In this paper, we study a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative method without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others.
MSC:46B10, 46B20, 47H10, 49J40.
Hinweise
Competing interests
The author declares that they have no competing interests.
1 Introduction
Let X be a real Banach space and be its dual space. Let C be a subset of X and let T be a self-mapping of C. We use to denote the set of fixed points of T. The duality mapping is defined by , . If X is a Hilbert space, then , where I is the identity mapping. It is well-known that if X is smooth, then J is single-valued, which is denoted by j.
Recall that a mapping is a contraction on C, if there exists a constant such that , . We use to denote the collection of all contractions on C. This is . A mapping is said to be nonexpansive if , . Let be a nonlinear mapping. Then A is called
(i)
L-Lipschitz continuous (or Lipschitzian) if there exists a constant such that
(ii)
accretive if there exists such that
(iii)
α-inverse strongly accretive if there exist and such that
(iv)
relaxed -cocoercive if there exist and two constants such that
Anzeige
Let C be a nonempty closed convex subset of a real Hilbert space H. Recall that the classical variational inequality is to find such that
(1.1)
where is a nonlinear mapping. Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. The variational inequality problem has been extensively studied in the literature; see [1‐8] and the references cited therein.
In 2006, Aoyama et al. [9] first considered the following generalized variational inequality problem in Banach spaces. Let be an accretive operator. Find a point such that
(1.2)
Problem (1.2) is very interesting as it is connected with the fixed point problem for a nonlinear mapping and the problem of finding a zero point of an accretive operator in Banach spaces; see [10‐13] and the references cited therein.
Anzeige
In 2010, Yao et al. [14] introduced the following system of general variational inequalities in Banach spaces. For given two operators , they considered the problem of finding such that
(1.3)
which is called the system of general variational inequalities in a real Banach space and the set of solutions of problem (1.3) denoted by . Yao et al. proved the following strong convergence theorem.
Theorem YNNLYLetCbe a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach spaceXwhich admits a weakly sequentially continuous duality mapping. Letbe the sunny nonexpansive retraction fromXontoC. Let the mappingsbeα-inverse-strongly accretive withandβ-inverse-strongly accretive with , respectively, with . For a given , let the sequencebe generated iteratively by
where , andare three sequences in . Suppose that the sequences , andsatisfy the following conditions:
(i)
, ;
(ii)
and ;
(iii)
.
Thenconverges strongly towhereis the sunny nonexpansive retraction ofConto .
In 2011, Katchang and Kumam [15] introduced the following system of general variational inequalities in Banach spaces. For given two operators , they considered the problem of finding such that
(1.4)
which is called the system of general variational inequalities in a real Banach space and the set of solutions of problem (1.4) denoted by . Katchang and Kumam proved the following strong convergence theorem.
Theorem KKLetCbe a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach spaceXwhich admits a weakly sequentially continuous duality mapping. Letbe a nonexpansive mapping andbe a sunny nonexpansive retraction fromXontoC. Let the mappingsbeβ-inverse-strongly accretive withandγ-inverse-strongly accretive with , respectively, and letKbe the 2-uniformly smooth constant ofX. Letfbe a contraction ofCinto itself with coefficient . Suppose that . For a given , let the sequencebe generated iteratively by
where , andare three sequences in . Suppose that the sequences , andsatisfy the following conditions:
(i)
, ;
(ii)
and ;
(iii)
.
Thenconverges strongly toandis a solution of problem (1.4), whereandis the sunny nonexpansive retraction ofContoF.
The problem of finding solutions of (1.4) by using iterative methods has been studied by many others; see [16‐19] and the references cited therein.
In this paper, we focus on the problem of finding such that
(1.5)
which is called a new general system of variational inequalities in Banach spaces, where are three mappings, for all . In particular, if and , then problem (1.5) reduces to problem (1.4). If we add up the requirement that for , then problem (1.5) reduces to problem (1.3).
In this paper, motivated and inspired by the idea of Katchang and Kumam [15] and Yao et al. [14], we introduce a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities in Banach spaces for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative algorithm without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others.
2 Preliminaries
In this section, we recall the well-known results and give some useful lemmas that are used in the next section.
Let X be a Banach space and let be a unit sphere of X. X is said to be uniformly convex if for each , there exists a constant such that for any ,
The norm on X is said to be Gâteaux differentiable if the limit
(2.1)
exists for each and in this case X is said to be smooth. X is said to have a uniformly Frechet differentiable norm if the limit (2.1) is attained uniformly for and in this case X is said to be uniformly smooth. We define a function , called the modulus of smoothness of X, as follows:
It is known that X is uniformly smooth if and only if . Let q be a fixed real number with . Then a Banach space X is said to be q-uniformly smooth if there exists a constant such that for all . For , the generalized duality mapping is defined by
In particular, if , the mapping is called the normalized duality mapping (or duality mapping), and usually we write . If X is a Hilbert space, then . Further, we have the following properties of the generalized duality mapping .
(1)
for all with .
(2)
for all and .
(3)
for all .
It is known that if X is smooth, then J is a single-valued function, which is denoted by j. Recall that the duality mapping j is said to be weakly sequentially continuous if for each with , we have weakly-∗. We know that if X admits a weakly sequentially continuous duality mapping, then X is smooth. For details, see [20].
Assume thatis a sequence of nonnegative real numbers such that
whereis a sequence inandis a sequence such that
(i)
;
(ii)
or .
Then .
Let C be a nonempty closed convex subset of a smooth Banach space X and let D be a nonempty subset of C. A mapping is said to be sunny if
whenever for and . A mapping is called a retraction if for all . 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.
It is well known that if X is a Hilbert space, then a sunny nonexpansive retraction is coincident with the metric projection from X onto C.
IfXis strictly convex and uniformly smooth and ifis a nonexpansive mapping having a nonempty fixed point set , then the setis a sunny nonexpansive retraction ofC.
LetXbe a real smooth and uniformly convex Banach space and let . Then there exists a strictly increasing, continuous and convex functionsuch thatandfor all .
LetXbe a uniformly smooth Banach space, letCbe a closed convex subset ofX, letbe a nonexpansive mapping withand let . Then the sequencedefined byconverges strongly to a point in F(T) as . If we define a mappingby , , thensolves the following variational inequality:
LetCbe a nonempty closed convex subset of a real 2-uniformly smooth Banach spaceX. Let the mappingbe relaxed -cocoercive and -Lipschitzian. Then we have
whereandKis the 2-uniformly smooth constant ofX. In particular, if , thenis a nonexpansive mapping.
In order to prove our main result, the next lemma is crucial for proving the main theorem.
Lemma 2.11LetCbe a nonempty closed convex subset of a real 2-uniformly smooth Banach spaceXwith the 2-uniformly smooth constantK. Letbe the sunny nonexpansive retraction fromXontoCand letbe a relaxed -cocoercive and -Lipschitzian mapping for . Letbe a mapping defined by
LetCbe a nonempty closed convex subset of a real smooth Banach space X. Letbe the sunny nonexpansive retraction fromXontoC. Letbe three possibly nonlinear mappings. For given , is a solution of problem (1.5) if and only if , and , whereGis the mapping defined as in Lemma 2.11.
3 Main results
We are now in a position to state and prove our main result.
Theorem 3.1LetXbe a uniformly convex and 2-uniformly smooth Banach space with the 2-uniformly smooth constantK, letCbe a nonempty closed convex subset ofXandbe a sunny nonexpansive retraction fromXontoC. Let the mappingsbe relaxed -cocoercive and -Lipschitzian withfor all . Letfbe a contractive mapping with the constantand letbe a nonexpansive mapping such that , whereGis the mapping defined as in Lemma 2.11. For a given , let , andbe the sequences generated by
(3.1)
whereandare two sequences insuch that
(C1) and ;
(C2) .
Thenconverges strongly to , which solves the following variational inequality:
ProofStep 1. We show that is bounded.
Let and . It follows from Lemma 2.12 that
Put and . Then and
From Lemma 2.10, we have () is nonexpansive. Therefore
(3.2)
and . It follows from (3.2) that
By induction, we have
Therefore, is bounded. Hence , , , , , , and are also bounded.
Step 2. We show that .
By nonexpansiveness of and (), we have
(3.3)
Let , . Then for all and
(3.4)
By (3.3), (3.4) and nonexpansiveness of S, we have
By this together with (C1) and (C2), we obtain that
Hence, by Lemma 2.6, we get as . Consequently,
(3.5)
Step 3. We show that .
Since
therefore
(3.6)
Next, we prove that . From Lemma 2.1 and nonexpansiveness of , we have
(3.7)
and
(3.8)
Similarly, we have
(3.9)
Substituting (3.7) and (3.8) into (3.9), we have
(3.10)
By the convexity of , we obtain
(3.11)
Substituting (3.10) into (3.11), we have
which implies
By the conditions (C1), (C2), (3.5) and for each , we obtain
(3.12)
Let . By Lemma 2.4(b) and Lemma 2.8, we obtain
which implies
(3.13)
And
which implies
(3.14)
Similarly, we have
which implies
(3.15)
From (3.11), (3.13), (3.14) and (3.15), we have
which implies
By the conditions (C1), (C2), (3.5) and (3.12), we obtain
It follows from the properties of g that
Therefore
(3.16)
By (3.6) and (3.16), we have
(3.17)
Define a mapping as
where η is a constant in . Then it follows from Lemma 2.7 that and W is nonexpansive. From (3.16) and (3.17), we have
(3.18)
Step 4. We claim that
(3.19)
where with being the fixed point of the contraction
From Lemma 2.9, we have and
Since , we have
It follows from (3.18) and Lemma 2.2 that
(3.20)
where as . It follows from (3.20) that
(3.21)
Let in (3.21), we obtain that
(3.22)
where is a constant such that for all and . Let in (3.22), we obtain
(3.23)
On the other hand, we have
It follows that
Noticing that j is norm-to-norm uniformly continuous on a bounded subset of C, it follows from (3.23) and that
Hence (3.19) holds.
Step 5. Finally, we show that as .
From (3.2), we have
which implies
It follows from Lemma 2.3, (3.19) and condition (C1) that converges strongly to q. This completes the proof. □
Example 3.2 Let and . Define the mappings and as follows:
Then it is obvious that S is nonexpansive, f is contractive with a constant , is relaxed -cocoercive and 1-Lipschitzian, is relaxed -cocoercive and 2-Lipschitzian and is relaxed -cocoercive and 3-Lipschitzian. In this case, we have . In the terms of Theorem 3.1, we choose the parameters , , . Then the sequence generated by (3.1) converges to , which solves the following variational inequality:
Let in Theorem 3.1, we obtain the following result.
Corollary 3.3LetXbe a uniformly convex and 2-uniformly smooth Banach space with the 2-uniformly smooth constantK, letCbe a nonempty closed convex subset ofXanda sunny nonexpansive retraction fromXontoC. Let the mappingsbe relaxed -cocoercive and -Lipschitzian with , for all . Letfbe a contractive mapping with the constantand letbe a nonexpansive mapping such that , whereis the set of solutions of problem (1.4). For a given , letandbe the sequences generated by
whereandare two sequences insuch that
(C1) and ;
(C2) .
Thenconverges strongly to , which solves the following variational inequality:
Remark 3.4 (i) Since for all is uniformly convex and 2-uniformly smooth, we see that Theorem 3.1 is applicable to for all .
(ii)
The problem of finding solutions for a finite number of variational inequalities can use the same idea of a new general system of variational inequalities in Banach spaces.
Acknowledgements
The author would like to thank Professor Dr. Suthep Suantai and the reviewer for careful reading, valuable comment and suggestions on this paper. This research is partially supported by the Center of Excellence in Mathematics, the Commission on Higher Education, Thailand. The author also thanks the Thailand Research Fund and Thaksin university for their financial support.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Competing interests
The author declares that they have no competing interests.