Nonlinear Systems of Second-Order ODEs. (10th December 2007) - Cerda, Patricio; Ubilla, Pedro
Abstract : We study existence of positive solutions of the nonlinear system− ( p 1 ( t, u, v ) u ′ ) ′ = h 1 ( t ) f 1 ( t, u, v ) in( 0, 1 ) ;− ( p 2 ( t, u, v ) v ′ ) ′ = h 2 ( t ) f 2 ( t, u, v ) in( 0, 1 ) ;u ( 0 ) = u ( 1 ) = v ( 0 ) = v ( 1 ) = 0, wherep 1 ( t, u, v ) = 1 / ( a 1 ( t ) + c 1 g 1 ( u, v ) ) andp 2 ( t, u, v ) = 1 / ( a 2 ( t ) + c 2 g 2 ( u, v ) ) . Here, it is assumed thatg 1, g 2 are nonnegative continuous functions, a 1 ( t ), a 2 ( t ) are positive continuous functions, c 1, c 2 ≥ 0, h 1, h 2 ∈ L 1 ( 0, 1 ), and that the nonlinearitiesf 1, f 2 satisfy superlinear hypotheses at zero and+ ∞ . The existence of solutions will be obtained using a combination among the method of truncation, a priori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent ofu andv .