On Krasnoselskii Fixed Point Theorem and fractal

Noncompact‐type Krasnoselskii fixed‐point theorems and their applications

“…Proof As in the preceding text, applying Lemmas 5.8-5.12 and Theorem 2.3 or 3.1 , we know that T C S has a fixed point u 21 2 K 21 , which is a solution of the problem (5.20) + (5.21). Because u 21 satisfies (5.23) and u 11 .1, y/ 2 C 1 .OE0, 1/ by Proposition 5.1, one obtains u 21…”

“…For the fixed-point problems for sum of two operators, many kinds of generalizations and variants of Krasnoselskii fixed-point theorem have been obtained (see [2,3,[16][17][18][19][20][21][22][23][24][25][26] and the references therein). It is well-known that, in most of the previous related works, the compactness of S plays a crucial role in their discussions.…”

“…It asserts that the sum T C S has at least one fixed point in a nonempty, closed convex subset K of a Banach space E, where S and T fulfill the following:(i) T is a contraction with constant < 1.(ii) S is continuous, and S.K/ resides in a compact subset of E. (iii) any x, y 2 K imply Tx C Sy 2 K.This capturing result has initiated numerous studies and has been extended in various directions by modifying assumption (i), (ii), (iii), or even the underlying spaces (e.g., [16][17][18]. It was mentioned by Burton [16] that the condition (iii) is too stringent and can be replaced by a mild one, in which he proposed the following improvement for (iii): if x D Tx C Sy with y 2 K, then x 2 K. Subsequently, Dhage [17] replaced (i) by the following requirement: T is a bounded linear operator on E, and T p is a nonlinear contraction for some p 2 N. More recently, in [18], the authors firstly replaced the contraction map by an expansion map and then replaced the compact operator S by a k-set contractive operator.For the fixed-point problems for sum of two operators, many kinds of generalizations and variants of Krasnoselskii fixed-point theorem have been obtained (see [2,3,[16][17][18][19][20][21][22][23][24][25][26] and the references therein). It is well-known that, in most of the previous related works, the compactness of S plays a crucial role in their discussions.…”

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 of some planar self-similar sets

“…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]). …”