Global existence for semilinear evolution equations with nonlocal conditions. (English) Zbl 0884.34069

Ntouyas, S. K.; Tsamatos, P. Ch.

J. Math. Anal. Appl. 210, No. 2, 679-687 (1997).

The authors apply Schauder’s fixed-point theory to differential equations in Banach spaces, \[ x'(t)= Ax(t)+ f(t,x(t)) \] with nonlocal initial conditions of the type \[ x(0)+ g(x)= x_0. \] Here, \(t\in[0, b]\), and some general assumptions on \(g: C([0,b], X)\to X\) and \(f:[0, b]\times X\to X\) are given which guarantee the existence of a solution \(x\in C([0,b], X)\).

Reviewer: R.Illner (Victoria)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34K30	Functional-differential equations in abstract spaces

Keywords:

differential equations in Banach spaces; initial conditions