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:
where , and are the left and right Riemann-Liouville fractional integrals of order respectively, satisfies the following assumptions:
(A) is measurable in t for every and locally Lipschitz in x for a.e. , and there exist and such that
for a.e. and all .
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:
where , and are the left and right Riemann-Liouville fractional integrals of order respectively, and F is continuously differentiable.
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)
Let f be a function defined on . The left and right Riemann-Liouville fractional integrals of order γ for function f, denoted by and respectively, are defined by
provided the right-hand sides are pointwise defined on , where Γ is the gamma function.
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
where , and .
Definition 2.3 (Left and right Caputo fractional derivatives)
Let and .
(i) If and , then the left and right Caputo fractional derivatives of order γ for function f, denoted by and respectively, exist almost everywhere on . and are represented by
respectively, where . In particular, if , then
(ii) If and , then and are represented by
In particular, , .
Property 2.1 ()
The left and right Riemann-Liouville fractional integral operators have the property of a semigroup, i.e.,
at any point for a continuous function f and for almost every point in if the function .
Definition 2.4 ()
Define and . The fractional derivative space is defined by the closure of with respect to the norm
where denotes the set of all functions with . It is obvious that the fractional derivative space is the space of functions having an α-order Caputo fractional derivative and .
Proposition 2.1 ()
Let and . The fractional derivative space is a reflexive and separable Banach space.
Proposition 2.2 ()
Let and . For all , we have
Moreover, if and , then
According to (2.1), we can consider with respect to the norm
Proposition 2.3 ()
Define and . Assume that and the sequence converges weakly touin , i.e., . Then in , i.e., , as .
In this paper, we treat BVP (1.1) in the Hilbert space with the equivalent norm defined in (2.3).
Proposition 2.4 ()
If , then for any , we have
In order to establish the variational structure for system (1.1), it is necessary to construct some appropriate function spaces. The Cartesian product space defined by
is also a reflexive and separable Banach space with respect to the norm
The space is a closed subset of under the norm (2.4) as is closed by Definition 2.4.
In this paper, we use the norm defined in (2.3), which is an equivalent norm in with norm (2.4).
Definition 2.5 Let denote the space of essentially bounded measurable functions from into under the norm
It is obvious that is a Banach space under the norm (2.5).
Remark 2.5 We use and to denote and respectively.
Definition 2.6 ()
Let f be Lipschitz near a given point x in a Banach space X, and v be any other vector in X. The generalized directional derivative of f at x in the direction v, denoted by , is defined as follows:
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 ()
Letxandybe points in a Banach spaceX, and suppose thatfis Lipschitz on an open set containing the line segment . Then there exists a pointuin such that
Definition 2.7 ()
A point is said to be a critical point of a locally Lipschitz f if , namely for every . A real number c is called a critical value of f if there is a critical point such that .
Definition 2.8 ()
If f is a locally Lipschitz function, we say that f satisfies the nonsmooth (P.S.) condition if each sequence in X such that is bounded and has a convergent subsequence, where .
Clarke considered the following abstract framework in :
• let be a σ-finite positive measure space, and let Y be a separable Banach space;
• let Z be a closed subspace of , where denotes the space of measure essentially bounded functions mapping S to Y, equipped with the usual supremum norm;
• define a functional f on Z via
where Z is a closed subspace of and is a given family of functions;
• suppose that the mapping is measurable for each v in Y, and that x is a point at which is defined (finitely);
• suppose that there exist and a function in such that
for all and all and in .
Under the conditions described above, f is Lipschitz in a neighborhood of x and one has
Further, if each is regular at for each t, then f is regular at x and the equality holds.
Remark 2.6 The interpretation of (2.7) is as follows: To every , there is a corresponding mapping from S to with
and having the property that for every v in Z, one has
Thus, every ζ in the left-hand side of (2.7) is an element of that can be written
where is a measurable selection of .
Lemma 2.2LetFsatisfy the condition (A) and be given by , then define a functionalfon by
Thenfis Lipschitz on , and one has
Proof Take an arbitrary element in , then it suffices to prove f is Lipschitz on .
When ( ), we conclude
by Proposition 2.2, where . In view of Lemma 2.1 and , one has
for a.e. , where .
By (2.9) and (2.10), we have
so f is also Lipschitz on .
For any ζ in , one has
for any in by Fatou’s lemma, and it is obvious that
for a.e. and all in . Then we conclude
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 .
We interpret (2.13) by saying that belongs to the subgradient at of the convex functional
which is defined in , where for all in . In view of condition (A) and (2.12), we have
for a.e. and all , in .
Now we can apply Clarke’s abstract framework to with the following cast of characters:
• with the Lebesgue measure, and let , which is a separable Banach space with the norm ;
• let , which is a closed subspace of , and denotes the space of measure essentially bounded functions mapping T to Y, equipped with the usual supremum norm by Definition 2.5;
• define a functional on Z by (2.14);
• 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
thus, every can be written as
for any to , where for a.e. .
When , it is obvious that and is dense in by Definition 2.4. So for each , we can choose such that
Combining (2.16) and (2.17), we have
for all . Then we conclude
and this completes the proof. □
Remark 2.7 The interpretation of expression (2.8) is as follows: If is an element in and , we deduce the existence of a measurable function such that
for a.e. and one has
and for any in .
Define a functional ϕ on by
if on , then we can define on by
for all , it is easy to verify .
Similarly, if on , then we can define on by
for all , and it is easy to verify .
Lemma 2.3The corresponding functionals and on are given by
whereFsatisfies the condition (A) and , then the functional defined by
is Lipschitz on , and , we have
where and .
Proof By direct computation, it is obvious that
In view of Lemma 2.2 and Remark 2.7, if , then we have
Since , (2.21) holds by (2.22) and (2.23), and this completes the proof. □
Making use of Property 2.1 and Definition 2.3, for any , BVP (1.1) is equivalent to the following problem:
where . Therefore, we seek a solution u of BVP (2.24), which corresponds to the solution u of BVP (1.1) provided that .
Let us denote by
then we are in a position to give the definition of the solution of BVP (2.24).
Definition 2.9 A function is called a solution of BVP (2.24) if
(i) is differentiable for almost every .
(ii) u satisfies (2.24).
Lemma 2.4Let , andφis defined by (2.20). If assumption (A) is satisfied and is 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
where for all and .
Let us define by
By the Fubini theorem and noting that , we obtain
Hence, by (2.26) we have, for every ,
If denotes the Canonical basis of , we can choose such that
The theory of Fourier series and (2.27) imply that
a.e. on for some . According to the definition of , we have
a.e. on for some .
In view of , we shall identify the equivalence class given by its continuous representant
Therefore, it follows from (2.28) and the classical result of the Lebesgue theory that is the classical derivative of a.e. on which means that (i) in Definition 2.9 is verified.
Since implies that , it remains to show that u satisfies (2.24). In fact, according to (2.29), we can get that
Moreover, implies that . □
Lemma 2.5 ()
LetXbe a real reflexive Banach space. If the functionalψ: is weakly lower semi-continuous and coercive, i.e., , then there exists such that . Moreover, then .
Lemma 2.6 ()
LetXbe a real reflexive Banach space, and is a locally Lipschitz function. If there exist and such that ,
andψsatisfies the nonsmooth (P.S.) condition with
Then andcis a critical value ofψ.
Definition 2.10 ()
Assume that the compact group G acts diagonally on , that is,
where V is a finite dimensional space. The action of G is admissible if every continuous equivariant map has a zero, where U is an open bounded invariant neighborhood of 0 in , .
Example 2.1 The antipodal action of on is admissible.
We consider the following situation:
(A1) The compact group G acts isometrically on the Banach space , the space is invariant and there exists a finite dimensional space V such that for each , and the action of G on V is admissible.
Lemma 2.7 ()
Suppose is an invariant locally Lipschitz functional. If, for every , there exist such that
(A2) , where ;
(A3) , as , where ;
(A4) φsatisfies the nonsmooth (P.S.)ccondition for every .
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
Theorem 3.1Let andFsatisfy the condition (A), and suppose the following conditions hold:
(B1) there exist and such that
for a.e. and all in ;
uniformly for a.e. ;
(B3) there exist and such that
uniformly for a.e. .
Then system (1.1) has at least one solution on .
Proof Let such that is bounded and as . First, we prove is a bounded sequence. Take such that , then there exists such that
for all . It follows from (3.1) that
where and .
By (A) and the nonsmooth (P.S.) condition, we have
is bounded, which combined with (3.2) implies that is bounded in since .
By Proposition 2.3, the sequence has a subsequence, also denoted by , such that
and is bounded, where is a positive constant.
Therefore, we have , where is the function from the nonsmooth (P.S.) condition, and such that
as , so
where and .
By (3.4) and (3.5), it is easy to verify that as , and hence that in . Thus, admits a convergent subsequence.
In view of (B3), there exist two positive constants and such that
for a.e. and . It follows from that
for all and a.e. . Therefore, we obtain
for all and a.e. .
For any with , , we have
by (3.6), where is a positive constant. Then there exists a sufficiently large such that .
By (B2), there exists and such that
for a.e. and .
Let and . Then it follows from (2.2) that
for all with . Therefore, we have
for all with . This implies all the conditions in Lemma 2.6 are satisfied, so there exists a critical point for φ and , and this completes the proof. □
Theorem 3.2LetFsatisfy (A), (B1), (B3) and the following conditions:
(B4) there exist and such that
uniformly for a.e. ;
(B5) for and allxin .
Then system (1.1) has an infinite number of solutions on for every positive integerksuch that , as .
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
For , denote
For any , let
and it is easy to verify that defined by (3.9) is a norm of . Since all the norms of a finite dimensional normed space are equivalent, there exists a positive constant such that
In view of (B3), there exist two positive constants and such that
for a.e. and . It follows from (3.10) and (3.11) that
where , . Since , there exists a positive constant such that
For any , let
then we conclude as . In fact, it is obvious that , so as . For every , there exists such that
As is reflexive, has a weakly convergent subsequence, still denoted by , such that . We claim . In fact, for any , we have , when , so
for any , therefore .
By Proposition 2.3, when in , then strongly in . So we conclude by (3.14). In view of (B4), there exist two positive constants and such that
for a.e. and . We conclude
by (3.15), where , .
Choosing , it is obvious that
that is, condition (A3) in Lemma 2.7 is satisfied. In view of (3.12), let , then
and this shows condition (A2) in Lemma 2.7 is satisfied.
We have proved the functional φ satisfies all the conditions of Lemma 2.7, then φ has an unbounded sequence of critical values by Lemma 2.7; we only need to show as .
In fact, since is a critical point of the functional φ, that is, , by Lemma 2.3 and Remark 2.7, we have
where . Hence, we have
since , it is obvious that
by (3.14). This completes the proof of Theorem 3.2. □
Theorem 3.3Let satisfy the condition (A) with . Then BVP (1.1) has at least one solution which minimizesφon .
Proof By (3.1), we obtain
where is defined in (2.9), and are constants. Hence, we get
If , we have
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
It is easy to verify all the conditions in Theorem 3.2, so BVP (1.1) has infinitely many solutions on and as .
Example 4.2 In BVP (1.1), let . It is easy to verify all the conditions in Theorem 3.3, so BVP (1.1) has at least one solution which minimizes φ on .
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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