## 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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