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Existence results for singular three point boundary value problems on time scales. (English) Zbl 1069.34012

Summary: We prove the existence of a positive solution for the three point boundary value problem on time scale \(\mathbb{T}\) given by \[ y^{\Delta\Delta}+ f(x, y)=0,\;x\in(0,1]\cap\mathbb{T},\;y(0)=0,\;y(p)=y \bigl(\sigma^2(1)\bigr), \] where \(p \in(0,1)\cup\mathbb{T}\) is fixed and \(f(x,y)\) is singular at \(y=0\) and possibly at \(x=0\), \(y=\infty\). We do so by applying a fixed point theorem due to J. A. Gatica, V. Oliker and P. Waltman [J. Differ. Equations 79, 62–78 (1989; Zbl 0685.34017)] for mappings that are decreasing with respect to a cone. We also prove the analogous existence results for the related dynamic equations \(y^{\Delta\Delta}+f(x,y)=0\), \(y^{\Delta\nabla}+f(x,y)=0\), and \(y^{\nabla\Delta}+f(x,y)=0\) satisfying similar three point boundary conditions.

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis

Citations:

Zbl 0685.34017
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References:

[1] Agarwal, R.; Bohner, M.; O’Regan, D.; Peterson, A., Dynamic equations on time scales: a survey, J. Comput. Appl. Math., 141, 1-26 (2002) · Zbl 1020.39008
[2] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044
[3] Anderson, D. R., Existence of solutions to higher-order discrete three-point problems, Electron. J. Differential Equations, 40, 1-7 (2003) · Zbl 1036.39002
[4] Anderson, D. R., Multiple positive solutions for a three-point boundary value problem, Math. and Comput. Modelling, 27, 49-57 (1998) · Zbl 0906.34014
[5] Anderson, D. R., Solutions to second-order three-point problems on time scales, J. Differ. Equations Appl., 8, 673-688 (2002) · Zbl 1021.34011
[6] Anderson, D. R., Taylor polynomials for nabla dynamic equations on time scales, Panamer. Math. J., 12, 17-27 (2002) · Zbl 1026.34011
[7] Anderson, D.; Avery, R.; Davis, J.; Henderson, J.; Yin, W., Positive solutions of boundary value problems, (Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston), 189-249
[8] Anderson, D.; Bullock, J.; Erbe, L.; Peterson, A.; Tran, H., Nabla dynamic equations, (Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston), 47-83 · Zbl 1032.39007
[9] Anderson, D. R.; Hoffacker, J., Green’s function for an even order mixed derivative problem on time scales, Dynam. Systems Appl., 12, 9-22 (2003) · Zbl 1049.39019
[10] Atici, F.; Eloe, P. W.; Kaymakcalan, B., The quasilinearization method for boundary value problems on time scales, J. Math. Anal. Appl., 276, 357-372 (2002) · Zbl 1021.34006
[11] Atici, F.; Guseinov, G., On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141, 75-99 (2002) · Zbl 1007.34025
[12] Baxley, J. V., Some singular nonlinear boundary value problems, SIAM J. Math. Anal., 22, 463-479 (1991) · Zbl 0719.34038
[13] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[14] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[15] Eloe, P.; Henderson, J., Singular nonlinear boundary value problems for higher ordinary differential equations, Nonlinear Anal., 17, 1-10 (1991) · Zbl 0731.34015
[16] Eloe, P.; Henderson, J., Singular nonlinear \((k,n\)−\(k)\) conjugate boundary value problems, J. Differential Equations, 133, 136-151 (1997) · Zbl 0870.34031
[17] Eloe, P.; Henderson, J., Singular nonlinear multipoint conjugate boundary value problems, Commun. Appl. Anal., 2, 497-511 (1998) · Zbl 0903.34016
[18] Eloe, P. W.; Sheng, Q.; Henderson, J., Notes on crossed symmetry solutions of the two-point boundary value problems on time scales, J. Differ. Equations Appl., 9, 29-48 (2003) · Zbl 1038.34013
[19] Erbe, L. H.; Kong, Q., Boundary value problems for singular second-order functional differential equations, J. Comput. Appl. Math., 53, 377-388 (1994) · Zbl 0816.34046
[20] Fink, A. M.; Gatica, J. A.; Hernández, G. E., Approximation of solutions of singular second order boundary value problems, SIAM J. Math. Anal., 22, 440-462 (1991) · Zbl 0722.34015
[21] Gatica, J. A.; Oliker, V.; Waltman, P., Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations, 79, 62-78 (1989) · Zbl 0685.34017
[22] Gupta, C. P., A sharper condition for the solvability of a three-point second order boundary value problem, J. Math. Anal. Appl., 205, 579-586 (1997) · Zbl 0874.34014
[23] Gupta, C. P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168, 540-551 (1992) · Zbl 0763.34009
[24] Henderson, J.; Yin, W., Focal boundary-value problems for singular ordinary differential equations, (Advances in Nonlinear Dynamics. Advances in Nonlinear Dynamics, Stability Control Theory Methods Appl., vol. 5 (1997), Gordon and Breach: Gordon and Breach Amsterdam), 283-295 · Zbl 0974.34014
[25] Henderson, J.; Yin, W., Singular \((k,n\)−\(k)\) boundary value problems between conjugate and right focal. Positive solutions of nonlinear problems, J. Comput. Appl. Math., 88, 57-69 (1998) · Zbl 0901.34026
[26] S. Hilger, Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis, Universität Würzburg, 1988; S. Hilger, Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD thesis, Universität Würzburg, 1988 · Zbl 0695.34001
[27] Krasnosel’skii, M. A., Positive Solutions to Operator Equations (1964), Noordhoff: Noordhoff Groningen, The Netherlands · Zbl 0121.10604
[28] Lomtatidze, A. G., A singular three-point boundary value problem (in Russian), Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy., 17, 122-134 (1986) · Zbl 0632.34011
[29] Ma, R., Positive Solutions of a nonlinear three-point boundary value problem, Electron. J. Differential Equations, 34, 1-8 (1998)
[30] P.K. Singh, A second order singular three-point boundary value problem, Appl. Math. Lett., in press; P.K. Singh, A second order singular three-point boundary value problem, Appl. Math. Lett., in press · Zbl 1062.34013
[31] Spedding, V., Taming nature’s numbers, New Scientist, 28-31 (July 2003)
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