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Multiple symmetric positive solutions for a second order boundary value problem. (English) Zbl 0949.34016

The paper is concerned with the existence of three solutions to the second-order boundary value problem \[ y''+ f(y)=0,\quad 0\leq t\leq 1,\quad y(0)= y(1)= 0,\tag{1} \] where \(f:\mathbb{R}\to [0,\infty)\) is continuous. Under growth conditions imposed on \(f\) the existence of at least three symmetric positive solutions on \((0,1)\) to (1) is proved via the Legett-Williams fixed point theorem. The results are an extension of the earlier ones by Avery, Leggett and Williams, and Guo and Lakshmikantham.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

[1] D. Anderson, Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling 27 (1998), no. 6, 49 – 57. · Zbl 0906.34014 · doi:10.1016/S0895-7177(98)00028-4
[2] R. Avery, Existence of multiple positive solutions to a conjugate boundary value problem, Math. Sci. Res. Hot-Line 2 (1998), no. 1, 1 – 6. · Zbl 0960.34503
[3] Richard I. Avery and Allan C. Peterson, Multiple positive solutions of a discrete second order conjugate problem, PanAmer. Math. J. 8 (1998), no. 3, 1 – 12. · Zbl 0959.39006
[4] Da Jun Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, vol. 5, Academic Press, Inc., Boston, MA, 1988. · Zbl 0661.47045
[5] J. Henderson and H.B. Thompson, Existence of multiple solutions for some \(n\)-th order boundary value problems, preprint. · Zbl 1108.34306
[6] Richard W. Leggett and Lynn R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), no. 4, 673 – 688. · Zbl 0421.47033 · doi:10.1512/iumj.1979.28.28046
[7] Yong Sun and Jing Xian Sun, Multiple positive fixed points of weakly inward mappings, J. Math. Anal. Appl. 148 (1990), no. 2, 431 – 439. · Zbl 0709.47052 · doi:10.1016/0022-247X(90)90011-4
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