Functional Differential Equations Research Articles

EXISTENCE RESULTS AND NUMERICAL SOLUTION OF FULLY FOURTH ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider a boundary value problem for a fully fourth-order nonlinear functional differential equation which contains all lower derivatives of proportional delay arguments.By the reduction of the problem to operator equation for the right-hand side nonlinear function, we establish the existence and uniqueness of the solution and construct iterative methods on both continuous and discrete levels for solving it. We obtain the total error estimate for the discrete iterative solution. Many examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

The equation G(x,x)=y˜, where G:X×X→Y, and X,Y are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones E+,M+ of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a “model” equation, namely, g(t,t)=0, where a continuous map g:E+×E+→M+ is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation.

On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

On a class of functional difference equations: explicit solutions, asymptotic behavior and applications

On the equation f n + (f″)m ≡ 1

Abstract Let n n and m m be two positive integers, and the second-order Fermat-type functional equation f n + ( f ″ ) m ≡ 1 {f}^{n}+{({f}^{^{\prime\prime} })}^{m}\equiv 1 does not have a nonconstant meromorphic solution in the complex plane, except ( n , m ) ∈ { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 1 ) , ( 3 , 1 ) } \left(n,m)\in \left\{\left(1,1),\left(1,2),\left(1,3),\left(2,1),\left(3,1)\right\} . The research gives a ready-to-use scheme to study certain Fermat-type functional differential equations in the complex plane by using the Nevanlinna theory, the complex method, and the Weierstrass factorization theorem.

Properties of Hopf bifurcation to a reaction-diffusion population model with nonlocal delayed effect

This paper is concerned with a reaction-diffusion population model with nonlocal delayed effect and zero-Dirichlet boundary condition. Under the condition when the delayed feedback control is dominant, the normal form for spatially nonhomogeneous Hopf bifurcation from the sufficiently small positive equilibrium is computed by means of the normal form method and the center manifold theorem for partial functional differential equations. It is revealed that Hopf bifurcations appearing at the small positive equilibrium are forward and all bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. To verify the validity of the obtained theoretical results, numerical simulations are also provided.

On Solvability Conditions for the Cauchy Problem for Non-Volterra Functional Differential Equations with Pointwise and Integral Restrictions on Functional Operators

Cauchy problems are considered for families of, generally speaking, non-Volterra functional differential equations of the second order. For each family considered, in terms of the parameters of this family, necessary and sufficient conditions for the unique solvability of the Cauchy problem for all equations of the family are obtained. Such necessary and sufficient conditions are obtained for the following four kinds of families: integral restrictions are imposed on positive and negative functional operators, namely, operator norms are specified; pointwise restrictions are imposed on positive and negative functional operators in the form of values of operators’ actions on the unit function; an integral constraint is imposed on a positive functional operator, a pointwise constraint is imposed on a negative functional operator; a pointwise constraint is imposed on a positive functional operator, an integral constraint is imposed on a negative functional operator. In all cases, effective conditions for the solvability of the Cauchy problem for all equations of the family are obtained, expressed through some inequalities regarding the parameters of the families. The set of parameters of families of equations for which Cauchy problems are uniquely solvable can be easily calculated approximately with any accuracy. The resulting solvability conditions improve the solvability conditions following from the Banach contraction principle. An example of the Cauchy problem for an equation with a coefficient changing sign is given. Taking into account various restrictions for the positive and negative parts of functional operators allows us to significantly improve the known solvability conditions.

An explicit approximation for super-linear stochastic functional differential equations

Since it is difficult to implement implicit schemes on the infinite-dimensional space, we aim to develop the explicit numerical method for approximating super-linear stochastic functional differential equations (SFDEs). Precisely, borrowing the truncation idea and linear interpolation we propose an explicit truncated Euler–Maruyama (EM) scheme for SFDEs, and obtain the boundedness and convergence in Lp(p≥2). We also prove the convergence rate with 1/2 order. Different from some previous works (Mao, 2003; Zhang et al., 2018), we release the global Lipschitz restriction on the diffusion coefficient. Furthermore, we reveal that numerical solutions preserve the underlying exponential stability. Moreover, we give several examples to support our theory.

Boundary-value problem for an elliptic functional differential equation with dilation and rotation of arguments

The paper is devoted to the Dirichlet problem in a flat bounded domain for a linear secondorder functional differential equation in the divergent form with dilation, contraction and rotation of the argument of the higher-order derivatives of the unknown function. We study the existence, the uniqueness and the smoothness of the generalized solution for all possible values of the coefficients and parameters of transformations in the equation.

Hybrid interpolative mappings for solving fractional Navier–Stokes and functional differential equations

The purpose of this study is to establish fixed-point results for new interpolative contraction mappings in the setting of Busemann space involving a convex hull. To illustrate our findings, we also offer helpful and compelling examples. Finally, the theoretical results are applied to study the existence of solutions to fractional Navier–Stokes and fractional-functional differential equations as applications.

Exponential ergodicity for stochastic functional differential equations with Markovian switching

This paper investigates exponential ergodicity for several kinds of stochastic functional differential equations (SFDEs) with Markovian switching. Firstly, we derive exponential ergodicity for SFDEs with Markovian switching using the Krylov-Bogoliubov theorem in the space C equipped with the uniform topology. Then, we obtain exponential ergodicity for neutral SFDEs with Markovian switching by making use of M-matrix theory and a generalized Razumikhin-type argument. Finally, we focus on SFDEs driven by Lévy processes with Markovian switching and prove exponential ergodicity through the Kurtz criterion of tightness for the space D endowed with the Skorohod topology. A concrete example is also given to illustrate our main results.

A System of Coupled Impulsive Neutral Functional Differential Equations: New Existence Results Driven by Fractional Brownian Motion and the Wiener Process

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper).

On measure of noncompactness in variable exponent Lebesgue spaces and applications to integral equations

A novel measure of noncompactness is defined in variable exponent Lebesgue spaces Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )}$\\end{document} on an unbounded domain R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{+}$\\end{document} and its properties are examined. Using the fixed point method, we apply that measure to study the existence theorem for nonlinear integral equations. Our results can be handily applied in studying various types of (differential, integral, functional, and partial differential) equations in Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )}$\\end{document}-spaces. The Lp(⋅)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{p(\\cdot )} $\\end{document}-spaces are natural extensions of classical constant exponent Lebesgue spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{p}$\\end{document}, which allows us to bypass several restrictions that were previously discussed in the literature.

More Effective Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of a Class of Third-Order Functional Differential Equations

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations.

Existence and Stability of Ulam–Hyers for Neutral Stochastic Functional Differential Equations

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of dxa(s)+g(s,xa(s-ω(s)))=Ixa(s)+f(s,xa(s-ϱ(s)))ds+ς(s)dϖH(s),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{d}}\\left[ {x}_{a}(s) + {\\mathfrak {g}}(s, {x}_{a}(s - \\omega (s)))\\right] =\\left[ {\\mathfrak {I}}{x}_a(s) + {\\mathfrak {f}}(s, {x}_a(s -\\varrho (s)))\\right] {\ extrm{d}}s + \\varsigma (s){\ extrm{d}}\\varpi ^{{\\mathbb {H}}}(s),$$\\end{document}0≤s≤T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le s\\le {\\mathcal {T}}$$\\end{document}, xa(s)=ζ(s),-ρ≤s≤0.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${x}_a(s) = \\zeta (s),\\ -\\rho \\le s\\le 0. $$\\end{document} The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples.

On exponential stability of switched functional differential equations with average dwell-time

We present a novel approach to the exponential stability of switched functional differential equations. Our approach does not involve Lyapunov functions. It is simple and based upon spectral properties of Metzler matrices, a comparison principle and the average dwell-time technique. Consequently, some new explicit criteria for the exponential stability of switched functional differential equations are derived. Some illustrative examples are given.

Statistical inference for discretely sampled stochastic functional differential equations with small noise

Estimating parameters of drift and diffusion coefficients for multidimensional stochastic delay equations with small noise are considered. The delay structure is written as an integral form with respect to a delay measure. Our contrast function is based on a local-Gauss approximation to the transition probability density of the process. We show consistency and asymptotic normality of the minimum-contrast estimator when a small dispersion coefficient ε→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0$$\\end{document} and sample size n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document} simultaneously.

Novel Criteria for Exponential Stability of Neutral Functional Differential Equations

We address the challenging problem of exponential stability of general neutral functional differential equations. By a novel approach, we present some new explicit criteria for the exponential stability of neutral functional differential equations. Some remarks and illustrative examples are given.

On a Class of Elliptic Functional–Differential Equations with Orthotropic Contractions–Expansions

On a Class of Elliptic Functional–Differential Equations with Orthotropic Contractions–Expansions

Asymptotic behaviour of the non-oscillating solutions of first order nonlinear impulsive functional-differential equations of neutral type

This paper deals with nonlinear first-order impulsive functional-differential equations of neutral type with variable coefficients. The asymptotic behaviour of the non-oscillating solutions of such equations is investigated as a function of the values of the variable coefficient in the neutral term to serve as a basis for further studies of the oscillation of the solutions of such equations.

On Some Nonlinear Functional-Differential Equations of Neutral Type with Linear Deviations of the Argument

On Some Nonlinear Functional-Differential Equations of Neutral Type with Linear Deviations of the Argument