Cyclic (noncyclic) phi-condensing operator and its application to a system of differential equations
We establish a best proximity pair theorem for noncyclic φ-condensing operators in strictly convex Banach spaces by using a measure of noncompactness. We also obtain a counterpart result for cyclic φ-condensing operators in Banach spaces to guarantee the existence of best proximity points, and so, an extension of Darbo’s fixed point theorem will be concluded. As an application of our results, we study the existence of a global optimal solution for a system of ordinary differential equations.
A note on best proximity point theory using proximal contractions
In this paper, a reduction technique is used to show that some recent results on the existence of best proximity points for various classes of proximal contractions can be concluded from the corresponding results in fixed point theory.
A New Extension of Darbo's Fixed Point Theorem Using Relatively Meir-Keeler Condensing Operators
We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an application of our main result, we investigate the existence of an optimal solution for a system of integrodifferential equations.
A New Approach to the Generalization of Darbo’s Fixed Point Problem by Using Simulation Functions with Application to Integral Equations
We investigate the existence of fixed points of self-mappings via simulation functions and measure of noncompactness. We use different classes of additional functions to get some general contractive inequalities. As an application of our main conclusions, we survey the existence of a solution for a class of integral equations under some new conditions. An example will be given to support our results.