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The concept of measure of noncompactness (MNC) allows us to select an important class of mappings which are more general than compact operators. The proposed work exploits the axiomatic definition of MNC and introduces the notion of relatively nonexpansive cyclic (noncyclic) SR-condensing operators along with the aid of SR-functions. The first phase of the paper concentrates on establishing the best proximity point (pair) theorems for such operators. The main results in this manuscript extend and generalize several state of art literature on Darbo type fixed point results. In the second phase, proposed results are applied to show the actuality of optimum solutions for system of second order differential equations with two initial conditions. [ABSTRACT FROM AUTHOR]

This paper introduces new kind of algorithms for multivalued non-self mapping to obtain the best proximity point without assuming the continuity of involved mapping. Some non-trivial examples are presented to illustrate the facts. Consequently, an application to finding an optimal approximate solution for the homotopy theory is also discussed. [ABSTRACT FROM AUTHOR]

Primarily this work intends to investigate the existence of best proximity points (pairs) for new classes of cyclic (noncyclic) mappings via simulation functions and measure of noncompactness. Use of different classes of additional functions make it possible to generalize the contractive inequalities in this work. As an application of the main conclusions, a survey for the existence of optimal solutions of a system of integro-differential equations under some new conditions is presented. As an application of our existence results, we establish the existence of a solution for the following system of integro-differential equations {(u'(t) = F1(t,u(t), ∫t0t k1(t,s,u(s))ds), u(t0) = u1, v'(t) = F2(t,v(t), ∫t0t k2(t,s,v(s))ds), v(t0) = u2, in the space of all bounded and continuous real functions on [0, + ∞) under suitable assumptions on F1, F2. [ABSTRACT FROM AUTHOR]

Best proximity results for cyclic and noncyclic F G -contractive operators in (strictly convex) Banach spaces are obtained via the measure of noncompactness. The work extends some of recent investigations in this direction. The obtained results are applied to investigation of optimum solutions for a system of second order differential equations with two initial conditions followed by an example. [ABSTRACT FROM AUTHOR]