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