Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE OF SOLUTION FOR A FRACTIONAL DIFFERENTIAL INCLUSION VIA NONSMOOTH CRITICAL POINT THEORY

  • YANG, BIAN-XIA (School of Mathematics and Statistics Lanzhou University) ;
  • SUN, HONG-RUI (School of Mathematics and Statistics Lanzhou University)
  • Received : 2015.08.24
  • Accepted : 2015.10.28
  • Published : 2015.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

This paper is concerned with the existence of solutions to the following fractional differential inclusion $$\{-{\frac{d}{dx}}\(p_0D^{-{\beta}}_x(u^{\prime}(x)))+q_xD^{-{\beta}}_1(u^{\prime}(x))\){\in}{\partial}F_u(x,u),\;x{\in}(0,1),\\u(0)=u(1)=0,$$ where $_0D^{-{\beta}}_x$ and $_xD^{-{\beta}}_1$ are left and right Riemann-Liouville fractional integrals of order ${\beta}{\in}(0,1)$ respectively, 0 < p = 1 - q < 1 and $F:[0,1]{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is locally Lipschitz with respect to the second variable. Due to the general assumption on the constants p and q, the problem does not have a variational structure. Despite that, here we study it combining with an iterative technique and nonsmooth critical point theory, we obtain an existence result for the above problem under suitable assumptions. The result extends some corresponding results in the literatures.

Keywords

References

  1. R.P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math. 109 (2010), 973-1033. https://doi.org/10.1007/s10440-008-9356-6
  2. B. Ahmad and V. Otero-Espinar, Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions, Bound. Value Probl. 2009 (2009). Article ID 625347.
  3. D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res. 36 (2000), 1403-1412. https://doi.org/10.1029/2000WR900031
  4. D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, The fractional-order governing equation of Levy motion, Water Resour. Res. 36 (2000), 1413-1423. https://doi.org/10.1029/2000WR900032
  5. M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential inclusions with infinite delay and applications to control theory, Fract. Calc. Appl. Anal. 11 (2008), 35-56.
  6. A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag Wien GmbH, 1997.
  7. B.A. Carreras, V.E. Lynch and G.M. Zaslavsky, Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model, Phys. Plasmas, 8 (2001), 5096-5103. https://doi.org/10.1063/1.1416180
  8. Y.K. Chang and J.J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Model. 49 (2009), 605-609. https://doi.org/10.1016/j.mcm.2008.03.014
  9. F. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
  10. K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. https://doi.org/10.1016/0022-247X(81)90095-0
  11. Y.K. Chang and J. J. Nieto, Some new existence results for fractional differential inclusions with boundary conditions, Math. Comput. Model. 49 (2009), 605-609. https://doi.org/10.1016/j.mcm.2008.03.014
  12. V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations, 22 (2006), 558-576. https://doi.org/10.1002/num.20112
  13. L. Gasinki and N. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman Hall/CRC, Boca Raton, 2005.
  14. J. Henderson and A. Ouahab, Fractional functional differential inclusions with finite delay, Nonlinear Anal. 70 (2009), 2091-2105. https://doi.org/10.1016/j.na.2008.02.111
  15. F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl. 62 (2011), 1181-1199. https://doi.org/10.1016/j.camwa.2011.03.086
  16. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science B.V., Amsterdam, 2006.
  17. A. Kristaly, Infinitely many solutions for a differential inclusion problem in ${\mathbb{R}}^N$, J. Differential Equations, 220 (2006), 511-530. https://doi.org/10.1016/j.jde.2005.02.007
  18. D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer Academic Publishers, Dordrecht, 1999.
  19. I. Podlubny, Fractional Differential Equations, Academic press, New York, 1999.
  20. M.F. Shlesinger, B.J. West and J. Klafter, Levy dynamics of enhanced diffusion: application to turbulence, Phys. Rev. Lett. 58 (1987), 1100-1103. https://doi.org/10.1103/PhysRevLett.58.1100
  21. E. Sayed, A.G. Ibrahim, Multivalued fractional differential equations of arbitrary orders, Appl. Math. Comput. 68 (1995), 15-25. https://doi.org/10.1016/0096-3003(94)00080-N
  22. H.R. Sun and Q.G. Zhang, Existence of solutions for a fractional boundary value problem via the Mountain Pass method and an iterative technique, Comput. Math. Appl. 64 (2012), 3436-3443. https://doi.org/10.1016/j.camwa.2012.02.023
  23. K.M. Teng, H.G. Jia and H.F. Zhang, Existence and multiplicity results for fractional differential inclusions with Dirichlet boundary conditions, Appl. Math. Comput. 220 (2013), 792-801. https://doi.org/10.1016/j.amc.2012.08.050