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Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay

Imene Soulahia, Abdelouaheb Ardjouni, Ahcene Djoudi (2015)

Commentationes Mathematicae Universitatis Carolinae

The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay x ' ( t ) = - ∫ t - τ ( t ) t a ( t , s ) g ( x ( s ) ) d s + d d t Q ( t , x ( t - τ ( t ) ) ) + G ( t , x ( t ) , x ( t - τ ( t ) ) ) . We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for τ , g , a , Q and G to show that this sum of mappings fits into the framework of a modification of Krasnoselskii’s...

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...

Existence of positive periodic solutions of an SEIR model with periodic coefficients

Tailei Zhang, Junli Liu, Zhidong Teng (2012)

Applications of Mathematics

An SEIR model with periodic coefficients in epidemiology is considered. The global existence of periodic solutions with strictly positive components for this model is established by using the method of coincidence degree. Furthermore, a sufficient condition for the global stability of this model is obtained. An example based on the transmission of respiratory syncytial virus (RSV) is included.

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