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Existence and Approximation of Fixed Points for Set-Valued Mappings
Fixed Point Theory and Applications volume 2010, Article number: 351531 (2010)
Abstract
Taking into account possibly inexact data, we study both existence and approximation of fixed points for certain set-valued mappings of contractive type. More precisely, we study the existence of convergent iterations in the presence of computational errors for two classes of set-valued mappings. The first class comprises certain mappings of contractive type, while the second one contains mappings satisfying a Caristi-type condition.
1. Introduction
The study of the convergence of iterations of mappings of contractive type has been an important topic in Nonlinear Functional Analysis since Banach's seminal paper [1] on the existence of a unique fixed point for a strict contraction [2–5]. Banach's celebrated theorem also yields convergence of iterates to the unique fixed point. During the last fifty years or so, many developments have taken place in this area. Interesting results have also been obtained regarding set-valued mappings, where the situation is more difficult and less understood. See, for example, [5–12] and the references cited therein. As already mentioned above, one of the methods used for proving the classical Banach theorem is to show the convergence of Picard iterations, which holds for any initial point. In the case of set-valued mappings, we do not have convergence of all trajectories of the dynamical system induced by the given mapping. Convergent trajectories have to be constructed in a special way. For instance, in [7], if at the moment we have reached a point
, then we choose an element of
(here
is the given mapping) such that
approximates the best approximation of
from
. Since our mapping acts on a general complete metric space, we cannot, in general, choose
as the best approximation of
by elements of
. Instead, we require
to approximate the best approximation up to a positive number
, such that the sequence
is summable. This method allowed Nadler [7] to obtain the existence of a fixed point of a strictly contractive set-valued mapping and the authors of [6] to obtain more general results. In view of this state of affairs, it is important to study convergence of the iterates of set-valued mappings in the presence of errors.
In this paper, we study the existence of convergent iterations in the presence of computational errors for two classes of set-valued mappings. The first class comprises certain mappings of contractive type, while the second one contains mappings satisfying a Caristi-type condition.
As we have already mentioned, the existence of a convergent iterative sequence for set-valued strict contractions was established by Nadler [7]. For a more general class of mappings satisfying a certain contractive condition, this was proved in [11]. In the present paper, we show that the existence result of [11] still holds even when possible computational errors are taken into account (see Theorems 2.2–2.4 below).
In Section 3, we obtain certain results regarding set-valued mappings satisfying a Caristi-type condition which complement the results in [6]. There we establish the existence of a fixed point for such mappings assuming that their graphs are closed. Here we first show that a set-valued mapping satisfies a Caristi-type condition if and only if there exists an iterative sequence such that the sum of the distances between
and
, when
runs from zero to infinity, is finite. Then we prove an analog of the Caristi-type result in [6], replacing the closedness of the graph of the mapping with a lower semicontinuity assumption as in Caristi's original theorem [13].
2. Set-Valued Mappings of Contractive Type
Let be a complete metric space. For each
and each nonempty set
, set

For each pair of nonempty sets , put

Let ,
satisfy

let be a decreasing function such that

and assume that

We begin with the following obvious fact.
Lemma 2.1.
Let ,
, and let a sequence of mappings
,
, satisfy

Then there exist sequences and
such that for any integer
,

Theorem 2.2.
Let and
be positive. Then there exist
and a natural number
such that for each sequence of mappings
,
, satisfying

and each satisfying

there is a sequence such that

and the inequality

holds for all integers .
This theorem is a consequence of Lemma 2.1 and our next result.
Theorem 2.3.
Let . Then there exists
so that for each
, there is a natural number
such that the following assertion holds.
Assume that a sequence of mappings ,
, satisfies

for all , a sequence
satisfies

and that for each integer , there is
such that


Then for all integers
.
Proof.
Choose a positive number such that

Let . Fix a natural number
such that

Assume that a sequence of mappings ,
, satisfies (2.12),
satisfies (2.13), and that for each integer
, there is

such that (2.14) and (2.15) hold.
Let be an integer. By (2.3), (2.5), and (2.14),

By (2.15), (2.18), and (2.12),

By (2.19) and (2.20),

for all integers .
We claim that there is an integer such that

Assume the contrary. Then

By (2.21), (2.23), and (2.16), we have, for all integers ,

When combined with (2.13), this implies that

This, however, contradicts (2.17).
Therefore, there is an integer such that (2.22) holds. Next, we assert that

Assume the contrary. Then there is an integer such that

There are two cases: either

or

Assume first that (2.28) holds. By (2.21), (2.28), and (2.16),

Now assume that (2.29) holds. By (2.29), (2.21), (2.27), and (2.16),

This contradicts (2.27).
The contradiction we have reached in both cases proves that (2.26) holds.
This completes the proof of Theorem 2.3.
We end this section with another consequence of Theorem 2.3.
Theorem 2.4.
Let ,
,
, and assume that, for all
,

Then for each , there is a sequence
such that

Proof.
Let . Put
and define a sequence
by induction so that for each integer
, there is
satisfying

In order to show that

we let and prove that for all sufficiently large natural numbers
,

Let be as guaranteed by Theorem 2.3.
There is a natural number such that

Choose such that

Let a natural number be as guaranteed by Theorem 2.3.
Then for all integers ,

Theorem 2.4 is proved.
3. Caristi-Type Theorems for Set-Valued Mappings
We begin this section by recalling [6, Theorem ].
Theorem 3.1.
Assume that is a complete metric space,
, graph
is closed,
is bounded from below, and that for each
,

Let ,
, and let
satisfy
. Assume that for each integer
,
and

Then converges to a fixed point of
.
In the proof of this theorem, we actually showed that .
It turns out that the existence of such a sequence is actually equivalent to the existence of a function which is bounded from below and such that for each
,

More precisely, we are going to prove the following result.
Theorem 3.2.
Let be a complete metric space and
. The following conditions are equivalent.
(A)There exists a function , which is bounded from below and not identically
, such that for each
, inequality (3.3) holds.
(B)There exists a sequence such that
for all integers
and
.
Proof.
The fact that (A) implies (B) was proved in [6]. To show that (B) implies (A), we define, for each ,

Let . Note that
if and only if there is a sequence
such that
,
,
, and

It is sufficient to show that (3.3) holds for all . To this end, let
. We may assume that
. Let
. There is
such that
,

Then

and so,

Since is any positive number, we conclude that (3.3) holds. This completes the proof of Theorem 3.2.
It should be mentioned that in Theorem 3.1 we pose an assumption on without assuming that
possesses lower semicontinuity properties, while in the original Caristi theorem no assumption was made on the mapping, but the function
was assumed to be lower semicontinuous. In the following result, we obtain a simple analog of Theorem 3.1 for this situation.
Theorem 3.3.
Assume that is a complete metric space,
,
is closed for each
,
is a lower semicontinuous function which is bounded from below and not identically
, and that for each
, inequality (3.3) holds. Then
has a fixed point.
Proof.
Choose such that

By Ekeland's variational principle [14], there is such that

Let . By (3.3) and (3.10), there exists
such that

Since is any positive number, it follows that
. Theorem 3.3 is proved.
Note added in proof
After our paper was accepted for publication, Pavel Semenov has kindly informed us that our Theorem 3.3 is almost identical with Corollary 1.7 on page 521 of Volume I of the Handbook of Multivalued Analysis by S. Hu and N. S. Papageorgiou, Kluwer Academic Publishers, Dordrecht, 1997. It may be of interest to note that the above authors deduce their Corollary from a set-valued version of Caristi's fixed point theorem, while we use Ekeland's variational principle (which is known to be equivalent to Caristi's fixed point theorem).
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Acknowledgments
This research was supported by the Israel Science Foundation (Grant no. 647/07), the Fund for the Promotion of Research at the Technion, and by the Technion President's Research Fund.
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Reich, S., Zaslavski, A. Existence and Approximation of Fixed Points for Set-Valued Mappings. Fixed Point Theory Appl 2010, 351531 (2010). https://doi.org/10.1155/2010/351531
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DOI: https://doi.org/10.1155/2010/351531