Journal of Inequalities and Applications (Jan 2011)
Positive solutions for Neumann boundary value problems of nonlinear second-order integro-differential equations in ordered Banach spaces
Abstract
Abstract The paper deals with the existence of positive solutions for Neumann boundary value problems of nonlinear second-order integro-differential equations - u ″ ( t ) + M u ( t ) = f ( t , u ( t ) , ( S u ) ( t ) ) , 0 < t < 1 , u ′ ( 0 ) = u ′ ( 1 ) = θ and u ″ ( t ) + M u ( t ) = f ( t , u ( t ) , ( S u ) ( t ) ) , 0 < t < 1 , u ′ ( 0 ) = u ′ ( 1 ) = θ in an ordered Banach space E with positive cone K, where M > 0 is a constant, f : [0, 1] × K × K → K is continuous, S : C([0, 1], K) → C([0, 1], K) is a Fredholm integral operator with positive kernel. Under more general order conditions and measure of noncompactness conditions on the nonlinear term f, criteria on existence of positive solutions are obtained. The argument is based on the fixed point index theory of condensing mapping in cones. Mathematics Subject Classification (2000): 34B15; 34G20.
