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Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces
Journal of Inequalities and Applications volume 2010, Article number: 293410 (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 [1–3]) 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 [4–7] 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
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
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:
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 [8–11] 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 :
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
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
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
for any , and .
(H2) Impulsive function is continuous, and for any , it satisfies
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
Definition 2.1.
A semigroup is said to be exponentially stable in if there exist constants and such that
Let . Denote by the Banach space of all continuous value functions on interval with the norm . It is wellknown ([12, Chapter 4, Theorem ]) that for any and , the initial value problem(IVP) of linear evolution equation
has a unique classical solution expressed by
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
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
where .
Proof.
For any , we first show that the initial value problem (IVP) of linear impulsive evolution equation
has a unique mild solution given by
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
Hence, on , can be expressed by
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,
Since is exponentially stable, we define an equivalent norm in by
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
Namely, . From the normality of cone in , we have
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
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
Let . We define an operator by
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
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
from Lemma 2.2 and (1.2), we have
for . Especially, we have
Combining this inequality with , it follows that
On the other hand, from (2.17), we have
Therefore, for all . It implies that . Similarly, we can prove that .
Now, we define sequences and by the iterative scheme
Then from the mixed monotonicity of operator , we have
Therefore, for any , and are monotone orderbounded 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
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
Hence, for any and , denote by the adjoint operator of , then , and
On the other hand, we have
From Lebesgue's dominated convergence theorem, we have
Hence, from (2.17), we have
On the other hand, it follows from (2.26) that . Hence . By the uniqueness of limits, we can deduce that
By the arbitrariness of , we have
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
for any , where ,
(H4) there exist positive constants with such that
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
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
By means of the normality of cone in , we have
Therefore
by Repeating the using of the above inequality, we can obtain that
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:
where is the Laplace operator, . Let equipped with the norm , . Then is a generating normal cone in . Consider the operator defined by
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 .

(iii)
For any with , we have
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
This implies that
for any and . Hence for any and , we have
Therefore, for any with , from the assumption , we have
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].
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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 NWNUKJCXGC347.
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Yang, H. Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces. J Inequal Appl 2010, 293410 (2010). https://doi.org/10.1155/2010/293410
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DOI: https://doi.org/10.1155/2010/293410
Keywords
 Mild Solution
 Initial Value Problem
 Lower Solution
 Impulsive Differential Equation
 Order Banach Space