# Lyapunov Inequalities for Two Dimensional Fractional Boundary-Value Problems with Mixed Fractional Derivatives

## Abstract

**:**

## 1. Introduction

**Theorem 1**

**Theorem 2**

- We obtain the Lyapunov-type inequalities, which provide the necessary conditions for the existence of nonzero positive solutions. Thanks to this, we can indicate when the nontrivial positive solution to the problem does not exist.
- Mixed fractional derivatives are considered, and because of that we can establish a connection to the fractional calculus of variations.

## 2. Preliminaries

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Theorem 3**

**Remark 1.**

## 3. Partial Differential Equation of the First Type

**Lemma 1.**

**Proof.**

**Theorem 4.**

**Proof.**

**Example 1.**

## 4. Partial Differential Equation of the Second Type

**Lemma 2.**

**Proof.**

**Theorem 5.**

**Proof.**

**Example 2.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ha, C.-W. Eigenvalues of a Sturm–Liouville problem and inequalities of Lyapunov-type. Proc. AMS
**1998**, 126, 3507–3511. [Google Scholar] [CrossRef] [Green Version] - Brown, R.C.; Hinton, D.B. Lyapunov inequalities and their applications. In Survey on Mathematical Inequalities, Mathematics and Its Applications; Rassias, T.M., Ed.; Springer: Berlin/Heidelberg, Germany, 2000; pp. 1–25. [Google Scholar]
- Medriveci, A.F.; Guseinov, G.S.; Kaymakcalan, B. On Lyapunov inequality in stability theory for Hill’s equation on time scales. J. Inequal. Appl.
**2000**, 5, 603–620. [Google Scholar] - Jleli, M.; Kirane, M.; Samet, B. On Lyapunov–type inequalities for a certain class of partial differential equations. Appl. Anal.
**2020**, 99, 40–49. [Google Scholar] [CrossRef] - Canada, A.C.; Monteiro, J.A.; Villegas, S. Lyapunov inequalitites for partial differential equations. J. Funct. Anal.
**2006**, 237, 176–193. [Google Scholar] [CrossRef] [Green Version] - de Nápoli, P.L.; Pinasco, J.P. Lyapunov–type inequalities for partial differential equations. J. Funct. Anal.
**2016**, 270, 1995–2018. [Google Scholar] [CrossRef] - Agarwal, R.P.; Bohner, M.; Özbekler, A. Lyapunov Inequalities and Applications; Springer: Cham, Switzerland, 2021. [Google Scholar]
- Agarwal, R.P.; Jleli, M.; Samet, B. On De La Vallée Poussin–type inequalities in higher dimension and applications. Appl. Math. Lett.
**2018**, 86, 264–269. [Google Scholar] [CrossRef] - Bohner, M.; Clark, S.; Ridenhour, J. Lyapunov inequalities for time scales. J. Inequal. Appl.
**2002**, 7, 61–67. [Google Scholar] [CrossRef] [Green Version] - De La Vallée Poussin, C. Sur l’équation différentielle linéqire du second order. Détermination d’une intégrale par deux valuers assignés. Extension aux équasions d’ordre n. J. Math. Pures Appl.
**1929**, 8, 125–144. [Google Scholar] - Saker, S.H.; Tunç, C.; Mahmoud, R.R. New Carlson–Bellman and Hardy–Littlewood dynamic inequalities. Math. Inequal. Appl.
**2018**, 21, 967–983. [Google Scholar] [CrossRef] [Green Version] - Ntouyas, S.K.; Ahmad, B.; Horikis, T.P. Recent developments of Lyapunov-type inequalities for fractional differential equations. In Differential and Integral Inequalities; Springer Optimization and Its, Applications; Andrica, D., Rassias, T.M., Eds.; Springer: Berlin/Heidelberg, Germany, 2019; pp. 619–686. [Google Scholar]
- Almeida, R.; Pooseh, S.; Torres, D.F.M. Computational Methods in the Fractional Calculus of Variations; Imperial College Press: Singapure, 2015. [Google Scholar]
- Bohner, M.; Hristova, S. Stability for generalized Caputo proportional fractional delay integro–differential equations. Bound. Value Probl.
**2022**, 14, 1–15. [Google Scholar] [CrossRef] - Klimek, M. On Solutions of Linear Fractional Differential Equations of a Variational Type; The Publishing Office of Czestochowa University of Technology: Czestochowa, Poland, 2009. [Google Scholar]
- Malinowska, A.B.; Odzijewicz, T.; Torres, D.F.M. Advanced Methods in the Fractional Calculus of Variations; Springer Briefs in Applied Sciences and Technology; Springer International Publishing: Cham, Switzerland, 2015. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, Ȯ.I. Fractional Integrals and Derivatives; Translated from the 1987 Russian original; Gordon and Breach: Yverdon, Switzerland, 1993. [Google Scholar]
- Tunç, O.; Tunxcx, C. Solution estimates to Caputo proportional fractional derivative delay integro—Differential equations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat.
**2023**, 12, 117. [Google Scholar] [CrossRef] - Ferreira, R.A.C. A Lyapunov-type inequality for a fractional boundary-value problem. Fract. Calc. Appl. Anal.
**2013**, 16, 978–984. [Google Scholar] [CrossRef] - Zayernouri, M.; Karniadakis, G.E. Fractional Sturm–Liouville eigen–problems: Theory and numerical approximation. J. Comput. Phys.
**2013**, 252, 495–517. [Google Scholar] [CrossRef] - Klimek, M.; Odzijewicz, T.; Malinowska, A.B. Variational methods for the fractional Sturm–Liouville problem. J. Math. Anal. Appl.
**2014**, 416, 402–426. [Google Scholar] [CrossRef] - Guezane-Lakoud, A.; Khaldi, R.; Torres, D.F.M. Lyapunov–type inequality for a fractional boundary-value problem with natural conditions. SeMA J.
**2018**, 75, 157–162. [Google Scholar] [CrossRef] [Green Version] - Eneeva, L.M. Lyapunov inequality for an equation with fractional derivatives with different origins. Vestnik KRAUNC. Fiz.-Mat. Nauki
**2019**, 28, 32–39. [Google Scholar] - Odzijewicz, T. Inequality criteria for existence of solutions to some fractional partial differential equations. Appl. Math. Lett.
**2020**, 101, 106075. [Google Scholar] [CrossRef] - Odzijewicz, T.; Malinowska, A.B.; Torres, D.F.M. Fractional calculus of variations of several independent variables. Eur. Phys. J.
**2013**, 222, 1813–1826. [Google Scholar] [CrossRef] [Green Version] - Odzijewicz, T. Variable o Fractional Isoperimetric Problem of Several Variables, Advances in the Theory and Applications of Non-integer Order Systems; Springer: Berlin/Heidelberg, Germany, 2013; Volume 257, pp. 133–139. [Google Scholar]

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**MDPI and ACS Style**

Odzijewicz, T.
Lyapunov Inequalities for Two Dimensional Fractional Boundary-Value Problems with Mixed Fractional Derivatives. *Axioms* **2023**, *12*, 301.
https://doi.org/10.3390/axioms12030301

**AMA Style**

Odzijewicz T.
Lyapunov Inequalities for Two Dimensional Fractional Boundary-Value Problems with Mixed Fractional Derivatives. *Axioms*. 2023; 12(3):301.
https://doi.org/10.3390/axioms12030301

**Chicago/Turabian Style**

Odzijewicz, Tatiana.
2023. "Lyapunov Inequalities for Two Dimensional Fractional Boundary-Value Problems with Mixed Fractional Derivatives" *Axioms* 12, no. 3: 301.
https://doi.org/10.3390/axioms12030301