Some generalizations of Darbo fixed point theorem and its application

Abstract

Abstract. Recently, B. Samet et al. (Fixed Point Theory and Appl. 2013:5, 2013, doi:10.1186/1687-1812 has introduced the concept of new contraction mappings and obtained results on the fixed point for nonlinear contractive mappings in a metric space. In the present paper, we introduce the concept of a new contraction via the measure of non-compactness on a Banach space, we obtain a few generalizations of Darbo fixed point theorem and extend some recent results of Aghajani et al. We also show the applicability of obtained results to the theory of integral equations.

Common fixed point theorems for two and three mappings

“…A contractive condition in terms of a measure of noncompactness, firstly used by Darbo [1], is one of the fruitful tools to obtain fixed point and common fixed point theorems. There are many extensions of this condition which are known as generalizations of Darbo's fixed point theorem; see for example [2][3][4][5][6][7][8][9][10][11] and the references therein.…”

Generalizations of Darbo’s fixed point theorem for new condensing operators with application to a functional integral equation

“…From Example 2.19, we take µ * is a measure of noncompactness on E 2 as follows: 17) where X and X 2 are the natural projections of X on E. Further, we define α : X 2 × X 2 → [0, +∞) as follows:…”

“…Due to this fact, researchers are always interested to find the extensions and generalizations of the Darbo's fixed point theorem. Up to now, several papers have been published on the generalization of Darbo's fixed point theorem (see [14][15][16][17][18] and references therein) and on the existence and behaviour of solutions of nonlinear differential and integral equations [19][20][21][22][23][24][25][26][27] using the technique of measure of noncompactnes.…”

Applications of measure of non-compactness and modified simulation function for solvability of nonlinear functional integral equations

“…Recently, there have been several successful efforts to apply the concept of a measure of noncompactness in the study of the existence and behavior of solutions of nonlinear differential and integral equations [2-6, 8-10, 17, 19, 20, 22-24]. In our investigations, we apply the method associated with the technique of measures of noncompactness to generalize the Darbo fixed point theorem [14] and to extend some recent results of Arab [7]. Moreover, as an application, we study the existence of solutions of the nonlinear integral equation of the form…”