an:02102565
Zbl 1083.34045
Liang, Jin; Liu, James; Xiao, Ti-Jun
Nonlocal Cauchy problems governed by compact operator families
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 57, No. 2, 183-189 (2004).
00107393
2004
j
34G20 47D06
nonlocal Cauchy problem; mild solution; compact operator families; equicontinuous family of functions; integrodifferential equation
Let \(A\) be the infinitesimal generator of a compact semigroup of linear operators on a Banach space \(X\). The authors establish the existence of mild solutions to the nonlocal Cauchy problem
\[
u'(t)=Au(t)+f(t,u(t)), \quad t\in[t_0,t_0+T], \quad u(t_0)+g(u)=u_0,
\]
under some conditions on \(f\) and \(g\), where \(f:[t_0,t_0+T]\times X\to X\) and \(g:C([t_0,t_0+T];X)\to X\) are given functions. They assume a Lipschitz condition on \(f\) with respect to \(u\), but they do not require any compactness assumption on \(g\), opposed to \textit{S. Aizicovici} and \textit{M. McKibben} [Nonlinear Anal., Theory Methods Appl. 39, No. 5(A), 649--668 (2000; Zbl 0954.34055)] and \textit{L. Byszewski} and \textit{H. Akca} [Nonlinear Anal., Theory Methods Appl. 34, No. 1, 65--72 (1998; Zbl 0934.34068)], where the authors assume a compactness property for \(g\), but do not require any Lipschitz condition on \(f\).
Behzad Djafari-Rouhani (El-Paso)
Zbl 0954.34055; Zbl 0934.34068