doi: 10.3934/mcrf.2021019

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 Published  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. IdczakR. 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. IdczakK. 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. RehbockS. 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. YeJ. 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. IdczakR. 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. IdczakK. 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. RehbockS. 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. YeJ. 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

[1]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[2]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[3]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[4]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[5]

Seddigheh Banihashemi, Hossein Jafaria, Afshin Babaei. A novel collocation approach to solve a nonlinear stochastic differential equation of fractional order involving a constant delay. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021025

[6]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[7]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[8]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020

[9]

Qiao Liu. Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021041

[10]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[11]

Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, 2021, 14 (2) : 389-406. doi: 10.3934/krm.2021009

[12]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016

[13]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017

[14]

Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021

[15]

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021033

[16]

Z. Reichstein and B. Youssin. Parusinski's "Key Lemma" via algebraic geometry. Electronic Research Announcements, 1999, 5: 136-145.

[17]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008

[18]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073

[19]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

[20]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107

2019 Impact Factor: 0.857

Article outline

[Back to Top]