On a Suzuki Type General Fixed Point Theorem with Applications
© S. L. Singh et al. 2010
Received: 29 October 2010
Accepted: 2 December 2010
Published: 13 December 2010
The main result of this paper is a fixed-point theorem which extends numerous fixed point theorems for contractions on metric spaces and recently developed Suzuki type contractions. Applications to certain functional equations and variational inequalities are also discussed.
The classical Banach contraction theorem has numerous generalizations, extensions, and applications. In a comprehensive comparison of contractive conditions, Rhoades  recognized that Ćirić's quasicontraction  (see condition (C) below) is the most general condition for a self-map of a metric space which ensures the existence of a unique fixed point. Pal and Maiti  proposed a set of conditions (see (PM.1)–(PM.4) below) as an extension of the principle of quasicontraction (C), under which may have more than one fixed point (see Example 2.7 below). Thus the condition (C) is independent of the conditions (PM.1)–(PM.4) (see also Rhoades [4, page 42]).
On the other hand, Suzuki  recently obtained a remarkable generalization of the Banach contraction theorem which itself has been extended and generalized on various settings (see, e.g, [6–15]). With a view of extending Suzuki's contraction theorem  and its several generalizations, we combine the ideas of Pal and Maiti , Suzuki , and Popescu  to obtain a very general fixed-point theorem. Subsequently, we use our results to solve certain functional equations and variational inequalities under different conditions than those considered in Bhakta and Mitra , Baskaran and Subrahmanyam , Pathak et al. [18, 19], Singh and Mishra [11, 12], and Pathak et al. [20, and references thereof].
2. Main Results
An orbit of a multivalued map , the collection of nonempty subsets of , at is a sequence . is called -orbitally complete if every Cauchy sequence of the form converges in , for all . For details, refer to Ćirić [2, 21].
The following theorem is our main result.
If only the condition (PM.4) is satisfied in Theorem 2.1, then the uniqueness of the fixed-point follows easily. Hence, we have the following (see also [10, Corollary 2.1]).
The following result is close in spirit to several generalizations of the Banach contraction theorem by Edelstein , Sehgal , Chatterjea , Rhoades [1, conditions (20) and (22)], and Suzuki [15, Theorem 3].
implies one of the following conditions:
An appropriate blend of the proof of Theorems 2.1 and 2 of Pal and Maiti  works.
Let and , . Then, the map satisfies all the requirements of Theorem 2.1 with , , and . Further, is not a Ćirić-Suzuki contraction, that is, does not satify the requirements of [10, Corollary 2.1]. Evidently, is not a quasicontraction.
We use Theorem 2.1 to obtain the following result for a multivalued map.
implies that at least one of the following conditions is true:
3.1. Application to Dynamic Programming
In multistage processes, some functional equations arise in a natural way (cf. Bellman  and Bellman and Lee ). The intent of this section is to study the existence of the solution of the functional equation (3.1) arising in dynamic programming.
Assume that the conditions (DP.1) and (DP.2) are satisfied. Then, the functional equation (3.1) has a unique bounded solution.
So Corollary 2.3 applies, wherein corresponds to the map . Therefore, has a unique fixed-point , that is, is the unique bounded solution of the functional equation (3.1).
3.2. Application to Variational Inequalities
As another application of Corollary 2.3, we show the existence of solutions of variational inequalities as in the work of Belbas and Mayergoyz . Variational inequalities arise in optimal stochastic control  as well as in other problems in mathematical physics, for examples, deformation of elastic bodies stretched over solid obstacles, elastoplastic torsion, and so forth, . The iterative method for solutions of discrete variational inequalities is very suitable for implementation on parallel computers with single-instruction, multiple-data architecture, particularly on massively parallel processors.
The corresponding problem of stochastic optimal control can be described as follows: is the generator of a diffusion process in , is a discount factor, is the continuous cost, and represents the cost incurred by stopping the process. The boundary condition " on " expresses the fact that stopping takes place either prior or at the time that the diffusion process exists from .
Note that the problem (3.17) arises in stochastic game theory.
These assumptions are related to the definition of " -matrices", arising from the finite difference discretization of continuous elliptic operators having the property (3.18) under the appropriate conditions and denotes the set of all discretized vectors in (see [31, 32]). Note that the matrix is an -matrix if and only if every off-diagonal entry of is nonpositive.
Now, we show the existence of iterative solutions of variational inequalities.
Notice that in two-person game, we have to determine the best strategies for each player on the basis of maximin and minimax criterion of optimality. This criterion will be well stated as follows: a player lists his/her worst possible outcomes, and then he/she chooses that strategy which corresponds to the best of these worst outcomes. Here, the problem (3.20) exhibits the situation in which two players are trying to control a diffusion process; the first player is trying to maximize a cost functional, and the second player is trying to minimize a similar functional. The first player is called the maximizing player and the second one the minimizing player. Here, represents the continuous rate of cost for both players, is the stopping cost for the maximizing player, and μ is the stopping cost for the minimizing player. This problem is fixed by inducting an operator implicitly defined for all as in (3.21).
Under the assumptions (3.18) and (3.19), a solution for (3.23) exists.
This research is supported by the Directorate of Research Development, Walter Sisulu University.
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