Abstract
In the present paper, we introduce a general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of strict pseudo-contractions and the set of solutions of the variational inequalities for finite family of strongly accretive mappings in a q-uniformly smooth Banach space. Furthermore, we prove strong convergence of the iterative sequence under suitable conditions. Our results generalize some recent results.
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1 Introduction
Throughout this paper, we always assume that X is a real Banach space with the dual \(X^{*}\). Let C be a subset of X, and T be a self-mapping of C. We use \(F(T)\) to denote the fixed points of T. For \(q>1\), the generalized duality mapping \(J_{q}: X\rightarrow2^{X^{*}}\) is defined by
where \(\langle\cdot,\cdot\rangle\) denotes the duality pairing between X and \(X^{*}\). In particular, \(J_{q} = J_{2}\) is called the normalized duality mapping and \(J_{q}(x)= \|x\|^{q-2}J_{2}(x)\) for \(x\neq0\). If \(X:= H\) is a real Hilbert space, then \(J = I\) where I is the identity mapping. It is well known that if X is smooth, then \(J_{q}\) is single-valued, which is denoted by \(j_{q}\) [1].
Let \(U=\{ x\in X : \| x \|=1\}\). A Banach space X is said to be strictly convex if \(\frac{\| x+y\|}{2} \leq1 \) for all \(x,y \in X \) with \(\| x\|=\| y\|=1\) and \(x\neq y \). It is also called uniformly convex if \(\lim\| x_{n} -y_{n} \|=0 \) for any two sequences \(\{ x_{n}\}\), \(\{ y_{n}\}\) in X such that \(\| x_{n}\|=\| y_{n}\|=1\) and \(\lim\|\frac{x_{n} + y_{n}}{2} \|=1 \). A Banach space X is said to be Gâteaux differentiable if the limit
exists for all \(x,y\in U\). In this case X is smooth. Also, we define a function \(\rho_{X}:[0, \infty) \rightarrow[0, \infty)\) called the modulus of smoothness of X as follows:
A Banach space X is said to be uniformly smooth if \(\frac{\rho_{X}(t)}{t}\rightarrow0\) as \(t\rightarrow0\). Suppose that \(q > 1\), then X is said to be q-uniformly smooth if there exists \(c > 0\) such that \(\rho_{X}(t)\leq ct^{q}\). It is easy to see that if X is q-uniformly smooth, then \(q\leq2\) and X is uniformly smooth.
Let C be a nonempty, closed, and convex subset of a Banach space X and D be a nonempty subset of C, then a mapping \(Q:C\rightarrow D\) is said to be sunny provided
whenever \(Qx+ t(x- Qx)\in C\) for \(x\in C\), and \(t\geq0\). A mapping \(Q:C\rightarrow D\) is called a retraction if \(Qx=x\) for all \(x\in D\). Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive.
A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. In real Hilbert space, a sunny nonexpansive retraction \(Q_{C}\) coincides with the metric projection from X onto C.
Definition 1.1
A mapping \(T:C\rightarrow C\) is said to be:
-
(i)
λ-strictly pseudo contractive [2], if for all \(x,y \in C \) there exist \(\lambda>0\) and \(j_{q}(x-y)\in J_{q}(x-y) \) such that
$$\bigl\langle Tx -Ty , j_{q}(x-y)\bigr\rangle \leq\| x-y \|^{q} -\lambda\bigl\| (I-T)x -(I-T)y\bigr\| ^{q}, $$or equivalently
$$\bigl\langle (I-T)x -(I-T)y , j_{q}(x-y)\bigr\rangle \geq\lambda\bigl\| (I-T)x -(I-T)y\bigr\| ^{q}. $$ -
(ii)
L-Lipschitzian if for all \(x,y\in C \), there exists a constant \(L>0 \) such that
$$\| Tx-Ty \|\leq L \| x-y \|. $$If \(0< L<1 \), then T is a contraction, and if \(L=1 \), then T is a nonexpansive mapping.
Remark 1.2
Let C be a nonempty subset of a real Hilbert space H and \(T: C \rightarrow C\) be a mapping. Then T is said to be k-strictly pseudocontractive [2], if for all \(x, y\in C\), there exists constant \(k\in[0,1)\) such that
Definition 1.3
A mapping \(F:C\rightarrow X\) is said to be accretive if for all \(x,y \in C \) there exists \(j_{q}(x-y)\in J_{q}(x-y) \) such that
For some \(\eta>0 \), \(F:C\rightarrow X \) is said to be η-strongly accretive if for all \(x,y \in C \) there exists \(j_{q}(x-y)\in J_{q}(x-y) \) such that
For some \(\mu>0\), the mapping \(F:C\rightarrow X\) is said to be μ-inverse strongly accretive if for all \(x,y \in C \) there exists \(j_{q}(x-y)\in J_{q}(x-y) \) such that
Note that if \(X:=H\) is a real Hilbert space, accretive and strongly accretive operators coincide with monotone and strongly monotone operators, respectively.
Let C be a nonempty, closed, and convex subset of X, and \(A:C\rightarrow X\) be a mapping. The classical variational inequality problem is to find \(x^{*}\in C\) such that
where \(j_{q}(x-x^{*})\in J_{q}(x-x^{*})\). The solution set of a variational inequality is denoted by \(VI(C,A)\). If \(X=:H\) is a real Hilbert space, the variational inequality problem reduces to find \(x^{*}\in C\) such that
For more details of the variational inequality and its applications, we recommend the reader [3, 4]. On the other hand, we note that the iterative approximations of fixed points for nonexpansive mappings have been extensively studied by many authors [5–9].
In order to find the common element of the solution set of a variational inclusion (3) and the set of fixed points of a nonexpansive mapping, Takahashi and Toyoda [10] introduced the following iterative scheme in a Hilbert space H. Starting with an arbitrary point \(x_{1}=x\in H\), define sequences \(\{x_{n}\}\) by
where \(A:H\rightarrow H\) is an α-inverse-strongly monotone mapping, \(S:C\rightarrow C\) is a nonexpansive mapping and \(\{\alpha_{n}\}\) is a sequence in \([0,1]\). Under mild conditions, they obtained a weak convergence theorem.
On the other hand, Aoyama et al. [11] considered the following algorithm in a uniformly convex and 2-uniformly smooth Banach spaces. For \(x_{1}=x\in C\),
where \(Q_{C}: X\rightarrow C\) is a sunny nonexpansive retraction, and A is a β-Lipschitzian and η-inverse strongly accretive operator. They proved that \(\{x_{n}\}\) generated by (5) converges weakly to a unique element z of \(VI(C,A)\).
Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth uniformly convex Banach space X. Assume the mapping \(A_{m}:C\rightarrow X\) be a \(\mu_{m}\)-inverse-strongly accretive mapping for each \(1\leq m\leq r\), where r is a positive integer. Let \(\{T_{n}\}_{n=1}^{\infty}:C\rightarrow C\) be a family of λ-strict pseudo-contractions with \(0 <\lambda< 1\). Define a mapping \(S_{n}x:=(1 -\gamma_{n})x +\gamma_{n}T_{n}x\) for all \(x\in C\) and \(n\geq1\).
In this paper, motivated by the works mentioned above, we consider the following iteration:
and we prove that the proposed iterative algorithm is strongly convergent under some mild conditions imposed on the algorithm parameters. The results proved in this paper represent a refinement and improvement of the previously found results in the earlier and recent literature.
2 Preliminaries
In order to prove our main results, we need the following lemmas.
Lemma 2.1
Let C be a closed convex subset of a smooth Banach space X. Let D be a nonempty subset of C. Let \(Q:C\rightarrow D\) be a retraction and J be the normalized duality mapping on X. Then the following are equivalent:
-
(a)
Q is sunny and nonexpansive.
-
(b)
\(\|Qx-Qy\|^{2}\leq\langle x-y,J(Qx-Qy)\rangle\), \(\forall x,y\in C\).
-
(c)
\(\langle x-Qx,J(y-Qx)\rangle\leq0\), \(\forall x\in C\), \(y\in D\).
-
(d)
\(\langle x-Qx,J_{q}(y-Qx)\rangle\leq0\), \(\forall x\in C\), \(y\in D\).
Lemma 2.2
[14]
Let C be a closed convex subset of a strictly convex Banach space X. Let \(T_{1}\) and \(T_{2}\) be two nonexpansive mappings from C into itself with \(F(T_{1})\cap F(T_{2})\neq\emptyset\). Define a mapping S by
where k is a constant in \((0, 1)\). Then S is nonexpansive and \(F(S)=F(T_{1})\cap F(T_{2})\).
Lemma 2.3
[15]
Let \(\{ s_{n}\}\) be a sequence of nonnegative real numbers satisfying
where \(\{a_{n}\}\), \(\{b_{n}\}\), \(\{c_{n}\}\) satisfy the restrictions:
-
(i)
\(\lim_{n\rightarrow\infty}a_{n}=0\), \(\sum_{n=1}^{\infty}a_{n} = \infty\),
-
(ii)
\(c_{n}\geq0\), \(\sum_{n=1}^{\infty}c_{n} < \infty\),
-
(iii)
\(\limsup_{n\rightarrow\infty} b_{n} \leq0\).
Then \(\lim_{n\rightarrow\infty} s_{n}=0\).
Lemma 2.4
[16]
Suppose that \(q>1\). Then the following inequality holds:
for arbitrary positive real numbers a, b.
Lemma 2.5
[17]
Let X be a real q-uniformly smooth Banach space, then there exists a constant \(C_{q}>0\) such that
for all \(x,y\in X \). In particular, if X is real 2-uniformly smooth Banach space, then there exists a best smooth constant \(K > 0\) such that
for all \(x,y\in C\).
Lemma 2.6
[18]
Let X a real smooth and uniformly convex Banach space and let \(r >0 \). Then there exists a strictly increasing, continuous, and convex function \(g:[0,2 r] \rightarrow R \) such that \(g(0)=0 \) and \(g(\| x-y\| )\leq\| x \|^{2}-2\langle x,Jy\rangle+\| y \|^{2} \), for all \(x,y \in B_{r}\), where \(B_{r}=\{ z\in X : \| z \|\leq r \}\).
Definition 2.7
[11]
Let \({T_{n}}\) be a family of mappings from a subset C of a Banach space X into itself with \(\bigcap_{n=1}^{\infty} F(T_{n})\neq\emptyset\). We say that \(\{T_{n}\}\) satisfies the AKTT-condition if for each bounded subset B of C,
Lemma 2.8
[11]
Suppose that \(\{T_{n}\}\) satisfies the AKTT-condition such that:
-
(i)
For each \(x\in C\), \(\{T_{n}x\}\) is converge strongly to some point in C.
-
(ii)
Let the mapping \(T:C\rightarrow C\) defined by \(Tx= \lim_{n\rightarrow\infty} T_{n}x\), for all \(x\in C\).
Then \(\lim_{n\rightarrow\infty}\sup_{\omega\in B}\|T\omega-T_{n}\omega\|=0\), for each bounded subset B of C.
Lemma 2.9
Let C be a closed and convex subset of a smooth Banach space X. Suppose that \(\{T_{n}\}_{n=1}^{\infty}:C\rightarrow X\) is a family of λ-strictly pseudocontractive mappings; \(\{\mu_{m}\}_{m=1}^{\infty}\) is a real sequence in \((0,1)\) such that \(\sum_{n=1}^{\infty}\mu_{m}=1\). Then the following conclusions hold:
-
(i)
A mapping \(G:C\rightarrow X\) defined by \(G:=\sum_{n=1}^{\infty}\mu_{n}T_{n}\) is a λ-strictly pseudocontractive mapping.
-
(ii)
\(F(G)=\bigcap_{n=1}^{\infty}F(T_{n})\).
Lemma 2.10
[19]
Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X which admits weakly sequentially continuous generalized duality mapping \(j_{q}\) from X into \(X^{*}\). Let \(T: C\rightarrow C\) be a nonexpansive mapping. Then, for all \(\{x_{n}\}\subset C\), if \(x_{n}\rightharpoonup x\) and \(x_{n}- Tx_{n}\rightarrow0\), then \(x=Tx\).
Lemma 2.11
[19]
Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X. Let \(F:C\rightarrow E\) be a k-Lipschitzian and η-strongly accretive operator with constants \(k, \eta>0\). Let \(0 < \mu< ( \frac{q\eta}{C_{q}k^{q}} )^{\frac{1}{q-1}}\) and \(\tau=\mu(\eta -\frac{C_{q}\mu^{q-1} k^{q}}{q})\). Then for \(t\in (0,\min\{1,\frac{1}{\tau}\})\), the mapping \(S:C\rightarrow E\) defined by \(S:=(I-t\mu F) \) is a contraction with a constant \(1-t\tau\).
Lemma 2.12
[19]
Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X. Let \(Q_{C}\) be a sunny nonexpansive retraction from X onto C. Let \(F:C\rightarrow X\) be a k-Lipschitzian and η-strongly accretive operator with constants \(k,\eta>0 \), \(f:C\rightarrow X\) be an L-Lipschitzian mapping with a constant \(L\geq0\) and \(S:C\rightarrow C\) be a nonexpansive mapping such that \(F(S)\neq \emptyset\). Let \(0<\mu<(\frac{q\eta}{C_{q} k^{q}})^{\frac{1}{q-1}} \) and \(0\leq\gamma L<\tau\), where \(\tau= \mu (\eta-\frac{C_{q} \mu^{q-1} k^{q} }{q})\). Then \(\{x_{t}\}\) defined by
has the following properties:
-
(i)
\(\{ x_{t} \}\) is bounded for each \(t\in(0,\min\{ 1,\frac{1}{\tau}\})\).
-
(ii)
\(\lim_{t\rightarrow0 }\| x_{t} -Sx_{t}\|=0 \).
-
(iii)
\(\{ x_{t} \}\) defines a continuous curve from \((0,\min\{1,\frac{1}{\tau}\})\) into C.
Lemma 2.13
[13]
Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space X which admits a weakly sequentially continuous generalized duality mapping \(j_{q}\) from X into \(X^{*}\). Let \(Q_{C}\) be a sunny nonexpansive retraction from X onto C. Let \(F:C\rightarrow X\) be a k-Lipschitzian and η-strongly accretive operator with constants \(k,\eta>0 \), \(f:C\rightarrow X \) be an L-Lipschitzian mapping with a constant \(L\geq0\), and \(S:C\rightarrow C \) be a nonexpansive mapping such that \(F(S)\neq\emptyset\). Suppose that \(0<\mu <(\frac{q\eta}{C_{q} k^{q}})^{\frac{1}{q-1}} \) and \(0\leq\gamma L<\tau\), where \(\tau= \mu(\eta-\frac{C_{q} \mu^{q-1} k^{q} }{q})\). For each \(t\in(0,\min\{1,\frac{1}{\tau}\})\), let \(\{ x_{t} \}\) be defined by (8), then \(\{ x_{t} \}\) converges strongly to \(x^{*}\in F(S) \) as \(t\rightarrow0\), in which \(x^{*}\) is the unique solution of the variational inequality
Lemma 2.14
[20]
Let X be a Banach space and J be a normality duality mapping. Then for any given \(x,y\in X\), the following inequality holds:
for all \(j(x+y)\in J(x+y)\).
3 Main results
Theorem 3.1
Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth, uniformly convex Banach space X. Let \(Q_{C}\) be a sunny nonexpansive retraction from X onto C. Assume that the mapping \(A_{m}:C\rightarrow H\) is a \(\mu_{m}\)-inverse-strongly accretive mapping for each \(1\leq m\leq r\), where r is a positive integer. Let \(F:C\rightarrow X\) be a k-Lipschitzian and η-strongly accretive operator with constants \(k,\eta>0 \), \(f:C\rightarrow X\) be an L-Lipschitzian mapping with a constant \(L\geq0\). Suppose that \(0<\mu<(\frac{q\eta}{C_{q} k^{q}})^{\frac{1}{q-1}} \) and \(0\leq\gamma L<\tau\), where \(\tau= \mu (\eta-\frac{C_{q} \mu^{q-1} k^{q} }{q})\). Let \(\{T_{n}\}_{n=1}^{\infty}:C\rightarrow C\) be a family of λ-strict pseudo-contractions with \(0 <\lambda< 1\). Define a mapping \(S_{n}x:=(1 -\gamma_{n})x +\gamma_{n}T_{n}x\), for all \(x\in C\) and \(n\geq1\). Assume that \(F:=(\bigcap_{m=1}^{r}VI(C,A_{m}))\cap(\bigcap_{n=1}^{\infty}F(T_{n}))\neq \emptyset\). Let \(\{x_{n}\}\) be a sequence generated by the following iterative algorithm:
where \(\{\alpha_{n}\}, \{\beta_{n}\}, \{\eta_{n}^{1}\}, \{\eta_{n}^{2}\},\ldots\) and \(\{\eta_{n}^{r}\}\) are sequences in \((0,1)\) and \(\lambda_{m}\) is a real number such that \(0<\lambda_{m}<(\frac{q\mu_{m}}{C_{q}})^{\frac{1}{q-1}}\), for each \(1\leq m\leq r\). Assume that the above control sequences satisfy the following restrictions:
-
(i)
\(\sum_{m=1}^{r}\eta_{n}^{m}=1\), \(\forall n\geq1\), \(\sum_{n=1}^{\infty}|\eta_{n+1}^{m}-\eta_{n}^{m}|<\infty\).
-
(ii)
\(\lim_{n\rightarrow\infty}\eta_{n}^{m}=\eta^{m}\in(0,1)\), for each m, where \(1\leq m\leq r\).
-
(iii)
\(\sum_{n=1}^{\infty}\beta_{n}=\infty\), \(\lim_{n\rightarrow \infty}\beta_{n}=0\), \(\sum_{n=1}^{\infty}|\beta_{n+1}-\beta_{n}|<\infty\).
-
(iv)
\(\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty\), \(\liminf_{n\rightarrow\infty}\alpha_{n}>0\).
-
(v)
\(0\leq\gamma_{n}\leq\delta\), \(\delta=\min\{1,(\frac{q\lambda}{C_{q}})^{\frac{1}{q-1}}\}\), and \(\sum_{n=1}^{\infty}|\gamma_{n+1}-\gamma_{n}|<\infty\).
Suppose in addition that \(\{ T_{n}\} _{n=0}^{\infty}\) satisfies the AKTT-condition. Let \(T:C\rightarrow C \) be the mapping defined by \(Tx=\lim_{n\rightarrow\infty}T_{n}x\) for all \(x\in C \) and suppose that \(F(T)=\bigcap_{n=0}^{\infty}F(T_{n})\). Then the sequence \(\{ x_{n} \}\) converges strongly to \(x^{*}\in F \) as \(n\rightarrow\infty\), in which \(x^{*}\) is the unique solution of the variational inequality,
Proof
We divide the proof into several steps.
Step 1. We show that \(I-\lambda_{m} A_{m}\) is nonexpansive for each m. Indeed, from Lemma 2.4, for all \(x,y\in C\) we have
It is clear that if \(0<\lambda_{m}\leq (\frac{q\mu_{m}}{C_{q}})^{\frac{1}{q-1}}\), then \(I-\lambda_{m} A_{m} \) is nonexpansive for each \(1\leq m\leq r\).
Now, for each \(1\leq m\leq r\), put
Let \(x^{*}\in F\), we have
On the other hand we have
From (10) and the fact that \(S_{n}\) is nonexpansive [19] we have
By induction, we find that
This shows that \(\{x_{n}\}\) is bounded. Hence by (10), \(\{y_{n}\}\) is also bounded.
Step 2: We show that \(\lim_{n\rightarrow\infty}\| x_{n+1} -x_{n} \|=0 \). Since
On the other hand, we have
where M is an appropriate constant such that
Observe that
It follows from (11) that
Note that
On the other hand,
Substituting (13) into (14), we obtain
where \(M_{1}=\sup_{n\geq 0}\{\gamma\|fx_{n-1}\|+\mu\|FS_{n-1}y_{n-1}\|, \|x_{n}-z_{n-1}\|,\|T_{n}y_{n-1}-y_{n-1}\|, M\}\).
Since \(\{T_{n}\}_{n=1}^{\infty}\) satisfies the AKTT-condition, we deduce that
From (14), (16), and Lemma 2.3, we deduce that
We observe that
From the condition (iii) and (17), we have
Step 3. We prove that \(\lim_{n\rightarrow \infty}\|T_{n}x_{n}-x_{n}\|=0\).
From Lemma 2.5, we have
and
By the convexity of \(\|\cdot\|\), for all \(q>1\), and Lemma 2.5, we obtain
where
By the fact that \(a^{r}-b^{r}\leq ra^{r-1}(a-b)\), \(\forall r\geq1\), we get
Since \(0<\lambda_{m}<(\frac{q\mu_{m}}{C_{q}})^{\frac{1}{q-1}}\), from (17) and (iii) and the fact that \(\{x_{n}\}\) is bounded we have
Setting \(r_{m}= \sup \{\| x_{n}-x^{*}\|,\| k_{n}^{m}-x^{*}\|\}\), we have from Lemmas 2.1 and 2.6
where \(g_{m}:[0,2r_{m})\rightarrow[0,\infty)\) is a continuous, strictly increasing, and convex function such that \(g_{m}(0)=0\) for all \(1\leq m\leq r\). Hence, we have
for all m, with \(1\leq m\leq r\). On the other hand, we have
Since \(g_{m}\) is increasing and convex by using (20) we have
Thus we have
Thanks to Lemma 2.5 we have
where \(M_{3}=\sup_{n\geq0}\{2\langle\gamma fx_{n} -\mu FS_{n}y_{n}, j_{q} (\beta_{n}(\gamma fx_{n}-\mu fS_{n}y_{n})+S_{n}y_{n}-x^{*} )\rangle\}\).
This in turn implies that
In view of (ii), (iii), (17), and (19) we have
By the properties of \(g_{m}\), we get
On the other hand,
It follows from (21), (18), and (iii) that
Next, we show that \(\|x_{n}-Sx_{n}\|\rightarrow0\) as \(n\rightarrow\infty\). For any bounded subset B of C, we observe that
where \(M_{3}=\sup_{n\geq1}\{\|\omega\|, \|T_{n}\omega\|\}\). By (v) and the fact that \(\{T_{n}\}\) satisfies the AKTT-condition, we have
that is, \(\{S_{n}\}\) satisfies the AKTT-condition. Now we define the nonexpansive mapping \(S:C\rightarrow C\) by \(Sx=\lim_{n\rightarrow \infty}S_{n}x\) for all \(x\in C\). Since \(\{\gamma_{n}\}\) is bounded, there exists a subsequence \(\{\gamma_{n_{i}}\}\) of \(\{\gamma_{n}\}\) such that \(\gamma_{n_{i}}\rightarrow\nu\) as \(i\rightarrow\infty\). It follows that
That is \(F(S)=F(T)\). Hence \(F(S)=\bigcap_{n=1}^{\infty}F(T_{n})=\bigcap_{n=1}^{\infty}F(S_{n})\). On the other hand we have
This implies by Lemma 2.8 and (22) that
Now we define a mapping \(h:C\rightarrow C\) by
where \(\eta^{m}=\lim_{n\rightarrow\infty}\eta_{n}^{m}\). From Lemma 2.9, h is nonexpansive such that
Next, we define a mapping \(U:C\rightarrow C\) by \(Ux=\delta Sx+(1-\delta)hx\), where \(\delta\in(0,1)\) is a constant. Then by Lemma 2.2, U is a nonexpansive and
Note that
In view of restriction (ii), we find from (21) that
Setting \(x_{t}=Q_{C}[t\gamma fx_{t}+(I-t\mu F)Ux_{t}]\), it follows from Lemma 2.13 that \(\{x_{t}\}\) converges strongly to a point \(x^{*}\in F(U)=F\), in which \(x^{*}\) is the unique solution of the variational inequality (9). From (23) and (24), we have
Step 4. We show that
where \(x^{*}\) is a solution of the variational inequality (9). To show this, we can choose a subsequence \(\{ x_{n_{j}}\}\) of \(\{ x_{n}\}\) such that
By reflexivity of a Banach space X and since \(\{ x_{n} \} \) is bounded, there exists a subsequence \(\{ x_{n_{j}}\}\) of \(\{x_{n}\}\) which converges weakly to z. Without loss of generality, we can assume that \(x_{n_{j}} \rightharpoonup z\). Since \(\| x_{n} -Ux_{n}\|\rightarrow0 \) by step 3, we obtain \(z=Uz\) and we have \(z\in F(U)\). Since Banach space X has a weakly sequentially continuous generalized duality mapping, we obtain
Step 5. Finally, we show that \(\lim_{n\rightarrow\infty} \| x_{n} -x^{*} \|=0 \). Setting \(h_{n}=\beta_{n}\gamma fx_{n}+(I-\beta_{n}\mu F)S_{n}y_{n}\), \(\forall n\geq1\). Then we can rewrite \(x_{n+1}=Q_{C}h_{n}\). It follows from Lemmas 2.1 and 2.4 that
which implies that
Put \(a_{n}=\beta_{n}(\tau-\gamma L)\) and \(b_{n}=\frac{q}{(1+(q-1)(\tau-\gamma L)\beta_{n})(\tau-\gamma L)}+\langle\gamma fx^{*}-\mu Fx^{*}, j_{q}(x_{n+1}-x^{*})\rangle\). Applying Lemma 2.3, we obtain \(x_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\). This completes the proof. □
Remark 3.2
Theorem 3.1 improves and extends Theorem 2.1; see Cho and Kang [21]. Especially, our results extend the above results from Hilbert space to a more general q-uniformly smooth Banach space.
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Nazari, E., Askari, S. & Ramezani, M. Approximation of common solutions for variational inequalities and fixed point of strict pseudo-contractions in q-uniformly smooth Banach spaces. Fixed Point Theory Appl 2015, 15 (2015). https://doi.org/10.1186/s13663-015-0264-y
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DOI: https://doi.org/10.1186/s13663-015-0264-y