Resumen de On Bounded Weak and Strong Solutions of Non Linear Differential Equations with and without Delay in Banach Spaces

Adel Mahmoud Gomaa

• Assume that E is a Banach space, Br={x∈E:∥x∥≤r} and C([−d,0],Br) is the Banach space of continuous functions from [−d,0] into Br. Consider f:R+×E→E; fd:[0,T]×C([−d,0],Br)→E; for each t∈[0,T] the mapping θt∈C([−d,0],Br) is defined by θtx(s)=x(t+s), s∈[−d,0] and let A(t) be a linear operator from E into itself. In this paper we give existence theorems for bounded weak and strong solutions of the nonlinear differential equation 26767 \dot{x}(t)=A(t)x+f(t,x),\qquad t\in \mathbf{R}^+, 26767 and we prove that, with certain conditions, the differential equation with delay 26767 \dot{x}(t)=L(t)x(t)+f^{d}(t,\theta_{t}x),\qquad \text{if}\quad t\in [0,T] \qquad\qquad(\mathrm{Q}) 26767 has at least one weak solution where L(t) is a linear operator from E into E. Moreover, under suitable assumptions, the problem (Q) has a solution. Furthermore under a generalization of the compactness assumptions, we show that (Q) has a solution too.