American Institute of Mathematical Sciences

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doi: 10.3934/dcdss.2020336

New results for oscillation of fractional partial differential equations with damping term

Liping Luo 1,2, , Zhenguo Luo 1,2,, and  Yunhui Zeng 1,2,

1. 

College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan 421002, P. R. China
2. 

Hunan Provincial Key Laboratory of Intelligent Information, Processing and Application, Hengyang, 421002, P. R. China

* Corresponding author: Zhenguo Luo

Received  March 2019 Revised  September 2019 Published  April 2020

Fund Project: The authors are supported by Hunan Provincial Natural Science Foundation of China (2019JJ40004, 2018JJ2006), the Project Supported by Scientific Research Fund of Hunan Provincial Education Department (17A030, 16A031), the Project of "Double First-Class" Applied Characteristic Discipline in Hunan Province (Xiangjiaotong[2018]469), the Project of Hunan Provincial Key Laboratory (2016TP1020) and the Open Fund Project of Hunan Provincial Key Laboratory of Intelligent Information Processing and Application for Hengyang Normal University (IIPA18K05), the training target of the young backbone teachers in Hunan colleges and Universities ([2015]361)

In this paper, we study the oscillatory behavior of solutions of a class of damped fractional partial differential equations subject to Robin and Dirichlet boundary value conditions. By using integral averaging technique and Riccati type transformations, we obtain some new sufficient conditions for oscillation of all solutions of this kind of fractional differential equations with damping term. Our results essentially enrich the ones in the existing literature. Finally, we also give two specific examples to illustrate our main results.

Keywords: Oscillation, fractional partial differential equation, damping term, integral averaging technique, Riccati transformation.
Mathematics Subject Classification: Primary: 35B05, 35R11; Secondary: 26A33.
Citation: Liping Luo, Zhenguo Luo, Yunhui Zeng. New results for oscillation of fractional partial differential equations with damping term. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020336
References:
[1]

S. Abbas, M. Benchohra and G. M. N'Guérékata, Topics in Fractional Differential Equations, Developments in Mathematics, 27. Springer, New York, 2012. doi: 10.1007/978-1-4614-4036-9.  Google Scholar

[2]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.  Google Scholar

[3]

C. C. Bernido and M. V. Carpio-Bernido, Analysis of Fractional Stochastic Processes: Advances and Applications, Conference Series, 36. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9257.  Google Scholar

[4]

S. T. Chen, X. H. Tang and J. S. Yu, Sign-changing ground state solutions for discrete nonlinear Schrodinger equations, J. Difference Equ. Appl., 25 (2019), 202-218.  doi: 10.1080/10236198.2018.1563601.  Google Scholar

[5]

S. S. Chen and J. S. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete and Continuous Dynamical Systems, 38 (2018), 43-62.  doi: 10.3934/dcds.2018002.  Google Scholar

[6]

R. Courant and D. Hilbert, Methods of Mathematical Physics. II: Partial Differential Equations, Interscience Publishers, New York-London, 1962.  Google Scholar

[7]

S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, 2008.  Google Scholar

[8]

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[9]

L. Erbe, B. G. Jia and Q. Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, J. Appl.Anal. Comput., 9 (2019), 271-294.  doi: 10.11948/2019.271.  Google Scholar

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Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506-515.  doi: 10.1007/BF02884022.  Google Scholar

[11]

Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430.  doi: 10.1112/S0024610703004563.  Google Scholar

[12]

I. Györi and G. Ladas, Oscillation Theory of Delay Differntial Equations: with Applications, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.  Google Scholar

[13]

S. Harikrishnan, P. Prakash and J. J. Nieto, Foreced oscillation of solutions of a nonlinear fractional partial differential equation, Appl. Math. Comput., 254 (2015), 14-19.  doi: 10.1016/j.amc.2014.12.074.  Google Scholar

[14] J. H. Huang, L. Xin and T. L. Shen, Dynamics of Fractional Partial Differential Equations, Science Press, Beijing, 2017.   Google Scholar
[15]

Y. X. Hui, G. H. Lin and Q. W. Sun, Oscillation threshold for a mosquito population suppression model with time delay, Mathematical Biosciences and Engineering, 16 (2019), 7362-7374.  doi: 10.3934/mbe.2019367.  Google Scholar

[16]

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

[17]

W. N. Li, On the forced oscillation of certain fractional partial differential equations, Appl. Math. Lett., 50 (2015), 5-9.  doi: 10.1016/j.aml.2015.05.016.  Google Scholar

[18]

W. N. Li, Forced oscillation criteria for a class of fractional partial differential equations with damping term, Mathematical Problems in Engineering, 2015 (2015), 1-6. doi: 10.1155/2015/410904.  Google Scholar

[19]

W. N. Li, Oscillation of solutions for certain fractional partial differential equations, Advances in Difference Equations, 2016 (2016), 1-8. doi: 10.1186/s13662-016-0756-z.  Google Scholar

[20]

W. N. Li and W. H. Sheng, Oscillation properties for solutions of a kind of partial fractional differential equations with damping term, J. Nonlinear Sci. Appl., 9 (2016), 1600-1608.  doi: 10.22436/jnsa.009.04.17.  Google Scholar

[21]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[22]

P. Prakash, S. Harikrishnan, J. J. Nieto and J.-H. Kim, Oscillation of a time fractional partial differential equation, Electron. J. Qual. Theory Differ. Equ., 15 (2014), 1-10.  Google Scholar

[23]

P. Prakash, S. Harikrishnan and M. Benchohra, Oscillation of certain nonlinear fractional partial differential equation with damping term, Appl. Math. Lett., 43 (2015), 72-79.  doi: 10.1016/j.aml.2014.11.018.  Google Scholar

[24]

A. Raheem and Md. Maqbul, Oscillation criteria for impulsive partial fractional differential equations, Computers and Mathematics with Applications, 73 (2017), 1781-1788.  doi: 10.1016/j.camwa.2017.02.016.  Google Scholar

[25]

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

[26]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dyn. Diff. Equat., 31 (2019), 369-383.  doi: 10.1007/s10884-018-9662-2.  Google Scholar

[27] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg, Higher Education Press, Beijing, 2010.  doi: 10.1007/978-3-642-14003-7.  Google Scholar
[28]

J. S. Yu, Modeling mosquito population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917.  Google Scholar

[29]

J. S. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical model, J. Difference Equ. Appl., 25 (2019), 1549-1567.  doi: 10.1080/10236198.2019.1669578.  Google Scholar

[30]

Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Communications on Pure and Applied Analysis, 18 (2019), 425-434.  doi: 10.3934/cpaa.2019021.  Google Scholar

[31]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 2014. Google Scholar

show all references

References:
[1]

S. Abbas, M. Benchohra and G. M. N'Guérékata, Topics in Fractional Differential Equations, Developments in Mathematics, 27. Springer, New York, 2012. doi: 10.1007/978-1-4614-4036-9.  Google Scholar

[2]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/9789814355216.  Google Scholar

[3]

C. C. Bernido and M. V. Carpio-Bernido, Analysis of Fractional Stochastic Processes: Advances and Applications, Conference Series, 36. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9257.  Google Scholar

[4]

S. T. Chen, X. H. Tang and J. S. Yu, Sign-changing ground state solutions for discrete nonlinear Schrodinger equations, J. Difference Equ. Appl., 25 (2019), 202-218.  doi: 10.1080/10236198.2018.1563601.  Google Scholar

[5]

S. S. Chen and J. S. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete and Continuous Dynamical Systems, 38 (2018), 43-62.  doi: 10.3934/dcds.2018002.  Google Scholar

[6]

R. Courant and D. Hilbert, Methods of Mathematical Physics. II: Partial Differential Equations, Interscience Publishers, New York-London, 1962.  Google Scholar

[7]

S. Das, Functional Fractional Calculus for System Identification and Controls, Springer, Berlin, 2008.  Google Scholar

[8]

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.  Google Scholar

[9]

L. Erbe, B. G. Jia and Q. Q. Zhang, Homoclinic solutions of discrete nonlinear systems via variational method, J. Appl.Anal. Comput., 9 (2019), 271-294.  doi: 10.11948/2019.271.  Google Scholar

[10]

Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506-515.  doi: 10.1007/BF02884022.  Google Scholar

[11]

Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430.  doi: 10.1112/S0024610703004563.  Google Scholar

[12]

I. Györi and G. Ladas, Oscillation Theory of Delay Differntial Equations: with Applications, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.  Google Scholar

[13]

S. Harikrishnan, P. Prakash and J. J. Nieto, Foreced oscillation of solutions of a nonlinear fractional partial differential equation, Appl. Math. Comput., 254 (2015), 14-19.  doi: 10.1016/j.amc.2014.12.074.  Google Scholar

[14] J. H. Huang, L. Xin and T. L. Shen, Dynamics of Fractional Partial Differential Equations, Science Press, Beijing, 2017.   Google Scholar
[15]

Y. X. Hui, G. H. Lin and Q. W. Sun, Oscillation threshold for a mosquito population suppression model with time delay, Mathematical Biosciences and Engineering, 16 (2019), 7362-7374.  doi: 10.3934/mbe.2019367.  Google Scholar

[16]

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

[17]

W. N. Li, On the forced oscillation of certain fractional partial differential equations, Appl. Math. Lett., 50 (2015), 5-9.  doi: 10.1016/j.aml.2015.05.016.  Google Scholar

[18]

W. N. Li, Forced oscillation criteria for a class of fractional partial differential equations with damping term, Mathematical Problems in Engineering, 2015 (2015), 1-6. doi: 10.1155/2015/410904.  Google Scholar

[19]

W. N. Li, Oscillation of solutions for certain fractional partial differential equations, Advances in Difference Equations, 2016 (2016), 1-8. doi: 10.1186/s13662-016-0756-z.  Google Scholar

[20]

W. N. Li and W. H. Sheng, Oscillation properties for solutions of a kind of partial fractional differential equations with damping term, J. Nonlinear Sci. Appl., 9 (2016), 1600-1608.  doi: 10.22436/jnsa.009.04.17.  Google Scholar

[21]

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[22]

P. Prakash, S. Harikrishnan, J. J. Nieto and J.-H. Kim, Oscillation of a time fractional partial differential equation, Electron. J. Qual. Theory Differ. Equ., 15 (2014), 1-10.  Google Scholar

[23]

P. Prakash, S. Harikrishnan and M. Benchohra, Oscillation of certain nonlinear fractional partial differential equation with damping term, Appl. Math. Lett., 43 (2015), 72-79.  doi: 10.1016/j.aml.2014.11.018.  Google Scholar

[24]

A. Raheem and Md. Maqbul, Oscillation criteria for impulsive partial fractional differential equations, Computers and Mathematics with Applications, 73 (2017), 1781-1788.  doi: 10.1016/j.camwa.2017.02.016.  Google Scholar

[25]

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

[26]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dyn. Diff. Equat., 31 (2019), 369-383.  doi: 10.1007/s10884-018-9662-2.  Google Scholar

[27] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg, Higher Education Press, Beijing, 2010.  doi: 10.1007/978-3-642-14003-7.  Google Scholar
[28]

J. S. Yu, Modeling mosquito population suppression based on delay differential equations, SIAM J. Appl. Math., 78 (2018), 3168-3187.  doi: 10.1137/18M1204917.  Google Scholar

[29]

J. S. Yu and B. Zheng, Modeling Wolbachia infection in mosquito population via discrete dynamical model, J. Difference Equ. Appl., 25 (2019), 1549-1567.  doi: 10.1080/10236198.2019.1669578.  Google Scholar

[30]

Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Communications on Pure and Applied Analysis, 18 (2019), 425-434.  doi: 10.3934/cpaa.2019021.  Google Scholar

[31]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Singapore, 2014. Google Scholar

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