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- W2086095236 abstract "Let ${bf y}(t,{bf x},{bf f})$ denote the solution of the Cauchy problem [ {bf y}'(t) + int_0^t {[d + a(t - s)]{bf Ly}(s)ds = {bf f}(t),quad t geqq 0,quad {bf y}(0) = {bf x},} ] where $d geqq 0$ and ${bf L}$ is a self-adjoint densely defined linear operator on a Hilbert space $mathcal{H}$ with ${bf L} geqq lambda _1 {bf I}$. Let ${bf U}(t){bf x} = {bf y}(t,{bf x},{bf 0}),{bf V} = {bf U}'$. By analyzing a related scalar equation with parameter, we find sufficient conditions on the kernel a in order that $int_0^infty {| {{bf V}(t)L^{ - gamma } } |} dt < infty (gamma > 0)$. These results and certain resolvent formulas can be used to study the asymptotic behavior of the solution ${bf y}(t,{bf x},{bf f})$ as $t to infty $. An application to a semilinear integro-partial differential equation is presented." @default.
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- W2086095236 date "1982-05-01" @default.
- W2086095236 modified "2023-09-26" @default.
- W2086095236 title "Resolvent Formulas for a Volterra Equation in Hilbert Space" @default.
- W2086095236 doi "https://doi.org/10.1137/0513032" @default.
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