1. Introduction
Banach’s contraction principle is a magnific tool in many fields of nonlinear analysis and in mathematical analysis. Applications in these fields are very interests and promise to other new applications. Banach’s contraction principle has been generalized and extended in many directions. The authors [1] have given an important result about contractions in multi-valued complete metric spaces. We have given the generalization of this result which is a particular case of author’s [2] paper. In this paper, we prove a new fixed point theorem for multi-valued mapping defined on complete metric spaces. To realize this result we give the proof of an intuitive lemma which is used to complete the proof of the main result.
Theorem 1.1 (Đorić and Lazović) Define a nonincreasing function from into by
(1)
Let be a complete metric space and T be a mapping from X into . Assuming that there exists such that implies
(2)
for all . Then, there exists , such that .
This theorem is a particular case of our main result when .
Theorem 1.2 (Kaliaj) Let be a complete metric space and let be a multi-valued mapping, assuming that T is -Suzuki integral contraction. Then, for any we have
(3)
Our main result is a particular case of this theorem when is absolutely continuous.
Definition 1.3 Let be a complete metric space and let be the family of all nonempty closed bounded subsets of X. Define the Hausdorff metric
(4)
for all .
It is well-known that, if is a complete metric space, then is also a complete metric space.
Definition 1.4 Let be a multi-valued mapping. We say that T is a -Suzuki contraction with
(5)
if there exists such that, the implication
(6)
holds whenever , where
(7)
Definition 1.5 The multi-valued mapping T is an -Suzuki integral contraction if is defined in conditions of Theorem 1.1 and there exist and , such that the implication
(8)
holds whenever , where
(9)
Definition 1.6 The multi-valued mapping T is said to be a -Suzuki contraction with
(10)
if there exists and such that, the implication
(11)
holds whenever , where
(12)
Definition 1.7 A function is said to be absolutely continuous in if, given , there exists some such that
(13)
whenever is a finite collection of mutually disjoint sub-intervals of with .
Lemma 1.8 (Kaliaj) Let be a complete metric space and let be a -Suzuki integral contraction and with
(14)
and for all .
Then, exists a such that
(15)
for all .
When is absolutely continuous, we have this
Corollary 1.9 Let be a complete metric space and let be a -Suzuki contraction and with
(16)
and for all .
Then, exists a such that
(17)
for all .
The main result is Theorem 2.1. First, we give the proof of this lemma:
Lemma 1.10 Let be a complete metric space and be a -Suzuki contraction. Then, the implication
holds for all .
Proof: First, since , it follows that . By the definition of distance of a point to the set we can obtain this inequality:
Because the function takes values in , it implies
Since is increase monotonic we have
and from the fact that T is a -Suzuki contraction, it follows that
where,
Substituting, the last inequality will be transformed in
which implies
(18)
Remember that, from the definition of Hausdorff distance, we can write:
which implies
Using the fact that is monotone increasing, it follows that
But
and since , it follows that
Indeed, using the fact that H is a Hausdorff distance, we can write:
and so,
If the maximum element of the set
it was then,
which is a contradiction, because . The last result with the result of inequality (18) implies
Since y was arbitrary, the last result holds .
Since is absolutely continuous, by the Corollary 1.9, we can write:
(19)
2. The Main Result
Theorem 2.1 Let be a complete metric space and let the mapping be a -Suzuki contraction. Then, T has a fixed point.
Proof. Let be an arbitrary fixed point in X. Choose a real number . If , then . Hence is a fixed point of T and the proof has finished.
Assume that . Then, there exists with
Since , using Lemma 1.10 we obtain:
(20)
We assume that since:
(21)
As before, if , for similarity, is a fixed point for the mapping T and the proof is done.
Assume that . Since is continuous at , given
there exists , such that
and, since there exists such that
it follows that
(22)
Inductively, assume now that we chose . The, by Lemma 1.10 we have
If , then is a fixed point of -Suzuki contraction T, and the proof is done.
Assume that . Since is continuous at , given
there exists , such that
and, since there exists such that
it follows that
(23)
Since , by Lemma 1.10, we have
By above construction, we obtain a sequence with terms in X such that
(24)
Hence, we get
and since it follows that
Summing side by side for to it follows that
Since the last result yields that is a Cauchy sequence in X, and by completeness of X it follows that converges to a point . We are going to prove that z is a fixed point of T. Suppose the contrary, i.e., . It follows that two cases are possibles:
1)
2)
We study each case as follows:
1) For an fixed arbitrary , since
we obtain
and since by Corollary 1.9 and Lemma 1.10 we have also
and
it follows that
(25)
Since was arbitrary, the last results yield
for all . From equality
it follows that there exists sequence such that
Then, by inequality (25) we obtain
and since it follows that
or
which is a contradiction and as consequence, .
2) Let be an arbitrary . Since
we get
(26)
Since , by Corollary 1.9, we have
(27)
If then
(28)
Otherwise, if then we obtain by inequality (26) that
and using inequality (26) again, we get
Since x was arbitrary, combining last result with inequality (28) yields
Then, by hypothesis, it follow that
whenever . Clearly, if then, the last inequality also holds. Thus, the last inequality holds for all . In particular, for , we have
Hence, by Theorem IX.4.1 in [3], it follows that
This contradiction shows that and the proof is done.
3. Conclusion
In this paper, we studied the -Suzuki contraction for multi-valued mappings in the complete metric spaces generated by the family of all nonempty closed bounded subsets of a set X, refereed as . We proved that this contraction has a fixed point.
[1] Ðorić and Lazović (2011) Some Suzuki-Type Fixed Point Theorems for Multi-Valued Mappings and Applications. Fixed Point Theory and Applications, 2011, Article Number: 40.
https://doi.org/10.1186/1687-1812-2011-40
[2] Kaliaj, S.B. (2020) An Integral Suzuki-Type Fixed Point Theorem with Application. 2009.08643, arXiv, math.FA.
[3] Natanason, I.P. (1961) Theory of Functions of a Real Variables. Frederick Ungar Publishing Co., New York.