A Class of Reaction-Diffusion Systems with Nonlocal Initial Conditions

Abstract

Abstract. We consider an abstract nonlinear multi-valued reaction-diffusion system with delay and, using some compactness arguments coupled with metric fixed point techniques, we prove some sufficient conditions for the existence of at least one C 0 -solution.

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

“…So, α has at least one fixed point (u α , h α ) which equal that the approximate problem (6) has at least one C 0 -solution (u α , v α ). Secondly, by using the same arguments in [19,20], we can show that: for each real number α ∈ (0, 1), the set of the C 0 -solutions {(u α , v α ), α ∈ (0, 1)} is compact and (1). Thirdly, by virtue of [11], we get that…”

“…We draw our results depending on the compactness method presented in [19,20]. In this study, we prove the approximate problem (6) has at least C 0 -solutions (1) if and only if the C 0 -solutions set {(u α , v α ), 0 ≤ α ≤ 1} is compact and closed.…”

“…(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.…”