A unique solution of the iterative boundary value problem for a second-order differential equation approached by fixed point results

Abstract The analysis of iterative differential equations is often related to the various applications of calculus, which support all mathematical sciences. These equations are critical when it comes to interpreting the infection models. Furthermore, the inclusion of self-mapping raises the complexity of determining the existence of solutions for the iterative differential equations. This paper considers a particular type of second-order iterative differential equations and uses the Banach fixed point theorem to find the existence and uniqueness of the proposed differential equation’s solution. We discuss the Hyers-Ulam and Hyers-Ulam-Rassias type stability of a solution to the proposed iterative boundary-value problem and present three illustrative examples to support our main results.

[1]  Elmar Eder,et al.  The functional differential equation x′(t) = x(x(t)) , 1984 .

[2]  S. Ulam.  A collection of mathematical problems , 1960 .

[3]  R. D. Driver,et al.  A Functional-Differential System of Neutral Type Arising in a Two-Body Problem of Classical Electrodynamics , 1963 .

[4]  R. Johnson,et al.  Functional equations, approximations, and dynamic response of systems with variable time delay , 1972 .

[5]  K. Nisar,et al.  Solutions to fractional neutral delay differential nonlocal systems , 2020 .

[6]  H. M. Baskonus,et al.  Results on approximate controllability results for second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay , 2020, Numerical Methods for Partial Differential Equations.

[7]  R. D. Driver,et al.  A two-body problem of classical electrodynamics: the one-dimensional case☆ , 1963 .

[8]  V. Vijayakumar,et al.  Controllability for a class of second-order evolution differential inclusions without compactness , 2019 .

[9]  Z. Hammouch,et al.  New results on controllability in the framework of fractional integrodifferential equations with nondense domain , 2019, The European Physical Journal Plus.

[10]  V Vijayakumar,et al.  Approximate controllability of second order nonlocal neutral differential evolution inclusions , 2020, IMA J. Math. Control. Inf..

[11]  S. Banach.  Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales , 1922 .

[12]  T. Rassias.  On the stability of the linear mapping in Banach spaces , 1978 .

[13]  V. Vijayakumar.  Approximate Controllability for a Class of Second-Order Stochastic Evolution Inclusions of Clarke’s Subdifferential Type , 2018, Results in Mathematics.

[14]  K. Nisar,et al.  Abundant solitary wave solutions to an extended nonlinear Schrödinger’s equation with conformable derivative using an efficient integration method , 2020 .

[15]  D. Baleanu,et al.  The new exact solitary wave solutions and stability analysis for the (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen , 2019, Advances in Difference Equations.

[16]  H. Henríquez,et al.  Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces , 2018 .

[17]  William Gurney,et al.  The systematic formulation of population models for insects with dynamically varying instar duration , 1983 .

[18]  Harsh V. S. Chauhan,et al.  Existence of solutions of non-autonomous fractional differential equations with integral impulse condition , 2020 .

[19]  Abderrazak Nabti,et al.  Global stability analysis of a fractional SVEIR epidemic model , 2021, Mathematical Methods in the Applied Sciences.

[20]  Nondecreasing and convex C2-solutions of an iterative functional-differential equation , 2000 .

[21]  An Existence Theorem for Iterative Functional Differential Equations , 2002 .

[22]  B. Ghanbari.  On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique , 2020, Mathematical Methods in the Applied Sciences.

[23]  Dumitru Baleanu,et al.  Existence and Hyers-Ulam type stability results for nonlinear coupled system of Caputo-Hadamard type fractional differential equations , 2021, AIMS Mathematics.

[24]  K. Nisar,et al.  Existence of solutions for some functional integrodifferential equations with nonlocal conditions , 2020, Mathematical Methods in the Applied Sciences.

[25]  Existence and approximation of solutions of some first order iterative differential equations , 2010 .

[26]  Weinian Zhang,et al.  Solutions of equivariance for iterative differential equations , 2004, Appl. Math. Lett..

[27]  N. Mahmudov,et al.  On the Approximate Controllability of Second-Order Evolution Hemivariational Inequalities , 2020, Results in Mathematics.

[28]  Marcus R. W. Martin,et al.  The escaping disaster: A problem related to state-dependent delays , 2004 .

[29]  K. Shah,et al.  Ulam stability results to a class of nonlinear implicit boundary value problems of impulsive fractional differential equations , 2019, Advances in Difference Equations.