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Positive solutions of three-point boundary value problems for the one-dimensional \(p\)-Laplacian with infinitely many singularities. (English) Zbl 1060.34006

Summary: We consider the singular three-point boundary value problem \[ (\phi(y'))' + a(t)f(y(t)) = 0,\quad 0 < t < 1,\quad y'(0) = 0,\quad y(1) = y(\zeta), \] where \(\phi(s) = |s|^{p-2}s\), \(p \geq 2\), \(0 < \beta < 1\), \(0 < \zeta < 1\), \(f\in C([0, +\infty )\), \([0, +\infty ))\), \(a: [0,1]\to [0, +\infty)\), and has countably many singularities in \([0, 1/2)\). We show that there exist countably many positive solutions by using the fixed-point index theory.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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