Korean J. Math.  Vol 23, No 4 (2015)  pp.537-555
DOI: https://doi.org/10.11568/kjm.2015.23.4.537

Existence of solution for a fractional differential inclusion via nonsmooth critical point theory

Bian-Xia Yang, Hong-Rui Sun

Abstract


This paper is concerned with the existence of solutions to the following fractional differential inclusion
\begin{equation*}
\left\{\begin{array}{l}- \frac{d}{dx}\left(p \ {}_0D_x^{-\beta}(u'(x))+q \ {}_xD_1^{-\beta}(u'(x))\right)\in \partial F_u(x,u),\hspace{0.5cm}x\in (0,1),\\
u(0)=u(1)=0,
\end{array}
\right.
\end{equation*}
where ${}_0D_x^{-\beta}$ and ${}_xD_1^{-\beta}$ are left and right Riemann-Liouville fractional integrals of order $\beta \in(0,1)$ respectively, $0<p=1-q<1$ and $F:[0,1]\times \Bbb{R}\rightarrow\Bbb{R}$ 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


Fractional differential inclusion, Nonsmooth critical point theory, Existence, Solutions

Subject classification

26A33, 34B15, 34B40

Sponsor(s)

This works was supported by the program for New Century Excellent Talents in University (NECT-12-0246), FRFCU(lzujbky-2013-k02) and FRFCU(lzujbky-2015-75).

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References


R.P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclu- sions, Acta Appl. Math. 109 (2010), 973–1033. (Google Scholar)

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. (Google Scholar)

D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, Application of a frac- tional advection-dispersion equation, Water Resour. Res. 36 (2000), 1403–1412. (Google Scholar)

D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, The fractional-order governing equation of L ́evy motion, Water Resour. Res. 36 (2000), 1413–1423. (Google Scholar)

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. (Google Scholar)

A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag Wien GmbH, 1997. (Google Scholar)

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. (Google Scholar)

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. (Google Scholar)

F. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983. (Google Scholar)

K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129. (Google Scholar)

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. (Google Scholar)

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. (Google Scholar)

L. Gasinki and N. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman Hall/CRC, Boca Raton, 2005. (Google Scholar)

J. Henderson and A. Ouahab, Fractional functional differential inclusions with finite delay, Nonlinear Anal. 70 (2009), 2091–2105. (Google Scholar)

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. (Google Scholar)

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. (Google Scholar)

A. Krist ́aly, Infinitely many solutions for a differential inclusion problem in RN , J. Differential Equations, 220 (2006), 511–530. (Google Scholar)

D. Motreanu and P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer Academic Publishers, Dordrecht, 1999. (Google Scholar)

I. Podlubny, Fractional Differential Equations, Academic press, New York, 1999. (Google Scholar)

M.F. Shlesinger, B.J. West and J. Klafter, L ́evy dynamics of enhanced diffusion: application to turbulence, Phys. Rev. Lett. 58 (1987), 1100–1103. (Google Scholar)

E. Sayed, A.G. Ibrahim, Multivalued fractional differential equations of arbitrary orders, Appl. Math. Comput. 68 (1995), 15–25. (Google Scholar)

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. (Google Scholar)

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. (Google Scholar)


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