Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 629468, 10 pages http://dx.doi.org/10.1155/2013/629468 Research Article Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces C. E. Chidume,1 C. O. Chidume,2 N. Djitté,3 and M. S. Minjibir1,4 1 Mathematics Institute, African University of Science and Technology, PMB 681, Garki, Abuja, Nigeria Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA 3 Université Gaston Berger, 234 Saint Louis, Senegal 4 Department of Mathematical Sciences, Bayero University, PMB 3011, Kano, Nigeria 2 Correspondence should be addressed to C. E. Chidume; Received 10 September 2012; Accepted 15 April 2013 Academic Editor: Josef Diblı́k Copyright © 2013 C. E. Chidume et al. 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. Let 𝐾 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Suppose that 𝑇 : 𝐾 → 2𝐾 is a multivalued strictly pseudocontractive mapping such that 𝐹(𝑇) ≠ 0. A Krasnoselskii-type iteration sequence {𝑥𝑛 } is constructed and shown to be an approximate fixed point sequence of 𝑇; that is, lim𝑛 → ∞ 𝑑(𝑥𝑛 , 𝑇𝑥𝑛 ) = 0 holds. Convergence theorems are also proved under appropriate additional conditions. 1. Introduction For several years, the study of fixed point theory of multivalued nonlinear mappings has attracted, and continues to attract, the interest of several well-known mathematicians (see, e.g., Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadler [6], and Downing and Kirk [7]). Interest in such studies stems, perhaps, mainly from the usefulness of such fixed point theory in real-world applications, such as in Game Theory and Market Economy, and in other areas of mathematics, such as in Nonsmooth Differential Equations. We describe briefly the connection of fixed point theory of multivalued 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 noncooperative 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 multivalued mapping 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 𝐺 = (𝑢𝑛 , 𝐾𝑛 ) with 𝑁 players denoted by 𝑛, 𝑛 = 1, . . . , 𝑁, where 𝐾𝑛 ⊂ R𝑚𝑛 is the set of possible strategies of the 𝑛th player and is assumed to be nonempty, compact, and convex, and 𝑢𝑛 : 𝐾 := 𝐾1 × 𝐾2 ⋅ ⋅ ⋅ × 𝐾𝑁 → R 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 𝜎𝑛 ∈ 𝐾𝑛 . All players together can take a collective action, which is a combined vector 𝜎 = (𝜎1 , 𝜎2 , . . . , 𝜎𝑁). For each 𝑛, 𝜎 ∈ 𝐾 and 𝑧𝑛 ∈ 𝐾𝑛 , we will use the following standard notations: 𝐾−𝑛 := 𝐾1 × ⋅ ⋅ ⋅ × 𝐾𝑛−1 × 𝐾𝑛+1 × ⋅ ⋅ ⋅ × 𝐾𝑁, 𝜎−𝑛 := (𝜎1 , . . . , 𝜎𝑛−1 , 𝜎𝑛+1 , . . . , 𝜎𝑁) , (1) (𝑧𝑛 , 𝜎−𝑛 ) := (𝜎1 , . . . , 𝜎𝑛−1 , 𝑧𝑛 , 𝜎𝑛+1 , . . . , 𝜎𝑁) . A strategy 𝜎𝑛 ∈ 𝐾𝑛 permits the 𝑛’th player to maximize his gain under the condition that the remaining players have chosen their strategies 𝜎−𝑛 if and only if 𝑢𝑛 (𝜎𝑛 , 𝜎−𝑛 ) = max 𝑢𝑛 (𝑧𝑛 , 𝜎−𝑛 ) . 𝑧𝑛 ∈𝐾𝑛 (2) 2 Abstract and Applied Analysis Now, let 𝑇𝑛 : 𝐾−𝑛 → 2𝐾𝑛 be the multivalued mapping defined by 𝑇𝑛 (𝜎−𝑛 ) := Arg max 𝑢𝑛 (𝑧𝑛 , 𝜎−𝑛 ) 𝑧𝑛 ∈𝐾𝑛 ∀𝜎−𝑛 ∈ 𝐾−𝑛 . (3) Definition 1. A collective action 𝜎 = (𝜎1 , . . . , 𝜎𝑁) ∈ 𝐾 is called a Nash equilibrium point if, for each 𝑛, 𝜎𝑛 is the best response for the 𝑛’th player to the action 𝜎−𝑛 made by the remaining players. That is, for each 𝑛, 𝑢𝑛 (𝜎) = max 𝑢𝑛 (𝑧𝑛 , 𝜎−𝑛 ) (4) 𝜎𝑛 ∈ 𝑇𝑛 (𝜎−𝑛 ) . (5) 𝑧𝑛 ∈𝐾𝑛 or, equivalently, This is equivalent to that 𝜎 is a fixed point of the multivalued mapping 𝑇 : 𝐾 → 2𝐾 defined by 𝑇 (𝜎) := 𝑇1 (𝜎−1 ) × 𝑇2 (𝜎−2 ) × ⋅ ⋅ ⋅ × 𝑇𝑁 (𝜎−𝑁) . (6) From the point of view of social recognition, game theory is perhaps the most successful area of application of fixed point theory of multivalued mappings. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a nonequilibrium point and convergent to equilibrium solution. This is part of the problem that is being addressed by iterative methods for fixed point of multivalued mappings. Nonsmooth Differential Equations. The mainstream of applications of fixed point theory for multivalued mappings 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 (DIs). Here is a simple model for this type of application. Consider the initial value problem 𝑑𝑢 = 𝑓 (𝑡, 𝑢) , 𝑑𝑡 a.e. 𝑡 ∈ 𝐼 := [−𝑎, 𝑎] , 𝑢 (0) = 𝑢0 . (7) If 𝑓 : 𝐼 × R → R is discontinuous with bounded jumps, measurable in 𝑡, one looks for solutions in the sense of Filippov [8, 9] which are solutions of the differential inclusion 𝑑𝑢 ∈ 𝐹 (𝑡, 𝑢) , 𝑑𝑡 a.e. 𝑡 ∈ 𝐼, 𝑢 (0) = 𝑢0 , (8) where 𝐹 (𝑡, 𝑥) = [lim inf 𝑓 (𝑡, 𝑦) , lim sup𝑓 (𝑡, 𝑦)] . 𝑦→𝑥 𝑦→𝑥 (9) Now set 𝐻 := 𝐿2 (𝐼) and let 𝑁𝐹 : 𝐻 → 2𝐻 be the multivalued NemyTskii operator defined by 𝑁𝐹 (𝑢) := {V ∈ 𝐻 : V (𝑡) ∈ 𝐹 (𝑡, 𝑢 (𝑡)) a.e. 𝑡 ∈ 𝐼} . (10) Finally, let 𝑇 : 𝐻 → 2𝐻 be the multivalued mapping defined by 𝑇 := 𝑁𝐹 𝑜𝐿−1 , where 𝐿−1 is the inverse of the derivative operator 𝐿𝑢 = 𝑢󸀠 given by 𝑡 𝐿−1 V (𝑡) := 𝑢0 + ∫ V (𝑠) 𝑑𝑠. 0 (11) One can see that problem (8) reduces to the fixed point problem: 𝑢 ∈ 𝑇𝑢. Finally, a variety of fixed point theorems for multivalued mappings with nonempty and convex values is available to conclude the existence of 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 about these topics, one can consult [10–13] and references therein as examples. We have seen that a Nash equilibrium point is a fixed point 𝜎 of a multivalued mapping 𝑇 : 𝐾 → 2𝐾 , that is, a solution of the inclusion 𝑥 ∈ 𝑇𝑥 for some nonlinear mapping 𝑇. This inclusion can be rewritten as 0 ∈ 𝐴𝑥, where 𝐴 := 𝐼 − 𝑇 and 𝐼 is the identity mapping on 𝐾. Many problems in applications can be modeled in the form 0 ∈ 𝐴𝑥, where, for example, 𝐴 : 𝐻 → 2𝐻 is a monotone operator, that is, ⟨𝑢 − V, 𝑥 − 𝑦⟩ ≥ 0 for all 𝑢 ∈ 𝐴𝑥, V ∈ 𝐴𝑦, 𝑥, 𝑦 ∈ 𝐻. Typical examples include the equilibrium state of evo (...truncated)