On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points

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October 2014, 19(8): 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points

Rafał Kamocki ^1, and Marek Majewski ^1,

1.	Department of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland, Poland

Received November 2013 Revised May 2014 Published August 2014

Abstract
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In the paper we consider a Dirichlet problem for a fractional differential equation. The main goal is to prove an existence and continuous dependence of solution on functional parameter $u$ for the above problem. To prove it we use a variational method.

Keywords: continuous dependence, Kuratowski-Painlevé limit., Fractional dirichlet problem, variational methods, saddle points, Riemann-Liouville derivative.

Mathematics Subject Classification: Primary: 34B08, 34A08; Secondary: 49J5.

Citation: Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2557-2568. doi: 10.3934/dcdsb.2014.19.2557

References:

[1]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990.
[2]	D. Bors, A. Skowron and S. Walczak, Optimal control and stability of elliptic systems with integral cost functional, Systems Science, 33 (2007), 13-26.
[3]	D. Bors and S. Walczak, Nonlinear elliptic systems with variable boundary data, Nonlinear Analysis: Theory, Methods and Applications, 52 (2003), 1347-1364. doi: 10.1016/S0362-546X(02)00179-7.
[4]	L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations, Journal of Mathematical Analysis and Applications, 399 (2013), 239-251. doi: 10.1016/j.jmaa.2012.10.008.
[5]	L. Debnath, Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 54 (2003), 3413-3442. doi: 10.1155/S0161171203301486.
[6]	D. Idczak, Fractional du Bois-Reymond Lemma of Order $\alpha\in(1/2,1)$, Proceedings of the 7th International Workshop on Multidimensional (nD) Systems (nDs), 2011, Poitiers, France.
[7]	R. Kamocki and M. Majewski, On a fractional Dirichlet problem, Proceedings of 17th International Conference Methods and Models in Automation and Robotics (MMAR), (2012), 60-63. doi: 10.1109/MMAR.2012.6347911.
[8]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[9]	L. Nirenberg, Topics in Nonlinear Functional Analysis, New York University - Courant Institute of Mathematical Sciences - AMS, New York, 1974.
[10]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 198, Academic Press, California, 1999.
[11]	S. Walczak, On the continuous dependance on parameters of solutions of the Dirichlet problem: Part I. Coercive case; Part II. The case of saddle points, Bulletin de la Classe des Sciences de l'Académie Royale de Beligique, 6 (1995), 247-261.

show all references

References:

[1]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990.
[2]	D. Bors, A. Skowron and S. Walczak, Optimal control and stability of elliptic systems with integral cost functional, Systems Science, 33 (2007), 13-26.
[3]	D. Bors and S. Walczak, Nonlinear elliptic systems with variable boundary data, Nonlinear Analysis: Theory, Methods and Applications, 52 (2003), 1347-1364. doi: 10.1016/S0362-546X(02)00179-7.
[4]	L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations, Journal of Mathematical Analysis and Applications, 399 (2013), 239-251. doi: 10.1016/j.jmaa.2012.10.008.
[5]	L. Debnath, Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 54 (2003), 3413-3442. doi: 10.1155/S0161171203301486.
[6]	D. Idczak, Fractional du Bois-Reymond Lemma of Order $\alpha\in(1/2,1)$, Proceedings of the 7th International Workshop on Multidimensional (nD) Systems (nDs), 2011, Poitiers, France.
[7]	R. Kamocki and M. Majewski, On a fractional Dirichlet problem, Proceedings of 17th International Conference Methods and Models in Automation and Robotics (MMAR), (2012), 60-63. doi: 10.1109/MMAR.2012.6347911.
[8]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[9]	L. Nirenberg, Topics in Nonlinear Functional Analysis, New York University - Courant Institute of Mathematical Sciences - AMS, New York, 1974.
[10]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Vol. 198, Academic Press, California, 1999.
[11]	S. Walczak, On the continuous dependance on parameters of solutions of the Dirichlet problem: Part I. Coercive case; Part II. The case of saddle points, Bulletin de la Classe des Sciences de l'Académie Royale de Beligique, 6 (1995), 247-261.

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