Best approximation results provide an approximate solution to the fixedpoint equation $Tx=x$, when the non-self mapping $T$ has no fixed point. Inparticular, a well-known best approximation theorem, due to Fan cite{5},asserts that if $K$ is a nonempty compact convex subset of a Hausdorfflocally convex topological vector space $E$ and $T:K\rightarrow E$ is acontinuous mapping, then there exists an element $x$ satisfying thecondition $d(x,Tx)=\inf \{d(y,Tx):y\in K\}$, where $d$ is a metric on $E$.Recently, Hussain et al. (Abstract and Applied Analysis, Vol. 2014, ArticleID 837943) introduced proximal contractive mappings and established certainbest proximity point results for these mappings in $G$-metric spaces. The aimof this paper is to introduce certain new classes of auxiliary functions andproximal contraction mappings and establish best proximity point theoremsfor such kind of mappings in $G$-metric spaces. As consequences of theseresults, we deduce certain new best proximity and fixed point results in$G$-metric spaces. Moreover, we present certain examples to illustrate theusability of the obtained results. تفاصيل المقالة

The main objective of the paper is to state newly fixed point theorems for set-valued mappings in the framework of 0-complete partial metric spaces which speak about a location of a fixed point with respect to an initial value of the set-valued mapping by using some $C$-class functions. The results proved herein generalize, modify and unify some recent results of the existing literature. As an application, we provide an existence theorem for a coupled elliptic system subject to various two-point boundary conditions. تفاصيل المقالة

‎In this paper‎, ‎we use the concept of $C$-class functions introduced‎‎by Ansari [4] to prove the existence and uniqueness of‎‎common fixed point for self-mappings in modular spaces of integral‎‎inequality‎. ‎Our results extended and generalized previous known‎‎results in this direction‎. تفاصيل المقالة