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Existence and approximations of fixed points for contractive mappings of integral type
Fixed Point Theory and Applications volume 2014, Article number: 138 (2014)
Abstract
The existence, uniqueness, and iterative approximations of fixed points for four classes of contractive mappings of integral type in complete metric spaces are established. The results presented in this paper generalize indeed several results of Branciari (J. Math. Math. Sci. 29(9):531-536, 2002), Rhoades (Int. J. Math. Math. Sci. 2003(63):4007-4013, 2003) and Liu et al. (Fixed Point Theory Appl. 2011:64, 2011). Four illustrative examples with uncountably many points are also included.
MSC:54H25.
1 Introduction
Over the past decade the researchers [1–18] introduced a lot of contractive mappings of integral type and discussed the existence of fixed points and common fixed points for these mappings in metric spaces and modular spaces, respectively. Branciari [5] was the first to study the existence of fixed points for the contractive mapping of integral type and proved the following result, which extends the Banach fixed point theorem.
Theorem 1.1 ([5])
Let f be a mapping from a complete metric space into itself satisfying
where is a constant and = { is Lebesgue integrable, summable on each compact subset of and for each }.
Then f has a unique fixed point such that for each .
Rhoades [16] and Liu et al. [10] extended the result of Branciari and proved the following fixed point theorems.
Theorem 1.2 ([16])
Let f be a mapping from a complete metric space into itself satisfying
where is a constant and . Then f has a unique fixed point such that for each .
Theorem 1.3 ([16])
Let f be a mapping from a complete metric space into itself satisfying
where is a constant and . Assume that f has a bounded orbit at some point . Then f has a unique fixed point such that .
Theorem 1.4 ([10])
Let f be a mapping from a complete metric space into itself satisfying
where and is a function with
Then f has a unique fixed point such that for each .
Theorem 1.5 ([10])
Let f be a mapping from a complete metric space into itself satisfying
where and are two functions with
Then f has a unique fixed point such that for each .
The purposes of this paper are both to study the existence, uniqueness, and iterative approximations of fixed points for four new classes of contractive mappings of integral type, which include the contractive mappings of integral type in [5, 10, 16] as special cases, and to construct four examples with uncountably many points to illustrate that the results obtained properly generalize Theorems 1.1-1.5 or are different from these theorems.
2 Preliminaries
Throughout this paper, we assume that , , , where ℕ denotes the set of all positive integers. Let be a metric space. For , and , put
, ,
, ,
,
.
The and are called the orbit and n th orbit of f at x, respectively.
Let
,
,
= { is a function such that for each nonempty bounded subset B in }.
The following lemma plays an important role in this paper.
Lemma 2.1 ([10])
Let and be a nonnegative sequence with . Then
3 Four fixed point theorems
In this section we show the existence, uniqueness and iterative approximations of fixed points for four classes of contractive mappings of integral type.
Theorem 3.1 Let f be a mapping from a complete metric space into itself satisfying
where . Then f has a unique fixed point such that for each .
Proof Let x be an arbitrary point in X. Note that
It follows from (3.1) and (3.2) that
Now we prove that
Suppose that (3.4) does not hold. That is, there exists some satisfying
Since and , it follows from (3.3) and (3.5) that
which is a contradiction and hence (3.4) holds. Clearly, (3.4) implies that there exists a constant c with .
Next we prove that . Otherwise . Taking the upper limit in (3.3) and using Lemma 2.1 and , we conclude that
which is absurd. Therefore, , that is,
Now we claim that is a Cauchy sequence. Suppose that is not a Cauchy sequence, which means that there is a constant such that for each positive integer k, there are positive integers and with such that
For each positive integer k, let denote the least integer exceeding and satisfying the above inequality. It follows that
Note that
Making use of (3.6)-(3.8), we obtain
It follows from (3.6) and (3.9) that
which together with (3.1), Lemma 2.1, and gives
which is a contradiction. Thus is a Cauchy sequence. Since is a complete metric space, it follows that there exists a point such that . Suppose that . It is clear that (3.6) implies that
which together with (3.1), Lemma 2.1, and yields
which is a contradiction. That is, .
Finally, we prove that a is a unique fixed point of f in X. Suppose that f has another fixed point . Note that
It follows from (3.1), and that
which is a contradiction. This completes the proof. □
Theorem 3.2 Let f be a mapping from a complete metric space into itself satisfying
where . Assume that f has a bounded orbit at some point . Then f has a unique fixed point such that .
Proof Without loss of generality we assume that . Now we prove that
Let . It is clear that there exist such that and . Suppose that for some with . In light of (3.10) and , we infer that
which is a contradiction. Thus (3.11) holds.
Next we prove that is a Cauchy sequence. Suppose that is not a Cauchy sequence. It follows that there exist an and two strictly increasing sequences and with for each satisfying
Put and . Clearly . Observe that ensures that
which implies that there exists some with
Using (3.10)-(3.13) and , we know that there exist , , and satisfying
which is impossible. Thus is a Cauchy sequence. Since is complete, it follows that there exists satisfying . Suppose that . Note that
Taking the upper limit in (3.10) and using (3.14), Lemma 2.1, and , we conclude that
which is absurd. Therefore, , that is, .
Suppose that f has another fixed point . Since
it follows from (3.10) that
which is a contradiction. That is, f has a unique fixed point in X. This completes the proof. □
As in the arguments of Theorems 3.1 and 3.2, we conclude similarly the following results and omit their proofs.
Theorem 3.3 Let f be a mapping from a complete metric space into itself satisfying
where . Then f has a unique fixed point such that for each .
Theorem 3.4 Let f be a mapping from a complete metric space into itself satisfying
where . Assume that f has a bounded orbit at some point . Then f has a unique fixed point such that .
4 Remarks and illustrative examples
Now we construct four examples with uncountably many points to show the fixed point theorems obtained in Section 3 generalize properly or are different from the known results in Section 1.
Remark 4.1 Theorem 3.1 generalizes Theorem 1.2, which, in turns, extends Theorem 1.1. The following example proves that Theorem 3.1 both extends substantially Theorem 1.1 and is different from Theorem 1.4.
Example 4.2 Let be endowed with the Euclidean metric , , and be defined by
Obviously, . Let with . In order to verify (3.1), we have to consider six possible cases as follows:
Case 1. . It is clear that
and
Case 2. and . Note that
and
Case 3. . It follows that
and
Case 4. . Notice that
and
Case 5. and . It is easy to see that
and
Case 6. and . It is easy to verify that
and
That is, (3.1) holds. It follows from Theorem 3.1 that f has a unique fixed point and for each . But we invoke neither Theorem 1.1 nor Theorem 1.4 to show that f possesses a fixed point in X.
Suppose that f satisfies the conditions of Theorem 1.1, that is, there exists satisfying
which means that
which is a contradiction.
Suppose that f satisfies the conditions of Theorem 1.4, that is, there exists satisfying
which implies that
which is a contradiction.
Remark 4.3 Theorem 3.2 is a generalization of Theorem 1.3. The below example demonstrates that Theorem 3.2 is different from Theorem 1.4.
Example 4.4 Let be endowed with the Euclidean metric , , and be defined by
It is easy to see that and is bounded for each . Let with . In order to verify (3.10), we have to consider six possible cases as follows:
Case 1. . It is clear that
Case 2. . Note that
and
Case 3. and . It follows that
and
Case 4. . It is easy to see that
Case 5. and . It follows that
and
Case 6. and . It is clear that
and
That is, the conditions of Theorem 3.2 are fulfilled. It follows from Theorem 3.2 that f has a unique fixed point and for each . However, Theorem 1.4 is useless in guaranteeing the existence of a fixed point of f in X. Suppose that f satisfies the conditions of Theorem 1.4, that is, there exists satisfying
which yields
which is impossible.
Remark 4.5 Theorem 3.3 extends Theorems 1.1 and 1.2. The example below is an application of Theorem 3.3.
Example 4.6 Let be endowed with the Euclidean metric , , and be defined by
Obviously, . Let with . In order to verify (3.15), we have to consider five possible cases as follows:
Case 1. . It follows that
and
Case 2. . It follows that
and
Case 3. and . It is clear that
and
Case 4. and . Notice that
and
Case 5. and . It is clear that
and
That is, the conditions of Theorem 3.3 are fulfilled. It follows from Theorem 3.3 that f has a unique fixed point and for each .
Remark 4.7 Theorem 3.4 extends Theorem 1.3. The following example shows that Theorem 3.4 both generalizes substantially Theorem 1.3 and differs from Theorem 1.5.
Example 4.8 Let be endowed with the Euclidean metric , , and be defined by
It is clear that and is bounded for each . Let with . In order to verify (3.16), we have to consider four possible cases as follows:
Case 1. . Note that
and
Case 2. . Clearly
and
Case 3. and . Obviously
and
Case 4. . It follows that
and
That is, the conditions of Theorem 3.4 are fulfilled. It follows from Theorem 3.4 that f has a unique fixed point and for each . But we do not invoke Theorems 1.3 and 1.5 to show the existence of a fixed point of f in X.
Suppose that f satisfies the conditions of Theorem 1.3, that is, there exists some satisfying
which yields
which is impossible.
Suppose that f satisfies the conditions of Theorem 1.5, that is, there exist satisfying
and
which means that
which is absurd.
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Acknowledgements
The authors would like to thank the referees for useful comments and suggestions. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380) and the fund of the Research Promotion Program, Gyeongsang National University, 2013 (RPP-2013-023).
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Liu, Z., Li, X. & Kang, S.M. Existence and approximations of fixed points for contractive mappings of integral type. Fixed Point Theory Appl 2014, 138 (2014). https://doi.org/10.1186/1687-1812-2014-138
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DOI: https://doi.org/10.1186/1687-1812-2014-138