On Bounded Weak and Strong Solutions of Non Linear Differential Equations with and without Delay in Banach Spaces - Gomaa, Adel Mahmoud (MATHEMATICA SCANDINAVICA, 2013)
Assume that E is a Banach space, B_r={xE:xr} and C([-d,0],B_r) is the Banach space of continuous functions from [-d,0] into B_r. Consider f:R^+× E→ E; f^d:[0,T]× C([-d,0],B_r)→ E; for each t[0,T] the mapping θ_tC([-d,0],B_r) is defined by θ_tx(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 x(t)=A(t)x+f(t,x),tR^+, 26767 and we prove that, with certain conditions, the differential equation with delay 26767 x(t)=L(t)x(t)+f^d(t,θ_tx),ift[0,T] (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.