Fixed point theorems for generalized contractions on GP-metric spaces

In this paper, we present two fixed point theorems on mappings, defined on GP-complete GP-metric spaces, which satisfy a generalized contraction property determined by certain upper semi-continuous 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 GP-complete GP-metric spaces.

MSC: 47H10, 54H25.

In 1922, Stefan Banach [1] stated his celebrated theorem on the existence and uniqueness of a fixed point of certain self-maps 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 well-known application is in ordinary differential equations, particularly, in the proof of the Picard-Lindelöf theorem which guarantees the existence and uniqueness of solutions of first-order 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., [3-27]) 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, D-metric spaces, and G-metric spaces (see, e.g., [28-44]) 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 GP-metric spaces which are a combination of the notions of partial metric spaces and G-metric 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 GP-metric spaces that we will need in this work.

Definition 1.1[43]

Let X be a non-empty set. A function is called a GP-metric if the following conditions are satisfied:

(GP1) if ;

(GP2) for all ;

(GP3)  , symmetry in all three variables;

(GP4) for any .

Then the pair is called a GP-metric space.

Example 1.1[43]

Let and define for all . Then is a GP-metric space.

Proposition 1.1[43]

Letbe aGP-metric space, then for anyx, y, zand, it follows that

(i) ;

(ii) ;

(iii) ;

(iv) .

Proposition 1.2[43]

EveryGP-metric spacedefines a metric spacewhere

Definition 1.2[43]

Let be a GP-metric 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 aGP-metric space. Then, for any sequenceinXand a point, the following are equivalent:

(A) isGP-convergent tox;

(B) as;

(C) as.

Definition 1.3[43]

Let be a GP-metric space.

(S1) A sequence is called a GP-Cauchy if and only if exists (and is finite);

(S2) A GP-partial metric space is said to be GP-complete if and only if every GP-Cauchy sequence in X is GP-convergent to such that .

Now, we introduce the following.

Definition 1.4 Let be a GP-metric space.

(M1) A sequence is called 0-GP-Cauchy if and only if ;

(M2) A GP-metric space is said to be 0-GP-complete if and only if every 0-GP-Cauchy sequence in XGP-converges to a point such that .

Example 1.2 Let and define for all . Then is a GP-complete GP-metric space. Moreover, if (where ℚ denotes a set of rational numbers), then is a 0-GP-complete GP-metric space.

Lemma 1.1 (See [45])

Letbe aGP-metric space. Then

(A) If, then;

(B) If, then.

In the rest of this paper, we will denote the positive natural numbers by and the natural numbers by ℕ.

In this section, we present our findings on fixed point theorems on 0-GP-complete GP-metric spaces. We first start with the following definition.

Definition 2.1 Let be a GP-metric space and be a map. Let denote the value

for all .

Lemma 2.1Ifis aGP-metric space andis a map, then, for each, we have

Proof Let . Then

The proof is complete. □

Lemma 2.2Letbe aGP-metric space and letbe a map such that

for all, whereis a function such thatfor all. Ifsatisfiesfor all, then the following hold:

(a) for all;

(b) for all.

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

that is, . Hence, (a) holds.

Clearly, (b) follows from (4), (a), and the fact that for all . □

Definition 2.2 A function is called upper semi-continuous from the right if for each and each sequence such that and , the equality holds .

Theorem 2.1Letbe aGP-completeGP-metric space and letbe a map such that

for all, whereis an upper semi-continuous 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 non-increasing 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 semi-continuity 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 semi-continuous 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 GP-complete GP-metric 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 aGP-completeGP-metric space and letbe a map such that

for all, whereis an upper semi-continuous 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 GP-complete GP-metric 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 GP-complete GP-metric space. Let be defined by

and for all .

Proof To prove this example, we need to consider the following cases:

• Let . Then

• Let . Then

• Let . Then

• Let and . Then

• Let and . Then

• Let and . Then

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 aGP-completeGP-metric 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

for all . Therefore,

for all . So,

and following this process,

for all and . Consequently,

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 GP-complete GP-metric space. Let be defined by

and for all .

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 GP-complete GP-metric space. Let be defined by

and for all .

Proof To prove this example, we need to examine the following cases:

• Let . Then and

• Let . Then and

• Let and . Then and

Then the condition of Theorem 2.2 holds and T has a unique fixed point 0 in . Moreover, . □

Corollary 2.2Letbe aGP-completeGP-metric 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 aGP-completeGP-metric 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 aGP-completeGP-metric space andbe a self-mapping. Assume that

Then the following statements are equivalent:

(i) there exist functionssuch that

for any;

(ii) there exists a functionsuch that for any bounded sequenceof positive reals, impliesand

for any;

(iii) there exists a continuous functionsuch thatand

for any;

(iv) there exist a functionand a nondecreasing, right-continuous functionwithand for allwith

for any;

(v) there exists a continuous and nondecreasing functionsuch thatand for allwith

for any.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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