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.
Keywords:fourth-order differential equations; impulsive conditions; weak solution; classical solution; Palais-Smale condition; mountain-pass theorem; Clarke’s theorem
The theory of impulsive boundary value problems (IBVPs) became an important area of studies in recent years. IBVPs appear in mathematical models of processes with sudden changes in their states. Such processes arise in population dynamics, optimal control, pharmacology, industrial robotics, etc. For an introduction to theory of IBVPs one is referred to . Some classical tools used in the study of impulsive differential equations are topological methods as fixed point theorems, monotone iterations, upper and lower solutions (see [2-4]). Recently, some authors have studied the existence of solutions of IBVPs using variational methods. The pioneering work in this direction is the paper of Nieto and O’Regan , where the second-order impulsive problem
(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.
In this paper, we consider the boundary value problem for fourth-order differential equation with impulsive effects
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:
A simple example of functions fulfilling the last condition is given by
The result to be proven is the following.
If we consider the problem
we introduce the following condition.
The obtained result is the following.
The proofs of the main results are given in Section 3.
and the corresponding norm.
By assumption (H1) an equivalent scalar product and norm in X are given by
We have the following compactness embedding, which can be proved in the standard way.
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.
Therefore, by (11) and (12), we have
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 8LetIbe a weakly lower semi-continuous operator that has a bounded minimizing sequence on a reflexive real Banach spaceE. ThenIhas a minimum. Ifis a differentiable functional, is a critical point ofI.
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.
Theorem 9 ([, p.4])
Theorem 10 ([, p.53])
ThenIpossesses, at least, mdistinct 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.
Then we have
Adding the last inequality with (14), by assumption (H2), we obtain
it follows that
Then by (16) it follows that
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 we are in a position to prove the next existence result for the problem (P).
Proof of Theorem 3 By assumption, we know that and are odd functions. So are are even functions and the functional ϕ is even. By Lemma 12 we know that ϕ is bounded from below and satisfies condition (PS). Let , be a natural number and define, for any fixed, the set
Arguing as in [, pp.16-18], one can prove that there exists , such that
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. □
Concerning the problem (P1), one can introduce similarly the notions of classical and weak solutions. In this case it is not difficult to verify that the weak solutions are critical points of the functional defined as
Since the functional
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 declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
The authors are thankful to the anonymous referees for the careful reading of the manuscript and suggestions.
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