Solvability of a neutral differential equation with deviated argument

Abstract

The paper contains theorem on the existence and asymptotic characterization of solutions of a differential equation of neutral type with deviated argument. The mentioned differential equation admits both delayed and advanced arguments. In our considerations we use the technique linking measures of noncompactness with the classical Schauder fixed point principle.

Continuous Dependence of the Solutions of Nonlinear Integral Quadratic Volterra Equation on the Parameter

“…From among papers cited here, majority of them was concerned with different kinds of equations from (22); see [5][6][7]13]. The authors of papers [8,9,11,13] examined equations similar to (22); however, their considerations were conducted in the Banach space (R + ) and therefore their assumptions were too restrictive. In the paper [12] we investigated similar equations to (22) with different kinds of assumptions than these given below.…”

“…[1][2][3][4]). Up to this time, a lot of papers have appeared on those equations [1][2][3][4][5][6][7][8][9]; however, to the best of our knowledge, there are no papers on continuous dependence of solutions of this kind of equations on parameter.…”

On existence of solutions of a neutral differential equation with deviating argument

“…The aim of the present paper is an improvement to the results contained in papers [4,11], concerning the existence and asymptotic behaviour of solutions of the following neutral differential equation with deviating argument…”

“…The problem (1)- (2) was considered in [4,11] (see also [1,2,3,6,7,8,9,12,13] under strong assumptions. Authors obtained their results by using two-component measure of noncompactness in the Banach space C p (R + ) consisting of all continuous functions on R + and tempered by p.…”

“…Let us observe that the function H(t) and h(t) do not satisfy the condition H(t) ≤ m exp(h(t)), m > 0 from [11, Theorem 1] …”