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Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings
Fixed Point Theory and Applications volume 2013, Article number: 58 (2013)
Abstract
Let and let K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let be a multi-valued strictly pseudo-contractive map with a nonempty fixed point set. A Krasnoselskii-type iteration sequence is constructed and proved to be an approximate fixed point sequence of T, i.e., . This result is then applied to prove strong convergence theorems for a fixed point of T under additional appropriate conditions. Our theorems improve several important well-known results.
MSC:47H04, 47H06, 47H15, 47H17, 47J25.
1 Introduction
For decades, the study of fixed point theory for multi-valued nonlinear mappings has attracted, and continues to attract, the interest of several well-known mathematicians (see, for example, Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadla [6], Downing and Kirk [7]).
Interest in the study of fixed point theory for multi-valued maps stems, perhaps, mainly from the fact that many problems in some areas of mathematics such as in Game Theory and Market Economy and in Non-Smooth Differential Equations can be written as fixed point problems for multi-valued maps. We describe briefly the connection of fixed point theory for multi-valued mappings and these applications.
Game theory and market economy
In game theory and market economy, the existence of equilibrium was uniformly obtained by the application of a fixed point theorem. In fact, Nash [3, 4] showed the existence of equilibria for non-cooperative static games as a direct consequence of Brouwer [1] or Kakutani [2] fixed point theorem. More precisely, under some regularity conditions, given a game, there always exists a multi-valued map whose fixed points coincide with the equilibrium points of the game. A model example of such an application is the Nash equilibrium theorem (see, e.g., [3]).
Consider a game with N players denoted by n, , where is the set of possible strategies of the n th player and is assumed to be nonempty, compact and convex and is the payoff (or gain function) of the player n and is assumed to be continuous. The player n can take individual actions, represented by a vector . All players together can take a collective action, which is a combined vector . For each n, and , we will use the following standard notations:
A strategy permits the n th player to maximize his gain under the condition that the remaining players have chosen their strategies if and only if
Now, let be the multi-valued map defined by
Definition A collective action is called a Nash equilibrium point if, for each n, is the best response for the n th player to the action made by the remaining players. That is, for each n,
or, equivalently,
This is equivalent to is a fixed point of the multi-valued map defined by
From the point of view of social recognition, game theory is perhaps the most successful area of application of fixed point theory of multi-valued mappings. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding of conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a non-equilibrium point and convergent to an equilibrium solution. This is part of the problem that is being addressed by iterative methods for a fixed point of multi-valued mappings.
Non-smooth differential equations
The mainstream of applications of fixed point theory for multi-valued maps has been initially motivated by the problem of differential equations (DEs) with discontinuous right-hand sides which gave birth to the existence theory of differential inclusion (DI). Here is a simple model for this type of application.
Consider the initial value problem
If is discontinuous with bounded jumps, measurable in t, one looks for solutions in the sense of Filippov [8] which are solutions of the differential inclusion
where
Now, set and let be the multi-valued Nemystkii operator defined by
Finally, let be a multi-valued map defined by , where is the inverse of the derivative operator given by
One can see that problem (1.4) reduces to the fixed point problem: .
Finally, a variety of fixed point theorems for multi-valued maps with nonempty and convex values is available to conclude the existence of a solution. We used a first-order differential equation as a model for simplicity of presentation, but this approach is most commonly used with respect to second-order boundary value problems for ordinary differential equations or partial differential equations. For more details about these topics, one can consult [9–12] and references therein as examples. Let E be a real normed linear space of dimension ≥2. The modulus of smoothness of E, , is defined by
A normed linear space E is called uniformly smooth if
It is well known (see, e.g., [13], p.16, [14]) that is nondecreasing. If there exist a constant and a real number such that , then E is said to be q-uniformly smooth. Typical examples of such spaces are the , and spaces for , where
Let denote the generalized duality mapping from E to defined by
where denotes the generalized duality pairing. is called the normalized duality mapping and is denoted by J. It is well known that if E is smooth, is single-valued.
Every uniformly smooth space has a uniformly Gâteaux differentiable norm (see, e.g., [13], p.17).
Let K be a nonempty subset of E. The set K is called proximinal (see, e.g., [15–17]) if for each , there exists such that
where for all . Every nonempty, closed and convex subset of a real Hilbert space is proximinal. Let and denote the families of nonempty, closed and bounded subsets and nonempty, proximinal and bounded subsets of K, respectively. The Hausdorff metric on is defined by
for all . Let be a multi-valued mapping on E. A point is called a fixed point of T if . The fixed point set of T is denoted by .
A multi-valued mapping is called L-Lipschitzian if there exists such that
When in (1.6), we say that T is a contraction, and T is called nonexpansive if .
Definition 1.1 Let K be a nonempty subset of a real Hilbert space H. A map is called k-strictly pseudo-contractive if there exists such that
Browder and Petryshyn [18] introduced and studied the class of strictly pseudo-contractive maps as an important generalization of the class of nonexpansive maps (mappings satisfying ). It is trivial to see that every nonexpansive map is strictly pseudo-contractive.
Motivated by this, Chidume et al.[19] introduced the class of multi-valued strictly pseudo-contractive maps defined on a real Hilbert space H as follows.
Definition 1.2 A multi-valued map is called k-strictly pseudo-contractive if there exists such that for all ,
They then proved convergence theorems for approximating fixed points of multi-valued strictly pseudo-contractive maps (see [19]) which extend recent results from the class of multi-valued nonexpansive maps to the more general and important class of multi-valued strictly pseudo-contractive maps.
Single-valued strictly pseudo-contractive maps have also been defined and studied in real Banach spaces, which are much more general than Hilbert spaces.
Definition 1.3 Let K be a nonempty subset of a real normed space E. A map is called k-strictly pseudo-contractive (see, e.g., [13], p.145, [18]) if there exists such that for all , there exists such that
In this paper, we define multi-valued strictly pseudo-contractive maps in arbitrary normed space E as follows.
Definition 1.4 A multi-valued map is called k-strictly pseudo-contractive if there exists such that for all ,
where and I is the identity map on E.
We observe that if T is single-valued, then inequality (1.10) reduces to (1.9).
Several papers deal with the problem of approximating fixed points of multi-valued nonexpansive mappings defined on Hilbert spaces (see, for example, Sastry and Babu [15], Panyanak [16], Song and Wong [17], Khan et al.[20], Abbas et al.[21] and the references contained therein) and their generalizations (see, e.g., Chidume et al.[19] and the references contained therein).
Chidume et al.[19] proved the following theorem for multi-valued k-strictly pseudo-contractive mappings defined on real Hilbert spaces.
Theorem CCDM (Theorem 3.2 [19])
Let K be a nonempty, closed and convex subset of a real Hilbert space H. Suppose thatis a multi-valued k-strictly pseudo-contractive mapping such that. Assume thatfor all. Letbe a sequence defined iteratively fromby
whereand. Then.
Using Theorem CCDM, Chidume et al. proved several convergence theorems for the approximation of fixed points of strictly pseudo-contractive maps under various additional mild compactness-type conditions either on the operator T or on the domain of T. The theorems proved in [19] are significant generalizations of several important results on Hilbert spaces (see, e.g., [19]).
Our purpose in this paper is to extend Theorem CCDM and other related results in [19], using Definition 1.4, from Hilbert spaces to the much more general class of q-uniformly smooth real Banach spaces. As we have noted, theses spaces include the , and spaces, and . Finally, we give important examples of multi-valued maps satisfying the conditions of our theorems.
2 Preliminaries
In the sequel, we need the following definitions and results.
Definition 2.1 Let E be a real Banach space and T be a multi-valued mapping. The multi-valued map is said to be strongly demiclosed at 0 (see, e.g., [22]) if for any sequence such that converges strongly to and converges to 0, then .
Lemma 2.2[19]
Let E be a reflexive real Banach space and let. Assume that B is weakly closed. Then, for every, there existssuch that
Proposition 2.3 Let K be a nonempty subset of a real Banach space E and letbe a multi-valued k-strictly pseudo-contractive mapping. Assume that for every, Tx is weakly closed. Then T is Lipschitzian.
Proof We first observe that for any , the set Tx is weakly closed if and only if the set Ax is weakly closed. Now, let and . From Lemma 2.2, there exists such that
Using the fact that T is k-strictly pseudo-contractive and inequality (2.2), we have
So,
From the definition of the Hausdorff distance, we have
Using (2.3) and (2.4), we obtain
Therefore, T is -Lipschitzian. □
Remark 1 We note that for a single-valued map T, for each , the set Tx is always weakly closed.
Lemma 2.4 Let, E be a q-uniformly smooth real Banach space, . Supposeis a multi-valued map with, and for all, ,
where, I is the identity map on E. Iffor all, then the following inequality holds:
Proof Let , , . Then, from inequality (2.5), the definition of the Hausdorff metric and the assumption that , we have
So,
Therefore, for all , , such that , using inequalities (2.5) and (2.6) and the fact that , we obtain
since . This completes the proof. □
Lemma 2.5 Let K be a nonempty closed subset of a real Banach space E and letbe a k-strictly pseudo-contractive mapping. Assume that for every, Tx is weakly closed. Thenis strongly demiclosed at zero.
Proof Let be such that and as . Since K is closed, we have that . Since, for every n, is proximinal, let such that . Using Lemma 2.2, for each n, there exists such that
We then have
Observing that , it then follows that
Taking limit as , we have that . Therefore . The proof is completed. □
Lemma 2.6[23]
Letand E be a smooth real Banach space. Then the following are equivalent:
-
(i)
E is q-uniformly smooth.
-
(ii)
There exists a constant such that for all ,
-
(iii)
There exists a constant such that for all and ,
where.
From now on, denotes the constant that appeared in Lemma 2.6. Let .
3 Main results
We prove the following theorem.
Theorem 3.1 Letbe a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Suppose thatis a multi-valued k-strictly pseudo-contractive mapping such thatand such thatfor all. For arbitraryand, letbe a sequence defined iteratively by
where. Then.
Proof Let . Then, using the recursion formula (3.1), Lemmas 2.6 and 2.4, we have
It follows that
Hence, . Since , we have that . □
A mapping is called hemicompact if, for any sequence in K such that as , there exists a subsequence of such that . We note that if K is compact, then every multi-valued mapping is hemicompact.
We now prove the following corollaries of Theorem 3.1.
Corollary 3.2 Letbe a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Letbe a multi-valued k-strictly pseudo-contractive mapping withand such thatfor all. Suppose that T is continuous and hemicompact. Letbe a sequence defined iteratively fromby
whereand. Then the sequenceconverges strongly to a fixed point of T.
Proof From Theorem 3.1, we have . Since T is hemicompact, there exists a subsequence of such that for some . Since T is continuous, we have . Therefore, and so . Setting in the proof of Theorem 3.1, it follows from inequality (3.2) that exists. So, converges strongly to p. This completes the proof. □
Corollary 3.3 Letbe a real number and K be a nonempty, compact and convex subset of a q-uniformly smooth real Banach space E. Letbe a multi-valued k-strictly pseudo-contractive mapping withand such thatfor all. Suppose that T is continuous. Letbe a sequence defined iteratively fromby
whereand. Then the sequenceconverges strongly to a fixed point of T.
Proof Observing that if K is compact, every map is hemicompact, the proof follows from Corollary 3.2. □
Remark 2 In Corollary 3.2, the continuity assumption on T can be dispensed if we assume that for every , the set Tx is proximinal and weakly closed. In fact, we have the following result.
Corollary 3.4 Letbe a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Letbe a multi-valued k-strictly pseudo-contractive mapping withand such that for every, Tx is weakly closed andfor all. Suppose that T is hemicompact. Letbe a sequence defined iteratively fromby
whereand. Then the sequenceconverges strongly to a fixed point of T.
Proof Following the same arguments as in the proof of Corollary 3.2, we have and . Furthermore, from Lemma 2.5, is strongly demiclosed at zero. It then follows that . Setting and following the same computations as in the proof of Theorem 3.1, we have from inequality (3.2) that exists. Since converges strongly to p, it follows that converges strongly to . The proof is completed. □
A mapping is said to satisfy Condition (I) if there exists a strictly increasing function with , for all such that
Convergence theorems have been proved in real Hilbert spaces for multi-valued nonexpansive mappings T under the assumption that T satisfies Condition (I) (see, e.g., [16, 24]). The following corollary extends such theorems to multi-valued strictly pseudo-contractive maps and to q-uniformly smooth real Banach spaces.
Corollary 3.5 Letbe a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Letbe a multi-valued k-strictly pseudo-contractive mapping withand such that for every, Tx is weakly closed andfor all. Suppose that T satisfies Condition (I). Letbe a sequence defined iteratively fromby
whereand. Then the sequenceconverges strongly to a fixed point of T.
Proof From Theorem 3.1, we have . Using the fact that T satisfies Condition (I), it follows that . Thus there exist a subsequence of and a sequence such that
By setting and following the same arguments as in the proof of Theorem 3.1, we obtain from inequality (3.2) that
We now show that is a Cauchy sequence in K. Notice that
This shows that is a Cauchy sequence in K and thus converges strongly to some . Using the fact that T is L-Lipschitzian and , we have
so that and thus . Therefore, and converges strongly to p. Setting in the proof of Theorem 3.1, it follows from inequality (3.2) that exists. So, converges strongly to p. This completes the proof. □
Corollary 3.6 Letbe a real number and K be a nonempty, compact and convex subset of a q uniformly smooth real Banach space E. Letbe a multi-valued k-strictly pseudo-contractive mapping withand such that for every, the set Tx is weakly closed andfor all. Letbe a sequence defined iteratively fromby
whereand. Then the sequenceconverges strongly to a fixed point of T.
Proof From Theorem 3.1, we have . Since and K is compact, has a subsequence that converges strongly to some . Furthermore, from Lemma 2.5, is strongly demiclosed at zero. It then follows that . Setting and following the same arguments as in the proof of Theorem 3.1, we have from inequality (3.2) that exists. Since converges strongly to q, it follows that converges strongly to . This completes the proof. □
Corollary 3.7 Letbe a real number and K be a nonempty compact convex subset of a q uniformly smooth real Banach space E. Letbe a multi-valued nonexpansive mapping. Assume thatfor all. Letbe a sequence defined iteratively from,
whereand. Then the sequenceconverges strongly to a fixed point of T.
Remark 3 The recursion formula (3.1) of Theorem 3.1 is of the Krasnoselkii type (see, e.g., [25]) and is known to be superior to the recursion formula of the Mann algorithm (see, e.g., Mann [26]) in the following sense: (i) The recursion formula (3.1) requires less computation time than the formula of the Mann algorithm because the parameter λ in formula (3.1) is fixed in , whereas in the algorithm of Mann, λ is replaced by a sequence in satisfying the following conditions: , . The must be computed at each step of the iteration process. (ii) The Krasnoselskii-type algorithm usually yields rate of convergence as fast as that of a geometric progression, whereas the Mann algorithm usually has order of convergence of the form .
Remark 4 In [24], the authors replace the condition with the following restriction on the sequence , i.e., and . We observe that if, for example, the set is a closed and convex subset of a real Hilbert space, then is unique and is characterized by
Since this has to be computed at each step of the iteration process, this makes the recursion formula difficult to use in any possible application.
Remark 5 The addition of bounded error terms to the recursion formula (3.1) leads to no generalization.
Remark 6 Our theorems in this paper are important generalizations of several important recent results in the following sense: (i) Our theorems extend results proved for multi-valued nonexpansive mappings in real Hilbert spaces (see, e.g., [15–17, 20, 21]) to a much larger class of multi-valued strictly pseudo-contractive mappings and in a much larger class of q-uniformly smooth real Banach spaces. (ii) Our theorems are proved with the superior Krasnoselskii-type algorithm.
We give examples of multi-valued maps where, for each , the set Tx is proximinal and weakly closed.
Example 1 Let be an increasing function. Define by
where and . For every , Tx is either a singleton or a closed and bounded interval. Therefore, Tx is always weakly closed and convex. Hence, for every , the set Tx is proximinal and weakly closed.
Example 2 Let H be a real Hilbert space and be a convex continuous function. Let be the multi-valued map defined by
where is the subdifferential of f at x and is defined by
It is well known that for every , is nonempty, weakly closed and convex. Therefore, since H is a real Hilbert space, it then follows that for every , the set Tx is proximinal and weakly closed. The subdifferential has deep connection with convex optimization problems.
The condition for all , which is imposed in all our theorems of this paper, can actually be replaced by another condition (see, e.g., Shahzad and Zegeye [24]). This is done in Theorem 3.9.
Let K be a nonempty, closed and convex subset of a real Hilbert space, be a multi-valued map and be defined by
We will need the following result.
Lemma 3.8 (Song and Cho [27])
Let K be a nonempty subset of a real Banach space andbe a multi-valued map. Then the following are equivalent:
-
(i)
;
-
(ii)
;
-
(iii)
. Moreover, .
Remark 7 We observe from Lemma 3.8 that if is any multi-valued map with , then the corresponding multi-valued map satisfies for all , the condition imposed in all our theorems and corollaries. Consequently, the examples of multi-valued maps satisfying the condition for all abound.
Theorem 3.9 Letbe a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Suppose thatis a multi-valued mapping such that. Assume thatis k-strictly pseudo-contractive. For arbitraryand, letbe a sequence defined iteratively by
where. Then.
We conclude this paper with an example of a multi-valued map T for which is k-strictly pseudo-contractive, the condition assumed in Theorem 3.9. Trivially, every nonexpansive map is strictly pseudo-contractive.
Example 3 Let with the usual metric and be the multi-valued map defined by
Then is strictly pseudo-contractive. In fact, for all .
References
Brouwer LEJ: Uber Abbildung von Mannigfaltigkeiten. Math. Ann. 1912, 71(4):598.
Kakutani S: A generalization of Brouwer’s fixed point theorem. Duke Math. J. 1941, 8(3):457–459. 10.1215/S0012-7094-41-00838-4
Nash JF: Non-cooperative games. Ann. Math. 1951, 54: 286–295. 10.2307/1969529
Nash JF: Equilibrium points in n -person games. Proc. Natl. Acad. Sci. USA 1950, 36(1):48–49. 10.1073/pnas.36.1.48
Geanakoplos J: Nash and Walras equilibrium via Brouwer. Econ. Theory 2003, 21: 585–603. 10.1007/s001990000076
Nadler SB Jr.: Multivalued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475
Downing D, Kirk WA: Fixed point theorems for set-valued mappings in metric and Banach spaces. Math. Jpn. 1977, 22(1):99–112.
Filippov AF: Differential equations with discontinuous right hand side. Sb. Math. 1960, 51: 99–128. English transl. Trans. Am. Math. Soc. 42, 199–232 (1964)
Chang KC: The obstacle problem and partial differential equations with discontinuous nonlinearities. Commun. Pure Appl. Math. 1980, 33: 117–146. 10.1002/cpa.3160330203
Erbe L, Krawcewicz W: Existence of solutions to boundary value problems for impulsive second order differential inclusions. Rocky Mt. J. Math. 1992, 22: 1–20. 10.1216/rmjm/1181072792
Frigon M, Granas A, Guennoun Z: A note on the Cauchy problem for differential inclusions. Topol. Methods Nonlinear Anal. 1993, 1: 315–321.
Deimling K: Multivalued Differential Equations. De Gruyter, Berlin; 1992.
Chidume C Lecture Notes in Mathematics 1965. In Geometric Properties of Banach Spaces and Nonlinear Iterations. Springer, Berlin; 2009.
Lindenstrauss J, Tzafriri L Ergebnisse Math. Grenzgebiete 97. In Classical Banach Spaces II: Function Spaces. Springer, Berlin; 1979.
Sastry KPR, Babu GVR: Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point. Czechoslov. Math. J. 2005, 55: 817–826. 10.1007/s10587-005-0068-z
Panyanak B: Mann and Ishikawa iteration processes for multi-valued mappings in Banach spaces. Comput. Math. Appl. 2007, 54: 872–877. 10.1016/j.camwa.2007.03.012
Song Y, Wang H: Erratum to “Mann and Ishikawa iterative processes for multi-valued mappings in Banach spaces” [Comput. Math. Appl. 54, 872–877 (2007)]. Comput. Math. Appl. 2008, 55: 2999–3002. 10.1016/j.camwa.2007.11.042
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Chidume, CE, Chidume, CO, Djitte, N, Minjibir, MS: Convergence theorems for fixed points of multi-valued strictly pseudo-contractive mappings in Hilbert spaces (2012, to appear)
Khan SH, Yildirim I, Rhoades BE: A one-step iterative scheme for two multi-valued nonexpansive mappings in Banach spaces. Comput. Math. Appl. 2011, 61: 3172–3178. 10.1016/j.camwa.2011.04.011
Abbas M, Khan SH, Khan AR, Agarwal RP: Common fixed points of two multi-valued nonexpansive mappings by one-step iterative scheme. Appl. Math. Lett. 2011, 24: 97–102. 10.1016/j.aml.2010.08.025
Garcia-Falset J, Lorens-Fuster E, Suzuki T: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 2011, 375: 185–195. 10.1016/j.jmaa.2010.08.069
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16(12):1127–1138. 10.1016/0362-546X(91)90200-K
Shahzad N, Zegeye H: On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces. Nonlinear Anal. 2009, 71: 838–844. 10.1016/j.na.2008.10.112
Krasnosel’skiĭ MA: Two observations about the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123–127.
Mann WR: Mean value methods in iterations. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Song Y, Cho YJ: Some notes on Ishikawa iteration for multi-valued mappings. Bull. Korean Math. Soc. 2011, 48(3):575–584. doi:10.4134/BKMS.2011.48.3.575 10.4134/BKMS.2011.48.3.575
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Chidume, C.E., Chidume, CC.O., Djitte, N. et al. Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings. Fixed Point Theory Appl 2013, 58 (2013). https://doi.org/10.1186/1687-1812-2013-58
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DOI: https://doi.org/10.1186/1687-1812-2013-58