Positive solutions of singular boundary value problems for systems of nonlinear fourth order differential equations

Abstract In this paper, we study the singular boundary value problems for systems of nonlinear fourth order differential equations { u ( 4 ) ( t ) = a 1 ( t ) f 1 ( t , u ( t ) , v ( t ) , u ″ ( t ) , v ″ ( t ) ) + b 1 ( t ) g 1 ( t , u ( t ) , v ( t ) , u ″ ( t ) , v ″ ( t ) ) , v ( 4 ) ( t ) = a 2 ( t ) f 2 ( t , u ( t ) , v ( t ) , u ″ ( t ) , v ″ ( t ) ) + b 2 ( t ) g 2 ( t , u ( t ) , v ( t ) , u ″ ( t ) , v ″ ( t ) ) , 0 t 1 , u ( 0 ) = u ( 1 ) = v ( 0 ) = v ( 1 ) = 0 , α 1 u ″ ( 0 ) − β 1 u ‴ ( 0 ) = 0 , γ 1 u ″ ( 1 ) + δ 1 u ‴ ( 1 ) = 0 , α 2 v ″ ( 0 ) − β 2 v ‴ ( 0 ) = 0 , γ 2 v ″ ( 1 ) + δ 2 v ‴ ( 1 ) = 0 . Under some weaker conditions, we show the existence of single and multiple positive solutions of the above problem by applying the fixed-point theorem of cone expansion and compression type due to Krasnosel’skill.