1 Introduction

The variational inequality theory is very effective. It is a powerful tool for current mathematical engineering problems^{[1, 2, 3, 4]},e.g.,economic problems,control problems,contact problems,mechanics,transportation,and equilibrium problems. Many variational inequalities have been introduced and studied^{[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]}. It is known that the accretionary of the under lying operator plays an indispensable role in the variational inequality theory and its generalizations. Huang and Fang^[26] introduced the generalized $m$-accretive mapping, and gave the definition of the resolvent operator for generalized $m$-accretive mappings in Banach spaces. Lan et al.^[27] introduced the concept of $(A,\eta)$-accretive mappings,which could generalize the existing $\eta$-subdifferential operators,the maximal $\eta$-monotone operator,the $H$-monotone operators,the $(H,\eta)$ monotone operators in Hilbert spaces,the $H$-accretive mapping,the generalized $m$-accretive mapping,and the $(H,\eta)$-accretive mappings in Banach spaces. Motivated by the research works going on in this field,in this paper,we suggest an iterative algorithm to find the common solutions of a system of general nonlinear variational inclusions involving different nonlinear operators and fixed point problems by use of the resolvent operator techniques in the framework of Banach spaces.

2 Preliminaries

Let $X$ be a real Banach space with the dual space $X^{*},$ $\langle \cdot ,\cdot \rangle$ be the dual pairing between $X$ and $X^{*},$ $2^X$ denotes the family of all nonempty subsets of $X$,and $CB(X)$ denotes the family of all nonempty closed bounded subsets of $X$. The generalized duality mapping $J_q:X\to 2^X$ is defined by

where $q$ is a constant,and $q>1$.

The smoothness modulus of $X$ is the function $\rho_{X}:[0,\infty) \to [0,\infty)$ defined by

A Banach space $X$ is called uniformly smooth if

$X$ is called $q$-uniformly smooth if there exists a constant $C>0$ such that

In particular,$J_2$ is the usual normalized duality mapping,and

If $X^{*} $ is strictly convex^[28] or $X$ is a uniformly smooth Banach space, then $J_q$ is single valued. We always denote the single valued generalized duality mapping by $j_q$ in the real uniformly smooth Banach space $X$ unless otherwise stated.

Lemma 1^[29] Let $X$ be a real uniformly smooth Banach space$.$ Then$,$ $X$ is $q$-uniformly smooth iff there exists a constant $C_q>0$ such that for all $x,y \in X,$

where $j_q :X \to X^{*}$ is the normalized duality mapping on $X.$

Definition 1 Let $X$ be a uniformly smooth Banach space$,$ and $T:X \to X$ be a single valued mapping. Then$,$ $T$ is said to be

(i) accretive if

(ii) strictly accretive if $T$ is accretive and

(iii) $\gamma$-strongly accretive if there exists a constant $\gamma>0$ such that

(iv) $\alpha$-relaxed accretive if there exists a constant $\alpha>0$ such that

Definition 2 A nonlinear operator $S:X\to X$ is said to be a $\tau$-Lipschitzian mapping if there exists a positive constant $\tau$ such that

If $\tau=1,$ the mapping $S$ is known as a nonexpansive mapping. We will denote by $E(S)$ the set of fixed points of $S,$ i.e.$,$

Definition 3 A single valued mapping $N: X \times X \to X$ is said to be $(\mu,\nu)$-Lipschitz continuous if there exist constants $\mu,\nu >0$ such that

Definition 4 Assume that $X$ is a $q$-uniformly smooth Banach Space. Let $T,F :X \to X$ be two single valued mappings. Then$,$ a nonlinear mapping $N:X\times X \to X$ is said to be

(i) $\gamma_{N}^{T}$-strongly accretive with respect to the first variable of $N$ and $T$ if there exists a constant $\gamma_{N}^{T}>0$ such that

(ii) $\alpha_{N}^{F}$-relaxed accretive with respect to the second variable of $N$ and $F$ if there exists a constant $\alpha_{N}^{F}>0$ such that

Throughout this work,we assume that $N_i :X \times X \to X$ is a nonlinear operator,$T_i,F_i,g_i:X\to X$ are single valued mappings, and $r_i$ is a fixed positive real number for each $i=1,2,3.$ Set

Let $\varphi:X\to \mathbb{R} \cup\{+ \infty \}$ be a proper convex lower semicontinuous function. The system of general nonlinear variational inclusions,involving the three different nonlinear operators generated by $r_1,r_2,r_3$,is defined as follows: Find $(x^{*},y^{*},z^{*}) \in X \times X\times X$ such that

where $g_i(x)\in X$ $(i=1,2,3)$. We denote by SGNVID$(\Sigma,\Xi,\wedge,\vee)$ the set of all solutions $(x^{*},y^{*},z^{*})$ of the problem (1).

Lemma 2^[2] If $M$ is a maximal accretive operator on $X,$ then for any $\rho>0,$ the resolvent operator associated with $M$ is defined by

$$J_M(u)=(I+\rho M)^{-1}(u),\quad \forall u \in X.$$

We note that an accretive operator is maximal iff its resolvent operator is defined everywhere. Furthermore,the resolvent operator is single valued and nonexpansive,and the subdifferential $\partial\varphi$ of a proper convex lower semicontinuous function $\varphi:X\to(-\infty,+\infty)$ is the maximal accretive operator.

Lemma 3^[2] For a given $x \in X,$ the point $z \in X$ satisfies the following inequality$:$

if and only if

$$x=J_{\varphi}^{\rho}(z),$$

where $\partial\varphi$ denotes the subdifferential of a proper convex lower semicontinuous function $\varphi,$ and

$$J_{\varphi}^{\rho}=(I+\rho \partial\varphi)^{-1}.$$

For any $x,y \in X$,$J_{\varphi}^{\rho}$ is nonexpansive,and

$$\|J_{\varphi}^{\rho}(x)-J_{\varphi}^{\rho}(y)\| \leqslant \|x-y\|.$$

From Lemma 3,we can see that the problem (1) is equivalent to the following system of equations:

Lemma 4^[28] Assume that $\{a_n\},$ $\{b_n\},$ and $\{c_n\}$ are three sequences of nonnegative real numbers such that

$$a_{n+1} \leqslant (1-\lambda_n)a_n +b_n \lambda_n +c_n,\quad \forall n>n_0,$$

where

$n_0$ is a nonnegative integer$,$ $\{\lambda_n\}$ is a sequence in $(0,1),$ and

Then$,$ $a_n\to 0 $ as $n \to +\infty.$

3 Main results

If $(x^{*},y^{*},z^{*}) \in$ SGNVID$(\Sigma,\Xi,\wedge,\vee)$, then by (2),we can see that

$$x^{*}=x^{*}-g_1(x^{*})+J_{\varphi}^{r_1}(g_1(y^{*})-r_1N_1(T_1y^{*},F_1y^{*})).$$

Consequently,if $S$ is a Lipschitz continuous mapping such that $x^{*} \in E(S),$ then

The formulation (3) is used to define the following iterative method for finding the common elements of the two different sets,i.e., the solution sets of the problem (1) and the set of the fixed point of the Lipschitz continuous mapping.

Algorithm 1\quad Let $r_1$,$r_2,$ and $r_3$ be the fixed positive real numbers. For the arbitrarily chosen initial $x_0 \in X$,compute the sequences $\{x_n\},$ $\{y_n \},$ and $\{z_n\}$ such that

where $\{\alpha_n\}$ is a sequence in $(0,1)$,and $ S:X\to X$ is a mapping.

{\bf Theorem 1} Let $X$ be a $q$-uniformly smooth Banach space. Let $N_i:X \times X \to X$ be the $(\mu_i, \nu_i)$-Lipschitz continuous mapping with the positive constants $\mu_i$ and $\nu_i$. Let $T_i :X \to X$ be the $\xi_i$-Lipschitz continuous mapping with the positive constant $\xi_i,$ and $F_i:X \to X$ be the $\beta_i$-Lipschitz continuous mapping with the positive constants $\beta_i$ $(i=1,2,3)$. Let $g_i:X \to X$ be the $\delta_i$-strongly accretive mapping with the positive constant $\delta_i$ and the $\rho_i$-Lipschitz continuous mapping with the positive constants $\rho_i$ $(i=1,2,3).$ Let $N_i:X\times X \to X$ be the ${\gamma_i}_{{N}_i}^{{T}_i}$-strongly accretive mapping with respect to the first variable and $T_i$ with the positive constant ${\gamma_i}_{{N}_i}^{{T}_i}$. Let $N_i$ be the ${\alpha_i}_{{N}_i}^{{F}_i}$-relaxed accretive mapping with respect to the second variable and $F_i$ with the positive constant ${\alpha_i}_{{N}_i}^{{ F}_i}.$ Let $ S:X \to X$ be a $\tau$-Lipschitz mapping such that

Assume that for each $x,y \in X,$

Put If the following conditions are satisfied:

(i) $(1-qr_1({\gamma_1}_{{N}_1}^{{T}_1}-{\alpha_1}_{{N}_1}^{{F}_1})+C_qr_{1}^q(\mu_1\xi_1+\nu_1\beta_1)^q)^{\frac{1}{q}} +2\Upsilon_1 < 1,$

(ii) $\Upsilon_1+\Upsilon_1\Omega + {\Phi_{{N}_{1}}^{{T}_1,{F}_1}(r_1)}\Omega < 1,$

(iii) $\tau \prod\limits_{i=1}^{3} \frac{\Phi_{{N}_i}^{{T}_i,{F}_i}(r_i)+\Upsilon_i}{1-\Upsilon_i} < 1,$

(iv) $\sum\limits_{n=0}^{\infty} \alpha_{n} = \infty,$

then the sequences $\{x_n\},$ $\{y_n\},$ and $\{z_n\}$ generated by Algorithm 1 converge strongly to $x^{*},$ $y^{*},$ \emph{and} $z^{*},$ respectively, such that $(x^{*},y^{*},z^{*}) \in $ SGNVID$(\Sigma,\Xi,\wedge,\vee)$ is the solution of the system of the general nonlinear variational inclusions and $x^{*} \in E(S).$

Proof Let $(x^{*},y^{*},z^{*}) \in $SGNVID$(\Sigma,\Xi,\wedge,\vee)$ be the solution of the system of the general nonlinear variational inclusions such that $x \in E(S)$. By (2) and (3),we have

Consequently,by (4),we obtain

Since $N_1(\cdot,\cdot)$ is $(\mu_1,\nu_1)$-Lipschitzian with the positive constants $\mu_1$ and $\nu_1$,$T_1$ is the $\xi_1$-Lipschitz continuous mapping with the positive constant $\xi_1$,and $F_1$ is the $\beta_1$-Lipschitz continuous mapping with the positive constant $\beta_1$,we have

Since $N_1(\cdot,\cdot)$ is the strongly accretive mapping with respect to the first variables of $N_1$ and $T_1$ with the positive constant ${\gamma_1}_{{N}_1}^{{T}_1}$,and is the relaxed accretive mapping with respect to the second variables of $N_1$ and $F_1$ with the positive constant ${\alpha_1}_{{N}_1}^{{F}_1}$,from Lemma 1,we have

where

$$\Phi_{{N}_1}^{{T}_1{F}_1}(r_1)=(1-qr_1{\gamma_1}_{{N}_1}^{{T}_1}+qr_1{\alpha_1}_{{N}_1}^{{F}_1}+C_qr{_1}^{q}(\mu_1 \xi_1+\nu_1\beta_1)^{q})^{\frac{1}{q}}.$$

Therefore, Note that Since $g_2$ is the $\delta_2$-strongly accretive mapping with respect to the positive constant $\delta_2$,and is the $\rho_2$-Lipschitz continuous mapping with respect to the positive constant $\rho_2$,we have

where

$$\Upsilon_{2}=(1-q\delta_2 +C_q\rho_{2}^{q})^{\frac{1}{q}}.$$

Therefore,

Now,we consider

By the assumptions of $N_2$,$T_2$,$F_2,$ and $g_2$ and (9) and (11),we have

From (12),(13),and (14),we have

Combining (10),(11),and (15),we have

We see that

Using the assumptions of $N_3$,$T_3$,$F_3,$ and $g_3,$ we have

Substituting (19) and (20) into (18),we have

Combining (17),(21),and (22),we have This implies that Substituting (24) into (16),we have That is

By (9) and (26),we get

Since $g_1$ is the $\delta_1$-strongly accretive mapping and $\rho_1$-Lipschitz continuous mapping,we have Substituting (26) into (29),we have Let Substitute (27),(28),and (30) into (8). Then,we get

$$\|x_{n+1}-x^{*}\|\leqslant (1-\alpha_n(1- \tau(\Upsilon_1+\Upsilon_1\Omega +\Phi_{{N}_{1}}^{{T}_1{F}_1}(r_1)\Omega)))\|x_n-x^{*}\|.$$

From the conditions (i) and (ii),we have

This shows that That is,

Put

Then,from (33),(34),and the assumption (iii),we have

$$\tau \ell=(0,1).$$

This implies that $\lambda_n\in (0,1).$ From the assumption (iv),we have

$$\sum_{n=0}^{\infty}\lambda_n =\infty.$$

Therefore,all the assumptions of Lemma 4 hold,and we can conclude that $x_n \to x^{*}$ as $n\to \infty.$ Consequently,from (24) and (26),we have $z_n \to z^{*}$ and $y_n \to y^{*}$ as $n\to \infty$,respectively. This completes the proof.