International Journal of AnalysisVolume 2013 (2013), Article ID 762831, 18 pageshttp://dx.doi.org/10.1155/2013/762831

Research Article

A General Approximation Scheme for Solutions of Various Problems in Fixed Point Theory

Eric U. Ofoedu

Department of Mathematics, Nnamdi Azikiwe University, PMB 5025, Anambra State, Awka, Nigeria

Received 28 August 2012; Accepted 15 October 2012

Academic Editor: Antoine Ayache

Copyright © 2013 Eric U. Ofoedu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

It is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; and the set of fixed points of a continuous pseudocontractive mapping. Moreover, the common element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize, improve, and unify several recently announced results.

1. Introduction

Let be a real normed space with dual . The normalized duality mapping from to is defined by where denotes the generalized duality pairing. It is well known (see, e.g., [1]) that if is strictly convex, then is single-valued and if is a Hilbert space, then is the identity mapping. In the sequel, we shall denote the single-valued normalized duality mapping by .

A mapping with domain , and range , in is called a strict contraction or simply a contraction if and only if there exists such that for all , and is called nonexpansive if and only if for all , A point is called a fixed point of an operator if and only if . The set of fixed points of an operator is denoted by , that is, .

The most important generalization of the class of nonexpansive mappings is, perhaps, the class of pseudocontractive mappings. These mappings are intimately connected with the important class of nonlinear accretive operators. This connection will be made precise in what follows.

A mapping with domain , and range , in is called pseudocontractive if and only if for all , the following inequality holds: for all . As a consequence of a result of Kato [2], the pseudocontractive mappings can also be defined in terms of the normalized duality mappings as follows: the mapping is called pseudocontractive if and only if for all , there exists such that It now follows trivially from (5) that every nonexpansive mapping is pseudocontractive. We note immediately that the class of pseudocontractive mappings is larger than that of nonexpansive mappings. For examples of pseudocontractive mappings which are not nonexpansive, the reader may see [1].

To see the connection between the pseudocontractive mappings and the accretive mappings, we introduce the following definition: a mapping with domain, , and range, , in is called accretive if and only if for all , the following inequality is satisfied: for all . Again, as a consequence of Kato [2], it follows that is accretive if and only if for all , there exists such that The operator is called -inverse strongly accretive  if and only if there exists such that for all , there exists such that It is easy to see from inequalities (4) and (6) that an operator is accretive if and only if the mapping is pseudocontractive. Consequently, the fixed point theory for pseudocontractive mappings is intimately connected with the mapping theory of accretive operators. For the importance of accretive operators and their connections with evolution equations, the reader may consult any of the references [1, 3].

Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors. It is of interest to note that if is a Hilbert space, accretive (-strongly accretive) operators coincide with the monotone (-strongly monotone) operators, where an operator with domain, , and range, , in is called monotone if and only if for all , we have that and is called -inverse strongly monotone if and only if for all , A bounded linear operator with domain, , and range, , in a Hilbert space is called a strongly positive operator if and only if there exists a constant such that for all .

Let be a closed convex nonempty subset of a real Hilbert space with inner product and norm . Let be a bifunction and be a proper extended real valued function, where denotes the set of real numbers. Let be a nonlinear monotone mapping. The generalized mixed equilibrium problem (GMEP) for , and is to find such that The set of solutions for GMEP (11) is denoted by If in (11), then (11) reduces to the classical equilibrium problem (EP), that is, the problem of finding such that and is denoted by , where If in (11), then GMEP (11) reduces to the classical variational inequality problem and is denoted by , where If , then GMEP (11) reduces to the following minimization problem: and GMEP is denoted by , where If , then (11) becomes the mixed equilibrium problem (MEP) and is denoted by , where If , then (11) reduces to the generalized equilibrium problem (GEP) and is denoted by , where If , then GMEP (11) reduces to the generalized variational inequality problem (abbreviated GVIP) and is denoted by , where The generalized mixed equilibrium problem (GMEP) includes special cases as the monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. Furthermore, there are several other problems, for example, the complementarity problems and fixed point problems, which can also be written in the form of the generalized mixed equilibrium problem. In other words, the generalized mixed equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, computer science, optimization theory, operations research, economics, and countless other fields. For the past 20 years or so, many existence results have been published for various equilibrium problems (see, e.g., [4–6]).

Iterative approximation of fixed points and zeros of nonlinear operators has been studied extensively by many authors to solve nonlinear operator equations as well as variational inequality problems and their generalizations (see, e.g., [7–24]). Most published results on nonexpansive mappings (e.g.) focus on the iterative approximation of their fixed points or approximation of common fixed points of a given family of this class of mappings.

Let be a closed convex nonempty subset of , let be a nonexpansive mapping such that . Given that and a real sequence in the interval , let a sequence be defined by Under appropriate conditions on the iterative parameter , it was shown by Bauschke [25], Halpern [26], Lions [27], and Wittmann [28] that converges strongly to , the projection of to the fixed point set, , of .

Marino and Xu [11], proved that the iteration scheme given by converges strongly to a unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for all ); provided is a strict contraction, is nonexpansive and the iterative parameter that satisfies appropriate conditions.

In [29], Yamada introduced the following hybrid iterative method: where is nonexpansive, is -Lipschitzian and an -strongly monotone operator. He proved that if and satisfy appropriate conditions, then (25) converges strongly to a unique solution of the varational inequality .

Tian [30] introduced the following iterative method: and proved that if is a contraction, is an -strongly monotone mapping, a nonexpansive mapping and the parameter satisfies appropriate conditions, then the sequence converges strongly to a unique solution of the variational inequality Let be a convex nonempty subset of a real Banach space. Let be a countable infinite family of nonexpansive mappings of into itself and let be a sequence of real numbers such that for all . For all , define a mapping by The mapping is called the -mapping generated by the countable infinite family of nonexpansive mappings (see, e.g., [31]).

In [32], Yao et al. studied the following problem. Let be a real Hilbert space. Consider the iterative scheme where is some constant, is a given contractive mapping, and is a strongly positive bounded linear operator on . Assuming that , and under appropriate conditions on the iterative parameters, Yao et al. proved that the sequence generated by (29) converges strongly to , where is a unique solution of the variational inequality Colao and Marino [33] proved that if and are sequences generated by and (where , , , and satisfy appropriate conditions), then both and converge strongly to an element , which is also a unique solution of the variational inequality Let be a nonempty closed and convex subset of real Hilbert space . Let be a continuous pseudocontractive mapping; be a continuous monotone mapping; an inverse strongly monoyone mapping. For each , let and be defined for each by Let be a sequence generated iteratively by , (where , , , and satisfy appropriate conditions), then it was recently proved by Chamnarnpan and Kumam [34] that the sequence defined by (34) converges strongly to an element , which is a unique solution of the variational inequality

In this paper, motivated by the results of the authors mentioned above, it is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; the set of fixed points of a continuous pseudocontractive mapping. Moreover, the element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize improve and unify several recently announced results.

2. Preliminary

In what follows, we shall make use of the following lemmas.

Lemma 1 (see, e.g., [31]). Let be a closed convex nonempty subset of a strictly convex real Banach space. Let be a countable infinite family of nonexpansive mappings of into itself and let be a real sequence such that for all , for some constant . Then, for all and exists (where is as defined in (28)).

In particular, for in Lemma 1, we define a mapping by

Lemma 2 (see, e.g., [31]). Let be a closed convex nonempty subset of a strictly convex real Banach space. Let be a countable infinite family of nonexpansive mappings of into itself such that and let be a real sequence such that for all , for some constant . Then, .

Lemma 3 (see [33]). Let be a closed convex nonempty subset of a strictly convex real Banach space. Let be a countable infinite family of nonexpansive mappings of into itself such that and let be a real sequence such that for all , for some constant . Suppose that is given by (28) for all , then

Lemma 4. Let be a real Hilbert space, then the following inequality holds:

Lemma 5 (see, e.g., [35–37]). Let be a sequence of nonnegative real numbers satisfying the following condition: where and are sequences of real numbers such that . Suppose that (i.e., ) or or , then as .

Lemma 6 (see, e.g., Suzuki [38]). Let and be two bounded sequences in a real Banach space such that , for all , where is a real sequence satisfying the condition . Suppose that then .

Lemma 7 (see Zegeye [39]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping, then for all and , there exists such that

Lemma 8 (see Zegeye [39]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping, then for all and , define a mapping by then the following hold: (1) is single-valued; (2) is firmly nonexpansive type mapping, that is, for all , (3) is closed and convex; and for all .

In the sequel, we shall require that the bifunction satisfies the following conditions: for all ; is monotone, in the sense that for all ; for all ; the function is convex and lower semicontinuous for all .

Lemma 9 (Compare with Lemma 2.4 of [5]). Let be a closed convex nonempty subset of a real Hilbert space . Let be a bifunction satisfying conditions – a continuous monotone mapping and a proper lower semicontinuous convex function. Then, for all and there exists such that Moreover, if for all we define a mapping by then the following hold: (1) is single-valued for all ; (2) is firmly nonexpansive, that is, for all , (3) for all ; (4) is closed and convex.

Let be a closed convex nonempty subset of a uniformly convex real Banach space , where is called uniformly convex if and only if the modulus of convexity is positive for all . Given a bounded sequence in , define It is well known (see, e.g., [40]) that there exists unique such that Clearly, and for all , so that exists. It is known (see [40]) that if , then the sequence converges.

Definition 10 (see [40]). If the sequence converges, then is called the asymptotic center of (with respect to ). Equivalently (see, e.g., [22]), is called the asymptotic center of (with respect to ) if and only if

Lemma 11 (see Theorem 1 of Edelstein [40]). Let be a closed convex nonempty subset of a uniformly convex real Banach space , Let be a bounded sequence in and be as in (49), then exists. In other words, the asymptotic center of the sequence (with respect to ) exists and is unique.

3. Main Results

Lemma 12. Let be a nonempty closed convex subset of a real Hilbert space . Let be a continuous pseudocontractive mapping. For , let be the mapping in Lemma 8, then for any and for any ,

Proof. Observe that for any , So, we obtain (in particular) for in (52) and in (53) that respectively. Adding (54) and (55) gives Using the fact that is pseudocontractive, we obtain from (56) that This implies that Equivalently, we obtain This implies that Thus,

Lemma 13. Let be a closed convex nonempty subset of a real Hilbert space . Let be a bifunction satisfying conditions – a monotone mapping and a proper lower semicontinuous convex function. Let and let be the mapping in Lemma 9, then for all and for all , one has that

Proof. We observe that In particular, we obtain from (63) and (64) (resp.) that Adding (65) (and using the monotonicity of and ), we get that This implies that Thus,

Lemma 14. Let be a bounded sequence of nonnegative real numbers, then

Proof. It is well known that for any two bounded sequences and of nonnegative real numbers, we have that It then follows that for any bounded sequence of nonnegative terms, To complete our proof, we show that Now, let be a subsequence of such that , then by the continuity of the function given by , we obtain that The last inequality follows from the fact that equals , where is the set of all subsequential limits of . This completes the proof.

Remark 15. We must note that Lemma 14 holds trivially if the sequence was an unbounded sequence of nonnegative real numbers.

Remark 16. In the sequel, we shall make the following assumptions: is a real Hilbert space; is a continuous pseudocontractive mapping; is a countable infinite family of nonexpansive mappings; is a bifunction satisfying conditions – a proper lower semicontinuous convex function; a continuous monotone mapping; is a fixed vector; is a strongly positive bounded linear operator with coefficient is an -inverse strongly monotone mapping. The sequences , , and are real sequences such that for all ,for all for all and for some constant is a real constant such that ; . For , and are as in Lemmas 8 and 9, respectively.

We shall study the strong convergence of the sequence generated iteratively from arbitrary by to a unique solution of the variational inequality

Lemma 17. Suppose that the conditions of Remark 16 are satisfied, then defined by (74) is bounded.

Proof. Observe that for all , Thus, since , we get from (76) that It thus follows that for , Next, we may (without loss of generality) assume that for all since . So, for each such that , we have Thus, we have that for all , the operator is a positive bounded linear operator, so that Therefore, for , we obtain from (74) using (78) and (80) that It is thus easy to show by mathematical induction (using (81)) that So, is bounded. Consequently, ,  ,   and are all bounded.

Lemma 18. Let be given by (74). Suppose that conditions of Remark 16 are satisfied, then

Proof. Set , then . Thus, Thus, for (using (84), Lemmas 3, 13, and 12), we obtain (for some constant ) the following: Since for all and for some constant , it is thus easy to see that . So, we obtain from (85) that Lemma 6 therefore gives but Hence,

Lemma 19. Let be given by (74). Suppose conditions of Remark 16 are satisfied, then

Proof. Observe that for some constant , So, we obtain from (91) that Using (89) and the fact that , we obtain from (92) that Next, observe that for fixed and using of Lemma 9, we obtain the following: so that Furthermore, using of Lemma 8, we obtain so that Moreover, using the recursion formula (74), Lemma 4, and convexity of , we have Thus, using (76) and (97), we obtain from (98) the following: So, from (99), we have (for some real constant ) From (100), we have and since , then using (101) and applying sandwich theorem, we obtain Furthermore, observe that Hence, using (93), (102), (103), and (104), we have Besides, we obtain from the second line of inequality (99) (using (95) and the fact that is firmly nonexpansive; thus nonexpansive) that for some constant , So, we obtain from (106) that for some constant , Hence, .

Lemma 20. Let be given by (74). Suppose that conditions of Remark 16 hold, then

Proof. Let , then we obtain This inequality implies the following: But from the first line of inequality (99), we have So, for some real constant , we obtain from (111) that Using (112) in (110) and rearranging terms, we obtain (for some real constant ) that So, (113) implies that Hence, The last inequality implies that

Lemma 21. Let be given by (74) and let be the unique solution of the variational inequality (75). Suppose that conditions of Remark 16 hold, then

Proof. Let be a subsequence of such that Since every real Hilbert space is a uniformly convex real Banach space and is a bounded sequence in , then we obtain by Lemma 11 that there exists a unique such that is the asymptotic center of the sequence . We first show that . Recall that and that by Lemma 1, exists for all and by Lemma 2 that ; thus, Thus, we obtain from (119) that so that This implies by Definition 10 (with ) and uniqueness of that ; that is, . Next, so that from (122), we obtain Thus, So, by Definition 10, we obtain that , that is, . Moreover, Thus, we obtain from (125) that Thus, Definition 10 again implies that . Hence . Furthermore, since we have shown that , then making use of this fact and Lemma 13, we have This implies that Inequality (129) thus gives Thus, we obtain from (130) that so that Definition 10 again gives and this implies that . Hence, . Now, let and be arbitrary, then, using Lemma 4 we have Since is the asymptotic center of , it is easy to see (using Definition 10 and Lemma 14) that Thus, But, So, Thus, since is arbitrary, we obtain from (136) that In particular, for and using the fact that , we obtain from (137) that This implies that Hence,

Theorem 22. Let be given by (74). Suppose that the conditins of Remark 16 are satisfied, then converges strongly to which is a unique solution of the variational inequality (75).

Proof. From the recursion formula (74) and Lemma 4, we have for some real constant . Thus, where and . It is easy to see that (using (93), Lemma 21 and the fact that ). Hence, we obtain by Lemma 5 that the sequence converges strongly to which is a unique solution of (75).

Remark 23. Let be a strict contraction, then our method of proof easily carries over to the iteration method given by , making use of the fact that is a contraction. Thus, we obatin the following theorem.

Theorem 24. Let be given by (143). Suppose that the conditions of Remark 16 are satisfied, then converges strongly to which is a unique solution of the variational inequality which is the optimality condition for the minimization problem , where is a potential function for (i.e., for all ), and one recalls that .

Remark 25. Our theorems extend and unify most of the results that have been proved for the class of nonlinear operators studied in this paper. In particular, it is easy to see that Theorem 24 extends, improves, generalizes, and unifies the corresponding results of Yao et al. [32], Colao and Marino [33], and Chamnarnpan and Kuman [34]. Furthermore, the setting of our theorems corrects the problem of well definedness which might arise in the results of Chamnarnpan and Kuman [34], and our method of proof is of independent interest.

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