Integral Solutions to a Class of Nonlocal Evolution Equations

Abstract

We study the existence of integral solutions to a class of nonlinear evolution equations of the form [Formula: see text] where A : D(A) ⊆ X → 2X is an m-accretive operator on a Banach space X, and f : [0, T] × X → X and [Formula: see text] are given functions. We obtain sufficient conditions for this problem to have a unique integral solution.

The existence and uniqueness of integral solutions to some nonlinear reaction–diffusion system with nonlocal retarded initial conditions

“…Lemma 5.4: Let us take Kρ ⊆ C b ([−τ , +∞); X) as the closed ball with a centre 0 and radiusρ = d/ω for each d ≥ 0 which is given by (15) and…”

“…(1) Here, τ 0 = 0 ≤ τ 1 ≤ τ 2 ≤ · · · ≤ τ n = τ to be fixed, f : The system (1) has been addressed by several authors, and they analyzed various cases, such as Alam and Alam [1], Burlică [2], Burlică and Roşu [3][4][5], Diaz and Vrabie [6], Meknani and Zhang [7], Neucala and Vrabie [8], Roşu [9,10] been studied by Burlică and Roşu [11], Vrabie [12][13][14] , Garcia and Reich [15] and Paicu and Vrabie [31]. Otherwise, the result for nonlinear evolution inclusions with nonlocal retarded initial conditions was investigated by Vrabie [16] and references therein.…”

Well-posedness of a nonlinear evolution equation arising in growing cell population

“…Since we are assuming that 0 = T K (0) and N satisfies condition (H4), we can apply Lemma 3.1 of [16] to obtain that u ∈ B r (0) then (u) = u and thus we may conclude that Equation (23) has a unique mild solution.…”

“…On the other hand, from the biological point of view it seems quite natural that if the density of the population of the cell cycle length l at mitosis is zero, i.e, f (t, l, l) = 0 , then the transition for the distribution of mothers cycle length to daughters cycle length should be also zero, this is mathematically described by assuming that the transition operator K satisfies that K(0) = 0. We have studied this case in Corollary 1 and thus we can replace the condition (H3) by assumptions (H3) and (H4) which allow us to consider more general functions describing the rate of cell mortality (see [16]). …”

Domains of accretive operators in Banach spaces

“…On the other hand, in quite a few applications [5,11,[14][15][16] it suffices to ascertain that the closure of the domain of an m-accretive operator is convex. For instance, in [11, theorem 3.5] this fact is used in the study of the existence of integral solutions to non-local evolution equations of the form…”