Sufficient conditions are obtained for the existence of solutions to a coupled system of nonlinear fourth-order differential equations.

The purpose of this paper is to emphasize the role of “common limit range property” to ascertain the existence of common fixed points in modified intuitionistic fuzzy metric spaces enjoying an implicit function utilized in Tanveer et al. (2012) and Imdad et al. (2012). As an application to our main result, we derive a fixed point theorem for finite families of self-mappings. We also give some examples which demonstrate the validity of the hypotheses and degree of generality of our main results. Our results improve and extend several previously known fixed point theorems of the existing literature.

We show that some recent results concerning the existence of best proximity points can be obtained from the same results in fixed point theory.

This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.

Given a self-mapping $g:A\to A$ and a non-self-mapping $T:A\to B$, the aim of this work is to provide sufficient conditions for the existence of a unique point $x\in A$, called g-best proximity point, which satisfies $d\left(gx,Tx\right)=d\left(A,B\right)$. In so doing, we provide a useful answer for the resolution of the nonlinear programming problem of globally minimizing the real valued function $x\to d\left(gx,Tx\right)$, thereby getting an optimal approximate solution to the equation $Tx=gx$. An iterative algorithm is also presented to compute a solution of such problems. Our results generalize a result due to Rhoades (2001) and hence such results provide an extension of Banach's contraction principle to the case of non-self-mappings.

We extend the concept of α-ψ-contractive mappings introduced recently by Samet et al. (2012) to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.

We prove some best proximity point results for relatively $u$-continuous mappings in Banach and hyperconvex metric spaces. Our results generalize and extend some recent results to relatively $u$-continuous mappings and to general spaces.

Recently, Basha (2011) established the best proximity point theorems for proximal contractions of the first and second kinds which are extension of Banach's contraction principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the notions of proximal contractions of the first and second kinds which are more general than the notion of self-contractions, establish the existence of an optimal approximate solution theorems for these non-self-mappings, and also give examples to validate our main results.

We first introduce certain new concepts of $\alpha$-$\eta$-proximal admissible and $\alpha$-$\eta$-$\psi$-rational proximal contractions of the first and second kinds. Then we establish certain best proximity point theorems for such rational proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity point theory. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach's contraction principle and some of its generalizations as special cases. Moreover, some examples are given to illustrate the usability of the obtained results.

Very recently, Abkar and Gabeleh (2013) observed that some best proximity point results under the $P$-property can be obtained from the same results in fixed-point theory. In this paper, motivated by this mentioned work, we show that the most best proximity point results on a metric space endowed with a partial order (under the $P$-property) can be deduced from existing fixed-point theorems in the literature. We present various model examples to illustrate this point of view.

Double differential transform method has been employed to compute double Laplace transform. To illustrate the method, four examples of different forms have been prepared.