Optimum solutions for a system of differential equations via measure of noncompactness

Cyclic (noncyclic) phi-condensing operator and its application to a system of differential equations

“…Definition 6. (See [10].) Let (A, B) be a nonempty and bounded pair in a normed linear space X and T : A ∪ B → A ∪ B a noncyclic (cyclic) mapping.…”

“…The main purpose of this section is to obtain the cyclic version of Theorem 6 for best proximity points in Banach spaces, which do not have the geometric property of strictly convexity. To this end, we need the following result of [10].…”

A New Extension of Darbo’s Fixed Point Theorem using Relatively Meir–keeler Condensing operators

“…In Section 3, we generalise Theorem 1.9 to a new class of mappings, called relatively Meir-Keeler condensing operators, in order to study best proximity pairs of points. Finally, in Section 4, we apply our results to the existence of an optimal solution for a system of integrodifferential equations, extending some of the recent results of [8].…”

“…Corollary 3.4 [8,Theorem 4.3]. Let (A, B) be a nonempty, bounded, closed and convex pair in a strictly convex Banach space X such that A 0 is nonempty and let µ be an MNC on X.…”

Condensing Mappings and Best Proximity Point Results

“…Sadovskii [3], in 1967, defined condensing mappings and extended Darbo's theorem. Since then a lot of work has been done using this concept, and several interesting results have appeared, see, for instance, [4][5][6][7][8][9].…”