3. Common Fixed Point Theorems for Multivalued Mappings on Metric Spaces with a Directed Graph

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Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 10 Issue 1(2018), Pages 26-37.

COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS ON METRIC SPACES WITH A DIRECTED GRAPH

JAMSHAID AHMAD,ABDULLAH EQAL AL-MAZROOEI

Abstract. The purpose of this article is to establish the existence of common fixed points of multivalued Θ-contraction mappings on a metric space endowed with a graph. Presented theorems are generalizations of recent fixed point theorems due to Hussain et al. [Fixed Point Theory and Applications (2015) 2015:185]. An example is also given to support our generalized result.

1. _{Introduction and Preliminaries}

The Banach Contraction Principle is one of the cornerstones in the development of Nonlinear Analysis, in general, and metric fixed point theory, in particular. The method of successive approximation introduced by Liouville in 1837 and systemati-cally developed by Picard in 1890 culminated in formulation of Banach Contraction Principle by Polish Mathematician Stefan Banach in 1922. This theorem provides an illustration of the unifying power of functional analytic methods and usefulness of fixed point theory in analysis. Extensions of the Banach contraction principle have been obtained either by generalizing the domain of the mapping or by extending the contractive condition on the mappings see [2, 3, 4, 13, 14, 15, 16, 17, 22, 23, 24]. Very recently, Jleli and Samet [18] introduced a new type of contraction and established some new fixed point theorems for such contraction in the context of generalized metric spaces.

Definition 1.1. Let Θ : (0,∞)→(1,∞)be a function satisfying:

(Θ1) Θ is nondecreasing;

(Θ2) for each sequence{αn} ⊆R+, limn→∞Θ(αn) = 1 if and only if limn→∞(αn) =

0;

(Θ3) there exists 0< k <1 andl∈(0,∞] such that lima→0+Θ(α_α)_k−1 =l;

A mappingT :X →X is said to be Θ-contraction if there exist the function Θ satisfying (Θ1)-(Θ3) and a constantα∈(0,1) such that for allx, y∈X,

d(T x, T y)6= 0 =⇒Θ(d(T x, T y))≤[Θ(d(x, y))]α_. _(1.1)

Theorem 1.2. [18] Let (X, d) be a complete metric space and T : X → X be a

Θ-contraction, thenT has a unique fixed point.

2000Mathematics Subject Classification. Primary 47H10; Secondary 54H25. Key words and phrases. Fixed point, Θ-contractions, metric spaces.

c

2018 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted February 20, 2017. Published April 30, 2017. Communicated by N. Hussain.

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To be consistent with Samet et al. [18], we denote by Ψ the set of all functions Θ : (0,∞)→(1,∞) satisfying the above conditions(Θ1−Θ3).

Hussain et al. [12] modified and extended the above family Θ of functions Θ : (0,∞) → (1,∞) and proved the following fixed point theorem for Θ-contractive condition in the setting of complete metric spaces.

(Θ0₁) Θ is nondecreasing and Θ(t) = 1 if and only ift= 0; (Θ4) Θ(a+b)≤Θ(a) + Θ(b) for alla, b >0.

To be consistent with Hussain et al. [12], we denote by Ω the set of all functions Θ : (0,∞)→(1,∞) satisfying the conditions Θ01 and (Θ2−Θ4).

Theorem 1.3. [12] Let (X, d) be a complete metric space and T : X → X be a self-mapping. If there exists a function Θ∈Ωand positive real numbers k1, k2, k3

andk4 with0≤k1+k2+k3+ 2k4<1 such that

Θ(d(T x, T y))≤[Θ(d(x, y))]k1_·_Θ(_d₍_{x, T x}_))]k2_·_[Θ(_d₍_{y, T y}_))]k3_·_[Θ((_d₍_{x, T y}_{) +}_d₍_{y, T x}_))]k4

(1.2)

for allx, y∈X, thenT has a unique fixed point.

For more details on Θ−contractions, we refer the reader to [5, 26, 33].

One of the generalization of the domain of the mapping is partially ordered metric spaces was first investigated in 2004 by Ran and Reurings [31], and then by Nieto and Lopez [29].

To extend this concept, Jachymski [20] introduced a new approach in metric fixed point theory by replacing order structure with a graph structure on a metric space. In this way, the results obtained in ordered metric spaces are generalized (see also [19] and the reference therein); in fact, Gwodzdz-lukawska and Jachymski [11] developed the Hutchinson-Barnsley theory for finite families of mappings on a metric space endowed with a directed graph.

Consistent with Jachymski [19], let (X, d) be a metric space and ∆ denotes the diagonal ofX×X. LetGbe a directed graph such that the setV(G) of its vertices coincides withXandE(G) be the set of edges of the graph which contains all loops, that is, ∆⊆E(G). LetE∗(G) denotes the set of all edges ofGthat are not loops i.e.,E∗(G) =E(G)−∆. Also assume that the graphGhas no parallel edges and, thus one can identifyGwith the pair (V(G), E(G)).

Definition 1.4. [19] An operator T :X →X is called a BanachG-contraction or simply aG-contraction if

(a) T preserves edges of G; for each x, y ∈ X with (x, y) ∈ E(G), we have (T(x), T(y))∈E(G),

(b) T decreases weights of edges ofG; there existsα∈(0,1) such that for all x, y∈X with (x, y)∈E(G), we have d(T(x), T(y))≤αd(x, y).

Ifx andy are vertices ofG, then a (directed) path inG fromx toy of length k∈Nis a finite sequence{xn} (n∈ {0,1,2, ..., k} ) of vertices such that x0=x,

xk=y and (xi−1, xi)∈E(G) fori∈ {1,2, ..., k}.

Notice that a graph G is connected if there is a (directed) path between any two vertices and it is weakly connected if Ge is connected, where Ge denotes the

undirected graph obtained fromG by ignoring the direction of edges. Denote by G−1 _{the graph obtained from}_G_{by reversing the direction of edges. Thus,}

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It is more convenient to treatGe as a directed graph for which the set of its edges

is symmetric, under this convention; we have that

E(Ge) =E(G)∪E(G−1).

IfGis such thatE(G) is symmetric, then forx∈V(G), [x]Gdenotes the equivalence

class of the relationRdefined onV(G) by the rule:

yRz if there is a path inGfromy to z.

IfT :X →X is an operator. Set

XT :={x∈X: (x, T(x))∈E(G)}.

Jachymski [20] used the following property:

(P) : for any sequence{xn}in X, ifxn→xas n→ ∞and (xn, xn+1)∈E(G),

then (xn, x)∈E(G).

Theorem 1.5. [20]Let (X, d)be a complete metric space and letGbe a directed graph such that V(G) = X. Let E(G) and the triplet (X, d, G) has property (P) andT :X →X aG-contraction. Then the following statements hold:

(1) T has a fixed point if and only ifXT 6=∅;

(2) if XT 6=∅ andGis weakly connected, thenT is a Picard operator;

(3) for anyx∈XT,T |[x]

e

G is a Picard operator; (4) if XT ⊆E(G), thenT is a weakly Picard operator.

For detailed discussion on Picard operators, we refer the reader to Berinde et. al [7, 8].

Latif and Beg [25] introduced a notion of K− multivalued mapping as an ex-tension of Kannan mapping to multivalued mappings. Rus [32] coined the term R−multivalued mapping which is a generalization of aK−multivalued mapping. Abbas and Rhoades [1] introduced the notion of a generalizedR−multivalued map-pings, which in turn generalize R−multivalued mappings, and obtained common fixed point results for such mappings.

Let (X, d) be a metric space. Denote byP(X) the family of all nonempty subsets ofX, byPcl(X) the family of all nonempty closed subset ofX.

A pointxinX is a fixed point of a multivalued mappingT:X→P(X) iffx∈T x. The set of all fixed points of multivalued mappingT is denoted by F ix(T).

Suppose thatT1, T2:X →Pcl(X). Set

XT1,T2 :={x∈X : (x, ux)∈E(G) whereux∈T1(x)∩T2(x)}.

A mappingT :X→Pcl(X) is said to be upper semicontinuous, if forxn∈X and yn ∈T xn withxn→x0andyn→y0, impliesy0∈T x0.

A clique in an undirected graphG= (V, E) is a subset of the vertex setW ⊂V, such that for every two vertices inW, there exists an edge connecting the two. This is equivalent to saying that the subgraph induced by W is complete i.e., for every x, y∈W(G), we have (x, y)∈E(G).

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2. Main Results

Definition 2.1. Let T1, T2 : X →Pcl(X) be two multivalued mappings. Suppose that for every vertexxinGand for everyux∈Ti(x), i∈ {1,2} we have(x, ux)∈ E(G). A pair (T1, T2) is said to be a graphic Θ−contraction if there exist some

Θ ∈ Ω and for any x, y ∈ X with (x, y) ∈ E(G) and ux ∈ Ti(x), there exists uy∈Tj(y)fori, j∈ {1,2} withi6=j such that(ux, uy)∈E∗(G)and

Θ (d(ux, uy))≤[Θ(d(x, y))]k1[Θ(d(x, ux))]k2[Θ(d(y, uy))]k3[Θ(d(x, uy)+d(y, ux))]k4

(2.1)

hold, wherek1, k2, k3 andk4 are non negative real numbers such that0≤k1+k2+

k3+ 2k4<1.

Now we state our main theorem.

Theorem 2.2. Let(X, d)be a complete metric space endowed with a directed graph

G such thatV(G) =X and E(G)⊇∆. If mappings T1, T2 :X →Pcl(X) form a graphicΘ-contraction pair, then following statement hold:

(i). F ix(T1)6=∅ orF ix(T2)6=∅ if and only ifF ix(T1) =F ix(T2)6=∅.

(ii). XT1,T2 6=∅ provided thatF ix(T1)∩F ix(T2)6=∅.

(iii). If XT1,T2 6=∅ and G is weakly connected, then F ix(T1) = F ix(T2) 6= ∅

provided that either (a) T1 or T2 is upper semicontinuous, or (b) Θ is

continuous,T1or T2 is bounded andGhas property (P).

(iv). F ix(T1)∩F ix(T2) is a clique of Ge if and only ifF ix(T1)∩F ix(T2) is a

singleton.

Proof. To prove (i), let x∗ ∈ T1(x∗). Assume x∗ ∈/ T2(x∗), then since the pair

(T1, T2) form a graphic Θ-contraction, so there exists anx∈T2(x∗) with (x∗, x)∈

E∗(G) such that

Θ (d(x∗, x)) ≤ [Θ(d(x∗, x∗))]k1_[Θ(_d₍_x∗_{, x}∗_))]k2_[Θ(_d₍_{x, x}∗_))]k3_[Θ(_d₍_x∗_{, x}_{) +}_d₍_x∗_{, x}∗_))]k4

= [Θ(d(x, x∗))]k3+k4

a contradiction becausek3+k4<1.Hencex∗∈T2(x∗) and soF ix(T1)⊆F ix(T2).

Similarly, F ix(T2) ⊆ F ix(T1) and therefore F ix(T1) = F ix(T2). Also, if x∗ ∈

T2(x∗), then we havex∗∈T1(x∗).The converse is straightforward.

To prove (ii), let F ix(T1)∩F ix(T2) 6= ∅. Then there exists x ∈ X such that

x∈T1(x)∩T2(x). As ∆⊆E(G), we conclude thatXT1,T2 6=∅.

To prove (iii), suppose that x0 is an arbitrary point of X. If x0 ∈ T1(x0) or

x0 ∈ T2(x0), then the proof is finished. So we assume that x0 ∈/ Ti(x0) for

i ∈ {1,2}. Now for i, j ∈ {1,2} with i 6= j, if x1 ∈ Ti(x0), then there exists

x2∈Tj(x1) with (x1, x2)∈E∗(G) such that

1 < Θ (d(x1, x2))≤[Θ(d(x0, x1))]k1[Θ(d(x0, x1))]k2[Θ(d(x1, x2))]k3[Θ(d(x0, x2) +d(x1, x1))]k4

≤ [Θ(d(x0, x1))]k1[Θ(d(x0, x1))]k2[Θ(d(x1, x2))]k3[Θ(d(x0, x1) +d(x1, x2))]k4

Using (Θ4), then we have

Θ (d(x1, x2)) ≤ [Θ(d(x0, x1))]k1[Θ(d(x0, x1))]k2[Θ(d(x1, x2))]k3[Θ(d(x0, x1)]k4[Θ(d(x1, x2))]k4

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Therefore, we write

1 < Θ (d(x1, x2))≤[Θ(d(x0, x1))]

k1 +k2 +k4 1−k3−k4

= [Θ(d(x0, x1))]λ. (2.2)

Similarly, for the pointx2inTj(x1),there existsx3∈Ti(x2) with (x2, x3)∈E∗(G)

such that

1 < Θ (d(x2, x3))≤[Θ(d(x1, x2))]k1[Θ(d(x1, x2))]k2[Θ(d(x2, x3))]k3[Θ(d(x1, x3) +d(x2, x2))]k4

≤ [Θ(d(x1, x2))]k1[Θ(d(x1, x2))]k2[Θ(d(x2, x3))]k3[Θ(d(x1, x2) +d(x2, x3))]k4

1 < Θ (d(x2, x3))≤[Θ(d(x1, x2))]k1[Θ(d(x1, x2))]k2[Θ(d(x2, x3))]k3[Θ(d(x1, x2)]k4[Θ(d(x2, x3))]k4

= [Θ(d(x1, x2))]k1+k2+k4[Θ(d(x2, x3))]k3+k4.

Therefore, we write

1 < Θ (d(x2, x3))≤[Θ(d(x1, x2))]

k1 +k2 +k4 1−k3−k4

= [Θ(d(x1, x2))]λ. (2.3)

Continuing this way, for x2n ∈ Tj(x2n−1), there exist x2n+1 ∈ Ti(x2n) with

(x2n, x2n+1)∈E∗(G) such that

1<Θ (d(x2n, x2n+1))≤[Θ (d(x2n−1, x2n))]λ. (2.4)

In a similar manner, forx2n+1 ∈Tj(x2n), there existx2n+2∈Ti(x2n+1) such that

for (x2n+1, x2n+2)∈E∗(G) implies

1<Θ (d(x2n+1, x2n+2))≤[Θ (d(x2n, x2n+1))]λ. (2.5)

Hence from (2.4) and (2.5), we obtain a sequence {xn} in X such that for xn ∈ Tj(xn−1), there existxn+1∈Ti(xn) with (xn, xn+1)∈E∗(G) and it satisfies

1<Θ (d(xn, xn+1))≤[Θ (d(xn−1, xn))]λ. (2.6)

Therefore

1 < Θ (d(xn, xn+1))≤[Θ (d(xn−1, xn))]λ

≤ [Θ (d(xn−2, xn−1))]λ 2

· · ·

≤ [Θ (d(x0, x1))]λ

n

From (2.7), we obtain lim

n→∞Θ (d(xn, xn+1)) = 1 that together with (Θ2) gives

lim

n→∞d(xn, xn+1) = 0. (2.7)

From the condition (Θ3), there exist 0< h <1 andl∈(0,∞] such that

lim

n→∞

Θ(d(xn, xn+1))−1

d(xn, xn+1)h

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Suppose that l <∞.In this case, let B= ₂l >0. From the definition of the limit, there existsn1∈Nsuch that

|Θ(d(xn, xn+1))−1 d(xn, xn+1)h

−l| ≤B

for alln > n1.This implies that

Θ(d(xn, xn+1))−1

d(xn, xn+1)h

≥l−B = l 2 =B

for alln > n1.Then

nd(xn, xn+1)h≤An[Θ(d(xn, xn+1))−1] (2.8)

for all n > n1, where A = _B1. Now we suppose that l = ∞. Let B > 0 be an

arbitrary positive number. From the definition of the limit, there exists n1 ∈ N such that

B ≤Θ(d(xn, xn+1))−1 d(xn, xn+1)h

for alln > n1.This implies that

nd(xn, xn+1)h≤An[Θ(d(xn, xn+1))−1]

for alln > n1,whereA= _B1.Thus, in all cases, there existA >0 andn1∈Nsuch that

nd(xn, xn+1)h≤An[Θ(d(xn, xn+1))−1] (2.9)

for alln > n1.Thus by (2.7) and (2.10), we get

nd(xn, xn+1)h≤An([(Θd(x0, x1))]h

n

−1). (2.10)

Lettingn→ ∞in the above inequality, we obtain

lim

n→∞nd(xn, xn+1) h_{= 0}_.

Thus, there existsn2∈Nsuch that

d(xn, xn+1)≤

1

n1/h (2.11)

for alln > n2.Now we prove that{xn} is a Cauchy sequence. Form > n > n2 we

have,

d(xn, xm)≤ m−1

X

i=n

d(xi, xi+1)≤

m−1 X

i=n

1

i1/h. (2.12)

By the convergence of the series P∞

i=1 1

i1/h, we getd(xn, xm)→0 asn, m → ∞. Therefore {xn} is a Cauchy sequence in X. Since X is complete, there exists an

elementx∗∈X such that xn→x∗ asn→ ∞.

Now, if Ti is upper semicontinuous, then as x2n ∈ X, x2n+1 ∈ Ti(x2n) with x2n → x∗ and x2n+1 → x∗ as n → ∞ implies that x∗ ∈ Ti(x∗). Using (i), we

getx∗∈Ti(x∗) =Tj(x∗).Similarly the result hold when Tj is upper

semicontin-uous.

Suppose that Θ is continuous. Sincex2nconverges tox∗asn→ ∞and (x2n, x2n+1)∈

E(G), we have (x2n, x∗)∈E(G).For x2n ∈Tj(x2n−1),there existsun∈Ti(x∗)

such that (x2n, un)∈E∗(G).As{un}is bounded, lim sup n→∞

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u∗both exist. Assume that u∗6=x∗.Since (T1, T2) is a graphic Θ−contraction, so

we have

1 < Θ (d(x2n, un))≤[Θ(d(x2n−1, x∗))]k1[Θ(d(x2n−1, x2n))]k2

×[Θ(d(x∗, un))]k3[Θ(d(x2n−1, un) +d(x∗, x2n))]k4.

1 < Θ (d(x2n, un))≤[Θ(d(x2n−1, x∗))]k1[Θ(d(x2n−1, x2n))]k2

×[Θ(d(x∗, un))]k3[Θ(d(x2n−1, un)]k4[Θ(d(x∗, x2n))]k4.

On taking lim sup

n→∞

and using the continuity of Θ,we get

Θ (d(x∗, u∗))≤[Θ(d(x∗, u∗))]k3+k4_,

a contradiction becausek3+k4 <1. Hence u∗ =x∗. Similarly, taking the lim inf

n→∞ givesu∗=x∗. Sinceun ∈Ti(x∗) for alln≥1 andTi(x∗) is a closed set, it follows

thatx∗∈Ti(x∗).Now from (i), we getx∗∈Ti(x∗) and henceF ix(T1) =F ix(T2).

Finally to prove (iv), suppose the setF ix(T1)∩F ix(T2) is a clique ofGe. We are

to show thatF ix(T1)∩F ix(T2) is singleton. Assume on contrary that there exist

u and v such that u, v ∈ F ix(T1)∩F ix(T2) but u 6=v. As (u, v) ∈ E∗(G) and

(T1, T2) form a graphic Θ−contraction, so for (ux, vy)∈E∗(G) implies

Θ (d(u, v))≤[Θ(d(u, v))]k1_[Θ(_d₍_{u, u}_))]k2_[Θ(_d₍_{v, v}_))]k3_[Θ(_d₍_{u, v}_{) +}_d₍_{v, u}_))]k4

Θ (d(u, v))≤[Θ(d(u, v))]k1_[Θ(_d₍_{u, u}_))]k2_[Θ(_d₍_{v, v}_))]k3_[Θ(_d₍_{u, v}_)]k4_[Θ(_d₍_{v, u}_))]k4

Θ (d(u, v))≤[Θ(d(u, v))]k1+2k4

a contradiction ask1+ 2k4<1. Hence u=v. Conversely, ifF ix(T1)∩F ix(T2) is

singleton, then it follows thatF ix(T1)∩F ix(T2) is a clique ofGe.

For specific choices of function Θ, we obtain some significant results. First, by taking Θ(t) =e

√

t _{in (2.1), we get the following result.}

G such that V(G) = X and E(G)⊇ ∆ and T1, T2 : X → Pcl(X) be multivalued mappings. If for any x, y ∈ X with (x, y) ∈ E(G) and ux ∈ Ti(x), there exists uy∈Tj(y)fori, j∈ {1,2} withi6=j such that(ux, uy)∈E∗(G)and

q

d(ux, uy)≤k1 p

d(x, y) +k2 p

d(x, ux) +k3 q

d(y, uy) +k4 q

d(x, uy) +d(y, ux)

(2.13)

wherek1, k2, k3 andk4 are non negative real numbers such that0≤k1+k2+k3+

2k4<1. Then following statement hold:

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singleton.

Remark 2.4. Notice that condition (2.14) is equivalent to

d(ux, uy) ≤ k12d(x, y) +k 2

2d(x, ux) +k23d(y, uy) +k24(d(x, uy) +d(y, ux))

+2k1k2 p

d(x, y)d(x, ux) + 2k1k3 q

d(x, y)d(y, uy)

+2k1k4 q

d(x, y)[d(x, uy) +d(y, ux)] + 2k2k3 q

d(x, ux)d(y, uy)

+2k2k4 q

d(x, ux)[d(x, uy) +d(y, ux)] + +2k3k4 q

d(y, uy)[d(x, uy) +d(y, ux)].

Next, in view of Remark 2.4 , by takingk1=k4= 0in Theorem 2.3, we obtain the

following result for multivalued mappings.

d(ux, uy)≤k22d(x, ux) +k32d(y, uy) + 2k2k3 q

d(x, ux)d(y, uy)

wherek2andk3are non negative real numbers such thatk2+k3<1,then following

statement hold:

singleton.

On the other hand, by takingk1=k2 =k3= 0 in Theorem 2.3, we obtain the

following Chatterjea type result for multivalued mappings.

Corollary 2.6. Let(X, d)be a complete metric space endowed with a directed graph

d(ux, uy)≤k24(d(x, uy) +d(y, ux))

wherek4 is non negative real number such2k4<1,then following statement hold:

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From Theorem 2.3, by takingk4= 0, we obtain the extension of Reich

contrac-tion for multivalued mappings.

d(ux, uy) ≤ k21d(x, y) +k 2

2d(x, ux) +k32d(y, uy)

+2k1k2 p

d(x, y)d(x, ux)

+2k1k3 q

d(x, y)d(y, uy) + 2k2k3 q

d(x, ux)d(y, uy)

wherek1, k2 andk3 are non negative real numbers such thatk1+k2+k3<1,then

following statement hold:

singleton.

If we take k2 =k3 =k4 = 0 in Theorem 2.3, then we get the following

result.

d(ux, uy)≤k21d(x, y) (2.14)

wherek1 is a non negative real number such that k1<1, then following statement

hold:

singleton.

Finally, by taking Θ(t) =en √

t_{in (2}_._{1) we have the following Corollary.}

n

q

d(ux, uy)≤k1n p

d(x, y) +k2n p

d(x, ux) +k3n q

d(y, uy) +k4n q

d(x, uy) +d(y, ux)

wherek1, k2, k3andk4are non negative real numbers such thatk1+k2+k3+2k4<1,

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singleton.

Example 2.10. Consider the sequence{xn} as follows: xn = 1 + 5 + 9 +...+ (4n−3) =n(2n−1).

LetX={xn:n∈N}=V(G),

E(G) ={(x, y) :x≤ywhere x, y∈V(G)}

and

E∗(G) ={(x, y) :x < ywherex, y∈V(G)}.

LetV (G) be endowed with usual metricd(x, y) =|x−y|. Then (X, d) is a com-plete metric space.DefineT1, T2:X →Pcl(X) as follows:

T1(x) ={x1}forx∈X

and

T2(x) =

{x1} , x=x1

{x1, xn−1} , x=xn, forn >1.

If we consider the mapping Θ : (0,∞)→(1,∞) defined by

Θ(t) =e √

tet .

We can easily show that Θ∈Ω and the pair of mappings (T1, T2) is a graphic

Θ-contraction.Indeed, the following holds:

e √

d(ux,uy)ed(ux,uy)) _≤ _ek1

√

d(x,y)ed(x,y)

for somek1∈(0,1).The above condition is equivalent to

d(ux, uy)ed(ux,uy)≤k21d(x, y)e

d(x,y)_.

So, we have to check that

d(ux, uy)ed(ux,uy)−d(x,y)≤k21d(x, y)

for somek1∈(0,1).For (ux, uy)∈E∗(G),we consider the following cases: Case 01. Ifx=x1, y=xm, form >1,then forux=x1∈T1(x),there exists

uy=xm−1∈T2(y),such that

d(ux, uy)ed(ux,uy)−d(x,y) = (2m2−5m+ 2)e−(4m−3)

< (2m2−m−1)e−1

= e−1d(x, y)

Case 02. Whenx=xn, y=xmwithm > n >1,then forux=x1∈T1(x),there

existsuy=xn−1∈T2(y),such that

d(ux, uy)ed(ux,uy)−d(x,y) = (2n2−5n+ 2)e−(2m

2₋₄_n2₊₆_n₋_m₋₂₎

< (2m2−2n2−m+n)e−1

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Now we show that forx, y∈X,ux∈T2(x); there existsuy ∈T1(y) such that

(ux, uy)∈E∗(G) and (2.15) is satisfied. For this, we consider the following cases: Case 01. Ifx=xn, y=x1 withn >1,we have forux=xn−1 ∈T2(x), there

existsuy=x1∈T1(y),such that

d(ux, uy)ed(ux,uy)−d(x,y) = (2n2−5n+ 2)e−(4n−3)

< (2n2−n−1)e−1

= e−1d(x, y)

Case 02. In casex=xn, y=xmwithm > n >1,then forux=xn−1∈T2(x),

there existsuy=x1∈T1(y),such that

d(ux, uy)ed(ux,uy)−d(x,y) = (2n2−5n+ 2)e−(2m

2₋₄_n2₊₆_n₋_m₋₂₎

< (2m2−2n2−m+n)e−1

= e−1d(x, y)

Thus for allx, yinV(G), (2.15) is satisfied. Hence all the conditions of Corollary 2.8 are satisfied. Moreover, x1 = 1 is the common fixed point of T1 and T2 with

F ix(T1) =F ix(T2).

Competing interests

The authors declare that they have no competing interests. Acknowledgement

This article was funded by the Deanship of Scientific Research (DSR), University of Jeddah. Therefore, authors acknowledge with thanks DSR, UJ for financial support.

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Jamshaid Ahmad

Department of Mathematics, University of Jeddah, P.O.Box 80327, Jeddah 21589, Saudi Arabia.

E-mail address:jamshaid [email protected]

Abdullah Eqal Al-Mazrooei

Department of Mathematics, University of Jeddah, P.O.Box 80327, Jeddah 21589, Saudi Arabia.