# American Institute of Mathematical Sciences

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

 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

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.
Citation: Rafał Kamocki, Marek Majewski. On the continuous dependence of solutions to a fractional Dirichlet problem. The case of saddle points. Discrete & 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).   Google Scholar [2] D. Bors, A. Skowron and S. Walczak, Optimal control and stability of elliptic systems with integral cost functional,, Systems Science, 33 (2007), 13.   Google Scholar [3] D. Bors and S. Walczak, Nonlinear elliptic systems with variable boundary data,, Nonlinear Analysis: Theory, 52 (2003), 1347.  doi: 10.1016/S0362-546X(02)00179-7.  Google Scholar [4] L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations,, Journal of Mathematical Analysis and Applications, 399 (2013), 239.  doi: 10.1016/j.jmaa.2012.10.008.  Google Scholar [5] L. Debnath, Recent applications of fractional calculus to science and engineering,, International Journal of Mathematics and Mathematical Sciences, 54 (2003), 3413.  doi: 10.1155/S0161171203301486.  Google Scholar [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).   Google Scholar [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.  doi: 10.1109/MMAR.2012.6347911.  Google Scholar [8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (2006).   Google Scholar [9] L. Nirenberg, Topics in Nonlinear Functional Analysis,, New York University - Courant Institute of Mathematical Sciences - AMS, (1974).   Google Scholar [10] I. Podlubny, Fractional Differential Equations,, Mathematics in Science and Engineering, (1999).   Google Scholar [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.   Google Scholar

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##### References:
 [1] J. P. Aubin and H. Frankowska, Set-Valued Analysis,, Birkhäuser, (1990).   Google Scholar [2] D. Bors, A. Skowron and S. Walczak, Optimal control and stability of elliptic systems with integral cost functional,, Systems Science, 33 (2007), 13.   Google Scholar [3] D. Bors and S. Walczak, Nonlinear elliptic systems with variable boundary data,, Nonlinear Analysis: Theory, 52 (2003), 1347.  doi: 10.1016/S0362-546X(02)00179-7.  Google Scholar [4] L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations,, Journal of Mathematical Analysis and Applications, 399 (2013), 239.  doi: 10.1016/j.jmaa.2012.10.008.  Google Scholar [5] L. Debnath, Recent applications of fractional calculus to science and engineering,, International Journal of Mathematics and Mathematical Sciences, 54 (2003), 3413.  doi: 10.1155/S0161171203301486.  Google Scholar [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).   Google Scholar [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.  doi: 10.1109/MMAR.2012.6347911.  Google Scholar [8] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (2006).   Google Scholar [9] L. Nirenberg, Topics in Nonlinear Functional Analysis,, New York University - Courant Institute of Mathematical Sciences - AMS, (1974).   Google Scholar [10] I. Podlubny, Fractional Differential Equations,, Mathematics in Science and Engineering, (1999).   Google Scholar [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.   Google Scholar
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