A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces

Abstract

The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.

1. Introduction

Let . A Banach space is said to be uniformly convex if, for any , there exists such that, for any ,

(1.1)

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit

(1.2)

exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for all . The norm of is said to be Fréchet differentiable if, for any , the above limit is attained uniformly for all . The modulus of smoothness of is defined by

(1.3)

where . It is known that is uniformly smooth if and only if . Let be a fixed real number with . A Banach space is said to be -uniformly smooth if there exists a constant such that for all .

From [1], we know the following property.

Let be a real number with and let be a Banach space. Then is -uniformly smooth if and only if there exists a constant such that

(1.4)

The best constant in the above inequality is called the -uniformly smoothness constant of (see [1] for more details).

Let be a real Banach space and the dual space of . Let denote the pairing between and . For , the generalized duality mapping is defined by

(1.5)

In particular, is called the normalized duality mapping. It is known that for all . If is a Hilbert space, then is the identity. Note the following.

(1) is a uniformly smooth Banach space if and only if is single-valued and uniformly continuous on any bounded subset of .

(2) All Hilbert spaces, (or ) spaces (), and the Sobolev spaces () are -uniformly smooth, while (or ) and spaces () are -uniformly smooth.

(3) Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is -uniformly smooth for any .

Further, we have the following properties of the generalized duality mapping :

(i) for all with ,

(ii) for all and ,

(iii) for all .

It is known that, if is smooth, then is single valued. Recall that the duality mapping is said to be weakly sequentially continuous if, for each sequence with weakly, we have weakly-. We know that, if admits a weakly sequentially continuous duality mapping, then is smooth. For the details, see [2].

Let be a nonempty closed convex subset of a smooth Banach space . Recall the following definitions of a nonlinear mapping , the following are mentioned.

Definition 1.1.

Given a mapping .

(i) is said to be accretive if

(1.6)

for all .

(ii) is said to be -strongly accretive if there exists a constant such that

(1.7)

for all .

(iii) is said to be -inverse-strongly accretive or -cocoercive if there exists a constant such that

(1.8)

for all .

(iv) is said to be -relaxed cocoercive if there exists a constant such that

(1.9)

for all .

(v) is said to be -relaxed cocoercive if there exist positive constants and such that

(1.10)

for all .

Remark 1.2.

() Every -strongly accretive mapping is an accretive mapping.

() Every -strongly accretive mapping is a -relaxed cocoercive mapping for any positive constant but the converse is not true in general. Then the class of relaxed cocoercive operators is more general than the class of strongly accretive operators.

() Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spaces (see, e.g., [3]).

() The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems using the auxiliary problem principle and projection methods [4]. Several classes of relaxed cocoercive variational inequalities have been studied in [5, 6].

Next, we consider a system of quasivariational inclusions as follows.

Find such that

(1.11)

where and are nonlinear mappings for each

As special cases of problem (1.11), we have the following.

(1) If and , then problem (1.11) is reduced to the following.

Find such that

(1.12)

(2) Further, if in problem (1.12), then problem (1.12) is reduced to the following

Find such that

(1.13)

In 2006, Aoyama et al. [7] considered the following problem.

Find such that

(1.14)

They proved that the variational inequality (1.14) is equivalent to a fixed point problem. The element is a solution of the variational inequality (1.14) if and only if satisfies the following equation:

(1.15)

where is a constant and is a sunny nonexpansive retraction from onto , see the definition below.

Let be a subset of , and be a mapping of into . Then is said to be sunny if

(1.16)

whenever for and . A mapping of into itself is called a retraction if . If a mapping of into itself is a retraction, then for all , where is the range of . A subset of is called a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction from onto .

The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 1.3 (see [8]).

Let be a smooth Banach space and a nonempty subset of . Let be a retraction and the normalized duality mapping on . Then the following are equivalent:

(1) is sunny and nonexpansive,

(2)

Recall that a mapping is called contractive if there exists a constant such that

(1.17)

A mapping is said to be -strictly pseudocontractive if there exists a constant such that

(1.18)

Note that the class of -strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings which are mappings on such that

(1.19)

for all . That is, is nonexpansive if and only if is -strict pseudocontractive. We denote by the set of fixed points of .

Proposition 1.4 (see [9]).

Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space and a nonexpansive mapping of into itself with . Then the set is a sunny nonexpansive retract of .

Definition 1.5.

A countable family of mapping is called a family of uniformly-strict pseudocontractions if there exists a constant such that

(1.20)

For the class of nonexpansive mappings, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [10, 11]. More precisely, take and define a contraction by

(1.21)

where is a fixed point and is a nonexpansive mapping. Banach's contraction mapping principle guarantees that has a unique fixed point in ; that is,

(1.22)

It is unclear, in general, what the behavior of is as even if has a fixed point. However, in the case of having a fixed point, Ceng et al. [12] proved that, if is a Hilbert space, then converges strongly to a fixed point of . Reich [11] extended Browder's result to the setting of Banach spaces and proved that, if is a uniformly smooth Banach space, then converges strongly to a fixed point of , and the limit defines the (unique) sunny nonexpansive retraction from onto .

Reich [11] showed that, if is uniformly smooth and is the fixed point set of a nonexpansive mapping from into itself, then there is a unique sunny nonexpansive retraction from onto and it can be constructed as follows.

Proposition 1.6 (see [11]).

Let be a uniformly smooth Banach space and a nonexpansive mapping such that . For each fixed and every , the unique fixed point of the contraction converges strongly as to a fixed point of . Define by . Then is the unique sunny nonexpansive retract from onto ; that is, satisfies the property.

(1.23)

Notation 1.

We use to denote strong convergence to of the net as .

Definition 1.7 (see [13]).

Let be a multivalued maximal accretive mapping. The single-valued mapping defined by

(1.24)

is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Recently, many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inequalities (1.11)–(1.14) by using different iterative methods (see, e.g., [7, 14–16]).

Very recently, Qin et al. [16] considered the problem of finding the solutions of a general system of variational inclusion (1.11) with -inverse strongly accretive mappings. To be more precise, they obtained the following results.

Lemma 1.8 (see [16]).

For any where , is a solution of the problem (1.11) if and only if is a fixed point of the mapping defined by

(1.25)

Theorem QCCK (see [16, Theorem ]).

Let be a uniformly convex and -uniformly smooth Banach space with the smoothness constant . Let be a maximal monotone mapping and a -inverse-strongly accretive mapping, respectively, for each . Let be a -strict pseudocontraction such that . Define a mapping by , . Assume that , where is defined as in Lemma 1.8. Let and let be a sequence generated by

(1.26)

where , , , and and are sequences in . If the control consequences and satisfy the following restrictions:

(C1),

(C2) and ,

then converges strongly to , where is the sunny nonexpansive retraction from onto and , where , is a solution to problem (1.11).

On the other hand, we recall the following well-known definitions and results.

In a smooth Banach space, a mapping is called strongly positive [17] if there exists a constant with property

(1.27)

where is the identity mapping and is the normalized duality mapping.

In [18], Moudafi introduced the viscosity approximation method for nonexpansive mappings (see [19] for further developments in both Hilbert and Banach spaces). Let be a contraction on . Starting with an arbitrary initial point , define a sequence recursively by

(1.28)

where is a sequence in . It is proved [18, 19] that, under certain appropriate conditions imposed on , the sequence generated by (1.28) strongly converges to the unique solution in of the variational inequality

(1.29)

Recently, Marino and Xu [20] introduced the following general iterative method:

(1.30)

where is a strongly positive bounded linear operator on a Hilbert space . They proved that, if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.30) converges strongly to the unique solution of the variational inequality

(1.31)

which is the optimality condition for the minimization problem

(1.32)

where is a potential function for for ).

Recently, Qin et al. [21] introduce the following iterative algorithm scheme:

(1.33)

where is nonself--strict pseudo-contraction, is a contraction, and is a strongly positive bounded linear operator on a Hilbert space . They proved, under certain appropriate conditions imposed on the sequences and , that defined by (1.33) converges strongly to a fixed point of , which solves some variational inequality.

In this paper, motivated by Qin et al. [16], Moudafi [18], Marino and Xu [20], and Qin et al. [21], we introduce a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions (1.11) with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. We prove the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in strictly convex and -uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Qin et al. [16], Moudafi [18], Marino and Xu [20], Qin et al. [21], and many others.

2. Preliminaries

Now we collect some useful lemmas for proving the convergence result of this paper.

Lemma 2.1 (see [22]).

The resolvent operator associated with is single valued and nonexpansive for all .

Lemma 2.2 (see [13]).

is a solution of variational inclusion (1.13) if and only if ; that is,

(2.1)

where denotes the set of solutions to problem (1.13).

Lemma 2.3 (see [23]).

Let be a strictly convex Banach space. Let and be two nonexpansive mappings from into itself with a common fixed point. Define a mapping by

(2.2)

where is a constant in . Then is nonexpansive and .

Lemma 2.4 (see [24]).

Let be a nonempty closed convex subset of reflexive Banach space which satisfies Opial's condition, and suppose that is nonexpansive. Then the mapping is demiclosed at zero, that is, imply that .

Lemma 2.5 (see [25]).

Assume that is a sequence of nonnegative real numbers such that

(2.3)

where is a sequence in and is a sequence such that

(a),

(b) or

Then

Lemma 2.6 (see [26]).

Let and be bounded sequences in a Banach space and a sequence in with . Suppose that for all and

(2.4)

Then

Definition 2.7 (see [27]).

Let be a family of mappings from a subset of a Banach space into with . We say that satisfies the AKTT-condition if, for each bounded subset of ,

(2.5)

Remark 2.8.

The example of the sequence of mappings satisfying AKTT-condition is supported by Example 3.11.

Lemma 2.9 (see [27, Lemma ]).

Suppose that satisfies AKTT-condition. Then, for each , converses strongly to a point in . Moreover, let the mapping be defined by

(2.6)

Then for each bounded subset of ,

Lemma 2.10 (see [28]).

Let be a real -uniformly smooth Banach space and a -strict pseudocontraction. Then is nonexpansive and .

Lemma 2.11 (see [29]).

Let be a real -uniformly smooth Banach space with the best smoothness constant . Then the following inequality holds:

(2.7)

Lemma 2.12 (see [17, Lemma ]).

Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and Then

3. Main Results

In this section, we prove that the strong convergence theorem for a countable family of uniformly -strict pseudocontractions in a strictly convex and -uniformly smooth Banach space admits a weakly sequentially continuous duality mapping. Before proving it, we need the following theorem.

Theorem 3.1 (see [17, Lemma ]).

Let be a nonempty closed convex subset of a reflexive, smooth Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonexpansive mapping such that is nonempty, let be a contraction with coefficient , and let be a strongly positive bounded linear operator with coefficient and . Then the net defined by

(3.1)

converges strongly as to a fixed point of which solves the variational inequality:

(3.2)

Lemma 3.2.

Let be a nonempty closed convex subset of a real -uniformly smooth Banach space with the smoothness constant . Let be an -Lipschitzian and relaxed -cocoercive mapping. Then

(3.3)

If , then is nonexpansive.

Proof.

Using Lemma 2.11 and the cocoercivity of the mapping , we have, for all ,

(3.4)

Hence (3.3) is proved. Assume that . Then, we have . This together with (3.3) implies that is nonexpansive.

Lemma 3.3.

Let be a strictly convex and -uniformly smooth Banach space admiting a weakly sequentially continuous duality mapping with the smoothness constant . Let be a maximal monotone mapping and a -Lipschitzian and relaxed -cocoercive mapping with , respectively, for each . Let be a countable family of uniformly -strict pseudocontractions. Define a mapping and by

(3.5)

where is defined as in Lemma 1.8. Assume that . Let be an -contraction; let be a strongly positive linear bounded self-adjoint operator with coefficient with . Then the following hold.

(i)For each , is nonexpansive such that

(3.6)

(ii)Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . The net defined by converges strongly as to a fixed point of , which solves the variational inequality

(3.7)

and is a solution of general system of variational inequality problem (1.11) such that .

Proof.

It follows from Lemma 2.10 that is nonexpansive such that for each . Next, we prove that is nonexpansive. Indeed, we observe that

(3.8)

The nonexpansivity of , , , and implies that is nonexpansive. By Lemma 2.3, we have that is nonexpansive such that

(3.9)

Hence (i) is proved. It is well known that, if is uniformly smooth, then is reflexive. Hence Theorem 3.1 implies that converges strongly as to a fixed point of , which solves the variational inequality

(3.10)

and is a solution of problem (1.11), where . This completes the proof of (ii).

Theorem 3.4.

Let be a strictly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant . Let be a maximal monotone mapping and a -Lipschitzian and relaxed -cocoercive mapping with , respectively, for each . Let be a countable family of uniformly -strict pseudocontractions. Define a mapping by

(3.11)

Assume that , where is defined as in Lemma 1.8. Let be an -contraction; let be a strongly positive linear bounded self adjoint operator with coefficient with . Let and let be a sequence generated by

(3.12)

where , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions

(C1),

(C2) and ,

then converges strongly to , which solves the variational inequality

(3.13)

and is a solution of general system of variational inequality problem (1.11) such that .

Proof.

First, we show that sequences , , and are bounded.

By the control condition (C2), we may assume, with no loss of generality, that .

Since is a linear bounded operator on , by (1.27), we have

(3.14)

Observe that

(3.15)

It follows that

(3.16)

Therefore, taking one has

(3.17)

Putting one sees that

(3.18)

It follows from Lemmas 2.1 and 3.2 that

(3.19)

This implies that

(3.20)

Setting and applying Lemma 2.10, we have that is a nonexpansive mapping such that for all and hence . Then

(3.21)

It follows from the last inequality that

(3.22)

By induction, we have

(3.23)

This shows that the sequence is bounded, and so are , , and .

On the other hand, from the nonexpansivity of the mappings , one sees that

(3.24)

In a similar way, one can obtain that

(3.25)

It follows that

(3.26)

This implies that

(3.27)

Setting

(3.28)

one sees that

(3.29)

and so it follows that

(3.30)

which, combined, with (3.27) yields that

(3.31)

Using the conditions (C1) and (C2) and AKTT-condition of , we have

(3.32)

Hence, from Lemma 2.6, it follows that

(3.33)

From (3.28), it follows that

(3.34)

By (3.33), one sees that

(3.35)

On the other hand, one has

(3.36)

It follows that

(3.37)

From the conditions (C1), (C2) and from (3.35), one sees that

(3.38)

Define the mapping by

(3.39)

where is defined as in Lemma 1.8. From Lemma 3.3(i), we see that is nonexpansive such that

(3.40)

From (3.38), it follows that

(3.41)

Since satisfies AKTT-condition and is the mapping defined by for all , we have that satisfies AKTT-condition. Let the mapping be the mapping defined by for all . It follows from the nonexpansivity of and

(3.42)

that is nonexpansive such that

(3.43)

Next, we prove that

(3.44)

where with be the fixed point of the contraction

(3.45)

Then solves the fixed point equation . It follows from Lemma 3.3(ii) that , which solves the variational inequality:

(3.46)

and is a solution of general system of variational inequality problem (1.11) such that . Let be a subsequence of such that

(3.47)

If follows from reflexivity of and the boundedness of sequence that there exists which is a subsequence of converging weakly to as . It follows from (3.41) and the nonexpansivity of , we have by Lemma 2.4. Since the duality map is single valued and weakly sequentially continuous from to , we get that

(3.48)

as required. Now from Lemma 2.11, we have

(3.49)

which implies that

(3.50)

where is an appropriate constant such that . Put

(3.51)

that is,

(3.52)

It follows from conditions (C1), (C2) and from (3.44) that

(3.53)

Apply Lemma 2.5 to (3.52) to conclude that as . This completes the proof.

Setting , and , we have the following result.

Theorem 3.5.

Let be a strictly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant . Let be a maximal monotone mapping and a -Lipschitzian and relaxed -cocoercive mapping with , respectively, for each . Let be a countable family of uniformly -strict pseudocontractions. Define a mapping by

(3.54)

Assume that , where is defined as in Lemma 1.8. Let and let be a sequence generated by

(3.55)

where , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions

(C1),

(C2) and ,

then converges strongly to , which solves the variational inequality

(3.56)

and is a solution of general system of variational inequality problem (1.11) such that .

Remark 3.6.

Theorem 3.4 mainly improves Theorem of Qin et al. [16], in the following respects:

(a)from the class of inverse-strongly accretive mappings to the class of Lipchitzian and relaxed cocoercive mappings,

(b)from a -strict pseudocontraction to the countable family of uniformly -strict pseudocontractions,

(c)from a uniformly convex and -uniformly smooth Banach space to a strictly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping.

Further, if is a countable family of nonexpansive mappings, then Theorem 3.4 is reduced to the following result.

Theorem 3.7.

Let be a strictly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant . Let be a maximal monotone mapping and a -Lipschitzian and relaxed -cocoercive mapping with , respectively, for each . Let be a countable family of nonexpansive mappings. Assume that , where is defined as in Lemma 1.8. Let be an -contraction; let be a strongly positive linear bounded self adjoint operator with coefficient with . Let and let be a sequence generated by

(3.57)

where , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions

(C1),

(C2) and ,

then converges strongly to which solves the variational inequality:

(3.58)

and is a solution of general system of variational inequality problem (1.11) such that .

Remark 3.8.

As in [27, Theorem ], we can generate a sequence of nonexpansive mappings satisfying AKTT-condition; that is, for any bounded subset of by using convex combination of a general sequence of nonexpansive mappings with a common fixed point. To be more precise, they obtained the following lemma.

Lemma 3.9 (see [27]).

Let be a closed convex subset of a smooth Banach space . Suppose that is a sequence of nonexpansive mappings of into inself with a common fixed point. For each , define by

(3.59)

where is a family of nonnegative numbers with indices with such that

(i) for all ,

(ii) for every ,

(iii).

Then the following are given.

()Each is a nonexpansive mapping.

() satisfies AKTT-condition.

()If is defined by

(3.60)

then and .

Theorem 3.10.

Let be a strictly convex and -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant . Let be a maximal monotone mapping and a -Lipschitzian and relaxed -cocoercive mapping with , respectively, for each . Let be a countable family of nonexpansive mappings. Assume that , where is defined as in Lemma 1.8. Let be an -contraction; let be a strongly positive linear bounded self adjoint operator with coefficient with . Let and let be a sequence generated by

(3.61)

where satisfies conditions (i)–(iii) of Lemma 3.9, , and and are sequences in . Suppose that satisfies AKTT-condition. Let be the mapping defined by for all and suppose that . If the control consequences and satisfy the following restrictions:

(C1),

(C2) and ,

then converges strongly to , which solves the variational inequality

(3.62)

and is a solution of general system of variational inequality problem (1.11) such that .

Proof.

We write the iteration (3.61) as

(3.63)

where is defined by (3.59). It is clear that each mapping is nonexpansive. By Theorem 3.7 and Lemma 3.9, the conclusion follows.

The following example appears in [27] shows that there exists satisfying the conditions of Lemma 3.9.

Example 3.11.

Let be defined by

(3.64)

for all with In this case, the sequence of mappings generated by is defined as follows: For .

(3.65)