Browsing by Author "Mohammed, Sizar Abid"

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On the Stability and Instability of Functional Volterra Integro-Differential Equations of First Order
(int Center Scientific Research & Studies, 2017) Tunc, Cemil; Mohammed, Sizar Abid
This paper is concerned with non-linear Volterra integro-differential equation (VIDE) with constant time-lag, tau : x'(t) = P(t)f(x(t)) - integral(t)(t) (tau) K(t, s)f(x(s))ds. Via Lyapunov functionals and basic inequalities, sufficient conditions are given for the exponential stability (ES) and instability (I) of the trivial solution of the former (VIDE). We introduce two new results for the above topics for the trivial solution of that (VIDE). Our conditions involve the nonlinear generalization and extensions of those found in the literature. The results to be obtained are new and complements that in the literature.
On the Stability and Uniform Stability of Retarded Integro-Differential Equations
(Elsevier Science inc, 2018) Tunc, Cemil; Mohammed, Sizar Abid
In this paper, the authors obtain new sufficient conditions for stability (S) and uniform stability (US) of solutions of the first order retarded Volterra integro-differential equations (VIDEs) in the form x' = A(t)x + integral(t)(t-tau) C(t, s)phi(s, x(s))ds + f(t, x, x(t - tau)). The analysis of the obtained (S) and (US) results mainly depend on the definition of an appropriate Lyapunov functional (LF). An example is provided to illustrate the effectiveness of the proposed results. MATLAB-Simulink is applied to show the behaviors of the paths of solutions of the considered (VIDEs) for a particular case. (C) 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
On the Asymptotic Analysis of Bounded Solutions To Nonlinear Differential Equations of Second Order
(Springeropen, 2019) Tunc, Cemil; Mohammed, Sizar Abid
In this paper, we consider two different models of nonlinear ordinary differential equations (ODEs) of second order. We construct two new Lyapunov functions to investigate boundedness of solutions of those nonlinear ODEs of second order. By using the Lyapunov direct or second method and inequality techniques, we prove two new theorems on the boundedness solutions of those ODEs of second order as t ->infinity. When we compare the conditions of the theorems of this paper with those of Meng in (J. Syst. Sci. Math. Sci. 15(1):50-57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58-64 2002), we can see that our theorems have less restrictive conditions than those in (Meng in J. Syst. Sci. Math. Sci. 15(1):50-57, 1995) and Sun and Meng in (Ann. Differ. Equ. 18(1):58-64 2002) because of the two new suitable Lyapunov functions. Next, in spite of the use of the Lyapunov second method here and in (Meng in J. Syst. Sci. Math. Sci. 15(1):50-57, 1995; Sun and Meng in Ann. Differ. Equ. 18(1):58-64 2002), the proofs of the results of this paper are proceeded in a very different way from that used in the literature for the qualitative analysis of ODEs of second order. Two examples are given to show the applicability of our results. At the end, we can conclude that the results of this paper generalize and improve the results of Meng in (J. Syst. Sci. Math. Sci. 15(1):50-57, 1995), Sun and Meng in (Ann. Differ. Equ. 18(1):58-64 2002), and some other that can be found in the literature, and they have less restrictive conditions than those in these references.