Generating iterated function systems of some planar self-similar sets

Abstract

We introduce a new type of nonlinear contraction and present some fixed point results without using continuity or semi-continuity. Our result complement, extend and generalize a number of fixed point theorems including the well-known Boyd and Wong theorem [On nonlinear contractions, Proc. Amer. Math. Soc. 20(1969), 458-464]. Also we discuss an application to iterated function systems.2010 MSC: Primary 47H10; 54H25.

A new fixed point theorem in the fractal space

“…20 Similarly, (A, B)). 22 Hence, (A, B)). Since ψ(t) is a increasing function, F f is ψ-contractive in the space H (X ) by Lemma 3.2.…”

“…16 They discussed the Banach contraction principle with some generalized contraction condi-17 tions and weakened the usual contraction condition. The main idea in the generalization of Ba- 18 nach's contraction theorem was to use the combining of the ideas in the contraction principle 19 (see [1][2][3][4][5][7][8][9][10]12,[14][15][16][17][18][19][20][21][22]). Here H (X ) denotes the space whose points are the compact 3 subsets of the complete metric space (X, d) other than the empty set.…”

“…Then H (X ) denotes the space whose points are the 18 compact subsets of X , other than the empty set. 22 We introduce the notation h d to show that d is the underlying metric. Proof.…”

Fixed point theorems for nonlinear contractions with applications to iterated function systems

“…In recent years this area received great attention of many mathematicians, scientists and a huge developments took place (cf. [1,2,3,4,5,7,10,11,14,18,22,23,24,25,27,28,29,31,32,33]).…”

Generating iterated function systems for a class of self-similar sets with complete overlap

“…Recently, Feng and Wang [5] studied the existence of a minimal presentation for some self-similar sets in R. Deng et al [3], [8], [11] discussed the form of the IFSs for self-similar sets produced by intersecting the linear homogeneous Cantor sets with their translations. Yao [10] explored the form of orthogonal matrices in generating IFSs for certain planar self-similar sets.…”