Global preservation of nodal structure in coupled systems of nonlinear Sturm-Liouville boundary value problems

Abstract

In this paper, we examine the solution set to the coupled system ( ∗ * ) \[ { − ( p 2 ( x ) υ ′ ( x ) ) ′ + q 2 ( x ) υ ( x ) = μ υ ( x ) + υ ( x ) ⋅ g ( u ( x ) , υ ( x ) ) , − ( p 1 ( x ) u ′ ( x ) ) ′ + q 1 ( x ) u ( x ) = λ u ( x ) + u ( x ) ⋅ f ( u ( x ) , υ ( x ) ) \left \{ {_{ - ({p_2}(x)\upsilon ’(x))’ + {q_2}(x)\upsilon (x) = \mu \upsilon (x) + \upsilon (x) \cdot g(u(x),\upsilon (x)),}^{ - ({p_1}(x)u’(x))’ + {q_1}(x)u(x) = \lambda u(x) + u(x) \cdot f(u(x),\upsilon (x))}} \right . \] where λ , μ ∈ R , x ∈ [ a , b ] \lambda ,\mu \in R,x \in [a,b] , and the system ( ∗ * ) is subject to zero Dirichlet boundary data on u u and υ \upsilon . We determine conditions on f f and g g which permit us to assert the existence of continua of solutions to ( ∗ * ) characterized by u u having n − 1 n - 1 simple zeros in ( a , b ) , υ (a,b),\upsilon having m − 1 m - 1 simple zeros in ( a , b ) (a,b) , where n n and m m are positive but not necessarily equal integers. Moreover, we also determine conditions under which these continua link solutions to ( ∗ * ) of the form ( λ , μ , u , 0 ) (\lambda ,\mu ,u,0) with u u having n − 1 n - 1 simple zeros in ( a , b ) (a,b) to solutions of ( ∗ * ) of the form ( λ , μ , 0 , υ ) (\lambda ,\mu ,0,\upsilon ) with υ \upsilon having m − 1 m - 1 simple zeros in ( a , b ) (a,b) .