On a class of abstract neutral functional differential equations

Existence results for some partial integrodifferential equations with nonlocal conditions

“…In the theory developed in [22,33], the internal energy and the heat flux are described as functionals of u and u x . The following system [7,28,23,12] has been frequently used to describe this phenomena, (1.2) ∂ ∂t u(t, x) + Here, Ω ⊂ R n is open, bounded and has smooth boundary, (t; x) ∈ [0; ∞) × Ω, u(t; x) represents the temperature in x at the time t, c is a physical constant and k i : R → R, i = 1, 2, are the internal energy and the heat flux relaxation respectively. By assuming that the solution u(.)…”

Decay integral solutions for neutral fractional differential equations with infinite delays

“…In most cases, the question of existence of integral solutions has been considered. Let us refer to some recent works, without being exhaustive with quotations: for˛D 1, we cite [1,2] for neutral functional differential equations with finite delays and [3] for a case of unbounded delays. Furthermore, in the case˛2 .0, 1/, we refer the reader to [4] for equation with finite delays and [5,6] for infinite delay ones.…”

“…Furthermore, in the case˛2 .0, 1/, we refer the reader to [4] for equation with finite delays and [5,6] for infinite delay ones. Although problem (1)- (2) has been an increasingly interesting subject, no attempt has been made to find its decay solutions with explicit decay rates, up to our knowledge. This is the main motivation for our study.…”

“…They noticed that this method helps in overcoming difficulties that appeared while using the Lyapunov method to deal with differential equations containing unbounded delays or unbounded parts. We make use of this idea to prove the existence of decay integral solutions to (1)- (2), but now with the question of determining decay rate of solutions. For this purpose, we will set up some features:…”

“…Our paper is organized as follows. In the next section, we recall some notions and facts concerning the fractional calculus, including the representation for integral solutions of (1)- (2). In addition, we construct a regular MNC on BC.0, 1; X/ (the space of continuous bounded functions from .0, 1/ to X) and give a fixed point principle, which will be used for our purpose.…”