Research Article | Open Access
He Yang, "Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces", Abstract and Applied Analysis, vol. 2010, Article ID 481648, 11 pages, 2010. https://doi.org/10.1155/2010/481648
Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces
This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach space , , , , , , where is a closed linear operator, and is a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearity , some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.
1. Introduction and Main Results
Differential equations involving impulse effects occur in many applications: physics, population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, and so forth. Therefore, it has been an object of intensive investigation in recent years; see, for instance, the monographs [1–5]. Correspondingly, the existence of mild solutions of impulsive evolution differential equations has also been studied by several authors; see [6–8]. However, the theory still remains to be developed.
In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of mild solutions for the initial value problem (IVP) of first-order nonlinear impulsive evolution equations in an ordered Banach space , where is a closed linear operator, generates a -semigroup in , is a nonlinear mapping, is a constant, , is an impulsive function, , , and is a Volterra integral operator with integral kernel , 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 norm . Let . An abstract function ( is a Banach space with norm ) is called a solution of IVP(1.1), if satisfies all the equalities of (1.1).
Let be an ordered Banach space with norm and partial order , whose positive cone is normal with a normal constant . If an abstract function satisfies we call it a lower solution of IVP(1.1). If all the inequalities of (1.3) are inverse, we call it an upper solution of IVP(1.1).
In 1999, Liu , by means of the semigroup theory, has proved the existence and uniqueness of mild solutions for IVP(1.1) when . He demands that the nonlinear term and the impulsive function satisfy the following conditions: where and are positive constants and satisfy where . Inequality (1.6) is a strongly restricted condition, and it is difficult to satisfy in applications.
Recently, Anguraj and Arjunan , under similar assumptions of , have obtained a unique mild solution for IVP(1.1) when . Cardinali and Rubbioni  have discussed the existence of mild solutions for the impulsive evolution differential inclusions under the measure of noncompactness conditions on every bounded set . However the assumptions in these papers are also difficult to satisfy in applications.
The purpose of this paper is to improve and extend the above mentioned results. We will delete the Lipschitz condition (1.5) for impulsive function and the restriction condition (1.6) and improve condition (1.4) for nonlinear term . Our main results are as follows.
Theorem 1.1. Let be an ordered Banach space, whose positive cone is normal, be a closed linear operator, generate a positive -semigroup , , and , . If IVP(1.1) has a lower solution and an upper solution with and the following conditions are satisfied: there exists a positive constant such that for any , , and ,for any with , , one has there exists a positive constant such that for any , and increasing or decreasing monotonic sequences and .Then IVP(1.1) has minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.
Clearly, condition greatly improves the measure of noncompactness condition in . Therefore, Theorem 1.1 greatly improves the main results in [6–8]. In Theorem 1.1, if Banach space is weakly sequentially complete, condition holds automatically; see [9, Theorem ]. Hence, from Theorem 1.1, we have the following.
Corollary 1.2. Let be an ordered and weakly sequentially complete Banach space, whose positive cone is normal, be a closed linear operator, generate a positive -semigroup , , and , . If IVP(1.1) has a lower solution and an upper solution with , and the conditions and are satisfied, then IVP(1.1) has minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.
The proof of Theorem 1.1 will be shown in the next section. In Section 2, we also discuss the uniqueness of mild solutions for IVP(1.1) between the lower solution and upper solution (see Theorem 2.4).
2. Proof of the Main Results
Let denote the Banach space of all continuous -value functions on interval with norm and let denote the Banach space of all continuously differentiable -value functions on interval with norm . Consider the initial value problem (IVP) of linear evolution equation without impulse It is well-known [10, chapter 4, Theorem ], when and , IVP(2.1) has a classical solution expressed by Generally, when and , the function given by (2.2) belongs to and it is called a mild solution of IVP(2.1).
Let us start by defining what we mean by a mild solution of problem
Definition 2.1. A function is called a mild solution of IVP(2.3), if is a solution of integral equation
To prove Theorem 1.1, for any , we consider the linear initial value problem (LIVP) of impulsive evolution equation where , , and , .
Lemma 2.2. For any , , and , LIVP(2.5) has a unique mild solution given by where is a -semigroup generated by .
Proof. Let . If is a mild solution of LIVP(2.5), then the restriction of on satisfies the initial value problem of linear evolution equation without impulse
Hence, on , can be expressed by
Iterating successively in the above equality with for , we see that satisfies (2.6).
Inversely, we can verify directly that the function defined by (2.6) satisfies all the equalities of LIVP(2.5).
Let denote the Kuratowskii measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see . For any and , set . If is bounded in , then is bounded in , and . In the proof of Theorem 1.1 we need the following lemma.
Lemma 2.3. Let be a bounded and countable set. Then is the Lebesgue integrable on , and
This lemma can be found in .
Evidently, is also an ordered Banach space with the partial order “” reduced by the positive function cone . is also normal with the same normal constant . For with , we use to denote the ordered interval in , and use to denote the ordered interval in .
Proof of Theorem 1.1. Let . We define a mapping by
Clearly, is continuous. By Lemma 2.2, the mild solution of IVP(1.1) is equivalent to the fixed point of operator . By assumptions and , is increasing in and maps any bounded set in into a bounded set.
We show that . Let . By the definition of the lower solution, we easily see that and for . Because is a solution of LIVP(2.5) for and , by Lemma 2.2 and the positivity of operator , we have namely, . Similarly, it can be shown that . Combining these facts with the increasing property of in , we see that maps into itself, and is a continuously increasing operator.
Now, we define two sequences and in by the iterative scheme Then from the monotonicity of , it follows that Next, we will show that and are uniformly convergent on .
For convenience, we denote . Let , , and let and . From , it follows that for . Let By Lemma 2.3, is Lebesgue integrable on . Going from to interval by interval we show that on .
For , there exists a such that . By (1.2) and Lemma 2.3, we have that and therefore,
For , from (2.10), (2.16), Lemma 2.3 and assumption , we have By this and the Gronwall-Bellman inequality, we obtain that on . In particular, , this means that is precompact in . Combining this with the continuity of , it follows that is precompact in , and .
Now, for , by (2.10) and the above argument for , we have Again by the Gronwall-Bellman inequality, we obtain that on , from which we obtain that and .
Continuing such a process interval by interval up to , we can prove that on every , . This means that is precompact in for every . Hence has a convergent subsequence in . Combining this fact with the monotonicity (2.13), we easily prove that itself is convergent in , that is, there exists such that as for every . On the other hand, for any , we have Let , then by the Lebesgue-dominated convergence theorem, for , we have and . Therefore, for any , we have Namely, , and . Similarly, we can prove that there exists such that . By the monotonicity of operator , it is easy to prove that and are the minimal and maximal fixed points of in , and they are the minimal and maximal mild solutions of IVP(1.1) in , respectively.
Now we discuss the uniqueness of mild solutions for IVP(1.1) in . If we replace assumption by the following assumption:there exist positive constants and such that for any , , and we have the following existence result.
Theorem 2.4. Let be an ordered Banach space, whose positive cone is normal, be a closed linear operator in , generate a positive -semigroup , and , . If IVP(1.1) has a lower solution and an upper solution with , such that the assumptions , and are satisfied, then IVP(1.1) has a unique mild solution between and , which can be obtained by a monotone iterative procedure starting from or .
Proof of Theorem 2.4. We firstly prove that and imply . For , let and be two increasing sequences. For with , by and , we have
By the normality of cone , we have
From this inequality and the definition of the measure of noncompactness, it follows that
where . If and are two decreasing sequences, the above inequality is also valid. Hence holds.
Therefore, by Theorem 1.1, IVP(1.1) has minimal solution and maximal solution in . By the proof of Theorem 1.1, (2.10)–(2.13) are valid. Going from to interval by interval we show that on every .
For , by (2.10) and assumption , we have From this inequality and the normality of cone , it follows that By the Gronwall-Bellman inequality, we obtain that on .
For , since , using (2.10) and the same argument as above for , we can prove that Again, by the Gronwall-Bellman inequality, we obtain that on .
Continuing such a process interval by interval up to , we see that over the whole . Hence, is the unique mild solution of IVP(1.1) in , which can be obtained by a monotone iterative procedure starting from or .
The author is very grateful to the reviewers for their helpful comments and suggestions. The paper was supported by NNSF of China (10871160), the NSF of Gansu Province (0710RJZA103), and the Project of NWNU-KJCXGC-3-47.
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