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Some existence and nonexistence principles for a class of singular boundary value problems. (English) Zbl 0860.34010

The author presents existence and nonexistence results to the following singular boundary value problem \[ \begin{aligned} (p(t)x'(t))'+ p(t)f(t,x(t)) &=0, \qquad 0<t<1,\\ \alpha x(0)- \beta \lim_{t\to 0}p(t)x'(t) &=0, \tag{1}\\ \gamma x(1)+ \delta \lim_{t\to1} p(t)x'(t) &=0, \end{aligned} \] where \(f:(0,1)\times (0,\infty)\to (0,\infty)\) is continuous, \(p\in C^1[0,1]\), and \(p(t)>0\) for \(t\in(0,1)\), and \(\alpha,\beta, \gamma,\delta\geq 0\), \(\beta\gamma+ \alpha\delta+ \alpha\gamma>0\). The nonlinear term \(f(t,u)\) may be singular at \(u=0\) and \(t=0,1\).
An example is \(f(t,u)= t^{-m} (1-t)^{-m} u^{-n}\), where \(m,n>0\). The author shows that, when \(m<2\), \(n>0\), problem (1) has a solution for general boundary conditions. When \(m>2\), no solutions exist.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

[1] O’Regan, D., Some existence principle and some general results for singular nonlinear two point boundary value problems, J. math. Analysis Applic., 166, 24-40 (1992) · Zbl 0756.34030
[2] Dunninger, D. R.; Kurtz, J. C., Existence of solutions for some nonlinear singular boundary value problems, J. math. Analysis Applic., 115, 396-405 (1986) · Zbl 0616.34012
[3] Bobisud, L. E.; O’Regan, D.; Rayalty, W. D., Solvability of some nonlinear BVP, Nonlinear Analysis, 12, 855-869 (1988) · Zbl 0653.34015
[4] Gatica, J. A.; Oliker, V.; Waltman, P., Singular nonlinear BVP for second order ODE, J. diff. Eqns, 79, 62-78 (1989) · Zbl 0685.34017
[5] O’Regan, D., Singular second order BVP, Nonlinear Analysis, 65, 1097-1109 (1990) · Zbl 0732.34021
[6] Guo, D.; Lakshmikantham, V., (Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, Vol. 5 (1988), Academic Press: Academic Press New York) · Zbl 0661.47045
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