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An existence result of a functional integral equation via Darbo type theorem and an iterative algorithm to solve it
Bhuban Chandra Deuri, Anupam Das
TL;DR
The paper considers the problem of establishing existence and obtaining a numerical solution for a functional integral equation. The authors generalize Darbo’s and coupled fixed-point theorems using C-class mappings, apply the Darbo-type result to the equation, and explore modified homotopy perturbation for its numerical solution. Under assumptions (H1)–(H4), equation (4.1) has at least one solution, and the stated conditions yield at least one coupled fixed point.
Problem
Method
The authors generalize Darbo’s and coupled fixed-point theorems using C-class mappings, apply the Darbo-type result to the equation, and explore modified homotopy perturbation for its numerical solution.
Source: §1 · p. 1, §2 · p. 3, §4 · p. 10Results
Under assumptions (H1)–(H4), equation (4.1) has at least one solution, and the stated conditions yield at least one coupled fixed point.
Source: §3 · p. 5, §4 · pp. 6, 8Takeaways & Limitations
The paper applies its Darbo-type result to establish solvability for a functional integral equation and examines its numerical solution using modified homotopy perturbation.
Source: §1 · p. 1The stated conditions require continuity of Δ and H, constants c1, c2, c3 ≥ 0, and an H2 radius r0 > 0.
Source: §4 · p. 6
Abstract
from arXiv · showhide
In our study, Darbo's fixed point theorem and the coupled fixed point theorem have been extended. The two theorems were generalized using $C-class$ mapping. Using the Darbo type theorem, we gave an existence result for a functional integral equation along with an appropriate illustration. In addition, we explored the numerical solution of a given functional integral problem using modified homotopy perturbation.
1. Introduction
The introduction reviews fixed-point theory and measures of noncompactness as tools for integral-equation problems, then situates the paper’s generalized theorems and applications within that work.
1. Introduction
- The introduction identifies Schauder and Darbo fixed-point results as important tools for solving functional integral problems.It also traces the measure-of-noncompactness concept to Kuratowski and Darbo’s theory to Darbo [2].Source: §1 · p. 1
- Earlier work applied measures of noncompactness and fixed-point theory to integral, functional integral, differential, and fractional integral equations.The cited background also notes prior discussion of functional-integral solvability in Banach spaces and generalizations of Darbo-type theorems.Source: §1 · p. 1
- The paper generalizes Darbo’s fixed-point theorem and the coupled fixed-point theorem using C-class mappings, then applies a Darbo-type theorem to a functional integral equation and explores an MHP numerical solution.The authors also give an example of the existence result.Source: §1 · p. 1
- Measures of noncompactness assign values to nonempty bounded subsets of a Banach space and are introduced through axioms including vanishing on relatively compact sets and monotonicity under inclusion.The listed axioms also include invariance under convex closure and a condition on decreasing sequences whose measure tends to zero.Source: §1 · p. 2
- Schauder’s theorem gives a fixed point for a continuous compact mapping on a nonempty, bounded, closed, convex subset of a Banach space.The introduction presents Darbo’s theorem as a related fixed-point result for continuous mappings on such a set when its stated contraction condition holds.Source: §1 · p. 2
2. Darbo Fixed point Results
The section extends Darbo-type fixed-point conditions through C-class mappings and associated functions, establishing fixed-point existence under the stated inequalities. It also derives simpler results and outlines a nested-set proof using Schauder’s theorem.
Theorem 2.1
- Theorem 2.1 establishes a fixed point under its generalized Darbo-type condition for a continuous mapping on the stated set.The proof constructs nested convex sets and uses Schauder’s theorem once the limiting set is compact.Source: §2 · p. 3
Theorem 2.2
- Theorem 2.2 guarantees a fixed point when the measure inequality is transformed by Q and E and includes the non-decreasing continuous function F.The condition applies for arbitrary measures of noncompactness and the stated Q and E classes.Source: §2 · p. 4
Theorem 2.3
- Theorem 2.3 gives a fixed point under an E-transformed measure inequality with contraction factor λ.It follows from Theorem 2.2 by setting Q(z)=z.Source: §2 · p. 4
Corollary 2.4
- Corollary 2.4 uses the sum of the measure and its F-transform in the contraction condition and still guarantees a fixed point.It follows by setting E(z1,z2)=z1+z2 in Theorem 2.3.Source: §2 · p. 4
Corollary 2.5
- Corollary 2.5 recovers Darbo’s fixed-point theorem by setting F(z)=0 in Corollary 2.4.The conclusion applies to an arbitrary measure of noncompactness.Source: §2 · p. 4
3. Coupled fixed point Results
The section extends fixed-point results to coupled mappings using an aggregate measure of noncompactness. Under the stated conditions, the coupled mapping has at least one coupled fixed point.
3. Coupled fixed point Results
- A coupled fixed point (C, D) satisfies Ψ(C, D) = C and Ψ(D, C) = D.Source: §3 · p. 4
- The section constructs a product-space measure of noncompactness by applying J to component measures, with an example using J(C, D) = C + D.The stated construction requires a convex L and a J that vanishes exactly when every component is zero.Source: §3 · p. 4
- Theorem 3.4 gives at least one coupled fixed point when its measure bound and subadditivity conditions on Q and F hold.The result assumes a continuous operator and a measure of noncompactness as in Theorem 2.1.Source: §3 · p. 5
- The proof symmetrizes Ψ as Ψcf(A, S) = (Ψ(A, S), Ψ(S, A)) and applies Theorem 2.1 to obtain a fixed point corresponding to a coupled fixed point.It uses the product-space measure ˜µ(s) = µ(s1) + µ(s2) and bounds its image through the component mappings.Source: §3 · p. 5
4. Application
The application establishes existence for a functional integral equation under assumptions H1–H4, then applies modified homotopy perturbation to approximate its solution. The reported approximation has zero absolute point errors on substitution and is identified as the exact solution.
4. Application
- Under assumptions (H1)–(H4), equation (4.1) has at least one solution in B = C̄(¯J).The assumptions include continuity and bounds on ∆ and H, as well as the stated Lipschitz condition on H.Source: §4 · p. 6
- By (H4), D maps Qr0 into itself and is continuous on Qr0; Corollary 2.5 gives a fixed point in ϖ ⊆ Qr0, implying equation (4.1) has a solution in B.Source: §4 · pp. 7–8
- In the illustration, the stated assumptions hold for r0 = 3, with the reported bound 0.5657 ≤ 3, so equation (4.2) has a solution in B.Source: §4 · p. 9
5. Establishing a convergent iterative algorithm by use of a MPH approach in order to solve an equation(4.2)
- For the numerical approach, the nonlinear fractional operator and known function are split into two operators and two functions, then joined through a homotopy parameter p from 0 to 1.At p = 1, the homotopy reaches the target equation.Source: §4 · p. 10
- The iterative algorithm is obtained by rearranging powers of p and applied with initial conditions ϑ0(x) = g1(x) = 0 and the stated g(x).The text then computes sequence terms for the cases shown.Source: §4 · p. 11
- The computed sequence terms are combined to form the reported approximation to the solution of equation (4.2).The passages give terms for two cases and then state the resulting approximation.Source: §4 · p. 11
4. Application
- By substituting (5.10) into equation (4.2) and comparing its two sides, the authors report zero absolute point errors and conclude that (5.10) is the exact solution to (4.2).Source: §4 · p. 11
Declarations
The declarations report that ethics approval and consent to participate were not applicable, all authors consented to publication, and no competing interests were declared.
Declarations
- Human and animal ethics, and ethics approval and consent to participate, are reported as not applicable.Source: Declarations · p. 12
- All authors gave consent for publication.Source: Declarations · p. 12
- The authors declare no competing interests.Source: Declarations · p. 12