Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations
© Cabada and Tersian; licensee Springer. 2014
Received: 28 January 2014
Accepted: 24 April 2014
Published: 8 May 2014
In this paper we study the existence and the multiplicity of solutions for an impulsive boundary value problem for fourth-order differential equations. The notions of classical and weak solutions are introduced. Then the existence of at least one and infinitely many nonzero solutions is proved, using the minimization, the mountain-pass, and Clarke’s theorems.
MSC: 34B15, 34B37, 58E30.
Keywordsfourth-order differential equations impulsive conditions weak solution classical solution Palais-Smale condition mountain-pass theorem Clarke’s theorem
(with ) is studied, using the minimization and the mountain-pass theorem. We mention also other papers for second-order impulsive equations as [6, 7]. In several recent papers [8–10], fourth-order impulsive problems are considered via variational methods.
Here, , the limits and exist and , .
We look for solutions in the classical sense, as given in the next definition.
Definition 1 A function and , is said to be a classical solution of the problem (P), if u satisfies the equation a.e. on , the limits and exist and satisfy the impulsive conditions , , , and boundary conditions .
To deduce the existence of solutions, we assume the following conditions:
(H1) The constant a is positive, b and c are continuous functions on and there exist positive constants , , , and such that and . The functions , , , are continuous functions.
where and , , are positive constants.
In the next section we will prove the following existence result for .
Theorem 2 Suppose that and conditions (H1) and (H2) hold. Then the problem (P) has at least one nonzero classical solution.
Having in mind the case , we introduce the following condition:
A simple example of this new situation is given by the functions and .
The result to be proven is the following.
Theorem 3 Suppose that , the functions , , , are odd and conditions (H1) and (H3) hold. Then the problem (P) has infinitely many nonzero classical solutions.
we introduce the following condition.
A simple example now is , .
The obtained result is the following.
Theorem 4 Suppose that and conditions (H1) and (H2′) hold. If , the problem (P1) has only the zero solution. If , the problem (P1) has at least one nonzero classical solution.
The proofs of the main results are given in Section 3.
and the corresponding norm.
where M is a positive constant depending on T, a and .
We have the following compactness embedding, which can be proved in the standard way.
Proposition 5 The inclusion is compact.
In the sequel we introduce the concept of a weak solution of our problem.
As a consequence, the critical points of ϕ are the weak solutions of the problem (P). Let us see that they are, actually, strong solutions too.
Lemma 7 If u is a weak solution of (P) then is classical solution of (P).
Then we obtain and . Similarly, we prove that , which shows that u is a classical solution of the problem (P). The lemma is proved. □
In the proofs of the theorems, we will use three critical point theorems which are the main tools to obtain weak solutions of the considered problems.
To this end, we introduce classical notations and results. Let E be a reflexive real Banach space. Recall that a functional is lower semi-continuous (resp. weakly lower semi-continuous (w.l.s.c.)) if (resp. ) in E implies (see , pp.3-5).
We have the following well-known minimization result.
Theorem 8 Let I be a weakly lower semi-continuous operator that has a bounded minimizing sequence on a reflexive real Banach space E. Then I has a minimum . If is a differentiable functional, is a critical point of I.
Note that a functional is w.l.s.c. on I if , is convex and continuous and is sequentially weakly continuous (i.e. in E implies ) (see , pp.301-303). The existence of a bounded minimizing sequence appears, when the functional I is coercive, i.e. as .
Next, recall the notion of the Palais-Smale (PS) condition, the mountain-pass theorem and Clarke’s theorem.
We say that I satisfies condition (PS) if any sequence for which is bounded and as possesses a convergent subsequence.
Theorem 9 ([, p.4])
there are constants such that if ,
there is an , such that .
Then I possesses a critical value . Moreover, c can be characterized as where .
Theorem 10 ([, p.53])
Let E be a real Banach space and with I even, bounded from below, and satisfying condition (PS). Suppose that , there is a set such that K is homeomorphic to by an odd map, and .
Then I possesses, at least, m distinct pairs of critical points.
3 Proofs of main results
This section is devoted to the proof of the three theorems enunciated in the introduction of this work.
First consider the case for which we prove that the functional ϕ satisfies the Palais-Smale condition.
Lemma 11 Suppose that and conditions (H1) and (H2) hold. Then the functional satisfies condition (PS).
which implies that is a bounded sequence in X.
i.e., strongly in X, which completes the proof. □
Now, we are in a position to prove the main results of this paper.
Since , we conclude that for sufficiently large λ. According to the mountain-pass Theorem 9, together with Lemmas 11 and 7, we deduce that there exists a nonzero classical solution of the problem (P). □
Now consider the case . In the next result we prove that the Palais-Smale condition is also valid.
Lemma 12 Suppose that and conditions (H1) and (H3) hold. Then the functional is bounded from below and satisfies condition (PS).
Further, if is a (PS) sequence, by (17) it follows that is a bounded sequence in X. Then, as in Lemma 11, we conclude that has a convergent subsequence. □
Now we are in a position to prove the next existence result for the problem (P).
where is the induced norm of on .
we obtain for any .
By Clarke’s Theorem 10, there exist at least m pairs of different critical points of the functional ϕ. Since m is arbitrary, there exist infinitely many solutions of the problem (P), which concludes the proof. □
Proof of Theorem 4 By the Poincaré inequalities (7) we find that is an equivalent norm to in X and the functional is convex.
is sequentially weakly continuous, from the fact that the inclusion is compact, we deduce that the functional is weakly lower semi-continuous.
Then, by Theorem 8, there exists a minimizer of , which is a critical point of .
which is a contradiction. Then, for , the problem (P1) has only the zero solution.
Suppose now that .
where . For it follows that . Then, since , by (25) it follows that for sufficiently small . In consequence we show that . So we ensure the existence of a nonzero minimizer of , which completes the proof of Theorem 4. □
The authors are thankful to the anonymous referees for the careful reading of the manuscript and suggestions.
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