1. Introduction
Let
Y be a normed linear space. A mapping
is said to be nonexpansive if
for all
. A mapping
is said to be asymptotically nonexpansive if there exists a sequence
with
and
such that
for all
and
. It is well known that every nonexpansive mapping or asymptotically nonexpansive mapping on a non-empty closed, bounded, convex subset of a uniformly convex Banach space has at least one fixed point, see [
1,
2,
3].
A sequence in a normed linear space Y is said to be an approximate fixed point sequence (AFPS) for if .
Many authors have studied the AFPS for different types of nonexpansive mappings (see [
4,
5,
6]). Most of the results have been established in uniformly convex Banach spaces, smooth reflexive Banach spaces or Hilbert spaces. These results have become a major tool in solving various problems such as integral equations, differential equations, optimization problems (see [
7,
8]).
Recently, Som et al. [
9] introduced two types of mappings, Reich type nonexpansive and Chatterjea type nonexpansive mappings, and gave sufficient conditions under which these classes of mappings possess an AFPS, in more general spaces, more specifically in Banach spaces. They checked some properties of the fixed point sets of these mappings: closedness, convexity, remotality, etc. and gave some sufficient conditions under which a Reich type nonexpansive mapping reduces to that of nonexpansive one. For more considerations on Reich contractions and Chatterjea contractions see [
10,
11,
12]. To obtain the desired AFPS, Som et al. used the Schäefer iteration method:
where
.
Definition 1 ([
9]).
Let Y be a normed linear space, E a non-empty subset of Y and be a mapping. Mapping S is said to be a Reich type nonexpansive mapping if there exist non-negative real numbers with such thatfor all . Definition 2 ([
9]).
Let Y be a normed linear space, E a non-empty subset of Y and be a mapping. Mapping S is said to be a Chatterjea type nonexpansive mapping if there exist non-negative real numbers with such thatfor all . To prove their main results, Som et al. used the following lemma:
Lemma 1 ([
13]).
Let and be two bounded sequences in a Banach space Y and . Let and suppose for all . Then . Theorem 1 ([
9]).
Let Y be a Banach space and E be a non-empty closed, convex, bounded subset of Y. Let be a Reich type nonexpansive mapping with coefficients such that . Furthermore, assume that for Then S has an AFPS in E. Moreover, the AFPS is asymptotically regular. Theorem 2 ([
9]).
Under the assumptions of Theorem 1, S has a fixed point, provided . Theorem 3 ([
9]).
Let Y be a Banach space and E be a non-empty subset of Y. If S is Reich type nonexpansive mapping on E with coefficients such that , then Fix is a closed subset of E. Theorem 4 ([
9]).
Let Y be a Hilbert space and E be a nonempty convex subset of Y. Let S be Reich type nonexpansive mapping on E with coefficients such that . Assume that . Then Fix is a convex subset of E. Theorem 5 ([
9]).
Let Y be a Banach space and E be a non-empty closed, convex, bounded subset of Y. Let be a Chatterjea type nonexpansive mapping with coefficients such that . Furthermore, assume that for Then S has an AFPS in E. Moreover, the AFPS is asymptotically regular. Theorem 6 ([
9]).
Suppose that all conditions of Theorem 5 are satisfied. Further, assume that for any , there exists such thatThen S has a fixed point in E. Theorem 7 ([
9]).
Let Y be a finite dimensional Banach space, and E be a non-empty subset of Y. Let be a Reich type nonexpansive mapping with coefficients and assume that . If is centerable and contains its Chebyshev center, then S becomes nonexpansive. In this paper, we show that in Theorem 1 and Theorem 5 S does not need to be a Reich type nonexpansive mapping or Chatterjea type nonexpansive mapping, and the additional conditions can be replaced with some weaker conditions. Moreover, our proofs are very simple. Next, we generalize Theorem 2, Theorem 3 and Theorem 6 and make some usefull remarks on Theorem 4 and Theorem 7. Some examples will validate our results. We mention that some of these results can be extended to Hardy-Rogers nonexpansive mapings.
2. Main Results
The first result is the following generalization of Theorem 1.
Theorem 8. Let Y be a Banach space, E be a non-empty closed, convex, bounded subset of Y and be a mapping. Assume that there exists such that for Then S has an AFPS in E and the AFPS is asymptotically regular. Proof. Fix
and consider the sequence
in
Y defined by
for all
. Obviously, since
E is convex and bounded, it follows that
is a bounded sequence in
E. We have
By hypothesis, taking
,
we get
Hence, using Lemma 1, we obtain
, i.e.,
is an AFPS of
S. Moreover, we have
Therefore, the AFPS
is asymptotically regular. □
The next result is a generalization of Theorem 2.
Theorem 9. Let Y be a Banach space and E be a non-empty closed, convex, bounded subset of Y. Let a mapping and p, q, r non-negative real numbers with and such that conditionholds for all u, . Under the assumption of Theorem 1, S has a unique fixed point. Proof. By Theorem 1, we know that S has an AFPS which is asymptotically regular.
Taking
and
in (1), we get
Since
, it follows that
which implies
as
n,
.
Hence,
is a Cauchy sequence in
E and then, is convergent to some
. Moreover, we have
Suppose . Then we have .
Now, taking
and
in (1), we get for all
Taking limit as
, we obtain
It follows that
, i.e.,
.
Similarly, if
taking
and
in (1) we obtain, for all
Letting
, we get
by where
, i.e.,
. □
The following theorem is a generalization of Theorem 3.
Theorem 10. Let Y be a Banach space and E be a non-empty subset of Y. If satisfies (1) with , then is a closed subset of E.
Proof. Due to symmetry, we can suppose
. Let
be a sequence in
such that
converges to
. Taking
and
in (1) we get
for all
.
It follows that
Taking limit as
, we obtain
which gives
, i.e.,
. Hence,
is a closed subset of
E. □
Our next result shows that in Theorem 5 it is not necessary that S be a Chatterjea type nonexpansive mapping. Moreover, we relax the additional condition satisfying by S.
Theorem 11. Let Y be a Banach space and E be a non-empty closed, convex, bounded subset of Y. Let be a mapping such that there exists with Then S has an AFPS in E. Moreover, the AFPS is asymptotically regular.
Proof. Fix
and consider the sequence
in
Y defined by
, with
. Since
and
E is convex and bounded, it follows that
is a bounded sequence in
E. We have for all
and
Hence,
By hypothesis, we get for all
Therefore, using Lemma 1, we obtain
, i.e.,
is an AFPS of
S and
Thus, the AFPS
is asymtotically regular. □
Now, using Theorem 11, we prove a generalization of Theorem 6.
Theorem 12. Suppose that all the conditions of Theorem 11 are satisfied. Further, assume that for any , there exists such thatThen S has a fixed point in E. Proof. By Theorem 11,
S has an AFPS
, where
and
We suppose that
is not a Cauchy sequence. Then there exists
such that
By hypothesis, there exists a such that (2) holds. We can assume .
By (3) we can find
M so that
for all
.
Pick m,, so that .
For
j in
we have
This implies that, since
and
, there exists
j in
with
Hence, we have
Therefore, we get
It follows that
which contradicts (4).
This contradiction proves that must be a Cauchy sequence and hence is convergent to some . Obviously, as .
By (4) we get for all
u,
Taking
and
in the above inequality we obtain
Letting
we get
which gives
, i.e.,
, so
z is a fixed point of
S. □
Remark 1. In Theorem 4 it is necessary that E be convex. If , it follows that which is convex, and if then S is nonexpansive mapping and it is well known that is convex.
Remark 2. In Theorem 7 by condition there exist , , . Since S is a Reich type nonexpansive mapping we get , , so S is nonexpansive mapping. The other hypotheses are superfluous.
Example 1. Let us consider the Banach space equipped with the usual norm. Let and define bySom et al. proved that S is a Chartterjea type nonexpansive mapping with coefficients (see Example 3.11, [9]). It is easy to prove that S is also a Reich type nonexpansive mapping with coefficients . If and we haveandso S does not satisfy the hypothesis of Theorem 1. Furthermore, if and we haveandso S does not satisfy the hypothesis of Theorem 5. Now we prove that S satisfies property If , or , we have and the property is obvious.
If and we have to prove thatSince we have If and , thenand For we haveso . For we have .
Hence .
Therefore, S satisfies the hypothesis of Theorem 8. Next we prove that If , or , we have and the property is obvious.
If and , we haveSince we haveso . If and we haveFor we have when . Hence ThusFor we have when . Then Therefore, S satisfies the hypothesis of Theorem 12.
Example 2. Let us consider the Banach space equipped with the usual norm and take . We define bywhere and . Choose .
Let be arbitrary. Then the following three cases may arise:
- Case I:
Let . Then, and .
- Case II:
Let . Then, .
- Case III:
Let and . Then, and .
Since , it follows that
Thus, we see thatfor all . Similarly, we can prove thatfor all . Suppose S is a Reich type nonexpansive mapping with coefficients .
If and we haveandSince it follows thatwhich is a contradiction. Therefore, S is not a Reich type nonexpansive mapping.
Now we can suppose that T is a Chatterjea type nonexpansive mapping.
If and we haveandSince it follows thatSimilarly, taking and we get By (5) and (6) we havewhich contradicts . Therefore, S is not a Chatterjea type nonexpansive mapping. Now, we prove thatWe distinguish four cases: - Case I:
- Case II:
- Case III:
Let , . Then, If , since it follows that which is a contradiction.
If it follows that or , which is a contradiction.
- Case IV:
Let , . Then, If it follows that which is a contradiction.
If it follows that or , which is a contradiction.
Therefore, S satisfies the hypotheses of Theorem 8 and Theorem 9. S has an AFPS and .