- Research Article
- Open Access
Mixed Monotone Iterative Technique for Abstract Impulsive Evolution Equations in Banach Spaces
© He Yang. 2010
Received: 29 December 2009
Accepted: 3 September 2010
Published: 14 September 2010
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
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).
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).
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:
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:
Our main result is as follows:
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
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
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.
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.
Then we have the following existence and uniqueness result in general ordered Banach space.
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:
3. An Example
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.
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.
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].
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.
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