Existence and approximation of solutions for system of generalized mixed variational inequalities

Abstract

The aim of this work is to study a system of generalized mixed variational inequalities, existence and approximation of its solution using the resolvent operator technique. We further propose an algorithm which converges to its solution and common fixed points of two Lipschitzian mappings. Parallel algorithms are used, which can be used to simultaneous computation in multiprocessor computers. The results presented in this work are more general and include many previously known results as special cases.

MSC:47J20, 65K10, 65K15, 90C33.

1 Introduction and preliminaries

Variational inequality theory was introduced by Stampacchia [1] in the early 1960s. The birth of variational inequality problem coincides with Signorini problem, see [[2], p.282]. The Signorini problem consists of finding the equilibrium of a spherically shaped elastic body resting on the rigid frictionless plane. Let H be a real Hilbert space whose inner product and norm are denoted by $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$, respectively. A variational inequality involving the nonlinear bifurcation, which characterized the Signorini problem with nonlocal friction is: find $x\in H$ such that

$$〈Tx,y-x〉+\phi (y,x)-\phi (x,x)\ge 0,\mathrm{\forall }y\in H,$$

where $T:H\to H$ is a nonlinear operator and $\phi (\cdot ,\cdot ):H\times H\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ is a continuous bifunction.

Inequality above is called mixed variational inequality problem. It is an useful and important generalization of variational inequalities. This type of variational inequality arise in the study of elasticity with nonlocal friction laws, fluid flow through porus media and structural analysis. Mixed variational inequalities have been generalized and extended in many directions using novel and innovative techniques. One interesting problem is to find common solution of a system of variational inequalities. The existence problem for solutions of a system of variational inequalities has been studied by Husain and Tarafdar [3]. System of variational inequalities arises in double porosity models and diffusion through a composite media, description of parallel membranes, etc.; see [4] for details.

In this paper, we consider the following system of generalized mixed variational inequalities (SGMVI). Find $x^{\ast },y^{\ast }\in H$ such that

$$\{\begin{array}{c}〈{\rho }_1{T}_1(y^{\ast },x^{\ast })+g_1(x^{\ast })-g_1(y^{\ast }),x-g_1(x^{\ast })〉+\phi (x)-\phi (g_1(x^{\ast }))\ge 0,\hfill \\ 〈{\rho }_2{T}_2(x^{\ast },y^{\ast })+g_2(y^{\ast })-g_2(x^{\ast }),x-g_2(y^{\ast })〉+\phi (x)-\phi (g_2(y^{\ast }))\ge 0\hfill \end{array}$$

(1.1)

for all $x\in H$ and ${\rho }_1,{\rho }_2>0$, where $T_1,T_2:H\times H\to H$ are nonlinear mappings and $g_1,g_2:H\to H$ are any mappings.

If $T_1,T_2:H\to H$ are univariate mappings then the problem (SGMVI) reduced to the following. Find $x^{\ast },y^{\ast }\in H$ such that

$$\{\begin{array}{c}〈{\rho }_1{T}_1(y^{\ast })+g_1(x^{\ast })-g_1(y^{\ast }),x-g_1(x^{\ast })〉+\phi (x)-\phi (g_1(x^{\ast }))\ge 0,\hfill \\ 〈{\rho }_2{T}_2(x^{\ast })+g_2(y^{\ast })-g_2(x^{\ast }),x-g_2(y^{\ast })〉+\phi (x)-\phi (g_2(y^{\ast }))\ge 0\hfill \end{array}$$

(1.2)

for all $x\in H$ and ${\rho }_1,{\rho }_2>0$.

If $T_1=T_2=T$ and $g_1=g_2=I$, then the problem (SGMVI) reduces to the following system of mixed variational inequalities considered by [5, 6]. Find $x^{\ast },y^{\ast }\in H$ such that

$$\{\begin{array}{c}〈{\rho }_{1}T(y^{\ast },x^{\ast })+x^{\ast }-y^{\ast },x-x^{\ast }〉+\phi (x)-\phi (x^{\ast })\ge 0,\hfill \\ 〈{\rho }_{2}T(x^{\ast },y^{\ast })+y^{\ast }-x^{\ast },x-y^{\ast }〉+\phi (x)-\phi (y^{\ast })\ge 0\hfill \end{array}$$

(1.3)

for all $x\in H$ and ${\rho }_1,{\rho }_2>0$.

If K is closed convex set in H and $\phi (x)={\delta }_K(x)$ for all $x\in K$, where ${\delta }_K$ is the indicator function of K defined by

$${\delta }_K(x)=\{\begin{array}{cc}0,\hfill & \text{if }x\in K;\hfill \\ +\mathrm{\infty },\hfill & \text{otherwise},\hfill \end{array}$$

then the problem (1.1) reduces to the following system of general variational inequality problem: Find $x^{\ast },y^{\ast }\in K$ such that

$$\{\begin{array}{c}〈{\rho }_1{T}_1(y^{\ast },x^{\ast })+g_1(x^{\ast })-g_1(y^{\ast }),x-g_1(x^{\ast })〉\ge 0,\hfill \\ 〈{\rho }_2{T}_2(x^{\ast },y^{\ast })+g_2(y^{\ast })-g_2(x^{\ast }),x-g_2(y^{\ast })〉\ge 0\hfill \end{array}$$

(1.4)

for all $x\in K$ and ${\rho }_1,{\rho }_2>0$. The problem (1.4) with $g_1=g_2$ has been studied by [7].

If $T_1=T_2=T$ and $g_1=g_2=I$, then the problem (1.4) reduces to the following system of general variational inequality problem. Find $x^{\ast },y^{\ast }\in K$ such that

$$\{\begin{array}{c}〈{\rho }_{1}T(y^{\ast },x^{\ast })+x^{\ast }-y^{\ast },x-x^{\ast }〉\ge 0,\hfill \\ 〈{\rho }_{2}T(x^{\ast },y^{\ast })+y^{\ast }-x^{\ast },x-y^{\ast }〉\ge 0\hfill \end{array}$$

(1.5)

for all $x\in K$ and ${\rho }_1,{\rho }_2>0$. The problem (1.5) is studied by Verma [8, 9] and Chang et al. [10].

In the study of variational inequalities, projection methods and its variant form has played an important role. Due to presence of the nonlinear term φ, the projection method and its variant forms cannot be extended to suggest iterative methods for solving mixed variational inequalities. If the nonlinear term φ in the mixed variational inequalities is a proper, convex and lower semicontinuous function, then the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. Hassouni and Moudafi [11] used the resolvent operator technique to study a new class of mixed variational inequalities.

For a multivalued operator $T:H\to H$, the domain of T, the range of T and the graph of T denote by

$$D(T)=\{u\in H:T(u)\ne \mathrm{\varnothing }\},R(T)=\bigcup _{u\in H}T(u)$$

and

$$Graph(T)=\{(u,u^{\ast })\in H\times H:u\in D(T)\text{ and }{u}^{\ast }\in T(u)\},$$

respectively.

Definition 1.1 T is called monotone if and only if for each $u\in D(T)$, $v\in D(T)$ and $u^{\ast }\in T(u)$, $v^{\ast }\in T(v)$, we have

$$〈v^{\ast }-u^{\ast },v-u〉\ge 0.$$

T is maximal monotone if it is monotone and its graph is not properly contained in the graph of any other monotone operator.

$T^{-1}$ is the operator defined by $v\in T^{-1}(u)\iff u\in T(v)$.

Definition 1.2 ([12])

For a maximal monotone operator T, the resolvent operator associated with T, for any $\sigma >0$, is defined as

$$J_T(u)={(I+\sigma T)}^{-1}(u),\mathrm{\forall }u\in H.$$

It is known that a monotone operator is maximal if and only if its resolvent operator is defined everywhere. Furthermore, the resolvent operator is single-valued and nonexpansive i.e., $\parallel J_T(x)-J_T(y)\parallel \le \parallel x-y\parallel$ for all $x,y\in H$. In particular, it is well known that the subdifferential ∂φ of φ is a maximal monotone operator; see [13].

Lemma 1.3 ([12])

For a given $z,u\in H$ satisfies the inequality

$$〈u-z,x-u〉+\sigma \phi (x)-\sigma \phi (u)\ge 0,\mathrm{\forall }x\in H$$

if and only if $u=J_{\phi }(z)$, where $J_{\phi }={(I+\sigma \partial \phi )}^{-1}$ is the resolvent operator and $\sigma >0$ is a constant.

Using Lemma 1.3, we will establish following important relation.

Lemma 1.4 The variational inequality problem (1.1) is equivalent to finding $x^{\ast },y^{\ast }\in H$ such that

$$\{\begin{array}{c}{x}^{\ast }=x^{\ast }-g_1(x^{\ast })+J_{\phi }(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast })),\hfill \\ y^{\ast }=y^{\ast }-g_2(y^{\ast })+J_{\phi }(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast })),\hfill \end{array}$$

(1.6)

where $J_{\phi }={(I+\partial \phi )}^{-1}$ is the resolvent operator and ${\rho }_1,{\rho }_2>0$.

Proof Let $x^{\ast },y^{\ast }\in H$ be a solution of (1.1). Then for all $x\in H$, we have

$$\{\begin{array}{c}〈{\rho }_1{T}_1(y^{\ast },x^{\ast })+g_1(x^{\ast })-g_1(y^{\ast }),x-g_1(x^{\ast })〉+\phi (x)-\phi (g_1(x^{\ast }))\ge 0,\hfill \\ 〈{\rho }_2{T}_2(x^{\ast },y^{\ast })+g_2(y^{\ast })-g_2(x^{\ast }),x-g_2(y^{\ast })〉+\phi (x)-\phi (g_2(y^{\ast }))\ge 0,\hfill \end{array}$$

which can be written as

$$\{\begin{array}{c}〈g_1(x^{\ast })-(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast })),x-g_1(x^{\ast })〉+\phi (x)-\phi (g_1(x^{\ast }))\ge 0,\hfill \\ 〈g_2(y^{\ast })-(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast })),x-g_2(y^{\ast })〉+\phi (x)-\phi (g_2(y^{\ast }))\ge 0,\hfill \end{array}$$

using Lemma 1.3 for $\sigma =1$, we get

$$\{\begin{array}{c}{g}_1(x^{\ast })=J_{\phi }(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast })),\hfill \\ g_2(y^{\ast })=J_{\phi }(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast })),\hfill \end{array}$$

i.e.,

$$\{\begin{array}{c}{x}^{\ast }=x^{\ast }-g_1(x^{\ast })+J_{\phi }(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast })),\hfill \\ y^{\ast }=y^{\ast }-g_2(y^{\ast })+J_{\phi }(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast })).\hfill \end{array}$$

This completes the proof. □

Definition 1.5 An operator $g:H\to H$ is said to be

• (1) ζ-strongly monotone if for each $x,x^{\prime }\in H$, there exists a constant $\zeta >0$ such that

  $$〈g(x)-g\left(x^{\prime }\right),x-x^{\prime }〉\ge \zeta {\parallel x-x^{\prime }\parallel }^2$$

  for all $y,y^{\prime }\in H$;

• (2) η-Lipschitz continuous if for each $x,x^{\prime }\in H$, there exists a constant $\eta >0$ such that

  $$\parallel g(x)-g\left(x^{\prime }\right)\parallel \le \eta \parallel x-x^{\prime }\parallel .$$

  An operator $T:H\times H\to H$ is said to be

• (3) relaxed $(\omega ,t)$-cocoercive with respect to the first argument if for each $x,x^{\prime }\in H$, there exist constants $t>0$ and $\omega >0$ such that

  $$〈T(x,\cdot )-T(x^{\prime },\cdot ),x-x^{\prime }〉\ge -\omega {\parallel T(x,\cdot )-T(x^{\prime },\cdot )\parallel }^2+t{\parallel x-x^{\prime }\parallel }^2;$$

• (4) μ-Lipschitz continuous with respect to the first argument if for each $x,x^{\prime }\in H$, there exists a constant $\mu >0$ such that

  $$\parallel T(x,\cdot )-T(x^{\prime },\cdot )\parallel \le \mu \parallel x-x^{\prime }\parallel ;$$

• (5) γ-Lipschitz continuous with respect to the second argument if for each $y,y^{\prime }\in H$, there exists a constant $\gamma >0$ such that

  $$\parallel T(\cdot ,y)-T(\cdot ,y^{\prime })\parallel \le \gamma \parallel y-y^{\prime }\parallel .$$

Lemma 1.6 ([14])

Let $\{a_n\}$ and $\{b_n\}$ be two nonnegative real sequences satisfying the following conditions:

$$a_{n+1}\le (1-d_n)a_n+b_n,\mathrm{\forall }n\ge n_0,$$

where $n_0$ is some nonnegative integer, $d_n\in (0,1)$ with ${\sum }_{n=0}^{\mathrm{\infty }}{d}_n=\mathrm{\infty }$ and $b_n=o(d_n)$, then $a_n\to 0$ as $n\to \mathrm{\infty }$.

Several iterative algorithms have been devised to study existence and approximation of different classes of variational inequalities. Most of them are sequential iterative methods, when we implement such algorithms on computers, then only one processor is used at a time. Availability of multiprocessor computers enabled researchers to develop iterative algorithms having the parallel characteristics. Lions [15] studied a parallel algorithm for a solution of parabolic variational inequalities. Bertsekas and Tsitsiklis [16, 17] developed parallel algorithm using the metric projection. Recently, Yang et al. [7] studied parallel projection algorithm for a system of nonlinear variational inequalities.

2 Existence and convergence

Lemma 1.4 established the equivalence between the fixed-point problem and the variational inequality problem (1.1). Using this equivalence in this section, we construct a parallel iterative algorithm to approximate the solution of the problem (1.1) and study the convergence of the sequence generated by the algorithm.

Algorithm 2.1 For arbitrary chosen points $x_0,y_0\in H$, compute the sequences $\{x_n\}$ and $\{y_n\}$ such that

$$\{\begin{array}{c}{x}_{n+1}=x_n-g_1(x_n)+J_{\phi }(g_1(y_n)-{\rho }_1{T}_1(y_n,x_n)),\hfill \\ y_{n+1}=y_n-g_2(y_n)+J_{\phi }(g_2(x_n)-{\rho }_2{T}_2(x_n,y_n)),\hfill \end{array}$$

(2.1)

where $J_{\phi }={(I+\partial \phi )}^{-1}$ is the resolvent operator and ${\rho }_1$, ${\rho }_2$ is positive real numbers.

Theorem 2.2 Let H be a real Hilbert space. Let $T_i:H\times H\to H$ and $g_i:H\to H$ be mappings such that $T_i$ is relaxed $({\omega }_i,t_i)$-cocoercive, ${\mu }_i$-Lipschitz continuous with respect to the first argument, ${\gamma }_i$-Lipschitz continuous with respect to the second argument and $g_i$ is ${\eta }_i$-Lipschitz continuous, ${\zeta }_i$-strongly monotone mapping for $i=1,2$. Assume that the following assumptions hold:

where $\kappa ={\sum }_{i=1}^2\sqrt{1-2{\zeta }_i+{\eta }_i^2}<1$.

Then there exist $x^{\ast },y^{\ast }\in H$, which solves the problem (1.1). Moreover, the iterative sequences $\{x_n\}$ and $\{y_n\}$ generated by the Algorithm 2.1 converges to $x^{\ast }$ and $y^{\ast }$, respectively.

Proof Using (2.1), we have

(2.2)

Since $T_1$ is relaxed $({\omega }_1,t_1)$-cocoercive and ${\mu }_1$-Lipschitz continuous in the first argument, we have

(2.3)

Since $g_1$ is ${\eta }_1$-Lipschitz continuous and ${\zeta }_1$-strongly monotone,

(2.4)

Similarly,

$${\parallel y_n-y_{n-1}-(g_1(y_n)-g_1(y_{n-1}))\parallel }^2\le (1-2{\zeta }_1+{\eta }_1^2){\parallel y_n-y_{n-1}\parallel }^2.$$

(2.5)

By ${\gamma }_1$-Lipschitz continuity of $T_1$ with respect to second argument,

$$\parallel T_1(y_{n-1},x_n)-T_1(y_{n-1},x_{n-1})\parallel \le {\gamma }_1\parallel x_n-x_{n-1}\parallel .$$

(2.6)

It follows from (2.2)-(2.6) that

$$\parallel x_{n+1}-x_n\parallel \le ({\psi }_1+{\rho }_1{\gamma }_1)\parallel x_n-x_{n-1}\parallel +({\psi }_1+{\theta }_1)\parallel y_n-y_{n-1}\parallel ,$$

(2.7)

where ${\psi }_1=\sqrt{1-2{\zeta }_1+{\eta }_1^2}$ and ${\theta }_1=\sqrt{1+2{\rho }_1{\omega }_1{\mu }_1^2-2{\rho }_1{t}_1+{\rho }_1^2{\mu }_1^2}$.

Similarly, we get

$$\parallel y_{n+1}-y_n\parallel \le ({\psi }_2+{\theta }_2)\parallel x_n-x_{n-1}\parallel +({\psi }_2+{\rho }_2{\gamma }_2)\parallel y_n-y_{n-1}\parallel ,$$

(2.8)

where ${\psi }_2=\sqrt{1-2{\zeta }_2+{\eta }_2^2}$ and ${\theta }_2=\sqrt{1+2{\rho }_2{\omega }_2{\mu }_2^2-2{\rho }_2{t}_2+{\rho }_2^2{\mu }_2^2}$.

Now (2.7) and (2.8) imply

$$\begin{array}{rcl}\parallel x_{n+1}-x_n\parallel +\parallel y_{n+1}-y_n\parallel & \le & ({\psi }_1+{\psi }_2+{\theta }_2+{\rho }_1{\gamma }_1)\parallel x_n-x_{n-1}\parallel \\ +({\psi }_1+{\psi }_2+{\theta }_1+{\rho }_2{\gamma }_2)\parallel y_n-y_{n-1}\parallel \\ \le & \mathrm{\Theta }(\parallel x_n-x_{n-1}\parallel +\parallel y_n-y_{n-1}\parallel ),\end{array}$$

where $\mathrm{\Theta }=max\{({\psi }_1+{\psi }_2+{\theta }_2+{\rho }_1{\gamma }_1),({\psi }_1+{\psi }_2+{\theta }_1+{\rho }_2{\gamma }_2)\}<1$ by assumption. Hence $\{x_n\}$ and $\{y_n\}$ are both Cauchy sequences in H, and $\{x_n\}$ converges to $x^{\ast }\in H$ and $\{y_n\}$ converges to $y^{\ast }\in H$. Since $g_1$, $g_2$, $T_1$, $T_2$ and $J_{\phi }$ are all continuous, we have

$$\{\begin{array}{c}{x}^{\ast }=x^{\ast }-g_1(x^{\ast })+J_{\phi }(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast })),\hfill \\ y^{\ast }=y^{\ast }-g_2(y^{\ast })+J_{\phi }(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast })).\hfill \end{array}$$

The result follows from Lemma 1.4. This completes the proof. □

If $T_1,T_2:H\to H$ are univariate mappings, then the Algorithm 2.1 reduces to the following.

Algorithm 2.3 For arbitrary chosen points $x_0,y_0\in H$, compute the sequences $\{x_n\}$ and $\{y_n\}$ such that

$$\{\begin{array}{c}{x}_{n+1}=x_n-g_1(x_n)+J_{\phi }(g_1(y_n)-{\rho }_1{T}_1(y_n)),\hfill \\ y_{n+1}=y_n-g_2(y_n)+J_{\phi }(g_2(x_n)-{\rho }_2{T}_2(x_n)),\hfill \end{array}$$

where $J_{\phi }={(I+\partial \phi )}^{-1}$ is the resolvent operator and ${\rho }_1$, ${\rho }_2$ is positive real numbers.

Theorem 2.4 Let H be a real Hilbert space. Let $T_i,g_i:H\to H$ be mappings such that $T_i$ is relaxed $({\omega }_i,t_i)$-cocoercive, ${\mu }_i$-Lipschitz continuous and $g_i$ is ${\eta }_i$-Lipschitz continuous, ${\zeta }_i$-strongly monotone mapping for $i=1,2$. Assume that the following assumptions hold:

where $\kappa ={\sum }_{i=1}^2\sqrt{1-2{\zeta }_i+{\eta }_i^2}<1$.

Then there exist $x^{\ast },y^{\ast }\in H$, which solves the problem (1.2). Moreover the iterative sequences $\{x_n\}$ and $\{y_n\}$ generated by the Algorithm 2.3 converges to $x^{\ast }$ and $y^{\ast }$, respectively.

3 Relaxed algorithm and approximation solvability

Lemma 1.4 implies that the system of general mixed variational inequality problem (1.1) is equivalent to the fixed-point problem. This alternative equivalent formulation is very useful for a numerical point of view. In this section, we construct a relaxed iterative algorithm for solving the problem (1.1) and study the convergence of the iterative sequence generated by the algorithm.

Algorithm 3.1 For arbitrary chosen points $x_0,y_0\in H$, compute the sequences $\{x_n\}$ and $\{y_n\}$ such that

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n(x_n-g_1(x_n)+J_{\phi }(g_1(y_n)-{\rho }_1{T}_1(y_n,x_n))),\hfill \\ y_{n+1}=(1-{\beta }_n)y_n+{\beta }_n(y_n-g_2(y_n)+J_{\phi }(g_2(x_n)-{\rho }_2{T}_2(x_n,y_n))),\hfill \end{array}$$

(3.1)

where $J_{\phi }={(I+\partial \phi )}^{-1}$ is the resolvent operator, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences in $[0,1]$ and ${\rho }_1$, ${\rho }_2$ is positive real numbers.

We first prove a result, which will be helpful to prove main result of this section.

Lemma 3.2 Let H be a real Hilbert space. Let $\{x_n\}$ and $\{y_n\}$ be sequences in H such that

$$\parallel x_{n+1}-x^{\ast }\parallel +\parallel y_{n+1}-y^{\ast }\parallel \le max\{(1-t_n),(1-s_n)\}(\parallel x_n-x^{\ast }\parallel +\parallel y_n-y^{\ast }\parallel )$$

(3.2)

for some $x^{\ast },y^{\ast }\in H$, where $\{s_n\}$ and $\{t_n\}$ are sequences in $(0,1)$ such that ${\sum }_{n=0}^{\mathrm{\infty }}{t}_n=\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{s}_n=\mathrm{\infty }$. Then $\{x_n\}$ and $\{y_n\}$ converges to $x^{\ast }$ and $y^{\ast }$, respectively.

Proof Now, define the norm ${\parallel \cdot \parallel }_1$ on $H\times H$ by

$${\parallel (x,y)\parallel }_1=\parallel x\parallel +\parallel y\parallel ,\mathrm{\forall }(x,y)\in H\times H.$$

Then $(H\times H,{\parallel \cdot \parallel }_1)$ is a Banach space. Hence, (3.2) implies that

$${\parallel (x_{n+1},y_{n+1})-(x^{\ast },y^{\ast })\parallel }_1\le max\{(1-t_n),(1-s_n)\}{\parallel (x_n,y_n)-(x^{\ast },y^{\ast })\parallel }_1.$$

Using Lemma 1.6, we get

$$\underset{n\to \mathrm{\infty }}{lim}{\parallel (x_n,y_n)-(x^{\ast },y^{\ast })\parallel }_1=0.$$

Therefore, sequences $\{x_n\}$ and $\{y_n\}$ converges to $x^{\ast }$ and $y^{\ast }$, respectively. This completes the proof. □

We now present the approximation solvability of the problem (1.1).

Theorem 3.3 Let H be a real Hilbert space H. Let $T_i:H\times H\to H$ and $g_i:H\to H$ be mappings such that $T_i$ is relaxed $({\omega }_i,t_i)$-cocoercive, ${\mu }_i$-Lipschitz continuous with respect to the first argument, ${\gamma }_i$-Lipschitz continuous with respect to the second argument and $g_i$ is ${\eta }_i$-Lipschitz continuous, ${\zeta }_i$-strongly monotone mapping for $i=1,2$. Suppose that $x^{\ast },y^{\ast }\in H$ be a solution of the problem (1.1) and $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences in $[0,1]$. Assume that the following assumptions hold:

• (i) $0<{\mathrm{\Theta }}_{1n}={\alpha }_n(1-({\psi }_1+{\rho }_1{\gamma }_1))-{\beta }_n({\psi }_2+{\theta }_2)<1$,

• (ii) $0<{\mathrm{\Theta }}_{2n}={\beta }_n(1-({\psi }_2+{\rho }_2{\gamma }_2))-{\alpha }_n({\psi }_1+{\theta }_1)<1$,

• (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{\mathrm{\Theta }}_{1n}=\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\mathrm{\Theta }}_{2n}=\mathrm{\infty }$,

where

$${\theta }_i=\sqrt{1+2{\rho }_i{\omega }_i{\mu }_i^2-2{\rho }_1{t}_i+{\rho }_i^2{\mu }_i^2},{\psi }_i=\sqrt{1-2{\zeta }_i+{\eta }_i^2},i=1,2.$$

Then the sequences $\{x_n\}$ and $\{y_n\}$ generated by the Algorithm 3.1 converges to $x^{\ast }$ and $y^{\ast }$, respectively.

Proof From Theorem 2.2 the problem (1.1) has a solution $(x^{\ast },y^{\ast })$ in H. By Lemma 1.4, we have

$$\{\begin{array}{c}{x}^{\ast }=x^{\ast }-g_1(x^{\ast })+J_{\phi }(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast })),\hfill \\ y^{\ast }=y^{\ast }-g_2(y^{\ast })+J_{\phi }(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast })).\hfill \end{array}$$

(3.3)

To prove the result, we first evaluate $\parallel x_{n+1}-x^{\ast }\parallel$ for all $n\ge 0$. Using (3.1), we obtain

(3.4)

Since $T_1$ is relaxed $({\omega }_1,t_1)$-cocoercive and ${\mu }_1$-Lipschitz mapping with respect to the first argument, we have

(3.5)

Since $g_1$ is ${\eta }_1$-Lipschitz continuous and ${\zeta }_1$-strongly monotone,

(3.6)

Similarly, we have

$${\parallel y_n-y^{\ast }-(g_1(y_n)-g_1\left(y^{\ast }\right))\parallel }^2\le (1-2{\zeta }_1+{\eta }_1^2){\parallel y_n-y^{\ast }\parallel }^2.$$

(3.7)

By ${\gamma }_1$-Lipschitz continuity of $T_1$ with respect to second argument,

$$\parallel T_1(y^{\ast },x_n)-T_1(y^{\ast },x^{\ast })\parallel \le {\gamma }_1\parallel x_n-x^{\ast }\parallel .$$

(3.8)

By (3.4)-(3.8), we have

$$\parallel x_{n+1}-x^{\ast }\parallel \le [1-{\alpha }_n+{\alpha }_n({\psi }_1+{\rho }_1{\gamma }_1)]\parallel x_n-x^{\ast }\parallel +{\alpha }_n({\psi }_1+{\theta }_1)\parallel y_n-y^{\ast }\parallel ,$$

(3.9)

where ${\psi }_1=\sqrt{1-2{\zeta }_1+{\eta }_1^2}$ and ${\theta }_1=\sqrt{1+2{\rho }_1{\omega }_1{\mu }_1^2-2{\rho }_1{t}_1+{\rho }_1^2{\mu }_1^2}$.

Similarly, we have

$$\parallel y_{n+1}-y^{\ast }\parallel \le {\beta }_n({\psi }_2+{\theta }_2)\parallel x_n-x^{\ast }\parallel +[1-{\beta }_n+{\beta }_n({\psi }_2+{\rho }_2{\gamma }_2)]\parallel y_n-y^{\ast }\parallel ,$$

(3.10)

where ${\psi }_2=\sqrt{1-2{\zeta }_2+{\eta }_2^2}$ and ${\theta }_2=\sqrt{1+2{\rho }_2{\omega }_2{\mu }_2^2-2{\rho }_2{t}_2+{\rho }_2^2{\mu }_2^2}$.

Now (3.9) and (3.10) imply

where

By the assumptions and Lemma 3.2, we get that the sequences $\{x_n\}$ and $\{y_n\}$ converges to $x^{\ast }$ and $y^{\ast }$, respectively. This completes the proof. □

Remark 3.4 Theorem 3.3 extend and generalize the main result in [5], which itself is a extension and improvement of the main result in Chang et al. [10].

If $T_1,T_2:H\to H$ are univariate mappings, then the Algorithm 3.1 reduces to the following.

Algorithm 3.5 For arbitrary chosen points $x_0,y_0\in H$, compute the sequences $\{x_n\}$ and $\{y_n\}$ such that

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n(x_n-g_1(x_n)+J_{\phi }(g_1(y_n)-{\rho }_1{T}_1(y_n))),\hfill \\ y_{n+1}=(1-{\beta }_n)y_n+{\beta }_n(y_n-g_2(y_n)+J_{\phi }(g_2(x_n)-{\rho }_2{T}_2(x_n))),\hfill \end{array}$$

where $J_{\phi }={(I+\partial \phi )}^{-1}$ is the resolvent operator, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences in $[0,1]$ and ${\rho }_1$, ${\rho }_2$ is positive real numbers.

As a consequence of Theorem 3.3, we have following result.

Corollary 3.6 Let H be a real Hilbert space H. Let $T_i,g_i:H\to H$ be mappings such that $T_i$ is relaxed $({\omega }_i,t_i)$-cocoercive, ${\mu }_i$-Lipschitz continuous and $g_i$ is ${\eta }_i$-Lipschitz continuous, ${\zeta }_i$-strongly monotone mapping for $i=1,2$. Suppose that $x^{\ast },y^{\ast }\in H$ be a solution of the problem (1.2) and $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences in $[0,1]$. Assume that the following assumptions hold:

• (i) $0<{\mathrm{\Theta }}_{1n}={\alpha }_n(1-{\psi }_1)-{\beta }_n({\psi }_2+{\theta }_2)<1$,

• (ii) $0<{\mathrm{\Theta }}_{2n}={\beta }_n(1-{\psi }_2)-{\alpha }_n({\psi }_1+{\theta }_1)<1$,

• (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{\mathrm{\Theta }}_{1n}=\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\mathrm{\Theta }}_{2n}=\mathrm{\infty }$,

where

$${\theta }_i=\sqrt{1+2{\rho }_i{\omega }_i{\mu }_i^2-2{\rho }_1{t}_i+{\rho }_i^2{\mu }_i^2},{\psi }_i=\sqrt{1-2{\zeta }_i+{\eta }_i^2},i=1,2.$$

Then the sequences $\{x_n\}$ and $\{y_n\}$ generated by the Algorithm 3.5 converges to $x^{\ast }$ and $y^{\ast }$, respectively.

4 Algorithms for common element

Now, we consider, the approximation solvability of the system (1.1) which is also a common fixed point of two Lipschitzian mappings. We propose a relaxed two-step algorithm, which can be applied to the approximation of solution of the problem (1.1) and common fixed point of two Lipschitzian mappings.

Algorithm 4.1 For arbitrary chosen points $x_0,y_0\in H$, compute the sequences $\{x_n\}$ and $\{y_n\}$ such that

$$\{\begin{array}{c}{x}_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{S}_1(x_n-g_1(x_n)+J_{\phi }(g_1(y_n)-{\rho }_1{T}_1(y_n,x_n))),\hfill \\ y_{n+1}=(1-{\beta }_n)y_n+{\beta }_n{S}_2(y_n-g_2(y_n)+J_{\phi }(g_2(x_n)-{\rho }_2{T}_2(x_n,y_n))),\hfill \end{array}$$

(4.1)

where $J_{\phi }={(I+\partial \phi )}^{-1}$ is the resolvent operator, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences in $[0,1]$ and ${\rho }_1$, ${\rho }_2$ be positive real numbers.

Let $F(S_i)$ denote the set of fixed points of the mapping $S_i$, i.e., $F(S_i)=\{x\in H:S_{i}x=x\}$, $Fix(S)={\bigcap }_{i=1}^{2}F(S_i)$ and the set of solutions of the problem (1.1).

Theorem 4.2 Let H be a real Hilbert space H. Let $T_i:H\times H\to H$ and $g_i:H\to H$ be mappings such that $T_i$ is relaxed $({\omega }_i,t_i)$-cocoercive, ${\mu }_i$-Lipschitz continuous with respect to the first argument, ${\gamma }_i$-Lipschitz continuous with respect to the second argument and $g_i$ is ${\eta }_i$-Lipschitz continuous, ${\zeta }_i$-strongly monotone mapping for $i=1,2$. Let $S_i:H\to H$ be ${\vartheta }_i$-Lipschitzian mapping for $i=1,2$ with $Fix(S)\ne \mathrm{\varnothing }$, $\{{\alpha }_n\}$, $\{{\beta }_n\}$ are sequences in $[0,1]$. Assume that the following assumptions hold:

• (i) $0<{\mathrm{\Theta }}_{1n}={\alpha }_n\vartheta (1-({\psi }_1+{\rho }_1{\gamma }_1))-{\beta }_n\vartheta ({\psi }_2+{\theta }_2)<1$,

• (ii) $0<{\mathrm{\Theta }}_{2n}={\beta }_n\vartheta (1-({\psi }_2+{\rho }_2{\gamma }_2))-{\alpha }_n\vartheta ({\psi }_1+{\theta }_1)<1$,

• (iii) ${\sum }_{n=0}^{\mathrm{\infty }}{\mathrm{\Theta }}_{1n}=\mathrm{\infty }$ and ${\sum }_{n=0}^{\mathrm{\infty }}{\mathrm{\Theta }}_{2n}=\mathrm{\infty }$,

where $\vartheta =max\{{\vartheta }_1,{\vartheta }_2\}$ and

$${\theta }_i=\sqrt{1+2{\rho }_i{\omega }_i{\mu }_i^2-2{\rho }_1{t}_i+{\rho }_i^2{\mu }_i^2},{\psi }_i=\sqrt{1-2{\zeta }_i+{\eta }_i^2},i=1,2.$$

If , then the sequences $\{x_n\}$ and $\{y_n\}$ generated by the Algorithm 4.1 converges to $x^{\ast }$ and $y^{\ast }$, respectively, such that and $\{x^{\ast },y^{\ast }\}\in Fix(S)$.

Proof Let us have and $\{x^{\ast },y^{\ast }\}\in Fix(S)$. By Lemma 1.4, we have

$$\{\begin{array}{c}{x}^{\ast }=x^{\ast }-g_1(x^{\ast })+J_{\phi }(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast })),\hfill \\ y^{\ast }=y^{\ast }-g_2(y^{\ast })+J_{\phi }(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast })).\hfill \end{array}$$

Also since $\{x^{\ast },y^{\ast }\}\in Fix(S)$, we have

$$\{\begin{array}{c}{x}^{\ast }=S_1(x^{\ast }-g_1(x^{\ast })+J_{\phi }(g_1(y^{\ast })-{\rho }_1{T}_1(y^{\ast },x^{\ast }))),\hfill \\ y^{\ast }=S_2(y^{\ast }-g_2(y^{\ast })+J_{\phi }(g_2(x^{\ast })-{\rho }_2{T}_2(x^{\ast },y^{\ast }))).\hfill \end{array}$$

To prove the result, we first evaluate $\parallel x_{n+1}-x^{\ast }\parallel$ for all $n\ge 0$. Using (4.1), we obtain

(4.2)

Using the arguments as in the proof of Theorem 3.3, from (4.2) we get that

$$\parallel x_{n+1}-x^{\ast }\parallel \le [1-{\alpha }_n+{\alpha }_n{\vartheta }_1({\psi }_1+{\rho }_1{\gamma }_1)]\parallel x_n-x^{\ast }\parallel +{\alpha }_n{\vartheta }_1({\psi }_1+{\theta }_1)\parallel y_n-y^{\ast }\parallel ,$$

(4.3)

where ${\psi }_1=\sqrt{1-2{\zeta }_1+{\eta }_1^2}$ and ${\theta }_1=\sqrt{1+2{\rho }_1{\omega }_1{\mu }_1^2-2{\rho }_1{t}_1+{\rho }_1^2{\mu }_1^2}$.

Similarly, we get

$$\parallel y_{n+1}-y^{\ast }\parallel \le {\beta }_n{\vartheta }_2({\psi }_2+{\theta }_2)\parallel x_n-x^{\ast }\parallel +[1-{\beta }_n+{\beta }_n{\vartheta }_2({\psi }_2+{\rho }_2{\gamma }_2)]\parallel y_n-y^{\ast }\parallel ,$$

(4.4)

where ${\psi }_2=\sqrt{1-2{\zeta }_2+{\eta }_2^2}$ and ${\theta }_2=\sqrt{1+2{\rho }_2{\omega }_2{\mu }_1^2-2{\rho }_2{t}_2+{\rho }_2^2{\mu }_2^2}$.

Adding (4.3) and (4.4), taking $\vartheta =max\{{\vartheta }_1,{\vartheta }_2\}$ we get

where

By the assumptions and Lemma 3.2, we get that the sequences $\{x_n\}$ and $\{y_n\}$ converges to $x^{\ast }$ and $y^{\ast }$, respectively. This completes the proof. □

A mapping $S:H\to H$ is said to be asymptotically λ-strictly pseudocontractive [18] if there exist a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ such that

$${\parallel S^{n}x-S^{n}y\parallel }^2\le k_n^2{\parallel x-y\parallel }^2+\lambda {\parallel (x-S^{n}x)-(y-S^{n}y)\parallel }^2$$

for some $\lambda \in (0,1)$, for all $x,y\in H$ and $n\ge 1$.

Kim and Xu [19] proved that, if $S:H\to H$ is an asymptotically λ-strictly pseudocontractive mapping, then $S^n$ is a Lipschitzian mapping with Lipschitz constant

$$L_n=\frac{\lambda +\sqrt{1+(k_n^2-1)(1-\lambda )}}{1-\lambda }$$

for each integer $n>1$.

Also if $x^{\ast }\in F(S)$, then $x^{\ast }\in F(S^n)$ for all integer $n\ge 1$.

Assume that $S_i:H\to H$ is asymptotically ${\lambda }_i$-strictly pseudocontractive mappings for $i=1,2$ with ${\bigcap }_{i=1}^{2}F(S_i)\ne \mathrm{\varnothing }$. Now generate sequence $\{x_n\}$ and $\{y_n\}$ by Algorithm 4.1 with $S_1:=S_1^j$ and $S_2:=S_2^k$ for some integer $j,k>1$. Theorem 4.2 can be applied to study approximate solvability of the problem (1.1) and common fixed points of two asymptotically strictly pseudocontractive mappings.

A mapping $S:H\to H$ is said to be asymptotically nonexpansive [20] if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ such that $\parallel S^{n}x-S^{n}y\parallel \le k_n\parallel x-y\parallel$ for all $x,y\in K$ and $n\ge 1$. Clearly every asymptotically nonexpansive mapping is an asymptotically 0-strictly pseudocontractive mapping. Theorem 4.2 can be applied to study approximate solvability of the problem (1.1) and common fixed points of two asymptotically nonexpansive mappings.

Remark 4.3 An important feature of the algorithms used in the paper is its suitability for implementing on multiprocessor computer. Assume that $\{x_n\}$ and $\{y_n\}$ are given, in order to get the new iterative point; we can set one processor of computer to compute $\{x_{n+1}\}$ and other processor to compute $\{y_{n+1}\}$, i.e., $\{x_{n+1}\}$ and $\{y_{n+1}\}$ are computed parallel, which will take less time then computing $\{x_{n+1}\}$ and $\{y_{n+1}\}$ in a sequence using a single processor; we refer [16, 17, 21–23] and references therein for more examples and ideas of the parallel iterative methods.