An existence result for second order functional differential inclusions in Banach spaces (Q2747430)

The author considers the existence of solutions to the second-order functional-differential inclusion on an infinite interval NEWLINE\[NEWLINEy''\in F(t, y_t),\quad t\in J= [0,\infty),\quad y_0= \phi,\quad y'(0)= \eta,NEWLINE\]NEWLINE in a real Banach space \(E\). Specifically, \(F(t,u): J\times C(J_0, E)\to 2^E\) is assumed to be nonempty, bounded, closed and convex-valued, measurable in \(t\), upper semicontinuous in \(u\) and satisfies a certain boundedness condition, with \(y_t(\theta)= y(t+\theta)\) for \(t\in J\), \(\theta\in J_0\) and \(y\) continuous on \([-r,\infty)\), \(\phi\in C(J_0, E)\), \(\eta\in E\), \(J_0= [-r,0]\), among other conditions. The proof is accomplished by defining a certain multivalued map \(N\) on the Fréchet space \(C([-r,\infty), E)\) and finding a fixed-point of \(N\) by applying a theorem due to Ma (Ma's theorem is an extension of Schaefer's well-known fixed-point theorem to multivalued maps between locally convex topological spaces).