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Convergence theorems for common solutions of various problems with nonlinear mapping
Journal of Inequalities and Applications volume 2014, Article number: 2 (2014)
Abstract
In this paper, motivated and inspired by Zegeye and Shahzad (Nonlinear Anal. 70:27072716, 2009), Qin et al. (J. Comput. Appl. Math. 225(1):2030, 2009) and Kimura and Takahashi (J. Math. Anal. Appl. 357:356363, 2009), we introduce a new hybrid projection iterative scheme that converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points for a family of hemirelatively nonexpansive mappings in a Banach space.
MSC:47H10, 47J20, 49J40.
1 Introduction
A real Banach space E is said to be strictly convex if \parallel \frac{x+y}{2}\parallel <1 for all x,y\in E with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. It is said to be uniformly convex if {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0 for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in E such that \parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1 and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1. It is known that a uniformly convex Banach space is reflexive and strictly convex. Let U=\{x\in E:\parallel x\parallel =1\} be the unit sphere of E. Then the Banach space E is said to be smooth if
exists for each x,y\in U. It is said to be uniformly smooth if the limit is attained uniformly for x,y\in E.
Let E be a real Banach space with the norm \parallel \cdot \parallel, and let {E}^{\ast} denote the dual space of E. We denote by J the normalized duality mapping from E to {2}^{{E}^{\ast}} defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing. It is well known that if {E}^{\ast} is strictly convex, then J is singlevalued, and if E is uniformly smooth, then J is uniformly normtonorm continuous on a bounded subset of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then {J}^{1} is singlevalued, onetoone, surjective and it is the duality mapping from {E}^{\ast} into E and thus J{J}^{1}={I}_{{E}^{\ast}} and {J}^{1}J={I}_{E} (see [1]). We note that J is the identity mapping in a Hilbert space.
A mapping A:D(A)\subset E\to {E}^{\ast} is said to be monotone if for each x,y\in D(A),
A mapping A is said to be γinverse strongly monotone if there exists a positive real number \gamma >0 such that
If a mapping A is γinverse strongly monotone, then it is Lipschitz continuous with constant \frac{1}{\gamma}, i.e.,
A mapping A is said to be strongly monotone, if for each x,y\in D(A), there exists k\in (0,1) such that
A monotone mapping A is said to be maximal if its graph G(A)=\{(x,y):y\in Ax\} is not properly contained in the graph of any other monotone mapping. It is known that the monotone mapping A is maximal if and only if for (x,{x}^{\ast})\in E\times {E}^{\ast},
for every (y,{y}^{\ast})\in G(A) implies that {x}^{\ast}\in Ax.
Let C be a nonempty, closed convex subset of a Banach space E. For a bifunction \theta :C\times C\to \mathbb{R}, we assume that θ satisfies the following conditions:
(E1) \theta (x,x)=0 for all x\in C;
(E2) θ is monotone, i.e., \theta (x,y)+\theta (y,x)\le 0 for all x,y\in C;
(E3) for each x,y,z\in C,
(E4) for each x\in C, the function y\mapsto \theta (x,y) is convex and lower semicontinuous.
The generalized mixed equilibrium problem is to find x\in C such that
where ψ is a lower semicontinuous and convex function. The set of solutions of problem (1.1) is denoted by GMEP. Recently, Zhang [2] considered this problem. Some special cases of problem (1.1) are stated as follows.
If A=0, then problem (1.1) reduces to the following mixed equilibrium problem of finding x\in C such that
which was considered by Ceng and Yao [3].
If \psi =0, then problem (1.1) reduces to the following generalized equilibrium problem of finding x\in C such that
which was studied in [4].
If \psi =0 and A=0, then problem (1.1) reduces to the following equilibrium problem of finding x\in C such that
The set of solutions of problem (1.2) is denoted by EP.
If \theta =0 and \psi =0, then problem (1.1) reduces to the following classical variational inequality problem of finding x\in C such that
The set of solutions of problem (1.3) is denoted by \mathit{VI}(C,A).
Equilibrium problems, which were introduced in [5] in 1994, have had great impact and influence on the development of several branches of pure and applied sciences. They include numerous problems in economics, finance, physics, network, elasticity, optimization, variational inequalities, minimax problems, and semigroups; see, for instance, [2–4, 6–17] and the references therein.
As well known, if C is a nonempty, closed and convex subset of a Hilbert space H and {P}_{C}:H\to C is the metric projection of H onto C, then {P}_{C} is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not true to more general Banach spaces. In this connection, Alber [18] introduced a generalized projection operator {\mathrm{\Pi}}_{C} in a Banach space E which is an analogue of the metric projection in Hilbert spaces. Consider the functional defined by
Observe that, in a Hilbert space H, (1.4) reduces to
The generalized projection {\mathrm{\Pi}}_{C}:E\to C is a mapping that assigns an arbitrary point x\in E to the minimum point of the functional \varphi (x,y), that is, {\mathrm{\Pi}}_{C}x=\overline{x}, where \overline{x} is the solution to the minimization problem
The existence and uniqueness of the mapping {\mathrm{\Pi}}_{C} follows from the properties of the functional \varphi (x,y) and strict monotonicity of the mapping J (see, for example, [18] and [19]). In a Hilbert space, {\mathrm{\Pi}}_{C}={P}_{C}. It is obvious from the definition of the function ϕ that:

(1)
{(\parallel x\parallel \parallel y\parallel )}^{2}\le \varphi (x,y)\le {(\parallel x\parallel +\parallel y\parallel )}^{2} for all x,y\in E.

(2)
\varphi (x,y)=\varphi (x,z)+\varphi (z,y)+2\u3008xz,JzJy\u3009 for all x,y,z\in E.

(3)
\varphi (x,y)=\u3008x,JxJy\u3009+\u3008yx,Jy\u3009\le \parallel x\parallel \parallel JxJy\parallel +\parallel yx\parallel \parallel y\parallel for all x,y\in E.

(4)
If E is a reflexive, strictly convex and smooth Banach space, then, for all x,y\in E,
\varphi (x,y)=0\phantom{\rule{1em}{0ex}}\text{if and only if}\phantom{\rule{1em}{0ex}}x=y.
Remark 1.1 In (4), it is sufficient to show that if \varphi (x,y)=0 then x=y. In fact, from (1) we have \parallel x\parallel =\parallel y\parallel. This implies that \u3008x,Jy\u3009={\parallel x\parallel}^{2}={\parallel Jy\parallel}^{2}. From the definition of J, we have Jx=Jy. Therefore, we have x=y. For more details, see [1].
Let C be a nonempty closed and convex subset of E, and let T be a mapping from C into itself. We denote by F(T) be the set of fixed points of T. A point p in C is said to be a weak asymptotic fixed point of T [20] if C contains a sequence \{{x}_{n}\} which converges weakly to p such that {lim}_{n\to \mathrm{\infty}}(T{x}_{n}{x}_{n})=0. The set of asymptotic fixed points of T will be denoted by \stackrel{\u02c6}{F}(T). A mapping T from C into itself is called relatively nonexpansive [21–23] if \stackrel{\u02c6}{F}(T)=F(T) and \varphi (p,Tx)\le \varphi (p,x) for all x\in C and p\in F(T). The asymptotic behavior of relatively nonexpansive mappings was studied in [21, 22] and [12].
A point p in C is said to be a strong asymptotic fixed point of T if C contains a sequence \{{x}_{n}\} which converges strongly to p such that {lim}_{n\to \mathrm{\infty}}(T{x}_{n}{x}_{n})=0. The set of strong asymptotic fixed points of T is denoted by \tilde{F}(T). A mapping T from C into itself is called relatively weak nonexpansive if \tilde{F}(T)=F(T) and \varphi (p,Tx)\le \varphi (p,x) for all x\in C and p\in F(T). A mapping T is called hemirelatively nonexpansive if F(T)\ne \mathrm{\varnothing} and \varphi (p,Tx)\le \varphi (p,x) for all x\in C and p\in F(T).
Remark 1.2 (1) It is obvious that a relatively nonexpansive mapping is a relatively weak nonexpansive mapping (see [14]). In fact, for any mapping T:C\to C, we have F(T)\subset \tilde{F}(T)\subset \stackrel{\u02c6}{F}(T). Therefore, if T is a relatively nonexpansive mapping, then F(T)=\tilde{F}(T)=\stackrel{\u02c6}{F}(T).

(2)
The class of hemirelatively nonexpansive mappings is more general than the class of relatively weak nonexpansive mappings.
The converse of Remark 1.2 is not true. In order to explain this better, we give the following example.
Example 1.1 ([24])
Let E be any smooth Banach space, and let {x}_{0}\ne 0 be any element of E. We define a mapping T:E\to E as follows:
for n=1,2,3,\dots . Then T is a hemirelatively nonexpansive mapping but not a relatively nonexpansive mapping.
Remark 1.3 There are other examples of hemirelatively nonexpansive mappings such as the generalized projections (or projections) from a smooth, strictly convex and reflexive Banach space, and others; see [25].
A mapping T:C\to C is said to be closed, if for any sequence \{{x}_{n}\}\subset C with {x}_{n}\to x and T{x}_{n}\to y, then Tx=y.
In 2009, Kimura and Takahashi [26] proposed the following hybrid iteration method with a generalized projection for a family of relatively nonexpansive mappings \{{T}_{\lambda}\} in a Banach space E:
They proved that \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathcal{F}}x\in C, where \mathcal{F}={\bigcap}_{\lambda \in \mathrm{\Lambda}}F({T}_{\lambda}) is the set of common fixed points of {T}_{\lambda}, and {\mathrm{\Pi}}_{K} is the generalized projection of E onto a nonempty closed convex subset K of E.
Recently, Zegeye and Shahzad [14] introduced the following iterative scheme for finding a common element of the solution set of a variational inequality problem and a fixed point of a relatively weak nonexpansive mapping with γinverse strongly monotone mapping satisfying \parallel Ax\parallel \le \parallel AxAp\parallel for all x\in C and p\in \mathit{VI}(C,A) (see, e.g., [27]):
On the other hand, Qin et al. [25] proposed the following hybrid iterative scheme:
where T:C\to C is a closed hemirelatively nonexpansive mapping. Under suitable conditions, they proved that the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{F(T)\cap \mathit{EP}(f)}{x}_{0}, where \mathit{EP}(f) is the solution of an equilibrium problem for a bifunction f:C\times C\to \mathbb{R}.
In this paper, we introduce a new hybrid projection iterative scheme that converges strongly to a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of common fixed points for a family of hemirelatively nonexpansive mappings in a Banach space.
2 Preliminaries
Let E be a normed linear space with dimE\ge 2. The modulus of smoothness of E is the function {\rho}_{E}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) defined by
The space E is said to be smooth if {\rho}_{E}(\tau )>0, \mathrm{\forall}\tau >0 and E is called uniformly smooth if and only if {lim}_{t\to 0}\frac{{\rho}_{E}(t)}{t}=0. The modulus of convexity of E is the function {\delta}_{E}:(0,2]\to [0,1] defined by
E is called uniformly convex if and only if {\delta}_{E}(\epsilon )>0 for every \epsilon \in (0,2]. Let p>1. Then E is said to be puniformly convex if there exists a constant c>0 such that \delta (\epsilon )\ge c\cdot {\epsilon}^{p} for all \epsilon \in [0,2]. Observe that every puniformly convex space is uniformly convex.
It is well known (see, for example, [28]) that
In the following, we shall need the following results.
Lemma 2.1 ([28])
Let E be a 2uniformly convex and smooth Banach space. Then, for all x,y\in E, we have
where J is the normalized duality mapping of E, and \frac{1}{c} (0<c\le 1) is the 2uniformly convex constant of E.
Let E be a real smooth, strictly convex and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then the following conclusions hold:

(i)
\varphi (y,{\mathrm{\Pi}}_{C}x)+\varphi ({\mathrm{\Pi}}_{C}x,x)\le \varphi (y,x), \mathrm{\forall}x\in E, y\in C.

(ii)
Suppose x\in E and z\in C. Then
z={\mathrm{\Pi}}_{C}x\phantom{\rule{1em}{0ex}}\iff \phantom{\rule{1em}{0ex}}\u3008zy,JxJz\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in C.
Lemma 2.3 ([30])
Let E be a strictly convex and smooth Banach space, C be a nonempty closed and convex subset of E and T:C\to C be a hemirelatively nonexpansive mapping. Then F(T) is a closed convex subset of C.
Lemma 2.4 ([29])
Let E be a real smooth and uniformly convex Banach space, and let \{{x}_{n}\}, \{{y}_{n}\} be two sequences of E. If {lim}_{n\to \mathrm{\infty}}\varphi ({x}_{n},{y}_{n})=0 and either \{{x}_{n}\} or \{{y}_{n}\} is bounded, then {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0.
Lemma 2.5 ([31])
Let E be a real smooth Banach space and A:E\to {E}^{\ast} be a maximal monotone mapping. Then {A}^{1}(0) is a closed and convex subset of E.
We denote the normal cone for C at a point v\in C by {N}_{C}(v), that is,
Lemma 2.6 ([32])
Let C be a nonempty closed convex subset of a Banach space E, and let A be a monotone and hemicontinuous mapping of C into {E}^{\ast} with C=D(A). Let T\subset E\times {E}^{\ast} be a mapping defined as follows:
Then T is maximal monotone and {T}^{1}(0)=\mathit{VI}(C,A).
Remark 2.1 It is well known that the monotone and hemicontinuous mapping A with D(A)=E is maximal (see, e.g., [1]). Note that Lemma 2.6 is for the monotone and hemicontinuous mapping.
Remark 2.2 Let C be a nonempty closed convex subset of a Banach space E, and let A be a monotone and hemicontinuous mapping from C into {E}^{\ast} with C=D(A). Then
It is obvious that the set \mathit{VI}(C,A) is a closed convex subset of C and the set {A}^{1}0=\mathit{VI}(E,A) is a closed convex subset of E (see [27]).
We make use of the function V:E\times {E}^{\ast}\to \mathbb{R} defined by
for all x\in E and {x}^{\ast}\in {E}^{\ast}, which was studied by Alber [18]. That is,
for all x\in E and {x}^{\ast}\in {E}^{\ast}. We know the following lemma.
Lemma 2.7 ([18])
Let E be a reflexive, strictly convex and smooth Banach space with {E}^{\ast} as its dual. Then
for all x\in E and {x}^{\ast},{y}^{\ast}\in {E}^{\ast}.
Lemma 2.8 ([2])
Let C be a closed subset of a smooth, strictly convex and reflexive Banach space E. Let B:C\to {E}^{\ast} be a continuous and monotone mapping, \psi :C\to \mathbb{R} be a lower semicontinuous and convex function, and θ be a bifunction from C\times C to ℝ satisfying (E1)(E4). Then, for r>0 and x\in E, there exists u\in C such that
Define a mapping {T}_{r}:E\to C by
for all x\in E. Then the following conclusions hold:

(i)
{T}_{r} is singlevalued;

(ii)
{T}_{r} is a firmly nonexpansive type mapping [33], i.e., for all x,y\in E,
\u3008{T}_{r}x{T}_{r}y,J{T}_{r}xJ{T}_{r}y\u3009\le \u3008{T}_{r}x{T}_{r}y,JxJy\u3009; 
(iii)
F({T}_{r})=\mathit{GMEP}=\stackrel{\u02c6}{F}({T}_{r});

(iv)
GMEP is a closed and convex subset of C;

(v)
\varphi (p,{T}_{r}z)+\varphi ({T}_{r}z,z)\le \varphi (p,z), \mathrm{\forall}p\in F({T}_{r}), x\in E.
Remark 2.3 ([2])
The mapping {T}_{r}:E\to C defined by (2.1) is a relatively nonexpansive mapping. Thus, it is a hemirelatively nonexpansive mapping.
3 An iterative scheme for a family of hemirelatively nonexpansive mappings
In this section, we introduce a new hybrid iterative scheme for a common element of the solution set of problem (1.1), the solution set of problem (1.3) for an inverse strongly monotone mapping and the set of common fixed points of a family of hemirelatively nonexpansive mappings.
Theorem 3.1 Let E be a real uniformly smooth and 2uniformly convex Banach space and C be a nonempty, closed and convex subset of E. Let A:C\to {E}^{\ast} be a γinverse strongly monotone mapping and B:C\to {E}^{\ast} be a continuous and monotone mapping. Let \psi :C\to \mathbb{R} be a lower semicontinuous and convex function and θ be a bifunction from C\times C to ℝ satisfying (E1)(E4). Let \{{T}_{\lambda}:\lambda \in \mathrm{\Lambda}\} be a family of closed hemirelatively nonexpansive mappings of C into itself having
where \mathcal{F}={\bigcap}_{\lambda \in \mathrm{\Lambda}}F({T}_{\lambda}) is the set of common fixed points of \{{T}_{\lambda}\}. Assume that \parallel Ax\parallel \le \parallel AxAp\parallel for all x\in C and p\in \mathit{VI}(C,A). Suppose that 0<a<{\mu}_{n}<b=\frac{{c}^{2}\gamma}{2}, where c is the constant in Lemma 2.1. Let \{{r}_{n}\}\subset [{c}^{\ast},+\mathrm{\infty}) for some {c}^{\ast}>0. Let \{{x}_{n}\} be the sequence generated by
where J is the normalized duality mapping, and \{{\alpha}_{n}\} is a sequence in [0,1] satisfying {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1{\alpha}_{n})>0. Then \{{x}_{n}\} converges to {\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}, where {\mathrm{\Pi}}_{\mathrm{\Omega}} is the generalized projection of E onto Ω.
Proof Step 1. We prove that {H}_{n,\lambda} and {W}_{n,\lambda} both are closed and convex subsets of C and \mathrm{\Omega}\subset {H}_{n,\lambda}\cap {W}_{n,\lambda} with n\ge 0, \lambda \in \mathrm{\Lambda}. In fact, it is obvious that {W}_{n,\lambda} is closed and convex, and {H}_{n,\lambda} is closed for each n\ge 0, \lambda \in \mathrm{\Lambda}. Since
and
{H}_{n,\lambda} is convex for each n\ge 0, \lambda \in \mathrm{\Lambda}. Hence {H}_{n,\lambda}\cap {W}_{n,\lambda} is closed and convex for all n\ge 0, \lambda \in \mathrm{\Lambda}.
Step 2. For any given p\in \mathrm{\Omega}, from (v) of Lemma 2.8 and that {T}_{\lambda} is hemirelatively nonexpansive, we have
for each \lambda \in \mathrm{\Lambda}. From Lemma 2.1, Lemma 2.7, and the assumption of A, we obtain
From (3.2) and (3.3),
for each \lambda \in \mathrm{\Lambda}. Thus
Therefore, p\in {H}_{0,\lambda} and p\in {H}_{0,\lambda}\cap {W}_{0,\lambda}. Suppose that \mathrm{\Omega}\subset {H}_{n1,\lambda}\cap {W}_{n1,\lambda}. Then, the methods in (3.2) and (3.3) imply that
which implies that p\in {H}_{n,\lambda}. Since {x}_{n}={\mathrm{\Pi}}_{{H}_{n1,\lambda}\cap {W}_{n1,\lambda}}{x}_{0}, it follows from Lemma 2.2 that
It implies that \u3008{x}_{n}p,J{x}_{0}J{x}_{n}\u3009\ge 0. Hence p\in {W}_{n,\lambda}. Therefore, \mathrm{\Omega}\subset {H}_{n,\lambda}\cap {W}_{n,\lambda}. Then, by induction on n, \mathrm{\Omega}\subset {H}_{n,\lambda}\cap {W}_{n,\lambda} for all n\ge 0, \lambda \in \mathrm{\Lambda}. From Lemma 2.5 and Lemma 2.6, we know that \mathit{VI}(C,A) is closed and convex set. Therefore, Ω is closed and convex. The sequence \{{x}_{n}\} generated by (3.1) is well defined.
Step 3. We prove that \{{x}_{n}\} is a Cauchy sequence. Let p\in \mathrm{\Omega}. From the definition of {H}_{n,\lambda}, {W}_{n,\lambda} and Lemma 2.2, we have {x}_{n}={\mathrm{\Pi}}_{{H}_{n1,\lambda}\cap {W}_{n1,\lambda}}{x}_{0} and
Thus \{{x}_{n}\} is bounded. Moreover, since
we have
which implies that \{\varphi ({x}_{n},{x}_{0})\} is nondecreasing. It follows that the limit of \{\varphi ({x}_{n},{x}_{0})\} exists. By the construction {H}_{n,\lambda}\cap {W}_{n,\lambda}, one has that
for any positive integer m\ge n. From Lemma 2.2, we have
Letting m,n\to \mathrm{\infty} in (3.5), we have
Thus, Lemma 2.4 implies that
This implies that \{{x}_{n}\} is a Cauchy sequence.
Step 4. Now, we prove that {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n}{T}_{\lambda}{y}_{n}\parallel =0 for each \lambda \in \mathrm{\Lambda} and q\in \mathcal{F}={\bigcap}_{\lambda \in \mathrm{\Lambda}}F({T}_{\lambda}). Since {x}_{n+1}\in {H}_{n,\lambda}, we obtain
for all \lambda \in \mathrm{\Lambda} and
By (3.5) and Lemma 2.4, we have
Hence,
for all \lambda \in \mathrm{\Lambda}. By the methods in (3.2) and (3.4), we have
for all n\ge 0, \lambda \in \mathrm{\Lambda}. Since J is uniformly continuous on the bounded sets, it follows from Lemma 2.8(v), (3.2) and (3.7) that for any given p\in \mathrm{\Omega},
for all \lambda \in \mathrm{\Lambda}. From Lemma 2.4,
Thus {lim}_{n\to \mathrm{\infty}}\parallel {z}_{n,\lambda}{x}_{n}\parallel =0 for each \lambda \in \mathrm{\Lambda}. Since J is uniformly continuous on bounded sets, we obtain
as n\to \mathrm{\infty}. Since {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1{\alpha}_{n})>0 and {J}^{1} is uniformly continuous on bounded sets, we obtain
for all \lambda \in \mathrm{\Lambda}. It follows from (3.6) and (3.9) that
for all \lambda \in \mathrm{\Lambda}. Since \{{x}_{n}\} is a Cauchy sequence, there exists a point q\in C such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=q. It follows from (3.6) that {y}_{n}\to q. Since {T}_{\lambda} is closed, from (3.10) we get q\in \mathcal{F}={\bigcap}_{\lambda \in \mathrm{\Lambda}}F({T}_{\lambda}).
Step 5. Now, we show that q\in \mathit{VI}(C,A)\cap \mathit{GMEP}. Let S\subset E\times {E}^{\ast} be a mapping as follows:
By Lemma 2.6, S is maximal monotone, and {S}^{1}(0)=\mathit{VI}(C,A). Let (v,w)\in G(S) (graph of S). Since w\in Sv=Av+{N}_{C}(v), we have wAv\in {N}_{C}(v). Moreover, {y}_{n}\in C implies that
On the other hand, from {y}_{n}={\mathrm{\Pi}}_{C}{J}^{1}(J{x}_{n}{\mu}_{n}A{x}_{n}) and Lemma 2.2, we obtain that
Hence,
From (3.11) and (3.12), we obtain
Since J is uniformly continuous on bounded sets, by (3.6) we have
Thus, since S is maximal monotone, we have q\in {S}^{1}(0) and q\in \mathit{VI}(C,A). Next, we show that q\in \mathit{GMEP}=F({T}_{r,\lambda}). Let
From (3.6) and (3.8), we obtain {lim}_{n\to \mathrm{\infty}}{u}_{n,\lambda}=q and {lim}_{n\to \mathrm{\infty}}{z}_{n,\lambda}=q for all \lambda \in \mathrm{\Lambda}. Since J is uniformly continuous, from (3.8) we have {lim}_{n\to \mathrm{\infty}}\parallel J{u}_{n,\lambda}J{z}_{n,\lambda}\parallel =0. Therefore, it follows from {r}_{n}\in [{c}^{\ast},\mathrm{\infty}) for some {c}^{\ast}>0 that {lim}_{n\to \mathrm{\infty}}\frac{\parallel J{u}_{n,\lambda}J{z}_{n,\lambda}\parallel}{{r}_{n}}=0. Since {u}_{n,\lambda}={T}_{{r}_{n},\lambda}{z}_{n,\lambda}, we have
Combining the above inequality and (E2), we get
Taking the limit as n\to \mathrm{\infty} in the above inequality and by (E4), we have {H}_{\lambda}(y,q)\le 0 for all y\in C, \lambda \in \mathrm{\Lambda}. For any t\in (0,1) and y\in C, define
Then {H}_{\lambda}({y}_{t},q)\le 0. From (E1) and (E4), we have
i.e., {H}_{\lambda}({y}_{t},y)\ge 0, for all \lambda \in \mathrm{\Lambda}. Thus, from (E3) and let t\downarrow 0, we have {H}_{\lambda}(q,y)\ge 0 for all y\in C, \lambda \in \mathrm{\Lambda}. This implies that q\in \mathit{GMEP}. Therefore q\in \mathrm{\Omega}.
Step 6. Finally, we prove that q={\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}. Since {x}_{n+1}={\mathrm{\Pi}}_{{H}_{n,\lambda}\cap {W}_{n,\lambda}}{x}_{0} and by Lemma 2.2, we have
Taking the limit in (3.13) and from \mathrm{\Omega}\subset {H}_{n,\lambda}\cap {W}_{n,\lambda} for all n\ge 0, \lambda \in \mathrm{\Lambda}, we obtain
Therefore, from Lemma 2.2, we have q={\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}. □
Remark 3.1 An iterative scheme for finding a solution of the variational inequality problem for a mapping A that satisfies the following conditions in a 2uniformly and uniformly smooth Banach space E:

(1)
A is inverse strongly monotone,

(2)
\mathit{VI}(C,A)\ne \mathrm{\varnothing},

(3)
\parallel Ax\parallel \le \parallel AxAu\parallel for all x\in C and u\in \mathit{VI}(C,A).
If condition (3) holds, then we can prove a convergence theorem for variational inequality problems. To consider the general variational inequality problem for inverse strongly monotone mappings, we have to assume condition (3) (see [27]).
For a practical case, we may apply this theorem to a finite number of mappings \{{T}_{1},{T}_{2},\dots ,{T}_{m}\} as follows.
Corollary 3.1 Let E be a real uniformly smooth and 2uniformly convex Banach space and C be a nonempty, closed and convex subset of E. Let A:C\to {E}^{\ast} be a γinverse strongly monotone mapping and B:C\to {E}^{\ast} be a continuous and monotone mapping. Let \psi :C\to \mathbb{R} be a lower semicontinuous and convex function and θ be a bifunction from C\times C to ℝ satisfying (E1)(E4). Let \{{T}_{1},{T}_{2},\dots ,{T}_{m}\} be a finite family of closed hemirelatively nonexpansive mappings of C into itself having
where \mathcal{F}={\bigcap}_{k=1}^{m}F({T}_{k}) is the set of common fixed points. Assume that \parallel Ax\parallel \le \parallel AxAp\parallel for all x\in C and p\in \mathit{VI}(C,A). Suppose that 0<a<{\mu}_{n}<b=\frac{{c}^{2}\gamma}{2}, where c is the constant in Lemma 2.1. Let \{{r}_{n}\}\subset [{c}^{\ast},+\mathrm{\infty}) for some {c}^{\ast}>0. Let \{{x}_{n}\} be the sequence generated by
where J is the normalized duality mapping, and \{{\alpha}_{n}\} is a sequence in [0,1] satisfying {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1{\alpha}_{n})>0. Then \{{x}_{n}\} converges to {\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}, where {\mathrm{\Pi}}_{\mathrm{\Omega}} is the generalized projection of E onto Ω.
If E=H is the Hilbert space, then J={J}^{1}=I is the identity mapping on H. Then Theorem 3.1 reduces to the following corollary.
Corollary 3.2 Let H be a real Hilbert space and C be a nonempty, closed and convex subset of H. Let A:C\to H be a continuous and monotone mapping. Let \psi :C\to \mathbb{R} be a lower semicontinuous and convex function and θ be a bifunction from C\times C to ℝ satisfying (E1)(E4). Let \{{T}_{\lambda}:C\to C:\lambda \in \mathrm{\Lambda}\} be a family of closed hemirelatively nonexpansive mappings with
where \mathcal{F}={\bigcap}_{\lambda \in \mathrm{\Lambda}}F({T}_{\lambda}) is the set of common fixed points of \{{T}_{\lambda}\}. Assume that \parallel Ax\parallel \le \parallel AxAp\parallel for all x\in C and p\in \mathit{VI}(C,A). Suppose that 0<a<{\mu}_{n}<b=\frac{{c}^{2}\gamma}{2}, where c is the constant in Lemma 2.1. Let \{{r}_{n}\}\subset [{c}^{\ast},+\mathrm{\infty}) for some {c}^{\ast}>0. Let \{{x}_{n}\} be the sequence generated by
where \{{\alpha}_{n}\} is a sequence in [0,1] satisfying {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}(1{\alpha}_{n})>0. Then \{{x}_{n}\} converges to {P}_{\mathrm{\Omega}}{x}_{0}, where {P}_{\mathrm{\Omega}} is the metric projection of H onto Ω.
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Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions which improved the presentation of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012R1A1A4A01010526).
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The main idea of this paper was proposed by KSK. KSK prepared the manuscript initially, JKK and WHL confirmed all the steps of proofs in this research. All authors read and approved the final manuscript.
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Kim, K.S., Kim, J.K. & Lim, W.H. Convergence theorems for common solutions of various problems with nonlinear mapping. J Inequal Appl 2014, 2 (2014). https://doi.org/10.1186/1029242X20142
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DOI: https://doi.org/10.1186/1029242X20142
Keywords
 common fixed point
 generalized mixed equilibrium problem
 generalized projection
 hemirelatively nonexpansive
 variational inequality problem