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Existence results for partial neutral integro-differential equation with unbounded delay. (English) Zbl 1168.47306

From the text: We study the existence and regularity of mild solution for a class of partial neutral integro-differential equation with unbounded delay of the form \[ \frac{d}{dt}D(t,u_ t)=AD(t,u_ t) +\int^ t_ 0B(t-s)D(s,u_ s)\,ds+g(t,u_ t),\tag{1} \] on \(t\in[0,a]\), with initial condition \[ x_ {\sigma}=\phi\in\beta,\tag{2} \] where \(A, B(t): X\to X\) are closed linear operators, \(t\geq 0\), \(X\) is a Banach space, the history \(x_ t: (-\infty,0]\to X\) is defined as \(x_ t(\theta):= x(t+\theta)\) and belongs to an abstract space \(\beta\), \(f,g: [0,a]\times \beta \to X\) are appropriately defined functions, and \(D(t,\phi):= \phi(0)+f(t,\phi)\) for \((t,\phi)\in [0,a]\times \beta\). This question is connected here with that of the existence of a linear bounded resolvent of the operator \((R(t))_ {t\geq 0}\) on \(X\) for the Cauchy problem \[ x'(t)=Ax(t)+\int^ t_ 0 B(t-s)x(s)\,ds,\quad t\geq 0, \]
\[ x(0)=x_ 0\in X. \] By a mild solution of (1),(2) on \([0,b]\) is meant a function \(u: (-\infty,b]\to X\), \(0<b\leq a\), \(u\in C([0,b], X)\) satisfying \(u_ 0=\phi\) and \[ u(t)=R(t)(\phi(0)+f(0,\phi))-f(t,u_ t)+\int_ 0^ tR(t-s)g(s,u_ s)\, ds, \] for \(t\in [0,b]\).

MSC:

47N20 Applications of operator theory to differential and integral equations
47D09 Operator sine and cosine functions and higher-order Cauchy problems
34K40 Neutral functional-differential equations
34G20 Nonlinear differential equations in abstract spaces
34K30 Functional-differential equations in abstract spaces
45K05 Integro-partial differential equations
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References:

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