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The existence of three positive solutions of \(m\)-point boundary value problems for some dynamic equations on time scales. (English) Zbl 1176.34027

Summary: We define a new operator which improves and generalizes a \(p\)-Laplacian operator for some \(p>1\). By using this operator, we consider the existence of triple positive solutions of \(m\)-point boundary value problems for some dynamic equations on time scales
\[ \begin{aligned} &[\varphi(p(t)u^\Delta(t))]^\nabla+a(t)f(u(t))=0,\quad t\in [0,T]_{{\mathbf T}^K\cap {\mathbf T}_k},\\ & u(0)-B_0\left(\sum^{m-2}_{i=1}\alpha_iu^\Delta(\xi_i)\right)=0,\quad u^\Delta(T)=0.\end{aligned} \]
where \(\varphi:\mathbb R\to\mathbb R\) is an increasing homeomorphism and positive homomorphism and \(\varphi)0)=0\). We show the existence of at least three positive solutions with suitable growth conditions imposed on the nonlinear term by using a new fixed-point theorem.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
39A10 Additive difference equations
47N20 Applications of operator theory to differential and integral equations
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[1] Hilger, S., Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[2] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales, An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[3] Agarwal, R. P.; Bohner, M.; Li, W. T., Nonoscillation and oscillation theory for functional differential equations, (Monographs and Textbooks in Pure and Applied Mathematics, vol. 267 (2004), Marcel Dekker: Marcel Dekker New York)
[4] Jones, M. A.; Song, B.; Thomas, D. M., Controlling wound healing through debridement, Math. Comput. Modelling, 40, 1057-1064 (2004) · Zbl 1061.92036
[5] Spedding, V., Taming nature’s numbers, New Sci., 28-32 (2003)
[6] Agarwal, R. P.; Bohner, M.; Rehak, P., Half-linear dynamic equations, (Nonlinear Analysis and Applications: To V. Lakshmikantham on His 80th Birthday, vol. 1 (2003), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 1-57 · Zbl 1056.34049
[7] Anderson, D. R.; Avery, R.; Henderson, J., Existence of solutions for a one-dimensional \(p\)-Laplacian on time scales, J. Difference Equ. Appl., 10, 889-896 (2004) · Zbl 1058.39010
[8] 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
[9] Agarwal, R. P.; O’Regan, D., Nonlinear boundary value problems on time scales, Nonlinear Anal., 44, 527-535 (2001) · Zbl 0995.34016
[10] Sang, Y.; Xi, H., Positive solutions of nonlinear \(m\)-point boundary value problem for \(p\)-Laplacian dynamic equations on time scales, Electron. J. Differential Equations, 2007, 34, 1-10 (2007) · Zbl 1118.34016
[11] He, Z., Double positive solutions of three-point boundary value problems for \(p\)-Laplacian dynamic equations on time scales, J. Comput. Appl. Math., 182, 304-315 (2005) · Zbl 1075.39011
[12] He, Z., Double positive solutions of three-point boundary value problems for \(p\)-Laplacian difference equations, Z. Anal. Anwendungen, 24, 2, 305-315 (2005) · Zbl 1093.39003
[13] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic Systems on Measure Chains (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0869.34039
[14] Anderson, D. R., Solutions to second-order three-point problems on time scales, J. Difference Equ. Appl., 8, 673-688 (2002) · Zbl 1021.34011
[15] Kaufmann, E. R., Positive solutions of a three-point boundary value problem on a time scale, Electron. J. Differential Equations, 2003, 82, 1-11 (2003) · Zbl 1047.34015
[16] Li, W. T.; Liu, X. L., Eigenvalue problems for second-order nonlinear dynamic equations on time scales, J. Math. Anal. Appl., 318, 578-592 (2006) · Zbl 1099.34026
[17] Sun, H. R.; Li, W. T., Existence theory for positive solutions to one-dimensional \(p\)-Laplacian boundary value problems on time scales, J. Differential Equations, 240, 217-248 (2007) · Zbl 1139.34047
[18] Sun, H. R., Existence of positive solutions to second-order time scale systems, Comput. Math. Appl., 49, 131-145 (2005) · Zbl 1075.34019
[19] Liang, S.; Zhang, J., The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on time scales, J. Comput. Appl. Math. (2008)
[20] Liu, B. F.; Zhang, J. H., The existence of positive solutions for some nonlinear boundary value problems with linear mixed boundary conditions, J. Math. Anal. Appl., 309, 505-516 (2005) · Zbl 1086.34022
[21] Hong, S., Triple positive solutions of three-point boundary value problems for \(p\)-Laplacian dynamic equations on time scales, J. Comput. Appl. Math., 206, 967-976 (2007) · Zbl 1120.39019
[22] (Bohner, M.; Peterson, A., Advances in Dynamic Equation on Time Scales (2003), Birkhäuser: Birkhäuser Boston) · Zbl 1025.34001
[23] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045
[24] Ren, J. L.; Ge, W. G.; Ren, B. X., Existence of positive solutions for quasi-linear boundary value problems, Acta Math. Appl. Sinica, 21, 3, 353-358 (2005), (in Chinese) · Zbl 1113.34016
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