Existence of solutions of neutral functional integrodifferential equation in Banach spaces

Existence Results for Abstract Degenerate Neutral Functional Differential Equations

“…Proceeding now as in the first part of the proof, we can show that k 2 (·) is a contraction for k large and that there exists a unique fixed point u 2 ∈ C([r 1 − r, 2r 1 ]; X ) of 2 (·). To finish, we note that the function u : [−r, 2r 1 ] → X defined by u(t) = u 1 (t) for t ∈ [−r, r 1 ] and u(t) = u 2 (t) for t ∈ [r 1 , 2r 1 ] is a mild solution of (1.1)-(1.2) on [−r, 2r 1 ].…”

“…(1.4) 330 E. Hernández and K. Balachandran [2] Datko [9] and Adimy and Ezzinbi [1], where the case in which A is a Hille-Yosida type operator is studied.…”

Existence of solutions for abstract neutral integro-differential equations with unbounded delay

“…It is convenient to note that in the papers [4], [3], [6], [7], [13], among several other works, an alternative assumption (which is really a particular case of condition (H g )) has been used to treat neutral systems. In these works it is assumed that the semigroup (T (t)) t 0 is compact and the set of operators {AT (t) : t ∈ (0, b]} is bounded.…”

Existence results for abstract neutral functional differential equations

“…We mention the papers [2,3,4,5,6] (among several works) where an alternative assumption has been used to treat some abstract neutral systems. In these works, is the generator of a 0 -semigroup, and it is assumed that the set { ( ) : ∈ (0, ]}, ( > 0), is bounded in the operator topology.…”