Positive solutions of first order boundary value problems with nonlinear nonlocal boundary conditions

We consider the existence of positive solutions of the nonlinear first order problem with a nonlinear nonlocal boundary condition given by x'(t) = r(t)x(t) + Sigma(m)(i=1) f(i) (t,x(t)), t epsilon [0,1] lambda x(0) = x(1) + Sigma(n)(i=1) Lambda j(Tj,x(T-j)), T-j epsilon [0, 1], where r : [0, 1] -> [0, infinity) is continuous, the nonlocal points satisfy 0 <= T1 < T2 < < T-n < 1, the nonlinear functions f(i) and Lambda j are continuous mappings from [0,1] x [0, infinity) [0, infinity) for i = 1,2,..., m and j = 1, 2,..., n respectively, and lambda > 1 is a positive parameter. The Leray-Schauder theorem and Leggett-Williams fixed point theorem were used to prove our results.