Existence and multiplicity results for a class of fractional differential inclusions with boundary conditions
© Zhang and Gong; licensee Springer 2012
Received: 7 May 2012
Accepted: 18 July 2012
Published: 31 July 2012
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.
- (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 .
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)
provided the right-hand sides are pointwise defined on , where Γ is the gamma function.
where , and .
Definition 2.3 (Left and right Caputo fractional derivatives)
- (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
- (ii)If and , then and are represented by
In particular, , .
Property 2.1 ()
at any point for a continuous function f and for almost every point in if the function .
Definition 2.4 ()
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 ()
Proposition 2.3 ()
Define and . Assume that and the sequence converges weakly to u in , 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 ()
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).
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 ()
the generalized gradient of f at x (the Clarke subdifferential).
Lemma 2.1 ()
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(2.6)
for all and all and in .
Further, if each is regular at for each t, then f is regular at x and the equality holds.
where is a measurable selection of .
Proof Take an arbitrary element in , then it suffices to prove f is Lipschitz on .
for a.e. , where .
so f is also Lipschitz on .
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 .
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).
for any to , where for a.e. .
and this completes the proof. □
and for any in .
for all , it is easy to verify .
for all , and it is easy to verify .
where and .
Since , (2.21) holds by (2.22) and (2.23), and this completes the proof. □
where . Therefore, we seek a solution u of BVP (2.24), which corresponds to the solution u of BVP (1.1) provided that .
then we are in a position to give the definition of the solution of BVP (2.24).
is differentiable for almost every .
u satisfies (2.24).
Lemma 2.4 Let , and φ is defined by (2.20). If assumption (A) is satisfied and is a solution of the corresponding Euler equation , then u is a solution of BVP (2.24) which, of course, corresponds to the solution of BVP (1.1).
where for all and .
a.e. on for some .
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.
Moreover, implies that . □
Lemma 2.5 ()
Let X be a real reflexive Banach space. If the functional ψ: is weakly lower semi-continuous and coercive, i.e., , then there exists such that . Moreover, then .
Lemma 2.6 ()
Then and c is a critical value of ψ.
Definition 2.10 ()
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) φ satisfies the nonsmooth (P.S.) c condition for every .
Then φ has 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 3.1 Let and F satisfy the condition (A), and suppose the following conditions hold:
for a.e. and all in ;
uniformly for a.e. ;
uniformly for a.e. .
Then system (1.1) has at least one solution on .
where and .
is bounded, which combined with (3.2) implies that is bounded in since .
and is bounded, where is a positive constant.
where and .
By (3.4) and (3.5), it is easy to verify that as , and hence that in . Thus, admits a convergent subsequence.
for all and a.e. .
by (3.6), where is a positive constant. Then there exists a sufficiently large such that .
for a.e. and .
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 3.2 Let F satisfy (A), (B 1), (B 3) and the following conditions:
uniformly for a.e. ;
(B5) for and all x in .
Then system (1.1) has an infinite number of solutions on for every positive integer k such 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.
for any , therefore .
by (3.15), where , .
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 .
by (3.14). This completes the proof of Theorem 3.2. □
Theorem 3.3 Let satisfy the condition (A) with . Then BVP (1.1) has at least one solution which minimizes φ on .
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.
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 thank the anonymous referees for valuable suggestions and useful hints from others.
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