Consider the forced higher-order nonlinear neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+C\left(t\right)x(t-\tau )]+\sum _{i=1}^m{Q}_i\left(t\right)f_i\left(x(t-{\sigma }_i)\right)=g\left(t\right),t\ge t_0,$$ where $n,m\ge 1$ are integers, $\tau ,{\sigma }_i\in {ℝ}^{+}=[0,\infty )$, $C,Q_i,g\in C([t_0,\infty ),ℝ)$, $f_i\in C(ℝ,ℝ)$, $(i=1,2,\cdots ,m)$. Some sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general $Q_i\left(t\right)$$(i=1...$

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.

Let $X$ be the Banach space of $C^0$-functions on $\langle 0,1\rangle$ with the sup norm and $\alpha ,\beta \in X\to R$ be continuous increasing functionals, $\alpha \left(0\right)=\beta \left(0\right)=0$. This paper deals with the functional differential equation (1) $x^{\text{'}\text{'}\text{'}}\left(t\right)=Q[x,x^{\text{'}},x^{\text{'}\text{'}}\left(t\right)]\left(t\right)$, where $Q:X^2\times R\to X$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $\alpha \left(x\right)=0=\beta \left(x^{\text{'}}\right)$, $x^{\text{'}\text{'}}\left(1\right)-x^{\text{'}\text{'}}\left(0\right)=0$ are given. The method of proof makes use of Schauder linearizatin technique and the Schauder fixed point theorem. The results are modified for 2nd order functional...

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...

In this paper, we study the existence of oscillatory and nonoscillatory solutions of neutral differential equations of the form $x\left(t\right)-cx(t-r)$’$P\left(t\right)x(t-\theta )-Q\left(t\right)x(t-\delta )$=0 where $c>0$, $r>0$, $\theta >\delta \ge 0$ are constants, and $P$, $Q\in C({ℝ}^{+},{ℝ}^{+})$. We obtain some sufficient and some necessary conditions for the existence of bounded and unbounded positive solutions, as well as some sufficient conditions for the existence of bounded and unbounded oscillatory solutions.

For a certain class of functional differential equations with perturbations conditions are given such that there exist solutions which converge to solutions of the equations without perturbation.