Second order nonlinear evolution equations in Banach spaces
The authors consider the Cauchy problem \((*)\) \(u''(t) \in A(t) u(t)\), \(t \geq 0\) a.e., \(u(0) = x\), \(\sup \{|u(t) |: t \geq 0\} < \infty\), where \(X\) is a uniformly smooth Banach space, with strongly monotone duality mapping, the solution \(u(t)\) is \(X\)-valued and for each \(t\), \(A(t)\) is a nonlinear multivalued and unbounded \(m\)-accreative operator. The main result of the paper is an existence and uniqueness theorem for the problem \((*)\) which includes some previous results (like for example: \textit{V. Barbu} [J. Fac. Sci., Univ. Tokyo, Sect. 1A 19, 295-319 (1972; Zbl 0256.47052)], \textit{H. Brezis} [Israel J. Math., 12, 51-60 (1972; Zbl 0248.47026)] and \textit{E. I. Poffald} and \textit{S. Reich} [J. Math. Anal. Appl. 113, 514-543 (1986; Zbl 0599.34078)].
nonlinear multivalued and unbounded \(m\)-accreative operator
scientific article; zbMATH DE number 811219