In [4] W. Li and S.S. Cheng prove a Picard type existence and uniqueness theorem for iterative differential equations of the form y'(x) = f(x,y(h(x)+g(y(x)))). In this article I show some analogue of this result for a larger class of functions f (also discontinuous), in which a unique differentiable solution of considered Cauchy's problem is obtained.

In this paper, some new fixed point theorems concerning the nonlinear alternative of Leray-Schauder type are proved in a Banach algebra. Applications are given to nonlinear functional integral equations in Banach algebras for proving the existence results. Our results of this paper complement the results that appear in Granas et. al. (Granas, A., Guenther, R. B. and Lee, J. W., Some existence principles in the Caratherodony theory of nonlinear differential system, J. Math. Pures Appl. 70 (1991),...

The aim of this paper is to study problems of the form: $in{f}_{(u\in V)}J\left(u\right)$ with $J\left(u\right):={\int }_0^{1}L(s,y_u\left(s\right),u\left(s\right))\mathrm{d}s+g\left(y_u\left(1\right)\right)$ where V is a set of admissible controls and yu is the solution of the Cauchy problem: $\dot{x}\left(t\right)=\langle f(.,x(.)),{\nu }_t\rangle +u\left(t\right),t\in (0,1)$, $x\left(0\right)=x_0$ and each ${\nu }_t$ is a nonnegative measure with support in [0,t]. After studying the Cauchy problem, we establish existence of minimizers, optimality conditions (in particular in the form of a nonlocal version of the Pontryagin principle) and prove some regularity results. We also consider the more general case where the control also enters the dynamics...

We consider a neutral type operator differential inclusion and apply the topological degree theory for condensing multivalued maps to justify the question of existence of its periodic solution. By using the averaging method, we apply the abstract result to an inclusion with a small parameter. As example, we consider a delay control system with the distributed control.

The authors establish some new sufficient conditions under which all solutions of a certain class of nonlinear neutral delay differential equations of the third order are stable, bounded, and square integrable. Illustrative examples are given to demonstrate the main results.

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$\dot{x}\left(t\right)-a\left(t\right)\dot{x}\left(g\left(t\right)\right)+b\left(t\right)x\left(h\left(t\right)\right)=0,$$ where $$\left|a\right(t\left)\right|<1,b\left(t\right)\ge 0,h\left(t\right)\le t,g\left(t\right)\le t,$$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

In this work, we propose the Shannon wavelets approximation for the numerical solution of a class of integro-differential equations which describe the charged particle motion for certain configurations of oscillating magnetic fields. We show that using the Galerkin method and the connection coefficients of the Shannon wavelets, the problem is transformed to an infinite algebraic system, which can be solved by fixing a finite scale of approximation. The error analysis of the method is also investigated....