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Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces
Journal of Inequalities and Applications volume 2011, Article number: 49 (2011)
Abstract
In order to unify some variational inequality problems, in this paper, a new system of generalized quasivariational inclusion (for short, (SGQVI)) is introduced. By using Banach contraction principle, some existence and uniqueness theorems of solutions for (SGQVI) are obtained in real Banach spaces. Two new iterative algorithms to find the common element of the solutions set for (SGQVI) and the fixed points set for Lipschitz mappings are proposed. Convergence theorems of these iterative algorithms are established under suitable conditions. Further, convergence rates of the convergence sequences are also proved in real Banach spaces. The main results in this paper extend and improve the corresponding results in the current literature.
2000 MSC: 47H04; 49J40.
1 Introduction
Variational inclusion problems, which are generalizations of variational inequalities introduced by Stampacchia [1] in the early sixties, are among the most interesting and intensively studied classes of mathematics problems and have wide applications in the fields of optimization and control, economics, electrical networks, game theory, engineering science, and transportation equilibria. For the past decades, many existence results and iterative algorithms for variational inequality and variational inclusion problems have been studied (see, for example, [2–13]) and the references cited therein). Recently, some new and interesting problems, which are called to be system of variational inequality problems, were introduced and investigated. Verma [6], and Kim and Kim [7] considered a system of nonlinear variational inequalities, and Pang [14] showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problem can be modeled as a system of variational inequalities. Ansari et al. [2] considered a system of vector variational inequalities and obtained its existence results. Cho et al. [8] introduced and studied a new system of nonlinear variational inequalities in Hilbert spaces. Moreover, they obtained the existence and uniqueness properties of solutions for the system of nonlinear variational inequalities. Peng and Zhu [9] introduced a new system of generalized mixed quasivariational inclusions involving (H, η)monotone operators. Very recently, Qin et al. [15] studied the approximation of solutions to a system of variational inclusions in Banach spaces and established a strong convergence theorem in uniformly convex and 2uniformly smooth Banach spaces. Kamraksa and Wangkeeree [16] introduced a general iterative method for a general system of variational inclusions and proved a strong convergence theorem in strictly convex and 2uniformly smooth Banach spaces. Wangkeeree and Kamraksa [17] introduced an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a general system of variational inequalities, and then proved the strong convergence of the iterative in Hilbert spaces. Petrot [18] applied the resolvent operator technique to find the common solutions for a generalized system of relaxed cocoercive mixed variational inequality problems and fixed point problems for Lipschitz mappings in Hilbert spaces. Zhao et al. [19] obtained some existence results for a system of variational inequalities by Brouwer fixed point theory and proved the convergence of an iterative algorithm infinite Euclidean spaces.
Inspired and motivated by the works mentioned above, the purpose of this paper is to introduce and investigate a new system of generalized quasivariational inclusions (for short, (SGQVI)) in quniformly smooth Banach spaces, and then establish the existence and uniqueness theorems of solutions for the problem (SGQVI) by using Banach contraction principle. We also propose two iterative algorithms to find the common element of the solutions set for (SGQVI) and the fixed points set for Lipschitz mappings. Convergence theorems with estimates of convergence rates are established under suitable conditions. The results presented in this paper unifies, generalizes, and improves some results of [6, 15–20].
2 Preliminaries
Throughout this paper, without other specifications, we denote by Z_{+} and R the set of nonnegative integers and real numbers, respectively. Let E be a real quniformly Banach space with its dual E*, q > 1, denote the duality between E and E* by 〈·, ·〉 and the norm of E by  · , and T: E → E be a nonlinear mapping. When {x_{ n }} is a sequence in E, we denote strong convergence of {x_{ n }} to x ∈ E by x_{ n } → x. A Banach space E is said to be smooth if \underset{t\to 0}{lim}\frac{x+tyx}{t} exists for all x, y ∈ E with x = y = 1. It is said to be uniformly smooth if the limit is attained uniformly for x = y = 1. The function
is called the modulus of smoothness of E. E is called quniformly smooth if there exists a constant c > 0 such that ρ_{ E }(t) ≤ ct^{q}.
Example 2.1.[20] All Hilbert spaces, L^{p}(or l^{p}) and the Sobolev spaces {W}_{m}^{p}, (p ≥ 2) are 2uniformly smooth, while L^{p}(or l^{p}) and {W}_{m}^{p} spaces (1 < p ≤ 2) are puniformly smooth.
The generalized duality mapping J_{ q }: E → 2E* is defined as
for all x ∈ E. Particularly, J = J_{2} is the usual normalized duality mapping. It is wellknown that J_{ q }(x) = x^{q2}J(x) for x ≠ 0, J_{ q }(tx) = t^{q1}J_{ q }(x), and J_{ q }(x) = J_{ q }(x) for all x ∈ E and t ∈ [0, +∞), and J_{ q } is singlevalued if E is smooth. If E is a Hilbert space, then J = I, where I is the identity mapping. Many properties of the normalized duality mapping J_{ q } can be found in (see, for example, [21]). Let ρ_{1}, ρ_{2} be two positive constants, A_{1}, A_{2} : E × E → E be two singlevalued mappings, M_{1}, M_{2} : E → 2^{E} be two setvalued mappings. The (SGQVI) problem is to find (x*, y*) ∈ E × E such that
The set of solutions to (SGQVI) is denoted by Ω.
Special examples are as follows:

(I)
If A_{1} = A_{2} = A, E = H is a Hilbert space, and M_{1}(x) = M_{2}(x) = ∂ϕ (x) for all x ∈ E, where ϕ: E → R ∪ {+∞} is a proper, convex, and lower semicontinuous functional, and ∂ϕ denotes the subdifferential operator of ϕ, then the problem (SGQVI) is equivalent to find (x*, y*) ∈ E × E such that
\left\{\begin{array}{cc}\hfill \u27e8{\rho}_{1}A\left({y}^{*},{x}^{*}\right)+{x}^{*}{y}^{*},x{x}^{*}\u27e9+\varphi \left(x\right)\varphi \left({x}^{*}\right)\ge 0,\hfill & \hfill \forall x\in E,\hfill \\ \hfill \u27e8{\rho}_{2}A\left({x}^{*},{y}^{*}\right)+{y}^{*}{x}^{*},x{y}^{*}\u27e9+\varphi \left(x\right)\varphi \left({y}^{*}\right)\ge 0,\hfill & \hfill \forall x\in E,\hfill \end{array}\right.(2.2)
where ρ_{1}, ρ_{2} are two positive constants, which is called the generalized system of relaxed cocoercive mixed variational inequality problem [22].

(II)
If A_{1} = A_{2} = A, E = H is a Hilbert space, and K is a closed convex subset of E, M_{1}(x) = M_{2}(x) = ∂ϕ (x) and ϕ (x) = δ_{ K } (x) for all x ∈ E, where δ_{ K } is the indicator function of K defined by
\varphi \left(x\right)={\delta}_{K}\left(x\right)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \mathsf{\text{if}}\phantom{\rule{0.3em}{0ex}}x\in K,\hfill \\ \hfill +\infty \hfill & \hfill \mathsf{\text{otherwise}},\hfill \end{array}\right.
then the problem (SGQVI) is equivalent to find (x*, y*) ∈ K × K such that
where ρ_{1}, ρ_{2} are two positive constants, which is called the generalized system of relaxed cocoercive variational inequality problem [23].

(III)
If for each i ∈ {1, 2}, z ∈ E, A_{ i }(x, z) = Ψ_{ i }(x), for all x ∈ E, where Ψ_{ i } : E → E, then the problem (SGQVI) is equivalent to find (x*, y*) ∈ E × E such that
\left\{\begin{array}{c}\hfill 0\in {x}^{*}{y}^{*}+{\rho}_{1}\left({\Psi}_{1}\left({y}^{*}\right)+{M}_{1}\left({x}^{*}\right)\right),\hfill \\ \hfill 0\in {y}^{*}{x}^{*}+{\rho}_{2}\left({\Psi}_{2}\left({x}^{*}\right)+{M}_{2}\left({y}^{*}\right)\right),\hfill \end{array}\right.(2.4)
where ρ_{1}, ρ_{2} are two positive constants, which is called the system of quasivariational inclusion [15, 16].

(IV)
If A_{1} = A_{2} = A and M_{1} = M_{2} = M then the problem (SGQVI) is reduced to the following problem: find (x*, y*) ∈ E × E such that
\left\{\begin{array}{c}\hfill 0\in {x}^{*}{y}^{*}+{\rho}_{1}\left(A\left({y}^{*},{x}^{*}\right)+M\left({x}^{*}\right)\right),\hfill \\ \hfill 0\in {y}^{*}{x}^{*}+{\rho}_{2}\left(A\left({x}^{*},{y}^{*}\right)+M\left({y}^{*}\right)\right),\hfill \end{array}\right.(2.5)
where ρ_{1}, ρ_{2} are two positive constants.

(V)
If for each i ∈ {1, 2}, z ∈ E, A_{ i }(x, z) = Ψ (x), and M_{1}(x) = M_{2}(x) = M, for all x ∈ E, where Ψ: E → E, then the problem (SGQVI) is equivalent to find (x*, y*) ∈ E × E such that
\left\{\begin{array}{c}\hfill 0\in {x}^{*}{y}^{*}+{\rho}_{1}\left(\Psi \left({y}^{*}\right)+M\left({x}^{*}\right)\right),\hfill \\ \hfill 0\in {y}^{*}{x}^{*}+{\rho}_{2}\left(\Psi \left({x}^{*}\right)+M\left({y}^{*}\right)\right),\hfill \end{array}\right.
where ρ_{1}, ρ_{2} are two positive constants, which is called the system of quasivariational inclusion [16].
We first recall some definitions and lemmas that are needed in the main results of this work.
Definition 2.1.[21] Let M: dom(M) ⊂ E → 2^{E} be a setvalued mapping, where dom(M) is effective domain of the mapping M. M is said to be

(i)
accretive if, for any x, y ∈ dom(M), u ∈ M(x) and v ∈ M(y), there exists j_{ q }(x  y) ∈ J_{ q }(x  y) such that
\u27e8uv,{j}_{q}\left(xy\right)\u27e9\ge 0. 
(ii)
maccretive (maximalaccretive) if M is accretive and (I + ρM)dom(M) = E holds for every ρ > 0, where I is the identity operator on E.
Remark 2.1. If E is a Hilbert space, then accretive operator and maccretive operator are reduced to monotone operator and maximal monotone operator, respectively.
Definition 2.2. Let T: E → E be a singlevalued mapping. T is said to be a γLipschitz continuous mapping if there exists a constant γ > 0 such that
We denote by F(T) the set of fixed points of T, that is, F(T) = {x ∈ E: Tx = x}. For any nonempty set Ξ ⊂ E × E, the symbol Ξ ∩ F(T) ≠ ∅ means that there exist x*, y* ∈ E such that (x*, y*) ∈ Ξ and {x*, y*} ⊂ F(T).
Remark 2.2. (1) If γ = 1, then a γLipschitz continuous mapping reduces to a nonexpansive mapping.

(2)
If γ ∈ (0, 1), then a γLipschitz continuous mapping reduces to a contractive mapping.
Definition 2.3. Let A: E × E → E be a mapping. A is said to be

(i)
τLipschitz continuous in the first variable if there exists a constant τ > 0 such that, for x,\stackrel{\u0303}{x}\in E,
A\left(x,y\right)A\left(\stackrel{\u0303}{x},\u1ef9\right)\le \tau x\stackrel{\u0303}{x},\phantom{\rule{1em}{0ex}}\forall y,\u1ef9\in E. 
(ii)
αstrongly accretive if there exists a constant α > 0 such that
\u27e8A\left(x,y\right)A\left(\stackrel{\u0303}{x},\u1ef9\right),{J}_{q}\left(x\stackrel{\u0303}{x}\right)\u27e9\ge \alpha x\stackrel{\u0303}{x}{}^{q},\phantom{\rule{1em}{0ex}}\forall \left(x,y\right),\left(\stackrel{\u0303}{x},\u1ef9\right)\in E\times E,
or equivalently,

(iii)
αinverse strongly accretive or αcocoercive if there exists a constant α > 0 such that
\u27e8A\left(x,y\right)A\left(\stackrel{\u0303}{x},\u1ef9\right),{J}_{q}\left(x\stackrel{\u0303}{x}\right)\u27e9\ge \alpha A\left(x,y\right)A\left(\stackrel{\u0303}{x},\u1ef9\right){}^{q},\phantom{\rule{1em}{0ex}}\forall \left(x,y\right),\left(\stackrel{\u0303}{x},\u1ef9\right)\in E\times E,
or equivalently,

(iv)
(μ, ν)relaxed cocoercive if there exist two constants μ ≤ 0 and ν > 0 such that
\u27e8A\left(x,y\right)A\left(\stackrel{\u0303}{x},\u1ef9\right),{J}_{q}\left(x\stackrel{\u0303}{x}\right)\u27e9\ge \left(\mu \right)A\left(x,y\right)A\left(\stackrel{\u0303}{x},\u1ef9\right){}^{q}+\nu x\stackrel{\u0303}{x}{}^{q},\phantom{\rule{1em}{0ex}}\forall \left(x,y\right),\left(\stackrel{\u0303}{x},\u1ef9\right)\in E\times E.
Remark 2.3. (1) Every αstrongly accretive mapping is a (μ, α)relaxed cocoercive for any positive constant μ. But the converse is not true in general.

(2)
The conception of the cocoercivity is applied in several directions, especially for solving variational inequality problems by using the auxiliary problem principle and projection methods [24]. Several classes of relaxed cocoercive variational inequalities have been investigated in [18, 23, 25, 26].
Definition 2.4. Let the setvalued mapping M: dom(M) ⊂ E → 2^{E} be maccretive. For any positive number ρ > 0, the mapping R_{(ρ, M)}: E → dom(M ) defined by
is called the resolvent operator associated with M and ρ, where I is the identity operator on E.
Remark 2.4. Let C ⊂ E be a nonempty closed convex set. If E is a Hilbert space, and M = ∂ϕ, the subdifferential of the indicator function ϕ, that is,
then R_{(ρ, M)}= P_{ C }, the metric projection operator from E onto C.
In order to estimate of convergence rates for sequence, we need the following definition.
Definition 2.5. Let a sequence {x_{ n }} converge strongly to x*. The sequence {x_{ n }} is said to be at least linear convergence if there exists a constant ϱ ∈ (0, 1) such that
Lemma 2.1.[27] Let the setvalued mapping M: dom(M) ⊂ E → 2^{E} be maccretive. Then the resolvent operator R_{(ρ, M)}is single valued and nonexpansive for all ρ > 0:
Lemma 2.2.[28] Let {a_{ n }} and {b_{ n }} be two nonnegative real sequences satisfying the following conditions:
for some n_{0} ∈ N, {λ_{ n }} ⊂ (0, 1) with {\sum}_{n=0}^{\infty}{\lambda}_{n}=\infty and b_{ n } = 0(λ_{ n }). Then lim_{n → ∞}a_{ n } = 0.
Lemma 2.3.[29] Let E be a real quniformly Banach space. Then there exists a constant c_{ q } > 0 such that
3 Existence and uniqueness of solutions for (SGQVI)
In this section, we shall investigate the existence and uniqueness of solutions for (SGQVI) in quniformly smooth Banach space under some suitable conditions.
Theorem 3.1. Let ρ_{1}, ρ_{2} be two positive constants, and (x*, y*) ∈ E × E. Then (x*, y*) is a solution of the problem (2.1) if and only if
Proof. It directly follows from Definition 2.4. This completes the proof. □
Theorem 3.2. Let E be a real quniformly smooth Banach space. Let M_{2} : E → 2^{E} be maccretive mapping, A_{2} : E × E → E be (μ_{2}, ν_{2})relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_{2}. Then, for each x ∈ E, the mapping {R}_{\left({\rho}_{2},{M}_{2}\right)}\left(x{\rho}_{2}{A}_{2}\left(x,\cdot \right)\right):E\to E has at most one fixed point. If
then the implicit function y(x) determined by
is continuous on E.
Proof. Firstly, we show that, for each x ∈ E, the mapping {R}_{\left({\rho}_{2},{M}_{2}\right)}\left(x{\rho}_{2}{A}_{2}\left(x,\cdot \right)\right):E\to E has at most one fixed point. Assume that y,\u1ef9\in E such that
Since A_{2} is Lipschitz continuous in the first variable with constant τ_{2}, then
Therefore, y=\u1ef9.
On the other hand, for any sequence {x_{ n }} ⊂ E, x_{0} ∈ E, x_{ n } → x_{0} as n → ∞: Since A_{2} : E × E → E is (μ_{2}, ν_{2})relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_{2}, one has
As a consequence, we have, by Lemma 2.1,
Together with (3.2), it yields that the implicit function y(x) is continuous on E. This completes the proof. □
Theorem 3.3. Let E be a real quniformly smooth Banach space. Let M_{2} : E → 2^{E} be maccretive mapping, A_{2} : E × E → E be α_{2}strong accretive and Lipschitz continuous in the first variable with constant τ_{2}. Then, for each x ∈ E, the mapping {R}_{\left({\rho}_{2},{M}_{2}\right)}\left(x{\rho}_{2}{A}_{2}\left(x,\cdot \right)\right):E\to E has at most one fixed point. If 1q{\rho}_{2}{\alpha}_{2}+{c}_{q}{\rho}_{2}^{q}{\tau}_{2}^{q}\ge 0, then the implicit function y(x) determined by
is continuous on E.
Proof. The proof is similar to Theorem 3.2 and so the proof is omitted. This completes the proof. □
Theorem 3.4. Let E be a real quniformly smooth Banach space. Let M_{ i }: E → 2^{E} be m accretive mapping, A_{ i }: E × E → E be (μ_{ i }, ν_{ i })relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_{ i } for i ∈ {1, 2}. If 1q{\rho}_{2}{\nu}_{2}+q{\rho}_{2}{\mu}_{2}{\tau}_{2}^{q}+{c}_{q}{\rho}_{2}^{q}{\tau}_{2}^{q}\ge 0, and
Then the solutions set Ω of (SGQVI) is nonempty. Moreover, Ω is a singleton.
Proof. By Theorem 3.2, we define a mapping P: E → E by
Since A_{ i } : E × E → E are (μ_{ i }, ν_{ i })relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_{ i } for i ∈ {1, 2}, one has, for any x,\stackrel{\u0303}{x}\in E,
and
From both Lemma 2.1 and Theorem 3.1, we get
Note that
Therefore, we obtain
From (3.3), this yields that the mapping P is contractive. By Banach contraction principle, there exists a unique x* ∈ E such that P(x*) = x*. Therefore, from Theorem 3.2, there exists an unique (x*, y*) ∈ Ω, where y* = y(x*). This completes the proof. □
Theorem 3.5. Let E be a real quniformly smooth Banach space. Let M_{ i } : E → 2^{E} be m accretive mapping, A_{ i } : E × E → E be α_{ i }strong accretive and Lipschitz continuous in the first variable with constant τ_{ i } for i ∈ {1, 2}. If 1q{\rho}_{2}{\alpha}_{2}+{c}_{q}{\rho}_{2}^{q}{\tau}_{2}^{q}\ge 0, and
Then the solutions set Ω of (SGQVI) is nonempty. Moreover, Ω is a singleton.
Proof. It is easy to know that Theorem 3.5 follows from Remark 2.3 and Theorem 3.4 and so the proof is omitted. This completes the proof. □
In order to show the existence of ρ_{ i }, i = 1, 2, we give the following examples.
Example 3.1. Let E be a 2uniformly smooth space, and let M_{1}, M_{2}, A_{1} and A_{2} be the same as Theorem 3.4. Then there exist ρ_{1}, ρ_{2} > 0 such that (3.3), where
or
Example 3.2. Let E be a 2uniformly smooth space, and let M_{1}, M_{2}, A_{1} and A_{2} be the same as Theorem 3.5. Then there exist ρ_{1}, ρ_{2} > 0 such that (3.4), where
or
4 Algorithms and convergence analysis
In this section, we introduce twostep iterative sequences for the problem (SGQVI) and a nonlinear mapping, and then explore the convergence analysis of the iterative sequences generated by the algorithms.
Let T: E → E be a nonlinear mapping and the fixed points set F(T) of T be a nonempty set. In order to introduce the iterative algorithm, we also need the following lemma.
Lemma 4.1. Let E be a real quniformly smooth Banach space, ρ_{1}, ρ_{2} be two positive constants. If (x*, y*) ∈ Ω and {x*, y*} ⊂ F(T), then
Proof. It directly follows from Theorem 3.1. This completes the proof. □
Now we introduce the following iterative algorithms for finding a common element of the set of solutions to a (SGQVI) problem (2.1) and the set of fixed points of a Lipschtiz mapping.
Algorithm 4.1. Let E be a real quniformly smooth Banach space, ρ_{1}, ρ_{2} > 0, and let T: E → E be a nonlinear mapping. For any given points x_{0}, y_{0} ∈ E, define sequences {x_{ n }} and {y_{ n }} in E by the following algorithm:
where {α_{ n }} and {β_{ n }} are sequences in [0, 1].
Algorithm 4.2. Let E be a real quniformly smooth Banach space, ρ_{1}, ρ_{2} > 0, and let T: E → E be a nonlinear mapping. For any given points x_{0}, y_{0} ∈ E, define sequences {x_{ n }} and {y_{ n }} in E by the following algorithm:
where {α_{ n }} is a sequence in [0, 1].
Remark 4.1. If A_{1} = A_{2} = A, E = H is a Hilbert space, and M_{1}(x) = M_{2}(x) = ∂ϕ(x) for all x ∈ E, where ϕ: E → R ∪ {+∞} is a proper, convex and lower semicontinuous functional, and ∂ϕ denotes the subdifferential operator of ϕ, then Algorithm 4.1 is reduced to the Algorithm (I) of [18].
Theorem 4.1. Let E be a real quniformly smooth Banach space, and A_{1}, A_{2}, M_{1} and M_{2} be the same as in Theorem 3.4, and let T be a κLipschitz continuous mapping. Assume that Ω ∩ F(T) ≠ ∅, {α_{ n }} and {β_{ n }} are sequences in [0, 1] and satisfy the following conditions:

(i)
{\sum}_{i=0}^{\infty}{\alpha}_{n}=\infty;

(ii)
lim_{n→ ∞}β_{ n } = 1;

(iii)
0<\kappa \sqrt[q]{1q{\rho}_{i}{\nu}_{i}+q{\rho}_{i}{\mu}_{i}{\tau}_{i}^{q}+{c}_{q}{\rho}_{i}^{q}{\tau}_{i}^{q}}<1,\phantom{\rule{0.3em}{0ex}}i=1,2.
Then the sequences {x_{ n }} and {y_{ n }} generated by Algorithm 4.1 converge strongly to x* and y*, respectively, such that (x*, y*) ∈ and {x*, y*} ⊂ F(T).
Proof. Let (x*, y*) ∈ Ω and {x*, y*} ⊂ F(T). Then, from (4.1), one has
Since T is a κLipschitz continuous mapping, and from both (4.2) and (4.3), we have
For each i ∈ {1, 2}, A_{ i } : E × E → E are (μ_{ i }, ν_{ i })relaxed cocoercive and Lipschitz continuous in the first variable with constant τ_{ i }, then
and so
Furthermore, by Lemma 2.1, one can obtain
and consequently,
Note that
Therefore, we have
Set \iota =max\left\{\sqrt[q]{1q{\rho}_{i}{\nu}_{i}+q{\rho}_{i}{\mu}_{i}\underset{i}{\overset{q}{\tau}}+{c}_{q}{\rho}_{i}^{q}{\tau}_{i}^{q}}:i=1,2\right\}. So the above inequality can be written as follows:
Taking a_{ n } = x_{ n }  x*, λ_{ n } = α_{ n }[1  κι(1  β_{ n }(1  κι))] and b_{ n } = α_{ n } κι(1 β_{ n }) x*  y*. By the condition (iii), we get
In addition, from the conditions (i) and (ii), it yields that b_{ n } = 0(λ_{ n }) and
Therefore, by Lemma 2.2, we obtain
that is, x_{ n } → x* as n → ∞. Again from lim_{n → ∞}β_{ n } = 1 and (4.6), one concludes
i.e., y_{ n } → y* as n → ∞. Thus (x_{ n }, y_{ n }) converges strongly to (x*, y*). This completes the proof. □
Theorem 4.2. Let E be a real quniformly smooth Banach space, and A_{1}, A_{2}, M_{1} and M_{2} be the same as in Theorem 3.5, and let T be a κLipschitz continuous mapping. Assume that Ω ∩ F(T) ≠ ∅, {α_{ n }} and {β_{ n }} are sequences in [0, 1] and satisfy the following conditions:

(i)
{\sum}_{i=0}^{\infty}{\alpha}_{n}=\infty;

(ii)
lim_{n→ ∞}β_{ n } = 1;

(iii)
0<\kappa \sqrt[q]{1q{\rho}_{i}{\alpha}_{i}+{c}_{q}{\rho}_{i}^{q}{\tau}_{i}^{q}}<1,\phantom{\rule{0.3em}{0ex}}i=1,2.
Then the sequences {x_{ n }} and {y_{ n }} generated by Algorithm 4.1 converge strongly to x* and y*, respectively, such that (x*, y*) ∈ Ω and {x*, y*} ⊂ F(T).
Proof. The proof is similar to the proof of Theorem 4.1 and so the proof is omitted. This completes the proof. □
Theorem 4.3. Let E be a real quniformly smooth Banach space, and A_{1}, A_{2}, M_{1} and M_{2} be the same as in Theorem 3.4, and let T be a κLipschitz continuous mapping. Assume that Ω ∩ F(T) ≠ ∅, {α_{ n }} is a sequence in (0, 1] and satisfy the following conditions:

(i)
{\sum}_{i=0}^{\infty}{\alpha}_{n}=\infty;

(ii)
0<\kappa \sqrt[q]{1q{\rho}_{i}{\nu}_{i}+q{\rho}_{i}{\mu}_{i}{\tau}_{i}^{q}+{c}_{q}{\rho}_{i}^{q}{\tau}_{i}^{q}}<1,\phantom{\rule{0.3em}{0ex}}i=1,2.
Then the sequences {x_{ n }} and {y_{ n }} generated by Algorithm 4.2 converge strongly to x* and y*, respectively, such that (x*, y*) ∈ Ω and {x*, y*} ⊂ F(T). Furthermore, sequences {x_{ n }} and {y_{ n }} are at least linear convergence.
Proof. From the proof of Theorem 4.1, it is easy to know that the sequences {x_{ n }} and {y_{ n }} generated by Algorithm 4.2 converge strongly to x* and y*, respectively, such that (x*, y*) ∈ Ω and {x*, y*} ⊂ F(T), and so,
Since {α_{ n }} is a sequence in (0, 1], we obtain, from (4.5),
and so,
Therefore, from (4.7)(4.10), it implies that sequences {x_{ n }} and {y_{ n }} are at least linear convergence. This completes the proof. □
Theorem 4.4. Let E be a real quniformly smooth Banach space, and A_{1}, A_{2}, M_{1} and M_{2} be the same as in Theorem 3.5, and let T be a κLipschitz continuous mapping. Assume that Ω ∩ F(T) ≠ ∅, {α_{ n }} is a sequence in (0, 1] and satisfy the following conditions:

(i)
{\sum}_{i=0}^{\infty}{\alpha}_{n}=\infty;

(ii)
lim_{n→∞}β_{ n } = 1;

(iii)
0<\kappa \sqrt[q]{1q{\rho}_{i}{\alpha}_{i}+{c}_{q}{\rho}_{i}^{q}{\tau}_{i}^{q}}<1,\phantom{\rule{0.3em}{0ex}}i=1,2.
Then the sequences {x_{ n }} and {y_{ n }} generated by Algorithm 4.2 converge strongly to x* and y*, respectively, such that (x*, y*) ∈ Ω and {x*, y*} ⊂ F(T). Furthermore, sequences {x_{ n }} and {y_{ n }} are at least linear convergence.
Proof. In a way similar to the proof of Theorem 4.2, with suitable modifications, we can obtain that the conclusion of Theorem 4.4 holds. This completes the proof. □
Remark 4.2. Theorem 4.1 generalizes and improves the main result in [18].
Abbreviations
 (SGQVI):

system of generalized quasivariational inclusion.
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Acknowledgements
The authors would like to thank two anonymous referees for their valuable comments and suggestions, which led to an improved presentation of the results, and grateful to Professor Siegfried Carl as the Editor of our paper. This work was supported by the Natural Science Foundation of China (Nos. 71171150,70771080,60804065), the Academic Award for Excellent Ph.D. Candidates Funded by Wuhan University and the Fundamental Research Fund for the Central Universities.
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JC carried out the (SGQVI) studies, participated in the sequence alignment and drafted the manuscript. ZW participated in the sequence alignment. All authors read and approved the final manuscript.
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Chen, Jw., Wan, Z. Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces. J Inequal Appl 2011, 49 (2011). https://doi.org/10.1186/1029242X201149
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DOI: https://doi.org/10.1186/1029242X201149