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

34B15Nonlinear boundary value problems for ODE
34A45Theoretical approximation of solutions of ODE
76S05Flows in porous media; filtration; seepage
Full Text: DOI
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