Browsing by Author "Raffoul, Youssef N."

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Citation - WoS: 5
Citation - Scopus: 4
Almost Automorphic Solutions of Delayed Neutral Dynamic Systems on Hybrid Domains
(Univ Belgrade, Fac Electrical Engineering, 2016) Adıvar, Murat; Koyuncuoglu, Halis Can; Raffoul, Youssef N.
We study the existence of almost automorphic solutions of the delayed neutral dynamic system on hybrid domains that are additively periodic. We use exponential dichotomy and prove the uniqueness of projector of exponential dichotomy to obtain some limit results leading to sufficient conditions for existence of almost automorphic solutions to neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of the coefficient matrices in the system. Hence, we significantly improve the results in the existing literature. Finally, we also provide an existence result for almost periodic solutions of the system.
Citation - WoS: 7
Citation - Scopus: 7
Existence of Periodic Solutions in Shifts Delta(+/-) for Neutral Nonlinear Dynamic Systems
(Elsevier Science Inc, 2014) Adıvar, Murat; Koyuncuoglu, Halis Can; Raffoul, Youssef N.
This paper focuses on the existence of a periodic solution of the delay neutral nonlinear dynamic systems x(Delta)(t) = A(t)x(t) + Q(Delta)(t, x(delta (-) (s, t))) + G(t, x(t), x(delta (-) (s, t))). In our analysis, we utilize a new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement t +/- T is an element of T for all t is an element of T and for a fixed T > 0, may not hold. More importantly, the new concept will easily handle time scales that are not periodic in the conventional way such as; (q(z)) over bar and boolean OR(infinity)(k-1) [3(+/- k), 2.3(+/- k)] boolean OR {0}. Hence, we will develop the tool that enables us to investigate the existence of periodic solutions of q-difference systems. Since we are dealing with systems, in order to convert our equation to an integral systems, we resort to the transition matrix of the homogeneous Floquet system y(Delta)(t) = A(t)y(t) and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point. (C) 2014 Elsevier Inc. All rights reserved.
Citation - WoS: 37
Citation - Scopus: 52
Existence of Resolvent for Volterra Integral Equations on Time Scales
(Cambridge Univ Press, 2010) Adıvar, Murat; Raffoul, Youssef N.
We introduce the concept of 'shift operators' in order to establish sufficient conditions for the existence of the resolvent for the Volterra integral equation x(t) = f(t) + integral(t)(t0)a(t, s)x(s)Delta s, t(0) is an element of T-kappa, on time scales. The paper will serve as the foundation for future research on the qualitative analysis of solutions of Volterra integral equations on time scales, using the notion of the resolvent.
Citation - WoS: 32
Citation - Scopus: 38
Existence Results for Periodic Solutions of Integro-Dynamic Equations on Time Scales
(Springer Heidelberg, 2009) Adıvar, Murat; Raffoul, Youssef N.
Using the topological degree method and Schaefer's fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. Furthermore, we provide several applications to scalar equations, in which we develop a time scale analog of Lyapunov's direct method and prove an analog of Sobolev's inequality on time scales to arrive at a priori bound on all periodic solutions. Therefore, we improve and generalize the corresponding results in Burton et al. (Ann Mat Pura Appl 161: 271-283, 1992)
Citation - WoS: 4
Citation - Scopus: 2
Inequalities and Exponential Decay in Time Varying Delay Differential Equations
(Pergamon-Elsevier Science Ltd, 2011) Adivar, Murat; Raffoul, Youssef N.
We use Lyapunov functionals to obtain sufficient conditions that guarantee exponential decay of solutions to zero of the time varying delay differential equation x'(t) = b(t) x(t) - a(t) x(t - h(t)). The highlights of the paper are allowing b( t) to change signs and the delay to vary with time. In addition, we obtain a criterion for the instability of the zero solution. Moreover, by comparison to existing literature we show effectiveness of our results. (C) 2011 Elsevier Ltd. All rights reserved.
Citation - WoS: 9
Citation - Scopus: 13
Inequalities and Exponential Stability and Instability in Finite Delay Volterra Integro-Differential Equations
(Springer-Verlag Italia Srl, 2012) Adıvar, Murat; Raffoul, Youssef N.
We use Liapunov functionals to obtain sufficient conditions that ensure exponential stability of the nonlinear Volterra integro- differential equation x(2) (t) = p(1)x(t) - integral(t)(t-iota) q(t,s)x(s)ds where the constant t is positive, the function p does not need to obey any sign condition and the kernel q is continuous. Our results improve the results obtained in literature even in the autonomous case. In addition, we give a new criteria for instability.
Citation - WoS: 9
Citation - Scopus: 8
A Note on Stability and Periodicity in Dynamic Delay Equations [comput. Math. Appl. 58 (2009) 264-273]
(Pergamon-Elsevier Science Ltd, 2010) Adıvar, Murat; Raffoul, Youssef N.
The purpose of this note is twofold: First we highlight the importance of an implicit assumption in [Murat Adivar, Youssef N. Raffoul, Stability and periodicity in dynamic delay equations, Computers and Mathematics with Applications 58 (2009) 264-272]. Second we emphasize one consequence of the bijectivity assumption which enables ruling out the commutativity condition delta circle sigma = sigma circle delta on the delay function. (C) 2010 Elsevier Ltd. All rights reserved.
Citation - WoS: 7
Citation - Scopus: 10
Periodic and Asymptotically Periodic Solutions of Systems of Nonlinear Difference Equations With Infinite Delay
(Taylor & Francis Ltd, 2013) Adıvar, Murat; Koyuncuoglu, Halis Can; Raffoul, Youssef N.
In this paper we study the existence of periodic and asymptotically periodic solutions of a system of nonlinear Volterra difference equations with infinite delay. By means of fixed point theory, we furnish conditions that guarantee the existence of such periodic solutions.
Citation - WoS: 3
Citation - Scopus: 7
Qualitative Analysis of Nonlinear Volterra Integral Equations on Time Scales Using Resolvent and Lyapunov Functionals
(Elsevier Science Inc, 2016) Adıvar, Murat; Raffoul, Youssef N.
In this paper we use the notion of the resolvent equation and Lyapunov's method to study boundedness and integrability of the solutions of the nonlinear Volterra integral equation on time scales x(t) = a(t) - integral(t)(t0) C(t, s)G(s, x(s)) Delta s, t is an element of[t(0), infinity) boolean AND T. In particular, the existence of bounded solutions with various L-P properties are studied under suitable conditions on the functions involved in the above Volterra integral equation. At the end of the paper we display some examples on different time scales. (C) 2015 Elsevier Inc. All rights reserved.
Citation - WoS: 21
Citation - Scopus: 28
Stability and Periodicity in Dynamic Delay Equations
(Pergamon-Elsevier Science Ltd, 2009) Adıvar, Murat; Raffoul, Youssef N.
Let T be an arbitrary time scale that is unbounded above. By means of a variation of Lyapunov's method and contraction mapping principle this paper handles asymptotic stability of the zero solution of the completely delayed dynamic equations x(Delta)(t) = -a(t)x(delta(t))delta(Delta)d(t). Moreover, if T is a periodic time scale, then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation. (c) 2009 Elsevier Ltd. All rights reserved.