Open Access

Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces

Journal of Inequalities and Applications20102010:293410

https://doi.org/10.1155/2010/293410

Received: 29 December 2009

Accepted: 3 September 2010

Published: 14 September 2010

Abstract

By constructing a mixed monotone iterative technique under a new concept of upper and lower solutions, some existence theorems of mild -periodic ( -quasi) solutions for abstract impulsive evolution equations are obtained in ordered Banach spaces. These results partially generalize and extend the relevant results in ordinary differential equations and partial differential equations.

1. Introduction and Main Result

Impulsive differential equations are a basic tool for studying evolution processes of real life phenomena that are subjected to sudden changes at certain instants. In view of multiple applications of the impulsive differential equations, it is necessary to develop the methods for their solvability. Unfortunately, a comparatively small class of impulsive differential equations can be solved analytically. Therefore, it is necessary to establish approximation methods for finding solutions. The monotone iterative technique of Lakshmikantham et al. (see [13]) is such a method which can be applied in practice easily. This technique combines the idea of method of upper and lower solutions with appropriate monotone conditions. Recent results by means of monotone iterative method are obtained in [47] and the references therein. In this paper, by using a mixed monotone iterative technique in the presence of coupled lower and upper -quasisolutions, we consider the existence of mild -periodic ( -quasi)solutions for the periodic boundary value problem (PBVP) of impulsive evolution equations
(1.1)

in an ordered Banach space , where is a closed linear operator and generates a -semigroup in ; only satisfies weak Carathé odory condition, , is a constant; ; is an impulsive function, ; denotes the jump of at , that is, , where and represent the right and left limits of at , respectively. Let is continuous at and left continuous at , and exists, . Evidently, is a Banach space with the norm . Let , . Denote by the Banach space generated by with the norm . An abstract function is called a solution of the PBVP(1.1) if satisfies all the equalities of (1.1).

Let be an ordered Banach space with the norm and the partial order " ", whose positive cone is normal with a normal constant . Let . If functions satisfy
(1.2)
(1.3)

we call coupled lower and upper -quasisolutions of the PBVP(1.1). Only choosing " " in (1.2) and (1.3), we call coupled -periodic -quasisolution pair of the PBVP(1.1). Furthermore, if , we call an -periodic solution of the PBVP(1.1).

Definition 1.1.

Abstract functions are called a coupled mild -periodic -quasisolution pair of the PBVP(1.1) if and satisfy the following integral equations:
(1.4)

where and for any , is an identity operator. If , then is called a mild -periodic solution of the PBVP(1.1).

Without impulse, the PBVP(1.1) has been studied by many authors, see [811] and the references therein. In particular, Shen and Li [11] considered the existence of coupled mild -periodic quasisolution pair for the following periodic boundary value problem (PBVP) in :
(1.5)

where is continuous. Under one of the following situations:

(i) is a compact semigroup,

(ii) is regular in and is continuous in operator norm for ,

they built a mixed monotone iterative method for the PBVP(1.5), and they proved that, if the PBVP(1.5) has coupled lower and upper quasisolutions (i.e., and without impulse in (1.2) and (1.3)) and with , nonlinear term satisfies one of the following conditions:

(F1) is mixed monotone,

(F2)There exists a constant such that

(1.6)

and is nonincreasing on .

Then the PBVP(1.5) has minimal and maximal coupled mild -periodic quasisolutions between and , which can be obtained by monotone iterative sequences from and . But conditions and are difficult to satisfy in applications except some special situations.

In this paper, by constructing a mixed monotone iterative technique under a new concept of upper and lower solutions, we will discuss the existence of mild -periodic ( -quasi) solutions for the impulsive evolution Equation(1.1) in an ordered Banach space . In our results, we will delete conditions and for the operator semigroup , and improve conditions and for nonlinearity . In addition, we only require that the nonlinear term satisfies weak Carathé odory condition:

(1)for each is strongly measurable.

(2)for a.e. is subcontinuous, namely, there exists with mes such that

(1.7)

for any , and .

Our main result is as follows:

Theorem 1.2.

Let be an ordered and weakly sequentially complete Banach space, whose positive cone is normal, be a closed linear operator and generate a positive -semigroup in . If the PBVP(1.1) has coupled lower and upper -quasisolutions and with , nonlinear term and impulsive functions 's satisfy the following conditions

(H1) There exist constants and such that

(1.8)

for any , and .

(H2) Impulsive function is continuous, and for any , it satisfies

(1.9)

for any , and .

then the PBVP(1.1) has minimal and maximal coupled mild -periodic -quasisolutions between and , which can be obtained by monotone iterative sequences starting from and .

Evidently, condition contains conditions and . Hence, even without impulse in PBVP(1.1), Theorem 1.2 still extends the results in [10, 11].

The proof of Theorem 1.2 will be shown in the next section. In Section 2, we also discuss the existence of mild -periodic solutions for the PBVP(1.1) between coupled lower and upper -quasisolutions (see Theorem 2.3). In Section 3, the results obtained will be applied to a class of partial differential equations of parabolic type.

2. Proof of the Main Results

Let be a Banach space, be a closed linear operator, and generate a -semigroup in . Then there exist constants and such that
(2.1)

Definition 2.1.

A -semigroup is said to be exponentially stable in if there exist constants and such that
(2.2)
Let . Denote by the Banach space of all continuous -value functions on interval with the norm . It is well-known ([12, Chapter 4, Theorem ]) that for any and , the initial value problem(IVP) of linear evolution equation
(2.3)
has a unique classical solution expressed by
(2.4)

If and , the function given by (2.4) belongs to . We call it a mild solution of the IVP(2.3).

To prove Theorem 1.2, for any , we consider the periodic boundary value problem (PBVP) of linear impulsive evolution equation in
(2.5)

where .

Lemma 2.2.

Let be an exponentially stable -semigroup in . Then for any and , the linear PBVP(2.5) has a unique mild solution given by
(2.6)

where .

Proof.

For any , we first show that the initial value problem (IVP) of linear impulsive evolution equation
(2.7)
has a unique mild solution given by
(2.8)

where and .

Let . Let . If is a mild solution of the linear IVP(2.7), then the restriction of on satisfies the initial value problem (IVP) of linear evolution equation without impulse
(2.9)
Hence, on , can be expressed by
(2.10)

Iterating successively in the above equality with for , we see that satisfies (2.8).

Inversely, we can verify directly that the function defined by (2.8) is a solution of the linear IVP(2.7). Hence the linear IVP(2.7) has a unique mild solution given by (2.8).

Next, we show that the linear PBVP(2.5) has a unique mild solution given by (2.6).

If a function defined by (2.8) is a solution of the linear PBVP(2.5), then , namely,
(2.11)
Since is exponentially stable, we define an equivalent norm in by
(2.12)

Then and , and especially, . It follows that has a bounded inverse operator , which is a positive operator when is a positive semigroup. Hence we choose . Then is the unique initial value of the IVP(2.7) in , which satisfies . Combining this fact with (2.8), it follows that (2.6) is satisfied.

Inversely, we can verify directly that the function defined by (2.6) is a solution of the linear PBVP(2.5). Therefore, the conclusion of Lemma 2.2 holds.

Evidently, is also an ordered Banach space with the partial order " " reduced by positive function cone . is also normal with the same normal constant . For with , we use to denote the order interval in , and to denote the order interval in . From Lemma 2.2, if is a positive -semigroup, and , then the mild solution of the linear PBVP(2.5) satisfies .

Proof of Theorem 1.2.

We first show that for any and . Since for any , from the assumption , we have
(2.13)
Namely, . From the normality of cone in , we have
(2.14)
Combining this fact with the fact that is strongly measurable, it follows that . Therefore, for any , we consider the periodic boundary value problem(PBVP) of impulsive evolution equation in
(2.15)
where . Let be large enough such that (otherwise, replacing by , the assumption still holds). Then generates an exponentially stable -semigroup . Obviously, is a positive -semigroup and for . From Lemma 2.2, the PBVP(2.15) has a unique mild solution given by
(2.16)
Let . We define an operator by
(2.17)

Then the coupled mild -periodic -quasisolution of the PBVP(1.1) is equivalent to the coupled fixed point of operator .

Next, we will prove that the operator has coupled fixed points on . For this purpose, we first show that is a mixed monotone operator and . In fact, for any , from assumptions and , we have
(2.18)
Since is a positive -semigroup, it follows that is a positive operator. Then . Hence from (2.17) we see that , which implies that is a mixed monotone operator. Since
(2.19)
from Lemma 2.2 and (1.2), we have
(2.20)
for . Especially, we have
(2.21)
Combining this inequality with , it follows that
(2.22)
On the other hand, from (2.17), we have
(2.23)

Therefore, for all . It implies that . Similarly, we can prove that .

Now, we define sequences and by the iterative scheme
(2.24)
Then from the mixed monotonicity of operator , we have
(2.25)
Therefore, for any , and are monotone order-bounded sequences in . Noticing that is a weakly sequentially complete Banach space, then and are relatively compact in . Combining this fact with the monotonicity of (2.25) and the normality of cone in , it follows that and are uniformly convergent in . Let
(2.26)

Then are strongly measurable, and for any . Hence, .

At last, we show that and are coupled mild -periodic -quasisolutions of the PBVP(1.1). For any , from subcontinuity of and continuity of 's, there exists with mes such that
(2.27)
Hence, for any and , denote by the adjoint operator of , then , and
(2.28)
On the other hand, we have
(2.29)
From Lebesgue's dominated convergence theorem, we have
(2.30)
Hence, from (2.17), we have
(2.31)
On the other hand, it follows from (2.26) that . Hence . By the uniqueness of limits, we can deduce that
(2.32)
By the arbitrariness of , we have
(2.33)

Similarly, we can prove that . Therefore, is coupled mild -periodic -quasisolution pair of the PBVP(1.1).

Now, we discuss the existence of mild -periodic solutions for the PBVP(1.1) on . We assume that the following assumptions are also satisfied:

(H3) there exists a constant with such that
(2.34)

for any , where ,

(H4) there exist positive constants with such that

(2.35)

for any

Then we have the following existence and uniqueness result in general ordered Banach space.

Theorem 2.3.

Let be an ordered Banach space, whose positive cone is normal, be a closed linear operator, and generate a positive -semigroup in . If the PBVP(1.1) has coupled lower and upper -quasisolution and with , nonlinear term and impulsive functions 's satisfy the following assumptions:

(H1)* there exist constants and such that

(2.36)

for any , and .

And , then the PBVP(1.1) has a unique mild -periodic solution on .

Proof.

From the proof of Theorem 1.2, when the conditions and are satisfied, the iterative sequences and defined by (2.24) satisfy (2.25). We show that there exists a unique such that . For any , from , (2.17), (2.24) and (2.25), we have
(2.37)
By means of the normality of cone in , we have
(2.38)
Therefore
(2.39)
by Repeating the using of the above inequality, we can obtain that
(2.40)

as . Then there exists a unique such that . Therefore, let in (2.24), from the continuity of operator , we have , which means that is a unique mild -periodic solution of the PBVP(1.1).

3. An Example

Let be a bounded domain with a sufficiently smooth boundary . Let , and , . Consider the existence of mild solutions for the boundary value problem of parabolic type:
(3.1)
where is the Laplace operator, . Let equipped with the -norm , . Then is a generating normal cone in . Consider the operator defined by
(3.2)

Then generates an analytic semigroup in . By the maximum principle of the equations of parabolic type, it is easy to prove that is a positive -semigroup in . Let be the first eigenvalue of operator and be a corresponding positive eigenvector. For solving the problem (3.1), the following assumptions are needed.

(i) There exists a constant such that

(a) , , ,

(b) , , , ,

(ii)

(a) The partial derivative of on is continuous on any bounded domain.

(b) The partial derivative of on has upper bound, and .

  1. (iii)

    For any with , we have

     
(3.3)
Let and be defined by and by . Then the problem (3.1) can be transformed into the PBVP(1.1). Assumption implies that and are coupled lower and upper -quasisolutions of the PBVP(1.1). From assumption (ii)(a), there exists a constant such that, for any , we have
(3.4)
This implies that
(3.5)
for any and . Hence for any and , we have
(3.6)
Therefore, for any with , from the assumption , we have
(3.7)

That is, assumption is satisfied. From , it is easy to see that assumption is satisfied. Therefore, the following result is deduced from Theorem 1.2.

Theorem 3.1.

If the assumptions are satisfied, then the problem (3.1) has coupled mild -periodic -quasisolution pair on .

Remark 3.2.

In applications of partial differential equations, we often choose Banach space as working space, which is weakly sequentially complete. Hence the result in Theorem 1.2 is more valuable in applications. In particular, we obtain a unique mild -periodic solution of the PBVP(1.1) in general ordered Banach space in Theorem 2.3.

Remark 3.3.

If , then the coupled lower and upper -quasisolutions are equivalent to coupled lower and upper quasisolutions of the PBVP(1.1). Since condition contains conditions and , even without impulse in PBVP(1.1), the results in this paper still extend the results in [10, 11].

Declarations

Acknowledgments

The author is very grateful to the reviewers for their helpful comments and suggestions. Research supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and Project of NWNUKJCXGC-3-47.

Authors’ Affiliations

(1)
Department of Mathematics, Northwest Normal University

References

  1. Du SW, Lakshmikantham V: Monotone iterative technique for differential equations in a Banach space. Journal of Mathematical Analysis and Applications 1982, 87(2):454–459. 10.1016/0022-247X(82)90134-2MathSciNetView ArticleMATHGoogle Scholar
  2. Lakshmikantham V, Vatsala AS: Quasisolutions and monotone method for systems of nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 1981, 79(1):38–47. 10.1016/0022-247X(81)90006-8MathSciNetView ArticleGoogle Scholar
  3. Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations, Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics. Volume 27. Pitman, Boston, Mass, USA; 1985:x+236.Google Scholar
  4. Lan H-Y: Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Journal of Computational and Applied Mathematics 2008, 222(2):531–543. 10.1016/j.cam.2007.11.021MathSciNetView ArticleMATHGoogle Scholar
  5. Ahmad B, Sivasundaram S: The monotone iterative technique for impulsive hybrid set valued integro-differential equations. Nonlinear Analysis: Theory, Methods & Applications 2006, 65(12):2260–2276. 10.1016/j.na.2006.01.033MathSciNetView ArticleMATHGoogle Scholar
  6. Li Y, Liu Z: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2007, 66(1):83–92. 10.1016/j.na.2005.11.013MathSciNetView ArticleMATHGoogle Scholar
  7. Ahmad B, Alsaedi A: Existence of solutions for anti-periodic boundary value problems of nonlinear impulsive functional integro-differential equations of mixed type. Nonlinear Analysis: Theory, Methods & Applications 2009, 3(4):501–509.MathSciNetMATHGoogle Scholar
  8. Lakshmikantham V, Leela S: Existence and monotone method for periodic solutions of first-order differential equations. Journal of Mathematical Analysis and Applications 1983, 91(1):237–243. 10.1016/0022-247X(83)90102-6MathSciNetView ArticleMATHGoogle Scholar
  9. Li YX: Existence and uniqueness of positive periodic solutions for abstract semilinear evolution equations. Journal of Systems Science and Mathematical Sciences 2005, 25(6):720–728.MathSciNetMATHGoogle Scholar
  10. Li YX: Periodic solutions of semilinear evolution equations in Banach spaces. Acta Mathematica Sinica 1998, 41(3):629–636.MathSciNetMATHGoogle Scholar
  11. Shen PL, Li FY: Coupled periodic solutions of nonlinear evolution equations in Banach spaces. Acta Mathematica Sinica 2000, 43(4):685–694.MathSciNetMATHGoogle Scholar
  12. Pazy A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences. Volume 44. Springer, New York, NY, USA; 1983:viii+279.View ArticleGoogle Scholar

Copyright

© He Yang. 2010

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.