In this paper, we study the existence and multiplicity results of solutions for some class of fractional differential inclusions with boundary conditions. Some existence and multiplicity results of solutions are given by using the least action principle and minmax methods in nonsmooth critical point theory. Recent results in the literature are generalized and improved. Some examples are given in the paper to illustrate our main results.
MSC: 26A33, 26A42, 58E05, 70H05.
Keywords:fractional differential inclusions; nonsmooth critical point theory; boundary value problem; variational methods
In this paper, we consider the fractional boundary value problem (BVP for short) for the following differential inclusion:
Differential equations with fractional order are generalization of ordinary differential equations to non-integer order. Fractional differential equations have received increasing attention during recent years, since the behavior of physical systems can be properly described by using fractional order system theory. So fractional differential equations got the attention of many researchers and considerable work has been done in this regard, see the monographs and articles of Kilbas et al., Miller and Ross , Podlubny , Samko et al., Agarwal , Lakshmikantham  and Vasundhara Devi  and the references therein.
Recently, fractional differential equations have been of great interest, and boundary value problems for fractional differential equations have been considered by the use of techniques of nonlinear analysis (fixed-point theorems [8-10], Leray-Schauder theory [11,12], lower and upper solution method, monotone iterative method [13-15]).
Variational methods have turned out to be a very effective analytical tool in the study of nonlinear problems. The classical critical point theory for functional was developed in the sixties and seventies (see [16,17]). The celebrated and important result in the last 30 years was the mountain pass theorem due to Ambrosetti and Rabinowitz  in 1973. The needs of specific applications (such as nonsmooth mechanics, nonsmooth gradient systems, etc.) and the impressive progress in nonsmooth analysis and multivalued analysis led to extensions of the critical point theory to nondifferentiable functions, locally Lipschitz functions in particular. The nonsmooth critical point theory for locally Lipschitz functions started with the work of Chang (see ). The theory of Chang was based on the subdifferential of locally Lipschitz functionals due to Clarke (see ). Using this subdifferential, Chang proposed a generalization of the well-known Palais-Smale condition and obtained various minimax principles concerning the existence and characterization of critical points for locally Lipschitz functions. Chang used his theory to study semilinear elliptic boundary value problems with a discontinuous nonlinearity. Later, in 2000, Kourogenis and Papageorgiou (see ) extended the theory of Chang and obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the p-Laplacian with discontinuous nonlinearities. Subsequently, many authors also studied the nonsmooth critical point theory (see [22-26]), then the nonsmooth critical point theory is also widely used to deal with nonlinear boundary value problems (see [27-31]). A good survey for nonsmooth critical point theory and nonlinear boundary value problems is the book of Gasinski and Papageorgiou .
There are some papers which are devoted to the boundary value problems for fractional differential inclusions (see [33-35]), and the main tools they use are fixed point theory for multi-valued contractions. However, to the best of the authors’ knowledge, there are few results on the solutions to fractional BVP which were established by the nonsmooth critical point theory, since it is often very difficult to establish a suitable space and variational functional for fractional differential equations with boundary conditions. Recently, Jiao and Zhou  introduced some appropriate function spaces as their working space and set up a variational functional for the following system:
They give two existence results of solutions for the above system by using the least action principle and mountain pass theorem in critical point theory. It is easy to see that system (1.1) is a generalization to system (1.2), and it is interesting to ask whether the results in  hold true when the potential F is just locally Lipschitz. But the main difficulty is the variational structure given in  cannot be applied to system (1.1) directly. So we have to find a new approach to solve this problem, and the main idea of the new approach comes from the inspiration of Theorem 2.7.3 and Theorem 2.7.5 in .
The structure of the paper is as follows. In the next section, for the convenience of readers, we present the mathematical background needed and the corresponding variational structure for system (1.1). In Section 3, using variational methods, we prove two existence theorems for the solutions of problem (1.1) which generalize the results in . Finally, in Section 4, two examples are presented to illustrate our results.
Definition 2.1 (Left and right Riemann-Liouville fractional integrals)
Definition 2.2 (Left and right Riemann-Liouville fractional derivatives) Let f be a function defined on . The left and right Riemann-Liouville fractional derivatives of order γ for function f, denoted by and respectively, are defined by
Definition 2.3 (Left and right Caputo fractional derivatives)
Property 2.1 ()
The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, i.e.,
Definition 2.4 ()
Proposition 2.1 ()
Proposition 2.2 ()
Proposition 2.3 ()
Proposition 2.4 ()
is also a reflexive and separable Banach space with respect to the norm
Definition 2.6 ()
where y is also a vector in X and λ is a positive scalar, and we denote by
the generalized gradient of f at x (the Clarke subdifferential).
Lemma 2.1 ()
Definition 2.7 ()
Definition 2.8 ()
Clarke considered the following abstract framework in :
• define a functional f on Z via
Under the conditions described above, f is Lipschitz in a neighborhood of x and one has
and having the property that for every v in Z, one has
By (2.9) and (2.10), we have
by (2.12) for any in and (2.13) remains true if we restrict to , which is a closed subspace of by Definition 2.4. The bounded linear functional ζ on restricted to is also a bounded linear functional, and we use to denote the functional restricted on .
• the mapping is measurable for each in (see ), and that is a point at which is defined (finitely);
• the condition (2.6) in Clarke’s abstract framework is satisfied by (2.15).
By (2.12), we get
Combining (2.16) and (2.17), we have
and this completes the proof. □
Proof By direct computation, it is obvious that
then we are in a position to give the definition of the solution of BVP (2.24).
(ii) u satisfies (2.24).
Lemma 2.4Let, andφis defined by (2.20). If assumption (A) is satisfied andis a solution of the corresponding Euler equation, thenuis a solution of BVP (2.24) which, of course, corresponds to the solution of BVP (1.1).
Proof By Lemma 2.3, we have
The theory of Fourier series and (2.27) imply that
Lemma 2.5 ()
Lemma 2.6 ()
andψsatisfies the nonsmooth (P.S.) condition with
Definition 2.10 ()
We consider the following situation:
Lemma 2.7 ()
Thenφhas an unbounded sequence of critical values.
Remark 2.8 The condition (A1) is needed for the proof of Lemma 2.7, see details in  and the references therein.
3 Main results and proofs of the theorems
By (A) and the nonsmooth (P.S.) condition, we have
Theorem 3.2LetFsatisfy (A), (B1), (B3) and the following conditions:
Proof The proof that the functional φ satisfies the nonsmooth (P.S.) condition is the same as that of Theorem 3.1, so we omit it. We only need to verify other conditions in Lemma 2.7.
Since is a separable and reflexive Banach space, there exist (see ) and such that
and this shows condition (A2) in Lemma 2.7 is satisfied.
by (3.14). This completes the proof of Theorem 3.2. □
Proof By (3.1), we obtain
According to the same arguments in , φ is weakly lower semi-continuous. By Lemma 2.5, the proof of Theorem 3.3 is completed. □
In this section, we give two examples to illustrate our results.
Example 4.1 In BVP (1.1), let
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
The authors thank the anonymous referees for valuable suggestions and useful hints from others.
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