text{lim}}_{n⟶\infty }{\psi }_n/{\phi }_n=0$, which amounts to say that ${\psi }_n=o{\phi }_n$. Thus, all the assumptions of Lemma 4 are fulfilled. Hence, ${\lim }_{n⟶\infty }{u}_n-p_n=0$ and $p_n-u^{\ast }\le u_n-p_n+u_n-u^{\ast }$. Thus, we have ${\lim }_{n⟶\infty }{p}_n-u^{\ast }=0$.

(ii) Next, we estimate that $p_n-u_n⟶0$. Since $p_n$ converges to $u^{\ast }\in FT\cap \text{Sol}\mathcal{C},f,g$, then following the same arguments as in (34) and (35), we obtain

$$\begin{array}{c}\text{(36)}& p_{n+1}-u_{n+1}\le \zeta 1-{\xi }_{n}1-\zeta p_n-u^{\ast }+\zeta {\zeta }^{2}1-a_{n}1-\zeta u_n-u^{\ast }\le {\zeta }^{3}1-a_{n}1-\zeta p_n-u_n+{\zeta }^{3}1-a_{n}1-\zeta p_n-u^{\ast }\\ +\zeta 1-{\xi }_{n}1-\zeta p_n-u^{\ast }\\ \le {\zeta }^{3}1-a_{n}1-\zeta p_n-u_n+{\zeta }^3{p}_n-u^{\ast }+p_n-u^{\ast }\\ \le {\zeta }^3{p}_n-u_n+1+{\zeta }^3{p}_n-u^{\ast }.\end{array}$$

Let ${\varphi }_n^{\prime }=p_n-u_n,{\psi }_n^{\prime }=1+{\zeta }^3{p}_n-u^{\ast },\forall n\in \mathbb{N}$. By the assumption $p_n$ converges to $u^{\ast }$ and utilizing the fact that $1+{\zeta }^3$ is bounded, we obtain that ${\psi }_n^{\prime }⟶0$ as $n⟶\infty$. Thus, all the assumptions of Lemma 3 are fulfilled. Hence, ${\text{lim}}_{n⟶\infty }{p}_n-u_n=0$. Also, we know that $u_n-u^{\ast }\le p_n-u_n+p_n-u^{\ast },\forall n\in \mathbb{N}$. Thus, ${\text{lim}}_{n⟶\infty }{u}_n-u^{\ast }=0$. Hence, $p_n-u_n⟶0$ as $n⟶\infty$.

Theorem 3.

Let $\mathrm{\mathscr{H}}$ be a real Hilbert space and $\mathcal{C}$ be a closed convex subset of $\mathrm{\mathscr{H}}$. Suppose $f,g,T$, and $\zeta$ are identical as in Theorem 1. Let $p_n$ and $u_n$ be sequences defined by (13) and (16), respectively. Suppose that assumption (17) holds and $FT\cap \text{Sol}\mathcal{C},f,g\ne \varphi$. If $p_0=u_0$, then $u_n$ converges faster than $p_n$ to $u^{\ast }$, such that $u^{\ast }\in FT\cap \text{Sol}\mathcal{C},f,g$.

Proof.

It follows from (27) that$$\begin{array}{}\text{(37)}& u_{n+1}-u^{\ast }\le {\zeta }^{3}1-a_{n}1-\zeta u_n-u^{\ast }.\end{array}$$

Since $a_n$ is a sequence in $\mathrm{0,1}$, we can choose a constant $a\in ℝ$, such that $0<a\le a_n<1,\forall n\in \mathbb{N}$. Then,$$\begin{array}{}\text{(38)}& u_{n+1}-u^{\ast }\le {\zeta }^{3}1-a1-\zeta u_n-u^{\ast }.\end{array}$$

By repeating the process, we obtain$$\begin{array}{}\text{(39)}& u_{n+1}-u^{\ast }\le {\zeta }^{3n+1}{1-a1-\zeta }^{n+1}{u}_0-u^{\ast },\hspace{1em}\forall n\in \mathbb{N}.\end{array}$$

Also, it follows from (13) that$$\begin{array}{c}\text{(40)}& p_{n+1}-u^{\ast }=T{q}_n-g{q}_n+{\mathrm{p}}_{\mathcal{C}}g{q}_n-\rho f{q}_n-T{u}^{\ast }-g{u}^{\ast }+{\mathrm{p}}_{\mathcal{C}}g{u}^{\ast }-\rho f{u}^{\ast }\le q_n-u^{\ast }-g{q}_n-g{u}^{\ast }+g{q}_n-g{u}^{\ast }-\rho f{q}_n-f{u}^{\ast }\\ \le 2q_n-u^{\ast }-g{q}_n-g{u}^{\ast }+q_n-u^{\ast }-\rho f{q}_n-f{u}^{\ast }.\end{array}$$

By following the arguments as discussed from (21)to(24), we have$$\begin{array}{}\text{(41)}& p_{n+1}-u^{\ast }\le \zeta q_n-u^{\ast }.\end{array}$$

Also,$$\begin{array}{c}\text{(42)}& q_n-u^{\ast }=1-{\xi }_n{p}_n+{\xi }_{n}T{p}_n-g{p}_n+{\mathrm{p}}_{\mathcal{C}}g{p}_n-\rho f{p}_n-u^{\ast }\le 1-{\xi }_n{p}_n-u^{\ast }+2{\xi }_n{p}_n-u^{\ast }-g{p}_n-g{u}^{\ast }\\ +{\xi }_n{p}_n-u^{\ast }-\rho f{p}_n-f{u}^{\ast }\\ \le 1-{\xi }_n{p}_n-u^{\ast }+{\xi }_n\zeta p_n-u^{\ast }\\ =1-{\xi }_{n}1-\zeta p_n-u^{\ast }.\end{array}$$

By combining (41) and (42), we get$$\begin{array}{}\text{(43)}& p_{n+1}-u^{\ast }\le \zeta 1-{\xi }_{n}1-\zeta p_n-u^{\ast }.\end{array}$$

Since ${\xi }_n$ is a sequence in $\mathrm{0,1}$, we can choose a constant $\xi \in ℝ$, such that $0<\xi \le {\xi }_n<1,\forall n\in \mathbb{N}$. Then,$$\begin{array}{}\text{(44)}& p_{n+1}-u^{\ast }\le \zeta 1-\xi 1-\zeta p_n-u^{\ast }.\end{array}$$

Thus, by repeating the process, we obtain$$\begin{array}{}\text{(45)}& p_{n+1}-u^{\ast }\le {\zeta }^{n+1}{1-\xi 1-\zeta }^{n+1}{p}_0-u^{\ast },\hspace{1em}\forall n\in \mathbb{N},\end{array}$$

Set $a_n={\zeta }^{3n+1}{1-a1-\zeta }^{n+1}{u}_0-u^{\ast }$ and $b_n={\zeta }^{n+1}{1-\xi 1-\zeta }^{n+1}{p}_0-u^{\ast }$; then,$$\begin{array}{c}\text{(46)}& A_n=\frac{a_n}{b_n}=\frac{{\zeta }^{3n+1}{1-a1-\zeta }^{n+1}{u}_0-u^{\ast }}{{\zeta }^{n+1}{1-\xi 1-\zeta }^{n+1}{p}_0-u^{\ast }}⟶0\text{as\hspace{0.17em}}n⟶\infty .\end{array}$$

Hence, $u_n$ converges faster that $p_n$.

3. Applications

3.1. Convex Minimization Problem

Now, we solve convex minimization problem as an application of Theorem 1.

Let $\mathcal{C}$ be a closed convex subset of a real Hilbert space $\mathrm{\mathscr{H}}$, ${\mathrm{\rho }}_{\mathcal{C}}:\mathrm{\mathscr{H}}⟶\mathcal{C}$ be a projection, and $F:\mathcal{C}⟶ℝ$ be a convex, Frechet differentiable mapping. We consider the following convex minimization problem:$$\begin{array}{}\text{(47)}& \underset{u^{\ast }\in \mathcal{C}}{\min }F{u}^{\ast }.\end{array}$$

Clearly, $u^{\ast }\in \mathcal{C}$ is a solution of ${\mathrm{p}}_{\mathcal{C}}I-\rho \nabla F$ if and only if$$\begin{array}{}\text{(48)}& \nabla F{u}^{\ast },u-u^{\ast }\ge 0,\hspace{1em}\forall u\in \mathcal{C}.\end{array}$$

More precisely, $u^{\ast }\in \mathcal{C}$ solves problem (47) if and only if $u^{\ast }$ is a fixed point of the projection mapping ${\mathrm{p}}_{\mathcal{C}}I-\rho \nabla F$, i e.,$$\begin{array}{}\text{(49)}& u^{\ast }={\mathrm{\rho }}_{\mathcal{C}}{u}^{\ast }-\rho \nabla F{u}^{\ast },\end{array}$$where $\nabla F$ is the gradient of mapping $F$. This formulation is known as gradient projection, which plays a key role in solving problem (47). So far, several iterative methods have been employed to solve minimization problems [7, 26, 32]. By considering $f\text{:}=\nabla F$ and assuming $T=g=I$, the identity mapping, we propose the following modified gradient projection algorithm for solving ${\mathrm{p}}_{\mathcal{C}}I-\rho \nabla F$ as follows:$$\begin{array}{}\text{(50)}& \begin{array}{c}{u}_1\in \mathcal{C},\hfill \\ w_n={\mathrm{p}}_{\mathcal{C}}{u}_n-\rho \nabla F{u}_n,\hfill \\ r_n=1-a_n{w}_n+a_n{\mathrm{p}}_{\mathcal{C}}{w}_n-\rho \nabla F{w}_n,\hfill \\ v_n={\mathrm{p}}_{\mathcal{C}}{r}_n-\rho \nabla F{r}_n,\hfill \\ u_{n+1}={\mathrm{p}}_{\mathcal{C}}{v}_n-\rho \nabla F{v}_n,\hfill \end{array}\end{array}$$where $a_n$ is a sequence in $\mathrm{0,1}$. Now, we approximate the proposed algorithm (50) to estimate the solution of (47).

Theorem 4.

Let $\mathcal{C}$ be a nonempty closed convex subset of real Hilbert space $\mathrm{\mathscr{H}}$. Let $F:\mathcal{C}⟶ℝ$ be a convex, Freschet differentiable mapping, and $\nabla F$ is $a$-inverse strongly monotone mapping. Suppose that the convex minimization problem (47) has a solution and condition (17) holds. Then, the sequence $u_n$ generated by (50) converges strongly to $u^{\ast }$ which solves convex minimization problem (47) with the following error estimates:$$\begin{array}{}\text{(51)}& u_{n+1}-u^{\ast }\le {\zeta }^{3n+1}{\displaystyle \prod _{k=0}^{n}1-a_{k}1-\zeta u_0-u^{\ast }},\hspace{1em}\forall n\in \mathbb{N},\end{array}$$where$$\begin{array}{}\text{(52)}& \zeta =\frac{a_1-\rho }{a_1}.\end{array}$$

Proof.

The desired conclusion is accomplished by taking $f=\nabla F$ and $T,g=I$ in Theorem 1.

Example 2.

Let $\mathrm{\mathscr{H}}=L^2\mathrm{0,1}=\mathcal{G}:\mathrm{0,1}⟶ℝ:{\displaystyle {\int }_0^1{\mathcal{G}}^{2}u\text{d}u}<\infty$. Then, $\mathrm{\mathscr{H}},{\cdot }_2$ is a Hilbert space given by$$\begin{array}{}\text{(53)}& {\mathcal{G}u}_2^2=\mathcal{G}u,\mathcal{G}u={\displaystyle {\int }_0^1{\mathcal{G}}^{2}u\text{d}u}.\end{array}$$

Consider a closed convex subset $\mathcal{C}=\mathcal{G}\in L^2\mathrm{0,1}:{\mathcal{G}u}_2^2\le 1$ of $\mathrm{\mathscr{H}}$. Define $F:\mathcal{C}⟶ℝ$ by $F\mathcal{G}={\mathcal{G}u}_2^2$. Then, $\mathcal{G}u=0$ is a unique minimum of a convex function $f$, and $f$ is the Frechet differentiable at $\mathcal{G}$. The gradient $\nabla F:\mathcal{C}⟶\mathrm{\mathscr{H}}$ is evaluated as $\nabla F\mathcal{G}=2\mathcal{G}$. Then, for all ${\mathcal{G}}_1,{\mathcal{G}}_2\in \mathcal{C}$, we get$$\begin{array}{c}\text{(54)}& \nabla F{\mathcal{G}}_1-\nabla F{\mathcal{G}}_2,{\mathcal{G}}_1-{\mathcal{G}}_2={\displaystyle {\int }_0^{1}2{\mathcal{G}}_{1}u-2{\mathcal{G}}_{2}u{\mathcal{G}}_1-{\mathcal{G}}_2\text{d}u}=2{\displaystyle {\int }_0^1{{\mathcal{G}}_{1}u-{\mathcal{G}}_{2}u}^2\text{d}u}\\ \ge -\frac{1}{4}{\displaystyle {\int }_0^1{2{\mathcal{G}}_{1}u-2{\mathcal{G}}_{2}u}^2\text{d}u}\\ =-\frac{1}{4}{\nabla F{\mathcal{G}}_1-\nabla F{\mathcal{G}}_2}_2^2,\end{array}$$i.e., $\nabla F$ is $1/4$ inverse strongly monotone. Also, $\zeta <1$ for $\rho =1/4$. Thus, all the assumptions of Theorem 4 are satisfied, and for $a_n=1/n+1$, the sequence $u_n$ generated by (50) is given as$$\begin{array}{}\text{(55)}& \begin{array}{c}{u}_0\in \mathcal{C},\hfill \\ w_n={\mathrm{p}}_{\mathcal{C}}\frac{1}{2}{u}_n,\hfill \\ r_n=1-\frac{1}{n+1}{w}_n+\frac{1}{n+1}{\mathrm{p}}_{\mathcal{C}}\frac{1}{2}{w}_n,\hfill \\ v_n={\mathrm{p}}_{\mathcal{C}}\frac{1}{2}{r}_n,\hfill \\ u_{n+1}={\mathrm{p}}_{\mathcal{C}}\frac{1}{2}{v}_n,\hfill \end{array}\end{array}$$where ${\mathrm{p}}_{\mathcal{C}}=\begin{array}{cc}\mathcal{G},\hfill & \mathcal{G}\in \mathcal{C},\hfill \\ \mathcal{G}/\mathcal{G},\hfill & \mathcal{G}\notin \mathcal{C}.\hfill \end{array}$ Then, the sequence $u_n$ generated by (50) converges to 0 function.

3.2. Split Feasibility Problem

This subsection is devoted to utilization of Theorem 1 to examine a split feasibility problem $\text{SFP}$. Let ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ be nonempty closed convex subsets of real Hilbert spaces ${\mathrm{\mathscr{H}}}_1$ and ${\mathrm{\mathscr{H}}}_2$, respectively. Let $\mathcal{A}:{\mathrm{\mathscr{H}}}_1⟶{\mathrm{\mathscr{H}}}_2$ be a bounded linear operator. The $\text{SFP}$ is to locate a point $u^{\ast }$, such that$$\begin{array}{}\text{(56)}& u^{\ast }\in {\mathcal{C}}_1:\mathcal{A}{u}^{\ast }\in {\mathcal{C}}_2.\end{array}$$

Let $\mathrm{\Gamma }$ denotes the solution set of SFP (56); then,$$\begin{array}{}\text{(57)}& \Gamma ≕u^{\ast }\in {\mathcal{C}}_1:\mathcal{A}{u}^{\ast }\in {\mathcal{C}}_2={\mathcal{C}}_1\cap {\mathcal{A}}^{-1}{\mathcal{C}}_2.\end{array}$$

A class of inverse problems has been solved by using $\text{SFP}$, for example, [6]. In [32], Xu established the relationship between SFP (56) and the fixed point of problem ${\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}$. More precisely, for $\rho >0$, $u^{\ast }\in {\mathcal{C}}_1$ solves SFP (56) if and only if ${\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}{u}^{\ast }=u^{\ast }$. Byrne [5] posed the following iterative algorithm for solving SFP (56) as follows:$$\begin{array}{}\text{(58)}& u_{n+1}={\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}{u}_n,\hspace{1em}\forall n\ge 0,\end{array}$$where $0<\rho <2/{\mathcal{A}}^2$, ${\mathcal{A}}^{\ast }$ is the adjoint of operator $\mathcal{A}$, and ${\mathrm{p}}_{{\mathcal{C}}_1}$ and ${\mathrm{p}}_{{\mathcal{C}}_2}$ are the projections onto ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$, respectively.

Note that the operator ${\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}$ with $0<\rho <2/{\mathcal{A}}^2$ is nonexpansive. Now, we propose following iterative algorithm to solve SFP (56):$$\begin{array}{}\text{(59)}& \begin{array}{c}{u}_1\in {\mathcal{C}}_1,\hfill \\ w_n={\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}{u}_n,\hfill \\ r_n=1-a_n{w}_n+a_n{\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}{w}_n,\hfill \\ v_n={\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}{r}_n,\hfill \\ u_{n+1}={\mathrm{p}}_{{\mathcal{C}}_1}I-\rho {\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}{v}_n,\hfill \end{array}\end{array}$$where $a_n$ is a sequence in $\mathrm{0,1}$ and $0<\rho <2/{\mathcal{A}}^2$.

Theorem 5.

Suppose that $\mathrm{\Gamma }\ne \varphi$ and condition (17) holds. Then, the sequence $u_n$ initiated in (59) converges weakly to $u^{\ast }$, which solves SFP (56) with following error estimates:$$\begin{array}{}\text{(60)}& u_{n+1}-u^{\ast }\le {\zeta }^{3n+1}{\displaystyle \prod _{k=0}^{n}1-a_{k}1-\zeta u_0-u^{\ast }},\hspace{1em}\forall n\in \mathbb{N},\end{array}$$where$$\begin{array}{}\text{(61)}& \zeta =\frac{a_1-\rho }{a_1}.\end{array}$$

Proof.

The desired conclusion follows by taking $\nabla F={\mathcal{A}}^{\ast }I-{\mathrm{p}}_{{\mathcal{C}}_2}\mathcal{A}$ and $T,g=I$ in Theorem 1.

4. Conclusion

In this study, a new iterative algorithm (16) has been proposed and employed to explore convergence analysis. Using this newly constructed iterative procedure, a common solution of the generalized variational inequality problem and fixed points of nonexpansive mapping is investigated, and theoretical findings are verified by a numerical example. Furthermore, we have shown that our iteration algorithm converges faster than the normal $S$-iteration process for contraction mapping. Finally, we applied our newly constructed iterative algorithm to investigate the convex optimization problem and split feasibility problem.

Acknowledgments

The first and fourth authors would like to thank the Deanship of Scientific Research, Prince Sattam bin Abdulaziz University, for supporting this work.

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