Abstract
Let C be a closed and convex subset of a real Hilbert space H. Let T be a Lipschitzian pseudocontractive mapping of C into itself, A be a γ-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H such that the domain of B is included in C. We introduce an iteration scheme for finding a minimum-norm point of . Application to a common element of the set of fixed points of a Lipschitzian pseudocontractive and solutions of variational inequality for α-inverse strongly monotone mappings is included. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings. To the best of our knowledge, approximating a common fixed point of pseudocontractive mappings with explicit scheme has not been possible and our result is even the first result that states the solution of a variational inequality in the set of fixed points of pseudocontractive mappings. Our scheme which is explicit is the best to use for the problem under consideration.
MSC:47H05, 47H09, 47J25.
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1 Introduction
Let C be a closed convex subset of a real Hilbert space H. A mapping is called a contraction mapping if there exists such that for all . If then T is called nonexpansive. T is called quasi-nonexpansive if for all and , where , the set of fixed points of T. A mapping T is called γ-strictly pseudocontractive [1] if and only if there exists such that
and T is called pseudocontractive if
where I is the identity mapping. We note that inequalities (1.1) and (1.2) can be equivalently written as
for some , and
respectively.
Clearly, the class of nonexpansive mappings is a subset of the class of γ-strictly pseudocontractive mappings and the class of γ-strictly pseudocontractive is contained in the class of pseudocontractive mappings. Moreover, this inclusion is strict due to the following example in [2].
Take , , , . If we define x⊥ to be . Define by
Then T is a Lipschitzian and pseudocontractive mapping but not a strictly pseudocontractive mapping.
Closely related to the class of pseudocontractive mappings is the class of monotone mappings. A mapping is called monotone if
and A is called γ-inverse strongly monotone if there exists a positive real number γ such that
If A is γ-inverse strongly monotone, then inequality (1.7) implies that A is Lipschitzian with constant , that is, , for all .
We remark the T is γ-strictly pseudocontractive if and only if is γ-inverse strongly monotone and T is pseudocontractive if and only if is monotone. Clearly, the class of monotone mappings includes the class of γ-inverse strongly monotone mappings. We note that the inclusion is proper. This can be seen from the example in [2]. Take , where T is as in (1.5). Then we see that A is monotone but not γ-inverse strongly monotone as T is not strictly pseudocontractive.
A mapping A is called maximal monotone if it is monotone and , the range of , is H for all . If A is maximal monotone, then to each and , there corresponds a unique element satisfying
We denote the resolvent of A by . That is, for all . If A is monotone then is nonexpansive single valued mapping from into and (see [3]).
It is now well known (see e.g. [4]) that if A is monotone then the solutions of the equation correspond to the equilibrium points of some evolution systems. Consequently, considerable research efforts, especially within the past 20 years or so, have been devoted to iterative methods for approximating the zeros of monotone mapping A or fixed point of pseudocontractive mapping T (see, for example, [5–11]).
Let A be a nonlinear mapping on H. Consider the problem of finding
When A is a maximal monotone mapping, a well-known methods for solving (1.8) is the proximal point algorithm: , and
where and , then Rockafellar [12] (also see [13]) proved that the sequence converges weakly to an element of .
In [14], Kamimura and Takahashi investigated the problem of finding a zero point of a maximal monotone mapping by considering the following iterative algorithm:
where is a sequence in , is a positive sequence, is a maximal monotone, and . They showed that the sequence generated by (1.9) converges weakly to some in the framework of real Hilbert spaces, provided that the control sequences satisfy some restrictions.
Let C be a nonempty, closed and convex subset of H and be a nonlinear mapping. The variational inequality problem which was introduced and studied by Stampacchia [15] is to:
The set of solutions of the variational inequality problem is denoted by .
Variational inequality theory has emerged as an important tool in studying a wide class of numerous problems in physics, optimization, variational inequalities, minimax problems, and the Nash equilibrium problems in noncooperative games (see, for instance, [16–22]).
In [23], Takahashi and Toyoda investigated the problem of finding a common point of solutions of the variational inequality problem (1.10) for a γ-inverse strongly monotone mapping and fixed points of a nonexpansive mapping by considering the following iterative algorithm:
where is a sequence in , is a positive sequence. They proved that the sequence generated by (1.11) converges weakly to some provided that the control sequences satisfy some restrictions.
It is worth to mention that the methods studied above give weak convergence theorems in the framework of Hilbert spaces.
Regarding iterative method for a common point of fixed points of nonexpansive and zeros of sum of two monotone mappings, Takahashi et al. [24] proved the following theorem.
Theorem TT [24]
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A be a γ-inverse strongly monotone mapping of C into H and let B be a maximal monotone mapping on H such that the domain of B is included in C. Let be the resolvent of B, for , and let T be a nonexpansive mapping of C into itself such that . Let and let be a sequence generated by
where , and satisfy certain conditions. Then converges strongly to a point of .
For other related results, we refer to [25–30].
A natural question arises: can we obtain an iterative scheme which converges strongly to a common point of fixed points of the pseudocontractive mapping T and zeros of two monotone mappings?
It is our purpose in this paper to introduce an iterative scheme which converges strongly to a common minimum-norm point of fixed points of a Lipschitzian pseudocontractive mapping and zeros of sum of two monotone mappings. Application to a common element of the set of fixed points of a Lipschitzian pseudocontractive mapping and solutions of variational inequality for γ-inverse strongly monotone mapping is included. The results obtained in this paper improve and extend the results of Kamimura and Takahashi [14], Takahashi and Toyoda [23], Takahashi et al. [24] and some other results in this direction.
2 Preliminaries
In what follows we shall make use of the following lemmas.
Lemma 2.1 [31]
Let C be a convex subset of a real Hilbert space H. Let . Then if and only if
We also remark that in a real Hilbert space H, the following identity holds:
Lemma 2.2 [32]
Let be a sequence of nonnegative real numbers satisfying the following relation:
where and satisfying the following conditions: , , and . Then .
Lemma 2.3 [33]
Let H be a real Hilbert space, C a closed convex subset of H and be a continuous pseudocontractive mapping, then
-
(i)
is closed convex subset of C;
-
(ii)
is demiclosed at zero, i.e., if is a sequence in C such that and , as , then .
Lemma 2.4 [34]
Let be sequences of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence such that and the following properties are satisfied by all (sufficiently large) numbers :
In fact, .
Lemma 2.5 [35]
Let H be a real Hilbert space. Then for all and for such that the following equality holds:
Lemma 2.6 [36]
Let C be a nonempty closed and convex subset of a real Hilbert space H. Let be γ-inverse strongly monotone mapping. Then, for , the mapping is nonexpansive.
Lemma 2.7 [37]
Let H be a Hilbert space. Let and be maximal monotone mappings. Suppose that . Then is a maximal monotone mapping.
3 Main result
Theorem 3.1 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a Lipschitzian pseudocontractive mapping with Lipschitz constant L. Let be a γ-inverse strongly monotone mapping and B be a maximal monotone mapping on H such that the domain of B is subset of C. Assume that is nonempty. Let be the sequence generated from an arbitrary by
where and , , , for some , satisfying the following conditions: (i) , (ii) , ; (iii) , . Then converges strongly to the minimum-norm point of ℱ.
Proof From Lemma 2.6 and the fact that is nonexpansive we see that is nonexpansive. Let . Then from (3.1), (1.2), Lemma 2.5 and using the fact that we have
and hence
In addition, from (3.1), Lemma 2.5, and (1.2) we get
and
Substituting (3.3) and (3.4) into (3.2) we obtain
and hence
Now, from (iii) of the hypotheses we have
and
Thus, inequality (3.5) implies that
Thus, by induction,
which implies that and hence are bounded.
Let . Then we see that . Let . Then, using (3.1), (2.1) and following the methods used to get (3.5), we obtain
and so
which implies that
Now, we consider two cases.
Case 1. Suppose that there exists such that is decreasing for all . Then we see that is convergent. Thus, from (3.9) and (3.6) we have
Moreover, from (3.1) and (3.11) we obtain
and hence Lipschitz continuity of T, (3.12), (3.11) imply that
In addition, from (3.13) and (3.11) we have
Furthermore, since is bounded subset of H which is reflexive, we can choose a subsequence of such that and . It follows from (3.14) that . Then, from (3.11) and Lemma 2.3, we have .
Next, we show that . Let
Then from (3.11) we get as . In addition, for any , we see that
This implies that
and hence we get
Now from (3.15) we obtain
That is,
Since B is monotone, we get for any , where is the graph of B defined by ,
On the other hand, since , and , as we have . Thus, letting , we obtain from (3.17)
Thus, maximality of B implies that , that is, . Hence, we get .
Therefore, by Lemma 2.1, we immediately obtain
Then it follows from (3.10), (3.18), and Lemma 2.2 that as . Consequently, .
Case 2. Suppose that there exists a subsequence of such that
for all . Then, by Lemma 2.4, there exists a nondecreasing sequence such that , and
for all . Now, from (3.9) and (3.6) we get , and as . Thus, like in Case 1, we obtain and
Now, from (3.10) we have
and hence (3.19) and (3.21) imply that
But using the fact that and (3.20) we obtain
This together with (3.21) implies that as . But for all and hence we obtain . Therefore, from the above two cases, we can conclude that converges strongly to the minimum-norm point of ℱ. The proof is complete. □
If, in Theorem 3.1, we assume that , then we get and hence we get the following corollary.
Corollary 3.2 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a Lipschtzian pseudocontractive mapping with Lipschitz constant L and be a maximal monotone mapping. Assume that is nonempty. Let be the sequence generated from an arbitrary by
where and , , , for some , satisfying the following conditions: (i) , (ii) , ; (iii) , . Then converges strongly to the minimum-norm point of ℱ.
We also have the following theorem for two maximal monotone mappings.
Theorem 3.3 Let C be a nonempty, closed and convex subset of a real Hilbert space H such that . Let be maximal monotone mappings. Let be a Lipschitzian pseudocontractive mapping with Lipschitz constant L such that is nonempty. Let be the sequence generated from an arbitrary by
where and , , , for some , satisfying the following conditions: (i) , (ii) , ; (iii) , . Then converges strongly to the minimum-norm point of ℱ.
Proof From Lemma 2.7 we find that is a maximal monotone and hence by Corollary 3.2 we get the required assertion. □
If, in Theorem 3.3, we assume that , the identity mapping on C, then we get the following corollary.
Corollary 3.4 Let C be a nonempty, closed and convex subset of a real Hilbert space H such that . Let be maximal monotone mappings such that is nonempty. Let be the sequence generated from an arbitrary by
where and , , , for some , satisfying the following conditions: and . Then converges strongly to the minimum-norm point of ℱ.
4 Applications
We next study the problem of finding a solution of a variational inequality. Let C be a nonempty closed convex subset of a real Hilbert space H. The normal cone for C at a point , denoted by , is defined by
Let be a proper lower semicontinuous convex function. Define the subdifferential
for all . Then from Rockafellar [38] we know that ∂f is maximal monotone mapping of H into itself. Let C be a nonempty closed convex subset of H and be the indicator function of C, that is,
Then is a proper lower semicontinuous convex function on H and is a maximal monotone mapping. Let for all and . From the fact that and , we get
Moreover,
and hence . Thus, the following corollary holds. Now, using Theorem 3.1, we obtain a strong convergence theorem for finding a common point of fixed points of Lipschtzian pseudocontractive mapping and solutions of the variational inequality problem for γ-inverse monotone mapping.
Theorem 4.1 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a Lipschitzian pseudocontractive mapping with Lipschitz constant L and let be a γ-inverse strongly monotone mapping such that . Let be the sequence generated from an arbitrary by
where , , , for some , satisfying the following conditions: (i) , (ii) , ; (iii) , . Then converges strongly to the minimum-norm point of ℱ.
If, in Theorem 4.1, we take , the identity mapping on C we have the following corollary for a solution of variational inequality for a γ-inverse strongly monotone mapping.
Corollary 4.2 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a γ-inverse strongly monotone mapping with . Let be the sequence generated from an arbitrary by
where , , , for some , satisfying the following conditions: , . Then converges strongly to the minimum-norm point of .
If, in Theorem 4.1, we take , where S is a nonexpansive self mapping of C into itself, then we get the following corollary.
Corollary 4.3 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let be a Lipschitzian pseudocontractive mapping with Lipschitz constant L and let be a nonexpansive mapping such that . Let be the sequence generated from an arbitrary by
where , , , for some , satisfying the following conditions: (i) , (ii) , ; (iii) , . Then converges strongly to the minimum-norm point of .
Proof Put in Theorem 4.1. Then we see that A is a -inverse strongly monotone mapping. Furthermore, for we have
and
Thus, we obtain . Therefore, the conclusion holds by Theorem 4.1 □
Remark 4.4 Theorem 3.1 provides convergence sequence to a common point of fixed points of a Lipschitzian pseudocontractive mapping and zeros of two monotone mappings in Hilbert spaces.
Remark 4.5 Theorem 3.1 improves Theorem 3.1 of Takahashi et al. [24] in the sense that our convergence is to the common minimum-norm point of fixed points of a Lipschitzian pseudocontractive mapping and zeros of sum of two monotone mappings. Corollary 3.4 improves Theorem 1 of Kamimura and Takahashi [14] in the sense that our convergence is for the a zero of sum of two maximal monotone mappings. Theorem 4.1 extends Theorem 3.1 of Takahashi and Toyoda [23] in the sense that our convergence is to the common minimum-norm point of fixed points of a Lipschitzian pseudocontractive mapping and solutions of variational inequality for a γ-inverse strongly monotone mapping.
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Acknowledgements
The authors are grateful to anonymous reviewers for their pertinent remarks and useful suggestions. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author, therefore acknowledges with thanks DSR for financial support.
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Shahzad, N., Zegeye, H. Approximating a common point of fixed points of a pseudocontractive mapping and zeros of sum of monotone mappings. Fixed Point Theory Appl 2014, 85 (2014). https://doi.org/10.1186/1687-1812-2014-85
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DOI: https://doi.org/10.1186/1687-1812-2014-85