Krasnosel'skii-Type Fixed-Set Results
© M. A. Al-Thagafi and N. Shahzad. 2010
Received: 8 February 2010
Accepted: 23 August 2010
Published: 26 August 2010
The Krasnosel'skii fixed-point theorem  is a well-known principle that generalizes the Schauder fixed-point theorem and the Banach contraction principle as follows.
Krasnosel'skii Fixed-Point Theorem
This theorem has been extensively used in differential and functional differential equations and was motivated by the observation that the inversion of a perturbed differential operator may yield the sum of a continuous compact map and a contraction map. Note that the conclusion of the theorem does not need to hold if the convexity of is relaxed even if is the zero operator. Ok  noticed that the Krasnosel'skii fixed-point theorem can be reformulated by relaxing or removing the convexity hypothesis of and by allowing the fixed-point to be a fixed-set. For variants or extensions of Krasnosel'skii-type fixed-point results, see [3–9], and for other interesting results see [10–13].
In this paper, we prove several new Krasnosel'skii-type fixed-set theorems for the sum , where is a multimap and is a self-map. The common domain of and is not convex. Our results extend, generalize, or improve several fixed-point and fixed-set results including that given by Ok . A positive answer to Ok's question  is provided. Applications to the theory of self-similarity are also given.
for every , where denotes the diameter of . Let and . We write . We say that (a) is a fixed point of if , and the set of fixed points of will be denoted by ; (b) is nonexpansive if for all ; (c) is -contraction if for all and some ; (d) is -condensing if it is continuous and, for every with , and ; (e) is -set-contractive if it is continuous and, for every , , and ; (f) is compact if is a compact subset of .
A mapping is said to be a -contraction in the sense of Garcia-Falset  if there exists a function satisfying either " is continuous and for " or "there exists with and nondecreasing such that " for which the inequality holds for all , . Our definition for -contraction is different in some sense from that of Garcia-Falset.
Lemma 2.3 (see ).
Theorem 2.4 (see ).
Theorem 2.6 (see ).
Theorem 2.7 (see ).
Let be a closed subset of a Banach space such that is bounded, open, and containing the origin. Suppose that is a 1-set-contractive map satisfying for all and . If is closed, then has a fixed point in .
3. Fixed-Set Results
However is a compact subset of [18, page 16], so .
Corollary 3.2 ([2, Theorem 2.4]).
The following statements are equivalent :
Note that Corollary 3.3 provides a positive answer to the following question of Ok . We do not know at present if the fixed-set can be taken to be a compact set in the statement of [2, Corollary ].
So . Note that is compact-valued and is a compact subset of . The continuity of follows from that of and . Moreover, is a compact subset of , and hence is a compact subset of . By Lemma 2.3, there exists a minimal such that . But, since is continuous and is compact-valued, is compact-valued and maps compact sets to compact sets. Then , is a compact subset of M, so . Thus , and hence .
Let . By (b), (d), and the closeness of , the map is a -contraction from into . So, by Theorem 2.5, there exists a unique such that . Then , and so . Since the map has a unique fixed-point, its fixed-point set is singleton. So is a single-valued map. To show that is continuous, let be a sequence in such that . Define and . Then , and . We claim that is convergent. First, notice that is bounded; otherwise, has a subsequence such that . As , (c) implies that has a convergent subsequence, a contradiction. Next, as is continuous and one-to-one, it follows from (c) that the sequence converges to . Therefore, is continuous. Now the result follows from Theorem 3.5.
Parts (i) and (ii) follow from Theorem 3.6. Part (iii) follows from Theorem 3.1.
which contradicts the second part of (b). It follows from Theorem 2.6 that there exists such that . Then , and so . Now parts (i) and (ii) follow from Theorem 3.5. Part (iii) follows from Theorem 3.1.
Definition 3.10 (self-similar sets).
It is well known that there exists a unique compact self-similar set with respect to any contractive IFS; see .
The authors thank the referee for his valuable suggestions. This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah under project no. 3-017/429.
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