NONLINEAR NEUTRAL CAPUTO-FRACTIONAL DIFFERENCE EQUATIONS WITH APPLICATIONS TO LOTKA-VOLTERRA NEUTRAL MODEL

In this paper, we consider a nonlinear neutral fractional difference equations. By applying Krasnoselskii's fixed point theorem, sufficient conditions for the existence of solutions are established, also the uniqueness of solutions is given. As an application of the main theorems, we provide the existence and uniqueness of the discrete fractional Lotka-Volterra model of neutral type. Our main results extend and generalize the results that are obtained in Azabut

A STUDY OF THE STABILITY IN NEUTRAL NONLINEAR DIFFERENTIAL EQUATIONS WITH FUNCTIONAL DELAY VIA FIXED POINTS

In this paper, we use a modification of Krasnoselskii's fixed point theorem introduced by Burton (see b0 Theorem 3) to obtain stability results of the zero solution of the totally nonlinear neutral differential equations with functional delayx′(t)=-a(t)h(x(t-τ(t)))+c(t)x′(t-τ(t))+G(t,x(t),x(t-τ(t))).The stability of the zero solution of this eqution provided that h(0)=G(t,0,0)=0. The Caratheodory condition is used for the function G

Stability in Nonlinear Neutral Caputo <i>q</i>-Fractional Difference Equations

In this article, we consider a nonlinear neutral q-fractional difference equation. So, we apply the fixed point theorem of Krasnoselskii to obtain the existence of solutions under sufficient conditions. After that, we use the fixed point theorem of Banach to show the uniqueness, as well as the stability of solutions. Our main results extend and generalize previous results mentioned in the conclusion

Asymptotic Behavior of Solutions in Nonlinear Neutral System with Two Volterra Terms

In this manuscript, we generalise previous results in the literature by providing sufficient conditions for the matrix measure to guarantee the stability, asymptotic stability and exponential stability of a neutral system of differential equations. This is achieved by constructing a suitable operator from our system and applying the Banach fixed point theorem

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation ddtx(t)=−a (t) h (x (t))+ddtQ (t, x (t−τ (t)))+G (t, x(t), x (t−τ (t))).$${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$ We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G

Exponential Stability of the Heat Equation with Boundary Time-Varying Delays

In this paper, we consider the heat equation with a time-varying delays term in the boundary condition in a bounded domain of Rn, the boundary Γ is a class C2 such that Γ = ΓD∪ΓN, with ΓD∩ ΓN = ∅, ΓD 6= ∅ and ΓN 6= ∅. Well-posedness of the problems is analyzed by using semigroup theory. The exponential stability of the problem is proved. This paper extends in n-dimensional the results of the heat equation obtained in [11]

Study of periodic and nonnegative periodic solutions of nonlinear neutral functional differential equations via fixed points

In this paper, we study the existence of periodic and non-negative periodic solutions of the nonlinear neutral differential equation ddtx(t)=−a (t) h (x (t))+ddtQ (t, x (t−τ (t)))+G (t, x(t), x (t−τ (t))).$${{\rm{d}} \over {{\rm{dt}}}}{\rm{x}}({\rm{t}}) = - {\rm{a}}\;({\rm{t}})\;{\rm{h}}\;({\rm{x}}\;({\rm{t}})) + {{\rm{d}} \over {{\rm{dt}}}}{\rm{Q}}\;({\rm{t}},\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))) + {\rm{G}}\;({\rm{t}},\;{\rm{x}}({\rm{t}}),\;{\rm{x}}\;({\rm{t}} - {\rm \tau} \;({\rm{t}}))).$$ We invert this equation to construct a sum of a completely continuous map and a large contraction which is suitable for applying the modificatition of Krasnoselskii’s theorem. The Caratheodory condition is used for the functions Q and G

Stability in linear neutral difference equations with variable delays

summary:In this paper we use the fixed point method to prove asymptotic stability results of the zero solution of a generalized linear neutral difference equation with variable delays. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Y. N. Raffoul (2006), E. Yankson (2009), M. Islam and E. Yankson (2005)

Periodic solutions for a system of nonlinear neutral functional difference equations with two functional delays

In this paper, we study the existence and uniqueness of periodic solutions of the system of nonlinear neutral difference equations ∆x (n) = A(n) x (n - t (n)) + ∆Q(n; x (n - g (n))) + G(n; x (n) ; x (n - g (n))). By using Krasnoselski's fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. An example is given to illustrate our result. Our results extend and generalize the work [13]