DOI: https://doi.org/10.11568/kjm.2015.23.4.537

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

#### 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

#### 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).#### Full Text:

PDF#### References

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