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The paper is devoted to proving an existence and uniqueness result for generalized solution to a class of nonlinear wave equations in space dimensions d = 1 , 2 , 3 . It is shown that for a nonlinearity of f and arbitrary growth and sign, which depends on the time variable and parameter, the initial value problem for the linear wave equation has a unique solution in a Colombeau subalgebra of bounded type. The proof is based on a fixed point theorem in an ultrametric topology on a closed subspace of these subalgebras. The results presented generalize and unify several existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, based on some geometrical properties of CAT p (0) spaces, for p ≥ 2 , we obtain two fixed point results for monotone multivalued nonexpansive mappings and proximally monotone nonexpansive mappings. Which under some assumptions, reduce to coincide and generalize a fixed point result for monotone nonexpansive mappings. This work is a continuity of the previous work of Ran and Reurings, Nieto and Rodríguez-López done for monotone contraction mappings. [ABSTRACT FROM AUTHOR]

In this paper, we define the concepts of ϕ-contraction in probabilistic generalized Menger space. We prove that ϕ-contraction mappings in probabilistic generalized Menger space have a unique fixed point. Also we prove some fixed point theorems related this concept. Some examples are given to support the obtained results. [ABSTRACT FROM AUTHOR]

In this paper it is shown that a classical continuation principle due to Granas for contractions holds under weaker contractive assumptions. This leads to a Leray-Schauder principle for such contractions in hyperbolic spaces. Some applications to nonexpansive mappings in hyperbolic geodesic spaces are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of $(\alpha,\psi)$-rational type contractive mappings and provide sufficient conditions for the existence and uniqueness of a fixed point for such class of generalized nonlinear contractive mappings in the setting of generalized metric spaces. We also deduce several interesting corollaries. [ABSTRACT FROM AUTHOR]

The paper contains a brief review of the author's results concerning the technique of constructing Fejér contraction mappings, which are used in iterative processes of solving linear and convex systems of inequalities as well as accompanying optimization problems. The general approach is based on the notion of M-Fejér step ' p → q' defined by the property . This property (postulate) assumes the existence of a point p ∈ $$ \overline {convM} $$ and of a proper sufficiently arbitrary point q. Some of the problems considered in the paper are illustrated by schemes reflecting the analytics of these problems. [ABSTRACT FROM AUTHOR]

Finding solutions to the Bellman equation often relies on restrictive boundedness assumptions. In this paper we develop a method of proof that allows to dispense with the assumption that returns are bounded from above. In applications our assumptions only imply that long run average (expected) growth is sufficiently discounted, in sharp contrast with classical assumptions either absolutely bounding growth or bounding each period (instead of long run) maximum (instead of average) growth. We discuss our work in relation to the literature and provide several examples. [ABSTRACT FROM AUTHOR]

In this paper, we present yet another new and novel variant of classical Banach contraction principle on a complete metric space endowed with a binary relation which, under universal relation, reduces to Banach contraction principle. In process, we observe that various kinds of binary relations, such as partial order, preorder, transitive relation, tolerance, strict order, symmetric closure, etc., utilized by earlier authors in several well-known metrical fixed point theorems can be weakened to the extent of an arbitrary binary relation. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to establish relation-theoretic metrical fixed point theorems via an implicit contractive condition which is general enough to yield a multitude of corollaries corresponding to several well known contraction conditions ( e.g. Banach (Fundam. Math. 3:133-181, ), Kannan (Am. Math. Mon. 76:405-408, ), Reich (Can. Math. Bull. 14:121-124, ), Bianchini (Boll. Unione Mat. Ital. 5:103-108, ), Chatterjea (C. R. Acad. Bulg. Sci. 25:727-730, ), Hardy and Rogers (Can. Math. Bull. 16:201-206, ), Ćirić (Proc. Am. Math. Soc. 45:267-273, ) and several others) wherein even such corollaries are new results on their own. As an example we utilize our main results, to prove a theorem on the existence and uniqueness of the solution of an integral equation besides providing an illustrative example. [ABSTRACT FROM AUTHOR]

There have been numerous attempts recently to extend many of the metric standard fixed point theorems to a more general semimetric context. In many instances a weakened form of the triangle inequality is involved and the space is assumed to be complete. Thus Cauchy sequences play a central role. One of the standard tests to determine when a sequence is Cauchy in a metric space ( X, d) is the summation criterion: If $${\{ p_{n} \} \subset X}$$ and $${{\sum_{i=1}^{\infty}} d(p_{i}, p_{i+1}) < \infty}$$ , then { p} is Cauchy. In this note we examine instances in which this criterion plays a critical role. [ABSTRACT FROM AUTHOR]

We study the finite-difference approximation for the quasi-variational inequalities for a stochastic game involving discrete actions of the players and continuous and discrete payoff. We prove convergence of iterative schemes for the solution of the discretized quasi-variational inequalities, with estimates of the rate of convergence (via contraction mappings) in two particular cases. Further, we prove stability of the finite-difference schemes, and convergence of the solution of the discrete problems to the solution of the continuous problem as the discretization mesh goes to zero. We provide a direct interpretation of the discrete problems in terms of finite-state, continuous-time Markov processes. [ABSTRACT FROM AUTHOR]