The Solutions of the Series-Like Iterative Equation with Variable Coefficients
© Yuzhen Mi et al. 2009
Received: 23 March 2009
Accepted: 6 July 2009
Published: 4 August 2009
By constructing a structure operator quite different from that ofZhang and Baker (2000) and using the Schauder fixed point theory, the existence and uniqueness of the solutions of the series-like iterative equations with variable coefficients are discussed.
Let , clearly is a Banach space, where , for in .
Let , then is a Banach space with the norm , where , for in .
Being a closed subset, defined by
is a complete space.
for any in , where denotes .
for where as and as .
3. Main Results
Theorem 3.1 (existence).
Given positive constants and if there exists constants and , such that
then (1.2) has a solution in .
For convenience, let
Define such that , where
For any , we have
Define as follows:
Thus is a self-diffeomorphism.
Now we prove the continuity of under the norm . For arbitrary ,
which gives continuity of .
It is easy to show that is a compact convex subset of . By Schauder's fixed point theorem, we assert that there is a mapping such that
Let we have as a solution of (1.2) in . This completes the proof.
Theorem 3.2 (Uniqueness).
Suppose that (P1) and (P2) are satisfied, also one supposes that
then for arbitrary function in , (1.2) has a unique solution .
The existence of (1.2) in is given by Theorem 3.1, from the proof of Theorem 3.1, we see that is a closed subset of , by (3.12) and , we see that is a contraction. Therefore has a unique fixed point in , that is, (1.2) has a unique solution in , this proves the theorem.
where It is easy to see that
thus By condition , we can choose and by condition , we can choose . Then by Theorem 3.1, there is a continuously differentiable solution of (4.1) in .
This work was supported by Guangdong Provincial Natural Science Foundation (07301595) and Zhan-jiang Normal University Science Research Project (L0804).
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