SavedAdd your journal

Journals / Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics / 2026 / Volume: 75 Issue: 3

A Novel Application of Expansive Mappings and Cone Theory for Third-Order Impulsive Dynamic Equations

Svetlin G. Georgiev(Sorbonne University)0000-0001-8015-4226
Sibel Doğru Akgöl(ÇUKUROVA UNIVERSITY)0000-0003-3513-1046
Pages
—
Publication Date
2026-09-26
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

The applicability and practical relevance of our theoretical results are validated through an illustrative example.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We establish novel existence criteria for solutions to a class of third-order impulsive nonlinear dynamic equations on time scales, subject to homogeneous boundary conditions.
📊
Key Findings & Data:This approach enables us to first prove the existence of at least one solution and subsequently demonstrate the presence of at least two distinct solutions.
💡
Conclusion & Impact:The applicability and practical relevance of our theoretical results are validated through an illustrative example.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:BVPs · dynamic equation · impulse effects

Abstract

We establish novel existence criteria for solutions to a class of third-order impulsive nonlinear dynamic equations on time scales, subject to homogeneous boundary conditions. Although first and second-order impulsive dynamic equations are well studied, the analysis of third-order problems remains comparatively underdeveloped. To address this gap, we employ very recent advancements in fixed point theory, utilizing innovative concepts involving expansive mappings and operator addition. This approach enables us to first prove the existence of at least one solution and subsequently demonstrate the presence of at least two distinct solutions. A significant advantage of our methodology is that it does not impose restrictions on boundedness on the nonlinear components of the equation, a requirement common in most existing studies. The applicability and practical relevance of our theoretical results are validated through an illustrative example.

🔖

Cite This Article

Georgiev, S. G., Akgöl, S. D. (2026). A Novel Application of Expansive Mappings and Cone Theory for Third-Order Impulsive Dynamic Equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Volume: 75 Issue: 3. https://doi.org/10.31801/cfsuasmas.1779784

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (26)

  1. Agarwal, R. P. and Bohner, M. (1999). Basic calculus on time scales and some of its applications. Results in Mathematics, 35(1):3–22.
  2. Agarwal, R. P., Hazarika, B., and Tikare, S. (2024). Dynamic Equations on Time Scales and Applications. Chapman and Hall/CRC, 1 edition.
  3. Avramescu, C. and Vladimirescu, C. (2003). Some remarks on Krasnoselskii’s fixed point theorem. Fixed Point Theory, 4:3–13.
  4. Bohner, M. and Peterson, A. (2001). Dynamic Equations on Time Scales: An Introduction with Applications. Springer Science & Business Media.
  5. Bohner, M. and Peterson, A. (2002). Advances in Dynamic Equations on Time Scales. Springer Science & Business Media.
  6. Djebali, S. and Mebarki, K. (2019). Fixed point index for expansive perturbation of $k$-set contraction mappings. Topological Methods in Nonlinear Analysis, 54(2):613–640.
  7. Dugundji, J. and Granas, A. (1982). Fixed Point Theory, volume 1 of Monografie Matematyczne. PNW, Warsaw.
  8. Georgiev, S., Kheloufi, A., and Mebarki, K. (2024). Existence of classical solutions for a class of impulsive hamilton-jacobi equations. Palestine Journal of Mathematics, 13(4):1084–1097.
  9. Georgiev, S. and Zennir, K. (2021). Boundary Value Problems on Time Scales, volume II. CRC Press.
  10. Georgiev, S. G. (2024). Existence of solutions for a class of wave dynamic equations on time scales. Discrete and Continuous Dynamical Systems – Series S, 17(10):2967–2981.
  11. Georgiev, S. G., Akgöl, S. D., and Kuş, M. E. (2022). Existence of solutions for third order multi point impulsive boundary value problems on time scales. Miskolc Mathematical Notes, 23(2):677–690.
  12. Georgiev, S. G., Tikare, S., and Kumar, V. (2023). Existence of solutions for two-point integral boundary value problems with impulses. Qualitative Theory of Dynamical Systems, 22:97.
  13. Guan, W. and Wang, D. B. (2014). Positive solutions for third-order $p$-Laplacian functional dynamic equations on time scales. Abstract and Applied Analysis, page 676052.
  14. Karaca, I. Y. and Fen, F. T. (2016a). Existence of positive solutions for nonlinear third-order $m$-point impulsive boundary-value problems on time scales. Ukrainian Mathematical Journal, 68:458–474.
  15. Karaca, I. Y. and Fen, F. T. (2016b). On positive solutions of nonlinear third-order impulsive boundary value problems on time scales. Mediterranean Journal of Mathematics, 13:4447–4461.
  16. Khemmar, K., Mebarki, K., and Georgiev, S. (2024). Existence of solutions for a class of boundary value problems for weighted $p(x)$-Laplacian impulsive systems. Filomat, 38(21):7563–7577.
  17. Krasnosel’skii, M. A. (1964). Positive Solutions of Operator Equations. Wolters-Noordhoff.
  18. Li, Y. and Li, Y. (2011). Existence of solutions of boundary value problems for a nonlinear third-order impulsive dynamic system on time scales. Differential Equations and Applications, 3(3):309–322.
  19. Mouhous, M., Georgiev, S. G., and Mebarki, K. (2022). Existence of solutions for a class of first order boundary value problems. Archivum Mathematicum, 58:141–158.
  20. Qiu, Y. C. (2014). Nonoscillatory solutions to third-order neutral dynamic equations on time scales. Advances in Difference Equations, 2014(1):309.
  21. Qiu, Y. C., Jadlovská, I., Chiu, K. S., and Li, T. (2020). Existence of nonoscillatory solutions tending to zero of third-order neutral dynamic equations on time scales. Advances in Difference Equations, 2020:231.
  22. Scindia, P., Tikare, S., and El-Deeb, A. A. (2023). Ulam stability of first-order nonlinear impulsive dynamic equations. Boundary Value Problems, 2023(1):86.
  23. Sun, J. P. (2006). Existence of solution and positive solution of bvp for nonlinear third-order dynamic equation. Nonlinear Analysis, 64(3):629–636.
  24. Tikare, S. and Tisdell, C. C. (2020). Nonlinear dynamic equations on time scales with impulses and nonlocal conditions. Journal of Classical Analysis, 16(2):125–140.
  25. Xu, H. (2010). Existence and uniqueness of positive solutions for a third-order three-point problem on time scales. Electronic Journal of Qualitative Theory of Differential Equations, 2010(56):1–12.
  26. Şenel, M. T. (2013). Behavior of solutions of a third-order dynamic equation on time scales. Journal of Inequalities and Applications, 2013(1):47.