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Existence of three positive pseudo-symmetric solutions for a one dimensional $p$-Laplacian. (English) Zbl 1028.34022
The five functionals fixed-point theorem due to {\it R. I. Avery} [Math. Sci. Res. Hot-Line 3, 9-14 (1999; Zbl 0965.47038)] is applied to obtain the existence of three positive pseudo-symmetric solutions to the three-point boundary value problem for a one-dimensional $p$-Laplacian $$(|u'|^{p-2}u')' + a(t)f(u) = 0, \quad u(0)=0,\ u(\theta) = u(1),$$ with $\theta \in (0,1)$.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
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References:
[1] Agarwal, R. P.; O’regan, D.; Wong, P. J. Y.: Positive solutions of differential, difference and integral equations. (1999)
[2] Avery, R. I.: Multiple positive solutions to boundary value problems, dissertation. (1997)
[3] Avery, R. I.: A generalization of the Leggett--Williams fixed point theorem. MSR hot-line 2, 9-14 (1998) · Zbl 0965.47038
[4] Avery, R. I.; Henderson, J.: Three symmetric positive solutions for a second order boundary value problem. Appl. math. Lett. 13, 1-7 (2000) · Zbl 0961.34014
[5] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040
[6] Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones. (1988) · Zbl 0661.47045
[7] X.-M. He, W.-G. Ge, A remark on some three-point boundary value problems for the one-dimensional p-Laplacian, ZAMM, in press
[8] Henderson, J.; Thompson, H. B.: Multiple symmetric positive solutions for a second order boundary value problem. Proc. amer. Math. soc. 128, 2373-2379 (2000) · Zbl 0949.34016
[9] Leggett, R. W.; Williams, L. R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces. Indiana univ. Math. J. 28, 673-688 (1979) · Zbl 0421.47033
[10] Zeidler, E.: Nonlinear functional analysis and its applications I: Fixed-point theorems. (1993) · Zbl 0794.47033