Abstract
We establish three major fixed-point theorems for functions satisfying generalized rational type almost contraction conditions. Firstly we consider the case of a single mapping, secondly we look at the case of a triplet of mappings and we conclude by the case of a family of mappings. The theorems we present generalize similar results already obtained by Abbas, Rhoades, Gaba, and others. The operators we consider are all of the weakly Picard type.
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Public Interest Statement
In this paper, we give fixed point results for a certain type of functions (called rational contractions). Roughly speaking, the biggest motivation comes from the fact that, using fixed point theory in metric spaces it is possible to obtain sufficient conditions for studying and solving differential and variational problems arising in the applied sciences. These problems from the applied sciences try to describe our daily activities as mathematical problems.
1. Introduction and preliminaries
Recently, applications of G-metric spaces, in the fields like optimization theory, differential and integral equations, have been discovered and this has generated a lot of interest for these type of spaces (see Mustafa, Obiedat, & Awawdeh, Citation2008; *Mustafa, Mustafa; Shoaib, Arshad, & Kazmi, Citation2017). Their relevance is no more to be demonstrated as it has been extensively discussed in the literature. In this paper, we prove three main fixed point results in that setting. We propose generalizations which ensure existence results for fixed points, and to this goal we investigate the character of the sequence of iterates (resp. ) where (resp. ) is (resp. are) the map (resp. maps) under consideration, and X a complete G-metric space. More precisely, we consider mappings that satisfy a rational type almost contraction and the results we present are comparable to previous ones already obtained in Gaba (Citation2017). The paper is divided in two major sections, a first section which gives an introduction and some preliminaries and a second section which deals with the statements of results. The second section contains three subsections of which the first two present proofs making use of classical arguments (already used in Gaba, Citation2017), and of which the third one presents a result based on -series, see Sihag et al. (Citation2014). The elementary facts about G-metric spaces can be found in Gaba (Citation2017), Mustafa (Mustafa) and the references therein. We give here a summary of these prerequisites.
Definition 1.1
(see [*Mustafa, Mustafa, Definition 3]) Let X be a nonempty set, and let the function satisfy the following properties:
(G1) | if whenever ; | ||||
(G2) | whenever with ; | ||||
(G3) | whenever with ; | ||||
(G4) | , (symmetry in all three variables); | ||||
(G5) | for any points |
Definition 1.2
(see [*Mustafa, Mustafa]) Let (X, G) be a G-metric space, and let be a sequence of points of X, therefore, we say that the sequence is G-convergent to if that is, for any , there exists such that , for all, . We call x the limit of the sequence and write or .
Proposition 1.3
(Compare [*Mustafa, Mustafa, Proposition 6]) Let (X, G) be a G-metric space. Define on X the metric by whenever . Then for a sequence , the following are equivalent
(i) | is G-convergent to | ||||
(ii) | |||||
(iii) | . | ||||
(iv) | |||||
(v) |
Definition 1.4
(See *Mustafa, Mustafa) Let (X, G) be a G-metric space. A sequence is called a G-Cauchy sequence if for any , there is such that for all , that is as .
Proposition 1.5
(Compare [*Mustafa, Mustafa, Proposition 9])
In a G-metric space (X, G), the following are equivalent
(i) | The sequence is G-Cauchy. | ||||
(ii) | For each there exists such that for all . |
Definition 1.6
(Compare [*Mustafa, Mustafa, Definition 4]) A G-metric space (X, G) is said to be symmetric if
Definition 1.7
(Compare [*Mustafa, Mustafa, Definition 9]) A G-metric space (X, G) is G-complete if every G-Cauchy sequence of elements of (X, G) is G-convergent in (X, G).
Theorem 1.8
(see *Mustafa, Mustafa) A G-metric G on a G-metric space (X, G) is continuous on its three variables.
We conclude this introductory part with:
Definition 1.9
(Compare [Sihag et al., Citation2014, Definition 2.1]) For a sequence of nonnegative real numbers, the series is an -series if there exist and such that
2. The results
This section on our main results begins with the case of a single map.
2.1. Single maps
Theorem 2.1
Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition:(2.1) (2.1)
for all , where a, b, c are non-negative reals.
Then
(a) | T has at least one fixed point | ||||
(b) | for any , the sequence G-converges to a fixed point of T; | ||||
(c) | if are two distinct fixed points, then |
Proof
We imitate the steps of the proof of [Gaba, Citation2017 Theorem 2.1].
Let be arbitrary and construct the sequence such that Moreover, we may assume, without loss of generality that for .
For the triplet , and by setting , we have:
since
If we set
we get, iterativelyClaim: The sequence is a non-increasing sequence of non-negative reals.
Indeed, since we have
it is very clear that for any natural number , and so . We then have the following consecutive equivalences:
Hence
Therefore
hence
For any , since we have
the above translates to
and we obtain
Put and observe that
Hence
therefore
In other words, is a G-Cauchy sequence so G-converges to some
Claim: is a fixed point of T.
For the triplet in (Equation2.1(2.1) (2.1) ), we get(2.2) (2.2)
On taking the limit on both sides of (Equation2.2(2.2) (2.2) ), and using the fact that the function G is continuous, we have
i.e. thus
If is a fixed point of T with , then
Therefore,
The following two corollaries, particular cases of Theorem 2.1, are of interest for us, due to our previous work in Gaba (Citation2017).
Corollary 2.2
Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition:(2.3) (2.3)
for all . Then
(a) | T has at least one fixed point | ||||
(b) | for any , the sequence G-converges to a fixed point; | ||||
(c) | if are two distinct fixed points, then |
Proof
Apply Theorem 2.1 with
Example 2.3
(Compare [Gaba, Citation2017, Example 2.2])
Let and let be defined by(X, G) is a symmetric G-complete G-metric space.
Let be defined by
We have
Again,
Finally,
Therefore T satisfies all the conditions of Theorem 2.2. Also, T has two distinct fixed points and
Corollary 2.4
(Compare [Gaba, Citation2017, Theorem 2.1]) Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition:(2.4) (2.4)
for all . Then
(a) | T has at least one fixed point | ||||
(b) | for any , the sequence G-converges to a fixed point; | ||||
(c) | if are two distinct fixed points, then |
Proof
Apply Theorem 2.1 with
The previous results naturally extend if we consider a partially ordered complex valued G-metric space. Moreover, one can replace the non-negative real constants a, b, c by non-negative real valued functions.
We can define a partial order on the set of complex numbers by setting, for any
Moreover, on partial ordered G-metric space, the convergence of a sequence is interpreted in the canonical way, i.e. for a sequence where is a partial ordered complex valued G-metric space,
Similarly for G-Cauchy sequences. Furthermore, a self mapping T defined on a partial ordered G-metric space is nondecreasing if whenever , for
We then state the result:
Theorem 2.5
Let be a symmetric, G-complete, complex valued G-metric space. Assume that if is a nondecreasing sequence of elements of X such that , then for all . Let be a nondecreasing mapping such that:(2.5) (2.5)
for all where are non-negative real valued functions.
If there exists with ,
then
(i) | T has at least one fixed point | ||||
(ii) | for any , the sequence G-converges to a fixed point; | ||||
(iii) | if are two distinct fixed points, then |
Proof
Following the steps of the proof of Theorem 2.1, it is very easy to see that the sequence of iterates is nondecreasing and G-converges to some . Therefore for all . Now applying (Equation2.5(2.5) (2.5) ) to the triplet we have:
Now taking the limit as , and using the fact that the function G is continuous, we have:
i.e. thus
If is a fixed point of T with , then
Therefore,
Another variant of Theorem 2.1 goes as follows:
Theorem 2.6
Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition:(2.6) (2.6)
for all , where are non-negative real valued functions and
Then T has at least one fixed point
Remark 2.7
In general, the self mapping T in Theorem 2.6 (as well as in Theorem 2.1) is a weaklyFootnote1 Picard operator. Moreover, the reader can convince him/her-self that if and are fixed points of T in X, a lower bound can be found for (see point (c) in Theorem 2.1). Furthermore, Theorem 2.6 can be expressed in a setting of a partially ordered complex valued G-metric space.
We conclude this subsection by proving the following result, which presents a reverse rational type contraction. Actually, this mapping can be classified as an expansion type mapping.
Theorem 2.8
Let (X, G) be a symmetric G-complete G-metric space and T be an onto self mapping on X. Suppose that T satisfies the following condition:(2.7) (2.7)
for all , where a, b, c are non-negative reals and
Then T has at least one fixed point
Proof
Let , then
Hence which implies that . So T is injective and invertible.
If H is the inverse mapping of T, then for , we have
Hence for all
From Theorem 2.6, the inverse mapping H has a fixed point i.e. But Thus u is also a fixed point of T.
In the next subsection, we consider the case of a triplet of functions and we state an analogue of Theorem 2.1.
2.2. Triplets of maps
Theorem 2.9
Let (X, G) be a symmetric G-complete G-metric space and T, P, Q be three self mappings on X. Suppose that T, P, Q satisfy the following condition:(2.8) (2.8)
for all where are non-negative functions. Then T, P and Q have a common fixed point, i.e. such that
Proof
For any initial point we construct the sequence by setting
Without loss of generality, assume that for .
Plugging in in (Equation2.8(2.8) (2.8) ), we have:
where
Each of the term in the numerator of can be bounded as follows:
By setting , is bounded as
i.e.
So if we denote
plugging in in (Equation2.8(2.8) (2.8) ) implies
If we inspire ourselves from the proof of Theorem 2.1, one can easily establish that the sequence is a non-increasing sequence of non-negative real numbers and that for any natural number , Moreover, it is readily seen that is a G-Cauchy sequence so G-converges to some
By substituting for (x, y, z) in (Equation2.8(2.8) (2.8) ), we get
where can easily be recovered from (Equation2.8(2.8) (2.8) ). Hence
Again, from (Equation2.8(2.8) (2.8) ), we can write that:(2.9) (2.9) (2.10) (2.10) (2.11) (2.11)
where and can easily be recovered from (Equation2.8(2.8) (2.8) ).
SinceFootnote2the relations (2.9), (Equation2.10(2.10) (2.10) ) and (Equation2.11(2.11) (2.11) ) respectively give that , and i.e.
This completes the proof.
Remark 2.10
The reader can convince him(her)-self that if we replace the condition (Equation2.8(2.8) (2.8) ) by(2.12) (2.12)
where the non-negative functions a, b, c, d, e and f are well chosen, then P, Q and T have a common fixed point.
We conclude this article with the case of a family of mappings.
2.3. Families of maps
Here, in this last subsection of the manuscript, we consider the case of a family of functions and we state an analogue of Theorem 2.9.
We make use of the following special class of homogeneous functions. Let be the class of continuous, non-decreasing, sub-additive and homogeneous functions such that and Footnote3.
Theorem 2.11
Let (X, G) be a symmetric G-complete G-metric space and be a family of self mappings on X such that(2.13) (2.13)
where and are non-negative functions, the constants are such that and some homogeneous with degree s.
If
is an -series, then has a common fixed point in X.
Proof
For any , we construct the sequence by setting We may assume without loss of generality that for all . We observe that, by setting , and plugging in the triplet we have
where
When we write the above for the triplet , we obtain
Also we get
Hence, we derive, iteratively, that
Therefore, for all , since
using the fact that F is sub-additive, we write
We already know that the sequence is a sequence of non-negative reals and that for any natural number , . Therefore for any natural number Hence
Now, let and as in Definition 1.9, then for and using the fact that the geometric mean of non-negative real numbers is at most their arithmetic mean, it follows that
As , we deduce that Thus is a G-Cauchy sequence and since X is complete there exists such that G-converges to .
Furthermore, for any
for some Now taking the limit using the fact that the function G is continuous, we obtain
This terminates the proof.
Additional information
Funding
Notes on contributors
Yaé Ulrich Gaba
Yaé Ulrich Gaba is a postdoctoral fellow at North-West University (South Africa) and also affiliated to the African Centre for Advanced Studies (ACAS). One of his research orientations deals with Fixed Point Theory in metric like spaces and their applications to physical sciences.
Notes
1 This means that the sequence of iterates , for any initial point , converges to a fixed and this fixed point might surely not be unique
2 See Theorem 1.8
3 The function is an example of such function.
References
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- Mustafa, Z., & Sims, B. (2006). A new approach to generalizedmetric spaces. Journal of Nonlinear Convex Analysis, 7, 289–297,
- Shoaib, A., Arshad, M., & Kazmi, S. H. (2017). Fixed point results for Hardy Roger type contraction in ordered complete dislocated Gd metric space. Turkish Journal of Analysis and Number Theory, 5(1), 5–12.
- Sihag, V., Vats, R. K., & Vetro, C. (2014). A fixed point theorem in G-metric spaces via α-series. Quaestiones Mathematicae, 37(3), 429–434.