Positive solutions for nonlinear fractional differential equations

Positivity - We study the existence and uniqueness of positive solutions of the nonlinear fractional differential equation $$\begin{aligned} \left\{ \begin{array}{l} ^{C}D^{\alpha }x\left( t\right)...     

Fixed points and stability in linear neutral differential equations with variable delays

In this paper we consider the asymptotic stability of a generalized linear neutral differential equation with variable delays by using the fixed point theory. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton (2003) [3], Zhang (2005) [14], Raffoul (2004) [13], and Jin and Luo (2008) [12]. Two examples are also given to illustrate our results.     

Existence of positive periodic solutions of nonlinear neutral differential systems with variable delays

In this work we use some mixed techniques of the Mawhin coincidence degree theory and fixed point theorem to prove the existence of positive periodic solutions of delay systems. As a consequence, we offer existence criteria and sufficient conditions for existence of periodic solutions to the systems with feedback control. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved.     

Periodicity and non‐negativity of solutions for nonlinear neutral differential equations with variable delay via fixed point theorems

We use a modification of Krasnoselskii's fixed point theorem introduced by Burton to show the periodicity and non‐negativity of solutions for the nonlinear neutral differential equation with variable delay We invert this equation to construct the sum of a compact map and a large contraction, which is suitable for applying the modification of Krasnoselskii's theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of positive solutions of delay dynamic equations

In this article we study the existence of positive solutions for a dynamic equations on time scales. The main tool employed here is the Schauder’s fixed point theorem. The asymptotic properties of solutions are also treated. The results obtained here extend the work of Dorociakova and Olach (Tatra Mt Math Publ 43:63–70, 2009). Three examples are also given to illustrate this work.     

Existence and Uniqueness of Solutions for Time-Fractional Oldroyd-B Fluid Equations with Generalized Fractional Derivatives

In this paper, we study the existence and uniqueness of solutions for time-fractional Oldroyd-B fluid equations with generalized fractional derivatives. We distinguish two cases. Firstly for the linear case, we get regularity results under some hypotheses of the source function and the initial data. Secondly for the nonlinear case, we use the Banach fixed point theorem to obtain the existence and uniqueness of solutions.     

Stability in totally nonlinear delay difference equations

In this paper we use fixed point method to prove asymptotic stability results of the zero solution of a nonlinear delay difference equation. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Raffoul (2006) \cite{r1}, Yankson (2009) \cite{y1}, Jin and Luo (2009) \cite{jin} and Chen (2013) \cite{c}     

Stability in nonlinear neutral integro-differential equations with variable delay using fixed point theory

The nonlinear neutral integro-differential equation $$\frac{d}{dt}x ( t ) =-\int_{t-\tau ( t ) }^{t}a ( t,s ) g \bigl( x ( s ) \bigr) ds+\frac{d}{dt}G \bigl( t,x \bigl( t-\tau ( t ) \bigr) \bigr) , $$ with variable delay τ(t)≥0 is investigated. We find suitable conditions for τ, a, g and G so that for a given continuous initial function ψ a mapping P for the above equation can be defined on a carefully chosen complete metric space $S_{\psi }^{0}$ in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient condition. The obtained theorem improves and generalizes previous results due to Burton (Proc. Am. Math. Soc. 132:3679–3687, 2004), Becker and Burton (Proc. R. Soc. Edinb., A 136:245–275, 2006) and Jin and Luo (Comput. Math. Appl. 57:1080–1088, 2009).