An \(L_2\)-approach to second-order nonlinear functional evolutions involving \(m\)-accretive operators in Banach spaces.
Let \(X\) be a real, uniformly smooth Banach space \(X\) with a strongly monotone duality map. The paper deals with the solvability of the incomplete Cauchy problem \(u^{\prime \prime}(t)\in A(t)u(t)+G(t,u_t)\), \(t\geq 0\); \(u(0)=x\), \(u_0=\phi\). Here, \(A(t)\) is an \(m\)-accretive (possibly multivalued) operator in \(X\), \(G(t,\psi)\) is a Lipschitz continuous operator on \([0,\infty )\times L^2((-r,0);X)\), \(\phi \in C^1([-r,0];X)\) with \(\phi (0)=x\in D(A(0))\), and \(u_t\) is defined by \(u_t(s):=u(t+s)\), \(s\in [-r,0]\) (\(r>0\) is a fixed contant which is called delay). This paper continues the study of the case \(G=0\) previously investigated by \textit{X.-M. Xue, G. Song} and {J.. Ma} [J. Math. Anal. Appl. 194, No. 1, 1--13 (1995; Zbl 0835.34084)].
scientific article; zbMATH DE number 1832845