An extension of Darbo’s theorem via measure of non-compactness with its application in the solvability of a system of integral equations

Abstract

In this work, we present a new extension of Darbo's theorem for two different classes of altering distance functions via measure of non-compactness. Using two-variable contractions we obtain the wellknown results in this literature (see [22]). We also use these results to discuss the existence of solutions for a system of integral equations. Finally, we provide an example to confirm the results obtained. Definition 1.1. ([11]) A function µ : M E → R + is called a measure of non-compactness in E if it satisfies the following hypothesis: (BM1) The family ker µ = X ∈ M E : µ (X) = 0 ∅ and ker µ ⊂ N E ; (BM2) X ⊂ Y ⇒ µ (X) ≤ µ (Y) ;

Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

“…They are very often used in the theory of functional equations, including ordinary differential equations, equations with partial derivatives, integral and integrodifferential equations, and optimal control theory. We also highlight that the interplay between fixed point theory and measures of noncompactness is very powerful and fruitful (see for instance [19,[23][24][25][26][27][28][29] and the references therein).…”

“…Now, for any ω ∈ Ω, either (a) pðTðω, ξðωÞÞ − x 0 Þ ≤ 1 or (b) pðTðω, ξðωÞÞ − x 0 Þ > 1. In case of (a), it follows by (29) that lðω, ξðωÞÞ = 1 and hence by (37), Tðω, ξðωÞÞ = ξðωÞ. If (b) holds, then by (29), we have…”