Existence and multiplicity results for a class of fractional differential inclusions with boundary conditions

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.

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.[1], Miller and Ross [2], Podlubny [3], Samko et al.[4], Agarwal [5], Lakshmikantham [6] and Vasundhara Devi [7] 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 [18] 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 [19]). The theory of Chang was based on the subdifferential of locally Lipschitz functionals due to Clarke (see [20]). 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 [21]) 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 [32].

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 [36] 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 [36] hold true when the potential F is just locally Lipschitz. But the main difficulty is the variational structure given in [36] 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 [20].

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 [36]. 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

and

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

and

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

and

respectively, where . In particular, if , then

and

(ii) If and , then and are represented by

In particular, , .

Property 2.1 ([36])

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 ([36])

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 ([36])

Letand. The fractional derivative spaceis a reflexive and separable Banach space.

Proposition 2.2 ([36])

Letand. For all, we have

Moreover, ifand, then

According to (2.1), we can consider with respect to the norm

Proposition 2.3 ([36])

Defineand. Assume thatand the sequenceconverges weakly touin, i.e., . Thenin, 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 ([36])

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

where .

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 ([13])

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 ([20])

Letxandybe points in a Banach spaceX, and suppose thatfis Lipschitz on an open set containing the line segment. Then there exists a pointuinsuch that

Definition 2.7 ([32])

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 ([32])

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 [20]:

• 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) andbe given by, then define a functionalfonby

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 [20]), 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 functionalsandonare given by

and

whereFsatisfies the condition (A) and, then the functional defined by

is Lipschitz on, and, we have

whereand.

Proof By direct computation, it is obvious that

In view of Lemma 2.2 and Remark 2.7, if , then we have

where .

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 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

where for all and .

Let us define by

so that

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

for .

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 ([32])

LetXbe a real reflexive Banach space. If the functionalψ: is weakly lower semi-continuous and coercive, i.e., , then there existssuch that. Moreover, then.

Lemma 2.6 ([32])

LetXbe a real reflexive Banach space, andis a locally Lipschitz function. If there existandsuch that,

andψsatisfies the nonsmooth (P.S.) condition with

where

Thenandcis a critical value ofψ.

Definition 2.10 ([37])

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 ([27])

Supposeis an invariant locally Lipschitz functional. If, for every, there existsuch 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 [27] and the references therein.

Theorem 3.1LetandFsatisfy the condition (A), and suppose the following conditions hold:

(B1) there existandsuch that

for a.e. and allin;

(B2) and

uniformly for a.e. ;

(B3) there existandsuch 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 existandsuch that

uniformly for a.e. ;

(B5) forand allxin.

Then system (1.1) has an infinite number of solutionsonfor 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 [38]) 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

then

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 3.3Letsatisfy 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 [36], 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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