Nonlinear Functional Equations Research Articles

A study on the solvability of fractional integral equation in a Banach algebra via Petryshyn's fixed point theorem

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra 0 $ ]]> C [ 0 , a ] , a > 0 . Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.

Solving generalized nonlinear functional integral equations with applications to epidemic models

Solving generalized nonlinear functional integral equations with applications to epidemic models

Three Positive Periodic Solutions of Nonlinear Functional Difference Equations

Three Positive Periodic Solutions of Nonlinear Functional Difference Equations

Adomian Decomposition Method and Block by Block Method for Solving Nonlinear Functional Volterra Integral Equation in Two-dimensions

This work proposes a new defintion of the nonlinear functional Volterra integral equation ib 2D of the second kind with continuous kernel (NT-DFVIE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions are obtained by two powerful methods Adomian Decomposition Method (ADM) and Block by block method (BBM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

A NEW CLASS OF GIBBS MEASURES FOR THREE-STATE SOS MODEL ON A CAYLEY TREE

The phase transition phenomenon is one of the central problems of statistical mechanics. It occurs when the model possesses multiple Gibbs measures. In this paper, we consider a three-state SOS (solid-on-solid) model on a Cayley tree. We reduce description of Gibbs measures to solving of a non-linear functional equation, each solution of which corresponds to a Gibbs measure.We give some sufficiency conditions on the existence of multiple Gibbs measures for the model. We give a review of some known (translation-invariant, periodic, non-periodic) Gibbs measures of the model and compare them with our new measures. We show that the Gibbs measures found in the paper differ from the known Gibbs measures, i.e, we show that these measures are new.

MIXED PROBLEM FOR A NONLINEAR IMPULSIVE DIFFERENTIAL EQUATION OF PARABOLIC TYPE

In this paper, we consider a nonlinear impulsive parabolic type partial differential equation with nonlinear impulsive conditions. Dirichlet type boundary value conditions with respect to spatial variable is used, and eigenvalues and eigenfunctions of the spectral problem are founded. The Fourier method of the separation of variables is applied. A countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of the unknown function. A theorem on a unique solvability of the countable system of nonlinear functional equations is proved by the method of successive approximations. A criteria of uniqueness and existence of a solution for the nonlinear impulsive mixed problem is obtained. A solution of the mixed problem is derived in the form of the Fourier series. The absolute and uniform convergence of the Fourier series is proved.

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods.

A constrained problem of a nonlinear functional integral equation subject to the pantograph problem

A constrained problem of a nonlinear functional integral equation subject to the pantograph problem

Ulam-Hyers and generalized Ulam-Hyers stability of fractional functional integro-differential equations

Unique solvability conditions for general fractional functional integro-differential equations with nonlocal initial conditions are established using fixed-point theory. Ulam-Hyers and generalized Ulam-Hyers stability conditions for a solution of these problems are formulated. An example of the nonlinear fractional functional integro-differential equation related to wheel shimmy models is considered as well.

Solving non-linear functional equations by relaxed new iterative method

For solving various equations of the form v=f+N(v), the new iterative method and the new algorithm proposed by V. Daftardar–Gejji et al. [Daftardar–Gejji V., Jafari H. J. Math. Anal. Appl. 316 (2), 753–763 (2006); Kumar M., Jhinga A., Daftardar–Gejji V. Int. J. Appl. Comp. Math. 6 (2), 26 (2020)] are been employed successfully and accurately. Our aim in this paper is to present a relaxed new iterative method by introducing a controlled parameter ω in order to extend these methods. According to the values of the parameter ω, we discuss and provide the convergence analysis. The proposed algorithm is fast, effective and simple to implement as compared to the existing one. Numerous non-linear equations are solved to show the applicability and efficiency of the algorithm compared to the other methods.

NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM FOR A SECOND ORDER IMPULSIVE SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONS WITH MIXED MAXIMA

A two-point nonlinear boundary value problem for a second order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. By applying some transformations is obtained a system of nonlinear functional integral equations. The existence and uniqueness of the solution of the nonperiodic two-point boundary value problem are reduced to the one valued solvability of the system of nonlinear functional integral equations in Banach space . 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.

Existence and Uniqueness of Solutions of Hammerstein-Type Functional Integral Equations

The authors deal with nonlinear and general Hammerstein-type functional integral equations (HTFIEs). The first objective of this work is to apply and extend Burton’s method to general and nonlinear HTFIEs in a Banach space via the Chebyshev norm and complete metric. The second objective of the paper is to extend and improve some earlier results to nonlinear HTFIEs. The authors prove two new theorems with regard to the existence and uniqueness of solutions (EUSs) of HTFIEs via a technique called progressive contractions, which belongs to T. A. Burton, and the Chebyshev norm and complete metric.

An infinite-dimensional nonlinear equation related to Gibbs measures of a SOS model

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation.

An Investigation of a Nonlinear Delay Functional Equation with a Quadratic Functional Integral Constraint

This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L1[0,T] and, secondly, the existence of nonincreasing solutions in unbounded interval L1(R+). Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ∈L1[0,T] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ∈L1(R+) on some parameters. Finally, some examples are provided to verify our investigation.

Kittel’s molecular zipper model on Cayley trees

Kittel’s 1D model represents a natural DNA with two strands as a (molecular) zipper, which may be separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a nonlinear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give a general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measures. Explicit critical temperature for occurrence of a phase transition (non-uniqueness of Gibbs measures) is obtained.

Asymptotic Synchronization of Nonlinear Functional Neutral Delay Difference Equations with Variable Coefficients

Asymptotic Synchronization of Nonlinear Functional Neutral Delay Difference Equations with Variable Coefficients

Existence, uniqueness, and numerical approximation of solutions of a nonlinear functional integral equation

A general nonlinear functional integral equation has been studied. The considered equation is of Fredholm type and includes some well-known integral or functional equations, as special cases. By assuming some smoothness conditions on the functions in the equation, we prove the existence and uniqueness of the solution by means of a fixed point method. Then, we apply the so-called Nyström method in order to approximate the solution, which leads to a system of nonlinear algebraic equations. In the next step, the Picard iteration method is utilized for approximating the solution of the resulting system of nonlinear algebraic equations. The convergence of the numerical scheme is proved, its computational complexity is computed and finally some illustrative examples are included to demonstrate the efficiency and applicability of the technique.

Common fixed point result for multivalued mappings with applications to systems of integral and functional equations

In this paper, we establish a common fixed point theorem for two multivalued mappings satisfying some dominated conditions on a complete metric space. This new rational type contractive inequality refines various results in the literature. The main theorem is illustrated with examples. As application, we found the existence of solutions of system nonlinear integral equations and functional equations.

Solutions of complex nonlinear functional equations including second order partial differential and difference in C^2

This article is devoted to exploring the existence and the form of finite order transcendental entire solutions of Fermat-type second order partial differential-difference equations $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2 +f(z_1+c_1,z_2+c_2)^2=e^{g(z_1,z_2)} $$ and $$ \Big(\frac{\partial^2 f}{\partial z_1^2}+\delta\frac{\partial^2 f}{\partial z_2^2} +\eta\frac{\partial^2 f}{\partial z_1\partial z_2}\Big)^2+(f(z_1+c_1,z_2+c_2) -f(z_1,z_2))^2=e^{g(z)}, $$ where \(\delta,\eta\in\mathbb{C}\) and \(g(z_1,z_2)\) is a polynomial in \(\mathbb{C}^2\). Our results improve the results of Liu and Dong [23] Liu et al. [24] and Liu and Yang [25] Several examples confirm that the form of tr

The stability of nonlinear delay integro-differential equations in the sense of Hyers-Ulam

Abstract In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.