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An algorithm for finding common solutions of various problems in nonlinear operator theory
Fixed Point Theory and Applications volume 2014, Article number: 9 (2014)
Abstract
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.
1 Introduction
Let H be a real Hilbert space. A mapping T with domain and range in H is called an L-Lipschitzian mapping (or simply a Lipschitz mapping) if and only if there exists such that for all ,
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, .
A mapping T with domain and range in H is called a quasi-nonexpansive mapping if and only if and for any , for any ,
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 [1])
Let be a subspace of the set of real numbers ℝ, endowed with the usual topology. Define by for all . Clearly, . Observe that
Thus, T is quasi-nonexpansive. The mapping T is, however, not a nonexpansive mapping since for and ,
But
Let be endowed with usual topology. Define by
It is easy to see that since for , implies that . Thus, for any , , which is not possible. So, . Next, observe that for any ,
So, the mapping T is quasi-nonexpansive. Finally, we show that T is not nonexpansive. To see this, let and , then
But,
So,
The concept of quasi-nonexpansive mappings was essentially introduced by Diaz and Metcalf [3]. 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.
A mapping T with domain and range in H is called pseudocontractive if and only if for all , the following inequality holds:
for all . As a consequence of a result of Kato [4], the pseudocontractive mappings can also be defined in terms of the normalized duality mappings as follows: the mapping T is called pseudocontractive if and only if for all , we have that
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 [5].
To see the connection between the pseudocontractive mappings and the monotone mappings, we introduce the following definition: a mapping A with domain and range in E is called monotone if and only if for all , the following inequality is satisfied:
The operator A is called η-inverse strongly monotone if and only if there exists such that for all , we have that
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.
Let C be a closed convex nonempty subset of a real Hilbert space H with inner product and norm . Let be a bifunction. The classical equilibrium problem (EP) for a bifunction f is to find such that
The set of solutions for EP (1.6) is denoted by
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.
Some attempts to modify the Mann iteration method so that strong convergence is guaranteed have recently been made (we should recall that Mann iteration method only guarantees weak convergence (see, for example, Bauschke et al. [20])). Nakajo and Takahashi [16] formulated the following modification of the Mann iteration method for a nonexpansive mapping T defined on a nonempty bounded closed and convex subset C of a Hilbert space H:
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 [21], Nilsrakoo and Saejung [22], Ofoedu et al. [23], Yang and Su [24], 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.
2 Preliminary
In what follows, we shall make use of the following lemmas.
Lemma 2.1 (see, e.g., Kopecka and Reich [28])
Let C be a nonempty closed and convex subset of a real Hilbert space. Let and be the metric projection of H onto C, then for any ,
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 2.3 Let H be a real Hilbert space, then for any , ,
Lemma 2.4 (see Zegeye [29])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a continuous pseudocontractive mapping, then for all and , there exists such that
Lemma 2.5 (see Zegeye [29])
Let C be a nonempty closed convex subset of a real Hilbert space H. 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, i.e., for all ,
-
(3)
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 .
Lemma 2.6 (see, e.g., [7, 30])
Let C be a closed convex nonempty subset of a real Hilbert space H. Let be a bifunction satisfying conditions (A1)-(A4), 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.
Lemma 2.7 (see Ofoedu [31])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a continuous pseudocontractive mapping. For , let be the mapping in Lemma 2.5, then for any and for any ,
Lemma 2.8 (Compare with Lemma 13 of Ofoedu [31])
Let C be a closed convex nonempty subset of a real Hilbert space H. Let be a bifunction satisfying conditions (A1)-(A4). Let and let be the mapping in Lemma 2.6, then for all and for all , we have that
3 Main results
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). For all , , let
and for all , , let
then in what follows we shall study the following iteration process:
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 .
Proof We first show that is closed and convex for each . From the definitions of it is obvious that is closed. Moreover, since is equivalent to , it follows that is convex for each . Next, we prove that for each . From the assumption, we see that . Suppose that for some , then for , we obtain that
Furthermore,
Thus,
Using (3.3) in (3.2) gives
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 .
Proof From Lemma 3.1, we obtain that , and is well defined for each . From and , we obtain that
Besides, by Lemma 2.1,
Thus, the sequence is a bounded nondecreasing sequence of real numbers. So, exists. Similarly, by Lemma 2.1, we have that for any positive integer μ,
Since exists, we have that and hence, is a Cauchy sequence in C. Therefore, there exists such that . Since , we have that
Thus,
and hence as , which implies that as .
Next, we observe that for and using Lemma 2.3,
But
So, using (3.7) in (3.6), we obtain that
Moreover, we obtain that
Using (3.8) in (3.9) we get that
Now, using the fact that , inequality (3.10) gives (for some constant ) that
Hence, we obtain from inequality (3.11) that
Moreover, from (3.10) we obtain that
which yields that
Now,
It follows from (3.13) and (3.14) that
and hence as .
We now show that . Observe that from (3.12) and (3.15) we obtain that
so that
Let be such that for all , then since as , we obtain from (3.17), using the continuity of , that
Similarly, if is such that for all , then we have again that
Continuing, we obtain that , . Hence, .
Next, we show that . Since is γ-inverse strongly monotone for , we have that is -Lipschitz continuous. Thus,
Hence, from (3.18) and (3.13), we obtain that
As a result, we get that
Let be such that for all . Then
Similarly, we have that for . Thus, .
Furthermore, we show that , . Using the fact that , as , we obtain that
Thus, we obtain from (3.19) that
This implies that . But by Lemma 2.7,
Thus,
So, the continuity of and the fact that as give
A similar argument gives
Hence,
Moreover, we show that . Observe that
Thus, we obtain from (3.20) that
This implies that . But by Lemma 2.8,
Thus,
So, the continuity of and the fact that as give
A similar argument gives
Hence,
Finally, we prove that . From , we obtain that
Since , we also have that
So,
Inequality (3.22) implies that
By taking limit as in (3.23), we obtain that
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.
Remark 3.4 Several authors (see, e.g., [8, 31] and references therein) have studied the following problem: Let C be a closed convex nonempty subset of a real Hilbert space H with inner product and norm . Let be a bifunction and be a proper extended real-valued function, where ℝ denotes the set of real numbers. Let be a nonlinear monotone mapping. The generalized mixed equilibrium problem (abbreviated GMEP) for f, Φ and Θ is to find such that
The set of solutions for GMEP (3.24) is denoted by
These authors always claim that if in (3.24), then (3.24) reduces to the classical equilibrium problem (abbreviated EP), that is, the problem of finding such that
and is denoted by , where
If in (3.24), then GMEP (1.6) reduces to the classical variational inequality problem and is denoted by , where
If , then GMEP (3.24) reduces to the following minimization problem:
and is denoted by , where
If , then (3.24) becomes the mixed equilibrium problem (abbreviated MEP) and is denoted by , where
If , then (1.6) reduces to the generalized equilibrium problem (abbreviated GEP) and is denoted by , where
If , then GMEP (3.24) reduces to the generalized variational inequality problem (abbreviated GVIP) and is denoted by , where
It is worthy to note that if we define by
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.
4 Application (convex differentiable optimization)
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 [39] 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 [39])
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.
Let K be a closed convex subset of a real Hilbert space H, consider the minimization problem given by
where ϕ is a Fréchet-differentiable convex functional. Let , the solution set of (4.1), be nonempty. It is known that a point if and only if the following optimality condition holds:
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.
Theorem 4.2 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 convex and Fréchet-differentiable functionals on H such that is -Lipschitz continuous for some , ; let be t bifunctions satisfying conditions (A1)-(A4). Let . Let be a sequence defined by
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. □
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Acknowledgements
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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Ofoedu, E.U., Odumegwu, J.N., Zegeye, H. et al. An algorithm for finding common solutions of various problems in nonlinear operator theory. Fixed Point Theory Appl 2014, 9 (2014). https://doi.org/10.1186/1687-1812-2014-9
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DOI: https://doi.org/10.1186/1687-1812-2014-9