The aim of this paper is to define generalized (αη)EB-contraction in extendedb-metric space to obtain some generalized fixed point theorems

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ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 12 Issue 4(2020) Pages 15-31.

FIXED POINT THEOREMS FOR GENERALIZED (αη)EB-CONTRACTIONS IN EXTENDED B-METRIC SPACES

WITH APPLICATIONS

MARYAM F. S. ALASMARI, AFRAH A.N. ABDOU

Abstract. The aim of this paper is to define generalized (αη)EB-contraction in extendedb-metric space to obtain some generalized fixed point theorems.

As application, we apply our fixed point theorem to prove the existence theorem for Fredholm integral equation

ϑ(t) =

b

Z

a

K(t, q, ϑ(q))dq+g(t),

for allt, q∈[a, b],wheref: [a, b]→RandK: [a, b]×[a, b]×R→Rare continuous functions. Our results generalize and extend several known results of literature.

1. Introduction and Preliminaries

In 1906, M. Frechet introduced the notion of metric space which is one of pillar of not only mathematics but also physical sciences. Because to its importance and simplicity, this notion has been extended, improved and generalized in many dif- ferent ways [1,2]. The famous extensions of the concept of metric spaces have been done by Bakhtin [3] which was formally defined by Czerwik [4] in 1993 with a view of generalizing Banach contraction principle.

Definition 1.1. (see.[4] )Let M be a nonempty set and s≥ 1 be a constant. A function d_b :M × M →[0,∞)is called a b-metric if the following assertions hold:

(b1) d_b(ϑ, θ) = 0⇔ϑ=θ;

(b2) d_b(ϑ, θ) =d_b(θ, ϑ) for allϑ, θ∈ M;

(b3) db(ϑ, ϕ)≤s[db(ϑ, θ) +db(θ, ϕ)],for allϑ, θ, ϕ∈ M.

The pair(M, db) is then said to be ab- metric space.

2000Mathematics Subject Classification. Primary 47H10; Secondary 54H25.

Key words and phrases. Fredholm integral equation, Fixed Point, Generalized (αη)eb- contraction, Extendedb-Metric Spaces.

c

2020 Universiteti i Prishtinës, Prishtinë, Kosovë.

Submitted February 15, 2020. Published April 1, 2020.

Communicated by guest editor Jamshaid Jasim.

15

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In 2017, Kamran et al. [5] introduced the notion of extendedb-metric spaces.

Definition 1.2. Let M be a non-empty set ands:M × M →[1,∞). A function deb : M × M → [0,∞) is called an extended b-metric if the following conditions hold:

(i)deb(ϑ, θ) = 0 ⇐⇒ ϑ=θ;

(ii)deb(ϑ, θ) =deb(θ, ϑ);

(iii)d_eb(ϑ, θ)≤s(ϑ, θ)[d_eb(ϑ, ϕ) +d_eb(ϕ, θ)], for allϑ, θ, ϕ∈ M.

The pair (M, d_eb) is called an extended b-metric space.

Note that ifs(ϑ, θ) =sfors≥1,then we getb-metric space from extended b-metric space.

Example 1.3. [5] Let M=C([a, b],R) be the space of all continuous real valued functions defined on[a, b].Note thatMis complete extendedb-metric space by con- sidering

deb(ϑ, θ) = sup

t∈[a,b]

|ϑ(t)−θ(t)|² withs(ϑ, θ) =|ϑ(t)|+|θ(t)|+ 2,wheres:M×M→[1,∞).

For more details in this direction, we refer the readers to [6, 7, 8].

In 2012, Samet et al. [9] introduced the concept ofα-admissible mapping on complete metric space in this way.

Definition 1.4. [9]Let Hbe a self-mapping onM andα:M × M →[0,+∞)be a function. We say thatHis an α-admissible mapping if

α(ϑ, θ)≥1 =⇒α(Hϑ,Hθ)≥1

∀ϑ, θ∈M.

Hussain et al. [10] extended the above notion ofα-admissible mapping as follows.

Definition 1.5. [10]LetHbe a self-mapping on Mandα, η:M × M →[0,+∞) be two functions. We say thatHis an α-admissible mapping with respect to η if

α(ϑ, θ)≥η(ϑ, θ) =⇒α(Hϑ,Hθ)≥η(Hϑ,Hθ)

∀ϑ, θ∈M.

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Ifη(ϑ, θ) = 1,then Definition 1.5 reduces to Definition 1.4.

Forfurther details in the direction ofF-contractions, we refer the following [10,11]

to the readers.

In this paper, we define the notion of generalized (αη)_EB-contraction and establish some new fixed point theorems in the context of extended b-metric spaces. We also furnish a notable example to describe the significance of established results.

2. Results and Discussions

Definition 2.1. Let (M, deb) be an extended b-metric space and H : M → M.

Then H is said to be generalized (αη)EB-contraction if there exists two functions α, η:M×M→[0,∞) andk∈[0,1) such that

α(ϑ,Hϑ)α(θ,Hθ) ≥ η(ϑ,Hϑ)η(θ,Hθ)

=⇒deb(Hϑ,Hθ) ≤ kmax{deb(ϑ, θ),min{deb(ϑ,Hϑ), deb(θ,Hθ)}} (2.1)

∀ϑ, θ∈ M.

Theorem 2.2. Let(M, deb)be a complete extendedb-metric space such thatdebis a continuous functional and letH:M → M. Suppose that the following assertions hold:

(i)His anα-admissible mapping with respect toη, (ii)His generalized (αη)_EB-contraction,

(iii)∃ϑ₀ ∈ Msuch thatα(ϑ₀,Hϑ₀)≥η(ϑ₀,Hϑ₀), (iv) lim_m,n→∞s(ϑ_n, ϑ_m)k <1,for eachϑ₀∈ M,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑn →ϑ, α(ϑn, ϑn+1)≥η(ϑn, ϑn+1),thenα(ϑ,Hϑ)≥η(ϑ,Hϑ)

ThenHhas a fixed point.

Proof. Letϑ₀ ∈ Msuch thatα(ϑ₀,Hϑ0)≥η(ϑ₀,Hϑ0) and construct{ϑn} in M byϑ_n+1=Hⁿϑ₀=Hϑn,∀n∈N.By (i), we have

α(ϑ₀, ϑ₁) =α(ϑ₀,Hϑ0)≥η(ϑ₀,Hϑ0) =η(ϑ₀, ϑ₁).

Continuing in this way, we get

α(ϑ_n−1, ϑn) =α(ϑ_n−1,Hϑ_n−1) ≥ η(ϑ_n−1,Hϑ_n−1) =η(ϑ_n−1, ϑn) (2.2)

∀n∈N.Then

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α(ϑ_n−1,Hϑn−1)α(ϑ_n,Hϑn)≥η(ϑ_n−1,Hϑn−1)η(ϑ_n,Hϑn)

∀n∈N.Clearly, if∃n0∈N for whichϑn₀+1=ϑn₀,thenHϑn₀ =ϑn₀ and the proof is completed. Hence, we assume thatϑn+16=ϑn ordeb(Hϑn−1,Hϑn)>0 for everyn∈N. Now asHis generalized (αη)EB-contraction, so we have

deb(ϑn, ϑn+1) = deb(Hϑ_n−1,Hϑn)

≤ kmax{deb(ϑ_n−1, ϑ_n),min{deb(ϑ_n−1,Hϑn−1), d_eb(ϑ_n,Hϑn)}}

∀n∈N.Now ifd_eb(ϑ_n−1,Hϑ_n−1)< d_eb(ϑ_n,Hϑ_n),then

max{deb(ϑ_n−1, ϑn),min{deb(ϑ_n−1,Hϑ_n−1), deb(ϑn,Hϑn)}}=deb(ϑ_n−1, ϑn)

∀n∈N.Ifdeb(ϑn,Hϑn)< deb(ϑ_n−1,Hϑ_n−1),then

max{deb(ϑn−1, ϑn),min{deb(ϑn−1,Hϑn−1), deb(ϑn,Hϑn)}}=deb(ϑn−1, ϑn)

∀n∈N.Thus in all case, we have

deb(ϑn, ϑn+1)≤kdeb(ϑ_n−1, ϑn)

∀n∈N.Continuing in this way, we get

deb(ϑn, ϑn+1) ≤ kⁿdeb(ϑ0, ϑ1) (2.3)

∀n∈N.For alln, m∈N(n < m), we get

d_eb(ϑ_n, ϑ_m) ≤ s(ϑ_n, ϑ_m) [d_eb(ϑ_n, ϑ_n+1) +d_eb(ϑ_n+1, ϑ_m)]

≤ s(ϑn, ϑm)deb(ϑn, ϑn+1) +s(ϑn, ϑm)s(ϑn+1, ϑm) [deb(ϑn+1, ϑn+2) +deb(ϑn+2, ϑm)]

≤ s(ϑ_n, ϑ_m)d_eb(ϑ_n, ϑ_n+1) +s(ϑ_n, ϑ_m)s(ϑ_n+1, ϑ_m)d_eb(ϑ_n+1, ϑ_n+2) +· · ·

+ s(ϑn, ϑm)s(ϑn+1, ϑm)s(ϑn+2, ϑm)· · ·s(ϑm−2, ϑm)s(ϑm−1, ϑm)deb(ϑm−1, ϑm)

≤ s(ϑ1, ϑm)s(ϑ2, ϑm)· · ·s(ϑn, ϑm)deb(ϑn, ϑn+1)

+ s(ϑ₁, ϑ_m)s(ϑ₂, ϑ_m)· · ·s(ϑ_n+1, ϑ_m)d_eb(ϑ_n+1, ϑ_n+2) +· · · + s(ϑ1, ϑm)s(ϑ2, ϑm)· · ·s(ϑ_m−1, ϑm)deb(ϑ_m−1, ϑm)

<

∞

X

n=1

d_eb(ϑ_n, ϑ_n+1)





n

Y

j=0

s(ϑ_j, ϑ_m)





≤

∞

X

n=1

kⁿdeb(ϑ0, ϑ1)





n

Y

j=0

s(ϑj, ϑm)



.

Thus

d_eb(ϑ_n, ϑ_m) ≤

∞

X

n=1

kⁿd_eb(ϑ₀, ϑ₁)





n

Y

j=0

s(ϑ_j, ϑ_m)



. (2.4)

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Since lim_m,n→∞s(ϑn, ϑm)k <1,so the series

∞

P

n=1

kⁿ

n

Q

j=0

s(ϑj, ϑm)

!

converges by ratio test for eachm∈N. Let

S=

∞

X

n=1

kⁿ





n

Y

j=0

s(ϑ_j, ϑ_m)





S_n =

n

X

i=1

kⁱ





i

Y

j=1

s(ϑ_j, ϑ_m)



.

Thus, form > n, (2.4) implies

d_eb(ϑ_n, ϑ_m)≤d_eb(ϑ₀, ϑ₁)[S_m−1−S_n−1].

Takingm, n→ ∞, we get that{ϑn}is a Cauchy. AsMis complete, so ϑn→ϑ^∗∈ M.Now ifHis continuous, then,ϑn+1 =Hϑn → Hϑ^∗ asn→ ∞.

Thus, Hϑ^∗ = ϑ^∗. Thus ϑ^∗ is a fixed point of H. Secondly as ϑn → ϑ^∗ and α(ϑn, ϑn+1)≥η(ϑn, ϑn+1),thenα(ϑ^∗,Hϑ^∗)≥η(ϑ^∗,Hϑ^∗).

Thus

α(ϑ^∗,Hϑ^∗)α(ϑn,Hϑn)≥η(ϑ^∗,Hϑ^∗)η(ϑn,Hϑn) By (1), we have

d_eb(Hϑ^∗, ϑ_n+1) = d_eb(Hϑ^∗,Hϑn)

≤ kmax{deb(ϑ^∗, ϑn),min{deb(ϑ^∗,Hϑ^∗), deb(ϑn,Hϑn)}}

= kmax{d_eb(ϑ^∗, ϑ_n),min{d_eb(ϑ^∗,Hϑ^∗), d_eb(ϑ_n, ϑ_n+1)}}. Letting n→ ∞and using the supposition thatdebis continuous functional, we have deb(Hϑ^∗, ϑ^∗) = 0.Thus,Hϑ^∗=ϑ^∗ andϑ^∗ is a fixed point ofH.

Ifη(ϑ, θ) = 1, then we have the following corollaries.

Corollary 2.3. Let(M, d_eb)be a complete extendedb-metric space such thatd_ebis a continuous functional and letH:M → M.Assume that the following assertions hold:

(i)His anα-admissible mapping, (ii) if forϑ, θ ∈ Msuch that∀ϑ, θ∈ M

α(ϑ,Hϑ)α(θ,Hθ)≥1 =⇒deb(Hϑ,Hθ)≤kmax{deb(ϑ, θ),min{deb(ϑ,Hϑ), deb(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

(iv) limm,n→∞s(ϑn, ϑm)k <1,for eachϑ0∈ M,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑ_n →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

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Corollary 2.4. Let(M, d_eb)be a complete extendedb-metric space such thatd_ebis a continuous functional and letH:M → M.Assume that the following assertions hold:

(i)His anα-admissible mapping,

(ii) if forϑ, θ ∈ Msuch that∀ϑ, θ∈ M. andl >0 (d_eb(Hϑ,Hθ) +l)α(ϑ,Hϑ)α(θ,Hθ)

≤kmax{d_eb(ϑ, θ),min{d_eb(ϑ,Hϑ), d_eb(θ,Hθ)}}+l.

(iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1, (iv) limm,n→∞s(ϑn, ϑm)k <1,for eachϑ0∈ M,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑn →ϑ, α(ϑn, ϑn+1)≥1,then α(ϑ,Hϑ)≥1.

Corollary 2.5. Let(M, deb)be a complete extendedb-metric space such thatdebis a continuous functional and letH:M → M.Assume that the following assertions hold:

(α(ϑ,Hϑ)α(θ,Hθ) + 1)^d^eb^(Hϑ,Hθ)≤2^k^max{d^eb(ϑ,θ),min{d_eb(ϑ,Hϑ),d_eb(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

(iv) lim_m,n→∞s(ϑ_n, ϑ_m)k <1,for eachϑ₀∈ M,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑ_n →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

α(ϑ,Hϑ)α(θ,Hθ)deb(Hϑ,Hθ)≤kmax{deb(ϑ, θ),min{deb(ϑ,Hϑ), deb(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑn →ϑ, α(ϑn, ϑn+1)≥1,then α(ϑ,Hϑ)≥1.

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Ifα(x, y) = 1, then we have the following corollaries.

(i)His anη-subadmissible mapping, (ii) if forϑ, θ ∈ Msuch that∀ϑ, θ∈ M

η(ϑ,Hϑ)η(θ,Hθ)≤1 =⇒d_eb(Hϑ,Hθ)≤kmax{deb(ϑ, θ),min{deb(ϑ,Hϑ), deb(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatη(ϑ0,Hϑ0)≤1,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑ_n →ϑ, η(ϑ_n, ϑ_n+1)≤1,thenη(ϑ,Hϑ)≤1.

(i)His anη-subadmissible mapping,

(ii) if forϑ, θ ∈ Msuch that∀ϑ, θ∈ M.andl >0

deb(Hϑ,Hθ)+l≤[kmax{deb(ϑ, θ),min{deb(ϑ,Hϑ), deb(θ,Hθ)}}+l]η(ϑ,Hϑ)η(θ,Hθ)

. (iii)∃ϑ₀ ∈ Msuch thatη(ϑ₀,Hϑ₀)≤1,

(iv) lim_m,n→∞s(ϑn, ϑm)k <1,for eachϑ0∈ M,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑn →ϑ, η(ϑn, ϑn+1)≤1,thenη(ϑ,Hϑ)≤1.

Corollary 2.9. Let(M, deb)be a complete extendedb-metric space such thatd_ebis a continuous functional and letH:M → M.Assume that the following assertions hold:

2^d^eb^(Hϑ,Hθ)≤(η(ϑ,Hϑ)η(θ,Hθ) + 1)^k^max{d^eb(ϑ,θ),min{deb(ϑ,Hϑ),deb(θ,Hθ)}}

. (iii)∃ϑ0 ∈ Msuch thatη(ϑ0,Hϑ0)≤1,

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Corollary 2.10. Let (M, deb) be a complete extended b-metric space such that deb is a continuous functional and let H : M → M. Assume that the following assertions hold:

deb(Hϑ,Hθ)≤kη(ϑ,Hϑ)η(θ,Hθ) max{deb(ϑ, θ),min{deb(ϑ,Hϑ), deb(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatη(ϑ0,Hϑ0)≤1,

(iv) limm,n→∞s(ϑn, ϑm)k <1,for eachϑ0∈ M,,

α(ϑ,Hϑ)α(θ,Hθ)≥1 =⇒deb(Hϑ,Hθ)≤kdeb(ϑ, θ).

(iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1, (iv) limm,n→∞s(ϑn, ϑm)k <1,for eachϑ0∈ M,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑn →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

(α(ϑ,Hϑ)α(θ,Hθ) + 1)^d^eb^(Hϑ,Hθ)≤2^kd^eb^(ϑ,θ). (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

(v) either His an continuous or if{ϑn} is a sequence in Msuch that ϑn →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

Corollary 2.13. Let (M, deb) be a complete extended b-metric space such that d_eb is a continuous functional and let H : M → M. Assume that the following assertions hold:

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α(ϑ,Hϑ)α(θ,Hθ)deb(Hϑ,Hθ)≤kdeb(ϑ, θ).

(iii)∃ϑ₀ ∈ Msuch thatα(ϑ₀,Hϑ0)≥1,

(iv) lim_m,n→∞s(ϑ_n, ϑ_m)k <1,for eachϑ₀∈ M,,

(v) either His an continuous or if{ϑ_n} is a sequence in Msuch that ϑ_n →ϑ, α(ϑn, ϑn+1)≥1,then α(ϑ,Hϑ)≥1.

3. Consequences

3.1. Fixed Point Results in b-Metric Spaces.

Theorem 3.1. Let (M, db) be a complete b-metric space such that db is a continuous functional and let H : M → M. Suppose that the following assertions hold:

(i)His anα-admissible mapping with respect toη,

(ii) there exist two functions α, η:M×M→[0,∞) and k∈[0,1) such that for allϑ, θ∈ M

α(ϑ,Hϑ)α(θ,Hθ)≥η(ϑ,Hϑ)η(θ,Hθ) =⇒db(Hϑ,Hθ)≤kmax{db(ϑ, θ),min{db(ϑ,Hϑ), db(θ,Hθ)}}

(iii)∃ϑ₀ ∈ Msuch thatα(ϑ₀,Hϑ₀)≥η(ϑ₀,Hϑ₀),

(iv) eitherHis an continuous or if{ϑ_n}is a sequence in Msuch thatϑ_n →ϑ, α(ϑn, ϑn+1)≥η(ϑn, ϑn+1),thenα(ϑ,Hϑ)≥η(ϑ,Hϑ)

Corollary 3.2. Let (M, db)be a completeb-metric space such thatdb is a continuous functional and letH:M → M.Assume that the following assertions hold:

α(ϑ,Hϑ)α(θ,Hθ)≥1 =⇒db(Hϑ,Hθ)≤kmax{db(ϑ, θ),min{db(ϑ,Hϑ), db(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑn →ϑ, α(ϑn, ϑn+1)≥1,then α(ϑ,Hϑ)≥1.

Corollary 3.3. Let (M, db)be a completeb-metric space such thatd_b is a continuous functional and letH:M → M.Assume that the following assertions hold:

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(ii) if forϑ, θ ∈ Msuch that∀ϑ, θ∈ M. andl >0 (db(Hϑ,Hθ) +l)α(ϑ,Hϑ)α(θ,Hθ)

≤kmax{db(ϑ, θ),min{db(ϑ,Hϑ), db(θ,Hθ)}}+l.

(iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑ_n →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

(α(ϑ,Hϑ)α(θ,Hθ) + 1)^d^b^(Hϑ,Hθ)≤2^k^max{d^b(ϑ,θ),min{d_b(ϑ,Hϑ),d_b(θ,Hθ)}}. (iii)∃ϑ₀ ∈ Msuch thatα(ϑ₀,Hϑ₀)≥1,

Corollary 3.5. Let (M, db)be a completeb-metric space such thatd_b is a continuous functional and letH:M → M.Assume that the following assertions hold:

α(ϑ,Hϑ)α(θ,Hθ)db(Hϑ,Hθ)≤kmax{db(ϑ, θ),min{db(ϑ,Hϑ), db(θ,Hθ)}}. (iii)∃ϑ₀ ∈ Msuch thatα(ϑ₀,Hϑ₀)≥1,

η(ϑ,Hϑ)η(θ,Hθ)≤1 =⇒d_b(Hϑ,Hθ)≤kmax{d_b(ϑ, θ),min{d_b(ϑ,Hϑ), d_b(θ,Hθ)}}.

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(iii)∃ϑ0 ∈ Msuch thatη(ϑ0,Hϑ0)≤1,

(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑn →ϑ, η(ϑn, ϑn+1)≤1,thenη(ϑ,Hϑ)≤1.

db(Hϑ,Hθ) +l≤[kmax{db(ϑ, θ),min{db(ϑ,Hϑ), db(θ,Hθ)}}+l]η(ϑ,Hϑ)η(θ,Hθ)

. (iii)∃ϑ₀ ∈ Msuch thatη(ϑ₀,Hϑ0)≤1,

(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑ_n →ϑ, η(ϑ_n, ϑ_n+1)≤1,thenη(ϑ,Hϑ)≤1.

2^d^b^(Hϑ,Hθ)≤(η(ϑ,Hϑ)η(θ,Hθ) + 1)^k^max{d^b(ϑ,θ),min{db(ϑ,Hϑ),db(θ,Hθ)}}

(iv) eitherHis an continuous or if{ϑ_n}is a sequence in Msuch thatϑ_n →ϑ, η(ϑ_n, ϑ_n+1)≤1,thenη(ϑ,Hϑ)≤1.

db(Hϑ,Hθ)≤kη(ϑ,Hϑ)η(θ,Hθ) max{db(ϑ, θ),min{db(ϑ,Hϑ), db(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatη(ϑ0,Hϑ0)≤1,

Corollary 3.10. Let (M, db)be a complete b-metric space such that db is a continuous functional and let H : M → M. Assume that the following assertions hold:

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α(ϑ,Hϑ)α(θ,Hθ)≥1 =⇒db(Hϑ,Hθ)≤kdb(ϑ, θ).

Corollary 3.11. Let (M, db)be a complete b-metric space such that db is a continuous functional and let H : M → M. Assume that the following assertions hold:

(α(ϑ,Hϑ)α(θ,Hθ) + 1)^d^b^(Hϑ,Hθ)≤2^kd^b^(ϑ,θ). (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

3.2. Fixed Point Results in Metric Spaces.

Theorem 3.12. Let (M, d) be a complete metric space and let H : M → M.

Suppose that the following assertions hold:

(i)His anα-admissible mapping with respect toη,

(ii) there exist two functionsα, η:M×M→[0,∞) and k∈[0,1) such that for allϑ, θ∈ M

α(ϑ,Hϑ)α(θ,Hθ)≥η(ϑ,Hϑ)η(θ,Hθ) =⇒d(Hϑ,Hθ)≤kmax{d(ϑ, θ),min{d(ϑ,Hϑ), d(θ,Hθ)}}

(iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥η(ϑ0,Hϑ0),

(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑn →ϑ, α(ϑn, ϑn+1)≥η(ϑn, ϑn+1),thenα(ϑ,Hϑ)≥η(ϑ,Hϑ)

Corollary 3.13. Let (M, d) be a complete metric space and let H : M → M.

Assume that the following assertions hold:

α(ϑ,Hϑ)α(θ,Hθ)≥1 =⇒d(Hϑ,Hθ)≤kmax{d(ϑ, θ),min{d(ϑ,Hϑ), d(θ,Hθ)}}.

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(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑn →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

(ii) if forϑ, θ ∈ Msuch that∀ϑ, θ∈ Mandl >0 (d(Hϑ,Hθ) +l)α(ϑ,Hϑ)α(θ,Hθ)

≤kmax{d(ϑ, θ),min{d(ϑ,Hϑ), d(θ,Hθ)}}+l.

(α(ϑ,Hϑ)α(θ,Hθ) + 1)^d(Hϑ,Hθ)≤2^kmax{d(ϑ,θ),min{d(ϑ,Hϑ),d(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

α(ϑ,Hϑ)α(θ,Hθ)d(Hϑ,Hθ)≤kmax{d(ϑ, θ),min{d(ϑ,Hϑ), d(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatα(ϑ0,Hϑ0)≥1,

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η(ϑ,Hϑ)η(θ,Hθ)≤1 =⇒d(Hϑ,Hθ)≤kmax{d(ϑ, θ),min{d(ϑ,Hϑ), d(θ,Hθ)}}. (iii)∃ϑ0 ∈ Msuch thatη(ϑ0,Hϑ0)≤1,

d(Hϑ,Hθ) +l≤[kmax{d(ϑ, θ),min{d(ϑ,Hϑ), d(θ,Hθ)}}+l]η(ϑ,Hϑ)η(θ,Hθ)

. (iii)∃ϑ0 ∈ Msuch thatη(ϑ0,Hϑ0)≤1,

2^d(Hϑ,Hθ)≤(η(ϑ,Hϑ)η(θ,Hθ) + 1)^kmax{d(ϑ,θ),min{d(ϑ,Hϑ),d(θ,Hθ)}}

d(Hϑ,Hθ)≤kη(ϑ,Hϑ)η(θ,Hθ) max{d(ϑ, θ),min{d(ϑ,Hϑ), d(θ,Hθ)}}. (iii)∃ϑ₀ ∈ Msuch thatη(ϑ₀,Hϑ0)≤1,

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α(ϑ,Hϑ)α(θ,Hθ)≥1 =⇒d(Hϑ,Hθ)≤kd(ϑ, θ).

(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑn →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

(α(ϑ,Hϑ)α(θ,Hθ) + 1)^d(Hϑ,Hθ)≤2^kd(ϑ,θ). (iii)∃ϑ₀ ∈ Msuch thatα(ϑ₀,Hϑ0)≥1,

(iv) eitherHis an continuous or if{ϑn}is a sequence in Msuch thatϑ_n →ϑ, α(ϑ_n, ϑ_n+1)≥1,then α(ϑ,Hϑ)≥1.

4. Applications

In this section, we present an application of Theorem 2.2 in establishing the existence of solutions for a Fredholm integral equation:

ϑ(t) = Z b

a

K(t, q, ϑ(q))dq+f(t), (4.1)

for allt, q∈[a, b],wheref : [a, b]→RandK: [a, b]×[a, b]×R→Rare continuous functions. LetMbe the set of all continuous real valued functions defined on [a, b].

i.e.,M=C([a, b],R).Define deb:M×M→Rby d_eb(ϑ, θ) = sup

t∈[a,b]

|ϑ(t)−θ(t)|², withs(ϑ, θ) =|ϑ(t) +θ(t)|+ 1

wheres:M × M →[1,∞).Then (M,d_eb) is a complete extendedb-metric space.

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Theorem 4.1. Suppose that (∀ϑ, θ∈ M), we have

|K(t, q, ϑ(q))−K(t, q, θ(q))| ≤√

k|ϑ(q)−θ(q)|

∀ t, q∈[a, b] andτ >0, then, the integral equation (21) has a solution.

Proof. DefineH:M → Mby H(ϑ(t)) =

Z b a

K(t, q, ϑ(q))dq+f(t),

for all t, q∈[a, b] andα, η: M×M→[0,∞) byα(ϑ, θ) =η(ϑ, θ) = 1, ∀ϑ, θ ∈ M and Q : R⁺

4 → R⁺ by Q(l1, l2, l3, l4) = τ, where τ > 0. Now we show that H satisfies all the assertions of Theorem 2.2. For anyϑ(t), θ(t)∈ M. Consider

|H(ϑ(t))− H(θ(t))|² = Z b

a

|K(t, q, ϑ(q))−K(t, q, θ(q))|

!² dq

≤ Z b

a

√k|ϑ(q)−θ(q)|

!2

dq

= k

Z b a

|ϑ(q)−θ(q)|

!2

dq

= kdeb(ϑ(t), θ(t))

which implies

deb(H(ϑ(t)),H(θ(t)))≤kdeb(ϑ, θ).

Thus all the assertions of the Theorem 2.2 are satisfied. Hence H has a unique fixed point and the Fredholm integral equation has a solution.

5. Conclusion

In this article, we have defined generazlied (αη)EB-contraction to obtain new fixed point theorems in the setting of complete extendedb-metric spaces. As application of our main theorems, the existence of solution for a Fredholm integral inclusion is also explored. We hope that the theorems proved in this paper will form new connections for those who are working in extendedb-metric space .

Conflict of Interests

The authors declare that they have no competing interests.

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Author’s Contribution

All authors contributed equally and significantly in writing this paper. Both authors read and approved the final paper.

Acknowledgement

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

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Maryam F. S. Alasmari

Department of Mathematics, University of Jeddah, P.O.Box 80327, Jeddah 21589, Saudi Arabia. [email protected]

Afrah A.N. Abdou

Department of Mathematics, University of Jeddah, P.O.Box 80327, Jeddah 21589, Saudi Arabia. [email protected]