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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 New York · Zbl 0559.47040
[8] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), 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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