© Rong Dong et al. 2010
Received: 5 October 2010
Accepted: 15 December 2010
Published: 21 December 2010
By the Leray-Schauder's degree, the existence of solutions for a weighted -Laplacian impulsive integro-differential system with multi-point and integral boundary value conditions is considered. The sufficient results for the existence are given under the resonance and nonresonance cases, respectively. Moreover, we get the existence of nonnegative solutions at nonresonance.
In recent years, there has been an increasing interest in the study of differential equations with nonstandard -growth conditions. These problems have many interesting applications (see [1–4]). Many results have been obtained on these kinds of problems, for example [5–17]. If (a constant), (1.1)–(1.4) becomes the well known -Laplacian problem. If is a general function, one can see easily that in general, while , so represents a non-homogeneity and possesses more nonlinearity, thus is more complicated than . For example, we have the following.
(a)In general, the infimum of eigenvalues for the -Laplacian Dirichlet problems is zero, and only under some special conditions (see ). When ( ) is an interval, the results in  show that if and only if is monotone. But the property of is very important in the study of -Laplacian problems, for example, in , the authors use this property to deal with the existence of solutions.
(b)If and (a constant) and , then is concave, this property is used extensively in the study of one-dimensional -Laplacian problems (see ), but it is invalid for . It is another difference between and .
Recently, there are many works devoted to the existence of solutions to the Laplacian impulsive differential equation boundary value problems, for example [20–28]. Many methods had been applied to deal with these problems, for example sub-super-solution method, fixed point theorem, monotone iterative method, coincidence degree, variational principles (see ), and so forth. Because of the nonlinearity of , results about the existence of solutions for -Laplacian impulsive differential equation boundary value problems are rare (see ). In , using coincidence degree method, the present author investigate the existence of solutions for -Laplacian impulsive differential equation with multipoint boundary value conditions. Integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow and population dynamics, there are many papers on the differential equations with integral boundary value problems, for example, [32–35].
where , the impulsive condition (1.8) is called linear impulsive condition (LI for short), and (1.3) is called nonlinear impulsive condition (NLI for short). In generaly, -Laplacian impulsive problems have two kinds of impulsive conditions, that is, LI and NLI.
This paper is organized as four sections. In Section 2, we present some preliminary and give the operator equation which has the same solutions of (1.1)–(1.4). In Section 3, we give the existence of solutions and nonnegative solutions for system (1.1)–(1.4) at nonresonance. Finally, in Section 4, we give the existence of solutions for system (1.1)–(1.4) at resonance.
Lemma 2.1 (see ).
We will discuss (2.4) with (1.4) in the cases of resonance and nonresonance, respectively.
2.1. The Case of Nonresonance
Hence the sequence is uniformly bounded and equicontinuous. By Ascoli-Arzela theorem, there exists a subsequence of (which we rename the same) which is convergent in . According to the bounded continuous of the operator , we can choose a subsequence of (which we still denote by ) which is convergent in , then is convergent in .
It is easy to see from (i) and Lemma 2.2.
This completes the proof.
It follows from (2.21) that (1.3) is satisfied.
It follows from (2.22) and (2.23) that (1.4) is satisfied.
2.2. The Case of Resonance
and hence, if (2.27) has a solution, then it is unique.
It only remains to prove the continuity of . Let is a convergent sequence in and , as . Since is a bounded sequence, it contains a convergent subsequence . Suppose as . Since , letting , we have , which together with (i) implies , it means is continuous. This completes the proof.
Similar to the proof of Lemma 2.3, we have the following lemma.
3. Existence of Solutions in the Case of Nonresonance
In this section, we will apply Leray-Schauder's degree to deal with the existence of solutions and nonnegative solutions for system (1.1)–(1.4) at nonresonance.
then problem (1.1)−(1.4) has at least one solution.
which implies that (1.1)–(1.4) has at least one solution. This completes the proof.
There are two cases.
Thus (3.16) is valid.
Thus (3.16) is valid. Thus problem (1.1) with (1.2), (1.4), and (1.8) has at least one solution. This completes the proof.
We have the following.
As it is similar to the proof of Theorems 3.2 and 3.3, we omit it here.
Then every solution of (1.1)–(1.4) is nonnegative.
Thus every solution of (1.1)–(1.4) is nonnegative. The proof is completed.
Under the conditions of Theorem 3.1, we also assume
Then (1.1)–(1.4) has a nonnegative solution.
It is not hard to check that
4. Existence of Solutions in the Case of Resonance
In the following, we will consider the existence of solutions for system (1.1)–(1.4) at resonance.
Suppose that the following conditions hold
Then problem (4.18) with (1.2), (1.3), and (1.4) has at least one solution.
If is Caratheodory, conditions (20), (30) and (40) of Theorem 4.2 are satisfied, condition (30) of Corollary 3.6 is also satisfied, , where are positive functions satisfying ; then (4.18) with (1.2), (1.3), and (1.4) has at least one solution.
According to Theorem 4.2, we get that (4.18) with (1.2), (1.3), and (1.4) has at least a solution. This completes the proof.
From Theorem 4.2, similar to the proof of Theorem 3.3, we have the following.
then problem (4.18) with (1.2), (1.3), and (1.8) has at least one solution.
Similar to the proof of Theorem 3.2, the condition (40) of Theorem 4.2 is satisfied. Thus problem (4.18) with (1.2), (1.3) and (1.8) has at least a solution.
Similar to the proof of Theorem 3.2 and Corollary 4.3, we have the following.
If is Caratheodory, (4.45), (4.46) and conditions (20) and (30) of Theorem 4.2 are satisfied, condition (30) of Corollary 3.6 is also satisfied, , where are positive functions satisfying ; then (4.43) with (1.2), (1.3), and (1.8) has at least one solution when the parameter is small enough.
This paper is partly supported by the National Science Foundation of China (10701066, 10926075, and 10971087), China Postdoctoral Science Foundation funded project (20090460969), and the Natural Science Foundation of Henan Education Committee (2008-755-65).
- Acerbi E, Mingione G: Regularity results for stationary electro-rheological fluids. Archive for Rational Mechanics and Analysis 2002, 164(3):213–259. 10.1007/s00205-002-0208-7MathSciNetView ArticleMATHGoogle Scholar
- Chen Y, Levine S, Rao M: Variable exponent, linear growth functionals in image restoration. SIAM Journal on Applied Mathematics 2006, 66(4):1383–1406. 10.1137/050624522MathSciNetView ArticleMATHGoogle Scholar
- Růžička M: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics. Volume 1748. Springer, Berlin, Germany; 2000:xvi+176.MATHGoogle Scholar
- Zhikov VV: Averaging of functionals of the calculus of variations and elasticity theory. Mathematics of the USSR-Izvestiya 1986, 29: 33–36.View ArticleMathSciNetMATHGoogle Scholar
- Acerbi E, Mingione G: Regularity results for a class of functionals with non-standard growth. Archive for Rational Mechanics and Analysis 2001, 156(2):121–140. 10.1007/s002050100117MathSciNetView ArticleMATHGoogle Scholar
- Coscia A, Mingione G: Hölder continuity of the gradient of -harmonic mappings. Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 1999, 328(4):363–368.MathSciNetMATHGoogle Scholar
- Deng S-G: A local mountain pass theorem and applications to a double perturbed -Laplacian equations. Applied Mathematics and Computation 2009, 211(1):234–241. 10.1016/j.amc.2009.01.042MathSciNetView ArticleMATHGoogle Scholar
- Fan X: Global regularity for variable exponent elliptic equations in divergence form. Journal of Differential Equations 2007, 235(2):397–417. 10.1016/j.jde.2007.01.008MathSciNetView ArticleMATHGoogle Scholar
- Fan X: Boundary trace embedding theorems for variable exponent Sobolev spaces. Journal of Mathematical Analysis and Applications 2008, 339(2):1395–1412. 10.1016/j.jmaa.2007.08.003MathSciNetView ArticleMATHGoogle Scholar
- Fan X, Zhang Q, Zhao D: Eigenvalues of -Laplacian Dirichlet problem. Journal of Mathematical Analysis and Applications 2005, 302(2):306–317. 10.1016/j.jmaa.2003.11.020MathSciNetView ArticleMATHGoogle Scholar
- Harjulehto P, Hästö P, Latvala V: Harnack's inequality for -harmonic functions with unbounded exponent . Journal of Mathematical Analysis and Applications 2009, 352(1):345–359. 10.1016/j.jmaa.2008.05.090MathSciNetView ArticleMATHGoogle Scholar
- Mihăilescu M, Rădulescu V: Continuous spectrum for a class of nonhomogeneous differential operators. Manuscripta Mathematica 2008, 125(2):157–167. 10.1007/s00229-007-0137-8MathSciNetView ArticleMATHGoogle Scholar
- Mihăilescu M, Pucci P, Rădulescu V: Nonhomogeneous boundary value problems in anisotropic Sobolev spaces. Comptes Rendus de l'Académie des Sciences—Series I 2007, 345(10):561–566.MathSciNetMATHGoogle Scholar
- Mihăilescu M, Rădulescu V, Repovš D: On a non-homogeneous eigenvalue problem involving a potential: an Orlicz-Sobolev space setting. Journal de Mathématiques Pures et Appliquées 2010, 93(2):132–148.View ArticleMathSciNetMATHGoogle Scholar
- Musielak J: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Volume 1034. Springer, Berlin, Germany; 1983:iii+222.Google Scholar
- Samko SG: Density in the generalized Sobolev spaces . Doklady Rossiĭskaya Akademiya Nauk 1999, 369(4):451–454.MathSciNetMATHGoogle Scholar
- Zhang Q: Existence of solutions for -Laplacian equations with singular coefficients in . Journal of Mathematical Analysis and Applications 2008, 348(1):38–50. 10.1016/j.jmaa.2008.06.026MathSciNetView ArticleMATHGoogle Scholar
- Kim I-S, Kim Yun-Ho: Global bifurcation of the -Laplacian in . Nonlinear Analysis: Theory, Methods & Applications 2009, 70(7):2685–2690. 10.1016/j.na.2008.03.055MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Nieto JJ: The monotone iterative technique for three-point second-order integrodifferential boundary value problems with -Laplacian. Boundary Value Problems 2007, -9.Google Scholar
- Jiao J, Chen L, Li L: Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations. Journal of Mathematical Analysis and Applications 2008, 337(1):458–463. 10.1016/j.jmaa.2007.04.021MathSciNetView ArticleMATHGoogle Scholar
- Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. Journal of Mathematical Analysis and Applications 2007, 325(1):226–236. 10.1016/j.jmaa.2005.04.005MathSciNetView ArticleMATHGoogle Scholar
- Liu L, Hu L, Wu Y: Positive solutions of two-point boundary value problems for systems of nonlinear second-order singular and impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):3774–3789. 10.1016/j.na.2007.10.012MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, O'Regan D: Variational approach to impulsive differential equations. Nonlinear Analysis: Real World Applications 2009, 10(2):680–690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters 2002, 15(4):489–493. 10.1016/S0893-9659(01)00163-XMathSciNetView ArticleMATHGoogle Scholar
- Di Piazza L, Satco B: A new result on impulsive differential equations involving non-absolutely convergent integrals. Journal of Mathematical Analysis and Applications 2009, 352(2):954–963. 10.1016/j.jmaa.2008.11.048MathSciNetView ArticleMATHGoogle Scholar
- Shen J, Wang W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(11):4055–4062. 10.1016/j.na.2007.10.036MathSciNetView ArticleMATHGoogle Scholar
- Yao M, Zhao A, Yan J: Periodic boundary value problems of second-order impulsive differential equations. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(1):262–273. 10.1016/j.na.2007.11.050MathSciNetView ArticleMATHGoogle Scholar
- Feng M, Du B, Ge W: Impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2009, 70(9):3119–3126. 10.1016/j.na.2008.04.015MathSciNetView ArticleMATHGoogle Scholar
- Kristály A, Rădulescu V, Varga C: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136. Cambridge University Press, Cambridge, UK; 2010.View ArticleMATHGoogle Scholar
- Cabada A, Tomeček J: Extremal solutions for nonlinear functional -Laplacian impulsive equations. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(3):827–841. 10.1016/j.na.2006.06.043MathSciNetView ArticleMATHGoogle Scholar
- Zhang Q, Qiu Z, Liu X: Existence of solutions and nonnegative solutions for weighted -Laplacian impulsive system multi-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(9):3814–3825. 10.1016/j.na.2009.02.040MathSciNetView ArticleMATHGoogle Scholar
- Yang Z: Existence of nontrivial solutions for a nonlinear Sturm-Liouville problem with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(1):216–225. 10.1016/j.na.2006.10.044MathSciNetView ArticleMATHGoogle Scholar
- Li Y, Li F: Sign-changing solutions to second-order integral boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(4):1179–1187. 10.1016/j.na.2007.06.024MathSciNetView ArticleMATHGoogle Scholar
- Ma R, An Y: Global structure of positive solutions for nonlocal boundary value problems involving integral conditions. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(10):4364–4376. 10.1016/j.na.2009.02.113MathSciNetView ArticleMATHGoogle Scholar
- Zhang X, Yang X, Ge W: Positive solutions of th-order impulsive boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(12):5930–5945. 10.1016/j.na.2009.05.016MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. 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.