Functional Integral Equations Research Articles

Solvability and numerical method for non-linear Volterra integral equations by using Petryshyn’s fixed point theorem

In this paper, utilizing the technique of Petryshyn’s fixed point theorem in Banach algebra, we analyze the existence of solution for functional integral equations, which includes as special cases many functional integral equations that arise in various branches of non-linear analysis and its application. Finally, we introduce the numerical method formed by modified homotopy perturbation approach to resolving the problem with acceptable accuracy.

Approximation of solutions for nonlinear functional integral equations

<abstract><p>In this article, we consider a class of nonlinear functional integral equations, motivated by an equation that offers increasing evidence to the extant literature through replication studies. We investigate the existence of solution for nonlinear functional integral equations on Banach space $ C[0, 1] $. We use the technique of the generalized Darbo's fixed-point theorem associated with the measure of noncompactness (MNC) to prove our existence result. Also, we have given two examples of the applicability of established existence result in the theory of functional integral equations. Further, we construct an efficient iterative algorithm to compute the solution of the first example, by employing the modified homotopy perturbation (MHP) method associated with Adomian decomposition. Moreover, the condition of convergence and an upper bound of errors are presented.</p></abstract>

On existence of solutions for some functional integral equations in Banach algebra by fixed point theorem

In this research, we analyze the existence of solution for some nonlinear functional integral equations using the techniques of measures of noncompactness and the Petryshyn's fixed point theorem in Banach space. The results obtained in this paper cover many existence results obtained by numerous authors under some weaker conditions. We also give an example satisfying the conditions of our main theorem but not satisfying the conditions described by other authors.

An integral functional equation on abelian semigroups

We investigate a generalization of many functional equations. Namely, we consider the following functional equation ?S f(x + y + t) d?(t) + ?S f(x + ?(y) + t) d?(t) = f(x) + h(y), x, y ? S, where (S,+) is an abelian semigroup, ? is a surjective endomorphism of S, E is a linear space over the field K ? {R, C} and ?,? are linear combinations of Dirac measures. Under appropriate conditions on ? and ? and based on Stetkar?s result [9], we find and characterize solutions of the previous functional equation.

On existence theorems for coupled systems of quadratic Hammerstein-Urysohn integral equations in Orlicz spaces

<abstract><p>We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $ L_\varphi $, namely when the generating $ N $-functions fulfill $ \Delta' $ and $ \Delta_3 $-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.</p></abstract>

On existence theorems for coupled systems of quadratic Hammerstein-Urysohn integral equations in Orlicz spaces

<abstract><p>We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $ L_\varphi $, namely when the generating $ N $-functions fulfill $ \Delta' $ and $ \Delta_3 $-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.</p></abstract>

Application of Petryshyn's fixed point theorem of existence result for non-linear 2D Volterra functional integral equations

Application of Petryshyn's fixed point theorem of existence result for non-linear 2D Volterra functional integral equations

A SCHAUDER TYPE HYBRID FIXED POINT THEOREM IN A PARTIALLY ORDERED METRIC SPACE WITH APPLICATIONS TO NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS

In this paper we prove some hybrid fixed point theorems for the monotonic nondecreasing mappings in a partially ordered metric space which includes the Schauder type fixed point theorems in an ordered Banach space proved by Dhage (2013,2014) and Dhage et al. (2022) in a partially ordered Banach space as the special cases. As an application, we discuss a nonlinear functional integral equation of Fredholm type and a nonlinear functional boundary value problem for proving the existence and approximation of solution by constructing the algorithms via Dhage monotone iteration method under some natural condtions.

Applying faster algorithm for obtaining convergence, stability, and data dependence results with application to functional-integral equations

<abstract><p>The goal of this manuscript is to create a new faster iterative algorithm than the previous writing's sober algorithms. In the setting of Banach spaces, this algorithm is used to analyze convergence, stability, and data-dependence results. Basic numerical examples are also provided to highlight the behavior and effectiveness of our approach. Ultimately, the proposed approach is used to solve the functional Volterra-Fredholm integral problem as an application.</p></abstract>

Periodic solutions for an impulsive system of integro-differential equations with maxima

A periodical boundary value problem for a first-order system of ordinary integro-differential equations with impulsive effects and maxima is investigated. The obtained system of nonlinear functional-integral equations and the existence and uniqueness of the solution of the periodic boundary value problem are reduced to the solvability of the system of nonlinear functional-integral equations. The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional-integral equations. We define the way with the aid of which we could prove the existence of periodic solutions of the given periodical boundary value problem.

An Implicit Hybrid Delay Functional Integral Equation: Existence of Integrable Solutions and Continuous Dependence

In this work, we are discussing the solvability of an implicit hybrid delay nonlinear functional integral equation. We prove the existence of integrable solutions by using the well known technique of measure of noncompactnes. Next, we give the sufficient conditions of the uniqueness of the solution and continuous dependence of the solution on the delay function and on some functions. Finally, we present some examples to illustrate our results.

On a nonlocal implicit problem under Atangana\u2013Baleanu\u2013Caputo fractional derivative

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.

Application of fixed point theorem to solvability of functional stochastic integral equations

In this paper, the existence of the solution of a class of nonlinear stochastic functional integral equations is investigated. Such that the concept of measures of noncompactness, and Petryshyn’s fixed point theorem in Banach space has been used. In addition to proving the theorems of the existence of the solution, by solving some examples, it is shown that the method is efficient. We also give an example that satisfies the conditions of our main theorem, while the conditions described in some other papers do not.

Numerical Solution for the 2D Linear Fredholm Functional Integral Equations

In this work, a numerical method is applied for obtaining numerical solutions of Fredholm two-dimensional functional linear integral equations based on the radial basis function (RBF). To find the approximate solutions of these types of equations, first, we approximate the unknown function as a finite series in terms of basic functions. Then, by using the proposed method, we give a formula for determining the unknown function. Using this formula, we obtain a numerical method for solving Fredholm two-dimensional functional linear integral equations. Using the proposed method, we get a system of linear algebraic equations which are solved by an iteration method. In the end, the accuracy and applicability of the proposed method are shown through some numerical applications.

Fixed Point Theorems for Weakly Asymptotically Regular Mappings in Banach Spaces with an Application

Using the technique of measures of weak non-compactness we obtain the existence of fixed points of a 2 × 2 block operator matrix involving nonlinear maps in non-separable Banach space. These theorems are created in terms of weak sequential continuity and the theory of countably condensing maps. Our results generalize, improve and complement a number of earlier works. The obtained results are then applied to a coupled system of nonlinear functional integral equations.

Stability and convergence analysis of hybrid algorithms for Berinde contraction mappings and its applications

In this paper, we construct a new hybrid iteration, called SR-iteration, and prove its stability and convergence analysis for weak contraction mappings in a Banach space. We compare rate of convergence between the SR-iteration and other iterations. Moreover, we provide numerical comparisons for supporting our main theorem and apply our main result to prove existence problem of mixed type Volterra-Fredholm functional nonlinear integral equation.

ON SOLUTIONS OF LINEAR FUNCTIONAL INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS VIA LAGRANGE POLYNOMIALS

In this study, a matrix-collocation method is developed numerically to solve the linear Fredholm-Volterra-type functional integral and integro-differential equations. The linear functional integro-differential equations are considered under initial conditions. The mentioned type problems often appear in various branches of science and engineering such as physics, biology, mechanics, electronics. The method essentially is a collocation method based on the Lagrange polynomials and matrix operations. By using presented method, the problem is reduced to a system of linear algebraic equations. The solution of this system gives the coefficients of assumed solution. An error analysis based on the residual function is studied. Some examples are solved to demonstrate the accuracy and efficiency of the method.

Solvability for generalized nonlinear two dimensional functional integral equations via measure of noncompactness

In this article, we provide the existence result for functional integral equations by using Petryshyn’s fixed point theorem connecting the measure of noncompactness in a Banach space. The results enlarge the corresponding results of several authors. We present fascinating examples of equations.

A NONLINEAR F-CONTRACTION FORM OF SADOVSKII'S FIXED POINT THEOREM AND ITS APPLICATION TO A FUNCTIONAL INTEGRAL EQUATION OF VOLTERRA TYPE

In this paper, we give a nonlinear F-contraction form of the Sadovskii fixedpoint theorem and we also investigate the existence of solutions for a functional integral equation of Volterra type.

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

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