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Abstract and Applied Analysis
Volume 2008, Article ID 636843, 13 pages
http://dx.doi.org/10.1155/2008/636843
Research Article

Differentiable Solutions of Equations Involving Iterated Functional Series

Deparment of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 12 October 2008; Accepted 28 December 2008

Academic Editor: John Rassias

Copyright © 2008 Wei Song. 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.

Abstract

The nonmonotonic differentiable solutions of equations involving iterated functional series are investigated. Conditions for the existence, uniqueness, and stability of such solutions are given. These extend earlier results due to Murugan and Subrahmanyam.

1. Introduction

Let be a self-mapping on a topological space and denote the mth iterate of , that is, Let be the set of all continuous self-mappings on . Equations having iteration as their main operation, that is, including iterates of the unknown mapping, are called iterative equations. It is one of the most interesting classes of functional equations [14], because it concludes the problem of iterative roots [1, 5, 6], that is, finding such that is identical to a given . As a natural generalization of the problem of iterative roots, a class of iterative equations named as polynomial-like iterative equation had fascinated many scholars, such as Dhombres [7], Zhao [8], Mukherjea and Ratti [9]. Despite their nice constructive proofs, the classical methods prevented them from obtaining more fruitful results. In 1986, Zhang [10] constructed an interesting operator called “structural operator” for (1.1) and used the fixed point theory in Banach space to get the solutions of (1.1). Hence he overcame the difficulties encountered by the formers. By means of this method, Zhang and Si made a series of work concerning these qualitative problems, such as [1115]. In 2002, Kulczycki and Tabor [16] improved Zhang's method and investigated the existence of Lipschitzian solutions of the iterative functional equation where is a compact convex subset of and are a given Lipschitz function. It is easy to see that (1.1) is the special case of (1.2) with , and

Recently Zhang et al. [17] and Xu et al. [18] developed this method and they have got the nonmonotonic, convex, and decreasing continuous solutions of (1.1). In fact they have answered the open problem 2 which was proposed by J. Zhang et al. [19].

The problem of differentiable solutions of iterative equation had also fascinated many scholars' attentions. In Zhang [12] and Si [15], the and solutions of (1.1) are considered. In Wang and Si [20] the differentiable solutions of the below equation are considered. Murugan and Subrahmanyam [21, 22] discussed the existence and uniqueness of solutions of the more general equations which involve iterated functional series. All the above references only got the increasing differentiable solutions for the above equations because they only considered the case that is increasing. Li and Deng [23] considered the solutions of the (1.2). In [24] solutions of the equation where is a compact convex subset of and are discussed. Li and Deng [23] and Li [24] work in higher dimensional case, they do not require monotonicity. It should be pointed out that Mai and Liu [25] made an important contribution to solutions of iterative equations. Mai and Liu proved the existence, uniqueness of solutions of a relatively general kind of iterative equations where is a connected closed subset of and Here denotes the set of all mappings from to .

Inspired by the above work, we will investigate (1.4) and extend earlier results due to Murugan and Subrahmanyam in two directions. In [21] the authors only get increasing solutions of (1.4), so the present paper will investigate the nonmonotonic differentiable solutions of (1.4) and give conditions for the existence, uniqueness, and stability of such solutions. In [21] the authors require not only all coefficients are nonnegative but also are all increasing, but we will find that those conditions are not necessary.

2. Preliminaries

Let and be two compact intervals. Let be the set of all continuously differentiable functions from to . Then is a closed subset of the Banach space consisting of all continuously differentiable functions from to with norm . Here the norm is defined by where and is the derivative of . Following Zhang [12], we define the families of functions where , are all constants.

Lemma 2.1. Both and are compact convex subsets of .

The Lemma above can be proved by a method which is contained in the proof of Theorem 3.1 in [12].

Lemma 2.2 (see [12]). Suppose that (or ). Then for where as and denotes .

We can get the following Lemma from [12]. In [12] the author proved that the lemma is valid for , but we find it is also valid for

Lemma 2.3 (see [12]). Suppose that satisfies that where , are positive constants. Then

Lemma 2.4 (see [18]). If both are homeomorphisms from to such that where is a positive constant. Then

3. Differentiable Solutions of (1.4)

3.1. Existence of Solutions

Let be coefficients of (1.4) and . For any define and . It is easy to see that both the convergence and the value of have nothing to do with the choice of .

Theorem 3.1. Suppose , are positive constants and for where and are positive constants for Assume further the following conditions: hold. Then for any given (1.4) has a solution where , and , .

As in [22], firstly we give the following three Lemmas which lead directly to the proof of Theorem 3.1. In the sequel we denote

Lemma 3.2. Under the assumptions of Theorem 3.1 the following series: are all convergent.

Proof. The convergence of , , , , and is easy to be verified. As mentioned in [21], the equality holds. We get that By the convergence of and , is also convergent.

Lemma 3.3. Under the assumptions of Theorem 3.1, for each the mapping defined by has the following properties: (i) ;(ii) , where and

Proof. For any , we have It is easy to see that for any We have for any and for any Thus is an orientation-preserving diffeomorphism.

By Lemma 2.2 we can see that for any , By (3.9), (3.11), and Lemma 2.3 we get for any :

Lemma 3.4. Under the assumptions of Theorem 3.1, for each the mappings satisfy the following inequalities: (i) ;(ii) .

Proof. Firstly we have and thus Secondly we get Notice that and for , then Finally we get

Proof of Theorem 3.1.. For any we define as follows: and denote for convenience. Clearly , , , (or ), and (3.10) yields that for any , Furthermore by (3.12) we get that for any , So , which means that
Secondly we prove that is continuous. For any we denote It is easy to see that By Lemma 3.4 we get By the discussion above we get where

Hence is continuous. By Schauder fixed point theorem, there exists a function such that That means is a solution of (1.4) in .

4. Uniqueness and Stability of Solutions

Theorem 4.1. Let be coefficient of (1.4) and , positive constants. Suppose that the conditions in (3.1) are valid. Further one assumes that Then for any , there exists a unique function satisfying (1.4), where . Furthermore the solution depends continuously on the given function

Proof. If , then by (3.26) the map defined in Theorem 3.1 becomes a strict contraction. The fix point of , which is a solution of (1.4), is unique by Banach's contraction principle.
Let be the solutions of (1.4) for the corresponding functions First, since we get Second, we have
By the above discussion we get which means that So the solution depends continuously on the given function

4.1. Examples

Example 4.2. Let and The equation where , has a unique solution

Proof. It is easy to see that , and . By simple calculation we get that by Theorem 3.1 the equation has a solution Further we get that this means . By Theorem 4.1 the solution is unique.

By similar discussion we have the following example.

Example 4.3. Let , and The equation where has a unique solution

Acknowledgements

The author would like to thank the referees for their detailed comments and helpful suggestions. He is supported by Project HITC200706 supported by Science Research Foundation in Harbin Institute of Technology.

References

  1. M. Kuczma, B. Choczewski, and R. Ger, Iterative Functional Equations, vol. 32 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1990. View at Zentralblatt MATH · View at MathSciNet
  2. K. Baron and W. Jarczyk, “Recent results on functional equations in a single variable, perspectives and open problems,” Aequationes Mathematicae, vol. 61, no. 1-2, pp. 1–48, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Nabeya, “On the functional equation f(p+qx+rf(x))=a+bx+cf(x),” Aequationes Mathematicae, vol. 11, pp. 199–211, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. Z. Zhang, L. Yang, and W. N. Zhang, Iterative Equations and Embedding Flow, Scientific and Technological Education, Shanghai, China, 1998.
  5. N. Abel, Oeuvres Complètes, vol. 2, Christiana, Christiana, Ten, USA, 1881.
  6. J. M. Dubbey, The Mathematical Work of Charles Babbage, Cambridge University Press, Cambridge, UK, 1978. View at Zentralblatt MATH · View at MathSciNet
  7. J. G. Dhombres, “Itération linéaire d'ordre deux,” Publicationes Mathematicae Debrecen, vol. 24, no. 3-4, pp. 277–287, 1977. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. L. R. Zhao, “Existence and uniqueness theorem for the solution of the functional equation λ1f(x)+λ2f2(x)=F(x),” Journal of University of Science and Technology of China, vol. 13, pp. 21–27, 1983. View at Google Scholar · View at MathSciNet
  9. A. Mukherjea and J. S. Ratti, “On a functional equation involving iterates of a bijection on the unit interval,” Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 8, pp. 899–908, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W. N. Zhang, “Discussion on the iterated equation i=1nλifi(x)=F(x),” Chinese Science Bulletin, vol. 32, no. 21, pp. 1444–1451, 1987. View at Google Scholar
  11. W. N. Zhang, “Stability of the solution of the iterated equation i=1nλifi(x)=F(x),” Acta Mathematica Scientia, vol. 8, no. 4, pp. 421–424, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. W. N. Zhang, “Discussion on the differentiable solutions of the iterated equation i=1nλifi(x)=F(x),” Nonlinear Analysis: Theory, Methods & Applications, vol. 15, no. 4, pp. 387–398, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. W. N. Zhang, “An application of Hardy-Boedewadt's theorem to iterated functional equations,” Acta Mathematica Scientia, vol. 15, no. 3, pp. 356–360, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. G. Si, “The existence and uniqueness of solutions to a class of iterative systems,” Pure and Applied Mathematics, vol. 6, no. 2, pp. 38–42, 1990. View at Google Scholar · View at MathSciNet
  15. J. G. Si, “The C2-solutions to the iterated equation i=1nλifi(x)=F(x),” Acta Mathematica Sinica, vol. 36, no. 3, pp. 348–357, 1993. View at Google Scholar · View at MathSciNet
  16. M. Kulczycki and J. Tabor, “Iterative functional equations in the class of Lipschitz functions,” Aequationes Mathematicae, vol. 64, no. 1-2, pp. 24–33, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. W. Zhang, K. Nikodem, and B. Xu, “Convex solutions of polynomial-like iterative equations,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 29–40, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. Xu and W. Zhang, “Decreasing solutions and convex solutions of the polynomial-like iterative equation,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 483–497, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Zhang, L. Yang, and W. Zhang, “Some advances on functional equations,” Advances in Mathematics, vol. 24, no. 5, pp. 385–405, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. X. Wang and J. G. Si, “Differentiable solutions of an iterative functional equation,” Aequationes Mathematicae, vol. 61, no. 1-2, pp. 79–96, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. V. Murugan and P. V. Subrahmanyam, “Existence of solutions for equations involving iterated functional series,” Fixed Point Theory and Applications, vol. 2005, no. 2, pp. 219–232, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. V. Murugan and P. V. Subrahmanyam, “Special solutions of a general iterative functional equation,” Aequationes Mathematicae, vol. 72, no. 3, pp. 269–287, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. X. Li and S. Deng, “Differentiability for the high dimensional polynomial-like iterative equation,” Acta Mathematica Scientia, vol. 25, no. 1, pp. 130–136, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. X. P. Li, “The C1 solution of the high dimensional iterative equation with variable coefficients,” College Mathematics, vol. 22, no. 3, pp. 67–71, 2006. View at Google Scholar
  25. J. Mai and X. Liu, “Existence, uniqueness and stability of Cm solutions of iterative functional equations,” Science in China Series A, vol. 43, no. 9, pp. 897–913, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet