Abstract
In this paper, we present two fixed point theorems on mappings, defined on GPcomplete GPmetric spaces, which satisfy a generalized contraction property determined by certain upper semicontinuous functions. Furthermore, we illustrate applications of our theorems with a number of examples. Inspired by the work of Jachymski, we also establish equivalences of certain auxiliary maps in the context of GPcomplete GPmetric spaces.
MSC: 47H10, 54H25.
Keywords:
fixed point; partial metric space; GPmetric space1 Introduction and preliminaries
In 1922, Stefan Banach [1] stated his celebrated theorem on the existence and uniqueness of a fixed point of certain selfmaps defined on certain metric spaces for the first time. Specifically, this elegant theorem, also known as the Banach contraction mapping principle, can be formulated as follows: any mapping has a unique point such that provided that there exists a constant satisfying the inequality for every , where is a complete metric space. A mapping T for which the inequality mentioned above holds is called a contraction.
Since its first appearance, the Banach contraction mapping principle has become the main tool to study contractions as they appear abundantly in a wide array of quantitative sciences. Its most wellknown application is in ordinary differential equations, particularly, in the proof of the PicardLindelöf theorem which guarantees the existence and uniqueness of solutions of firstorder initial value problems. It is worth emphasizing that the remarkable strength of the Banach principle originates from the constructive process it provides to identify the fixed point. This notable strength further attracted the attention of not only many prominent mathematicians studying in many branches of mathematics related to nonlinear analysis, but also many researchers who are interested in iterative methods to examine the quantitative problems involving certain mappings and space structures required in their work in various areas such as social sciences, biology, economics, and computer sciences.
Indeed, in 1994, Matthews, a computer scientist who is an expert on semantics, announced in [2] an analog of Banach’s principle in a new space he called a partial metric space. Matthews’s innovative approach was quickly adopted and improved by fixed point theorists (see, e.g., [327]) with the aim of discovering analogs of Banach’s principle in the context of partial metric spaces to broaden its applications and enrich the fixed point theory as a result.
A closer look to the work of these distinguished mathematicians after Matthews’s studies reveals that their discoveries can be categorized in terms of the techniques implemented to produce the analogs of Banach’s principle. The first technique is to introduce new space structures with certain properties which guarantee the existence and/or uniqueness of fixed points of contractions. In addition to Matthews’s investigations, cone metric spaces, Dmetric spaces, and Gmetric spaces (see, e.g., [2844]) constitute a few of the examples to the first approach. The second technique is to introduce mappings defined on metric spaces satisfying certain new contractive conditions. For example, cyclic contractions and weak ϕ contractions can be listed as a few.
As another example to the first approach mentioned above, Zand and Nezhad [43] recently introduced GPmetric spaces which are a combination of the notions of partial metric spaces and Gmetric spaces. Then they proved a number of fixed point theorems on these new spaces for certain type of contractions. In this paper, we exercise the second approach by using the space structure they initiated to prove certain fixed point theorems for generalized contractions. First, we review the necessary notation, definitions, and fundamental results produced on GPmetric spaces that we will need in this work.
Definition 1.1[43]
Let X be a nonempty set. A function is called a GPmetric if the following conditions are satisfied:
(GP3) , symmetry in all three variables;
Then the pair is called a GPmetric space.
Example 1.1[43]
Let and define for all . Then is a GPmetric space.
Proposition 1.1[43]
Letbe aGPmetric space, then for anyx, y, zand, it follows that
Proposition 1.2[43]
EveryGPmetric spacedefines a metric spacewhere
Definition 1.2[43]
Let be a GPmetric space and let be a sequence of points of X. A point is said to be the limit of the sequence or if
Proposition 1.3[43]
Letbe aGPmetric space. Then, for any sequenceinXand a point, the following are equivalent:
Definition 1.3[43]
(S1) A sequence is called a GPCauchy if and only if exists (and is finite);
(S2) A GPpartial metric space is said to be GPcomplete if and only if every GPCauchy sequence in X is GPconvergent to such that .
Now, we introduce the following.
Definition 1.4 Let be a GPmetric space.
(M1) A sequence is called 0GPCauchy if and only if ;
(M2) A GPmetric space is said to be 0GPcomplete if and only if every 0GPCauchy sequence in XGPconverges to a point such that .
Example 1.2 Let and define for all . Then is a GPcomplete GPmetric space. Moreover, if (where ℚ denotes a set of rational numbers), then is a 0GPcomplete GPmetric space.
Lemma 1.1 (See [45])
In the rest of this paper, we will denote the positive natural numbers by and the natural numbers by ℕ.
2 Main results
In this section, we present our findings on fixed point theorems on 0GPcomplete GPmetric spaces. We first start with the following definition.
Definition 2.1 Let be a GPmetric space and be a map. Let denote the value
Lemma 2.1Ifis aGPmetric space andis a map, then, for each, we have
The proof is complete. □
Lemma 2.2Letbe aGPmetric space and letbe a map such that
for all, whereis a function such thatfor all. Ifsatisfiesfor all, then the following hold:
Proof
(a) From Lemma 2.1, we have
Since for all , then by Lemma 1.1(B), we get for all . Consequently, . Now by condition (4), we deduce that
Clearly, (b) follows from (4), (a), and the fact that for all . □
Definition 2.2 A function is called upper semicontinuous from the right if for each and each sequence such that and , the equality holds .
Theorem 2.1Letbe aGPcompleteGPmetric space and letbe a map such that
for all, whereis an upper semicontinuous function from the right such thatfor all. ThenThas a unique fixed point. Moreover, .
Proof Let . If there is such that , then is a fixed point of T and the uniqueness of follows as in the last part of the proof below. Hence, we assume that for all . Put and construct the sequence , where for all . Thus, and for all . Define the sequence by for all . From Lemma 2.2(b) we know that is a nonincreasing sequence. Hence, there exists such that as . We will show that c must be equal to 0. Let . By taking limitsup as in condition (b) of Lemma 2.2, we get that
and so, by upper semicontinuity from the right of the function ϕ, we deduce
which is a contradiction. Hence, . Consequently, . Next we show that . Assume the contrary. Then there exist and sequences , in with and such that for all . From the fact that , we can suppose, without loss of generality, that . For each , we have
and hence . Now, let be such that and for all . Then
for all . So, . Since for all and ϕ is upper semicontinuous from the right, we deduce that . On the other hand, for each , we have
so , a contradiction because . Consequently, and thus is a Cauchy sequence in the GPcomplete GPmetric space . Hence, there is such that
We show that z is a fixed point of T. To this end, we first note that , so . On the other hand, since for each , , it follows that
Therefore, and thus . Finally, let be such that . Then
Hence, , i.e., . This concludes the proof. □
Definition 2.3 Let be a given function. Then will denote the value
Then we obtain the following statement.
Corollary 2.1Letbe aGPcompleteGPmetric space and letbe a map such that
for all, whereis an upper semicontinuous from the right function such thatfor all. ThenThas a unique fixed point. Moreover, .
Proof Clearly, by taking in the hypothesis, we have
for all . Then the conditions of Theorem 2.1 hold. This concludes the proof. □
Example 2.1 Let , be defined by . Then is a GPcomplete GPmetric space. Let be defined by and for all .
Proof Without loss of generality, we assume that . Then
Then the condition of Theorem 2.1 holds and T has a unique fixed point 0 in . Moreover, . □
Example 2.2 Let , be defined by . Then is a GPcomplete GPmetric space. Let be defined by
Proof To prove this example, we need to consider the following cases:
Then the condition of Theorem 2.1 holds and T has a unique fixed point 0 in . Moreover, . □
Lemma 2.3Letbe nondecreasing and let. If, then.
Theorem 2.2Letbe aGPcompleteGPmetric space andbe a map such that
wherefor all, andis a nondecreasing function such thatfor all. ThenThas a unique fixed point. Moreover, .
Proof Let . If there is such that , then is a fixed point of T and the uniqueness of follows as in the last part of the proof below. Hence, we will assume that for all . Put and construct the sequence , where for all . Thus, and for all . By Lemma 2.2(b),
for all . Then, since ϕ is nondecreasing, we deduce that
for all . Hence, . Now, choose an arbitrary . Since , it follows from Lemma 2.3 that , so there is such that
and following this process,
and thus is a Cauchy sequence in the complete metric space . Hence, there is such that
We show that z is a fixed point of T. Assume the contrary. Then . For each , we have
From our assumption that , it easily follows that there is such that for all . So,
for all . Taking limits as , we obtain that
a contradiction. Consequently, . Finally, the uniqueness of z follows as in Theorem 2.1. □
Example 2.3 Let , be defined by . Then is a GPcomplete GPmetric space. Let be defined by
Proof Without loss of generality, we assume that . Then
And so
Then the condition of Theorem 2.2 holds and T has a unique fixed point 0 in . Moreover, . □
Example 2.4 Let , be defined by . Then is a GPcomplete GPmetric space. Let be defined by
Proof To prove this example, we need to examine the following cases:
Then the condition of Theorem 2.2 holds and T has a unique fixed point 0 in . Moreover, . □
Corollary 2.2Letbe aGPcompleteGPmetric space andbe a map such that
wherefor all, andis a nondecreasing function such thatfor all. ThenThas a unique fixed point. Moreover, .
Similarly, we have the corollary below.
Corollary 2.3Letbe aGPcompleteGPmetric space andbe a map such that
wherefor all, andis a nondecreasing function such thatfor all. ThenThas a unique fixed point. Moreover, .
In [46], Jachymski proved the equivalence of auxiliary functions (see Lemma 1). Inspired by the results from this remarkable paper of Jachymski, we finish this paper by stating the following theorem.
Theorem 2.3 (See [46])
Letbe aGPcompleteGPmetric space andbe a selfmapping. Assume that
Then the following statements are equivalent:
(i) there exist functionssuch that
(ii) there exists a functionsuch that for any bounded sequenceof positive reals, impliesand
(iii) there exists a continuous functionsuch thatand
(iv) there exist a functionand a nondecreasing, rightcontinuous functionwithand for allwith
(v) there exists a continuous and nondecreasing functionsuch thatand for allwith
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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