Nonlinear Differential and Integral Equations and Their Infinite Systems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 March 2024 | Viewed by 2379

Special Issue Editors

Department of Nonlinear Analysis, Rzeszów University of Technology, Rzeszów, Poland
Interests: nonlinear functional analysis; ordinary differential equations; nonlinear integral equations
Department of Nonlinear Analysis, Rzeszów University of Technology, Rzeszów, Poland
Interests: nonlinear functional analysis; ordinary differential equations; nonlinear integral equations
Department of Nonlinear Analysis, Rzeszów University of Technology, Rzeszów, Poland
Interests: nonlinear functional analysis; ordinary differential equations; nonlinear integral equations

Special Issue Information

Dear Colleagues,

From the point of view of applications differential and integral equations form one of the most important subjects of mathematical sciences. Those equations are frequently applied in the description of numerous events of the real world and in modelling of several phenomena appearing in engineering, economics, physics, biology and other branches of exact and natural sciences. It is worth mentioning that the theory of integral equations creates a complement of the theory of differential equations and provides a lot of handy tools used in that theory. In the present Special Issue we focus on some qualitative aspects of both mentioned theories. The particular attention is paid to the properties of solutions of nonlinear differential and integral equations connected with behaviour of those solutions, such as stability and asymptotic stability of solutions and their behaviour at infinity. Moreover, we study the solutions of infinite systems of nonlinear differential and integral equations treating such systems as realizations of differential and integral equations in sequence Banach spaces. We consider mainly classical sequence spaces such as c0, c, l. We investigate properties of solutions of infinite systems of differential and integral equations defined on a finite interval as well as an infinite one. In the case of an infinite interval we can study the behaviour of solutions of infinite systems with respect to various compactness conditions in the mentioned sequence Banach spaces.This Special Issue will cover the symmetries appearing in the theory of nonlinear differential and integral equations such as example of the symmetry of behaviour of solutions of differential equations with respect to initial conditions, the symmetry of asymptotic behaviour of solutions in + ∞ and ─∞ and many other properties of solutions of equations in question. Possible topics include but are not limited to the following:

  • Nonlinear differential equations and various properties of solutions of those equations;
  • Stability and asymptotic stability of solutions of differential and integral equations;
  • Existence of solutions of infinite systems of semilinear differential equations in connection of their nonlinear perturbations;
  • Behaviour of solutions of infinite systems of integral equations in various sequence Banach spaces;
  • Solutions of infinite systems of differential and integral equations and their behaviour with respect to different compactness conditions.

Prof. Dr. Józef Banaś
Dr. Agnieszka Chlebowicz
Prof. Dr. Beata Rzepka
Guest Editors

Manuscript Submission Information

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Keywords

  • nonlinear differential equation
  • nonlinear integral equation
  • stability of solutions
  • asymptotic stability of solutions
  • asymptotic behaviour of solutions at infinity
  • infinite system of nonlinear differential equations
  • infinite system of nonlinear integral equations
  • sequence Banach space
  • compactness conditions

Published Papers (3 papers)

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Research

19 pages, 325 KiB  
Article
Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations
Symmetry 2024, 16(1), 107; https://doi.org/10.3390/sym16010107 - 16 Jan 2024
Viewed by 493
Abstract
In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real [...] Read more.
In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper. Full article
16 pages, 313 KiB  
Article
On Ulam Stabilities of Delay Hammerstein Integral Equation
Symmetry 2023, 15(9), 1736; https://doi.org/10.3390/sym15091736 - 11 Sep 2023
Cited by 4 | Viewed by 579
Abstract
In this paper, we consider a Hammerstein integral equation (Hammerstein IE) in two variables with two variables of time delays. The aim of this paper is to investigate the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of the considered IE via Banach’s fixed [...] Read more.
In this paper, we consider a Hammerstein integral equation (Hammerstein IE) in two variables with two variables of time delays. The aim of this paper is to investigate the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of the considered IE via Banach’s fixed point theorem (Banach’s FPT) and the Bielecki metric. The proofs of the new outcomes of this paper are based on these two basic tools. As the new contributions of the present study, here, for the first time, we develop the outcomes that can be found in the earlier literature on the Hammerstein IE, including variable time delays. The present study also includes complementary outcomes for the symmetry of Hammerstein IEs. Finally, a concrete example is given at the end of this study for illustrations. Full article
13 pages, 1997 KiB  
Article
On the Solitary Waves and Nonlinear Oscillations to the Fractional Schrödinger–KdV Equation in the Framework of the Caputo Operator
Symmetry 2023, 15(8), 1616; https://doi.org/10.3390/sym15081616 - 21 Aug 2023
Cited by 1 | Viewed by 582
Abstract
The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. Motivated by the several applications of the mentioned evolution equation, in this investigation, [...] Read more.
The fractional Schrödinger–Korteweg-de Vries (S-KdV) equation is an important mathematical model that incorporates the nonlinear dynamics of the KdV equation with the quantum mechanical effects described by the Schrödinger equation. Motivated by the several applications of the mentioned evolution equation, in this investigation, the Laplace residual power series method (LRPSM) is employed to analyze the fractional S-KdV equation in the framework of the Caputo operator. By incorporating both the Caputo operator and fractional derivatives into the mentioned evolution equation, we can account for memory effects and non-local behavior. The LRPSM is a powerful analytical technique for the solution of fractional differential equations and therefore it is adapted in our current study. In this study, we prove that the combination of the residual power series expansion with the Laplace transform yields precise and efficient solutions. Moreover, we investigate the behavior and properties of the (un)symmetric solutions to the fractional S-KdV equation using extensive numerical simulations and by considering various fractional orders and initial fractional values. The results contribute to the greater comprehension of the interplay between quantum mechanics and nonlinear dynamics in fractional systems and shed light on wave phenomena and symmetry soliton solutions in such equations. In addition, the proposed method successfully solves fractional differential equations with the Caputo operator, providing a valuable computational instrument for the analysis of complex physical systems. Moreover, the obtained results can describe many of the mysteries associated with the mechanism of nonlinear wave propagation in plasma physics. Full article
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