Abstract
In this article, the author discusses the existence of solutions for a class of impulsive differential equations by means of a variational approach different from earlier approaches.
MSC: 34B37, 45G10, 47H30, 47J30.
Keywords:
impulsive differential equation; integral equation; variational method; critical point theory1 Introduction
The theory of impulsive differential equations has been emerging as an important area of investigation in recent years [13]. There is a vast literature on the existence of solutions by using topological methods, including fixed point theorems, LeraySchauder degree theory, and fixed point index theory [415]. But it is quite difficult to apply the variational approach to an impulsive differential equation; therefore, there was no result in this area for a long time. Only in the recent five years, there appeared a few articles which dealt with some impulsive differential equations by using variational methods [1620]. Motivated by [17], in this article we shall use a different variational approach to discuss the existence of solutions for a class of impulsive differential equations and we only deal with classical solutions.
Consider the boundary value problem (BVP) for the secondorder nonlinear impulsive differential equation:
where
for any
(denoted by
where
where
Let us list some conditions.
(H_{1}) There exist
(H_{2}) There exist
Lemma 1
where
and
Proof For
So, if
It is clear, by (5), that
so
Substituting (9) into (7), we get
By virtue of (5), we see that
so, letting
Substituting (11) into (10), we get
so
Conversely, suppose that
By (4), it is clear that
Differentiating again, we get
From (13) we see that
It follows from (4), (5), (12), (14), and (15) that
Lemma 2Let condition (H_{1}) be satisfied. If
Proof It is clear, for function
By (4), (5), (16), and condition (H_{1}), we have
so,
where
It is clear that
is bounded and continuous from
Let
which implies by virtue of the uniform continuity of
2 Variational approach
Theorem 1If conditions (H_{1}) and (H_{2}) are satisfied, then BVP (1) has at least one solution
Proof By Lemma 1 and Lemma 2, we need only to show that the integral equation (2) has a
solution
where G is the linear integral operator defined by
and the nonlinear operator g is defined by (18), which is bounded and continuous from
so [22,23] the linear operator G defined by (20) is completely continuous from
has a solution
is a
Hence we need only to show that there exists a
By (4), (5), (16), and condition (H_{1}), we have
and
So, (25), (26), and condition (H_{2}) imply
It is well known [24],
where G is defined by (20) and is regarded as a positivedefinite operator from
which implies by virtue of
So, there exists a
It is well known [22,23] that the ball
It follows from (31) and (32) that
Hence
Example 1 Consider the BVP
where
Conclusion BVP (33) has at least one solution
Proof Evidently, (33) is a BVP of the form (1) with
It is clear that
Moreover, it is well known that
So, (35) and (36) imply that
and consequently, condition (H_{1}) is satisfied for
For
By (35), we have
It follows from (38) and (39) that
Since, by virtue of (37),
we see that (40) implies that condition (H_{2}) is satisfied for
By using the Mountain Pass Lemma and the Minimax Principle established by Ambrosetti and Rabinowitz [25,26], we have obtained in [23] the existence of a nontrivial solution and the existence of infinitely many nontrivial solutions for a class of nonlinear integral equations. Since (2) is a special case of such nonlinear integral equations, we get the following result for (2).
Lemma 3 (Special case of Theorem 1 and Theorem 2 in [23])
Suppose the following.
(a) There exist
(b) There exist
(c)
Then the integral equation (2) has at least one nontrivial solution in
(d)
Then the integral equation (2) has infinite many nontrivial solutions in
Let us list more conditions for the function
(H_{3}) There exist
(H_{4})
(H_{5})
Theorem 2Suppose that conditions (H_{1}), (H_{3}), and (H_{4}) are satisfied. Then BVP (1) has at least one solution
Proof In the proof of Lemma 2, we see that condition (H_{1}) implies condition (a) of Lemma 3 (see (17)). On the other hand, it is clear that conditions (H_{3}), (H_{4}), (H_{5}) are the same as conditions (b), (c), (d) in Lemma 3, respectively. Hence the conclusion of Theorem 2 follows from Lemma 3, Lemma 2, and Lemma 1. □
Example 2 Consider the BVP
Conclusion BVP (41) has infinite many solutions
Proof Obviously, (41) is a BVP of form (1). In this situation,
It is clear that
so, condition (H_{1}) is satisfied for
so,
and, consequently, (H_{3}) is satisfied for
Competing interests
The author declares that they have no competing interests.
Acknowledgements
Research was supported by the National Nature Science Foundation of China (No. 10671167).
References

Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)

Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations, World Scientific, Singapore (1995)

Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York (2006)

Agarwal, RP, O’Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput.. 114, 51–59 (2000). Publisher Full Text

Yan, B: Boundary value problems on the half line with impulses and infinite delay. J. Math. Anal. Appl.. 259, 94–114 (2001). Publisher Full Text

Agarwal, RP, O’Regan, D: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput.. 161, 433–439 (2005). Publisher Full Text

Kaufmann, ER, Kosmatov, N, Raffoul, YN: A secondorder boundary value problem with impulsive effects on an unbounded domain. Nonlinear Anal.. 69, 2924–2929 (2008). Publisher Full Text

Guo, D: Positive solutions of an infinite boundary value problem for nthorder nonlinear impulsive singular integrodifferential equations in Banach spaces. Nonlinear Anal.. 70, 2078–2090 (2009). Publisher Full Text

Guo, D: Multiple positive solutions for first order impulsive superlinear integrodifferential equations on the half line. Acta Math. Sci. Ser. B. 31(3), 1167–1178 (2011). Publisher Full Text

Guo, D, Liu, X: Multiple positive solutions of boundary value problems for impulsive differential equations. Nonlinear Anal.. 25, 327–337 (1995). Publisher Full Text

Guo, D: Multiple positive solutions for first order nonlinear impulsive integrodifferential equations in a Banach space. Appl. Math. Comput.. 143, 233–249 (2003). Publisher Full Text

Guo, D: Multiple positive solutions of a boundary value problem for nthorder impulsive integrodifferential equations in Banach spaces. Nonlinear Anal.. 63, 618–641 (2005). Publisher Full Text

Xu, X, Wang, B, O’Regan, D: Multiple solutions for sublinear impulsive threepoint boundary value problems. Appl. Anal.. 87, 1053–1066 (2008). Publisher Full Text

Jankowski, J: Existence of positive solutions to second order fourpoint impulsive differential problems with deviating arguments. Comput. Math. Appl.. 58, 805–817 (2009). Publisher Full Text

Liu, Y, O’Regan, D: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul.. 16, 1769–1775 (2011)

Tian, Y, Ge, W: Applications of variational methods to boundary value problem for impulsive differential equations. Proc. Edinb. Math. Soc.. 51, 509–527 (2008)

Nieto, JJ, O’Regan, D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl.. 10, 680–690 (2009). Publisher Full Text

Zhang, Z, Yuan, R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl.. 11, 155–162 (2010). Publisher Full Text

Chen, H, Sun, J: An application of variational method to secondorder impulsive differential equations on the half line. Appl. Math. Comput.. 217, 1863–1869 (2010). Publisher Full Text

Bai, L, Dai, B: Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory. Math. Comput. Model.. 53, 1844–1855 (2011). Publisher Full Text

Guo, D: A class of secondorder impulsive integrodifferential equations on unbounded domain in a Banach space. Appl. Math. Comput.. 125, 59–77 (2002). Publisher Full Text

Krasnoselskii, MA: Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Oxford (1964)

Guo, D: The number of nontrivial solutions to Hammerstein nonlinear integral equations. Chin. Ann. Math., Ser. B. 7(2), 191–204 (1986)

Ambrosetti, A, Rabinowitz, PH: Dual variational method in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text

Rabinowitz, PH: Variational methods for nonlinear eigenvalue problems. Varenna, Italy. (1974)