### A boundary value problem for non-linear differential equations with a retarded argument

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Conditions for the existence and uniqueness of a solution of the Cauchy problem $${u}^{\text{'}}\left(t\right)=p\left(t\right)u\left(\tau \left(t\right)\right)+q\left(t\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}u\left(a\right)=c\phantom{\rule{0.166667em}{0ex}},$$ established in [2], are formulated more precisely and refined for the special case, where the function $\tau $ maps the interval $]a,b[$ into some subinterval $[{\tau}_{0},{\tau}_{1}]\subseteq [a,b]$, which can be degenerated to a point.

This paper deals with the periodic boundary value problem for nonlinear impulsive functional differential equation $$\left\{\begin{array}{c}{x}^{\text{'}}\left(t\right)=f(t,x\left(t\right),x\left({\alpha}_{1}\left(t\right)\right),\cdots ,x\left({\alpha}_{n}\left(t\right)\right))\text{for}\phantom{\rule{4.0pt}{0ex}}\text{a.e.}\phantom{\rule{4pt}{0ex}}t\in [0,T],\Delta x\left({t}_{k}\right)={I}_{k}\left(x\left({t}_{k}\right)\right),\phantom{\rule{4pt}{0ex}}k=1,\cdots ,m,x\left(0\right)=x\left(T\right).\hfill \end{array}\right.$$ We first present a survey and then obtain new sufficient conditions for the existence of at least one solution by using Mawhin’s continuation theorem. Examples are presented to illustrate the main results.

The existence of at least two solutions for nonlinear equations close to semilinear equations at resonance is obtained by the degree theory methods. The same equations have no solutions if one slightly changes the right-hand side. The abstract result is applied to boundary value problems with specific nonlinearities.

We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the...

MSC 2010: 34A37, 34B15, 26A33, 34C25, 34K37In this paper we prove the existence of solutions for fractional impulsive differential equations with antiperiodic boundary condition in Banach spaces. The results are obtained by using fractional calculus' techniques and the fixed point theorems.

In the paper [13] we proved a fixed point theorem for an operator $\mathcal{A}$, which satisfies a generalized Lipschitz condition with respect to a linear bounded operator $A$, that is: $$m(\mathcal{A}x-\mathcal{A}y)\prec Am(x-y).$$ The purpose of this paper is to show that the results obtained in [13], [14] can be extended to a nonlinear operator $A$.

In the present paper we study the approximate solutions of a certain difference-differential equation under the given initial conditions. The well known Gronwall-Bellman integral inequality is used to establish the results. Applications to a Volterra type difference-integral equation are also given.