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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 707163, 19 pages
Existence of Solutions for Sturm-Liouville Boundary Value Problem of Impulsive Differential Equations
1School of Mathematics and Statistics, Lanzhou University, Gansu, Lanzhou 730000, China
2Departamento de Análisis Matemático, Universidad de Santiago de Compostela, Santiago de Compostela 15782, Spain
3Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 20 January 2012; Revised 13 March 2012; Accepted 3 April 2012
Academic Editor: D. O'Regan
Copyright © 2012 Hong-Rui Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with the existence of solutions for Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations, under different assumptions on the nonlinearity and impulsive functions, existence criteria of single and multiple solutions are established. The main tools are variational method and critical point theorems. Some examples are also given to illustrate the main results.
Impulsive differential equation is one of the main tools to study the dynamics of processes in which sudden changes occur. The theory of impulsive differential equation has recently received considerable attention, see [1–9]. Some classical tools such as fixed-point theorems in cones and the method of upper and lower solutions with monotone iterative technique have been widely used to study impulsive differential equations. In the last few years, variational method has been used to determine the existence of solutions for impulsive differential equation possessing a variational structure under certain boundary condition, one can refer to [10–26] and the references therein for detailed discussions.
Especially, in , Nieto and O’Regan studied the nonlinear Dirichlet impulsive problem: where and . By the least action principle, the existence of a solution was obtained by assuming sublinear growth on the nonlinearity and impulses.
In , Zhou and Li discussed the problem: with impulse conditions (1.2), and Dirichlet boundary condition (1.3), where and . By using a symmetric Mountain Pass theorem, the existence result of an infinite number of solutions was obtained.
Tian and Ge in  studied the existence of multiple solutions for the following equation with impulsive effect: By applying variational methods and upper and lower solutions methods, they obtained the existence of at least four solutions and gave some accurate characteristics of the solutions. In the special case of and , in  they studied the corresponding linear and nonlinear problem and established some existence results of positive solutions via critical point theory and variational methods. In , Sun and Chen considered the equation: with impulsive condition (1.6) and Neumann boundary condition , they got existence criteria of at least one solution, two solutions and infinitely many solutions under some different conditions.
In the above-cited articles, analogous results are also given when the impulses are absent, so they cannot reflect the impact of the impulses on the existence of the solutions. In , Zhang and Li studied the periodic solutions generated by impulses for a second-order impulsive differential equation. As defined in , a solution is called a solution generated by impulses if this solution is nontrivial only if impulsive terms are not zero.
Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Many systems involve the impulsive condition which depends not only on the derivative of a function but also the function itself. A few papers discussed the solutions of impulsive differential equations involving impulses both on the function and on its derivative by critical point theory; see for example [21, 22].
Inspired by the above results, in this paper we consider the existence and multiplicity of solution to the following Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations: where , , , . , , , (or ) denotes the right limit (or left limit) of at , (or ) denotes the right limit (or left limit) of at , , , . For convenience, we denote (1.9)–(1.12) as problem ().
We begin by establishing the corresponding variational framework of problem () in an appropriate space of functions. Then under the assumption that the nonlinearity and the impulsive functions are superlinear, we get the existence of at least one nontrivial solution. Furthermore, when is odd about the second variable and are odd, the existence of an infinite number of solutions is obtained. Moreover, we study the existence of solutions generated by impulses. Under suitable hypothesis, existence criteria of at least one solution and infinitely many solutions generated by impulses of problem () are established. The main tools are the Mountain Pass theorem, symmetric Mountain Pass theorem, and the least action principle.
Note that when , the impulsive conditions (1.10) and (1.11) of problem () degenerate to the impulsive condition (1.2), and when , (1.12) is Dirichlet boundary condition; however, our assumptions on are different from the assumptions on in . The construction of a suitable space of functions and corresponding energy functional become more complicated with the impulse effects of (1.10), (1.11), and Sturm-Liouville boundary conditions taken into consideration.
The rest of this paper is organized as follows. In Section 2, we list some preliminary results and several lemmas, which are important in proving the existence of solutions. In Section 3, we state and prove the main results and give some examples to illustrate them.
We define the space: Its norm is induced by the inner product: that is,
If , then , and thus the limits and exist, we define then .
Now we state some lemmas, which are needed in the proof of the main results.
Lemma 2.1. is a Hilbert space.
Proof. Clearly, is a linear space equipped with the inner product (2.2), so is an inner space. Next, we will show that is complete.
Suppose is an arbitrary Cauchy sequence, from the definition of , for , is also a Cauchy sequence in . Then, there exists with in ; therefore, converges to in , that is, as . So we have that is, Set Then we have So and .
Observe that since converge to in , then the Cauchy sequence converge to in , that is, is complete. So is a Hilbert space.
For , we denote then we can get the following result.
Lemma 2.2. Let , then for every .
Proof. Let , by mean value theorem, there exists , such that For , from Hölder inequality, we have
For convenience, denote For , set From the assumption of the continuity of and , we can see that and for .
Lemma 2.4. If the function is a critical point of , then u is a classical solution of problem ().
Proof. Suppose that is a critical point of , then for every . By (2.16), we have
For , choose on and , then and
By the definition of and since , (2.18) gives us , thus and , so the limits exist, then integrating (2.18) by parts, we obtain So Thus, we can get that satisfies (1.9) for .
In view of (2.17), (2.19), and , we get that If we choose , such that , then it follows from the above equality that and the impulsive condition (1.11) holds.
If , then from the definition of , we know that , and the boundary condition (1.12) holds. If , , in view of (2.21) and (2.22), we can conclude that Since and are arbitrary, we deduce that and satisfies the boundary condition (1.12). If one of and is equal to 0, similarly we can obtain the conclusion.
Therefore, is a classical solution of problem ().
Lemma 2.5 (Mountain Pass theorem [10, Theorem 2.2]). Let be a real Banach space and satisfying the (PS) condition. Suppose and() there are constants such that , where ;() there is an such that .Then possesses a critical value . Moreover c can be characterized as where
Lemma 2.6 (see [10, Theorem 9.12]). Let be an infinite dimensional Banach space, and let be even functional which satisfies the (PS) condition, and . If , where is finite dimensional, and satisfies() there are constants such that , where ;() for each finite dimensional subspace , there is an such that on .Then possesses an unbounded sequence of critical values.
3. Main Results
Lemma 3.1. If there exist and such that then the functional satisfies the (PS) condition.
Proof. Let be a sequence in satisfying that is bounded and as . Now we shall prove that is a bounded sequence in . By (2.15), (2.16), (3.1), and (3.2), we have
where is a constant. So is bounded in .
From the reflexivity of , we may extract a weakly convergent subsequence that, for simplicity, we call , . In what follows, we will prove that strongly converges to . By , we see that, for , in , and converges to in . From the continuity of and , we know that In view of (2.16), we get By and , we obtain that From the previous discussions, it is easy to see that as . Therefore, satisfies the (PS) condition.
Proof. In order to show that problem () has at least one nontrivial solution, that is, has at least one nontrivial critical point, it is sufficient to check the conditions in Lemma 2.5. Clearly , , and satisfies the (PS) condition from Lemma 3.1.
First, it follows from (3.7) and (3.8) that there exists a constant , such that for : where is given in Lemma 2.2. Choose with , then , and by (3.9) and (2.11), we have If we choose , , then . Thus, condition of Lemma 2.5 holds.
By (3.1) and (3.2), we know that there are positive constants such that If we choose , then for any , we have Since , we can find such that and . Hence the condition of Lemma 2.5 is satisfied.
Theorem 3.3. Under the same assumptions as Theorem 3.2, if is odd in , and are odd, then problem () has infinitely many solutions.
Proof. The proof follows the analogous ideas as that we have developed for Theorem 3.2. By the assumption we know that is even, , and satisfies the (PS) condition from Lemma 3.1.
If we let be any finite dimensional subspace of , and , then for any with , it follows from (3.10) that where . So satisfies the condition . On the other hand, fix , the same as (3.13), we can find some such that then of Lemma 2.6 is true. Thus, possesses an infinite number of critical points, that is, problem () has infinitely many solutions.
When and for , problem degenerates to the following problem : Now, we show the nonexistence of solutions for problem .
Lemma 3.4. Assume that satisfies Then, has no nontrivial solution.
Proof. If is a nontrivial solution of , then we have Since and (3.17), we obtain .
Remark 3.5. By Lemma 3.4, if we get a solution of problem () under the assumption of (3.17), then the solution is generated by impulses.
The next theorem gives some sufficient conditions that problem () has at least one solution and infinitely many solutions generated by impulses under the assumption of (3.17), compared with Theorem 3.2, we change the conditions on nonlinearity and make certain requirements about .
Theorem 3.6. Suppose for , (3.2), (3.8), and (3.17) hold, and there exist and such that Then problem () has at least one nontrivial solution generated by impulses. Furthermore, if is odd in , and are odd, then problem () has infinitely many solutions generated by impulses.
Proof. Firstly, we show that satisfies the (PS) condition. Let be a sequence in satisfying that is bounded and as . Now, we shall prove that is a bounded sequence in .
Equation (3.17) implies that By (3.2), (3.19), and (3.20), we have where is a constant. So is bounded in , since . Similar to the proof in Lemma 3.1, we can see that has a convergent subsequence in . So, satisfies the (PS) condition.
From (3.8), there exists a constant such that for , we have where is given in Lemma 2.2. Hence combining this with (3.20), for with , we have If we choose , then . Thus, condition in Lemma 2.5 holds.
For some constant , let Clearly, and for . Then for any , by (3.19), we have In view of (3.25) and (3.12), one can get Since , we can find such that and . Hence, the condition in Lemma 2.5 is satisfied.
Finally, by Remark 3.5 and Lemma 2.5, the problem () has at least one nontrivial solution generated by impulses. Analogous to the proof of Theorem 3.3, we know that when is odd about , and ) are odd, problem () has an infinite number of solutions which are generated by impulses.
Theorem 3.7. Suppose (3.17) holds, for , and there exist positive constants and which satisfy and such that(i) for ;(ii) for , and for . Then problem () has at least one nontrivial solution generated by impulses.
Proof. Firstly, we show that is weakly lower semi continuous. Let be a sequence which weakly converges to some in , then
On the other hand, we have that weakly converges to in , then uniformly converges to on , then
From (3.27) and (3.28), we conclude that .
Next, we verify that is coercive.
From (3.17), the hypothesis and Lemma 2.2, we have which implies that is coercive since for .
Therefore by the least action principle [27, Theorem 1.1], we obtain that is a critical value of , this means that there exists such that .
Finally, we show that is nontrivial.
In (3.24), if we choose positive constant such that , then for , and for any , by the assumption and with same calculation of (3.25), we have Then For small enough, we have that since . Hence, , and by Remark 3.5, is a nontrivial solution generated by impulses.
Example 3.8. Let , consider the system:
Compare with problem (P), .
If we choose , where and are odd numbers and , then the conditions of Theorem 3.3 are satisfied. So, the impulsive boundary value problem (3.32) has an infinite number of nontrivial solutions.
Example 3.9. Consider (3.32) with defined by If we choose , the assumptions of Theorem 3.6 are satisfied, then the impulsive boundary value problem (3.32) has at least one nontrivial solution generated by impulses.
Example 3.11. Let , we consider the following problem: here . In view of and Lemma 3.4, we get that the above problem without impulsive, that is, the problem: has no nontrivial solution. In addition, it is easy to verify that the hypotheses of Theorem 3.7 are fulfilled. Thus, via Theorem 3.7, the impulsive boundary value problem (3.35) has at least one nontrivial solutions generated by impulses.
The research of H. R. Sun was supported by NSF of China (10801065), FRFCU (lzujbky-2011-43, lzujbky-2012-k25) and SRF for ROCS, and SEM. The research of J. J. Nieto was partially supported by Ministerio de Ciencia e lnnovación and FEDER, Project MTM2010-15314.
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