Existence Results to Some Integrodifferential Equations

Toka Diagana
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Toka Diagana: Howard University, Department of Mathematics

Chapter Chapter 8 in Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, 2013, pp 209-219 from Springer

Abstract: Abstract In this chapter we study the existence of asymptotically almost automorphic mild solutions to the abstract partial neutral integrodifferential equation 8.1 $$\displaystyle\begin{array}{rcl} \frac{d} {dt}D(t,u_{t})& =& AD(t,u_{t}) +\int _{ 0}^{t}B(t - s)D(s,u_{ s})ds + g(t,u_{t}),\ t \in [\sigma,\sigma +a),{}\end{array}$$ 8.2 $$\displaystyle\begin{array}{rcl} u_{\sigma }& =& \varphi \in \mathcal{B},{}\end{array}$$ where $$A,B(t): D(A) \subset \mathcal{X} \rightarrow \mathcal{X}$$ are densely defined closed linear operators with a common domain D(A), which is independent of t; the history $$\displaystyle{u_{t}: (-\infty,0] \rightarrow \mathcal{X},\ \ \mbox{ defined by}\ \ u_{t}(\theta ):= u(t+\theta )}$$ belongs to an abstract phase space $$\mathcal{B}$$ defined axiomatically, f, g are functions subject to some additional conditions, and $$\displaystyle{D(t,\varphi ) =\varphi (0) + f(t,\varphi ).}$$ For that, we will make extensive use of the concept of compact asymptotically almost automorphy and the so-called resolvent of operators.

Keywords: Mild Solution; Partial Integrodifferential Equations; Common Domain; Automorphic Solutions; Nonautonomous Version (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-00849-3_8

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DOI: 10.1007/978-3-319-00849-3_8

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