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Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay

Abdelouaheb Ardjouni, Ahcène Djoudi (2014)

Commentationes Mathematicae Universitatis Carolinae

We use a modification of Krasnoselskii’s fixed point theorem due to Burton (see [Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9 (2002), 181–190], Theorem 3) to show that the totally nonlinear neutral differential equation with variable delay x ' ( t ) = - a ( t ) h ( x ( t ) ) + c ( t ) x ' ( t - g ( t ) ) Q ' ( x ( t - g ( t ) ) ) + G ( t , x ( t ) , x ( t - g ( t ) ) ) , has a periodic solution. We invert this equation to construct a fixed point mapping expressed as a sum of two mappings such that one is compact and the other is a large contraction. We show that the mapping fits...

Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale

Abdelouaheb Ardjouni, Ahcène Djoudi (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

Let 𝕋 be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay x ▵ t = - a t h x σ t + c ( t ) x ▵ ˜ t - r t + G t , x t , x t - r t , t ∈ 𝕋 , where f ▵ is the ▵ -derivative on 𝕋 and f ▵ ˜ is the ▵ -derivative on ( i d - r ) ( 𝕋 ) . We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show...

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