The purpose of this paper is to give theorems on continuity and differentiability with respect to (h,t) of the solution of the initial value problem du/dt = A(h,t)u + f(h,t), u(0) = u₀(h) with parameter $h\in \Omega \subset {ℝ}^m$ in the “hyperbolic” case.

This paper is concerned with the existence and approximate controllability for impulsive fractional-order stochastic infinite delay integro-differential equations in Hilbert space. By using Krasnoselskii's fixed point theorem with stochastic analysis theory, we derive a new set of sufficient conditions for the approximate controllability of impulsive fractional stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided...

In this paper, we prove existence and controllability results for first and second order semilinear neutral functional differential inclusions with finite or infinite delay in Banach spaces, with nonlocal conditions. Our theory makes use of analytic semigroups and fractional powers of closed operators, integrated semigroups and cosine families.

Our paper deals with the following nonlinear neutral differential equation with variable delay $$\frac{d}{dt}D{u}_t\left(t\right)=p\left(t\right)-a\left(t\right)u\left(t\right)-a\left(t\right)g\left(u(t-\tau \left(t\right))\right)-h(u\left(t\right),u(t-\tau \left(t\right))).$$ By using Krasnoselskii’s fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. A sufficient condition is established for the positivity of the above equation. Stability results of this equation are analyzed. Our results extend and complement some results obtained in the work [Yuan, Y., Guo, Z.: On the existence and stability of...

In this paper we investigate the existence of mild solutions to second order initial value problems for a class of delay integrodifferential inclusions with nonlocal conditions. We rely on a fixed point theorem for condensing maps due to Martelli.

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 $$x^{\text{'}}\left(t\right)=-{\int }_{t-\tau \left(t\right)}^{t}a(t,s)g\left(x\left(s\right)\right)ds+\frac{d}{dt}Q(t,x(t-\tau \left(t\right)))+G(t,x\left(t\right),x(t-\tau \left(t\right))).$$ 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 $\tau$, $g$, $a$, $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of Krasnoselskii’s...

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 $x^{▵}\left(t\right)=-a\left(t\right)h\left(x^{\sigma }\left(t\right)\right)+c\left(t\right)x^{\widetilde{▵}}\left(t-r\left(t\right)\right)+G\left(t,x\left(t\right),x\left(t-r\left(t\right)\right)\right)$, $t\in 𝕋$, where $f^{▵}$ is the $▵$-derivative on $𝕋$ and $f^{\widetilde{▵}}$ is the $▵$-derivative on $(id-r)\left(𝕋\right)$. 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...