2018 International Conference on Physics, Computing and Mathematical Modeling (PCMM 2018) ISBN: 978-1-60595-549-0
Fixed Point Theorems for Generalized Ψ-φ-Khan-contractions in
b-rectangular Metric Spaces
Jing-shuai LIU and Pei-sheng JI
*1School of Mathematical Sciences, Qingdao University, Shangdong 266071, P. R. China
*Corresponding author
Keywords: Fixed point, Generalized Ψ-φ-Khancontraction, b-rectangular metric space.
Abstract. In this paper, we introduce the concept of generalized Ψ-φ-Khan contraction in
b-rectangular metric spaces. By using the method of the fixed point theory, we obtain some fixed
point theorems for such mapping in complete b-rectangular metric space.
Introduction and Preliminaries
The contraction principle introduced by Banach [1] is one of the most important results in mathematical analysis. Indeed, it is widely used as the source of metric fixed point theory. In 2009, D. Doric [2] introduced a new concept of contraction called generalized (Ψ-φ)-weak contraction and proved a common fixed point theorem which generalized the Banach contraction principle. Afterward, some authors have obtained fixed point theorems for some kinds of (Ψ-φ)-weak contractive mappings (see [3,4,5,8]).
In 2016, H. Piri et al. [9] proved Khan type fixed point theorems in a generalized metric space. In [10], they also studied some results of existence and uniqueness of fixed points for a class of mappings satisfying an inequality of rational expressions. Hossein Piri et al. [11] introduced the concept of F-Khan-contractions and proved a fixed point theorem in complete metric spaces. Moreover, Ansari et al. [12] proved C-class function on Khan type common fixed point theorems in generalized metric space.
In 2015, George et al. [13] introduced the concept of rectangular b-metric space and proved an analogue of Banach contraction principle and Kannan's fixed point theorem in this space. Since then many fixed point theorems for various contractions on rectangular b-metric spaces (see [6,14,15]).
In this paper, we will introduce the generalized Ψ-φ-Khan-contraction which satisfy different
conditions in b rectangular metric space and then we shall prove some fixed point theorems for these
contractions in a partially ordered complete b-rectangular metric space.
Through this paper, we denote by the set of positive integers.
Definition 1.1. ([13]) Let be a nonempty set, s≥1 be a given real number and let
be a mapping such that for all and distinct points , each
distinct from x and y:
(1) if and only if x and y;
(2) ;
(3) (b-rectangular inequality).
Then (X,d) is called a b-rectangular metric space or a b-generalized metric space (b-g.m.s.).
In 1984, Khan et al. [7] introduced altering distance functions as follows:
Definition 1.2. ([7]) A function is called an altering distance function if the
following properties are satisfied:
(a) Ψ is non-decreasing and continuous;
(b) Ψ (t)=0 if and only if .
The family of all altering distance functions is denoted by .
Definition 1.3. We denote by ϕ the set of function satisfying the following
(a) φ is lower semi-continuous;
(b) φ (t)=0 for all t=0.
The following lemma will be used for proving our main results.
Lemma 1. ([6]) Let (X,d) be a b-rectangular metric space and let {xn} be a Cauchy sequence in X
such that xn ≠ xm whenever n≠m. Then {xn} can converge to at most one point.
Lemma 2. ([6]) Let (X,d) be a b-g.m.s. Suppose that sequences {xn} and {yn} in X are such that
xn→x and yn→y as n→∞, with x≠y, and xn≠x, yn≠y for n∈N. Then we have
.
If y∈X and {xn} is a Cauchy sequence in X with xn≠xm for infinitely many m,n∈N with n≠m,
converging to x≠y, then
, for all x∈X.
Main Results
In this section, we introduce the concept of generalized Ψ-φ-Khan-contraction and prove the fixed
point theorems for such mapping.
Definition 2.1. Let (X,d) be a b-rectangular metric space with s>1, and let T:X→X be a self
mapping. A mapping T is said to be a generalized Ψ-φ-Khan-contraction, if there exists
such that for all distinct x,y∈X, the following conditions hold:
where
Theorem 2.1. Let be a partially ordered complete b-rectangular metric space with
parameter s>1. Let T:X→X be a non-decreasing mapping with respect to such that there exists x0∈X
with . Suppose that following conditions are satisfied:
(i) T is a generalized Ψ-φ-Khan-contraction; (ii) T is continuous.
Then T has a fixed point. Moreover, the set of fixed points T is well ordered only if T has one and
only one fixed point.
Proof. Let x0∈X such that and {xn} be a sequence in X with , for all
. If for some n∈N, then xn=Txn, xn is a fixed point of T. Therefore, we will
assume that xn for all n∈N. Since and is non-decreasing, we have
Continuing this process, we obtain that for all n∈N.
We shall divide the proof into two cases.
Case 1. Assume that for all m∈N and . Then
from (2.1) we have
where
Assume that , for all . From (2.2), (2.3) and the definition of ѱ we
Since ѱ is non-decreasing, so from (2.4), implies that
for all n∈N.
It follows that the sequence is non-increasing. Hence, there exists r>0 such that
. Then, from (2.4), we have ,
which is a contradiction. We have
.
Assume first that xn=xm for some n>m. Similarly as in the proof of Theorem 1 of [6], we get a
contradiction. So we may assume that xn≠xm for all n,m∈N with n≠m.
Analogously, we can prove that , by substituting and
in (2.1), we obtain
where
Using (2.7) and (2.8) we have
Since is non-decreasing and , then it is easy to see from (2.9), we get
.
Taking limit on both sides of above inequality and use (2.6), we have
.
Next, we will show that {xn} is a b-g.m.s Cauchy sequence in X. We will prove by using a
contradiction technique. Assume that is not a Cauchy sequence. Therefore there exists for
we can find two subsequences {xmi} and {xni} of {xn} such that ni is the smallest index for which
and . Then,
Taking the upper limit in (2.10) as i→∞, we get
Using b-rectangular inequality, we have
In addition, we have . By taking
the upper limit in the above inequality and using (2.6), we get
Now putting x=xmi and y=xni-2 in (2.1), then we obtain
where
Taking the upper limit in (2.15), by using (2.6) and (2.11) we get
In the same way, by taking the lower limit as i→∞ in (2.15) and using (2.6), we get
Next, taking the upper limit as i→∞ in (2.14) and using (2.12), (2.16) and (2.17) we have
Since ѱ is non-decreasing function, we have sε≤ε, which is a contradiction. Hence {xn} is a Cauchy
sequence in X. Since (X,d) is complete, there exists u∈X such that . We show
that u is a fixed point of T. Since T is continuous, then
.
Thus by Lemma 1, it follows that xn differs from both u and Tu for n sufficiently large. Using the
b-rectangular inequality, we obtain
Taking limit as n͢∞, we have
.
So we have Tu=u. Then, u is a fixed point of T. Finally, suppose that the set of fixed points of T is
well ordered. Assume, on the contrary, that u,v are two fixed points of T such that u≠v. Hence, from
(2.1) with x=u and y=v we have
By (2.18) and (2.19), we get
By virtue of (2.20) and the monotonicity of ѱ, it follows that .
Therefore, , i.e. u=v. Then T has a unique fixed point. Conversely, suppose that T has a
unique fixed point, then the set of fixed points of T is singleton. Hence, it is well ordered.
Case 2. Assume that there exists m∈N such that .
By condition (2.1), it follows that and hence . This completes the
existence of a fixed point of T. The uniqueness follows as in Case 1.
Theorem 2.2 Under the hypotheses of Theorem 2.1, if the continuity of T is replaced by the
condition that , for all n∈N whenever {xn} is a non-decreasing sequence in X such that
.
Then T has a fixed point in X.
Proof. Following similar arguments to those given in the proof of Theorem 2.1, we construct an
increasing sequence {xn} in X such that xn→u for some u∈X. Using the assumption on X, we have that
for all n∈N. Now, we show that Tu=u. Suppose, on the contrary, that Tu≠u. By (2.1), we have
where
Letting n→∞ in (2.22), we obtain that .
Next, we take the upper limit as n→∞ in (2.21) and use Lemma 2 and (2.22) we get
By virtue of (2.23) and monotonicity of ѱ, it follows that , which is a
contradiction. Therefore, we get u=Tu and then u is a fixed point of T.
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