On the existence of bounded solutions of differential equations in Banach spaces. (English) Zbl 0606.34039

We give sufficient conditions for the existence of bounded solutions of the differential equation \(y'=f(t,y)\), \(y(0)=x_ 0\), on the half-line \(t\geq 0\). Here f is a function with values in a Banach-space satisfying some conditions expressed in terms of an axiomatic measure of noncompactness \(\mu\). The proof of our theorem is suggested by the paper of A. Stokes [Proc. Natl. Acad. Sci. USA 45, 231-235 (1959; Zbl 0086.073)] concerning finite dimensional vector differential equations.


34G20 Nonlinear differential equations in abstract spaces
34C11 Growth and boundedness of solutions to ordinary differential equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.


Zbl 0086.073
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