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Youssef N. Raffoul

Publications and source records attributed to Youssef N. Raffoul.

4 recordsLinked to original sources
arXiv2026

Multiparameter $C$-semigroups and multiparameter $C$-cosine functions

In this paper, we present several new results concerning multiparameter $C$-semigroups. We introduce and systematically analyze the class of multiparameter $C$-cosine functions, providing several new structural results and applications to abstract multiparameter Cauchy problems of first/second order in locally convex spaces. We also consider automatic extensions of multiparameter $C$-semigroups and multiparameter $C$-cosine functions.

math.FA↗
arXiv2014

Existence of periodic solutions in shifts $δ_{\pm}$ for neutral nonlinear dynamic systems

In this study, we focus on the existence of a periodic solution for the neutral nonlinear dynamic systems with delay% \[ x^Δ(t)=A(t)x(t)+Q^Δ\left(t,x\left(δ_{-}(s,t)\right) \right) +G\left(t,x(t),x\left(δ_{-}(s,t)\right) \right) . \] We utilize the new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement $t\pm T\in\mathbb{T}$ for all $t\in\mathbb{T}$ 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; $\overline{q^{\mathbb{Z}}}$ and $\cup_{k=1}^{\infty}\left[ 3^{\pm k},2.3^{\pm k}\right] \cup\left\{0\right\} .$ Hence, we develop a tool that enables the investigation 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 $y^Δ(t)=A(t)y(t)$ and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point.

math.CA↗
arXiv2011

Shift operators and stability in delayed dynamic equations

In this paper, we use what we call the shift operator so that general delay dynamic equations of the form \[ x^Δ(t)=a(t)x(t)+b(t)x(δ_{-}(h,t))δ_{-}^Δ% (h,t),\ \ \ t\in\lbrack t_{0},\infty)_{\mathbb{T}}% \] can be analyzed with respect to stability and existence of solutions. By means of the shift operators we define a general delay function opening an avenue for the construction of Lyapunov functional on time scales. Thus, we use the Lyapunov's direct method to obtain inequalities that lead to stability and instability. Therefore, we extend and unify stability analysis of delay differential, delay difference, delay $h-$difference, and delay $q-$difference equations which are the most important particular cases of our delay dynamic equation. \textbf{Keywords}: Delay dynamic equation, instability, shift operators, stability, time scales.

math.CA↗