Fixed Point Results for Multivalued Maps in Metric Spaces with Generalized Inwardness Conditions
© M. Frigon 2010
Received: 19 November 2009
Accepted: 12 February 2010
Published: 21 February 2010
We establish fixed point theorems for multivalued mappings defined on a closed subset of a complete metric space. We generalize Lim's result on weakly inward contractions in a Banach space. We also generalize recent results of Azé and Corvellec, Maciejewski, and Uderzo for contractions and directional contractions. Finally, we present local fixed point theorems and continuation principles for generalized inward contractions.
1. Introduction and Preliminaries
The well known Nadler fixed point Theorem  says that a multivalued contraction on to itself has a fixed point. However, to insure the existence of a fixed point to a multivalued contraction defined on a closed subset of , extra assumptions are needed.
In 2000, Lim  obtained the following fixed point theorem for weakly inward multivalued contractions in Banach spaces using the transfinite induction.
From this observation, generalizations of this result to complete metric spaces were recently obtained with simpler proofs by Azé and Corvellec , and by Maciejewski . They generalized the inwardness condition using the metric left-open segment
which should be nonempty for every and "close enough" of . They also obtained results for directional -contractions in the sense of Song . In 2005, Uderzo  established a local fixed point theorem for directional -contractions.
In this paper, we generalize their results. More precisely, we first generalize the inwardness conditions used in [2–4]. In particular, for with , one can have . Also, we slightly generalize the notion of -directional contractions.
Here is the well known Caristi Theorem  which will play a crucial role in the following.
Theorem 1.3 (Caristi ).
This result, which is equivalent to the Ekeland variational Principle [8, 9], can also be deduced from the Bishop-Phelps theorem. The following formulation appeared in  (see also ) while the original formulation appeared in a different form in  (see also ).
Theorem 1.4 (Bishop and Phelps).
The interested reader can find a multivalued version of Caristi's fixed point theorem in an article of Mizoguchi and Takahashi .
2. Generalizations of Inward Contractions
In this section, we obtain fixed point results for contractions defined on a closed subset of a metric space satisfying a generalized inwardness condition.
Corollary 2.2 (Maciejewski ).
From the proof of Theorem 2.1, one sees that one can weaken the assumption that is a contraction, and hence one can generalize a result due to Azé and Corvellec .
Corollary 2.4 (Azé and Corvellec ).
Therefore, (2.8) and (2.12) are not satisfied.
A careful look at their proofs permits to realize that a wider class of maps can be considered. Indeed, it is easy to see that the previous results are corollaries of the following theorem which is a direct consequence of Theorem 1.3.
3. Intersection Conditions
Observe that even though Theorem 2.7 generalizes Theorems 2.1 and 2.3, Condition (2.17) is quite restrictive in the multivalued context since every has to satify a suitable condition. Here is a fixed point result where at least one element of has to be in a suitable set.
The previous theorem generalizes a result of Downing and Kirk .
Corollary 3.4 (Downing and Kirk ).
So, (3.7) is satisfied and the conclusion follows from Corollary 3.3.
Thus, Condition (3.1) is satisfied.
An analogous condition in the multivalued context leads to the following result.
Theorems 3.1 and 3.6 are equivalent.
It is clear that if (3.1) is satisfied, then (3.14) is also satisfied. Thus, Theorem 3.6 implies Theorem 3.1.
Corollary 3.9 (Song ).
Uderzo  generalized Song's result introducing the notion of directional multi-valued -contraction (this means that satisfies the following condition (ii)). This notion generalizes the notion of directional contractions used by Song  (Condition (ii) in Corollary 3.9). We show how Uderzo's result can be obtained from Theorem 3.8.
Corollary 3.10 (Uderzo ).
4. Local Fixed Point Theorems for Generalized Inward Contractions
In this section, we present local versions of fixed point theorems for generalized inward contractions.
As corollaries, we obtain local versions of Theorems 2.1 and 2.3.
We obtain as corollary the following result due to Maciejewski .
The conclusion follows from Corollary 4.4.
5. Continuation Principle for Generalized Inward Contractions
In this section, we obtain continuation principles for families of contractions satisfying a generalized inwardness condition. For and open in , we denote the boundary of relative to . Here is a generalization of Theorem in . The proof is analogous.
The previous theorem applied inductively to insures the existence of such that . The sequence has a weakly converging subsequence still denoted such that . The demi-closedness of (see [18, Theorem ]) implies that has a fixed point.
Similarly to Theorem 5.1, we can prove the following continuation principles using Theorems 4.1 and 4.6, respectively.
This work was partially supported by CRSNG Canada.
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