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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 $\bbfT$ given by $$y^{\Delta\Delta}+ f(x, y)=0,\ x\in(0,1]\cap\bbfT,\ y(0)=0,\ y(p)=y \bigl(\sigma^2(1)\bigr),$$ where $p \in(0,1)\cup\bbfT$ 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 {\it J. A. Gatica, V. Oliker} and {\it 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:
34B10Nonlocal and multipoint boundary value problems for ODE
34B16Singular nonlinear boundary value problems for ODE
39A12Discrete version of topics in analysis
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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, 189-249 (2003)
[8] Anderson, D.; Bullock, J.; Erbe, L.; Peterson, A.; Tran, H.: Nabla dynamic equations. Advances in dynamic equations on time scales, 47-83 (2003) · 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) · Zbl 1025.34001
[14] Bohner, M.; Peterson, A.: Dynamic equations on time scales: an introduction with applications. (2001) · 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. Stability control theory methods appl. 5, 283-295 (1997) · 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 · Zbl 0695.34001
[27] Krasnosel’skii, M. A.: Positive solutions to operator equations. (1964)
[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 · Zbl 1062.34013
[31] Spedding, V.: Taming nature’s numbers. New scientist, 28-31 (July 2003)