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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)

##### MSC:
 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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##### References:
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