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doi: 10.3934/dcdss.2020443

Oscillation criteria for kernel function dependent fractional dynamic equations

Bahaaeldin Abdalla 1, and  Thabet Abdeljawad 1,2,3,,

1. 

Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia
2. 

Department of Medical Research, China Medical University, Taichung 40402, Taiwan
3. 

Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

* Corresponding author: tabdeljawad@psu.edu.sa

Received  November 2019 Revised  March 2020 Published  November 2020

In this work, we examine the oscillation of a class fractional differential equations in the frame of generalized nonlocal fractional derivatives with function dependent kernel type. We present sufficient conditions to prove the oscillation criteria in both of the Riemann-Liouville (RL) and Caputo types. Taking particular cases of the nondecreasing function appearing in the kernel of the treated fractional derivative recovers the oscillation of several proven results investigated previously in literature. Two examples, where the kernel function is quadratic and cubic polynomial, have been given to support the validity of the proven results for the RL and Caputo cases, respectively.

Keywords: General fractional integrals, derivative of a function with respect to a function, fractional differential equations, oscillation theory, Young's inequality, singular function dependent kernel.
Mathematics Subject Classification: Primary: 34A08, 34C10; Secondary: 26A33.
Citation: Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020443
References:
[1]

B. Abdalla, On the oscillation of q-fractional difference equations, Advances of Difference Equations, 2017 (2017), Paper No. 254, 11 pp. doi: 10.1186/s13662-017-1316-x.  Google Scholar

[2]

B. Abdalla, Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives, Advances of Difference Equations, 2018 (2018), Paper No. 107, 15 pp. doi: 10.1186/s13662-018-1554-6.  Google Scholar

[3]

B. Abdalla and T. Abdeljawad, On the oscillation of Hadamard fractional differential equations, Advances of Difference Equations, 2018 (2018), Paper No. 409, 13 pp. doi: 10.1186/s13662-018-1870-x.  Google Scholar

[4]

B. Abdalla and T. Abdeljawad, On the oscillation of Caputo fractional differential equations with Mittag-Leffler nonsingular kernel, Chaos, Solitons Fractals, 127 (2019), 173-177.  doi: 10.1016/j.chaos.2019.07.001.  Google Scholar

[5]

Y. Adjabi, F. Jarad, D. Baleanu and T. Abdeljawad, On Cauchy problems with Caputo Hadamard fractional derivatives, Journal of Computational Analysis and Applications, 21 (2016), 661-681.   Google Scholar

[6]

J. Alzabut and T. Abdeljawad, Sufficient conditions for the oscillation of nonlinear fractional difference equations, Journal of Fractional Calculus and Applications, 5 (2014), 177-187.   Google Scholar

[7]

A. Aphithana, S. K. Ntouyas and J. Tariboon, Forced oscillation of fractional differential equations via conformable derivatives with damping term, Boundary Value Problems, 2019 (2019), Paper No. 47, 16 pp. doi: 10.1186/s13661-019-1162-8.  Google Scholar

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A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Thermal Science, 20 (2016), 757-763.   Google Scholar

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Y. Bolat, On the oscillation of fractional order delay differential equations with constant coefficients, Commun Nonlinear Sci Numer. Simul., 19 (2014), 3988-3993.  doi: 10.1016/j.cnsns.2014.01.005.  Google Scholar

[10]

D. X. Chen, Oscillation criteria of fractional differential equations, Advances in Difference Equations, 2012 (2012), Art. No. 33, 10 pp. doi: 10.1186/1687-1847-2012-33.  Google Scholar

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D. Chen, P. Qu and Y. Lan, Forced oscillation of certain fractional differential equations, Advances in Difference Equations, 2013 (2013), Art No. 125, 10 pp. doi: 10.1186/1687-1847-2013-125.  Google Scholar

[12]

S. R. Grace, R. P. Agarwal, P. J. Y. Wong and A. Zafer, On the oscillation of fractional differential equations, Fractional Calculus Applied Analysis, 15 (2012), 222-231.  doi: 10.2478/s13540-012-0016-1.  Google Scholar

[13]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd edition, Cambridge University Press, Cambridge, 1988.  Google Scholar

[14]

F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems - S, 13 (2020), 709-722.  doi: 10.3934/dcdss.2020039.  Google Scholar

[15]

F. Jarad, T. Abdeljawad and D. Baleanu, Captuto-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012 (2012), Art No. 142, 8 pp. doi: 10.1186/1687-1847-2012-142.  Google Scholar

[16]

A. A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[17]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[18]

J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas–Milovic model with Mittag–Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), Paper No. 303, 23 pp. doi: 10.1051/mmnp/2018068.  Google Scholar

[19]

Y. Zhou, B. Ahmmad, F. Chen and A. Alsaedi, Oscialltion of fractional partial differential equations, Bull. Malays. Math. Soc., 42 (2017), 449-465.  doi: 10.1007/s40840-017-0495-7.  Google Scholar

[20]

P. Zhu and Q. Xiang, Oscillation criteria for a class of fractioal delay differential equations, Advances in Difference Equations, 2018 (2018), Paper No. 403, 11 pp. doi: 10.1186/s13662-018-1813-6.  Google Scholar

show all references

References:
[1]

B. Abdalla, On the oscillation of q-fractional difference equations, Advances of Difference Equations, 2017 (2017), Paper No. 254, 11 pp. doi: 10.1186/s13662-017-1316-x.  Google Scholar

[2]

B. Abdalla, Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives, Advances of Difference Equations, 2018 (2018), Paper No. 107, 15 pp. doi: 10.1186/s13662-018-1554-6.  Google Scholar

[3]

B. Abdalla and T. Abdeljawad, On the oscillation of Hadamard fractional differential equations, Advances of Difference Equations, 2018 (2018), Paper No. 409, 13 pp. doi: 10.1186/s13662-018-1870-x.  Google Scholar

[4]

B. Abdalla and T. Abdeljawad, On the oscillation of Caputo fractional differential equations with Mittag-Leffler nonsingular kernel, Chaos, Solitons Fractals, 127 (2019), 173-177.  doi: 10.1016/j.chaos.2019.07.001.  Google Scholar

[5]

Y. Adjabi, F. Jarad, D. Baleanu and T. Abdeljawad, On Cauchy problems with Caputo Hadamard fractional derivatives, Journal of Computational Analysis and Applications, 21 (2016), 661-681.   Google Scholar

[6]

J. Alzabut and T. Abdeljawad, Sufficient conditions for the oscillation of nonlinear fractional difference equations, Journal of Fractional Calculus and Applications, 5 (2014), 177-187.   Google Scholar

[7]

A. Aphithana, S. K. Ntouyas and J. Tariboon, Forced oscillation of fractional differential equations via conformable derivatives with damping term, Boundary Value Problems, 2019 (2019), Paper No. 47, 16 pp. doi: 10.1186/s13661-019-1162-8.  Google Scholar

[8]

A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Thermal Science, 20 (2016), 757-763.   Google Scholar

[9]

Y. Bolat, On the oscillation of fractional order delay differential equations with constant coefficients, Commun Nonlinear Sci Numer. Simul., 19 (2014), 3988-3993.  doi: 10.1016/j.cnsns.2014.01.005.  Google Scholar

[10]

D. X. Chen, Oscillation criteria of fractional differential equations, Advances in Difference Equations, 2012 (2012), Art. No. 33, 10 pp. doi: 10.1186/1687-1847-2012-33.  Google Scholar

[11]

D. Chen, P. Qu and Y. Lan, Forced oscillation of certain fractional differential equations, Advances in Difference Equations, 2013 (2013), Art No. 125, 10 pp. doi: 10.1186/1687-1847-2013-125.  Google Scholar

[12]

S. R. Grace, R. P. Agarwal, P. J. Y. Wong and A. Zafer, On the oscillation of fractional differential equations, Fractional Calculus Applied Analysis, 15 (2012), 222-231.  doi: 10.2478/s13540-012-0016-1.  Google Scholar

[13]

G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd edition, Cambridge University Press, Cambridge, 1988.  Google Scholar

[14]

F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems - S, 13 (2020), 709-722.  doi: 10.3934/dcdss.2020039.  Google Scholar

[15]

F. Jarad, T. Abdeljawad and D. Baleanu, Captuto-type modification of the Hadamard fractional derivatives, Advances in Difference Equations, 2012 (2012), Art No. 142, 8 pp. doi: 10.1186/1687-1847-2012-142.  Google Scholar

[16]

A. A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, vol. 204, Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[17]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[18]

J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas–Milovic model with Mittag–Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), Paper No. 303, 23 pp. doi: 10.1051/mmnp/2018068.  Google Scholar

[19]

Y. Zhou, B. Ahmmad, F. Chen and A. Alsaedi, Oscialltion of fractional partial differential equations, Bull. Malays. Math. Soc., 42 (2017), 449-465.  doi: 10.1007/s40840-017-0495-7.  Google Scholar

[20]

P. Zhu and Q. Xiang, Oscillation criteria for a class of fractioal delay differential equations, Advances in Difference Equations, 2018 (2018), Paper No. 403, 11 pp. doi: 10.1186/s13662-018-1813-6.  Google Scholar

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