On new fixed point results for -contractive multi-valued mappings onα-complete metric spaces and their consequences

Abstract

The purpose of this paper is to establish new fixed point results formulti-valued mappings satisfying an -contractive condition, and viaBianchini-Grandolfi gauge functions, on α-complete metric spaces.Our results unify, generalize, and complement various results from theliterature. We also give examples which support our main result while previousresults in the literature are not applicable. Some of the fixed point results inmetric spaces endowed with an arbitrary binary relation and endowed with a graphare given here to illustrate the usability of the obtained results.

MSC: 47H10, 54H25.

1 Introduction and preliminaries

Throughout this paper, we denote by ℕ, , and ℝ the sets of positive integers,non-negative real numbers, and real numbers, respectively.

We recollect some essential notations, required definitions, and primary resultscoherent with the literature. For a nonempty set X, we denote by the class of all nonempty subsets of X. Let be a metric space, we denote by the class of all nonempty closed subsets ofX, by the class of all nonempty closed bounded subsets ofX. For , let the functional be defined by

for every , where is the distance from a to. Such a functional is called the generalizedPompeiu-Hausdorff metric induced by d.

In this paper, we denote by Ψ the class of functions satisfying the following conditions:

() ψ is a nondecreasing function;

() , for all , where is the nth iterate of ψ.

These functions are known in the literature as Bianchini-Grandolfi gauge functions insome sources (see e.g.[1–3]).

Remark 1.1 For each , we see that the following assertions hold:

• 1. , for all .

• 2. for each ;

• 3. ;

Example 1.2 The function defined by , where , is a Bianchini-Grandolfi gauge function.

Example 1.3 The function defined by

is a Bianchini-Grandolfi gauge function.

In [4], Samet et al. introduced the concepts of anα-admissible mapping and anα-ψ-contractive mapping as follows.

Definition 1.4 ([4])

Let T be a self mapping on a nonempty set X and be a mapping. We say that T isα-admissible if the following condition holds:

Definition 1.5 ([4])

Let be a metric space and be a given mapping. We say that T is anα-ψ-contractive mapping if there exist two functions and such that

for all .

One also proved some fixed point theorems for such mappings on complete metric spacesand showed that these results can be utilized to derive fixed point theorems inpartially ordered metric spaces.

Afterwards, Asl et al.[5] introduced the concept of an -admissible mapping which is a multi-valued version ofthe α-admissible mapping provided in [4].

Definition 1.6 ([5])

Let X be a nonempty set, and be two mappings. We say that T is-admissible if the following condition holds:

where .

They extended the α-ψ-contractive condition of Sametet al.[4] from a single-valued version to a multi-valued version as follows.

Definition 1.7 ([5])

Let be a metric space, be a multi-valued mapping and be a given mapping. We say that T is anα-ψ-contractive multi-valued mapping if there exists such that

for all .

Asl et al.[5] also established a fixed point result for multi-valued mappings oncomplete metric spaces satisfying an α-ψ-contractivecondition.

Recently, Ali et al.[6] introduced the notion of -contractive multi-valued mappings, where and Ξ is the family of functions satisfying the following conditions:

() ξ is continuous;

() ξ is nondecreasing on;

() if and only if ;

() ξ is subadditive.

Remark 1.8 From () and (), we have , for all .

Example 1.9 Let be a mapping which is defined by

for each , where is a Lebesgue integrable mapping which is summable oneach compact subset of and satisfies the following conditions:

• for each , we have ;

• for each , we have

  

Then .

Lemma 1.10Letbe a metric space. If, thenis a metric space.

Lemma 1.11 ([6])

Letbe a metric space, , and. If there existssuch that, then there existssuch that

where.

Definition 1.12 ([6])

Let be a metric space. A multi-valued mapping is called an -contractive mapping if there exist three functions, , and such that

(1.1)

where .

In the case when is strictly increasing, the -contractive mapping is called a strictly-contractive mapping.

Ali et al.[6] also prove fixed point results for -contractive multi-valued mapping on complete metricspaces.

Question 1 Is it possible to prove fixed point results for-contractive multi-valued mapping T undersome weaker condition for T?

Question 2 Is it possible to prove fixed point results for-contractive multi-valued mapping in some space whichis more general than complete metric spaces?

Question 3 Is it possible to find some consequences or applications of thefixed point results?

On the other hand, Mohammadi et al.[7] extended the concept of an -admissible mapping to α-admissible asfollows.

Definition 1.13 ([7])

Let X be a nonempty set, and be two given mappings. We say that T isα-admissible whenever for each and with , we have , for all .

Remark 1.14 It is clear that -admissible mapping is alsoα-admissible, but the converse may not be true as shown inExample 15 of [8].

Recently, Hussain et al.[9] introduced the concept of α-completeness for metric spacewhich is a weaker than the concept of completeness.

Definition 1.15 ([9])

Let be a metric space and be a mapping. The metric space X is said tobe α-complete if and only if every Cauchy sequence in X with , for all , converges in X.

Remark 1.16 If X is complete metric space, then X is alsoα-complete metric space. But the converse is not true.

Example 1.17 Let and the metric defined by , for all . Define by

It is easy to see that is not a complete metric space, but is an α-complete metric space. Indeed,if is a Cauchy sequence in X such that, for all , then , for all . Since is a closed subset of ℝ, we see that is a complete metric space and then there exists such that as .

In this paper, we establish new fixed point results for -contractive multi-valued mappings onα-complete metric spaces by using the idea ofα-admissible multi-valued mapping due to Mohammadi et al.[7]. These results are real generalization of main results of Ali etal.[6] and many results in literature. We furnish some interesting exampleswhich support our main theorems while results of Ali et al.[6] are not applicable. We also obtain fixed point results in metric spaceendowed with an arbitrary binary relation and fixed point results in metric spaceendowed with graph.

2 Main results

First, we introduce the concept of α-continuity for multi-valuedmappings in metric spaces.

Definition 2.1 Let be a metric space, and be two given mappings. We say T is anα-continuous multi-valued mapping on if, for all sequences with as , and , for all , we have as , that is,

Note that the continuity of T implies the α-continuity ofT, for all mappings α. In general, the converse is nottrue (see in Example 2.2).

Example 2.2 Let , and the metric defined by , for all . Define and by

and

Clearly, T is not a continuous multi-valued mapping on. Indeed, for sequence in X, we see that , but .

Next, we show that T is an α-continue multi-valued mapping on. Let be a sequence in X such that as and , for all . Then we have , for all . Therefore, . This shows that T is anα-continuous multi-valued mapping on .

Now we give first main result in this paper.

Theorem 2.3Letbe a metric space andbe a strictly-contractive mapping. Suppose that thefollowing conditions hold:

(S₁) is anα-complete metric space;

(S₂) Tis anα-admissible multi-valued mapping;

(S₃) there existandsuch that;

(S₄) Tis anα-continuous multi-valued mapping.

ThenThas a fixed point.

Proof Starting from and in (S₃), we have . If , then we see that is a fixed point of T. Assume that. If , we obtain that is a fixed point of T. Then we have nothingto prove. So we let . From -contractive condition, we get

(2.1)

If , then we get

(2.2)

which is a contradiction. Therefore, . From (2.1), we get

(2.3)

For fixed by using Lemma 1.11, there exists such that

(2.4)

From (2.3) and (2.4), we have

(2.5)

Since ψ is strictly increasing function, we have

(2.6)

Put and then .

If or , then we find that is a fixed point of T and thus we havenothing to prove. Therefore, we may assume that and . Since , , , and T is an α-admissiblemulti-valued mapping, we have . Applying from -contractive condition, we have

(2.7)

Suppose that . From (2.7), we get

(2.8)

which is a contradiction. Therefore, we may let . From (2.7), we have

(2.9)

By using Lemma 1.11 with , there exists such that

(2.10)

From (2.9) and (2.10), we get

(2.11)

It follows from ψ being a strictly increasing function that

(2.12)

Continuing this process, we can construct a sequence in X such that ,

(2.13)

and

(2.14)

for all .

Let such that . By the triangle inequality, we have

Since , we have . Using , we get . This implies that is a Cauchy sequence in . From (2.13) and the α-completeness of, there exists such that as .

By α-continuity of the multi-valued mapping T, we get

(2.15)

Now we obtain

Therefore, and hence T has a fixed point. Thiscompletes the proof. □

Corollary 2.4Letbe a metric space andbe a strictly-contractive mapping. Suppose that thefollowing conditions hold:

(S₁) is anα-complete metric space;

() Tis an-admissible multi-valuedmapping;

(S₃) there existandsuch that;

(S₄) Tis anα-continuous multi-valued mapping.

ThenThas a fixed point.

Corollary 2.5 (Theorem 2.5 in [6])

Letbe a complete metric space andbe a strictly-contractive mapping. Suppose that thefollowing conditions hold:

(A₁) Tis an-admissible multi-valuedmapping;

(A₂) there existandsuch that;

(A₃) Tis a continuous multi-valued mapping.

ThenThas a fixed point.

Next, we give second main result in this work.

Theorem 2.6Letbe a metric space andbe a strictly-contractive mapping. Suppose that thefollowing conditions hold:

(S₁) is anα-complete metric space;

(S₂) Tis anα-admissible multi-valued mapping;

(S₃) there existandsuch that;

() ifis a sequence inXwithasand, for all, then we have, for all.

ThenThas a fixed point.

Proof Following the proof of Theorem 2.3, we know that is a Cauchy sequence in X such that as and

(2.16)

for all . From condition (), we get

(2.17)

for all . By using the -contractive condition of T, we have

(2.18)

for all . Suppose that . Let . Since as , we can find such that

(2.19)

for all . Furthermore, we obtain that

(2.20)

for all . Also, as is a Cauchy sequence, there exists such that

(2.21)

for all . It follows from as that we can find such that

(2.22)

for all . Using (2.19)-(2.22), we get

(2.23)

for all . For , by (2.18) and the triangle inequality, we have

Letting in the above inequality, we get

This implies that , which is a contradiction. Therefore,, that is, . This completes the proof. □

Corollary 2.7Letbe a metric space andbe a strictly-contractive mapping. Suppose that thefollowing conditions hold:

(S₁) is anα-complete metric space;

() Tis an-admissible multi-valuedmapping;

(S₃) there existandsuch that;

() ifis a sequence inXwithasand, for all, then we have, for all.

ThenThas a fixed point.

Corollary 2.8 (Theorem 2.6 in [6])

Letbe a complete metric space andbe a strictly-contractive mapping. Suppose that thefollowing conditions hold:

(A₁) Tis an-admissible multi-valuedmapping;

(A₂) there existandsuch that;

() ifis a sequence inXwithasand, for all, then we have, for all.

ThenThas a fixed point.

Remark 2.9 Theorems 2.3 and 2.6 generalize many results in the followingsense:

• The condition (1.1) is weaker than some kinds of the contractiveconditions such as Banach’s contractive condition [10], Kannan’s contractive condition [11], Chatterjea’s contractive condition [12], Nadler’s contractive condition [13], etc.;

• the condition of being α-admissible of amulti-valued mapping T is weaker than the condition of being-admissible of T;

• for the existence of fixed point, we merely require thatα-continuity of T and α-completeness ofX, whereas other result demands stronger than these conditions.

Consequently, Theorems 2.3 and 2.6 extend and improve the following results:

• Theorem 2.5 and Theorem 2.6 of Ali et al.[6];

• Theorem 2.1 and Theorem 2.2 of Samet et al.[4];

• Theorem 2.3 of Asl et al.[5];

• Theorem 2.1 and Theorem 2.2 of Amiri et al.[14];

• Theorem 2.1 of Salimi et al.[15];

• Theorem 3.1 and Theorem 3.4 of Mohammadi et al.[7].

Next, we give an example to show that our result is more general than the results ofAli et al.[6] and many known results in the literature.

Example 2.10 Let and the metric defined by , for all . Define and by

and

Clearly, is not complete metric space. Therefore, the resultsof Ali et al.[6] are not applicable here.

Next, we show that by Theorem 2.6 can be guaranteed the existence of a fixedpoint of T. Define functions by and , for all . It is easy to see that and .

Firstly, we will show that T is a strictly -contractive mapping. For and , we have and then

It is to be observed that ψ is strictly increasing function.Therefore, T is a strictly -contractive mapping.

Moreover, it is easy to see that T is an α-admissiblemulti-valued mapping and there exists and such that

Also, T is an α-continuous mapping.

Finally, for each sequence in X with as and , for all , we have , for all . Thus the condition () in Theorem 2.6 holds.

Therefore, by using Theorem 2.3 or 2.6, we get T has a fixed point inX. In this case, T has infinitely fixed points such as−2, −1, and 0.

3 Consequences

3.1 Fixed point results in metric spaces endowed with an arbitrary binaryrelation

It has been pointed out in some studies that some results in metric spacesendowed with an arbitrary binary relation can be concluded from the fixed pointresults related with α-admissible mappings on metric spaces. Inthis section, we give some fixed point results on metric spaces endowed with anarbitrary binary relation which can be regarded as consequences of the resultspresented in the previous section. The following notions and definitions areneeded.

Let be a metric space and ℛ be a binaryrelation over X. Denote

i.e.,

Definition 3.1 Let X be a nonempty set and ℛ be a binaryrelation over X. A multi-valued mapping is said to be a weakly comparative iffor each and with , we have , for all .

Definition 3.2 Let be a metric space and ℛ be a binaryrelation over X. The metric space X is said to be-complete if and only if every Cauchy sequence in X with , for all , converges in X.

Definition 3.3 Let be a metric space and ℛ be a binaryrelation over X. We say that is a -continuousmapping to if for given and sequence with

Definition 3.4 Let be a metric space and ℛ be a binaryrelation over X. A mapping is called an -contractive mapping if there exist two functions and such that

(3.1)

where .

In the case when is strictly increasing, the-contractive mapping is called a strictly-contractive mapping.

Theorem 3.5Letbe a metric space, ℛ be a binary relation overXandbe a strictly-contractive mapping. Suppose thatthe following conditions hold:

(S₁) is an-complete metric space;

(S₂) Tis a weakly comparative mapping;

(S₃) there existandsuch that;

(S₄) Tis a-continuous multi-valued mapping.

ThenThas a fixed point.

Proof This result can be obtain from Theorem 2.3 by define amapping by

This completes the proof. □

By using Theorem 2.6, we get the following result.

Theorem 3.6Letbe a metric space, ℛ be a binary relation overXandbe a strictly-contractive mapping. Suppose thatthe following conditions hold:

(S₁) is an-complete metric space;

(S₂) Tis a weakly comparative mapping;

(S₃) there existandsuch that;

() ifis a sequence inXwithasand, for all, then we have, for all.

ThenThas a fixed point.

3.2 Fixed point results in metric spaces endowed with graph

In 2008, Jachymski [16] obtained a generalization of Banach’s contraction principle formappings on a metric space endowed with a graph. Afterwards, Dinevari and Frigon [17] extended some results of Jachymski [16] to multi-valued mappings. For more fixed point results on a metricspace with a graph, one can refer to [18–20].

In this section, we give fixed point results on a metric space endowed with agraph. Before presenting our results, we give the following notions anddefinitions.

Throughout this section, let be a metric space. A set is called a diagonal of the Cartesian product and is denoted by Δ. Consider a graphG such that the set of its vertices coincides with X and theset of its edges contains all loops, i.e.,. We assume G has no parallel edges, sowe can identify G with the pair . Moreover, we may treat G as a weightedgraph by assigning to each edge the distance between its vertices.

Definition 3.7 Let X be a nonempty set endowed with a graphG and be a multi-valued mapping, where X is anonempty set X. We say that Tweakly preserves edges if for each and with , we have , for all .

Definition 3.8 Let be a metric space endowed with a graphG. The metric space X is said to be -complete if and only if every Cauchysequence in X with , for all , converges in X.

Definition 3.9 Let be a metric space endowed with a graphG. We say that is an -continuous mapping to if for given and sequence with

Definition 3.10 Let be a metric space endowed with a graphG. A mapping is called an -contractive mapping if there exist two functions and such that

(3.2)

where .

In the case when is strictly increasing, the-contractive mapping is called a strictly-contractive mapping.

Theorem 3.11Letbe a metric space endowed with a graphG, andbe a strictly-contractive mapping. Suppose thatthe following conditions hold:

(S₁) is an-complete metric space;

(S₂) Tweakly preserves edges;

(S₃) there existandsuch that;

(S₄) Tis an-continuous multi-valuedmapping.

ThenThas a fixed point.

Proof This result can be obtained from Theorem 2.3 by defining amapping by

This completes the proof. □

By using Theorem 2.6, we get the following result.

Theorem 3.12Letbe a metric space endowed with a graphGandbe a strictly-contractive mapping. Suppose thatthe following conditions hold:

(S₁) is an-complete metric space;

(S₂) Tweakly preserves edges;

(S₃) there existandsuch that;

() ifis a sequence inXwithasand, for all, then we havefor all.

ThenThas a fixed point.

Remark 3.13

• 1. If we assume G is such that , then clearly G is connected and our Theorems 3.11 and 3.12 improve Nadler’s contraction principle [13] and in the case of a single-valued mapping, we improve Banach’s contraction principle [10], Kannan’s contraction theorem [11], Chatterjea’s contraction theorem [12], and Bianchini and Grandolfi’s fixed point theorem.

• 2. Theorems 3.11 and 3.12 are partial some generalized fixed point results endowed with a graph of Jachymski [16] and Dinevari and Frigon [17].

• 3. Theorems 3.11 and 3.12 are generalizations of fixed point results of Theorem 2.5 and Theorem 2.6 of Ali et al. [6] in a graph version.