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## Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function

#### Jie BAİ [1] , Ling BAİ [2] , Zhijun ZENG [3]

In this paper, we study the calculus of variations of the nabla notion on time scales including $\nabla$-derivative, $\nabla$-integral, and $\nabla$-derivatives of exponential function. The Euler-Lagrange equations of the first-order both single-variable problem and multivariable problem with nabla derivatives of exponential function on time scales are obtained. In particular, we show that the calculus of variations with multiple variables could solve the problem of conditional extreme value. Moreover, we verify the solution to the multivariable problem is exactly the extremum pair. As applications of these results, an example of conditional extremum is provided.
time scales, the Euler-Lagrange equation, calculus of variations, conditional extremum, $\nabla$-derivatives of exponential function
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Primary Language en Mathematics Mathematics Orcid: 0000-0002-0127-4623Author: Jie BAİ Institution: Liaoning UniversityCountry: China Orcid: 0000-0001-6837-9983Author: Ling BAİ Institution: Beijing Municipal Commission Of Health And Family Planning Information CenterCountry: China Orcid: 0000-0002-7114-0559Author: Zhijun ZENG (Primary Author)Institution: Northeast Normal UniversityCountry: China Publication Date : February 6, 2020
 Bibtex @research article { hujms504587, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {68 - 77}, doi = {10.15672/HJMS.2018.653}, title = {Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function}, key = {cite}, author = {BAİ, Jie and BAİ, Ling and ZENG, Zhijun} } APA BAİ, J , BAİ, L , ZENG, Z . (2020). Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function. Hacettepe Journal of Mathematics and Statistics , 49 (1) , 68-77 . DOI: 10.15672/HJMS.2018.653 MLA BAİ, J , BAİ, L , ZENG, Z . "Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 68-77 Chicago BAİ, J , BAİ, L , ZENG, Z . "Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 68-77 RIS TY - JOUR T1 - Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function AU - Jie BAİ , Ling BAİ , Zhijun ZENG Y1 - 2020 PY - 2020 N1 - doi: 10.15672/HJMS.2018.653 DO - 10.15672/HJMS.2018.653 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 68 EP - 77 VL - 49 IS - 1 SN - 2651-477X-2651-477X M3 - doi: 10.15672/HJMS.2018.653 UR - https://doi.org/10.15672/HJMS.2018.653 Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function %A Jie BAİ , Ling BAİ , Zhijun ZENG %T Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 1 %R doi: 10.15672/HJMS.2018.653 %U 10.15672/HJMS.2018.653 ISNAD BAİ, Jie , BAİ, Ling , ZENG, Zhijun . "Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function". Hacettepe Journal of Mathematics and Statistics 49 / 1 (February 2020): 68-77 . https://doi.org/10.15672/HJMS.2018.653 AMA BAİ J , BAİ L , ZENG Z . Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 68-77. Vancouver BAİ J , BAİ L , ZENG Z . Calculus of Variations on Time Scales with Nabla Derivatives of Exponential Function. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 77-68.

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