From (3.7) and (3.8) we get
, since , by Lemma 2.6, we get is a Cauchy sequence.
Since X is complete, there exists z in X, such that , that is , and .
Now let us give that
, for every . For is convergent to 0, and by Lemma 2.5, we get
, thus we have , thus from the above relation, there exists a point in X such that
For such , (3.2) implies that
therefore by Lemma 3.5,
therefore we get
, for each . (3.9)
Now we show that for each ,
It is obvious that the above inequality is true for , assume that the relation holds for some . We get (3.10) is true when we have if , then if , we get the following relation from (3.9) and induction hypothesis, and that is
then (3.10) is proved.
Now we consider the following two possible cases in order to prove that f has a fixed point z in X, and that is .
Case 1 , therefore, . First, we prove the following relation
, for . (3.11)
When it is obvious, and it follows from (3.6) when , from (3.10) and take we have
, then we get .
Now suppose that (3.11) holds for some ,
Therefore, we get
, that is , (3.11.1)
then by taking in (3.6)
using the above two relations, (3.11.1) and (3.11.2) we have
From (3.2) and (3.10) with and , we have
So by induction we prove the relation of (3.11).
Now (3.11) and show that for every , thus, (3.9) shows that
Therefore . Furthermore by using Lemma 2.5, we get
In the same way,
, thus we have , that is , and by using Lemma 2.5 in (3.12), we get
, which claims that , and that is a contraction.
Case 2. , and that is when . We now prove that we can find a subsequence of such that
, for . (3.13)
The contraries of the above relation are as follows
for . If n is even we have
if n is odd then we get
for . By (3.8) we have
this is impossible. Therefore, one of the following relations is true for every ,
That means there exists a subsequence of such that (3.13) is true for every . Thus (3.2) shows that
From Lemma 2.5, we have
Therefore , which is impossible unless . hence z in X is a fixed point of f. From the process of the above proof, we know , then by
this proves that . By (3.2) we can prove the uniqueness of the common fixed point z,
, so (3.2) shows that
which is impossible unless . □
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Banach, S. (1922) Sur les opérations dans les ensembles abtraits et leur applications aux équations intégrales. Fundamenta Mathematicae, 3, 133-181.
|||Ekeland, I. (1974) On the Variational Principle. Journal of Mathematical Analysis and Applications, 47, 324-353. https://doi.org/10.1016/0022-247X(74)90025-0|
Meir, A. and Keeler, E. (1969) A Theorem on Contraction Mappings. Journal of Mathematical Analysis and Applications, 28, 326-329.
|||Nadler Jr., S.B. (1969) Multi-Valued Contraction Mappings. Pacific Journal of Mathematics, 30, 475-488. https://doi.org/10.2140/pjm.1969.30.475|
Caristi, J. (1976) Fixed Point Theorems for Mappings Satisfying Inwardness Conditions. Transactions of the American Mathematical Society, 215, 241-251.
Caristi, J. and Kirk, W.A. (1975) Geometric Fixed Point Theory and Inwardness Conditions. Lecture Notes in Mathematics, 499, 74-83.
|||Subrahmanyam, P.V. (1974) Remarks on Some Fixed Point Theorems Related to Banach’s Contraction Principle. Electronic Journal of Mathematical and Physical Sciences, 8, 445-457.|
Suzuki, T. (2004) Generalized Distance and Existence Theorems in Complete Metric Spaces. Journal of Mathematical Analysis and Applications, 253, 440-458.
|||Mustafa, Z., Parvaech, V., Roshan, J.R. and Kadelburg, Z. (2014) b2-Metric Spaces and Some Fixed Point Theorems. Fixed Point Theory and Applications, 2014, Article Number: 144. https://doi.org/10.1186/1687-1812-2014-144|
|||Fadail, Z.M., Ahmad, A.G.B., Ozturk, V. and Radenovi?, S. (2015) Some Remarks on Fixed Point Results of b2-Metric Spaces. Far East Journal of Mathematical Sciences, 97, 533-548. https://doi.org/10.17654/FJMSJul2015_533_548|
Copyright © 2020 by authors and Scientific Research Publishing Inc.
This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.