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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 |
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.
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Nonlinear continuous Fornasini-Marchesini model of fractional order with nonzero initial conditions, Journal of Integral Equations and Applications, 32 (2020), 19-34.
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A fractional multiterm Gronwall lemma and its application to stability of fractional integrodifferential systems, Journal of Integral Equations and Applications, 32 (2020), 447-456.
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Computing optimal control with hyperbolic partial differential equation, J. Aust. Math. Soc. Ser. B, 40 (1998), 266-287.
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A. N. Tikhonov and A. A. Samarski, Equations of Mathematical Physics, Dover Publications, Inc., New York, 1990. |
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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. |
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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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [8] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1993. |
| [9] |
A. N. Tikhonov and A. A. Samarski, Equations of Mathematical Physics, Dover Publications, Inc., New York, 1990. |
| [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. |
| [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. |
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