Skip to main content
SpringerLink
Log in

Solutions for a class of Hamiltonian systems on time scales with non-local boundary conditions

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

In this work, the solvability of a class of second-order Hamiltonian systems on time scales is generalized to non-local boundary conditions. The measurements obtained by non-local conditions are more accurate than those given by local conditions in some problems. Compared with the known results, this work establishes the variational structure in an appropriate Sobolev’s space. Then, by applying the mountain pass theorem and symmetric mountain pass theorem, the existence and multiplicity of the solutions are obtained. Finally, some examples with numerical simulation results are given to illustrate the correctness of the results obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. HILGER, S. Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph. D. dissertation, Universität Würzburg (1988)

  2. AULBACH, B. and HILGER, S. A unified approach to continuous and discrete dynamics. Qualitative Theory of Differential Equations, 53, 37–56 (1990)

    MathSciNet  MATH  Google Scholar 

  3. BOHNER, M. and PETERSON, A. Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser Basel, Boston (2001)

    Book  Google Scholar 

  4. AGARWAL, R. P., VICTORIA, O., KANISHKA, P., and VIVERO, D. R. Basic properties of Sobolev’s spaces on time scales. Advances in Difference Equations, 2006, 038121 (2006)

    MathSciNet  MATH  Google Scholar 

  5. ZHOU, J. W. and LI, Y. K. Sobolev’s spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales. Nonlinear Analysis: Theory, Methods and Applications, 73(5), 1375–1388 (2010)

    Article  MathSciNet  Google Scholar 

  6. CABADA, A. and VIVERO, D. R. Criterions for absolute continuity on time scales. Journal of Difference Equations and Applications, 11(11), 1013–1028 (2005)

    Article  MathSciNet  Google Scholar 

  7. GUSEINOV, G. S. Integration on time scales. Journal of Mathematical Analysis and Applications, 285(1), 107–127 (2003)

    Article  MathSciNet  Google Scholar 

  8. SU, Y. H. and FENG, Z. S. Homoclinic orbits and periodic solutions for a class of Hamiltonian systems on time scales. Journal of Mathematical Analysis and Applications, 411(1), 37–62 (2014)

    Article  MathSciNet  Google Scholar 

  9. LI, Y. K., ZHAO, L., and CHEN, X. R. Positive periodic solutions of functional differential equations with impulse on time scales. Journal of Applied Mathematics and Computing, 34, 495–510 (2010)

    Article  MathSciNet  Google Scholar 

  10. AGARWAL, R., BOHNER, M., O’REGAN, D., and PETERSON, A. Dynamic equations on time scales: a survey. Journal of Applied Mathematics and Computing, 141(1/2), 1–26 (2002)

    MathSciNet  MATH  Google Scholar 

  11. AGARWAL, R. P., BOHNER, M., and O’REGAN, D. Time scale boundary value problems on infinite intervals. Journal of Applied Mathematics and Computing, 141(1/2), 27–34 (2002)

    Article  MathSciNet  Google Scholar 

  12. GUNENDI, M. and YASLAN, I. Existence of positive solutions for multi-point time scale boundary value problems on infinite intervals. Journal of Nonlinear Functional Analysis, 2017, 45 (2017)

    Article  Google Scholar 

  13. KUZNETSOVA, M. A. A uniqueness theorem on inverse spectral problems for the Sturm-Liouville differential operators on time scales. Results in Mathematics, 75, 44 (2020)

    Article  MathSciNet  Google Scholar 

  14. CARL, S. and HEIKKILÄ, S. Fixed Point Theory in Ordered Sets and Applications, Springer, New York (2011)

    Book  Google Scholar 

  15. AGARWAL, R. P., VICTORIA, O., KANISHKA, P., and VIVERO, D. R. Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. Nonlinear Analysis: Theory, Methods and Applications, 67(2), 368–381 (2007)

    Article  MathSciNet  Google Scholar 

  16. ERBE, L., PETERSON, A., and WU, H. W. Oscillation of solution to second-order half-linear delay dynamic equations on time scales. Electronic Journal of Differential Equations, 71, 1–15 (2016)

    MathSciNet  MATH  Google Scholar 

  17. SU, Y. H. and FENG, Z. S. A non-autonomous Hamiltonian system on time scales. Nonlinear Analysis: Theory, Methods and Applications, 75(10), 4126–4136 (2012)

    Article  MathSciNet  Google Scholar 

  18. DOGAN, A. Positive solutions of the p-Laplacian dynamic equations on time scales with sign changing nonlinearity. Electronic Journal of Differential Equations, 39, 1–17 (2018)

    MathSciNet  MATH  Google Scholar 

  19. LIU, H. T. and ZHOU, Z. G. Dynamic behavior of rectangular crack in three-dimensional orthotropic elastic medium by means of non-local theory. Applied Mathematics and Mechanics (English Edition), 38(2), 173–190 (2017) https://doi.org/10.1007/s10483-017-2161-9

    Article  MathSciNet  Google Scholar 

  20. PACHPATTE, D. B. Properties of certain iterated dynamic integrodiffetential equation on time scales. Applied Mathematics and Computation, 346, 767–775 (2019)

    Article  MathSciNet  Google Scholar 

  21. LI, Y. K. and ZHOU, J. W. Existence of solutions for a class of damped vibration problems on time scales. Advances in Difference Equations, 2010, 727486 (2010)

    Article  MathSciNet  Google Scholar 

  22. LUO, Z. G., XIE, J. L., and CHEN, G. P. Existence of solutions of a second-order impulsive differential equation. Advances in Difference Equations, 2014, 118 (2014)

    Article  MathSciNet  Google Scholar 

  23. GUO, B. L. and ZHOU, G. L. Exponential stability of stochastic generalized porous media equations with jump. Applied Mathematics and Mechanics (English Edition), 35(8), 1067–1078 (2014) https://doi.org/10.1007/s10483-014-1845-7

    Article  MathSciNet  Google Scholar 

  24. BAI, L., DAI, B. X., and NIETO, J. J. Necessary and sufficient conditions for the existence of nonconstant solutions generated by impulses of second order BVPs with convex potential. Electronic Journal of Qualitative Theory of Differential Equations, 1, 1–13 (2018)

    Article  Google Scholar 

  25. WEI, Y. F. and BAI, Z. B. Multiple solutions for some nonlinear impulsive differential equations with three-point boundary conditions via variational approach. Journal of Applied Analysis and Computation, 11(6), 3031–3043 (2021)

    Article  MathSciNet  Google Scholar 

  26. ZHOU, H. Complex variable solution for boundary value problem with X-shaped cavity in plane elasticity and its application. Applied Mathematics and Mechanics (English Edition), 38(9), 1329–1346 (2017) https://doi.org/10.1007/s10483-017-2235-8

    Article  MathSciNet  Google Scholar 

  27. WANG, C., AGARWAL, R. P., O’REGAN, D., and SAKTHIVEL, R. Theory of Translation Closedness for Time Scales: with Applications in Translation Functions and Dynamic Equations, Springer, Cham (2020)

    Book  Google Scholar 

  28. ZHANG, P. and QING, H. Two-phase non-local integral models with a bi-Helmholtz averaging kernel for nanorods. Applied Mathematics and Mechanics (English Edition), 42(10), 1379–1396 (2021) https://doi.org/10.1007/s10483-021-2774-9

    Article  MathSciNet  Google Scholar 

  29. MABOOD, F., MACKOLIL, J., MAHANTHESH, B., RAUF, A., and SHEHZAD, S. A. Dynamics of Sutterby fluid flow due to a spinning stretching disk with non-Fourier/Fick heat and mass flux models. Applied Mathematics and Mechanics (English Edition), 42(9), 1247–1258 (2021) https://doi.org/10.1007/s10483-021-2770-9

    Article  MathSciNet  Google Scholar 

  30. MAWHIN, J. and WILLEM, M. Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)

    Book  Google Scholar 

  31. RABINOWITZ, P. H. Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, Rhode Island (1986)

    Book  Google Scholar 

  32. LIAN, W., BAI, Z. B., and DU, Z. J. Existence of solution of a three-point boundary value problem via variational approach. Applied Mathematics Letters, 104, 106283 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Funding

Project supported by the National Natural Science Foundation of China (No. 11571207), the Natural Science Foundation of Shandong Province of China (Nos. ZR2021MA064 and ZR2020MA017), and the Taishan Scholar Project of Shandong Province of China

Author information

Authors and Affiliations

  1. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, Shandong Province, 266590, China

    Yongfang Wei, Suiming Shang & Zhanbing Bai

Authors
  1. Yongfang Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suiming Shang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhanbing Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhanbing Bai.

Additional information

Citation: WEI, Y. F., SHANG, S. M., and BAI, Z. B. Solutions for a class of Hamiltonian systems on time scales with non-local boundary conditions. Applied Mathematics and Mechanics (English Edition), 43(4), 587–602 (2022) https://doi.org/10.1007/s10483-022-2832-9

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wei, Y., Shang, S. & Bai, Z. Solutions for a class of Hamiltonian systems on time scales with non-local boundary conditions. Appl. Math. Mech.-Engl. Ed. 43, 587–602 (2022). https://doi.org/10.1007/s10483-022-2832-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-022-2832-9

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation