In this paper, the concept of pseudoconvexity and quasiconvexity for continuous~-time functions are studied and an equivalence condition for pseudoconvexity is obtained. Moreover, under pseudoconvexity assumptions, some relationships between Minty and Stampacchia vector variational inequalities and continuous-time programming problems are presented. Finally, some characterizations of the solution sets of a single-valued continuous-time programming problem are ‎obtained.‎ تفاصيل المقالة

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