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Positive solutions for nonlinear three-point boundary value problems on time scales. (English) Zbl 1070.34029
The authors generalize some known results on positive solutions of a three-point boundary value problem related to the dynamic equation $u^{\Delta\nabla} (t) + a(t) f(t,u(t)) = 0.$ The symbol $$\Delta\nabla$$ denotes the natural Laplace operator on time scales (arbitrary closed subsets of the reals). The methods used in the proofs are based on Leggett–Williams/ Krasnoselskii fixed-point techniques in cones.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 39A12 Discrete version of topics in analysis
##### Keywords:
time scales; positive solutions; cone fixed-point theorem
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##### References:
 [1] Agarwal, R.P.; Bohner, M.; Li, W.T., Nonoscillation and oscillation theory for functional differential equations, Pure appl. math., (2004), Dekker Florida [2] Agarwal, R.P.; Bohner, M.; Wong, P.J., Sturm – liouville eigenvalue problems on time scales, Appl. math. comput., 99, 153-166, (1999) · Zbl 0938.34015 [3] Agarwal, R.P.; O’Regan, D., Triple solutions to boundary value problems on time scales, Appl. math. lett., 13, 7-11, (2000) · Zbl 0958.34021 [4] Agarwal, R.P.; O’Regan, D., Nonlinear boundary value problems on time scales, Nonlinear anal., 44, 527-535, (2001) · Zbl 0995.34016 [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] Atici, F.M.; Guseinov, G.Sh., 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 [7] Avery, R.I.; Anderson, D.R., Existence of three positive solutions to a second-order boundary value problem on a measure chain, J. comput. appl. math., 141, 65-73, (2002) · Zbl 1032.39009 [8] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser Boston Cambridge, MA · Zbl 0978.39001 [9] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhäuser Boston Cambridge, MA · Zbl 1025.34001 [10] Chyan, C.J.; Henderson, J., Twin solutions of boundary value problems for differential equations on measure chains, J. comput. appl. math., 141, 123-131, (2002) · Zbl 1134.39301 [11] Chyan, C.J.; Henderson, J., Eigenvalue problems for nonlinear differential equations on a measure chain, J. math. anal. appl., 245, 547-559, (2000) · Zbl 0953.34068 [12] Erbe, L.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dynam. contin. discrete impuls. systems, 6, 121-137, (1999) · Zbl 0938.34027 [13] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Math. comput. modelling, 32, 5-6, 571-585, (2000) · Zbl 0963.34020 [14] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045 [15] He, X.; Ge, W., Triple solutions for second-order three-point boundary value problems, J. math. anal. appl., 268, 256-265, (2002) · Zbl 1043.34015 [16] Henderson, J., Multiple solutions for 2mth order sturm – liouville boundary value problems on a measure chain, J. differ. equations appl., 6, 417-429, (2000) · Zbl 0965.39008 [17] Hilger, S., Analysis on measure chains—A unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001 [18] Hong, C.H.; Yeh, C.C., Positive solutions for eigenvalue problems on a measure chain, Nonlinear anal., 51, 499-507, (2002) · Zbl 1017.34018 [19] Kaufmann, E.R., Positive solutions of a three-point boundary value problem on a time scale, Electron. J. differential equations, 82, 1-11, (2003) · Zbl 1047.34015 [20] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic systems on measure chains, (1996), Kluwer Academic Boston · Zbl 0869.34039 [21] Lan, K.Q., Multiple positive solutions of semilinear differential equations with singularities, J. London math. soc., 63, 690-704, (2001) · Zbl 1032.34019 [22] 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 [23] Li, W.T.; Sun, H.R., Multiple positive solutions for nonlinear dynamic systems on a measure chain, J. comput. appl. math., 162, 421-430, (2004) · Zbl 1045.39007 [24] Liu, B., Positive solutions of a nonlinear three-point boundary value problem, Appl. math. comput., 132, 11-28, (2002) · Zbl 1032.34020 [25] Liu, B., Positive solutions of a nonlinear three-point boundary value problem, Comput. math. appl., 44, 201-211, (2002) · Zbl 1008.34014 [26] Ma, R., Positive solutions of a nonlinear three-point boundary value problem, Electron. J. differential equations, 34, 1-8, (1999) [27] H.R. Sun, W.T. Li, Existence of positive solutions to second-order time scale systems, Comput. Math. Appl., in press [28] H.R. Sun, W.T. Li, Positive solutions of nonlinear three-point boundary value problems on time scales, Nonlinear Anal. TMA, in press [29] J.P. Sun, W.T. Li, Positive solutions of a second order nonlinear three-point boundary value problem, Indian J. Pure Appl. Math. (2003), submitted for publication [30] Webb, J.R.L., Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear anal., 47, 4319-4332, (2001) · Zbl 1042.34527
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