We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation x(t)=L(t)x(t)+f(t,x(t)), t R where {L(t) in R}$ is a family of linear operators from a Banach space E into itself and f R E to E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a<b and d>0, we let C([-d,0],E) be the Banach space of continuous functions from [-d,0] into E and f^d [a,b] C([-d,0],E) E. Let L: [a,b] to L(E) be a strongly measurable and Bochner integrable operator on [a,b] and for t in [a,b] define tau_tx(s)=x(t+s) for each s in[-d,0]. We prove that, under certain conditions, the differential equation with delay x(t)=L(t)x(t)+f^d(t,tau_tx) if t in [a,b], Q has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too., Adel Mahmoud Gomaa., and Seznam literatury
The positive solution is studied for a (k, n - k) conjugate boundary value problem. The nonlinear term is allowed to be singular with respect to both the time and space variables. By applying the approximation theorem for completely continuous operators and the Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type, an existence theorem for a positive solution is established.