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An algorithm for finding common solutions of various problems in nonlinear operator theory
© Ofoedu et al.; licensee Springer. 2014
- Received: 27 August 2013
- Accepted: 21 November 2013
- Published: 9 January 2014
In this paper, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone operators; the set of common fixed points of a finite family of quasi-nonexpansive mappings; and the set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets.
MSC:47H06, 47H09, 47J05, 47J25.
- classical equilibrium problem
- generalized mixed equilibrium problem
- η-inverse strongly monotone mapping
- maximal monotone operator
- nonexpansive mappings
- real Hilbert space
- pseudocontractive mappings
- variational inequality problem
If , then T is called strict contraction or simply a contraction; and if , then T is called nonexpansive. A point is called a fixed point of an operator T if and only if . The set of fixed points of an operator T is denoted by , that is, .
Every nonexpansive mapping with a nonempty fixed point set is quasi-nonexpansive. The following examples show that the converse is not true.
Example 1.1 (see )
The concept of quasi-nonexpansive mappings was essentially introduced by Diaz and Metcalf . Although Examples 1.1 and 1.2 guarantee the existence of a quasi-nonexpansive mapping which is not nonexpansive, we must note that a linear quasi-nonexpansive mapping defined on a subspace of a given vector space is nonexpansive on that subspace.
Another important generalization of the class of nonexpansive mappings is 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.
It now follows trivially from (1.3) 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 .
It is easy to see from inequalities (1.3) and (1.4) that an operator A is monotone if and only if the mapping is pseudocontractive. Consequently, the fixed point theory for pseudocontractive mappings is intimately connected with the zero of monotone mappings. For the importance of monotone mappings and their connections with evolution equations, the reader may consult any of the references [5, 6].
Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors.
The classical equilibrium problem (EP) includes as special cases the monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, 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 classical equilibrium problem. In other words, the classical 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., [7–10]). Approximation methods for such problems thus become a necessity.
Iterative approximation of fixed points and zeros of nonlinear mappings has been studied extensively by many authors to solve nonlinear mapping equations as well as variational inequality problems and their generalizations (see, e.g., [11–19]). Most published results on nonexpansive mappings (for example) focus on the iterative approximation of their fixed points or approximation of common fixed points of a given family of this class of mappings.
where denotes the metric projection from H onto a closed convex subset C of H. They proved that if the sequence is bounded away from 1, then defined by (1.7) converges strongly to .
Formulations similar to (1.7) for different classes of nonlinear problems had been presented by Kim and Xu , Nilsrakoo and Saejung , Ofoedu et al. , Yang and Su , Zegeye and Shahzad [25–27].
In this paper, motivated by the results of the authors mentioned above, it is our aim to prove strong convergence of a new iterative algorithm to a common element of the set of solutions of a finite family of classical equilibrium problems; a common set of zeros of a finite family of inverse strongly monotone mappings; a set of common fixed points of a finite family of quasi-nonexpansive mappings; and a set of common fixed points of a finite family of continuous pseudocontractive mappings in Hilbert spaces on assumption that the intersection of the aforementioned sets is not empty. Moreover, the common element is shown to be the metric projection of the initial guess on the intersection of these sets. Our theorems complement the results of the authors mentioned above and those of several other authors.
In what follows, we shall make use of the following lemmas.
Lemma 2.1 (see, e.g., Kopecka and Reich )
Lemma 2.2 Let C be a closed convex nonempty subset of a real Hilbert space H; and let be the metric projection of H onto C. Let , then if and only if for all .
Lemma 2.4 (see Zegeye )
Lemma 2.5 (see Zegeye )
- (2)is firmly nonexpansive type mapping, i.e., for all ,
is closed and convex; and for all .
In the sequel, we shall require that the bifunction satisfies the following conditions:
(A1) , ;
(A2) f is monotone in the sense that for all ;
(A3) for all ;
(A4) the function is convex and lower semicontinuous for all .
is single-valued for all ;
- (2)is firmly nonexpansive, that is, for all ,
for all ;
is closed and convex.
Lemma 2.7 (see Ofoedu )
Lemma 2.8 (Compare with Lemma 13 of Ofoedu )
where , ; such that ; a sequence in such that ; such that ; and is a sequence in for some such that , .
Lemma 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be m continuous pseudocontractive mappings; let be l continuous quasi-nonexpansive mappings; let be d -inverse strongly monotone mappings with constants , ; let be t bifunctions satisfying conditions (A1)-(A4). Let . Let be a sequence defined by (3.1), then the sequence is well defined for each .
So, . This implies, by induction, that so that the sequence generated by (3.1) is well defined for all . □
Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be m continuous pseudocontractive mappings; let be l continuous quasi-nonexpansive mappings; let be d -inverse strongly monotone mappings with constants , ; let be t bifunctions satisfying conditions (A1)-(A4). Let . Let be a sequence defined by (3.1). Then the sequence converges strongly to the element of F nearest to .
and hence as , which implies that as .
and hence as .
Continuing, we obtain that , . Hence, .
Similarly, we have that for . Thus, .
Now, by Lemma 2.2 we have that . This completes the proof. □
Remark 3.3 We note that makes sense since it could be easily shown that F is closed and convex. In fact, it is enough to show that the set of zeros of γ-inverse monotone mappings and a fixed point set of continuous quasi-nonexpansive mappings are convex sets. Closure of the two sets simply follows from the continuity of the mappings involved.
then it could be easily checked that Γ is a bifunction and satisfies properties (A1)-(A4). Thus, the so-called generalized mixed equilibrium problem reduces to the classical equilibrium problem for the bifunction Γ. Thus, consideration of the so-called generalized mixed equilibrium problem in place of the classical equilibrium problem studied in this paper leads to no further generalization.
In Section 1, we defined a Lipschitz continuous mapping and an inverse strongly monotone mapping. Inverse strongly monotone mappings arise in various areas of optimization and nonlinear analysis (see, for example, [32–38]). It follows from the Cauchy-Schwarz inequality that if a mapping is -inverse strongly monotone, then A is L-Lipschitz continuous. The converse of this statement, however, fails to be true. To see this, take for instance , where I is the identity mapping on H, then A is L-Lipschitz continuous (with ) but not -inverse strongly monotone (that is, not firmly nonexpansive in this case).
Baillon and Haddad  showed in 1977 that if and A is the gradient of a convex functional on H, then A is -inverse strongly monotone if and only if A is L-Lipschitz continuous. This remarkable result, which has important applications in optimization theory (see, for example, [40–42]), has become known as the Baillon-Haddad theorem. In fact, we have the following theorem.
Theorem 4.1 (Baillon-Haddad) (see Corollary 10 of )
Let be a convex Fréchet-differentiable functional on H such that ∇ϕ is L-Lipschitz continuous for some , then ∇ϕ is a -inverse strongly monotone mapping (where ∇ϕ denotes the gradient of the functional ϕ).
Now, let us turn to the problem of minimizing a continuously Fréchet-differentiable convex functional with minimum norm in Hilbert spaces.
It is easy to see that if , then optimality condition (4.2) is equivalent to if and only if .
Thus, we obtain the following as a corollary of Theorem 3.2.
where , ; such that ; a sequence in such that ; such that ; and is a sequence in for some such that , . Then the sequence converges strongly to the element of F nearest to .
Proof Since, by our hypothesis, is -Lipschitz continuous for some , , we obtain from Theorem 4.1 that is -inverse strongly monotone, ; and since , it is then easy to see that is -inverse strongly monotone, . The rest, therefore, follows as in the proof of Theorem 3.2 with . This completes the proof. □
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support.
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