Browsing by Author "Bohner, Martin"

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Existence of Solutions for Nonlinear Impulsive Multiple Retarded Differential and Impulsive Integro-Differential Equations of Second Order
(Yokohama Publ, 2024) Bohner, Martin; Tunc, Osman; Tunc, Cemil
. In this paper, we deal with the existence of solutions of certain impulsive multiple retarded differential equations (ImMRDEs) and impulsive multiple retarded integro-differential equations (ImMRIDEs) of second order. We prove two new results on the existence of solutions of the considered ImMRDE and ImMRIDE of second order. The technique of the proofs depends on the Schaefer fixed point theorem (Schaefer FPT) and fixed moments of impulse effects. The outcomes of this paper have more general forms and improve the known results in the relevant literature, and they have new contributions to the theory of impulsive retarded differential equations (ImMRDEs). The outcomes of this paper have also new complementary properties for the works in relation to the symmetry of impulsive differential equations (ImDEs) of second order with or without delay, impulsive retarded integro-differential equations (ImMRIDEs) of second order, and some others.
On the Fundamental Qualitative Properties of Integro-Delay Differential Equations
(Elsevier, 2023) Bohner, Martin; Tunc, Osman; Korkmaz, Erdal
This paper discusses qualitative properties of solutions of certain unperturbed and perturbed systems of nonlinear integro-delay differential equations (IDDEs), namely asymptotic stability, uniform stability, integrability and boundedness. Here, four new theorems are proved on these properties of solutions by using Lyapunov-Krasovskii? functional (LKF) technique. As illustrations and applications of our results, we also pro-vide two examples, solve them numerically, and plot the trajectories of their solutions. The results of this paper include weaker sufficient conditions than the ones found in the literature, e.g., some superfluous conditions are removed here, and the results have also new contributions to the qualitative theory of integro-differential equations (IDEs) and IDDEs.(C) 2023 Elsevier B.V. All rights reserved.
Qualitative Analysis of Caputo Fractional Integro-Differential Equations With Constant Delays
(Springer Heidelberg, 2021) Bohner, Martin; Tunc, Osman; Tunc, Cemil
In this paper, a nonlinear Volterra integro-differential equation with Caputo fractional derivative, multiple kernels, and multiple constant delays is considered. The aim of this paper is to investigate qualitative properties of solutions of this equation such as uniform stability, asymptotic stability, and Mittag-Leffler stability of the zero solution as well as boundedness of nonzero solutions. Here, we prove four new theorems on the mentioned properties of the solutions of the considered fractional integro-differential equation. The technique used in the proofs of these theorems includes defining an appropriate Lyapunov function and applying the Lyapunov-Razumikhin method. To illustrate the obtained results, two examples are provided, one of them related to an RLC circuit, to illustrate and show applications of the given results. The obtained results are new, original, and they can be useful for applied researchers in sciences and engineering.
Qualitative Analysis of Integro-Differential Equations With Variable Retardation
(Amer inst Mathematical Sciences-aims, 2022) Bohner, Martin; Tunc, Osman
The paper is concerned with a class of nonlinear time-varying re-tarded integro-differential equations (RIDEs). By the Lyapunov-Krasovskii functional method, two new results with weaker conditions related to uniform stability (US), uniform asymptotic stability (UAS), integrability, boundedness, and boundedness at infinity of solutions of the RIDEs are given. For illustrative purposes, two examples are provided. The study of the results of this paper shows that the given theorems are not only applicable to time-varying linear RIDEs, but also applicable to time-varying nonlinear RIDEs.