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...

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x\left(t\right)-L\left(x_t\right)]=A[x\left(t\right)-L\left(x_t\right)]+G\left(x_t\right)+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{-\infty }^t{(t-s)}^{\alpha -1}\left({\int }_{-\infty }^{s}a(s-\xi )x\left(\xi \right)d\xi \right)ds+f\left(t\right)$, ($\alpha >0$) with the periodic condition $x\left(0\right)=x\left(2\pi \right)$, where $a\in L^1\left({ℝ}_{+}\right)$ . Our approach is based on the R-boundedness of linear operators $L^p$-multipliers and UMD-spaces.

We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral...

We use a modification of Krasnoselskii’s fixed point theorem due to Burton (see [Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9 (2002), 181–190], Theorem 3) to show that the totally nonlinear neutral differential equation with variable delay $$x^{\text{'}}\left(t\right)=-a\left(t\right)h\left(x\left(t\right)\right)+c\left(t\right)x^{\text{'}}(t-g\left(t\right))Q^{\text{'}}\left(x(t-g\left(t\right))\right)+G(t,x\left(t\right),x(t-g\left(t\right))),$$ has a periodic solution. We invert this equation to construct a fixed point mapping expressed as a sum of two mappings such that one is compact and the other is a large contraction. We show that the mapping fits...

In this paper we discuss the existence of mild and strong solutions of abstract nonlinear mixed functional integrodifferential equation with nonlocal condition by using Sadovskii’s fixed point theorem and theory of fractional power of operators.