Semilinear Volterra integrodifferential equations with nonlocal initial conditions

Abstract

We establish the global existence of mild solutions to a class of nonlocal Cauchy problems associated with semilinear Volterra integrodifferential equations in a Banach space.

Abstract semilinear stochastic Itó‐Volterraintegrodifferential equations

“…It is known that A is a positive definite, self-adjoint operator in H (see [3]). Moreover, using the properties of A and (H13), it follows from [25] and [29, …”

Controllability of McKean–Vlasov Stochastic Integrodifferential Evolution Equation in Hilbert Spaces

“…The deterministic related equations coupled with a classic initial condition have been studied extensively both when A is linear and when A is nonlinear (we refer to [26] and the references therein). Since then, many authors have continued this work in several directions and established existence theories with nonlocal initial conditions for first-order Volterra integral and integrodifferential equations [3,7] and differential inclusions [1]. Recently in [9], the authors further extended the existence results for semilinear stochastic delay evolution equations with nonlocal conditions.…”

Functional integro‐differential stochastic evolution equations inHilbert space

“…Byszewski [13] introduced nonlocal initial conditions into such abstract initial-value problems and argued that the corresponding models more accurately describe the phenomena since more information was taken into account at the onset of the experiment, thereby reducing the ill effects incurred by a single (possibly erroneous) initial measurement. Since then, many authors have continued this work in several directions and established existence theories for first-order nonlinear evolution equations [2,4,29], second-order equations [7], delay equations [7,28], Volterra integral and integro-differential equations [5,25], and differential inclusions [1]. Concrete nonlocal parabolic and elliptic partial (integro-) differential equations arising in the mathematical modeling of various physical, biological, and ecological phenomena, as well as a discussion of the advantages of replacing the classical initial condition with a more general functional, can be found in [13,21] and the references contained therein.…”