Existence, uniqueness and qualitative properties of heteroclinic solutions to nonlinear second-order ordinary differential equations

Abstract

By means of the shooting method together with the maximum principle and the Kneser–Hukahara continuum theorem, the authors present the existence, uniqueness and qualitative properties of solutions to nonlinear second-order boundary value problem on the semi-infinite interval of the following type: $$ \begin{cases} y''=f(x,y,y'),& x\in[0,\infty), \\ y'(0)=A,& y(\infty)=B \end{cases} $$ and $$ \begin{cases} y''=f(x,y,y'),& x\in[0,\infty), \\ y(0)=A,&y(\infty)=B, \end{cases} $$ where $A,B\in \mathbb{R}$, $f(x,y,z)$ is continuous on $[0,\infty)\times\mathbb{R}^2$. These results and the matching method are then applied to the search of solutions to the nonlinear second-order non-autonomous boundary value problem on the real line $$ \begin{cases} y''=f(x,y,y'), & x\in\mathbb{R} ,\\ y(-\infty)=A,& y(\infty)=B, \end{cases} $$ where $A\not=B$, $f(x,y,z)$ is continuous on $\mathbb{R}^3$. Moreover, some examples are given to illustrate the main results, in which a problem arising in the unsteady flow of power-law fluids is included.

Keywords

• semi-infinite interval
• heteroclinic solution
• shooting method
• maximum principle
• kneser–hukahara continuum theorem
• matching method