Positive solutions to fractional boundary value problems with nonlinear boundary conditions

Abstract

Abstract We consider the existence of at least one positive solution of the problem − D 0 + α u ( t ) = f ( t , u ( t ) ) , 0 < t < 1 , under the circumstances that u ( 0 ) = 0 , u ( 1 ) = H 1 ( φ ( u ) ) + ∫ E H 2 ( s , u ( s ) ) d s , where 1 < α < 2 , D 0 + α is the Riemann-Liouville fractional derivative, and u ( 1 ) = H 1 ( φ ( u ) ) + ∫ E H 2 ( s , u ( s ) ) d s represents a nonlinear nonlocal boundary condition. By imposing some relatively mild structural conditions on f , H 1 , H 2 , and φ , one positive solution to the problem is ensured. Our results generalize the existing results and an example is given as well. MSC: 34A08, 34B18.