Existence and uniqueness of solutions for a second order differential equation with integral boundary conditions
Abstract In this paper, we consider the following second order differential equation ( ϕ ( u ′ ( t ) ) ) ′ + f ( t , u ( t ) , u ′ ( t ) ) = 0 , 0 t 1 , subject to the integral boundary conditions: u ( 0 ) - k 1 u ′ ( 0 ) = ∫ 0 1 h 1 ( u ( s ) ) ds , u ( 1 ) + k 2 u ′ ( 1 ) = ∫ 0 1 h 2 ( u ( s ) ) ds , where f : [ 0 , 1 ] × R 2 → R and h i : R → R ( i = 1 , 2 ) are continuous and k i ( i = 1 , 2 ) are nonnegative constant. ϕ ( u ) is continuous and strictly increasing function with ϕ ( 0 ) = 0 and ϕ ( R ) = R . The existence and uniqueness result to the above boundary value problem is obtained by applying the method of upper and lower solutions and Leray–Schauder degree theory.