Ordinary Differential Equations With Integral Boundary Of Positive Solutions

The existence of positive solutions are considered to the boundary value problemsandwhere constantsλ, a, b and functions f, g0, g1 satisfy the following conditions:(H1) 1 -λ- aλ~2≥0, and one of the following two conditions is satisfied(H1-1) a > 0;(H1-2) 1 + a + aλ< 0.(H2) 1 +λ+ bλ~2≥0, and one of the following two conditions is satisfied(H2-1) b - 1 - bλ> 0;(H2-2) b < 0.(H3) k = 1 + a - b + abλ~2 + (a + b)λ> 0.(H4) g0,g1 : [0,1]â†'(-∞,+∞) are continuous, andwhere(H5) f : [0,1]×[0,∞)â†'[0,∞) is continuous and nonnegative.Let whereγ0(t) andγ1(t) are given by :when conditions (H1-1) and (H2-1) are satisfied,when conditions (H1-1) and (H2-2) are satisfied,when conditions (H1-2) and (H2-1) are satisfied,when conditions (H1-2) and (H2-2) are satisfied,The main results of this paper as follows:Assume that (H1)–(H5) hold, and there exist L1 > L0 > 0 such that(i) If conditions (H1-1) and (H2-1) hold, or conditions (H1-2) and (H2-2) hold, thenthe boundary value problems (1.1) has at least one positive solution.(ii) If conditions (H1-1) and (H2-2) hold, or conditions (H1-2) and (H2-1) hold,then the boundary value problems (1.2) has at least one positive solution.