Existence of Positive Solution to System of Nonlinear Third-Order Three-Point BVPs

Abstract EN

We are concerned with the following system of third-order three-point boundary value problems: u ′ ′ ′ ( t ) + f ( t , v ( t ) ) = 0 , t ∈ ( 0,1 ) , v ′ ′ ′ ( t ) + g ( t , u ( t ) ) = 0 , t ∈ ( 0,1 ) , u ( 0 ) = u ′ ′ ( 0 ) = 0 , u ′ ( 1 ) = α u ( η ) , v ( 0 ) = v ′ ′ ( 0 ) = 0 , and v ′ ( 1 ) = α v ( η ) , where 0 < η < 1 and 0 < α < 1 / η .

By imposing some suitable conditions on f and g , we obtain the existence of at least one positive solution to the above system.

The main tool used is the theory of the fixed-point index.