C- class function on fixed point theorems for contractive mappings of integral type in n-Banach spaces

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Adv. Fixed Point Theory, 8 (2018), No. 4, 384-400 https://doi.org/10.28919/afpt/3920

ISSN: 1927-6303

C- CLASS FUNCTION ON FIXED POINT THEOREMS FOR CONTRACTIVE MAPPINGS OF INTEGRAL TYPE IN n− BANACH SPACES

RASHWAN A. RASHWAN1, D. DHAMODHARAN2,∗, HASANEN A. HAMMAD3AND R. KRISHNAKUMAR4

1_{Department of Mathematics, Faculty of Science, Assuit University, Assuit 71516, Egypt}

2_{Department of Mathematics, Jamal Mohamed College (Autonomous),Tiruchirapplli-620020, India}

3_{Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt}

4_{Department of Mathematics,Urumu Dhanalakshmi College, Tiruchirappalli-620019, India}

Copyright c2018 R. A. Rashwan et al. This is an open access article distributed under the Creative Commons Attribution License, which

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.In this paper, we present some common fixed point theorems for class of mappings satisfying contractive

condition of integral type inn−Banach spaces viaC-Class function. Our results are version of some known results.

Keywords:n−Banach space; subadditive integrable function; Lebesgue integrable mapping; common fixed point.

2010 AMS Subject Classification:Primary 47H10; Secondary 54H25.

1. Introduction

In [8, 9] G¨ahler introduced an attractive theory of 2−norm and n−norm on a linear space. Raymond W. Freese and Y.J. Cho [7] gave as a survey of the latest results on the relations

between linear 2−normed spaces and normed linear spaces and completion of linear 2−normed

∗_{Corresponding author}

E-mail address: dharan [email protected]

Received December 7, 2017

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spaces. Later on Misiak [18] had also developed the notion of ann−norm in1989. The concept on n−inner product spaces is also due to Misiak who had studied the same as early as 1980. A systematic development of linearn−normed spaces has been extensively made by S.S. Kim and Y.J. Cho [15] and R. Malceski [17], A. Misiak [18] and Hendra Gunawan and Mashadi

[13]. For related works ofn−metric spaces andn−inner product spaces see for example [18], [11] and [13]. In 2002, Branciari [5] obtained a fixed point theorem for a single mapping

satisfying an analogue of a Banach contraction principle for integral type inequality [4]. After

the paper of Branciari, a lot of research works have been carried out on generalizing contractive

conditions of integral type for diferent contractive mappings satisfying some properties, see for

example [1, 3, 4, 12, 14, 19, 16]. The aim of this paper is to transform the concept of fixed

point inn−Banach spaces into fixed point inn−Banach spaces of integral type by using known contractive type mapping.We recall some preliminary definitions.

Definition 1.1. [10] letnbe a natural number, letX be a real vector space of dimensiond≥n

(dmay be infinity). A real valued functionk., ..., .konXnsatisfying four properties, (1) kx₁, ...,x_nk=0 if and only ifx₁, ...,x_nare linearly dependent inX,

(2) kx1, ...,xnkis invariant under permutation ofx1,x2, ...,xn,

(3) kx₁, ...,x_n₋₁,αxnk=|α| kx1, ...,xn−1,xnkfor everyα∈R,

(4) kx₁, ...,x_n₋₁,y+zk ≤ kx₁, ...,x_n₋₁,yk+kx₁, ...,x_n₋₁,zkfor allyandzinX,is called an

n−norm overX and the pair(X,k., ..., .k)is called a linearn−normed spaces.

Example 1.1. [13]. LetX=Rnwith the normk., ..., .konX by

kx₁, ...,x_nk=x_{i j} =



      

x₁₁ x₁₂ .. x₁_n x₂₁ x₂₂ .. x₂_n

.. .. .. ..

x_n₁ x_n₂ .. x_nn



      

wherex_i=x_i₁,x_i₂, ...,x_in∈Rn for eachi=1,2, ...,n.Then(X,k., ..., .k)is a linearn−normed space.

Definition 1.2. [10]. A sequence x_k in an n−normed space (X,k., ..., .k) is said to con-verge to an element x∈X (in the n−norm) whenever lim

k→∞

ku₁, ...,u_n₋₁,x_k−xk=0 for every

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Definition 1.3. [19]. A sequencex_k in ann−normed space(X,k., ..., .k)is said to be Cauchy sequence with respect ton−norm if lim

k→∞

ku₁, ...,u_n₋₁,x_k−xk=0 for everyu₁, ...,u_n₋₁∈X.

Definition 1.4. [10]. If every Cauchy sequence in X converges to an element,x∈X then X

is said to be complete (with respect to then−norm). A complete n−normed space is called an

n−Banach space.

Definition 1.5. [10]. LetX be an−Banach space andT be a self mapping ofX.T is said to be continuous atxif for every sequencex_k inX,x_k−→xask→∞impliesT xk−→T xask→∞

inX.

Definition 1.6. A function ψ :[0,∞)→[0,∞) is called an altering distance function if the

following properties are satisfied:

(i) ψ is non-decreasing and continuous, (ii)ψ(t) =0 if and only ift=0.

Definition 1.7. An ultra altering distance function is a continuous, nondecreasing mapping

ϕ:[0,∞)→[0,∞)such thatϕ(t)>0 ,t>0 andϕ(0)≥0.

We denote this set withΦu

Definition 1.8. [21] A mapping F :[0,∞)2 →[0,∞) is called coneC-class function if it is

continuous and satisfies following axioms:

(1) F(s,t)≤s;

(2) F(s,t) =simplies that eithers=0 ort=0; for alls,t∈P.

We denote coneC-class functions asC.

Example 1.2. [21] The following functions F :[0,∞)2 →[0,∞) are elements of C, for all

s,t ∈[0,∞):

(1) F(s,t) =s−t,

(2) F(s,t) =ks, where 0<k<1,

(3) F(s,t) =sβ(s), whereβ :[0,∞)→[0,1),

(4) F(s,t) =Ψ(s), where Ψ:[0,∞)→[0,∞) ,Ψ(0) =0 ,Ψ(s)>0 for alls∈[0,∞)with

s6=0 andΨ(S)≤sfor alls∈[0,∞).,

(5) F(s,t) =s−ϕ(s), whereϕ :[0,∞)→[0,∞)is a continuous function such thatϕ(t) =

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(6) F(s,t) =s−h(s,t),whereh:[0,∞)×[0,∞)→[0,∞)is a continuous function such that

h(s,t) =0⇔t=0 for allt,s>0.

(7) F(s,t) =ϕ(s),F(s,t) =s⇒s=0, hereϕ :[0,∞)→[0,∞)is a upper semi continuous

function such thatϕ(0) =0 andϕ(t)<tfort>0.

Lemma 1.1. Letψ andϕare altering distance and ultra altering distance functions respectively

,F∈C and{sn}a decreasing sequence in[0,∞)such that

ψ(sn+1)≤F(ψ(sn),ϕ(sn)) (3.1)

for alln≥1. Then lim

n→∞

s_n=0.

Definition 1.9. [2]. ζ :[0,∞)−→[0,∞)is subadditive on each[a,b]⊂[0,∞]if,

a+b

Z

0

ζ(t)dt=

a

Z

0

ζ(t)dt+

b

Z

0

ζ(t)dt

2. Main results

Theorem 2.1.LetX be anBanach space. Suppose f be a self mapping ofX such that

ψ(

kf x−f y,u1,...,un−1k Z

0

ζ(t)dt)≤F(ψ(α

kx−y,u1,...,un−1k Z

0

ζ(t)dt+β

kx−f x,u1,...,un−1k+ky−f y,u1,...,un−1k Z

0

ζ(t)dt

+γ

kx−f y,u1,...,un−1k+ky−f x,u1,...,un−1k Z

0

ζ(t)dt),ϕ(α

0

ζ(t)dt

+β

kx−f x,u1,...,un−1k+ky−f y,u1,...,un−1k Z

0

ζ(t)dt

+γ

kx−f y,u1,...,un−1k+ky−f x,u1,...,un−1k Z

0

ζ(t)dt)) (2.1)

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subadditive on each[a,b]⊂[0,∞),non-negative and for eachε >0,

ε

Z

0

ζ(t)dt>0. (2.2)

Then f has a unique fixed point inX,with lim

k→∞

fkx_◦=zfor eachx_◦∈X.

Proof.Letx_◦∈X,and define the iterate sequence{x_k}by

x_k₊₁= f x_k= fk+1x, (2.3)

then by (2.1) and (2.3), we get

ψ(

kxk−xk−1,u1,...,un−1k

Z

0

ζ(t)dt) = ψ(

kf xk−1−f xk,u1,...,un−1k

Z

0

ζ(t)dt)

≤ F(ψ(α

kxk−1−xk,u1,...,un−1k

Z

0

ζ(t)dt

+β

kxk−1−f xk−1,u1,...,un−1k+kxk−f xk,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kxk−1−f xk,u1,...,un−1k+kxk−f xk−1,u1,...,un−1k

Z

0

ζ(t)dt),

ϕ(α

Z

0

ζ(t)dt

+β

kxk−1−f xk−1,u1,...,un−1k+kxk−f xk,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kxk−1−f xk,u1,...,un−1k+kxk−f xk−1,u1,...,un−1k

Z

0

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≤ F(ψ((α+β+γ)

Z

0

ζ(t)dt

+(β+γ)

Z

0

ζ(t)dt),

ϕ((α+β+γ)

Z

0

ζ(t)dt

+(β+γ)

Z

0

ζ(t)dt))

≤ ψ((α+β+γ)

Z

0

ζ(t)dt

+(β+γ)

Z

0

ζ(t)dt)

⇒

Z

0

ζ(t)dt ≤q

Z

0

ζ(t)dt (2.4)

whereq= (α₁₋+β+γ

β−γ),for eachkand foru1, ...,un−1∈X.

implies that the sequence {

R

0

ζ(t)dt}is monotonic decreasing and continuous. There exists a real number, sayr≥0 such that

lim

n→∞

Z

0

ζ(t)dt=r

asn→∞equation (2.4)⇒

ψ(r)≤F(ψ(r),ϕ(r))

so,ψ(r) =0 orϕ(r) =0 which is only possible ifr=0. Thus

lim

n→∞

Z

0

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Claim:{x_k}is a Cauchy sequence inX.Suppose{x_k}is a Cauchy sequence inX. Then there exist anε >0 and sub sequence{ni}and{mi}such thatmi<ni<mi+1

kx_mi−x_ni,u1,...,un−1k

Z

0

ζ(t)dt ≥ε and

Z

0

ζ(t)dt ≤ε

ε≤

Z

0

ζ(t)dt≤

kx_mi−x_ni₋₁,u1,...,un−1k

Z

0

ζ(t)dt+

kx_ni₋₁−x_ni,u1,...,un−1k

Z

0

ζ(t)dt

therefore lim

i→∞

R

0

ζ(t)d=ε now

ε≤

kx_mi₋₁−x_ni₋₁,u1,...,un−1k

Z

0

ζ(t)dt≤

kx_mi₋₁−x_mi,u1,...,un−1k

Z

0

ζ(t)dt+

Z

0

ζ(t)dt

by taking limiti→∞we get,

lim

i→∞

kx_mi₋₁−x_ni₋₁,u1,...,un−1k

Z

0

ζ(t)dt =ε

from (2.4) and (2.1) we get,

ψ(ε)≤ψ(

Z

0

ζ(t)dt) =ψ(

kf x_mi₊₁−f x_ni₊₁,u1,...,un−1k

Z

0

ζ(t)dt)

≤F(ψ(J(x_m_i−x_n_i,u₁, ...,u_n₋₁)),ϕ(J(x_m_i₋₁−x_n_i₋₁,u₁, ...,u_n₋₁)))

where implies

ψ(ε)≤F(ψ(J(xmi−xni,u1, ...,un−1)),ϕ(J(xmi−1−xni−1,u1, ...,un−1))) (2.5)

J(x_m_i−x_n_i,u₁, ...,u_n₋₁) =α

Z

0

ζ(t)dt

+β

kx_mi−f x_mi,u1,...,un−1k+kxni−f xni,u1,...,un−1k

Z

0

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+γ

kx_mi−f x_ni,u1,...,un−1k+kx_ni−f x_mi,u1,...,un−1k

Z

0

ζ(t)dt

=α

Z

0

ζ(t)dt

+β

kx_mi−x_mi₋₁,u1,...,un−1k+kx_ni−x_ni₋₁,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kx_mi−x_ni₋₁,u1,...,un−1k+kx_ni−x_mi₋₁,u1,...,un−1k

Z

0

ζ(t)dt

Taking limit asi→∞we get,

lim

i→∞

J(x_m_i−x_n_i,u₁, ...,u_n₋₁) =α ε+0+γ ε

lim

i→∞

J(xmi−xni,u1, ...,un−1)≤ε

Therefore from (2.5) we have, ψ(ε)≤F(ψ(ε),ϕ(ε)) so, ψ(ε) =0 or ϕ(ε) =0. That is a contraction becauseε >0. Therefore{xn}is a Cauchy sequence inX. Hence{xn}is a Cauchy

sequence. There exists a pointzinX such that lim

k→∞

fkx_◦=z∈X andu₁, ...,u_n₋₁∈X

To prove the uniqueness ofz,suppose that(z6=w)be another fixed point of f,then from (2.1), we get

ψ(

kz−w,u1,...,un−1k Z

0

ζ(t)dt) = ψ(

kf z−f w,u1,...,un−1k Z

0

ζ(t)dt)

≤ F(ψ(α

0

ζ(t)dt

+β

kz−f z,u1,...,un−1k+kw−f w,u1,...,un−1k Z

0

ζ(t)dt

+γ

kz−f w,u1,...,un−1k+kw−f z,u1,...,un−1k Z

0

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ϕ(α

0

ζ(t)dt+β

kz−f z,u1,...,un−1k+kw−f w,u1,...,un−1k Z

0

ζ(t)dt

+γ

kz−f w,u1,...,un−1k+kw−f z,u1,...,un−1k Z

0

ζ(t)dt)

≤ F(ψ((α+2γ)

0

ζ(t)dt),ϕ((α+2γ)

0

ζ(t)dt)

≤ ψ((α+2γ)

0

ζ(t)dt)

⇒

0

ζ(t)dt ≤ (α+2γ)

0

ζ(t)dt

which is a contradiction.Sinceα+2γ <1. Thereforez=w.Hencezis a unique fixed point of f.

Remark 2.1.

(1) On settingζ(t) =1 overR+,the contractive condition of integral type transform into a general contractive condition not involving integral.

(2) From Condition 2.1 of integral type several contractive mappings of integral type can

be obtained. Now, our next theorem is the extension of the Theorem 2.1 for a pair of

mappings

Theorem 2.2.LetX be anBanach space. Let f andgbe a self mappings ofX ψ(

kf x−gy,u1,...,un−1k R

0

ζ(t)dt)≤F(ψ(α

kx−y,u1,...,un−1k R

0

ζ(t)dt+β

kx−f x,u1,...,un−1k+ky−gy,u1,...,un−1k R

0

ζ(t)dt

+γ

kx−gy,u1,...,un−1k+ky−f x,u1,...,un−1k Z

0

ζ(t)dt),ϕ(α

0

ζ(t)dt

+β

kx−f x,u1,...,un−1k+ky−gy,u1,...,un−1k Z

0

ζ(t)dt+γ

kx−gy,u1,...,un−1k+ky−f x,u1,...,un−1k Z

0

ζ(t)dt))

(2.6)

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on each[a,b]⊂[0,∞),non-negative and for eachε>0,

ε

Z

0

ζ(t)dt>0.

Then f andghave a unique common fixed point inX,with lim

k→∞

fkx_◦=zfor eachx_◦∈X.

Proof.Letx◦∈X,and define the iterate sequencexkby

x₂_k₊₁= f x_kandx₂_k₊₂=gx_k₊₁

Then by (2.6), we have

ψ(

kx2k+1−x2k+2,u1,...,un−1k

Z

0

ζ(t)dt) = ψ(

kf x2k−gx2k+1,u1,...,un−1k

Z

0

ζ(t)dt)

≤ F(ψ(α

kx2k−x2k+1,u1,...,un−1k

Z

0

ζ(t)dt

+β

kx2k−x2k+1,u1,...,un−1k+kx2k+1−x2k+2,u1,...,un−1k

Z

0

ζ(t)dt

+γ

Z

0

ζ(t)dt),

ϕ(α

kx2k−x2k+1,u1,...,un−1k

Z

0

ζ(t)dt

+β

Z

0

ζ(t)dt

+γ

Z

0

ζ(t)dt))

by simple calculation, we get

kx2k+1−x2k+2,u1,...,un−1k

Z

0

ζ(t)dt≤(α+β+γ 1−β−γ)

kx2k−x2k+1,u1,...,un−1k

Z

0

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by exactly the same argument we produce

kx2k−x2k+1,u1,...,un−1k

Z

0

ζ(t)dt ≤q

kx2k−1−x2k,u1,...,un−1k

Z

0

ζ(t)dt

for allu₁, ...,u_n₋₁∈X and for allk,we get kxk−xk+1,u1,...,un−1k

Z

0

ζ(t)dt ≤ q

Z

0

ζ(t)dt ≤q2

kxk−2−xk−1,u1,...,un−1k

Z

0

ζ(t)dt

≤...≤qk

kx◦−x1,u1,...,un−1k

Z

0

ζ(t)dt,

by the same steps of Theorem 2.1 one can reaches to conclude thatx_kis a Cauchy inX,and let lim

k→∞

x_k=z∈X.Now, foru1, ...,un−1∈X,by (2.6), we have

ψ(

kx2k+1−gz,u1,...,un−1k

Z

0

ζ(t)dt) = ψ(

kf x2k−gz,u1,...,un−1k

Z

0

ζ(t)dt)

≤ F(ψ(α

kx2k−z,u1,...,un−1k

Z

0

ζ(t)dt

+β

kx2k−x2k+1,u1,...,un−1k+kz−gz,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kx2k−gz,u1,...,un−1k+kz−x2k+1,u1,...,un−1k

Z

0

ζ(t)dt),

ϕ(α

kx2k−z,u1,...,un−1k

Z

0

ζ(t)dt

+β

kx2k−x2k+1,u1,...,un−1k+kz−gz,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kx2k−gz,u1,...,un−1k+kz−x2k+1,u1,...,un−1k

Z

0

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Taking the limit ask→∞,gives

kz−gz,u1,...,un−1k

Z

0

ζ(t)dt≤(β+γ)

Z

0

ζ(t)dt

since(β+γ <1)we get

R

0

ζ(t)dt=0,. gz=z.Similarly, we can show that, f z=z and hence zis a common fixed point of f and g. Next, suppose that (z6=w) be another fixed point of f andgthen from (2.6), we get

ψ(

kz−w,u1,...,un−1k

R

0

ζ(t)dt) =ψ(

kf z−gw,u1,...,un−1k

R

0

ζ(t)dt)

≤F(ψ(α

R

0

ζ(t)dt+β

kz−f z,u1,...,un−1k+kw−gw,u1,...,un−1k

R

0

ζ(t)dt

+γ

kz−gw,u1,...,un−1k+kw−f z,u1,...,un−1k

R

0

ζ(t)dt),ϕ(α

R

0

ζ(t)dt

+β

kz−f z,u1,...,un−1k+kw−gw,u1,...,un−1k

R

0

ζ(t)dt+γ

kz−gw,u1,...,un−1k+kw−f z,u1,...,un−1k

R

0

ζ(t)dt))

ψ(

R

0

ζ(t)dt)≤(α+2γ)

R

0

ζ(t)dt,

sinceα+2γ <1,we get a contradiction, therefore

R

0

ζ(t)dt =0,, we obtain that z=w. Thereforezis a unique common fixed point of f andg.The proof is complete.Finally, we extend the result for a sequence of mappings.

Theorem 2.3.Let X ben−Banach space with f :X −→X and f_k:X −→X,be a sequence of mappings such that

(i)

kfkx−fky,u1,...,un−1k R

0

ζ(t)dt≤F(ψ(α

kx−y,u1,...,un−1k R

0

ζ(t)dt+β

kx−fkx,u1,...,un−1k+ky−fky,u1,...,un−1k R

0

ζ(t)dt

+γ

kx−fky,u1,...,un−1k+ky−fkx,u1,...,un−1k Z

0

ζ(t)dt),ϕ(α

0

ζ(t)dt

+β

kx−fkx,u1,...,un−1k+ky−fky,u1,...,un−1k Z

0

ζ(t)dt

+γ

kx−fky,u1,...,un−1k+ky−fkx,u1,...,un−1k Z

0

ζ(t)dt))

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(ii) lim

k→∞

fkx= f xfor eachx∈X,for eachx,y,u1, ...,un−1∈Xwith non negative realsα+2β+2γ< 1,whereζ :[0,∞]−→[0,∞]is a Lebesgue integrable mapping which is summable, subadditive

on each[a,b]⊂[0,∞],non-negative and for eachε>0,

ε

R

0

ζ(t)dt>0.

Then f has a unique fixed point inX,such that lim

k→∞

z_k=z,z_kbeing the unique fixed point of

f_k,k=1,2, ...

Proof.If we take the limit in (2.7), we have

ψ(

kf x−f y,u1,...,un−1k

Z

0

ζ(t)dt) ≤ F(ψ(α

kx−y,u1,...,un−1k

Z

0

ζ(t)dt

+β

kx−f x,u1,...,un−1k+ky−f y,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kx−f y,u1,...,un−1k+ky−f x,u1,...,un−1k

Z

0

ζ(t)dt),

ϕ(α

kx−y,u1,...,un−1k

Z

0

ζ(t)dt

+β

kx−f x,u1,...,un−1k+ky−f y,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kx−f y,u1,...,un−1k+ky−f x,u1,...,un−1k

Z

0

ζ(t)dt))

for allx,y,u₁, ...,u_n₋₁∈X and hence f satisfies (2.7). Hence by Theorem (2.1), f has a unique fixed point sayz∈X.Now for allx,y,u₁, ...,u_n₋₁∈X

kz−zk,u1,...,un−1k

Z

0

ζ(t)dt=

kf z−fkzk,u1,...,un−1k

Z

0

ζ(t)dt

≤

kf z−fkz,u1,...,un−1k

Z

0

ζ(t)dt+

kfkz−fkzk,u1,...,un−1k

Z

0

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again from (2.7), we obtain

ψ(

Z

0

ζ(t)dt) ≤ F(ψ(α

Z

0

ζ(t)dt

+β

kz−fkz,u1,...,un−1k+kzk−fkzk,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kz−fkzk,u1,...,un−1k+kzk−fkz,u1,...,un−1k

Z

0

ζ(t)dt),

ϕ(α

Z

0

ζ(t)dt

+β

kz−fkz,u1,...,un−1k+kzk−fkzk,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kz−fkzk,u1,...,un−1k+kzk−fkz,u1,...,un−1k

Z

0

ζ(t)dt))

then, by the condition (ii) of this theorem and taking the limit ask→∞,we get

ψ(

Z

0

ζ(t)dt)≤ F(ψ(α

Z

0

ζ(t)dt+β

kz−fkz,u1,...,un−1k

Z

0

ζ)(t)dt,

+γ

kz−zk,u1,...,un−1k+kzk−fkz,u1,...,un−1k

Z

0

ζ(t)dt)

ϕ(α

Z

0

ζ(t)dt+β

kz−fkz,u1,...,un−1k

Z

0

ζ(t)dt

+γ

kz−zk,u1,...,un−1k+kzk−fkz,u1,...,un−1k

Z

0

(15)

from (2.7) in (2.6), we have

ψ(

kz−zk,u1,...,un−1k Z

0

ζ(t)dt) = ψ(

kf z−fkzk,u1,...,un−1k Z

0

ζ(t)dt)

≤ F(ψ(

kf z−fkz,u1,...,un−1k Z

0

ζ(t)dt+α

0

ζ(t)dt

+β

kz−fkz,u1,...,un−1k Z

0

ζ(t)dt+γ

kz−zk,u1,...,un−1k+kzk−fkz,u1,...,un−1k Z

0

ζ(t)dt),

ϕ(

kf z−fkz,u1,...,un−1k Z

0

ζ(t)dt+α

0

ζ(t)dt

+β

0

ζ(t)dt+γ

kz−zk,u1,...,un−1k+kzk−fkz,u1,...,un−1k Z

0

ζ(t)dt))

≤ F(ψ((1+β)

0

ζ(t)dt+ (α+γ)

0

ζ(t)dt

+γ

kzk−fkz,u1,...,un−1k Z

0

ζ(t)dt),ϕ((1+β)

0

ζ(t)dt

+(α+γ)

0

ζ(t)dt+γ

0

ζ(t)dt))

≤ ψ((1+β)

0

ζ(t)dt+ (α+γ)

0

ζ(t)dt

+γ

0

ζ(t)dt)

0

ζ(t)dt ≤ ( 1+β 1−α−γ)

0

ζ(t)dt

+( γ

1−α−γ)

0

ζ(t)dt.

So by condition (ii) of this theorem we get a contradiction, therefore kz−zk,u1,...,un−1k

Z

0

(16)

we get lim

k→∞

z_k=z(This complete the proof).

Conflict of Interests

The authors declare that there is no conflict of interests.

REFERENCES

[1] I. Altun, D. Turkoglu and B.E.Rhoades, Fixed points of weakly compatible maps satisfying a general

con-tractive condition of integral type, Fixed Point Theory Appl. 2007 (2007), Article ID 17301.

[2] H. Aydi, A fixed point theorem for a contractive condition of integral type involving altering distances, Int. J.

Nonlinear Anal. Appl. 3(1) (2012), 42-53.

[3] H. Aydi, A common fixed point result by altering distances involving a contractive condition of integral type

in partial metric spaces, Demonstratio Math. 46(2) (2013), 383-394.

[4] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund.

Math. 3 (1922), 133-181.

[5] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type,

Int. J. Math. Math. Sci. 29(9) (2002), 531-536.

[6] D. Dey, A. Gangyly and M.Saha, Fixed point theorems for mappings under general contractive condition of

integral type, Bull. Math. Anal. Appl. 3(1) (2011), 27-34.

[7] R. W. Freese and Y. J. Cho., Geometry of Linear 2-normed Spaces, Nova Science Publishers, Inc. New York,

NY, 2001.

[8] S. Gahler, Lineare 2-Normierte Raume, Math. Nachr. 28(7) (1964), 1-43.

[9] S. Gahler, Unter Suchungen Uber Veralla gemeinerte m-mertische raume, I. Math. Nachr. 40 (1969), 165-189.

[10] M. Gangopadhyay, M. Saha and A. P. Baisnab, Some fixed point theorems for contractive type mapping in

n-Banach spaces, Int. J. Stat. Math. 1(3) (2011), 58-64.

[11] H. Ganawan and M. Mashadi, On finite dimensional 2-normed spaces, Soochow J. Math. 27 (2001), 321-329.

[12] U. C. Gairola and A. S. Rawat, A fixed point theorem for integral type inequality, Int. J. Math. Anal. 2(15)

(2008), 709-712.

[13] H. Ganawan and M. Mashadi, On n-normed spaces, Int. J. Math. Math. Sci. 27(10) (2001), 631-639.

[14] V. R. Hosseini and N. Hosseini, Common fixed point theorem by altering distance involving under a

contrac-tive condition of integral type, Int. Math. Forum 5(40) (2010), 1951-1957.

[15] S. S. Kim and Y. J. Cho., Strict convexity in Linear n-normed Spaces, Demonstr. Math. 29(4) (1996), 739-744.

[16] S. Kumar, R. Chugh and R. Kumar, Fixed point theorem for compatible mappings satisfying a contractive

condition of integrale type, Soochow J. Math. 33(2) (2007), 181-185.

(17)

[18] A. Misiak, n-inner product spaces, Math. Nachr. 140 (1989), 299-319.

[19] B. E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral

type, Int. J. Math. Math. Sci. 63 (2003), 4007-4013.

[20] A. H.Ansari, Note onϕ−ψ-contractive type mappings and related fixed point, The 2nd Regional Conference

on Mathematics And Applications, PNU, September 2014, 377-380.

[21] Arslan Hojat Ansari,Sumit Chandok, Nawab Hussin and Ljiljana Paunovic, Fixed points of(ψ,φ)- weak

contractions in regular cone metric spaces via new function, J. Adv. Math. Stud. 9(2016), 72-82.

[22] Krishnakumar.R and Dhamodharan. D, Common fixed point of contractive modulus on 2-cone Banach space,