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doi: 10.3934/mcrf.2021019
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## A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order

 Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

* Corresponding author: Dariusz Idczak

Received  May 2020 Revised  January 2021 Early access March 2021

In the paper, a new Gronwall lemma for functions of two variables with singular integrals is proved. An application to weak relative compactness of the set of solutions to a fractional partial differential equation is given.

Citation: Dariusz Idczak. A Gronwall lemma for functions of two variables and its application to partial differential equations of fractional order. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021019
##### References:
 [1] N. Dunford and B. J. Pettis, Linear operators on summable functions, Trans. Amer. Math. Soc., 47 (1940), 323-392.  doi: 10.1090/S0002-9947-1940-0002020-4.  Google Scholar [2] D. Idczak, R. Kamocki and M. Majewski, Nonlinear continuous Fornasini-Marchesini model of fractional order with nonzero initial conditions, Journal of Integral Equations and Applications, 32 (2020), 19-34.  doi: 10.1216/JIE.2020.32.19.  Google Scholar [3] D. Idczak, K. Kibalczyc and S. Walczak, On an optimization problem with cost of rapid variation of control, J. Aust. Math. Soc. Ser. B., 36 (1994), 117-131.  doi: 10.1017/S0334270000010286.  Google Scholar [4] D. Idczak, A fractional multiterm Gronwall lemma and its application to stability of fractional integrodifferential systems, Journal of Integral Equations and Applications, 32 (2020), 447-456.  doi: 10.1216/jie.2020.32.447.  Google Scholar [5] R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Mathematical Methods in the Applied Sciences, 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.  Google Scholar [6] M. Majewski, Stability analysis of an optimal control problem for a hyperbolic equation, J. Optim. Theory Appl., 141 (2009), 127-146.  doi: 10.1007/s10957-008-9504-1.  Google Scholar [7] V. Rehbock, S. Wangs and K. L. Teo, Computing optimal control with hyperbolic partial differential equation, J. Aust. Math. Soc. Ser. B, 40 (1998), 266-287.  doi: 10.1017/S0334270000012510.  Google Scholar [8] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993.  Google Scholar [9] A. N. Tikhonov and A. A. Samarski, Equations of Mathematical Physics, Dover Publications, Inc., New York, 1990.  Google Scholar [10] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar [11] Z. Zhang and Z. Wei, A generalized Gronwall inequality and its application to fractional neural evolution inclusions, Journal of Inequalities and Applications, (2016), Paper No. 45, 18 pp. doi: 10.1186/s13660-016-0991-6.  Google Scholar

show all references

##### References:
 [1] N. Dunford and B. J. Pettis, Linear operators on summable functions, Trans. Amer. Math. Soc., 47 (1940), 323-392.  doi: 10.1090/S0002-9947-1940-0002020-4.  Google Scholar [2] D. Idczak, R. Kamocki and M. Majewski, Nonlinear continuous Fornasini-Marchesini model of fractional order with nonzero initial conditions, Journal of Integral Equations and Applications, 32 (2020), 19-34.  doi: 10.1216/JIE.2020.32.19.  Google Scholar [3] D. Idczak, K. Kibalczyc and S. Walczak, On an optimization problem with cost of rapid variation of control, J. Aust. Math. Soc. Ser. B., 36 (1994), 117-131.  doi: 10.1017/S0334270000010286.  Google Scholar [4] D. Idczak, A fractional multiterm Gronwall lemma and its application to stability of fractional integrodifferential systems, Journal of Integral Equations and Applications, 32 (2020), 447-456.  doi: 10.1216/jie.2020.32.447.  Google Scholar [5] R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Mathematical Methods in the Applied Sciences, 37 (2014), 1668-1686.  doi: 10.1002/mma.2928.  Google Scholar [6] M. Majewski, Stability analysis of an optimal control problem for a hyperbolic equation, J. Optim. Theory Appl., 141 (2009), 127-146.  doi: 10.1007/s10957-008-9504-1.  Google Scholar [7] V. Rehbock, S. Wangs and K. L. Teo, Computing optimal control with hyperbolic partial differential equation, J. Aust. Math. Soc. Ser. B, 40 (1998), 266-287.  doi: 10.1017/S0334270000012510.  Google Scholar [8] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993.  Google Scholar [9] A. N. Tikhonov and A. A. Samarski, Equations of Mathematical Physics, Dover Publications, Inc., New York, 1990.  Google Scholar [10] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081.  doi: 10.1016/j.jmaa.2006.05.061.  Google Scholar [11] Z. Zhang and Z. Wei, A generalized Gronwall inequality and its application to fractional neural evolution inclusions, Journal of Inequalities and Applications, (2016), Paper No. 45, 18 pp. doi: 10.1186/s13660-016-0991-6.  Google Scholar
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