Second-order periodic boundary value problems on time scales. (English) Zbl 1068.34016

The author considers the existence of solutions of periodic boundary value problem for second-order nonlinear dynamic equations on time scales \[ -y^{\Delta\nabla}(t)+q(t)y(t)=f(t,y(t)),\quad t\in [a,b], \]
\[ y(\rho(a))=y(b),\quad y^\Delta(\rho(a))=y^\Delta(b), \] where \([a,b]\) is a subset of the time sale \(T\); \(h^\Delta(t)\) is the delta derivative of \(h\), \(h^\nabla(t)\) is the nabla derivative of \(h\).


34B15 Nonlinear boundary value problems for ordinary differential equations
39A12 Discrete version of topics in analysis
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[1] Erbe, L.; Peterson, A., Green’s functions and comparison theorems for differential for differential equations on measure chains, Dynam. Contin. Discrete Impuls. Systems, 6, 1, 121-137 (1999) · Zbl 0938.34027
[3] Leela, S.; Sivasundaram, S., Dynamic systems on time scales and superlinear convergence of iterative process, WSSIAA, 3, 431-436 (1994) · Zbl 0884.58084
[4] Kaymakcalan, B., Monotone iterative method for dynamic systems on time scales, Dynamic Systems and Appl., 2, 213-220 (1993) · Zbl 0783.34005
[5] Akin, E., Boundary value problems for a differential equation on measure chain, Panam. Math. J., 10, 3, 17-30 (2000) · Zbl 0973.39010
[7] Merdivenci Atici, F.; Guseinov, G. Sh., Positive periodic solutions for nonlinear difference equations with periodic coefficients, J. Math. Anal. Appl., 232, 166-182 (1999) · Zbl 0923.39010
[8] Cabada, A.; Lois, S., An existence result for nonlinear second order periodic boundary value problems with lower and upper solutions in inverse order, J. Comput. Appl. Math., 110, 1, 105-114 (1999)
[9] Cabada, A.; Nieto, J., A generalization of the monotone iterative technique for nonlinear second order periodic boundary value problems, J. Math. Appl., 151, 181-189 (1990) · Zbl 0719.34039
[10] Cabada, A., The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl., 185, 302-320 (1994) · Zbl 0807.34023
[12] Rudolf, B.; Kubadek, Z., Remarks on J.J. Nieto’s paper: Nonlinear second-order periodic boundary value problems, J. Math. Anal. Appl., 146, 203-206 (1990) · Zbl 0713.34015
[13] Zhuang, W.; Chen, Y.; Cheng, S. S., Monotone methods for a discrete boundary problem, Computers Math. Applic, 32, 12, 41-49 (1996) · Zbl 0872.39005
[14] Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results Math., 35, 1-2, 3-22 (1999) · Zbl 0927.39003
[15] Erbe, L.; Peterson, A., Positive solutions for a nonlinear differential equation on a measure chain, Mathl. Comput. Modelling, 32, 5/6, 571-586 (2000) · Zbl 0963.34020
[16] Kaymakcalan, B.; Leela, S., A survey of dynamic systems on time scales, Nonlinear Times and Digest, 1, 37-60 (1994) · Zbl 0811.34012
[17] Kaymakcalan, B., Existence and comparison results for dynamic systems on a time scale, J. Math. Anal. Appl., 172, 243-255 (1991) · Zbl 0791.34007
[18] Bohner, M.; Peterson, A., Dynamic Equations on time scales, An Introduction with Applications (2001), Birkhauser: Birkhauser New York · Zbl 0978.39001
[19] Kaymakcalan, B.; Laksmikantham, V.; Sivasundaram, S., Dynamical systems on measure chains, (Math. and its Appl., 370 (1996), Kluwer Acedemic) · Zbl 0869.34039
[20] Hilger, S., Differential and difference calculus-unified!, Nonlinear Analysis, 30, 5, 2683-2694 (1997) · Zbl 0927.39002
[21] Krasnosel’skii, M. A., Positive Solutions of Operator Equations (1964) · Zbl 0121.10604
[22] Topal, S. G., Calculus and Dynamic Equations on Time Scales, Ph.D. Thesis (2001)
[23] Cabada, A.; Espinar, V. O., Optimal existence results for nth order periodic boundary value difference equations, J. Math. Anal. App., 247, 1, 67-86 (2000) · Zbl 0962.39006
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