Fixed point results for dominated mappings in rectangular $b

http://www.aimspress.com/journal/Math

DOI:10.3934/math.2020335 Received: 06 March 2020 Accepted: 08 June 2020 Published: 17 June 2020

Research article

Fixed point results for dominated mappings in rectangular b-metric spaces with applications

Abdullah Shoaib¹, Tahair Rasham², Giuseppe Marino³, Jung Rye Lee^4,∗ and Choonkil Park^5,∗

1 Department of Mathematics and Statistics, Riphah International University, Islamabad 44000, Pakistan

2 Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan

3 Dipartimento di Matematica e Informatica, Universita della Calabria, 87036, Arcavacata di Rende (CS), Italy

4 Department of Mathematics, Daejin University, Kyunggi 11159, Korea

5 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

* Correspondence: Email: jrlee@daejin.ac.kr; baak@hanyang.ac.kr.

Abstract: In this paper, we establish some fixed point results forα-dominated mappings fulfilling new generalized locally ´Ciri´c type rational contraction conditions in complete rectangularb-metric space.

As an application, we establish the existence of fixed point of-dominated mappings in an ordered complete rectangularb-metric space. The notion of graph dominated mappings is introduced. Fixed point results with graphic contractions for such mappings are established.

Keywords:fixed point; complete rectangularb-metric space;α-dominated mapping; ´Ciri´c type rational contraction condition; partial order;-dominated mapping; graph dominated mapping Mathematics Subject Classification:46S40, 47H10, 54H25

1. Introduction and preliminaries

Let W be a set and H : W −→ W be a mapping. A pointw ∈ W is called a fixed point of H if w = Hw. Fixed point theory plays a fundamental role in functional analysis (see [15]). Shoaib [17]

introduced the concept ofα-dominated mapping and obtained some fixed point results (see also [1, 2]).

Georgeet al.[11] introduced a new space and called it rectangularb-metric space (r.b.m. space). The triangle inequality in theb-metric space was replaced by rectangle inequality. Useful results onr.b.m.

spaces can be seen in ( [5, 6, 8–10]). ´Ciri´c introduced new types of contraction and proved some metrical fixed point results (see [4]). In this article, we introduce ´Ciri´c type rational contractions for

α-dominated mappings inr.b.m.spaces and proved some metrical fixed point results. New interesting results in metric spaces, rectangular metric spaces andb-metric spaces can be obtained as applications of our results.

Definition 1.1. [11] Let U be a nonempty set. A function dlb : U × U → [0,∞) is said to be a rectangularb-metric if there existsb≥1 such that

(i)dlb(θ, ν)=dlb(ν, θ);

(ii)dlb(θ, ν)=0 if and only ifθ =ν;

(iii)dlb(θ, ν) ≤ b[dlb(θ,q)+dlb(q,l)+dlb(l, ν)] for allθ, ν ∈ U and all distinct pointsq,l ∈U r{θ, ν}.

The pair (U,dlb) is said a rectangularb-metric space (in short,r.b.m.space) with coefficientb.

Definition 1.2. [11] Let (U,dlb) be anr.b.m.space with coefficientb.

(i) A sequence{θn}in (U,dlb) is said to be Cauchy sequence if for eachε >0, there correspondsn0 ∈N such that for alln,m≥n₀we havedlb(θm, θn)< εor lim_n,m→+∞dlb(θn, θm)= 0.

(ii) A sequence{θn}is rectangularb-convergent (for short, (dlb)-converges) toθif limn→+∞dlb(θn, θ)= 0.

In this caseθis called a (dlb)-limit of{θn}.

(iii) (U,dlb) is complete if every Cauchy sequence inU dlb-converges to a pointθ∈U.

Let $b, whereb ≥ 1, denote the family of all nondecreasing functions δb : [0,+∞) → [0,+∞) such that P₊∞

k=1b^kδ^k_b(t) < +∞ and bδb(t) < t for all t > 0, where δ^k_b is the k^th iterate of δb. Also bⁿ⁺¹δⁿ_b⁺¹(t)=bⁿbδb(δⁿ_b(t))< bⁿδⁿ_b(t).

Example 1.3. [11] Let U = N. Definedlb : U×U → R⁺∪ {0} such thatdlb(u,v) = dlb(v,u) for all u,v∈Uandα >0

dlb(u,v)=























0, ifu= v;

10α, ifu=1, v=2;

α, ifu∈ {1,2}andv∈ {3};

2α, ifu∈ {1,2,3}andv∈ {4};

3α, ifuorv<{1,2,3,4}andu,v.

Then (U,dlb) is anr.b.m.space withb=2> 1.Note that

d(1,4)+d(4,3)+d(3,2)= 5α <10α= d(1,2).

Thusdlb is not a rectangular metric.

Definition 1.4. [17] Let (U,dlb) be anr.b.m. space with coefficientb. LetS : U → U be a mapping andα:U×U → [0,+∞). IfA⊆U,we say that theS isα-dominated onA,wheneverα(i,S i)≥ 1 for alli∈A.IfA=U, we say thatS isα-dominated.

Forθ, ν∈U,a>0,we defineDlb(θ, ν) as

Dlb(θ, ν)= max{dlb(θ, ν), dlb(θ,Sθ).dlb(ν,Sν)

a+dlb(θ, ν) ,dlb(θ,Sθ),dlb(ν,Sν)}.

2. Main result

Now, we present our main result.

Theorem 2.1. Let (U,dlb) be a complete r.b.m. space with coefficient b, α : U ×U → [0,∞), S : U → U,{θn}be a Picard sequence and S be aα-dominated mapping on{θn}.Suppose that, for some δb ∈$b, we have

dlb(Sθ,Sν)≤δb(Dlb(θ, ν)), (2.1) for all θ, ν ∈ {θn} with α(θ, ν) ≥ 1. Then {θn} converges to θ^∗ ∈ U. Also, if (2.1) holds for θ^∗ and α(θn, θ^∗)≥1for all n∈N∪ {0}, then S has a fixed pointθ^∗in U.

Proof. Letθ₀ ∈Ube arbitrary. Define the sequence{θn}byθn+1 =Sθnfor alln∈N∪{0}.We shall show that{θn}is a Cauchy sequence. Ifθn = θn+1,for somen∈N,thenθnis a fixed point ofS. So, suppose that any two consecutive terms of the sequence are not equal. SinceS : U → U be an α-dominated mapping on{θn},α(θn,Sθn)≥1 for alln∈N∪ {0}and thenα(θn, θn+1)≥1 for alln∈N∪ {0}.Now by using inequality (2.1), we obtain

dlb(θn+1, θn+2) = dlb(Sθn,Sθn+1)≤δb(Dlb(θn, θn+1))

≤ δb(max{dlb(θn, θn+1),dlb(θn, θn+1).dlb(θn+1, θn+2) a+dlb(θn, θn+1) , dlb(θn, θn+1),dlb(θn+1, θn+2)})

≤ δb(max{dlb(θn, θn+1),dlb(θn+1, θn+2)}).

If max{dlb(θn, θn+1),dlb(θn+1, θn+2)}= dlb(θn+1, θn+2),then

dlb(θn+1, θn+2) ≤ δb(dlb(θn+1, θn+2))

≤ bδb(dlb(θn+1, θn+2)).

This is the contradiction to the fact thatbδb(t)<tfor allt>0.So

max{dlb(θn, θn+1),dlb(θn+1, θn+2)}=dlb(θn, θn+1).

Hence, we obtain

dlb(θn+1, θn+2)≤ δb(dlb(θn, θn+1))≤δ²_b(dlb(θn−1, θn)) Continuing in this way, we obtain

dlb(θn+1, θn+2)≤δⁿ_b⁺¹(dlb(θ0, θ1)). (2.2) Suppose for somen,m∈Nwithm> n,we haveθn= θm. Then by (2.2)

dlb(θn, θn+1) = dlb(θn,Sθn)= dlb(θm,Sθm)=dlb(θm, θm+1)

≤ δ^m−n_b (dlb(θn, θn+1))< bδb(dlb(θn, θn+1))

As dlb(θn, θn+1) > 0, so this is not true, because bδb(t) < t for all t > 0. Therefore, θn , θm for any n,m ∈N. Since P₊∞

k=1b^kδ^k_b(t) < +∞, for someν ∈ N, the seriesP₊∞

k=1b^kδ^k_b(δ^ν−_b ¹(dlb(θ₀, θ₁))) converges.

Asbδb(t)< t,so

bⁿ⁺¹δⁿ_b⁺¹(δ^ν−_b ¹(dlb(θ₀, θ₁)))< bⁿδⁿ_b(δ^ν−_b ¹(dlb(θ₀, θ₁))), for alln∈N.

Fixε >0.Then ^ε₂ =ε⁰ >0.Forε⁰,there existsν(ε⁰)∈Nsuch that

bδb(δ^ν(ε_b ⁰⁾⁻¹(dlb(θ0, θ1)))+b²δ²_b(δ^ν(ε_b ⁰⁾⁻¹(dlb(θ0, θ1)))+· · · < ε⁰ (2.3) Now, we suppose that any two terms of the sequence{θn}are not equal. Letn,m ∈ N withm > n >

ν(ε⁰).Now, ifm>n+2,

dlb(θn, θm) ≤ b[dlb(θn, θn+1)+dlb(θn+1, θn+2)+dlb(θn+2, θm)]

≤ b[dlb(θn, θn+1)+dlb(θn+1, θn+2)]+b²[dlb(θn+2, θn+3) +dlb(θn+3, θn+4)+dlb(θn+4, θm)]

≤ b[δⁿ_b(dlb(θ₀, θ₁))+δⁿ_b⁺¹(dlb(θ₀, θ₁))]+b²[δⁿ_b⁺²(dlb(θ₀, θ₁))

+δⁿ_b⁺³(dlb(θ₀, θ₁))]+b³[δⁿ_b⁺⁴(dlb(θ₀, θ₁))+δⁿ_b⁺⁵(dlb(θ₀, θ₁))]+· · ·

≤ bδⁿ_b(dlb(θ₀, θ₁))+b²δⁿ_b⁺¹(dlb(θ₀, θ₁))+b³δⁿ_b⁺²(dlb(θ₀, θ₁))+· · ·

= bδb(δⁿ⁻¹_b (dlb(θ₀, θ₁)))+b²δ²_b(δⁿ⁻¹_b (dlb(θ₀, θ₁)))+· · · . By using (2.3), we have

dlb(θn, θm)

< bδb(δ^ν(ε_b ⁰⁾⁻¹(dlb(θ₀, θ₁)))+b²δ²_b(δ^ν(ε_b ⁰⁾⁻¹(dlb(θ₀, θ₁)))+· · ·< ε⁰ < ε.

Now, ifm= n+2, then we obtain dlb(θn, θn+2)

≤ b[dlb(θn, θn+1)+dlb(θn+1, θn+3)+dlb(θn+3, θn+2)]

≤ b[dlb(θn, θn+1)+b[dlb(θn+1, θn+2)+dlb(θn+2, θn+4)+dlb(θn+4, θn+3)]

+dlb(θn+3, θn+2)]

≤ bdlb(θn, θn+1)+b²dlb(θn+1, θn+2)+bdlb(θn+2, θn+3)+b²dlb(θn+3, θn+4) +b³[dlb(θn+2, θn+3)+dlb(θn+3, θn+5)+dlb(θn+5, θn+4)]

≤ bdlb(θn, θn+1)+b²dlb(θn+1, θn+2)+ b+b³

dlb(θn+2, θn+3)+b²dlb(θn+3, θn+4) +b³dlb(θn+5, θn+4)+b⁴[dlb(θn+3, θn+4)+dlb(θn+4, θn+6)+dlb(θn+6, θn+5)]

≤ bdlb(θn, θn+1)+b²dlb(θn+1, θn+2)+ b+b³

dlb(θn+2, θn+3) +

b²+b⁴

dlb(θn+3, θn+4)+b³dlb(θn+5, θn+4)+b⁴dlb(θn+6, θn+5) +b⁵[dlb(θn+4, θn+5)+dlb(θn+5, θn+7)+dlb(θn+7, θn+6)]

≤ bdlb(θn, θn+1)+b²dlb(θn+1, θn+2)+ b+b³

dlb(θn+2, θn+3) +

b²+b⁴

dlb(θn+3, θn+4)+

b³+b⁵

dlb(θn+4, θn+5)+· · ·

< 2[bdlb(θn, θn+1)+b²dlb(θn+1, θn+2)+b³dlb(θn+2, θn+3) +b⁴dlb(θn+3, θn+4)+b⁵dlb(θn+4, θn+5)+· · ·]

≤ 2[bδⁿ_b(dlb(θ₀, θ₁))+b²δⁿ_b⁺¹(dlb(θ₀, θ₁))+b³δⁿ_b⁺²(dlb(θ₀, θ₁))+· · ·]

< 2[bδb(δ^ν(ε_b ⁰⁾⁻¹(dlb(θ₀, θ₁)))+b²δ²_b(δ^ν(ε_b ⁰⁾⁻¹(dlb(θ₀, θ₁)))+· · ·]<2ε⁰= ε.

It follows that

n,m→lim+∞dlb(θn, θm)=0. (2.4) Thus{θn}is a Cauchy sequence in (U,dlb). As (U,dlb) is complete, so there existsθ^∗inU such that{θn} converges toθ^∗,that is,

n→lim+∞dlb(θn, θ^∗)= 0. (2.5) Now, suppose thatdlb(θ^∗,Sθ^∗)>0. Then

dlb(θ^∗,Sθ^∗) ≤ b[dlb(θ^∗, θn)+dlb(θn, θn+1)+dlb(θn+1,Sθ^∗)

≤ b[dlb(θ^∗, θn+1)+dlb(θn, θn+1)+dlb(Sθn,Sθ^∗).

Sinceα(θn, θ^∗)≥ 1,we obtain

dlb(θ^∗,Sθ^∗) ≤ bdlb(θ^∗, θn+1)+bdlb(θn, θn+1)+bδb(max{dlb(θn, θ^∗), dlb(θ^∗,Sθ^∗).dlb(θn, θn+1)

a+dlb(θn, θ^∗) , dlb(θn, θn+1)dlb(θ^∗,Sθ^∗)}).

Lettingn→ +∞, and using the inequalities (2.4) and (2.5), we obtaindlb(θ^∗,Sθ^∗) ≤ bδb(dlb(θ^∗,Sθ^∗)).

This is not true, becausebδb(t)< tfor allt >0 and hencedlb(θ^∗,Sθ^∗) = 0 orθ^∗ = Sθ^∗. HenceS has a

fixed pointθ^∗inU.

Remark 2.2. By taking fourteen different proper subsets of Dlb(θ, ν), we can obtainvnew results as corollaries of our result in a completer.b.m.space with coefficientb.

We have the following result without usingα-dominated mapping.

Theorem 2.3. Let(U,dlb)be a complete r.b.m. space with coefficient b, S : U → U,{θn}be a Picard sequence. Suppose that, for someδb ∈$b, we have

dlb(Sθ,Sν)≤δb(Dlb(θ, ν)) (2.6) for allθ, ν∈ {θn}. Then{θn}converges toθ^∗ ∈U.Also, if (2.6) holds forθ^∗, then S has a fixed pointθ^∗ in U.

We have the following result by takingδb(t)=ct,t∈R⁺with 0<c< ¹_b without usingα-dominated mapping.

Theorem 2.4. Let(U,dlb)be a complete r.b.m. space with coefficient b, S : U → U,{θn}be a Picard sequence. Suppose that, for some0<c< ¹_b, we have

dlb(Sθ,Sν)≤ c(Dlb(θ, ν)) (2.7)

for allθ, ν∈ {θn}. Then{θn}converges toθ^∗ ∈U.Also, if (2.7) holds forθ^∗, then S has a fixed pointθ^∗ in U.

Ran and Reurings [16] gave an extension to the results in fixed point theory and obtained results in partially ordered metric spaces. Arshad et al.[3] introduced -dominated mappings and established some results in an ordered complete dislocated metric space. We apply our result to obtain results in ordered completer.b.m.space.

Definition 2.5. (U,,dlb) is said to be an ordered completer.b.m.space with coefficientbif (i) (U,) is a partially ordered set.

(ii) (U,dlb) is anr.b.m.space.

Definition 2.6. [3] LetU be a nonempty set, is a partial order on θ.A mapping S : U → U is said to be-dominated onA ifa S afor each a ∈ A ⊆ θ.If A = U, thenS : U → U is said to be -dominated.

We have the following result for-dominated mappings in an ordered completer.b.m.space with coefficientb.

Theorem 2.7. Let(U,,dlb)be an ordered complete r.b.m. space with coefficient b, S : U → U, {θn} be a Picard sequence and S be a-dominated mapping on{θn}.Suppose that, for someδb ∈ $b, we have

dlb(Sθ,Sν)≤δb(Dlb(θ, ν)), (2.8) for allθ, ν ∈ {θn}withθ ν. Then{θn}converges toθ^∗ ∈U. Also, if (2.8) holds forθ^∗andθn θ^∗for all n∈N∪ {0}. Then S has a fixed pointθ^∗in U.

Proof. Letα: U×U → [0,+∞) be a mapping defined byα(θ, ν) = 1 for allθ, ν ∈U withθ νand α(θ, ν) = ₁₁⁴ for all other elementsθ, ν∈U. AsS is the dominated mappings on{θn}, soθ Sθfor all θ∈ {θn}.This implies thatα(θ,Sθ)= 1 for allθ∈ {θn}.SoS :U → U is theα-dominated mapping on {θn}.Moreover, inequality (2.8) can be written as

dlb(Sθ,Sν)≤δb(Dlb(θ, ν))

for all elementsθ, νin{θn}withα(θ, ν) ≥ 1. Then, as in Theorem 2.1,{θn}converges to θ^∗ ∈ U.Now, θn θ^∗ impliesα(θn, θ^∗) ≥ 1. So all the conditions of Theorem 2.1 are satisfied. Hence, by Theorem

2.1,S has a fixed pointθ^∗inU.

Now, we present an example of our main result. Note that the results of Georgeet al. [11] and all other results in rectangularb-metric space are not applicable to ensure the existence of the fixed point of the mapping given in the following example.

Example 2.8. LetU = A∪B,whereA={¹_n :n∈ {2,3,4,5}}andB= [1,∞].Definedl :U×U →[0,∞) such thatdl(θ, ν)= dl(ν, θ) forθ, ν∈U and



















dl(¹₂, ¹₃)= dl(¹₄, ¹₅)= 0.03 dl(¹₂, ¹₅)= dl(¹₃, ¹₄)= 0.02 dl(¹₂,¹₄)=dl(¹₅,¹₃)=0.6 dl(θ, ν)=|θ−ν|² otherwise

be a complete r.b.m. space with coefficient b = 4 > 1 but (U,dl) is neither a metric space nor a rectangular metric space. Takeδb(t)= ₁₀^t ,thenbδb(t)<t.LetS :U →U be defined as

Sθ =















1

5 ifθ∈A

1

3 ifθ= 1

9θ¹⁰⁰+85 otherwise.

Letθ₀ =1.Then the Picard sequence{θn}is{1, ¹₃, ¹₅,¹₅,¹₅,· · · }. Define α(θ, ν)=

( ₈

5 ifθ, ν∈ {θn}

4

7 otherwise.

ThenS is anα-dominated mapping on{θn}.Now,S satisfies all the conditions of Theorem 2.1. Here ¹₅ is the fixed point inU.

3. Fixed point results for graphic contractions

Jachymski [13] proved the contraction principle for mappings on a metric space with a graph. Let (U,d) be a metric space and4 represents the diagonal of the cartesian product U ×U. Suppose that G be a directed graph having the vertices setV(G) along with U,and the set E(G) denoted the edges of U included all loops, i.e., E(G) ⊇ 4. If G has no parallel edges, then we can unify G with pair (V(G),E(G)). Iflandmare the vertices in a graphG,then a path inGfromltomof lengthN(N ∈N) is a sequence{θi}^N_i=oofN+1 vertices such thatlo =l,lN =mand (l_n−1,ln)∈ E(G) wherei=1,2,· · ·N (see for detail [7, 8, 12, 14, 18, 19]). Recently, Younis et al. [20] introduced the notion of graphical rectangularb-metric spaces (see also [5, 6, 21]). Now, we present our result in this direction.

Definition 3.1. Let θ be a nonempty set andG = (V(G),E(G)) be a graph such thatV(G) = U and A⊆ U. A mappingS :U → Uis said to be graph dominated onAif (θ,Sθ)∈E(G) for allθ∈A.

Theorem 3.2. Let(U,dlb)be a complete rectangular b-metric space endowed with a graph G,{θn}be a Picard sequence and S :U →U be a graph dominated mapping on{θn}. Suppose that the following hold:

(i) there existsδb ∈$bsuch that

dlb(Sθ,Sν)≤δb(Dlb(θ, ν)), (3.1) for allθ, ν ∈ {θn}and (θn, ν) ∈ E(G). Then(θn, θn+1) ∈ E(G) and{θn}converges to θ^∗. Also, if (3.1) holds forθ^∗and(θn, θ^∗)∈E(G)for all n∈N∪ {0}, then S has a fixed pointθ^∗in U.

Proof. Defineα:U×U → [0,+∞) by α(θ, ν)=

( 1, ifθ, ν∈U, (θ, ν)∈E(G)

1

4, otherwise.

SinceS is a graph dominated on{θn},forθ∈ {θn},(θ,Sθ)∈E(G). This implies thatα(θ,Sθ)=1 for all θ∈ {θn}.SoS :U →U is anα-dominated mapping on{θn}.Moreover, inequality (3.1) can be written as

dlb(Sθ,Sν)≤δb(Dlb(θ, ν)),

for all elementsθ, ν in {θn} withα(θ, ν) ≥ 1. Then, by Theorem 2.1, {θn} converges to θ^∗ ∈ U. Now, (θn, θ^∗)∈E(G) implies thatα(θn, θ^∗)≥ 1.So all the conditions of Theorem 2.1 are satisfied. Hence, by

Theorem 2.1,S has a fixed pointθ^∗inU.

Acknowledgments

The authors would like to thank the Editor, the Associate Editor and the anonymous referees for sparing their valuable time for reviewing this article. The thoughtful comments of reviewers are very useful to improve and modify this article.

Conflict of interest

The authors declare that they have no competing interests.

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