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Existence of multiple solutions for second order boundary value problems. (English) Zbl 1013.34017

The authors prove the existence of at least three solutions to nonlinear two-point boundary value problems \[ y''+ f(x,y,y')= 0,\quad x\in [0,1],\quad y(0)= 0= y(1), \] where \(f:[0,1]\times \mathbb{R}^2\to\mathbb{R}\) is continuous and satisfies the Bernstein-Nagumo condition. The proofs are based on the method of lower and upper solutions and the theory of topological degree. For earlier work, see R. Avery [Math. Sci. Res. Hot-Line 2, No. 1, 1-6 (1998; Zbl 0960.34503)].
Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators

Citations:

Zbl 0960.34503
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References:

[1] Ako, K., Subfunctions for ordinary differential equations II, Funckial. Ekvac., 10, 145-162 (1967) · Zbl 0162.11601
[2] Ako, K., Subfunctions for ordinary differential equations III, Funckial. Ekvac., 11, 111-129 (1968) · Zbl 0224.34015
[3] Avery, R. I.; Peterson, A. C., Multiple positive solutions of a discrete second order conjugate problem, PanAmer. Math. J., 8, 1-12 (1998) · Zbl 0959.39006
[4] Anderson, D., Multiple positive solutions for a three-point boundary value problem, Math. Comput. Modelling, 27, 49-57 (1998) · Zbl 0906.34014
[5] Avery, R., Existence of multiple positive solutions to a conjugate boundary value problem, MSR Hot-Line, 2 (1998) · Zbl 0960.34503
[6] Jackson, L. K., Subfunctions and second-order ordinary differential inequalities, Adv. in Math., 2, 307-363 (1968) · Zbl 0197.06401
[7] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0559.47040
[8] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045
[9] Henderson, J.; Thompson, H. B., Existence of multiple solutions for some n th order boundary value problems, Comm. Appl. Nonlinear Anal., 7, 57-62 (2000) · Zbl 1108.34306
[10] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math., 28, 673-688 (1979) · Zbl 0421.47033
[11] Yong, S.; Jingxian, S., Multiple positive fixed points of weakly inward mappings, J. Math. Anal. Appl., 148, 431-439 (1990) · Zbl 0709.47052
[12] Thompson, H. B., Existence of solutions for a two point boundary value problem, Rend. Circ. Mat. Palermo (2), XXXV, 261-275 (1986) · Zbl 0608.34018
[13] Thompson, H. B., Minimal solutions for two point boundary value problems, Rend. Circ. Mat. Palermo (2), XXXVII, 261-281 (1988) · Zbl 0698.34019
[14] Thompson, H. B., Second order ordinary differential equations with fully nonlinear two point boundary conditions, Pacific J. Math., 172, 255-277 (1996) · Zbl 0855.34024
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