Existence and asymptotic properties of solutions of an integro-differential evolution equation with nonlocal conditions on infinite interval
This good paper studies the existence, regularity and asymptotic properties of solutions for the following semilinear neutral integrodifferential evolution equation with nonlocal condition in Banach space \(X\). \[ \begin{cases} \frac{d}{dt}\left[x(t) + \int_0^t N(t - s)x(s)ds\right] = -A x(t) + \int_0^t \gamma(t - s)x(s)ds + F(t, x(r(t)) , t \geq 0,\\ x(0) = x_0 + H(\sigma(x), x), \end{cases} \] where \(N(t)\in \mathcal{L}(X)\) in the neutral term, the operator \(-A\) generates an analytic semigroup on \(X\), \(\gamma(t)\) is a family of closed linear operators in \(X\) with domain \(D(\gamma(t))\subseteq D(-A)\) for every \(t \geq 0.\) The function \(r(\cdot) : [0, \infty) \to [0, \infty)\) is continuous with \(0 \leq r(t) \leq t\) and it is regarded as a delay function. One of objectives is to develop and extend the results achieved in [\textit{J. Zhu} and \textit{X. Fu}, J. Integral Equations Appl. 32, No. 2, 239--258 (2020; Zbl 1465.45011)]. The obtained results in this paper develop and extend the recently results for this kind of evolution equation. Denote by \(X_\alpha\) the a Banach space for each \(0 < \alpha \leq 1\) consist all of functions \(x\in D(A^\alpha) \) with the norm \(\|x_\alpha\| = \|A^\alpha x\| \). The authors discuss the existence of mild solutions in the space \(X_\alpha\), not merely in the state space \(X.\) Thus the results are still valid when the nonlinear function \(F(\cdot, \cdot)\) involves derivatives of spacial variables for this case \(F\) is defined on \([0,+\infty ) \times X_\alpha,\) not on \( [0, +\infty) \times X \)(which makes the discussion more complicated). So, in this way, the achieved existence conclusions may be more applicable. The regularity and asymptotic properties of solutions are discussed in the space \(X.\)
scientific article; zbMATH DE number 7633380
scientific article