JOURNAL BROWSE
Search
Advanced SearchSearch Tips
CONVEX SOLUTIONS OF THE POLYNOMIAL-LIKE ITERATIVE EQUATION ON OPEN SET
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
CONVEX SOLUTIONS OF THE POLYNOMIAL-LIKE ITERATIVE EQUATION ON OPEN SET
Gong, Xiaobing;
  PDF(new window)
 Abstract
Because of difficulty of using Schauder's fixed point theorem to the polynomial-like iterative equation, a lots of work are contributed to the existence of solutions for the polynomial-like iterative equation on compact set. In this paper, by applying the Schauder-Tychonoff fixed point theorem we discuss monotone solutions and convex solutions of the polynomial-like iterative equation on an open set (possibly unbounded) in . More concretely, by considering a partial order in defined by an order cone, we prove the existence of increasing and decreasing solutions of the polynomial-like iterative equation on an open set and further obtain the conditions under which the solutions are convex in the order.
 Keywords
iterative equation;open set;order;increasing operator and decreasing operator;
 Language
English
 Cited by
1.
On a Zoltán Boros’ problem connected with polynomial-like iterative equations, Nonlinear Analysis: Real World Applications, 2015, 26, 56  crossref(new windwow)
 References
1.
R. P. Agarwal, M. Meehan, and D. ORegan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge, 2001.

2.
H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), no. 4, 620-709. crossref(new window)

3.
K. Baron and W. Jarczyk, Recent results on functional equations in a single variable, perspectives and open problems, Aequationes Math. 61 (2001), no. 1-2, 1-48. crossref(new window)

4.
J. G. Dhombres, Iteration lineaire dordre deux, Publ. Math. Debrecen 24 (1977), no. 3-4, 277-287.

5.
X. Gong and W. Zhang, Convex solutions of the polynomial-like iterative equation in Banach spaces, Publ. Math. Debrecen 82 (2013), no. 2, 341-358. crossref(new window)

6.
W. Jarczyk, On an equation of linear iteration, Aequationes Math. 51 (1996), no. 3, 303-310. crossref(new window)

7.
M. Kuczma, B. Choczewski, and R. Ger, Iterative Functional Equations, Encyclopedia Math. Appl., vol. 32, Cambridge Univ. Press, Cambridge, 1990.

8.
M. Kuczma and A. Smajdor, Fractional iteration in the class of convex functions, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 16 (1968), 717-720.

9.
M. Kulczycki and J. Tabor, Iterative functional equations in the class of Lipschitz functions, Aequationes Math. 64 (2002), no. 1-2, 24-33. crossref(new window)

10.
J. Mai and X. Liu, Existence, uniqueness and stability of $C^m$ solutions of iterative functional equations, Sci. China Ser. A 43 (2000), no. 9, 897-913. crossref(new window)

11.
M. Malenica, On the solutions of the functional equation ${\phi}(x)+{\phi}^2(x)=F(x)$, Mat. Vesnik 6(19)(34) (1982), no. 3, 301-305.

12.
J. Matkowski and W. Zhang, On linear dependence of iterates, J. Appl. Anal. 6 (2000), no. 1, 149-157.

13.
A. Mukherjea and J. S. Ratti, On a functional equation involving iterates of a bijection on the unit interval, Nonlinear Anal. 7 (1983), no. 8, 899-908. crossref(new window)

14.
A. Mukherjea and J. S. Ratti, A functional equation involving iterates of a bijection on the unit interval. II, Nonlinear Anal. 31 (1998), no. 3-4, 459-464. crossref(new window)

15.
W. Rudin, Functional Analysis, Second edition, McGraw Hill, New York, 1991.

16.
J. Si, Existence of locally analytic solutions of the iterated equation ${\sum}_{i=1}^{n}{\lambda}_if^i(x)=F(x)$, Acta Math. Sinica. 37 (1994), no. 5, 590-600.

17.
J. Tabor and J. Tabor, On a linear iterative equation, Results Math. 27 (1995), no. 3-4, 412-421. crossref(new window)

18.
J. Tabor and M. Zoldak, Iterative equations in Banach spaces, J. Math. Anal. Appl. 299 (2004), no. 2, 651-662. crossref(new window)

19.
G. Targonski, Topics in Iteration Theory, Studia Mathematica: Skript, 6. Vandenhoeck & Ruprecht, Gttingen, 1981.

20.
T. Trif, Convex solutions to polynomial-like iterative equations on open intervals, Aequationes Math. 79 (2010), no. 3, 315-325. crossref(new window)

21.
B. Xu and W. Zhang, Decreasing solutions and convex solutions of the polynomial-like iterative equation, J. Math. Anal. Appl. 329 (2007), no. 1, 483-497. crossref(new window)

22.
D. Yang and W. Zhang, Characteristic solutions of polynomial-like iterative equations, Aequationes Math. 67 (2004), no. 1-2, 80-105. crossref(new window)

23.
E. Zeidler and P. R. Wadsack, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New york, 1986.

24.
J. Zhang, L. Yang, and W. Zhang, Some advances on functional equations, Adv. Math. (China) 24 (1995), no. 5, 385-405.

25.
W. Zhang, Discussion on the iterated equation ${\sum}_{i=1}^{n}{\lambda}_if^i(x)=F(x)$, Chinese Sci. Bull. 32 (1987), no. 21, 1444-1451.

26.
W. Zhang, Discussion on the differentiable solutions of the iterated equation ${\sum}_{i=1}^{n}{\lambda}_if^i(x)=F(x)$, Nonlinear Anal. 15 (1990), no. 4, 387-398. crossref(new window)

27.
W. Zhang, Solutions of equivariance for a polynomial-like iterative equation, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 5, 1153-1163. crossref(new window)

28.
W. Zhang, K. Nikodem, and B. Xu, Convex solutions of polynomial-like iterative equations, J. Math. Anal. Appl. 315 (2006), no. 1, 29-40. crossref(new window)

29.
L. Zhao, A theorem concerning the existence and uniqueness of solutions of the functional equation ${\lambda}_1f(x)+{\lambda}_2f^2(x)=F(x)$, J. Univ. Sci. Tech. 32 (1983), 21-27 (in Chinese).