Existence and uniqueness for nonlinear boundary value problems on infinite intervals. (English) Zbl 0719.34037

The author considers the boundary value problem \((1)\quad y''=f(x,y,y'),\quad 0\leq x<\infty,\) \((2)\quad a_ 0y(0)=a_ 1y'(0)=A,\) \(a_ 0\geq 0\), \(a_ 1\geq 0\), \(a_ 0+a_ 1>0\), \((3)\quad y(\infty)=B.\) The basic assumptions on the function f(x,y,z) are: f(x,y,z) is continuous on \(I\times {\mathbb{R}}^ 2\), \(I=[a,b]\), for \(0<b<\infty\); f(x,y,z) is nondecreasing in y for each fixed pair (x,z)\(\in I\times {\mathbb{R}}\); f(x,y,z) satisfies a uniform Lipschitz condition on each compact subset of \(I\times {\mathbb{R}}^ 2\) with respect to z; and zf(x,y,z)\(\leq 0\) for \((x,y,z)\in I\times {\mathbb{R}}^ 2\), \(z\neq 0\). Using the shooting method, and with additional assumptions on f(x,y,z) and supposing that \(a_ 0\), \(a_ 1\) are both positive, he proves that the boundary value problem (1)-(3) has a unique solution.
The following example \(y''=-2xy'/(1-\alpha y)^{1/2},\quad 0\leq x<\infty,\) \(y(0)=1,\quad y(\infty)=0,\) which arises in nonlinear mechanics in the problem of unsteady flow of gas through a semi-infinite porous medium, \(0<\alpha \leq 1\), is given.
Reviewer: M.Shahin (Dallas)


34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
76S05 Flows in porous media; filtration; seepage
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