Existence of homoclinic solutions for a class of second order p-Laplacian systems with impulsive effects
Boundary Value Problemsvolume 2014, Article number: 220 (2014)
This paper is concerned with the existence of homoclinic solutions for a class of second order p-Laplacian systems with impulsive effects. A new result is obtained under more relaxed conditions by using the mountain pass theorem, a weak convergence argument, and a weak version of Lieb’s lemma.
MSC: 34C37, 35A15, 37J45, 47J30.
Consider homoclinic solutions of the following problem:
where , is of class , , , , and as . ℤ denotes the sets of integers, and () are impulsive points. Moreover, there exist a positive integer m and a positive constant T such that , , , . and represent the right and left limits of at , respectively.
When and , problem (1.1) becomes the following problem:
By using the mountain pass theorem, a weak convergence argument, and a weak version of Lieb’s methods, Fang and Duan  investigated homoclinic solutions of problem (1.2) and obtained the following main result.
Assume that the following conditions hold:
(V1): there exists a positive number T such that
(V2): uniformly for;
(V3): there exists a constantsuch that
(V4): there exist constantsandsuch that
(I): there exists a constantwithsuch that
Then problem (1.2) possesses a nontrivial weak homoclinic orbit.
For , problem (1.1) involves impulsive effects. It is well known that impulsive differential equations are used in various fields of science and technology, for example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, pharmacokinetics, and frequency modulated systems, and so on. For more details of impulsive differential equations, we refer the readers to the books , .
Recently, the existence and multiplicity of solutions for impulsive differential equations via variational methods have been investigated by some researchers. See for example – and references therein. However, there are few papers , – concerning homoclinic solutions of impulsive differential equations by variational methods. So it is a novel method to employ variational methods to investigate the existence of homoclinic solutions for impulsive differential equations.
Motivated by the above papers, we will establish a new result for (1.1).
where denotes the space of sequences whose second powers are summable on ℤ, that is,
The space is equipped with the following norm:
Similar to , we can check that E is a Banach space with the norm given by
It is obvious that
with the embedding being continuous. Here () denotes the Banach spaces of functions on ℝ with values in ℝ under the norm
Here and in subsequence, and denote the inner product and norm in ℝ, respectively. () denote different positive constants. Now, we state our main result.
Suppose that a, I, and V satisfy (V1) and the following conditions:
(A): , , andas;
(V2)′: , , and there exists a constantsuch that
(V3)′: there is a constantsuch that
(V5): and there exists a constantsuch that
(I)′: , and there exists a constant c withsuch that
Then problem (1.1) has a nontrivial homoclinic solution.
Let E be a real Banach space and, , be such thatand
Then, for each, , there existssuch that
Assume that (V3)′ and (V5) hold. Then for every,
is nondecreasing on ;
is nonincreasing on .
For any, the following inequalities hold:
where, from (A).
Ifis bounded in E anddoes not converge to 0 in measure, then there exist a sequenceand a subsequenceofsuch thatin E.
The functional φ corresponding to (1.1) on E is given by
If (V1), (A), (I)′, and (V2)′ hold, thenand
Furthermore, the critical points of φ in E are classical solutions of (1.1) with.
Firstly, we show that . By (V2)′, for any given , there exists such that
Then, by and (2.3), we have
From (2.4), we have
From (I)′ and Lemma 2.3, we have
It is easy to check that and
Next we prove that and
By (2.3), we have
Let in E, so for almost every , we have
Therefore, for any and for any function , from (2.10), we have
Now, we prove that . From (2.11), in E and , we have
Let such that for any , , and . Hence, we have
which implies that
The proof is complete. □
3 Proof of Theorem 1.1
Proof of Theorem 1.1
Firstly, we prove that under the assumptions of Theorem 1.1, there exist and such that and
It is easy to see that . From (V2)′, there exists such that
By and (3.1), we have
Since , we know that . From Lemma 2.2(ii), we have for any
where , . Take such that
and for . For , from Lemma 2.2(i) and (3.5), we get
Since and , it follows from (3.7) that there exists such that and . Set , then , and .
Secondly, we prove that under the assumptions of Theorem 1.1, there exists a bounded sequence in E such that
where , . Furthermore, does not converge to 0 in measure.
From (I)′ and Lemma 2.3, we have
Since , the above inequalities implies that there exists a constant such that
By (V2)′, we have
which implies that
For any , there exists such that
If converges to 0 in measure, from (I)′ and (3.14), we have
This is a contradiction. Hence, (3.8) holds and does not converge to 0 in measure.
Finally, from (3.10), we know that in E, what we need to do is to prove that . By (3.8), does not converge to 0 in measure and Lemma 2.4, there exists a sequence in ℤ such that in E. For any fixed , set and . Then () are impulsive points and
For any with , by (V1) and (A), we have
By (3.17), we have
where is the dual space of E. Equation (3.18) implies that
For any , let for and for . Then from (2.2), we have
Since as and in E, it follows from (3.8) that
Assume that for some , . Since
For any and , take sufficiently large such that
Since in E, in , therefore uniformly converges to u in . By the continuity of I, there exists such that, for , we have
From (I)′, we have
where . Similarly, we have
It follows from (3.36) that
Therefore, and u is a nontrivial homoclinic solution of φ. □
Fang H, Duan H: Existence of nontrivial weak homoclinic orbits for second-order impulsive differential equations. Bound. Value Probl. 2012., 2012: 10.1186/1687-2770-2012-138
Bainov DD, Simeonov PS: Impulsive Differential Equations: Periodic Solutions and Applications. Longman, New York; 1993.
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Chen P, Tang XH, Agarwal RP: Fast homoclinic solutions for a class of damped vibration problems. Appl. Math. Comput. 2013, 219: 6053-6065. 10.1016/j.amc.2012.10.103
Chen P, Tang XH: New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Math. Comput. Model. 2012, 55: 723-739. 10.1016/j.mcm.2011.08.046
Chen H, Sun J: An application of variational method to second-order impulsive differential equation on the half-line. Appl. Math. Comput. 2010, 217: 1863-1869. 10.1016/j.amc.2010.06.040
Gong W, Zhang Q, Tang XH: Existence of subharmonic solutions for a class of second-order p -Laplacian systems with impulsive effects. J. Appl. Math. 2012., 2012: 10.1155/2012/434938
Luo Z, Xiao J, Xu J: Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations. Nonlinear Anal. 2012, 75: 2249-2255. 10.1016/j.na.2011.10.023
Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680-690. 10.1016/j.nonrwa.2007.10.022
Sun J, Chen H: Variational method to the impulsive equation with Neumann boundary conditions. Bound. Value Probl. 2009., 2009:
Sun J, Chen H, Yang L: The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method. Nonlinear Anal. 2010, 73: 440-449. 10.1016/j.na.2010.03.035
Sun J, Chen H, Nieto JJ, Otero-Novoa M: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 2010, 72: 4575-4586. 10.1016/j.na.2010.02.034
Sun J, Chen H, Nieto JJ: Infinitely many solutions for second-order Hamiltonian system with impulsive effects. Math. Comput. Model. 2011, 54: 544-555. 10.1016/j.mcm.2011.02.044
Zhang D, Dai B: Existence of solutions for nonlinear impulsive differential equations with Dirichlet boundary conditions. Math. Comput. Model. 2011, 53: 1154-1161. 10.1016/j.mcm.2010.11.082
Zhang Z, Yuan R: An application of variational methods to Dirichlet boundary value problem with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 155-162. 10.1016/j.nonrwa.2008.10.044
Zhou J, Li Y: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. 2009, 71: 2856-2865. 10.1016/j.na.2009.01.140
Zhou J, Li Y: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. TMA 2010, 72: 1594-1603. 10.1016/j.na.2009.08.041
Han X, Zhang H: Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system. J. Comput. Appl. Math. 2011, 235: 1531-1541. 10.1016/j.cam.2010.08.040
Zhang H, Li ZX: Periodic and homoclinic solutions generated by impulses. Nonlinear Anal., Real World Appl. 2011, 1: 39-51. 10.1016/j.nonrwa.2010.05.034
Zhang QF: Existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Abstr. Appl. Anal. 2014., 2014:
Brezis H, Nirenberg L: Remarks on finding critical points. Commun. Pure Appl. Math. 1991, 64: 939-963. 10.1002/cpa.3160440808
Tang XH, Lin XY: Homoclinic solutions for a class of second-order Hamiltonian systems. J. Math. Anal. Appl. 2009, 354: 539-549. 10.1016/j.jmaa.2008.12.052
Tang XH, Lin XY: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Proc. R. Soc. Edinb., Sect. A, Math. 2011, 141: 1103-1119. 10.1017/S0308210509001346
Lieb EH: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 1983, 74: 441-448. 10.1007/BF01394245
This work is supported by the Scientific Research Foundation of Guangxi Education Office of China (No. 2013LX171) and the Scientific Research Foundation of Guilin University of Aerospace Technology (No. YJ1301).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.