Implicit Extragradient-Like Method for Fixed Point Problems and Variational Inclusion Problems in a Banach Space

Throughout this article, one always supposes that H is a real infinite dimensional Hilbert space endowed with norm $\parallel ·\parallel$ and inner product $\langle ·,·\rangle$. Let the nonempty subset $C\subset H$ be convex and closed, and let the mapping $P_C$ be the nearest point (metric) projection of H onto C. Let $A:C\to H$ be a nonself mapping. The celebrated variational inequality problem (VIP) is to find an element $x^{*}\in C$ s.t. $\langle y-x^{*},A{x}^{*}\rangle \ge 0$, for all $y\in C$. One denotes by VI($C,A$) the set of all solutions of the VIP. The VIP acts as a unified framework for lots of real industrial and applied problems, such as, machine learning, transportation, image processing and economics; see, e.g., [1,2,3,4,5,6,7]. One knows that Korpelevich’s extragradient algorithm [8] is now one of the most popular algorithms to numerically solve the VIP. It reads as follows: with $\ell \in (0,\frac{1}{L})$, where L is the Lipschitz constant of the mapping A. If its solution set is not empty, it was obtained that sequence $\{x_j\}$ is weakly convergent. The price for the weak convergence is that A must be a Lipschitz and monotone mapping. To date, now, the extragradient method and relaxed extragradient methods have received much attention and has been studied extensively; see, e.g., [9,10,11,12,13]. Next, one assumes that B is monotone set-valued mapping defined on H and A is a monotone single-valued mapping defined on H, respectively. The so-called variational inclusion problem is to find an element $x^{*}\in H$ s.t. $0\in (A+B)x^{*}$ and it has recently been studied by many authors based on splitting-based approximation methods; see, e.g., [14,15,16,17]. This model provides a unified framework for a lot of theoretical and practical problems and was investigated via different methods [18,19,20,21,22]. To the best of the authors’ knowledge, there are no few associated results obtained in Banach spaces. Next, one will turn our attention to Banach spaces.

Next, $E^{*}$ will be used to present be the dual space of Banach space E. Let $\varnothing \ne C\subset E$ be a convex and closed set. Given a nonlinear mapping T on C, one uses the symbol $\mathrm{Fix}(T)$ to denote the set of all the fixed points of T. Recall that the mapping T is said to be a Lipschitz mapping if and only if, $\forall x,y\in C$, $\parallel Tx-Ty\parallel \le L\parallel x-y\parallel$. If the Lipschitz constant L is just one, one say that T is a nonexpansive mapping whose complementary mappings are monotone.

Recall that the duality mapping $J_q:E\to 2^{E^{*}}$ is defined by For a particular case, one uses J to stand for $J_2$, which is commonly called the normalized duality. Recall that a mapping T defined on C is said to be a strict pseudocontraction of order q if, $\forall x,y\in C$, there is $j_q(x-y)\in J_q(x-y)$ such that the following inequality holds $\langle Tx-Ty,j_q(x-y)\rangle \le {\parallel x-y\parallel }^q-\varsigma {\parallel (I-T)x-(I-T)y\parallel }^q$ for some $\varsigma >0.$

The convexity modulus of space E, ${\delta }_E$, which maps the interval $(0,2]$ to the interval $[0,1]$, is defined as follows The smoothness modulus of space E, ${\rho }_E$, which maps the interval $[0,\infty )$ to the interval $[0,\infty )$, is defined as follows Recall that a space E is said to be uniformly convex if ${\delta }_E(e)>0$, $\forall e\in (0,2]$. Recall that a space E is said to be uniformly smooth if ${lim}_{\tau \to 0^{+}}\frac{{\rho }_E(\tau )}{\tau }=0$. Further it is said to be q-uniformly ($q>1$) smooth if $\exists c>0$ s.t. $c\ge s^{-q}{\rho }_E(s)$, $\forall s>0$. In q-uniformly ($q>1$) smooth spaces, one has the following celebrated inequality, for some $j_q(x+y)\in J_q(x+y)$,

One says that an operator $\Pi :C\to D$, where D is convex and closed subset of C, is said to be a sunny mapping if, $(1-\xi )\Pi (x)+\xi x\in C$ for $\xi \ge 0$ and $x\in C$, $\Pi (x)=\Pi [\Pi (x)+\xi (x-\Pi (x))]$. If $\Pi$ is both nonexpansive and sunny, then $\langle \overline{x}-x^{\prime },J(\Pi (\overline{x})-\Pi (x^{\prime }))\rangle \ge {\parallel \Pi (\overline{x})-\Pi (x^{\prime })\parallel }^2$, $\forall \overline{x},x^{\prime }\in C$.

In the setting of q-uniformly smooth spaces, we recall that an operator B is said to be accretive if, for some $j_q(x^{\prime }-\overline{x})\in J_q(x^{\prime }-\overline{x})$, $\langle u-v,j_q(x^{\prime }-\overline{x})\rangle \ge 0,$ for all $u\in B{x}^{\prime },$ $v\in B\overline{x}$. An accretive operator B is said to be inverse-strongly of order q if, for each $x^{\prime },\overline{x}\in C$, $\exists j_q(x^{\prime }-\overline{x})\in J_q(x^{\prime }-\overline{x})$ s.t. $\langle u-v,j_q(x^{\prime }-\overline{x})\rangle \ge \alpha {\parallel u-v\parallel }^q,$ for all $u\in B{x}^{\prime },$$v\in B\overline{x}$ for some $\alpha >0$. An accretive operator B is said to be m-accretive if $(I+\lambda B)C=E$ for all $\lambda >0$. Furthermore, one can define a mapping $J_{\lambda }^B:(I+\lambda B)C\to C$ by $J_{\lambda }^B={(I+\lambda B)}^{-1}$ for each $\lambda >0$. Such $J_{\lambda }^B$ is called the resolvent of B for $\lambda >0$. In the sequel, we will use the notation $T_{\lambda }:=J_{\lambda }^B(I-\lambda A)={(I+\lambda B)}^{-1}(I-\lambda A)\forall \lambda >0$. From [9], we have $\mathrm{Fix}(T_{\lambda })={(A+B)}^{-1}0\forall \lambda >0$ and $\parallel y-T_{\lambda }y\parallel \le 2\parallel y-T_{r}y\parallel$ for $0<\lambda \le r$ and $y\in C$. From [23], one has $J_{\lambda }^{B}x=J_{\mu }^B(\frac{\mu }{\lambda }x+(1-\frac{\mu }{\lambda })J_{\lambda }^{B}x)$ for all $\lambda ,\mu >0,x\in E$ and $\mathrm{Fix}(J_{\lambda }^B)=B^{-1}0$, where $B^{-1}0=\{x\in C:0\in Bx\}$.

Let $A_1,A_2:C\to E$ and $B_1,B_2:C\to 2^E$ be nonlinear mappings with $B_{k}x\ne \varnothing \forall x\in C,k=1,2$. Consider the symmetrical system of variational inclusions, which consists of finding the pair $(x^{*},y^{*})$ in $C\times C$ s.t. where ${\zeta }_k$ is a positive constant for $k=1,2$. Ceng, Postolache and Yao [24] obtain the fact that problem (2) is equivalent to a fixed point problem.

Based on the equivalent relation, Ceng et al. [24] suggested a composite viscosity implicit rule for solving the GSVI (2) as follows: where $\mu \in (0,1)$, $S:=(1-\alpha )I+\alpha T$ with $0<\alpha <min\{1,\frac{2\lambda }{{\kappa }_2}\}$, and the sequences $\{{\alpha }_j\},\{{\delta }_j\},\{{\beta }_j\},\{{\gamma }_j\}\subset (0,1)$ are such that (i) ${\alpha }_j+{\delta }_j+{\beta }_j+{\gamma }_j=1\forall j\ge 1$; (ii) ${lim}_{j\to \infty }{\alpha }_j=0$, ${lim}_{j\to \infty }\frac{{\beta }_j}{{\alpha }_j}=0$; (iii) ${lim}_{j\to \infty }{\gamma }_j=1$; (iv) ${\sum }_{j=0}^{\infty }{\alpha }_j=\infty$. They proved that $\{x_j\}$ converges strongly to a point of $\mathrm{Fix}(G)\cap \mathrm{Fix}(T)$, which solves a certain VIP.

In this article, we introduce and investigate an implicitly composite solution method to solve the GSVI (2) with certain VIP and CFPP constraints. We then analyze convergence of the suggested method in the setting of real Hilbert spaces under some mild conditions. An application is also considered.

From now on, one always uses ${\kappa }_q>0$ to denote the smoothness coefficient; see [25,26]. One also lists some essential lemmas for the strong convergence theorem next.

The following lemmas was proved in [29], which is a simple extension of the inequality established in [26].

Throughout this section, suppose that C is a convex closed set in a Banach space E, which is both uniformly convex and q-uniformly smooth with $q\in (1,2]$. Let both $B_1:C\to 2^E$ and $B_2:C\to 2^E$ be m-accretive operators. Let $A_k:C\to E$ be single-valued ${\sigma }_k$-inverse strongly accretive mapping for each $k=1,2$. Further, one assumes that $G:C\to C$ is a self mapping defined as $G:=J_{{\zeta }_1}^{B_1}(I-{\zeta }_1{A}_1)J_{{\zeta }_2}^{B_2}(I-{\zeta }_2{A}_2)$ with constants ${\zeta }_1,{\zeta }_2>0$. Let $S_n$ be a $\varsigma$-uniformly strictly pseudocontractive mapping for each $n\ge 1$. Let $A:C\to E$ and $B:C\to 2^E$ be a $\sigma$-inverse-strongly accretive mapping of order q and an m-accretive operator, respectively. Assume that the feasibility set $\Omega :={\bigcap }_{n=0}^{\infty }\mathrm{Fix}(S_n)\cap {(A+B)}^{-1}0\cap \mathrm{Fix}(G)$ is nonempty.

Algorithm 1: Composite extragradient implicit method for the GSVI (2) with VIP and CFPP constraints.

Initial Step. Given $\xi \in (0,1),\alpha \in (0,min\{1,{(\frac{\varsigma q}{{\kappa }_q})}^{\frac{1}{q-1}}\})$. Let $x_0\in C$ be an arbitrary initial.  Iteration Steps. Compute $x_{n+1}$ from the current $x_n$ as follows:  Step 1. Calculate $$\left\{\begin{array}{c}{w}_n=s_n((1-\xi )x_n+\xi G{w}_n)+(1-s_n)((1-\alpha )x_n+\alpha S_n{x}_n),\hfill \\ u_n=J_{{\zeta }_1}^{B_1}(I-{\zeta }_1{A}_1)J_{{\zeta }_2}^{B_2}(w_n-{\zeta }_2{A}_2{w}_n);\hfill \end{array}\right.$$  Step 2. Calculate $y_n=J_{{\lambda }_n}^B(u_n-{\lambda }_{n}A{u}_n)$;  Step 3. Calculate $z_n=J_{{\lambda }_n}^B(u_n-{\lambda }_{n}A{y}_n+r_n(y_n-u_n))$;  Step 4. Calculate $x_{n+1}={\alpha }_{n}u+{\beta }_n{u}_n+{\gamma }_n((1-\alpha )z_n+\alpha S_n{z}_n)$, where u is a fixed element in C, $\{r_n\},\{s_n\},\{{\alpha }_n\}$, $\{{\beta }_n\},\{{\gamma }_n\}\subset (0,1]$ with $\{{\lambda }_n\}\subset (0,\infty )$ and ${\alpha }_n+{\beta }_n+{\gamma }_n=1$.  Set $n:=n+1$ and go to Step 1.

We next give two possible cases.

Case 1. We assume that there is an integer $n_0\ge 1$ with the restriction that $\{{\Gamma }_n\}$ is non-increasing. From (4), we get ${\eta }_n\le {\Gamma }_n-{\alpha }_n{\Gamma }_n-{\Gamma }_{n+1}+{\delta }_n.$ From ${lim}_{n\to \infty }{\alpha }_n={lim}_{n\to \infty }{\delta }_n=0$, one sees that ${lim}_{n\to \infty }{\eta }_n=0$. It is easy to see from Lemma 4 that ${lim}_{n\to \infty }g(\parallel u_n-S_{\alpha ,n}{z}_n\parallel )=0$, and From the fact that g is a strictly increasing, continuous and convex function with $g(0)=0$, one has By use of Lemma 5, we get where ${\tilde{h}}_1$ is a convex, strictly increasing, continuous function as in Lemma 5. This hence entails In a similar way, one concludes which hence entails Note that which leads us to From (8), one has which immediately yields that Since ${\tilde{h}}_1(0)={\tilde{h}}_2(0)=h_1(0)=0$ and the fact that ${\tilde{h}}_1,{\tilde{h}}_2$ and $h_1$ are strictly increasing, continuous and convex functions, one concludes that ${lim}_{n\to \infty }\parallel w_n-v_n-x^{*}+y^{*}\parallel ={lim}_{n\to \infty }\parallel v_n-u_n+x^{*}-y^{*}\parallel ={lim}_{n\to \infty }\parallel u_n-y_n\parallel$. This immediately implies that Furthermore, borrowing $w_n=s_n((1-\xi )x_n+\xi G{w}_n)+(1-s_n)S_{\alpha ,n}{x}_n$ defined as the first step in the iteration procedure, we obtain that which yields This further implies that In a similar way, one further concludes Using (10) leads us to Hence, Note that $h_2(0)=h_3(0)={\tilde{h}}_3(0)=0$ and the fact that $h_2,h_3$ and ${\tilde{h}}_3$ are strictly increasing, continuous and convex functions. From (6) and (9) we have By using (9) and (11), one concludes ${lim}_{n\to \infty }\parallel x_n-u_n\parallel ={lim}_{n\to \infty }\parallel x_n-z_n\parallel =0.$ It follows that Thanks to (11), we get ${lim}_{n\to \infty }\parallel S_{\alpha ,n}{x}_n-x_n\parallel =0,$ which, together with $S_{\alpha ,n}{x}_n-x_n=\alpha (S_n{x}_n-x_n)$, leads to From the boundedness of $\{x_n\}$ and setting $D=\overline{\mathrm{conv}}\{x_n:n\ge 0\}$, we have ${\sum }_{n=1}^{\infty }{sup}_{x\in D}\parallel S_{n}x-S_{n-1}x\parallel <\infty$. Lemma 6 yields that ${lim}_{n\to \infty }{sup}_{x\in D}\parallel S_{n}x-Sx\parallel =0$. So, ${lim}_{n\to \infty }\parallel S_n{x}_n-S{x}_n\parallel =0.$ Further, from (13), we have Letting $S_{\alpha }x:=(1-\alpha )x+\alpha Sx\forall x\in C$, we deduce from Lemma 1 that $S_{\alpha }:C\to C$ is a nonexpansive mapping. It is easy to see from (14) that ${lim}_{n\to \infty }\parallel x_n-S_{\alpha }{x}_n\parallel =0.$ For each $n\ge 0$, set $T_{{\lambda }_n}:=J_{{\lambda }_n}^B(I-{\lambda }_{n}A)$. It follows that In light of $0<\lambda \le {\lambda }_n$ for all $n\ge 0$, we obtain We define a mapping $\Psi :C\to C$ by $\Psi x:={\theta }_1{S}_{\alpha }x+{\theta }_{2}Gx+(1-{\theta }_1-{\theta }_2)T_{\lambda }x$ $\forall x\in C$ with ${\theta }_1+{\theta }_2<1$ for constants ${\theta }_1,{\theta }_2\in (0,1)$. Lemma 7 guarantees that $\Psi$ is nonexpansive and Taking into account that we deduce from (12) and (15) that Let $z^{\lambda }=\lambda u+(1-\lambda )\Psi z^{\lambda },$ $\forall \lambda \in (0,1)$. Lemma 8 guarantees that $\{z^{\lambda }\}$ converges to a point $x^{*}\in \mathrm{Fix}(\Psi )=\Omega$ in norm, and $x^{*}$ further solves the VIP: $\langle x^{*}-u,J(x^{*}-p)\rangle \le 0$, $\forall p\in \Omega .$ From (1), we have Further, from (16), one has where M is a constant such that $\parallel z_t-x_n{\parallel }^q\le M$ for all $n\ge 0$ and $t\in (0,1)$. From the properties of $J_q$ and the fact that $z_t\to x^{*}$ as $t\to 0$, one gets ${lim}_{t\to 0}\parallel J_q(x_n-x^{*})-J_q(x_n-z_t)\parallel =0.$ A simple calculation indicates that and then Using (17), we have ${lim\; sup}_{n\to \infty }\langle u-x^{*},J_q(x_{n+1}-x^{*})\rangle \le 0.$ An application of Lemma 9 yields that ${\Gamma }_n\to 0$ as $n\to \infty$. Thus, $x_n\to x^{*}$ as $n\to \infty$.