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Twin solutions of boundary value problems for ordinary differential equations and finite difference equations. (English) Zbl 1006.34022

The existence of at least two positive solutions is proved to the right focal boundary value problem \[ y''+ f(y)= 0,\quad y(0)= y'(1)= 0. \] The same is also proved to its discrete finite difference analogy. A fixed-point theorem by Avery and Henderson, which is presented without proof, is applied for this goal.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis
34A45 Theoretical approximation of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
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References:

[1] Zeidler, E., Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems (1993), Springer-Verlag: Springer-Verlag New York
[2] Avery, R. I.; Henderson, J., Three symmetric positive solutions for a second-order boundary value problem, Appl. Math. Lett., 13, 3, 1-7 (2000) · Zbl 0961.34014
[3] J. Henderson and H.B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proceedings American Mathematical Society.; J. Henderson and H.B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proceedings American Mathematical Society. · Zbl 0949.34016
[4] J. Henderson and H.B. Thompson, Existence of multiple solutions for second order boundary value problems, J. Differential Equations.; J. Henderson and H.B. Thompson, Existence of multiple solutions for second order boundary value problems, J. Differential Equations. · Zbl 1013.34017
[5] Kaufmann, E., Multiple positive solutions for higher order boundary value problems, Rocky Mountain J. Math., 28, 1017-1028 (1998) · Zbl 0930.34010
[6] P.J.Y. Wong and R.P. Agarwal, Existence of multiple solutions of discrete two-point right focal boundary value problems, J. Difference Equations and Applications.; P.J.Y. Wong and R.P. Agarwal, Existence of multiple solutions of discrete two-point right focal boundary value problems, J. Difference Equations and Applications. · Zbl 0964.39004
[7] Agarwal, R. P.; O’Regan, D.; Wong, P. J.Y., Positive Solutions of Differential, Difference and Integral Equations (1999), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0923.39002
[8] Leggett, R.; Williams, L., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033
[9] Avery, R. I., A generalization of the Leggett-Williams fixed point theorem, MSR Hotline, 2, 9-14 (1998) · Zbl 0965.47038
[10] R.I. Avery and A.C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, (Preprint).; R.I. Avery and A.C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, (Preprint). · Zbl 1005.47051
[11] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964), P. Noordhoff: P. Noordhoff Groningen, The Netherlands · Zbl 0121.10604
[12] Eloe, P. W.; Henderson, J., Twin solutions for nonlinear multipoint conjugate boundary value problems, Dynamics of Continuous, Discrete & Impulsive Systems, 5, 283-293 (1999) · Zbl 0942.34016
[13] Eloe, P. W.; Henderson, J.; Kaufmann, E., Multiple positive solutions for difference equations, J. Difference Equations and Applications, 3, 219-229 (1998) · Zbl 1005.39502
[14] Merdivenci, F., Two positive solutions for a boundary value problem for difference equations, J. Difference Equations and Applications, 1, 262-270 (1995) · Zbl 0854.39001
[15] Wong, P. J.Y.; Agarwal, R. P., Eigenvalue intervals and double positive solutions of certain discrete boundary value problems, Communications in Applied Analysis, 3, 189-217 (1999) · Zbl 0923.39002
[16] R.I. Avery and J. Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces, (Preprint).; R.I. Avery and J. Henderson, Two positive fixed points of nonlinear operators on ordered Banach spaces, (Preprint). · Zbl 1014.47025
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