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A fundamental solution of a second-order differential equation in a Banach space. (English) Zbl 0855.34073

The author considers existence and uniqueness results for the abstract second order problem \[ {d^2u \over dt^2} = A(t) u + f,\;t \in (0,T], \quad u(0) = u_0,\;{du \over dt} (0) = u_1. \] Here for each \(t \in [0,T]\), \(A(t)\) is a closed densely defined linear operator from \(D(A(t)) = D(0) \subset X\) to a real Banach space \(X\), \(f : \mathbb{R} \mapsto X\), and both \(u_0\), \(u_1 \in X\). Under the assumption that the homogeneous problem has a fundamental solution, existence of a unique solution is first proved for the above problem when \(f = f(t)\) is Lipschitz continuous and \(X\) is a reflexive Banach space. Then for a nonlinear Lipschitz continuous forcing term, that is for \(f = f(t,u)\) and \(u_0\), \(u_1 \in D(A)\), the Banach contraction mapping theorem is applied to discuss existential results. Finally the technique is extended to cover semilinear problems with \(f\) depending on \(t,u\) and \({du \over dt}\).
Reviewer: A.K.Pani (Bombay)

MSC:

34G20 Nonlinear differential equations in abstract spaces
34G10 Linear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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