Existence of solutions of nonlinear \(m\)-point boundary-value problems. (English) Zbl 0988.34009

Let \(\xi_i \in (0,1)\), and let \(a_i, b_i, i=1,\dots ,m-2\), be positive numbers satisfying the conditions \(\sum_{i=1}^{m-2} a_i < 1\), \(\sum_{i=1}^{m-2} b_i < 1\). Here, the existence of a positive solution to the multipoint boundary value problem \[ u''(t) + a(t) f(u) = 0, \quad t\in (0,1), \qquad u'(0) = \sum_{i=1}^{m-2}b_i u'(\xi_i), \quad u(1)=\sum_{i=1}^{m-2} a_i u(\xi_i), \] is studied. By using a fixed-point theorem, the existence of a solution is proved if \(f(u)\) is either superlinear or sublinear.


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Feng, W.; Webb, J.R.L., Solvability of a three-point nonlinear boundary value problems at resonance, Nonlinear anal., 30, 3227-3238, (1997) · Zbl 0891.34019
[2] Feng, W.; Webb, J.R.L., Solvability of a m-point boundary value problems with nonlinear growth, J. math. anal. appl., 212, 467-480, (1997) · Zbl 0883.34020
[3] Feng, W., On a m-point nonlinear boundary value problem, Nonlinear anal., 30, 5369-5374, (1997) · Zbl 0895.34014
[4] Gupta, C.P., A generalized multi-point boundary value problem for second order ordinary differential equations, Appl. math. comput., 89, 133-146, (1998) · Zbl 0910.34032
[5] Il’in, V.A.; Moiseev, E.I., Nonlocal boundary value problem of the second kind for a sturm – liouville operator, Differential equations, 23, 979-987, (1987) · Zbl 0668.34024
[6] Ma, Ruyun, Existence theorems for a second order m-point boundary value problem, J. math. anal. appl., 211, 545-555, (1997) · Zbl 0884.34024
[7] Ma, Ruyun, Positive solutions of a nonlinear three-point boundary value problem, Electron. J. differential equations, 34, 1-8, (1999) · Zbl 0926.34009
[8] Mawhin, J., Topological degree methods on nonlinear boundary value problems, NSF-CBMS regional conference series in math., 40, (1979), Amer. Math. Soc Providence
[9] Stanĕk, S., On some boundary value problems for second order functional differential equations, Nonlinear anal., 28, 539-546, (1997) · Zbl 0873.34053
[10] Guo, D.H.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045
[11] Wang, H., On the existence of positive solutions for semilinear elliptic equations in annulus, J. differential equations, 109, 1-7, (1994) · Zbl 0798.34030
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