Kozak, Monika A fundamental solution of a second-order differential equation in a Banach space. (English) Zbl 0855.34073 Zesz. Nauk. Uniw. Jagiell. 1169, Univ. Iagell. Acta Math. 32, 275-289 (1995). 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) Cited in 2 ReviewsCited in 39 Documents 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 Keywords:existence; uniqueness; abstract second order problem; Banach space; semilinear problems PDFBibTeX XMLCite \textit{M. Kozak}, Zesz. Nauk. Uniw. Jagiell., Univ. Iagell. Acta Math. 1169(32), 275--289 (1995; Zbl 0855.34073)