Some New Generalization of Darbo’s Fixed Point Theorem and Its Application on Integral Equations

Abstract

In this article, we propose some new fixed point theorem involving measure of noncompactness and control function. Further, we prove the existence of a solution of functional integral equations in two variables by using this fixed point theorem in Banach Algebra, and also illustrate the results with the help of an example.

Solution of a Fractional Integral Equation Using the Darbo Fixed Point Theorem

“…Definition 3 (see [9]). Let Q be the set of all functions Q: R + × R + ⟶ R + that fulfills the axioms:…”

Solvability for two dimensional functional integral equations via Petryshyn’s fixed point theorem

“…By unifying and enlarging the earlier results of [12,15,29,34] and using Petryshyn's fixed point, we obtained a new method to prove the existence of solutions for some functional integral equations. The merit of Theorem 3.1 among the others (Darbo's and Schauder fixed point theorem) lies in that in applying this theorem, here one does not need to confirm the involved operator maps is on a closed convex subset onto itself.…”

“…They are often applied to the theories of Hadamard-type fractional integral operators and they find relations of the obtained results with earlier results about integral operators on different function spaces as well as the operator theory and geometry of Banach spaces (see [9,23,24,[30][31][32]). Numerous papers have been attached to the problem for the existence of solution of FIEs and infinite systems of integral equations in two variables in the different space (see [4,5,[12][13][14]21,29]). Here, we will study the following equation z(s, t) = f (s, t, z(ϕ 1 (s, t)), ..., z(ϕ k (s, t)) + F s, t, s 0 t 0 g(s, t, u, v, z(β 1 (u, v)), ..., z(β n (u, v)))dvdu, z(γ 1 (s, t)), ..., z(γ m (s, t)) × h(s, t, z(θ 1 (s, t)), ..., z(θ r (s, t))…”

A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications

“…In their research published in [16], Omran et al made a significant discovery by establishing Banach fixed-point theorems within the context of generalized metric spaces equipped with the Hadamard product, offering fresh perspectives on this fundamental concept. It is worth noting that a plethora of other noteworthy results related to fixed point theory and its myriad of applications can be found in references [17][18][19][20][21][22][23][24][25], collectively contributing to the ever-evolving landscape of fixed-point theory and its multifaceted implications.…”