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

show all references

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

[1]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

[2]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[3]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[4]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[5]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[6]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[7]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[8]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[9]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[10]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[11]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[12]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[13]

Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122

[14]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[15]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[16]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[17]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[18]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073

[19]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[20]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]