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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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[1] Baxley, J.V.; Brown, S.E., Existence and uniqueness for two-point boundary value problems, (), 219-234 · Zbl 0461.34011
[2] Baxley, J.V., Nonlinear second order boundary value problems on [0, ∞), (), 50-58
[3] Baxley, J.V., A singular boundary value problem: membrane response of a spherical cap, SIAM J. appl. math., 48, 497-505, (1988) · Zbl 0642.34014
[4] Cronin, J., Fixed points and topological degree in nonlinear analysis, (1964), American Mathematical Society Providence, RI · Zbl 0117.34803
[5] Kidder, R.E., Unsteady flow of gas through a semi-infinite porous medium, J. appl. mech., 24, 329-332, (1957) · Zbl 0078.40903
[6] Granas, A.; Guenther, R.B.; Lee, J.W.; O’Regan, D., Boundary value problems on infinite intervals and semiconductor devices, J. math. anal. appl., 116, 335-348, (1986) · Zbl 0594.34019
[7] Granas, A.; Guenther, R.; Lee, J., Nonlinear boundary value problems for ordinary differential equations, Dissertationes mathematicae, CCXLIV, (1985), Warsaw
[8] Muldowney, J.S.; Willett, D., An elementary proof of the existence of solutions to second order nonlinear boundary value problems, SIAM J. math. anal., 5, 701-707, (1974) · Zbl 0285.34012
[9] Na, T.Y., Computational methods in engineering boundary value problems, (1979), Academic Press New York · Zbl 0456.76002
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