New contractive conditions of integral type on complete S-metric spaces

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O R I G I N A L R E S E A R C H

New contractive conditions of integral type on complete S-metric

spaces

Nihal Yilmaz O¨ zgu¨r1 •_{Nihal Tas¸}1

Received: 21 November 2016 / Accepted: 24 May 2017 / Published online: 6 June 2017 Ó The Author(s) 2017. This article is an open access publication

Abstract An S-metric space is a three-dimensional general-ization of a metric space. In this paper our aim is to examine some fixed-point theorems using new contractive conditions of integral type on a complete S-metric space. We give some illustrative examples to verify the obtained results. Our find-ings generalize some fixed-point results on a complete metric space and on a complete S-metric space. An application to the Fredholm integral equation is also obtained.

Keywords Integral-type contractive conditions Fixed point S-metric

Mathematics Subject Classification Primary 47H10 Secondary 54H25

Introduction

Recently, the notion of an S-metric has been introduced and studied as a generalization of a metric. This notion has been defined by Sedghi et al. [13] as follows:

Definition 1.1 [13] Let X6¼ ; be any set and S : X X X ! ½0; 1Þ be a function satisfying the following conditions for all u; v; z; a2 X.

(S1) Sðu; v; zÞ ¼ 0 if and only if u ¼ v ¼ z. (S2) Sðu; v; zÞ Sðu; u; aÞ þ Sðv; v; aÞ þ Sðz; z; aÞ.

Then the function S is called an S-metric on X and the pair (X, S) is called an S-metric space.

Some fixed-point theorems have been given for self-mappings satisfying various contractive conditions on an S-metric space (see [4,6,8,9,13,14]). One of the important results among these studies is the Banach’s contraction principle on a complete S-metric space.

Theorem 1.2 [13] Let (X, S) be a complete S-metric space, h2 ð0; 1Þ and T : X ! X be a self-mapping of X such that

SðTu; Tu; TvÞ hSðu; u; vÞ;

for all u; v2 X: Then T has a unique fixed point in X. On the other hand some generalizations of the well-known C´ iric´’s and Nemytskii-Edelstein fixed-point theo-rems obtained on S-metric spaces via some new fixed point results (see [8,9,13,14] for more details).

Later, different applications of some contractive condi-tions have been constructed on an S-metric space such as differential equations, complex valued functions etc. (see [5,7,10,11]).

In recent years, fixed-point theory has been examined for various contractive conditions. For example, contrac-tive conditions of integral type were adapted into some studied fixed-point results. So more general fixed-point theorems were obtained.

Through the whole paper we assume that 1 :½0; 1Þ ! ½0; 1Þ is a Lebesgue-integrable mapping which is sum-mable ( i.e., with finite integral) on each compact subset of ½0; 1Þ; nonnegative and such that for each e [ 0;

Ze

1ðtÞdt [ 0: ð1Þ

& Nihal Yilmaz O¨zgu¨r nihal@balikesir.edu.tr Nihal Tas¸

nihaltas@balikesir.edu.tr DOI 10.1007/s40096-017-0226-0

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Branciari [1] studied a fixed-point theorem for a general contractive condition of integral type on a complete metric space as seen in the following theorem.

Theorem 1.3 [1] LetðX; qÞ be a complete metric space, h2 ð0; 1Þ; the function 1 : ½0; 1Þ ! ½0; 1Þ be defined as in (1) and T : X! X be a self-mapping of X such that

Z qðTu;TvÞ 0 1ðtÞdt h Z qðu;vÞ 0 1ðtÞdt;

for all u; v2 X; then T has a unique fixed point w 2 X such that

lim

n!1T

n_u_{¼ w;}

for each u2 X:

After the study of Branciari, some researchers have investigated new generalized contractive conditions of integral type using different known inequalities on various metric spaces (see [2,3,12]).

The purpose of this paper is to give new contractive conditions of integral type satisfying some new generalized inequalities given in [6] on a complete S-metric space. Our results generalize some known fixed-point results on a complete metric space and on a complete S-metric space.

Fixed-point results under some contractive

conditions of integral type

In this section we obtain new fixed-point theorems using some contractive conditions of integral type on a complete S-metric space. We construct three examples to show the validity of our results. At first we recall some basic results about S-metric spaces.

Lemma 2.1 [13] Let (X, S) be an S-metric space. Then we have

Sðu; u; vÞ ¼ Sðv; v; uÞ:

The above Lemma2.1can be considered as a symmetry condition on an S-metric space. The following definition is related to convergent sequences on an S-metric space. Definition 2.2 [13] Let (X, S) be an S-metric space.

(1) A sequencefung in X converges to u if and only if

Sðun; un; uÞ ! 0 as n ! 1. That is, there exists n02

N such that for all n n0, Sðun; un; uÞ\e for each

e [ 0. We denote this by lim

n!1un¼ uor limn!1Sðun; un; uÞ ¼ 0:

(2) A sequencefung in X is called a Cauchy sequence if

Sðun; un; umÞ ! 0 as n; m ! 1. That is, there exists

n02 N such that for all n; m n0, Sðun; un; umÞ\e

for each e [ 0.

(3) The S-metric space (X, S) is called complete if every Cauchy sequence is convergent.

In the following lemma we see the relationship between a metric and an S-metric.

Lemma 2.3 [4] Let ðX; qÞ be a metric space. Then the following properties are satisfied :

(1) Sqðu; v; zÞ ¼ qðu; zÞ þ qðv; zÞ for all u; v; z 2 X is an

S-metric on X.

(2) un! u in ðX; qÞ if and only if un! u in ðX; SqÞ:

(3) fung is Cauchy in ðX; qÞ if and only if fung is

Cauchy inðX; SqÞ:

(4) ðX; qÞ is complete if and only if ðX; SqÞ is complete.

We call the function Sqdefined in Lemma2.3(1) as the

S-metric generated by the metric q: It can be found an example of an S-metric which is not generated by any metric in [4,9].

Now we give the following theorem.

Theorem 2.4 Let (X, S) be a complete S-metric space, h2 ð0; 1Þ; the function 1 : ½0; 1Þ ! ½0; 1Þ be defined as in (1) and T : X ! X be a self-mapping of X such that

Z SðTu;Tu;TvÞ 0 1ðtÞdt h Z Sðu;u;vÞ 0 1ðtÞdt; ð2Þ

for all u; v2 X: Then T has a unique fixed point w 2 X and we have

lim

n!1T

n_u_{¼ w;}

for each u2 X:

Proof Let u02 X and the sequence fung be defined as

Tnu0¼ un:

Suppose that un6¼ unþ1 for all n. Using the inequality (2),

we obtain Z Sðun;un;unþ1Þ 0 1ðtÞdt h Z Sðun1;un1;unÞ 0 1ðtÞdt hn Z Sðu0;u0;u1Þ 0 1ðtÞdt: ð3Þ If we take limit for n! 1, using the inequality (3) we get

lim n!1 Z Sðun;un;unþ1Þ 0 1ðtÞdt ¼ 0;

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since h2 ð0; 1Þ. The condition (1) implies lim

n!1Sðun; un; unþ1Þ ¼ 0:

Now we show that the sequence fung is a Cauchy

sequence. Assume that fung is not Cauchy. Then there

exists an e [ 0 and subsequencesfmkg and fnkg such that

mk\nk\mkþ1 with

Sðumk; umk; unkÞ e ð4Þ

and

Sðumk; umk; unk1Þ\e:

Hence using Lemma2.1, we have

Sðumk1; umk1; unk1Þ 2Sðumk1; umk1; umkÞ þ Sðunk1; unk1; umkÞ \ 2Sðumk1; umk1; umkÞ þ e and lim k!1 Z Sðu_{mk 1};u_{mk 1};u_{nk 1}Þ 0 1ðtÞdt Ze 0 1ðtÞdt: ð5Þ

Using the inequalities (2), (4) and (5) we obtain Ze 0 1ðtÞdt Z Sðu_mk;u_mk;u_nkÞ 0 1ðtÞdt h Z Sðu_{mk 1};u_{mk 1};u_{nk 1}Þ 0 1ðtÞdt h Ze 0 1ðtÞdt;

which is a contradiction with our assumption since h2 ð0; 1Þ. So the sequence fung is Cauchy. Using the

completeness hypothesis, there exists w2 X such that lim

n!1T

n_u

0¼ w:

From the inequality (2) we find Z SðTw;Tw;unþ1Þ 0 1ðtÞdt ¼ Z SðTw;Tw;TunÞ 0 1ðtÞdt h Z Sðw;w;unÞ 0 1ðtÞdt:

If we take limit for n! 1, we get Z

SðTw;Tw;wÞ

0

1ðtÞdt ¼ 0;

which implies Tw¼ w:

Now we show the uniqueness of the fixed point. Suppose that w1 is another fixed point of T. Using the

Z Sðw;w;w1Þ 0 1ðtÞdt ¼ Z SðTw;Tw;Tw1Þ 0 1ðtÞdt h Z Sðw;w;w1Þ 0 1ðtÞdt; which implies Z Sðw;w;w1Þ 0 1ðtÞdt ¼ 0;

since h2 ð0; 1Þ. Using the inequality (1) we get w¼ w1.

Consequently, the fixed point w is unique. h Remark 2.5

(1) If we set the function 1 :½0; 1Þ ! ½0; 1Þ in Theo-rem2.4as

1ðtÞ ¼ 1;

for all t2 ½0; 1Þ, then we obtain the Banach’s contraction principle on a complete S-metric space. (2) Since an S-metric space is a generalization of a

metric space, Theorem2.4is a generalization of the classical Banach’s fixed-point theorem.

(3) If we set the S-metric as S : X X X ! C and take the function 1 :½0; 1Þ ! ½0; 1Þ as

1ðtÞ ¼ 1;

for all t2 ½0; 1Þ in Theorem2.4, then we get Theorem 3.1 in [10] and Corollary 2.5 in [5] for n¼ 1:

Example 2.6 Let X¼ R, k [ 1 be a fixed real number and the function S : X X X ! ½0; 1Þ be defined as

Sðu; v; zÞ ¼ k

kþ 1ðjv zj þ v þ z 2uj jÞ;

for all u; v; z2 R. It can be easily seen that the function S is an S-metric. Now we show that this S-metric can not be generated by any metric q. On the contrary, we assume that there exists a metric q such that

Sðu; v; zÞ ¼ qðu; zÞ þ qðv; zÞ; ð6Þ

for all u; v; z2 R. Hence we find Sðu; u; zÞ ¼ 2qðu; zÞ ¼ 2k kþ 1ju zj and qðu; zÞ ¼ k kþ 1ju zj: ð7Þ Similarly, we get Sðv; v; zÞ ¼ 2qðv; zÞ ¼ 2k jv zj

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and

qðv; zÞ ¼ k

kþ 1jv zj: ð8Þ

Using the equalities (6), (7) and (8), we obtain k kþ 1ðjv zj þ v þ z 2uj jÞ ¼ k kþ 1ju zj þ k kþ 1jv zj; which is a contradiction. Consequently, S is not generated by any metric andðR; SÞ is a complete S-metric space.

Let us define the self-mapping T :R! R as Tu¼u

6;

for all u2 R and the function 1 : ½0; 1Þ ! ½0; 1Þ as 1ðtÞ ¼ 3t2_;

for all t2 ½0; 1Þ. Then we get Ze 0 1ðtÞdt ¼ Ze 0 3t2dt¼ e3_{[ 0;}

for each e [ 0. Therefore T satisfies the inequality (2) in Theorem2.4for h¼1 2: Indeed, we have k3 27ðk þ 1Þ3ju vj 3 4k 3 ðk þ 1Þ3ju vj 3 ;

for all u; v2 R. Consequently, T has a unique fixed point u¼ 0:

Now we give the first generalization of Theorem2.4. Theorem 2.7 Let (X, S) be a complete S-metric space, the function 1 :½0; 1Þ ! ½0; 1Þ be defined as in (1) and T: X! X be a self-mapping of X such that

Z SðTu;Tu;TvÞ 0 1ðtÞdt h1 Z Sðu;u;vÞ 0 1ðtÞdt þ h2 Z SðTu;Tu;vÞ 0 1ðtÞdt þ h3 Z SðTv;Tv;uÞ 0 1ðtÞdt þ h4 Z maxfSðTu;Tu;uÞ;SðTv;Tv;vÞg 0 1ðtÞdt; ð9Þ for all u; v2 X with nonnegative real numbers hi ði 2

f1; 2; 3; 4gÞ satisfying maxfh1þ 3h3þ 2h4; h1þ h2þ

h3g\1: Then T has a unique fixed point w 2 X and we

have lim n!1T n u¼ w; for each u2 X.

Tnu0¼ un:

Suppose that un6¼ unþ1 for all n. Using the inequality (9),

the condition (S2) and Lemma2.1we get Z

Sðun;un;unþ1Þ

0

1ðtÞdt ¼ Z

SðTun1;Tun1;TunÞ

0 1ðtÞdt h1 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h2 Z Sðun;un;unÞ 0 1ðtÞdt þ h3 Z Sðunþ1;unþ1;un1Þ 0 1ðtÞdt þ h4 Z maxfSðun;un;un1Þ;Sðunþ1;unþ1;unÞg 0 1ðtÞdt ¼ h1 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h3 Z Sðunþ1;unþ1;un1Þ 0 1ðtÞdt þ h4 Z maxfSðun;un;un1Þ;Sðunþ1;unþ1;unÞg 0 1ðtÞdt h1 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h3 Z 2Sðunþ1;unþ1;unÞ 0 1ðtÞdt þ h3 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h4 Z Sðun;un;un1Þ 0 1ðtÞdt þ h4 Z Sðunþ1;unþ1;unÞ 0 1ðtÞdt ¼ ðh1þ h3þ h4Þ Z Sðun1;un1;unÞ 0 1ðtÞdt þ ð2h3þ h4Þ Z Sðun;un;unþ1Þ 0 1ðtÞdt; which implies Z Sðun;un;unþ1Þ 0 1ðtÞdt h1þ h3þ h4 1 2h3 h4 Sðun1Z;un1;unÞ 0 1ðtÞdt: ð10Þ If we put h¼h1þh3þh4

12h3h4 then we find h\1 since

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Z Sðun;un;unþ1Þ 0 1ðtÞdt hn Z Sðu0;u0;u1Þ 0 1ðtÞdt: ð11Þ

If we take limit for n! 1, using the inequality (11) we get lim n!1 Z Sðun;un;unþ1Þ 0 1ðtÞdt ¼ 0;

since h2 ð0; 1Þ. The condition (1) implies lim

By the similar arguments used in the proof of Theo-rem2.4, we see that the sequence fung is Cauchy. Then

there exists w2 X such that lim

n!1T

n_u

0¼ w;

since (X, S) is a complete S-metric space. From the inequality (9) we find Z Sðun;un;TwÞ 0 1ðtÞdt ¼ Z SðTun1;Tun1;TwÞ 0 1ðtÞdt h1 Z Sðun1;un1;wÞ 0 1ðtÞdt þ h2 Z Sðun;un;wÞ 0 1ðtÞdt þ h3 Z SðTw;Tw;un1Þ 0 1ðtÞdt þ h4 Z maxfSðun;un;un1Þ;SðTw;Tw;wÞg 0 1ðtÞdt:

Taking limit for n! 1 and using Lemma2.1we get Z SðTw;Tw;wÞ 0 1ðtÞdt ðh3þ h4Þ Z SðTw;Tw;wÞ 0 1ðtÞdt;

which implies Tw¼ w since h3þ h4\1.

Now we show the uniqueness of the fixed point. Let w1

be another fixed point of T. Using the inequality (9) and Lemma2.1, we get Z Sðw;w;w1Þ 0 1ðtÞdt ¼ Z SðTw;Tw;Tw1Þ 0 1ðtÞdt h1 Z Sðw;w;w1Þ 0 1ðtÞdt þ h2 Z Sðw;w;w1Þ 0 1ðtÞdt þ h3 Z Sðw1;w1;wÞ 0 1ðtÞdt þ h4 Z maxfSðw;w;wÞ;Sðw1;w1;w1Þg 0 1ðtÞdt; Z Sðw;w;w1Þ 0 1ðtÞdt ðh1þ h2þ h3Þ Z Sðw;w;w1Þ 0 1ðtÞdt: Then we obtain Z Sðw;w;w1Þ 0 1ðtÞdt ¼ 0;

that is, w¼ w1since h1þ h2þ h3\1: Consequently, T has

a unique fixed point w2 X: h

Remark 2.8

1ðtÞ ¼ 1;

for all t2 ½0; 1Þ, then we obtain Theorem 3 in [6]. (2) Theorem2.7is a generalization of Theorem2.4on a complete S-metric space. Indeed, if we take h1¼ h

and h2¼ h3¼ h4¼ 0 in Theorem2.7, then we get

Theorem2.4.

(3) Since Theorem2.7 is a generalization of Theo-rem2.4, Theorem 2.7 generalizes the classical Banach’s fixed-point theorem.

1ðtÞ ¼ 1;

for all t2 ½0; 1Þ in Theorem2.7, then we get Theorem 3.1 in [7].

Now we give the second generalization of Theorem2.4. Theorem 2.9 Let (X, S) be a complete S-metric space, the function 1 :½0; 1Þ ! ½0; 1Þ be defined as in (1) and T : X! X be a self-mapping of X such that

Z SðTu;Tu;TvÞ 0 1ðtÞdt h1 Z Sðu;u;vÞ 0 1ðtÞdt þ h2 Z SðTu;Tu;uÞ 0 1ðtÞdt þ h3 Z SðTu;Tu;vÞ 0 1ðtÞdt þ h4 Z SðTv;Tv;uÞ 0 1ðtÞdt þ h5 Z SðTv;Tv;vÞ 0 1ðtÞdt þ h6 Z maxfSðu;u;vÞ;SðTu;Tu;uÞ;SðTu;Tu;vÞ;SðTv;Tv;uÞ;SðTv;Tv;vÞg 0 1ðtÞdt; ð12Þ

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for all u; v2 X with nonnegative real numbers hi ði 2

f1; 2; 3; 4; 5; 6gÞ satisfying maxfh1þ h2þ 3h4þ h5þ

3h6; h1þ h3þ h4þ h6g\1: Then T has a unique fixed

point w2 X and we have lim

n!1T

n_u_{¼ w;}

for each u2 X:

Tnu0¼ un:

Suppose that un 6¼ unþ1for all n. Using the inequality (12),

the condition (S2) and Lemma2.1we get

which implies Z Sðun;un;unþ1Þ 0 1ðtÞdt h1þ h2þ h4þ h6 1 2h4 h5 2h6 Sðun1Z;un1;unÞ 0 1ðtÞdt: ð13Þ If we put h¼ h1þh2þh4þh6

12h4h52h6 then we find h\1 since h1þ

h2þ 3h4þ h5þ 3h6\1: Using the inequality (13) we have

Z Sðun;un;unþ1Þ 0 1ðtÞdt hn Z Sðu0;u0;u1Þ 0 1ðtÞdt: ð14Þ

If we take limit for n! 1; using the inequality (14) we get lim n!1 Z Sðun;un;unþ1Þ 0 1ðtÞdt ¼ 0;

since h2 ð0; 1Þ: The condition (1) implies lim

By the similar arguments used in the proof of Theorem2.4, we see that the sequencefung is Cauchy. Then there exists

w2 X such that lim

n!1T

n_u

0¼ w;

since (X, S) is a complete S-metric space. From the inequality (12) we find Z Sðun;un;TwÞ 0 1ðtÞdt ¼ Z SðTun1;Tun1;TwÞ 0 1ðtÞdt h1 Z Sðun1;un1;wÞ 0 1ðtÞdt þ h2 Z Sðun;un;un1Þ 0 1ðtÞdt þ h3 Z Sðun;un;wÞ 0 1ðtÞdt þ h4 Z SðTw;Tw;un1Þ 0 1ðtÞdt þ h5 Z SðTw;Tw;wÞ 0 1ðtÞdt þ h6 Z

maxfSðun1;un1;wÞ;Sðun;un;un1Þ;Sðun;un;wÞ;SðTw;Tw;un1Þ;SðTw;Tw;wÞg

0

1ðtÞdt:

If we take limit for n! 1, using Lemma2.1we get Z

Sðun;un;unþ1Þ

0

1ðtÞdt ¼ Z

SðTun1;Tun1;TunÞ

0 1ðtÞdt h1 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h2 Z Sðun;un;un1Þ 0 1ðtÞdt þ h3 Z Sðun;un;unÞ 0 1ðtÞdt þ h4 Z Sðunþ1;unþ1;un1Þ 0 1ðtÞdt þ h5 Z Sðunþ1;unþ1;unÞ 0 1ðtÞdt þ h6 Z

maxfSðun1;un1;unÞ;Sðun;un;un1Þ;Sðun;un;unÞ;Sðunþ1;unþ1;un1Þ;Sðunþ1;unþ1;unÞg

0 1ðtÞdt ðh1þ h2þ h4þ h6Þ Z Sðun1;un1;unÞ 0 1ðtÞdt þ ð2h4þ h5þ 2h6Þ Z Sðunþ1;unþ1;unÞ 0 1ðtÞdt;

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Z SðTw;Tw;wÞ 0 1ðtÞdt ðh4þ h5þ h6Þ Z SðTw;Tw;wÞ 0 1ðtÞdt;

which implies Tw¼ w since h4þ h5þ h6\1.

be another fixed point of T. Using the inequality (12) and Lemma2.1, we get Z Sðw;w;w1Þ 0 1ðtÞdt ¼ Z SðTw;Tw;Tw1Þ 0 1ðtÞdt h1 Z Sðw;w;w1Þ 0 1ðtÞdt þ h2 Z Sðw;w;wÞ 0 1ðtÞdt þ h3 Z Sðw;w;w1Þ 0 1ðtÞdt þ h4 Z Sðw1;w1;wÞ 0 1ðtÞdt þ h5 Z Sðw1;w1;w1Þ 0 1ðtÞdt þ h6 Z maxfSðw;w;w1Þ;Sðw;w;wÞ;Sðw;w;w1Þ;Sðw1;w1;wÞ;Sðw1;w1;w1Þg 0 1ðtÞdt; which implies Z Sðw;w;w1Þ 0 1ðtÞdt ðh1þ h3þ h4þ h6Þ Z Sðw;w;w1Þ 0 1ðtÞdt: Then we obtain Z Sðw;w;w1Þ 0 1ðtÞdt ¼ 0;

that is, w¼ w1since h1þ h3þ h4þ h6\1: Consequently,

T has a unique fixed point w2 X: h

Remark 2.10

(1) In Theorem2.9, if we set the function 1 :½0; 1Þ ! ½0; 1Þ as

1ðtÞ ¼ 1;

for all t2 ½0; 1Þ, then we obtain Theorem 4 in [6]. (2) Theorem2.9is a generalization of Theorem2.4on a complete S-metric space. Indeed, if we take h1¼ h

and h2¼ h3¼ h4¼ h5¼ h6¼ 0 in Theorem2.9,

then we get Theorem2.4.

(3) Since Theorem2.9 is another generalization of Theorem2.4, Theorem 2.9generalizes the classical Banach’s fixed-point theorem.

1ðtÞ ¼ 1;

In the following example we give a self-mapping sat-isfying the conditions of Theorems2.7 and 2.9, respec-tively, but does not satisfy the condition of Theorem2.4. Example 2.11 LetR be the complete S-metric space with the S-metric defined in Example 1 given in [9]. Let us define the self-mapping T :R! R as

Tu¼ uþ 80 if u2 f0; 2g 75 if otherwise

;

for all u2 R and the function 1 : ½0; 1Þ ! ½0; 1Þ as 1ðtÞ ¼ 2t;

for all t2 ½0; 1Þ. Then we get Ze 0 1ðtÞdt ¼ Ze 0 2tdt¼ e2 [ 0;

for each e [ 0. Therefore T satisfies the inequality (9) in Theorem2.7 for h1¼ h2¼ h3¼ 0, h4 ¼1₂ and the

inequality (12) in Theorem 2.9for h1¼ h3¼ h4¼ h5¼ 0,

h2¼ h6¼1₃: Hence T has a unique fixed point u¼ 75: But

T does not satisfy the inequality (2) in Theorem2.4. Indeed, if we take u¼ 0 and v ¼ 1; then we obtain

Z10 0 2tdt¼ 100 h Z2 0 2tdt¼ 4h;

which is a contradiction since h2 ð0; 1Þ:

Finally, we give another generalization of Theorem2.4. Theorem 2.12 Let (X, S) be a complete S-metric space, the function 1 :½0; 1Þ ! ½0; 1Þ be defined as in (1) and T : X! X be a self-mapping of X such that

Z SðTu;Tu;TvÞ 0 1ðtÞdt h1 Z Sðu;u;vÞ 0 1ðtÞdt þ h2 Z SðTu;Tu;uÞ 0 1ðtÞdt þ h3 Z SðTv;Tv;vÞ 0 1ðtÞdt þ h4 Z maxfSðTu;Tu;vÞ;SðTv;Tv;uÞg 0 1ðtÞdt; ð15Þ for all u; v2 X with nonnegative real numbers hi ði 2

f1; 2; 3; 4gÞ satisfying h1þ h2þ h3þ 3h4\1: Then T has

a unique fixed point w2 X and we have lim

n!1T

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Tnu0¼ un:

Suppose that un 6¼ unþ1for all n. Using the inequality (15),

the condition (S2) and Lemma2.1we get Z

Sðun;un;unþ1Þ

0

1ðtÞdt ¼ Z

SðTun1;Tun1;TunÞ

0 1ðtÞdt h1 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h2 Z Sðun;un;un1Þ 0 1ðtÞdt þ h3 Z Sðunþ1;unþ1;unÞ 0 1ðtÞdt þ h4 Z maxfSðun;un;unÞ;Sðunþ1;unþ1;un1Þg 0 1ðtÞdt h1 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h2 Z Sðun1;un1;unÞ 0 1ðtÞdt þ h3 Z Sðun;un;unþ1Þ 0 1ðtÞdt þ h4 Z 2Sðun;un;unþ1ÞþSðun1;un1;unÞ 0 1ðtÞdt ðh1þ h2þ h4Þ Z Sðun1;un1;unÞ 0 1ðtÞdt þ ðh3þ 2h4Þ Z Sðun;un;unþ1Þ 0 1ðtÞdt; which implies Z Sðun;un;unþ1Þ 0 1ðtÞdt h1þ h2þ h4 1 h3 2h4 Sðun1Z;un1;unÞ 0 1ðtÞdt: ð16Þ If we put h¼h1þh2þh4

1h32h4 then we find h\1 since

h1þ h2þ h3þ 3h4\1. Using the inequality (16) and

mathematical induction, we have Z Sðun;un;unþ1Þ 0 1ðtÞdt hn Z Sðu0;u0;u1Þ 0 1ðtÞdt: ð17Þ

Taking limit for n! 1 and using the inequality (17) we find lim n!1 Z Sðun;un;unþ1Þ 0 1ðtÞdt ¼ 0;

since h2 ð0; 1Þ. The condition (1) implies

lim

By the similar arguments used in the proof of Theorem2.4, we see that the sequencefung is Cauchy. Then there exists

w2 X such that lim

n!1T

n_u

0¼ w;

since (X, S) is a complete S-metric space. From the inequality (15) we find Z Sðun;un;TwÞ 0 1ðtÞdt ¼ Z SðTun1;Tun1;TwÞ 0 1ðtÞdt h1 Z Sðun1;un1;wÞ 0 1ðtÞdt þ h2 Z Sðun;un;un1Þ 0 1ðtÞdt þ h3 Z SðTw;Tw;wÞ 0 1ðtÞdt þ h4 Z maxfSðun;un;wÞ;SðTw;Tw;un1Þg 0 1ðtÞdt:

If we take limit for n! 1, using Lemma2.1we get Z SðTw;Tw;wÞ 0 1ðtÞdt ðh3þ h4Þ Z SðTw;Tw;wÞ 0 1ðtÞdt;

which implies Tw¼ w since h3þ h4\1:

be another fixed point of T. Using the inequality (15) and Lemma2.1, we get Z Sðw;w;w1Þ 0 1ðtÞdt ¼ Z SðTw;Tw;Tw1Þ 0 1ðtÞdt h1 Z Sðw;w;w1Þ 0 1ðtÞdt þ h2 Z Sðw;w;wÞ 0 1ðtÞdt þ h3 Z Sðw1;w1;w1Þ 0 1ðtÞdt þ h4 Z maxfSðw;w;w1Þ;Sðw1;w1;wÞg 0 1ðtÞdt; which implies Z Sðw;w;w1Þ 0 1ðtÞdt ðh1þ h4Þ Z Sðw;w;w1Þ 0 1ðtÞdt: Then we obtain Z Sðw;w;w1Þ 0 1ðtÞdt ¼ 0;

(9)

that is, w¼ w1 since h1þ h4\1: Consequently, T has a

unique fixed point w2 X: h

Remark 2.13

1ðtÞ ¼ 1;

for all t2 ½0; 1Þ, then we obtain Theorem 2 in [6]. (2) Theorem2.12 is another generalization of Theo-rem2.4on a complete S-metric space. Indeed, if we take h1¼ h and h2¼ h3¼ h4 ¼ 0 in Theorem2.12,

then we get Theorem2.4.

(3) Since Theorem2.12 is another generalization of Theorem2.4, Theorem2.12generalizes the classical Banach’s fixed-point theorem.

Let us consider the self-mapping T :R! R and the function 1 :½0; 1Þ ! ½0; 1Þ defined in Example 2.11. Then T satisfy the contractive condition (15) in Theo-rem2.12and so u¼ 75 is a unique fixed point of T. Notice that T does not satisfy the inequality (2) in Theorem2.4.

An application to the Fredholm integral equation

In this section, we give an application of the contraction condition (2) to the Fredholm integral equation

yðuÞ ¼ lðuÞ þ k Zb

a

kðu; tÞyðtÞdt; ð18Þ

where y : a; b½ ! R with 1\a\b\1, k(u, t) which is called the kernel of the integral equation (18) is continuous on the squared region ½a; b a; b½ with kðu; tÞj j M (M [ 0) and l(u) is continuous on a; b½ :

Let C a; b½ ¼ f j f : a; bf ½ ! Risacontinuousfunctiong: Now we define the function S: C a; b½ C a; b½ C a; b½ ! 0; 1½ Þ by Sðf ; g; hÞ ¼ sup u2 a;b½ fðuÞ hðuÞ j j þ sup u2 a;b½

fðuÞ þ hðuÞ 2gðuÞ

j j;

ð19Þ for all f ; g; h2 C a; b½ : Then the function S is an S-metric. Now we show that this S-metric can not be generated by any metric q: We assume that this S-metric is generated by any metric q; that is, there exists a metric q such that

Sðf ; g; hÞ ¼ qðf ; hÞ þ qðg; hÞ; ð20Þ for all f ; g; h2 C a; b½ . Then we get

Sðf ; f ; hÞ ¼ 2qðf ; hÞ ¼ 2 sup u2 a;b½ fðuÞ hðuÞ j j qðf ; hÞ ¼ sup u2 a;b½ fðuÞ hðuÞ j j: _ð21Þ Similarly, we obtain Sðg; g; hÞ ¼ 2qðg; hÞ ¼ 2 sup u2 a;b½ gðuÞ hðuÞ j j and qðg; hÞ ¼ sup u2 a;b½ gðuÞ hðuÞ j j: _ð22Þ

Using the equalities (20), (21) and (22), we find sup

u2 a;b½

fðuÞ hðuÞ

j j þ sup

u2 a;b½

fðuÞ þ hðuÞ 2gðuÞ

j j ¼ sup u2 a;b½ fðuÞ hðuÞ j j þ sup u2 a;b½ gðuÞ hðuÞ j j;

which is a contradiction. Hence this S-metric is not gen-erated by any metric q: Consequently, ðC a; b½ ; SÞ is a complete S-metric space.

Proposition 3.1 Let C a; bð ½ ; SÞ be a complete S-metric space with the S-metric defined in (19) and k be a real number with

k

j j\ 1

Mðb aÞ:

Then the Fredholm integral equation (18) has a unique solution y: a; b½ ! R:

Proof Let us define the function T : C a; b½ ! C a; b½ as

TyðuÞ ¼ lðuÞ þ k Zb

a

kðu; tÞyðtÞdt:

Now we show that T satisfies the contractive condition (2). We get SðTy1; Ty1; Ty2Þ ¼ 2 sup u2 a;b½ Ty1ðuÞ Ty2ðuÞ j j ¼ 2 sup u2 a;b½ k Zb a

kðu; tÞ yð 1ðuÞ y2ðuÞÞdt

2 kj jM sup u2 a;b½ Zb a y1ðuÞ y2ðuÞ ð Þdt 2 kj jM sup u2 a;b½ Zb a y1ðuÞ y2ðuÞ j jdt 2 kj jM sup u2 a;b½ y1ðuÞ y2ðuÞ j j Zb a dt kj jMðb aÞSðy1; y1; y2Þ

(10)

which implies Z

SðTy1;Ty1;Ty2Þ

0 1ðtÞdt\ Z Sðy1;y1;y2Þ 0 1ðtÞdt:

Consequently, the contractive condition (2) is satisfied and the Fredholm integral equation (18) has a unique solution

y. h

Now we give an example of Proposition3.1.

Example 3.2 Let us consider the Fredholm integral equation defined as yðuÞ ¼ e þ k Ze 1 ln u t yðtÞdt: ð23Þ

Now we find a solution of the Fredholm integral equation (23) with the initial condition y0ðuÞ ¼ 0. We solve this

equation for kj j\ 1 e1since ln u t \1 for all 1 u; t e. We obtain y1ðuÞ ¼ e; y2ðuÞ ¼ e þ k Ze 1 ln u t edt¼ e þ ke ln u; y3ðuÞ ¼ e þ k Ze 1 ln u t ðe þ ke ln tÞdt ¼ e þ ke ln u þk 2 2e ln u; y4ðuÞ ¼ e þ k Ze 1 ln u t eþ ke ln t þ k2 2 e ln t dt ¼ e þ ke ln u þk 2 2e ln uþ k3 2 e ln u; ynðuÞ ¼ e þ ke ln u 1 þ k 2þ k2 4 þ þ kn 2n ! e þ 2k 2 ke ln u:

Consequently, this is a solution of the Fredholm integral equation (18) for kj j\ 1

e1\1.

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