Existence results on semiinfinite intervals for functional-differential and integrodifferential inclusions in Banach spaces with nonlocal conditions (Q2777551)

The authors first consider the existence of mild solutions for the problem NEWLINE\[NEWLINEy'- Ay\in F(t,y_t)\text{ for }t\in J= [0,\infty),\;y(t)+ (f(y_{t_1},\dots, y_{t_p}))(t)= \phi(t)\text{ for }t\in [-r,0],NEWLINE\]NEWLINE where \(A\) is a linear closed operator from a dense subspace \(D(A)\) of the real Banach space \(E\) into \(E,F: J\times C([- r,0],E)\to 2^E\) is bounded, closed and convex valued, \(y_t(\theta)= y(t+\theta)\) for \((t,\theta)\in N\times [-r,0]\), \(f:[C([- r,0], E)]^p\to C([-r, 0], E)\), \(p\in\mathbb{N}\), \(0< t_1<\cdots< t_p<\infty\) and \(\phi\in C([-r,0], E)\). The existence of a mild solution is proven under the additional hypotheses that \(A\) is the infinitesimal generator of a semigroup of compact bounded linear operators \(T(t)\), \(t\geq 0\) in \(E\), \(F\) is measurable with respect to its first variable, upper semicontinuous with respect to its second variable, has integrable selections and satisfies a boundedness condition and \(f\) is completely continuous and is bounded. Then, the following problem is considered NEWLINE\[NEWLINE\begin{multlined} y'- Ay\in \int^t_0 K(t,s) F(s,y_s) ds\quad\text{for }t\in J,\\ y(t)+ (f(y_{t_1},\dots, y_{t_p}))(t)= \phi(t)\quad\text{for }t\in [-r,0].\end{multlined}NEWLINE\]NEWLINE The authors prove the existence of a mild solution for this problem under the conditions stated above, plus the additional assumptions that \(K:\{(t,s)\in J\times J: t\geq s\}\to \mathbb{R}\), for each \(t\in J_m= [0,m]\), \(m\in\mathbb{N}\), \(K(t,s)\) is measurable on \([0,t]\) and satisfies a boundedness condition on \(J_m\), and the map \(t\to K(t,s)\) from \(J\) into \(L^\infty(J_m,\mathbb{R})\) is continuous. Both proofs are accomplished by applying a fixed-point theorem due to \textit{T.-W. Ma} [Dissertations Math., Warszawa 92 (1972)].