Abstract
This paper investigates the existence of solutions for a weighted Laplacian impulsive integrodifferential system with multipoint and integral mixed boundary value problems via LeraySchauder’s degree; sufficient conditions for the existence of solutions are given. Moreover, we get the existence of nonnegative solutions.
MSC: 34B37.
Keywords:
weighted Laplacian; impulsive integrodifferential system; LeraySchauder’s degree1 Introduction
In this paper, we consider the existence of solutions and nonnegative solutions for the following weighted Laplacian integrodifferential system:
where , , , , with the following impulsive boundary value conditions:
where and , is called the weighted Laplacian; , ; (); is nonnegative, ; , ; ; T and S are linear operators defined by , , , where .
If and , we say the problem is nonresonant, but if or , we say the problem is resonant.
Throughout the paper, means functions which are uniformly convergent to 0 (as ); for any , will denote the jth component of v; the inner product in will be denoted by , will denote the absolute value and the Euclidean norm on . Denote , , , , , where , . Denote by the interior of , . Let
Obviously, is a Banach space with the norm , and is a Banach space with the norm . Denote with the norm
In the following, and will be simply denoted by PC and , respectively. We denote
The study of differential equations and variational problems with nonstandard growth conditions has attracted more and more interest in recent years (see [14]). The applied background of these kinds of problems includes nonlinear elasticity theory [4], electrorheological fluids [1,3], and image processing [2]. Many results have been obtained on these kinds of problems; see, for example, [515]. Recently, the applications of variable exponent analysis in image restoration have attracted more and more attention [1619]. If (a constant), (1)(4) becomes the wellknown pLaplacian problem. If is a general function, one can see easily in general, but , so represents a nonhomogeneity and possesses more nonlinearity, thus is more complicated than . For example:
(a) If is a bounded domain, the Rayleigh quotient
is zero in general, and only under some special conditions (see [9]), when () is an interval, the results show that if and only if is monotone. But the property of is very important in the study of pLaplacian problems, for example, in [20], the authors use this property to deal with the existence of solutions.
(b) If and (a constant) and , then u is concave, this property is used extensively in the study of onedimensional pLaplacian problems (see [21]), but it is invalid for . It is another difference between and .
In recent years, many results have been devoted to the existence of solutions for the Laplacian impulsive differential equation boundary value problems; see, for example, [2229]. There are some methods to deal with these problems, for example, subsupersolution method, fixed point theorem, monotone iterative method, coincidence degree. Because of the nonlinear property of , results on the existence of solutions for pLaplacian impulsive differential equation boundary value problems are rare (see [3033]). In [34], using the coincidence degree method, the present author investigates the existence of solutions for Laplacian impulsive differential equation with multipoint boundary value conditions, when the problem is nonresonant. 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; see, for example, [3538].
In this paper, when is a general function, we investigate the existence of solutions and nonnegative solutions for the weighted Laplacian impulsive integrodifferential system with integral and multipoint boundary value conditions. Results on these kinds of problems are rare. Our results contain both of the cases of resonance and nonresonance. Our method is based upon LeraySchauder’s degree. The homotopy transformation used in [34] is unsuitable for this paper. Moreover, this paper will consider the existence of (1) with (2), (4) and the following impulsive condition:
where , the impulsive condition (5) is called a linear impulsive condition (LI for short), and (3) is called a nonlinear impulsive condition (NLI for short). In general, pLaplacian impulsive problems have two kinds of impulsive conditions, including LI and NLI; but Laplacian impulsive problems only have LI in general. It is another difference between pLaplacian impulsive problems and Laplacian impulsive problems. Moreover, since the Rayleigh quotient in general and the Laplacian is nonhomogeneity, when we deal with the existence of solutions of variable exponent impulsive problems like (1)(4), we usually need the nonlinear term that satisfies the sub growth condition, but for the pLaplacian impulsive problems, the nonlinear term only needs to satisfy the sub growth condition.
Let , the function is assumed to be Caratheodory, by which we mean:
(i) For almost every , the function is continuous;
(ii) For each , the function is measurable on J;
(iii) For each , there is a such that, for almost every and every with , , , , one has
We say a function is a solution of (1) if with absolutely continuous on , , which satisfies (1) a.e. on J.
In this paper, we always use to denote positive constants, if it cannot lead to confusion. Denote
We say f satisfies the sub growth condition if f satisfies
We will discuss the existence of solutions for system (1)(4) or (1) with (2), (4) and (5) in the following three cases:
This paper is organized as five sections. In Section 2, we present some preliminaries and give the operator equation which has the same solutions of (1)(4) in the three cases, respectively. In Section 3, we give the existence of solutions for system (1)(4) or (1) with (2), (4) and (5) when , . In Section 4, we give the existence of solutions for system (1)(4) or (1) with (2), (4) and (5) when , . Finally, in Section 5, we give the existence of solutions and nonnegative solutions for system (1)(4) or (1) with (2), (4) and (5) when , .
2 Preliminary
For any , denote . Obviously, φ has the following properties.
Lemma 2.1 (see [34])
φis a continuous function and satisfies:
(i) For any, is strictly monotone, i.e.,
(ii) There exists a function, assuch that
It is well known that is a homeomorphism from to for any fixed . Denote
It is clear that is continuous and sends bounded sets to bounded sets.
In this section, we will do some preparation and give the operator equation which has the same solutions of (1)(4) in three cases, respectively. At first, let us now consider the following simple impulsive problem with boundary value condition (4):
We will discuss it in three cases, respectively.
2.1 Case (i)
Suppose that and . If u is a solution of (6) with (4), we have
Denote . It is easy to see that is dependent on a, b and . Define the operator as
By solving for in (7) and integrating, we find
which together with boundary value condition (4) implies
and
then W is a Banach space.
Throughout the paper, we denote
Lemma 2.2Suppose thaton, () andon. Then the functionhas the following properties:
(i) For any fixed, the equation
(ii) The function, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any, we have
Proof (i) From Lemma 2.1, it is immediate that
and hence, if (8) has a solution, then it is unique.
Suppose that , it is easy to see that there exists some such that the absolute value of the th component of satisfies
Thus the th component of keeps sign on J, namely, for any , we have
Obviously, we have
then it is easy to see that the th component of keeps the same sign of . Thus,
Let us consider the equation
According to the preceding discussion, all the solutions of (9) belong to . Therefore
it means the existence of solutions of .
In this way, we define a function , which satisfies .
(ii) By the proof of (i), we also obtain sends bounded sets to bounded sets, and
It only remains to prove the continuity of . Let be a convergent sequence in W and , as . Since is a bounded sequence, it contains a convergent subsequence . Suppose that as . Since , letting , we have , which together with (i) implies , it means is continuous. This completes the proof. □
Now we denote by the Nemytskii operator associated to f defined by
It is clear that is continuous and sends bounded sets of to bounded sets of , and hence it is compact continuous.
If u is a solution of (6) with (4), we have
Lemma 2.3 (i) The operatoris continuous and sends equiintegrable sets into relatively compact sets in.
(ii) The operatoris continuous and sends bounded sets into relatively compact sets in.
Proof (i) It is easy to check that , , . Since and
it is easy to check that is a continuous operator from to .
Let now U be an equiintegrable set in , then there exists such that
We want to show that is a compact set.
Let be a sequence in , then there exists a sequence such that . For any , we have
Hence the sequence is uniformly bounded and equicontinuous. By the AscoliArzela theorem, there exists a subsequence of (which we rename the same) which is convergent in PC. According to the bounded continuity of the operator , we can choose a subsequence of (which we still denote ) which is convergent in PC, then is convergent in PC.
Since
it follows from the continuity of and the integrability of in that is convergent in PC. Thus is convergent in .
(ii) It is easy to see from (i) and Lemma 2.2.
This completes the proof. □
It is easy to see that is compact continuous.
Lemma 2.4Suppose that, ; on, () andon. Thenuis a solution of (1)(4) if and only ifuis a solution of the following abstract operator equation:
Proof Suppose that u is a solution of (1)(4). By integrating (1) from 0 to t, we find that
It follows from (13) and (4) that
Combining the definition of , we can see
Conversely, if u is a solution of (12), then (2) is satisfied. It is easy to check that
and
By the condition of the mapping , we have
Thus
It follows from (15) and (16) that (4) is satisfied.
From (12), we have
It follows from (17) that (3) is satisfied.
Hence u is a solution of (1)(4). This completes the proof. □
2.2 Case (ii)
Suppose that and . If u is a solution of (6) with (4), we have
Denote . It is easy to see that is dependent on a, b and . Boundary value condition (4) implies that
Throughout the paper, we denote .
Lemma 2.5The functionhas the following properties:
(i) For any fixed, the equationhas a unique solution.
(ii) The function, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any, we have
Proof Similar to the proof of Lemma 2.2, we omit it here. □
It is clear that is continuous and sends bounded sets of to bounded sets of , and hence it is compact continuous.
Similar to the proof of Lemma 2.3, we have the following.
Lemma 2.6 (i) The operatoris continuous and sends equiintegrable sets into relatively compact sets in.
(ii) The operatoris continuous and sends bounded sets into relatively compact sets in.
It is easy to see thatis compact continuous.
Lemma 2.7Suppose that, , thenuis a solution of (1)(4) if and only ifuis a solution of the following abstract operator equation:
Proof Similar to the proof of Lemma 2.4, we omit it here. □
2.3 Case (iii)
Suppose that and . If u is a solution of (6) with (4), we have
Denote . It is easy to see that is dependent on a, b and .
Obviously, can be rewritten as
Denote . Moreover, we also have
Lemma 2.8Suppose that, g, hsatisfy one of the following:
Then the functionhas the following properties:
(i) For any fixed, the equationhas a unique solution.
(ii) The function, defined in (i), is continuous and sends bounded sets to bounded sets. Moreover, for any, we have
Proof Similar to the proof of Lemma 2.2, we omit it here. □
It is clear that is continuous and sends bounded sets of to bounded sets of , and hence it is compact continuous.
Similar to the proof of Lemma 2.3, we have
Lemma 2.9 (i) The operatoris continuous and sends equiintegrable sets into relatively compact sets in.
(ii) The operatoris continuous and sends bounded sets into relatively compact sets in.
It is easy to see thatis compact continuous.
Lemma 2.10Suppose that, and, g, hsatisfy one of the following:
Thenuis a solution of (1)(4) if and only ifuis a solution of the following abstract operator equation:
Proof Similar to the proof of Lemma 2.4, we omit it here. □
3 Existence of solutions in Case (i)
In this section, we apply LeraySchauder’s degree to deal with the existence of solutions for system (1)(4) or (1) with (2), (4) and (5) when , .
When f satisfies the sub growth condition, we have the following theorem.
Theorem 3.1Suppose that, ; on, () andon; fsatisfies the subgrowth condition; and operatorsAandBsatisfy the following conditions:
then problem (1)(4) has at least a solution.
Proof First we consider the following problem:
Denote
Obviously, () has the same solution as the following operator equation when :
It is easy to see that the operator is compact continuous for any . It follows from Lemma 2.2 and Lemma 2.3 that is compact continuous from to for any .
We claim that all the solutions of (21) are uniformly bounded for . In fact, if it is false, we can find a sequence of solutions for (21) such that as and for any .
From Lemma 2.2, we have
Thus
It follows from (11) and Lemma 2.2 that
Denote . If the above inequality holds then
It follows from (14), (20) and (22) that
which implies that
Thus
It follows from (23) and (24) that is uniformly bounded.
Thus, we can choose a large enough such that all the solutions of (21) belong to . Therefore the LeraySchauder degree is well defined for , and
It is easy to see that u is a solution of if and only if u is a solution of the following usual differential equation:
Obviously, system () possesses a unique solution . Since , we have
which implies that (1)(4) has at least one solution. This completes the proof. □
Theorem 3.2Suppose that, ; on, () andon; fsatisfies the subgrowth condition; and operatorsAandsatisfy the following conditions:
Then problem (1) with (2), (4) and (5) has at least a solution.
From Theorem 3.1, it suffices to show that
(a) Suppose that , where is a large enough positive constant. From the definition of D, we have
Since , we have . Thus (25) is valid.
(b) Suppose that , we can see that
There are two cases: Case (i): ; Case (ii): .
Case (i): Since , we have , and
Thus (25) is valid.
Case (ii): Since , we have , and
Thus (25) is valid.
Thus problem (1) with (2), (4) and (5) has at least a solution. This completes the proof. □
Let us consider
where ε is a parameter, and
where are Caratheodory. We have the following theorem.
Theorem 3.3Suppose that, ; on, () andon; fsatisfies the subgrowth condition; and we assume that
then problem (26) with (2)(4) has at least one solution when parameterεis small enough.
Proof Denote
We consider the existence of solutions of the following equation with (2)(4)
Denote
We know that (27) with (2)(4) has the same solution of .
Obviously, . So . As in the proof of Theorem 3.1, we know that all the solutions of are uniformly bounded, then there exists a large enough such that all the solutions of belong to . Since is compact continuous from to , we have
Since f and h are Caratheodory, we have
Thus
Thus, when ε is small enough, from (28), we can conclude that
Thus has no solution on for any , when ε is small enough. It means that the LeraySchauder degree is well defined for any , and
Since , from the proof of Theorem 3.1, we can see that the righthand side is nonzero. Thus (26) with (2)(4) has at least one solution when ε is small enough. This completes the proof. □
Theorem 3.4Suppose that, ; on, () andon; fsatisfies the subgrowth condition; and we assume that
where, and, , then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.
Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □
4 Existence of solutions in Case (ii)
In this section, we apply LeraySchauder’s degree to deal with the existence of solutions for system (1)(4) or (1) with (2), (4) and (5) when , .
When f satisfies the sub growth condition, we have the following.
Theorem 4.1Suppose that, ; fsatisfies the subgrowth condition; and operatorsAandBsatisfy the following conditions:
then problem (1)(4) has at least a solution.
Proof Similar to the proof of Theorem 3.1, we omit it here. □
Theorem 4.2Suppose that, ; fsatisfies the subgrowth condition; and operatorsAandsatisfy the following conditions:
where
then problem (1) with (2), (4) and (5) has at least a solution.
Proof Similar to the proof of Theorem 3.2, we omit it here. □
Theorem 4.3Suppose that, ; fsatisfies the subgrowth condition; and we assume that
then problem (26) with (2)(4) has at least one solution when parameterεis small enough.
Proof Similar to the proof of Theorem 3.3, we omit it here. □
Theorem 4.4Suppose that, ; fsatisfies the subgrowth condition; and we assume that
where, and, , then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.
Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □
5 Existence of solutions in Case (iii)
In this section, we apply LeraySchauder’s degree to deal with the existence of solutions and nonnegative solutions for system (1)(4) or (1) with (2), (4) and (5) when , .
When f satisfies the sub growth condition, we have the following theorem.
Theorem 5.1Suppose that, and, g, hsatisfy one of the following:
whenfsatisfies the subgrowth condition; and operatorsAandBsatisfy the following conditions:
then problem (1)(4) has at least a solution.
Proof Similar to the proof of Theorem 3.1, we omit it here. □
Theorem 5.2Suppose that, and, g, hsatisfy one of the following:
whenfsatisfies the subgrowth condition; and operatorsAandsatisfy the following conditions:
where
then problem (1) with (2), (4) and (5) has at least a solution.
Proof Similar to the proof of Theorem 3.2, we omit it here. □
Theorem 5.3Suppose that, and, g, hsatisfy one of the following:
whenfsatisfies the subgrowth condition; and we assume that
then problem (26) with (2)(4) has at least one solution when parameterεis small enough.
Proof Similar to the proof of Theorem 3.3, we omit it here. □
Theorem 5.4Suppose that, and, g, hsatisfy one of the following:
whenfsatisfies the subgrowth condition; and we assume that
where, and, , then problem (26) with (2), (4) and (5) has at least one solution when parameterεis small enough.
Proof Similar to the proof of Theorem 3.2 and Theorem 3.3, we omit it here. □
In the following, we will consider the existence of nonnegative solutions. For any , the notation means for any .
Theorem 5.5Suppose that, , , . We also assume:
Then every solution of (1)(4) is nonnegative.
Proof Let u be a solution of (1)(4). From Lemma 2.10, we have
We claim that . If it is false, then there exists some such that .
It follows from (1^{0}) and (2^{0}) that
Thus (29) and condition (3^{0}) hold
Similar to the proof before Lemma 2.8, from the boundary value conditions, we have
From (29) and (30), we get a contradiction to (31). Thus .
We claim that
If it is false, then there exists some such that
It follows from (1^{0}) and (2^{0}) that
Thus (33) and condition (3^{0}) hold
From (33), (34), we get a contradiction to (31). Thus (32) is valid.
Obviously, , , and is decreasing, i.e., for any with . For any , there exist such that
It follows from condition (3^{0}) that is increasing on and is decreasing on . Thus , .
from (4) and (35), we have . Then .
If
from (4), (36) and condition (4^{0}), we have . Then .
Thus , . The proof is completed. □
Corollary 5.6Under the conditions of Theorem 5.1, we also assume:
Then (1)(4) has a nonnegative solution.
Denote
then satisfies the Caratheodory condition, and for any .
then and are continuous and satisfy
It is not hard to check that
(2^{0})′ for uniformly, where , and ;
Let us consider
It follows from Theorem 5.1 and Theorem 5.5 that (37) has a nonnegative solution u. Since , we have , and then
Thus u is a nonnegative solution of (1)(4). This completes the proof. □
Note (i) Similarly, we can get the existence of nonnegative solutions of (26) with (2)(4).
(ii) Similarly, under the conditions of Case (ii), we can discuss the existence of nonnegative solutions.
6 Examples
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors typed, read and approved the final manuscript.
Acknowledgements
Partly supported by the National Science Foundation of China (10701066 & 10971087).
References

Acerbi, E, Mingione, G: Regularity results for a class of functionals with nonstandard growth. Arch. Ration. Mech. Anal.. 156, 121–140 (2001). Publisher Full Text

Chen, Y, Levine, S, Rao, M: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math.. 66, 1383–1406 (2006). Publisher Full Text

Růžička, M: Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin (2000)

Zhikov, VV: Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR, Izv.. 29, 33–36 (1987). Publisher Full Text

Deng, SG: A local mountain pass theorem and applications to a double perturbed Laplacian equations. Appl. Math. Comput.. 211, 234–241 (2009). Publisher Full Text

Diening, L, Harjulehto, P, Hästö, P, Růžička, M: Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin (2011)

Fan, XL: Global regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ.. 235, 397–417 (2007). Publisher Full Text

Fan, XL: Boundary trace embedding theorems for variable exponent Sobolev spaces. J. Math. Anal. Appl.. 339, 1395–1412 (2008). Publisher Full Text

Fan, XL, Zhang, QH, Zhao, D: Eigenvalues of Laplacian Dirichlet problem. J. Math. Anal. Appl.. 302, 306–317 (2005). Publisher Full Text

Harjulehto, P, Hästö, P, Latvala, V: Harnack’s inequality for harmonic functions with unbounded exponent p. J. Math. Anal. Appl.. 352, 345–359 (2009). PubMed Abstract  Publisher Full Text

Harjulehto, P, Hästö, P, Lê, ÚV, Nuortio, M: Overview of differential equations with nonstandard growth. Nonlinear Anal. TMA. 72, 4551–4574 (2010). Publisher Full Text

Mihăilescu, M, Rădulescu, V: Continuous spectrum for a class of nonhomogeneous differential operators. Manuscr. Math.. 125, 157–167 (2008). Publisher Full Text

Musielak, J: Orlicz Spaces and Modular Spaces, Springer, Berlin (1983)

Samko, SG: Density of in the generalized Sobolev spaces . Dokl. Akad. Nauk. 369, 451–454 (1999)

Zhang, QH: Existence of positive solutions to a class of Laplacian equations with singular nonlinearities. Appl. Math. Lett.. 25, 2381–2384 (2012). Publisher Full Text

Guo, ZC, Liu, Q, Sun, JB, Wu, BY: Reactiondiffusion systems with growth for image denoising. Nonlinear Anal., Real World Appl.. 12, 2904–2918 (2011). Publisher Full Text

Guo, ZC, Sun, JB, Zhang, DZ, Wu, BY: Adaptive PeronaMalik model based on the variable exponent for image denoising. IEEE Trans. Image Process.. 21, 958–967 (2012). PubMed Abstract  Publisher Full Text

Harjulehto, P, Hästö, P, Latvala, V, Toivanen, O: Critical variable exponent functionals in image restoration. Appl. Math. Lett.. 26, 56–60 (2013). Publisher Full Text

Li, F, Li, ZB, Pi, L: Variable exponent functionals in image restoration. Appl. Math. Comput.. 216, 870–882 (2010). Publisher Full Text

Kim, IS, Kim, YH: Global bifurcation of the pLaplacian in . Nonlinear Anal.. 70, 2685–2690 (2009). Publisher Full Text

Ahmad, B, Nieto, JJ: The monotone iterative technique for threepoint secondorder integrodifferential boundary value problems with pLaplacian. Bound. Value Probl.. 2007, (2007) Article ID 57481

Chen, P, Tang, XH: New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Math. Comput. Model.. 55, 723–739 (2012). Publisher Full Text

Li, J, Nieto, JJ, Shen, J: Impulsive periodic boundary value problems of firstorder differential equations. J. Math. Anal. Appl.. 325, 226–236 (2007). Publisher Full Text

Luo, ZG, Xiao, J, Xu, YL: Subharmonic solutions with prescribed minimal period for some secondorder impulsive differential equations. Nonlinear Anal.. 75, 2249–2255 (2012). Publisher Full Text

Ma, RY, Sun, JY, Elsanosi, M: Signchanging solutions of second order Dirichlet problem with impulsive effects. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal.. 20, 241–251 (2013)

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

Di Piazza, L, Satco, B: A new result on impulsive differential equations involving nonabsolutely convergent integrals. J. Math. Anal. Appl.. 352, 954–963 (2009). Publisher Full Text

Xiao, JZ, Zhu, XH, Cheng, R: The solution sets for second order semilinear impulsive multivalued boundary value problems. Comput. Math. Appl.. 64, 147–160 (2012). Publisher Full Text

Yao, MP, Zhao, AM, Yan, JR: Periodic boundary value problems of secondorder impulsive differential equations. Nonlinear Anal.. 70, 262–273 (2009). Publisher Full Text

Bai, L, Dai, BX: Three solutions for a pLaplacian boundary value problem with impulsive effects. Appl. Math. Comput.. 217, 9895–9904 (2011). Publisher Full Text

Bogun, I: Existence of weak solutions for impulsive pLaplacian problem with superlinear impulses. Nonlinear Anal., Real World Appl.. 13, 2701–2707 (2012). Publisher Full Text

Cabada, A, Tomeček, J: Extremal solutions for nonlinear functional ϕLaplacian impulsive equations. Nonlinear Anal.. 67, 827–841 (2007). Publisher Full Text

Feng, MQ, Du, B, Ge, WG: Impulsive boundary value problems with integral boundary conditions and onedimensional pLaplacian. Nonlinear Anal.. 70, 3119–3126 (2009). Publisher Full Text

Zhang, QH, Qiu, ZM, Liu, XP: Existence of solutions and nonnegative solutions for weighted Laplacian impulsive system multipoint boundary value problems. Nonlinear Anal.. 71, 3814–3825 (2009). Publisher Full Text

Ding, W, Wang, Y: New result for a class of impulsive differential equation with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul.. 18, 1095–1105 (2013). Publisher Full Text

Hao, XN, Liu, LS, Wu, YH: Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul.. 16, 101–111 (2011). Publisher Full Text

Liu, ZH, Han, JF, Fang, LJ: Integral boundary value problems for first order integrodifferential equations with impulsive integral conditions. Comput. Math. Appl.. 61, 3035–3043 (2011). Publisher Full Text

Zhang, XM, Yang, XZ, Ge, WG: Positive solutions of nthorder impulsive boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal.. 71, 5930–5945 (2009). Publisher Full Text