Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional -Laplacian

By using the fixed point theory for completely continuous operator, this paper investigates the existence of positive solutions for a class of fourth-order impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Moreover, we offer some interesting discussion of the associated boundary value problems. Upper and lower bounds for these positive solutions also are given, so our work is new.

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. [1]; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al. [2]. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see, e.g., [3–18] and references cited therein).

Moreover, the theory of boundary-value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For boundary-value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [19], Karakostas and Tsamatos [20], Lomtatidze and Malaguti [21], and the references therein. For more information about the general theory of integral equations and their relation with boundary-value problems, we refer to the book of Corduneanu [22] and Agarwal and O'Regan [23].

On the other hand, boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary-value problems as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention in the literature. To identify a few, we refer the reader to [24–46] and references therein. In particular, we would like to mention some results of Zhang et al. [34], Kang et al. [44], and Webb et al. [45]. In [34], Zhang et al. studied the following fourth-order boundary value problem with integral boundary conditions

where is a positive parameter, , is the zero element of , and . The authors investigated the multiplicity of positive solutions to problem (1.1) by using the fixed point index theory in cone for strict set contraction operator.

In [44], Kang et al. have improved and generalized the work of [34] by applying the fixed point theory in cone for a strict set contraction operator; they proved that there exist various results on the existence of positive solutions to a class of fourth-order singular boundary value problems with integral boundary conditions

where and may be singular at or ; are continuous and may be singular at , and ; , and , and , and are nonnegative, .

More recently, by using a unified approach, Webb et al. [45] considered the widely studied boundary conditions corresponding to clamped and hinged ends and many nonlocal boundary conditions and established excellent existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam

subject to various boundary conditions

where denotes a linear functional on given by involving a Stieltjes integral, and is a function of bounded variation.

At the same time, we notice that there has been a considerable attention on -Laplacian BVPs [18, 32, 35, 36, 38, 42] as -Laplacian appears in the study of flow through porous media (), nonlinear elasticity (), glaciology (), and so forth. Here, it is worth mentioning that Liu et al. [43] considered the following fourth-order four-point boundary value problem:

where , , and . By using upper and lower solution method, fixed-point theorems, and the properties of Green's function and , the authors give sufficient conditions for the existence of one positive solution.

Motivated by works mentioned above, in this paper, we consider the existence of positive solutions for a class of boundary value problems with integral boundary conditions of fourth-order impulsive differential equations:

Here is -Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the right-hand limit and left-hand limit of at , respectively, and is nonnegative.

For the case of , problem (1.6) reduces to the problem studied by Zhang et al. in [33]. By using the fixed point theorem in cone, the authors obtained some sufficient conditions for the existence and multiplicity of symmetric positive solutions for a class of -Laplacian fourth-order differential equations with integral boundary conditions.

For the case of , and , problem (1.6) is related to fourth-order two-points boundary value problem of ODE. Under this case, problem (1.6) has received considerable attention (see, e.g., [40–42] and references cited therein). Aftabizadeh [40] showed the existence of a solution to problem (1.6) under the restriction that is a bounded function. Bai and Wang [41] have applied the fixed point theorem and degree theory to establish existence, uniqueness, and multiplicity of positive solutions to problem (1.6). Ma and Wang [42] have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that is either superlinear or sublinear on by employing the fixed point theorem of cone extension or compression.

Being directly inspired by [18, 34, 43], in the present paper, we consider some existence results for problem (1.6) in a specially constructed cone by using the fixed point theorem. The main features of this paper are as follows. Firstly, comparing with [39–43], we discuss the impulsive boundary value problem with integral boundary conditions, that is, problem (1.6) includes fourth-order two-, three-, multipoint, and nonlocal boundary value problems as special cases. Secondly, the conditions are weaker than those of [33, 34, 46], and we consider the case . Finally, comparing with [33, 34, 39–43, 46], upper and lower bounds for these positive solutions also are given. Hence, we improve and generalize the results of [33, 34, 39–43, 46] to some degree, and so, it is interesting and important to study the existence of positive solutions of problem (1.6).

The organization of this paper is as follows. We shall introduce some lemmas in the rest of this section. In Section 2, we provide some necessary background. In particular, we state some properties of the Green's function associated with problem (1.6). In Section 3, the main results will be stated and proved. Finally, in Section 4, we offer some interesting discussion of the associated problem (1.6).

To obtain positive solutions of problem (1.6), the following fixed point theorem in cones is fundamental, which can be found in [47, page 94].

Lemma 1.1.

Let and be two bounded open sets in Banach space , such that and . Let be a cone in and let operator be completely continuous. Suppose that one of the following two conditions is satisfied:

(a), and ;

(b), and .

Then, has at least one fixed point in .

In order to define the solution of problem (1.6), we shall consider the following space.

Let , and

Then is a real Banach space with norm

where .

A function is called a solution of problem (1.6) if it satisfies (1.6).

To establish the existence of multiple positive solutions in of problem (1.6), let us list the following assumptions:

;

with

Write

From , it is clear that .

We shall reduce problem (1.6) to an integral equation. To this goal, firstly by means of the transformation

we convert problem (1.6) into

Lemma 2.1.

Assume that and hold. Then problem (2.6) has a unique solution given by

where

Proof.

The proof follows by routine calculations.

Write . Then from (2.9) and (2.10), we can prove that have the following properties.

Proposition 2.2.

If holds, then we have

Proposition 2.3.

For , we have

Proposition 2.4.

If holds, then for , we have

where

Proof.

By (2.6) and (2.12), we have

On the other hand, noticing , we obtain

The proof of Proposition 2.4 is complete.

Remark 2.5.

From (2.9) and (2.13), we can obtain that

Lemma 2.6.

If and hold, then problem (2.7) has a unique solution and can be expressed in the following form:

where

and is defined in (2.10).

Proof.

First suppose that is a solution of problem (2.7).

If it is easy to see by integration of problem (2.7) that

If then integrate from to ,

Similarly, if we have

Integrating again, we can get

Letting in (2.23), we find

Substituting and (2.24) into (2.23), we obtain

where

Therefore, we have

Let

Then,

and the proof of sufficient is complete.

Conversely, if is a solution of (2.18).

Direct differentiation of (2.18) implies, for ,

Evidently,

The Lemma is proved.

Remark 2.7.

From (2.19), we can prove that the properties of are similar to that of .

Suppose that is a solution of problem (1.6). Then from Lemmas 2.6 and 2.1, we have

For the sake of applying Lemma 1.1, we construct a cone in via

where

It is easy to see that is a closed convex cone of .

Define an operator by

From (2.35), we know that is a solution of problem (1.6) if and only if is a fixed point of operator .

Definition 2.8 (see [1]).

The set is said to be quasi-equicontinuous in if for any there exist such that if then

We present the following result about relatively compact sets in which is a consequence of the Arzela-Ascoli Theorem. The reader can find its proof partially in [1].

Lemma 2.9.

is relatively compact if and only if is bounded and quasi-equicontinuous on .

Write

where .

Lemma 2.10.

Suppose that and hold. Then and is completely continuous.

Proof.

For , it is clear that , and

From (2.35) and Remark 2.5, we obtain the following cases.

Case 1.

if , noticing , then we have

Case 2.

if , noticing and , then we have

Therefore, , that is, . Also, we have since . Hence we have .

Next, we prove that is completely continuous.

It is obvious that is continuous. Now we prove is relatively compact.

Let be a bounded set. Then, for all , we have

Therefore is uniformly bounded.

On the other hand, for all with , we have

and by the continuity of , we have

and then is quasi-equicontinuous. It follows that is relatively compact on by Lemma 2.9. So is completely continuous.

In this section, we apply Lemma 1.1 to establish the existence of positive solutions of problem (1.6). We begin by introducing the following conditions on and .

There exist numbers such that

where is defined in (2.4), are defined in (2.14), respectively, and write

where denotes or

Theorem 3.1.

Assume that hold. Then problem (1.6) has at least one positive solution with

Proof.

Let be the cone preserving, completely continuous operator that was defined by (2.35). For with , (2.13), (2.19), and (3.1) imply

where

Now if we let , then (3.5) shows that

Further, let

Then, and implies

that is,

Hence, for all . Therefore, for all , (3.2) implies

that is, implies

Applying (b) of Lemma 1.1 to (3.7) and (3.12) yields that has a fixed point with . Hence, since for we have , it follows that (3.4) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.1, we can prove the following results.

Corollary 3.2.

Assume that and hold. If , and , then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.4).

Proof.

The proof is similar to that of Theorem of [6].

In Theorem 3.3, we assume the following condition on and .

There exist numbers such that

where are defined in (2.14), and write

Theorem 3.3.

Assume that , and hold. Then problem (1.6) has at least one positive solution with

Proof.

For with , (3.13) implies

that is, implies

Next, we turn to (3.14) and (3.15). From (3.14), (3.15), and (3.16), we have

Let

Thus, for , we have

Then, (3.22) and (3.23) imply

Applying (a) of Lemma 1.1 to (3.19) and (3.24) yields that has a fixed point with . Hence, since for we have , it follows that (3.17) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.3, we can prove the following results.

Corollary 3.4.

Assume that and hold. If and ; then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.17).

Proof.

The proof is similar to that of Theorem of [6].

Theorem 3.5.

Assume that (3.1) of and (3.14) and (3.15) of hold. In addition, letting and satisfy the following condition:

There is a such that and implies

Then, problem (1.6) has at least two positive solutions and with

where and satisfy

Proof.

If (3.1) of holds, similar to the proof of (3.7), we can prove that

If (3.14) and (3.15) of hold, similar to the proof of (3.23), we have

Finally, we show that

In fact, for with then by (2.18), we have

and it follows from that

which implies that (3.30) holds.

Applying Lemma 1.1 to (3.28), (3.29), and (3.30) yields that has two fixed point with , and. Hence, since for we have , it follows that (3.26) holds. This and Lemma 2.9 complete the proof.

Remark 3.6.

Similar to the proof of that of [5], we can prove that problem (1.6) can be generalized to obtain many positive solutions.

In this section, we offer some interesting discussions associated with problem (1.6).

Discussion.

Generally, it is difficult to obtain the upper and lower bounds of positive solutions for nonlinear higher-order boundary value problems (see, e.g., [33, 34, 39–43, 46, 48, 49] and their references).

For example, we consider the following problems:

Here is -Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the right-hand limit and left-hand limit of at , respectively, and is nonnegative.

By means of the transformation (2.5), we can convert problem (4.1) into

Using the similar proof of that of Lemmas 2.1 and 2.6, we can obtain the following results. In addition, if we replace by in , respectively, then we obtain , where

Lemma 4.1.

If and hold, then BVP (4.2) has a unique solution and can be expressed in the form

where

is defined in (2.10).

Lemma 4.2.

If and hold, then BVP (4.3) has a unique solution and can be expressed in the form

where

It is not difficult to prove that and have the similar properties to that of and . But for , and have no property (2.13). In fact, if , then we can prove that and have the following properties.

Proposition 4.3.

If holds, then for , we have

where

Proof.

We only consider (4.9). By (4.3), we have

On the other hand, noticing , we obtain

Similarly, we can prove that (4.10) holds, too.

Remark 4.4.

Since , .

From (4.9) and (4.10), we can only define a cone by

which implies that we cannot obtain the lower bounds for the positive solutions of problem (4.1).

To illustrate how our main results can be used in practice, we present an example.

Example 5.1.

Consider the following boundary value problem:

where , and

Conclusion.

Equation(5.1) has at least one positive solution for with

Proof.

By simple computation, we have . Select then for , we have

By Theorem 3.1, (5.1) has a positive solution with .

The authors thank the referee for his/her careful reading of the manuscript and useful suggestions. This work is sponsored by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201010772018), and Beijing Municipal Education Commission (71D0911003).

1. Lakshmikantham, V, Baĭnov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics,p. xii+273. World Scientific, Singapore (1989)

2. Benchohra, M, Henderson, J, Ntouyas, S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications,p. xiv+366. Hindawi Publishing Corporation, New York, NY, USA (2006)

3. Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems. 4(1), 134–141 (2010). Publisher Full Text

4. Baĭnov, DD, Simeonov, PS: Systems with Impulse Effect,p. 255. Ellis Horwood, Chichester, UK (1989)

5. Samoĭlenko, AM, Perestyuk, NA: Impulsive Differential Equations,p. x+462. World Scientific, Singapore (1995)

6. Yan, J: Existence of positive periodic solutions of impulsive functional differential equations with two parameters. Journal of Mathematical Analysis and Applications. 327(2), 854–868 (2007). Publisher Full Text

7. Feng, M, Xie, D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. Journal of Computational and Applied Mathematics. 223(1), 438–448 (2009). Publisher Full Text

8. Nieto, JJ: Basic theory for nonresonance impulsive periodic problems of first order. Journal of Mathematical Analysis and Applications. 205(2), 423–433 (1997). Publisher Full Text

9. Nieto, JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters. 15(4), 489–493 (2002). Publisher Full Text

10. Guo, D: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space. Applied Mathematics and Computation. 143(2-3), 233–249 (2003). Publisher Full Text

11. Liu, X, Guo, D: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. Journal of Mathematical Analysis and Applications. 216(1), 284–302 (1997). Publisher Full Text

12. Agarwal, RP, O'Regan, D: Multiple nonnegative solutions for second order impulsive differential equations. Applied Mathematics and Computation. 114(1), 51–59 (2000). Publisher Full Text

13. Liu, B, Yu, J: Existence of solution of -point boundary value problems of second-order differential systems with impulses. Applied Mathematics and Computation. 125(2-3), 155–175 (2002). Publisher Full Text

14. Agarwal, RP, Franco, D, O'Regan, D: Singular boundary value problems for first and second order impulsive differential equations. Aequationes Mathematicae. 69(1-2), 83–96 (2005). Publisher Full Text

15. Lin, X, Jiang, D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. Journal of Mathematical Analysis and Applications. 321(2), 501–514 (2006). Publisher Full Text

16. Jankowski, T: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments. Applied Mathematics and Computation. 197(1), 179–189 (2008). Publisher Full Text

17. Jankowski, T: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Applied Mathematics and Computation. 202(2), 550–561 (2008). Publisher Full Text

18. Feng, M, Du, B, Ge, W: Impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Nonlinear Analysis: Theory, Methods & Applications. 70(9), 3119–3126 (2009). PubMed Abstract | Publisher Full Text

19. Gallardo, JM: Second-order differential operators with integral boundary conditions and generation of analytic semigroups. Rocky Mountain Journal of Mathematics. 30(4), 1265–1291 (2000). Publisher Full Text

20. Karakostas, GL, Tsamatos, PCh: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electronic Journal of Differential Equations. 2002(30), 1–17 (2002)

21. Lomtatidze, A, Malaguti, L: On a nonlocal boundary value problem for second order nonlinear singular differential equations. Georgian Mathematical Journal. 7(1), 133–154 (2000)

22. Corduneanu, C: Integral Equations and Applications,p. x+366. Cambridge University Press, Cambridge, UK (1991)

23. Agarwal, RP, O'Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations,p. x+341. Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)

24. Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems involving integral conditions. Nonlinear Differential Equations and Applications. 15(1-2), 45–67 (2008). Publisher Full Text

25. Webb, JRL, Infante, G: Non-local boundary value problems of arbitrary order. Journal of the London Mathematical Society. 79(1), 238–258 (2009)

26. Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society. 74(3), 673–693 (2006). Publisher Full Text

27. Ahmad, B, Nieto, JJ: The monotone iterative technique for three-point second-order integrodifferential boundary value problems with -Laplacian. Boundary Value Problems. 2007, (2007)

28. Ahmad, B, Nieto, JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems. 2009, (2009)

29. Ahmad, B, Alsaedi, A, Alghamdi, BS: Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 9(4), 1727–1740 (2008). PubMed Abstract | Publisher Full Text

30. Feng, M, Ji, D, Ge, W: Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces. Journal of Computational and Applied Mathematics. 222(2), 351–363 (2008). Publisher Full Text

31. Zhang, X, Feng, M, Ge, W: Multiple positive solutions for a class of -point boundary value problems. Applied Mathematics Letters. 22(1), 12–18 (2009). Publisher Full Text

32. Feng, M, Ge, W: Positive solutions for a class of -point singular boundary value problems. Mathematical and Computer Modelling. 46(3-4), 375–383 (2007). Publisher Full Text

33. Zhang, X, Feng, M, Ge, W: Symmetric positive solutions for -Laplacian fourth-order differential equations with integral boundary conditions. Journal of Computational and Applied Mathematics. 222(2), 561–573 (2008). Publisher Full Text

34. Zhang, X, Feng, M, Ge, W: Existence results for nonlinear boundary-value problems with integral boundary conditions in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications. 69(10), 3310–3321 (2008). PubMed Abstract | Publisher Full Text

35. Yang, Z: Positive solutions of a second-order integral boundary value problem. Journal of Mathematical Analysis and Applications. 321(2), 751–765 (2006). Publisher Full Text

36. Ma, R: Positive solutions for multipoint boundary value problem with a one-dimensional -Laplacian. Computational & Applied Mathematics. 42, 755–765 (2001). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

37. Bai, Z, Huang, B, Ge, W: The iterative solutions for some fourth-order -Laplace equation boundary value problems. Applied Mathematics Letters. 19(1), 8–14 (2006). Publisher Full Text

38. Liu, B, Liu, L, Wu, Y: Positive solutions for singular second order three-point boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 66(12), 2756–2766 (2007). PubMed Abstract | Publisher Full Text

39. Zhang, X, Liu, L: A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with -Laplacian. Nonlinear Analysis: Theory, Methods & Applications. 68(10), 3127–3137 (2008). PubMed Abstract | Publisher Full Text

40. Aftabizadeh, AR: Existence and uniqueness theorems for fourth-order boundary value problems. Journal of Mathematical Analysis and Applications. 116(2), 415–426 (1986). Publisher Full Text

41. Bai, Z, Wang, H: On positive solutions of some nonlinear fourth-order beam equations. Journal of Mathematical Analysis and Applications. 270(2), 357–368 (2002). Publisher Full Text

42. Ma, R, Wang, H: On the existence of positive solutions of fourth-order ordinary differential equations. Applicable Analysis. 59(1–4), 225–231 (1995)

43. Liu, L, Zhang, X, Wu, Y: Positive solutions of fourth order four-point boundary value problems with -Laplacian operator. Journal of Mathematical Analysis and Applications. 326(2), 1212–1224 (2007). Publisher Full Text

44. Kang, P, Wei, Z, Xu, J: Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces. Applied Mathematics and Computation. 206(1), 245–256 (2008). Publisher Full Text

45. Webb, JRL, Infante, G, Franco, D: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions. Proceedings of the Royal Society of Edinburgh. 138(2), 427–446 (2008)

46. Ma, H: Symmetric positive solutions for nonlocal boundary value problems of fourth order. Nonlinear Analysis: Theory, Methods & Applications. 68(3), 645–651 (2008). PubMed Abstract | Publisher Full Text

47. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones,p. viii+275. Academic Press, Boston, Mass, USA (1988)

48. Eloe, PW, Ahmad, B: Positive solutions of a nonlinear th order boundary value problem with nonlocal conditions. Applied Mathematics Letters. 18(5), 521–527 (2005). Publisher Full Text

49. Hao, X, Liu, L, Wu, Y: Positive solutions for nonlinear th-order singular nonlocal boundary value problems. Boundary Value Problems. 2007, (2007)