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

An S-metric space is a three-dimensional generalization 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 findings 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.


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 X 6 ¼ ; be any set and S : X Â X Â X ! ½0; 1Þ be a function satisfying the following conditions for all u; v; z; a 2 X.
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 selfmappings satisfying various contractive conditions on an Smetric 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, h 2 ð0; 1Þ and T : X ! X be a self-mapping of X such that SðTu; Tu; TvÞ hSðu; u; vÞ; for all u; v 2 X: Then T has a unique fixed point in X.
On the other hand some generalizations of the wellknown Ć irić's and Nemytskii-Edelstein fixed-point theorems 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 conditions 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, contractive 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 summable ( i.e., with finite integral) on each compact subset of ½0; 1Þ; nonnegative and such that for each e [ 0; 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. for all u; v 2 X; then T has a unique fixed point w 2 X such that lim n!1 T n u ¼ w; for each u 2 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 Lemma 2.1 can be considered as a symmetry condition on an S-metric space. The following definition is related to convergent sequences on an S-metric space.
(1) A sequence fu n g in X converges to u if and only if Sðu n ; u n ; uÞ ! 0 as n ! 1. That is, there exists n 0 2 N such that for all n ! n 0 , Sðu n ; u n ; uÞ\e for each e [ 0. We denote this by lim n!1 u n ¼ uor lim n!1 Sðu n ; u n ; uÞ ¼ 0: (2) A sequence fu n g in X is called a Cauchy sequence if Sðu n ; u n ; u m Þ ! 0 as n; m ! 1. That is, there exists n 0 2 N such that for all n; m ! n 0 , Sðu n ; u n ; u m Þ\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) S q ðu; v; zÞ ¼ qðu; zÞ þ qðv; zÞ for all u; v; z 2 X is an S-metric on X. We call the function S q defined in Lemma 2.3 (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. for all u; v 2 X: Then T has a unique fixed point w 2 X and we have lim n!1 T n u ¼ w; for each u 2 X: Proof Let u 0 2 X and the sequence fu n g be defined as Suppose that u n 6 ¼ u nþ1 for all n. Using the inequality (2), we obtain If we take limit for n ! 1, using the inequality (3) we get lim n!1 Z Sðu n ;u n ;u nþ1 Þ 0 1ðtÞdt ¼ 0; since h 2 ð0; 1Þ. The condition (1) implies lim n!1 Sðu n ; u n ; u nþ1 Þ ¼ 0: Now we show that the sequence fu n g is a Cauchy sequence. Assume that fu n g is not Cauchy. Then there exists an e [ 0 and subsequences fm k g and fn k g such that m k \n k \m kþ1 with and Sðu m k ; u m k ; u n k À1 Þ\e: Hence using Lemma 2.1, we have Using the inequalities (2), (4) and (5) we obtain which is a contradiction with our assumption since h 2 ð0; 1Þ. So the sequence fu n g is Cauchy. Using the completeness hypothesis, there exists w 2 X such that lim n!1 T n u 0 ¼ w: From the inequality (2) we find If we take limit for n ! 1, we get Z SðTw;Tw;wÞ 0 1ðtÞdt ¼ 0; which implies Tw ¼ w: Now we show the uniqueness of the fixed point. Suppose that w 1 is another fixed point of T. Using the inequality (2) we have (1) If we set the function 1 : ½0; 1Þ ! ½0; 1Þ in Theorem 2.4 as for all t 2 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, Theorem 2.4 is a generalization of the classical Banach's fixed-point theorem.
(3) If we set the S-metric as S : X Â X Â X ! C and take the function 1 : ½0; 1Þ ! ½0; 1Þ as for all t 2 ½0; 1Þ in Theorem 2.4, then we get Theorem 3.1 in [10] and Corollary 2.5 in [5] for n ¼ 1: be a fixed real number and the function S : X Â X Â X ! ½0; 1Þ be defined as for all u; v; z 2 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 for all u; v; z 2 R. Hence we find Similarly, we get Using the equalities (6), (7) and (8), we obtain which is a contradiction. Consequently, S is not generated by any metric and ðR; SÞ is a complete S-metric space. Let us define the self-mapping T : R ! R as for all u 2 R and the function 1 : ½0; 1Þ ! ½0; 1Þ as for all t 2 ½0; 1Þ. Then we get for each e [ 0. Therefore T satisfies the inequality (2) in for all u; v 2 R. Consequently, T has a unique fixed point u ¼ 0: Now we give the first generalization of Theorem 2.4.
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 for all u; v 2 X with nonnegative real numbers h i ði 2 f1; 2; 3; 4gÞ satisfying maxfh 1 þ 3h 3 þ 2h 4 ; h 1 þ h 2 þ h 3 g\1: Then T has a unique fixed point w 2 X and we have lim n!1 T n u ¼ w; for each u 2 X.
Proof Let u 0 2 X and the sequence fu n g be defined as Suppose that u n 6 ¼ u nþ1 for all n. Using the inequality (9) which implies If we put h ¼ h 1 þh 3 þh 4 1À2h 3 Àh 4 then we find h\1 since h 1 þ 3h 3 þ 2h 4 \1. Using the inequality (10) we have Z Sðu n ;u n ;u nþ1 Þ 0 1ðtÞdt h n Z Sðu 0 ;u 0 ;u 1 Þ 0 1ðtÞdt: If we take limit for n ! 1, using the inequality (11)  Sðu n ; u n ; u nþ1 Þ ¼ 0: By the similar arguments used in the proof of Theorem 2.4, we see that the sequence fu n g is Cauchy. Then there exists w 2 X such that lim n!1 T n u 0 ¼ w; since (X, S) is a complete S-metric space. From the inequality (9) we find Then we obtain 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 for all u; v 2 X with nonnegative real numbers h i ði 2 f1; 2; 3; 4; 5; 6gÞ satisfying maxfh 1 þ h 2 þ 3h 4 þ h 5 þ 3h 6 ; h 1 þ h 3 þ h 4 þ h 6 g\1: Then T has a unique fixed point w 2 X and we have lim n!1 T n u ¼ w; for each u 2 X: Proof Let u 0 2 X and the sequence fu n g be defined as Suppose that u n 6 ¼ u nþ1 for all n. Using the inequality (12), the condition (S2) and Lemma 2.1 we get which implies Z Sðu n ;u n ;u nþ1 Þ If we put h ¼ h 1 þh 2 þh 4 þh 6 1À2h 4 Àh 5 À2h 6 then we find h\1 since h 1 þ h 2 þ 3h 4 þ h 5 þ 3h 6 \1: Using the inequality (13) we have Z Sðu n ;u n ;u nþ1 Þ 0 1ðtÞdt h n Z Sðu 0 ;u 0 ;u 1 Þ 0 1ðtÞdt: If we take limit for n ! 1; using the inequality (14) we get lim n!1 Z Sðu n ;u n ;u nþ1 Þ 0 1ðtÞdt ¼ 0; since h 2 ð0; 1Þ: The condition (1) implies lim n!1 Sðu n ; u n ; u nþ1 Þ ¼ 0: By the similar arguments used in the proof of Theorem 2.4, we see that the sequence fu n g is Cauchy. Then there exists w 2 X such that lim n!1 T n u 0 ¼ w; since (X, S) is a complete S-metric space. From the inequality (12)  If we take limit for n ! 1, using Lemma 2.1 we get Z Sðu n ;u n ;u nþ1 Þ Z maxfSðu nÀ1 ;u nÀ1 ;u n Þ;Sðu n ;u n ;u nÀ1 Þ;Sðu n ;u n ;u n Þ;Sðu nþ1 ;u nþ1 ;u nÀ1 Þ;Sðu nþ1 ;u nþ1 ;u n Þg for all t 2 ½0; 1Þ, then we obtain Theorem 4 in [6]. (2) Theorem 2.9 is a generalization of Theorem 2.4 on a complete S-metric space. Indeed, if we take for all t 2 ½0; 1Þ in Theorem 2.9, then we get Theorem 3.4 in [7].
In the following example we give a self-mapping satisfying the conditions of Theorems 2.7 and 2.9, respectively, but does not satisfy the condition of Theorem 2.4.

Example 2.11
Let R 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 for all u 2 R and the function 1 : ½0; 1Þ ! ½0; 1Þ as 1ðtÞ ¼ 2t; for all t 2 ½0; 1Þ. Then we get for each e [ 0. Therefore T satisfies the inequality (9) in Theorem 2.7 for Hence T has a unique fixed point u ¼ 75: But T does not satisfy the inequality (2) in Theorem 2.4. Indeed, if we take u ¼ 0 and v ¼ 1; then we obtain which is a contradiction since h 2 ð0; 1Þ: Finally, we give another generalization of Theorem 2.4. 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 Proof Let u 0 2 X and the sequence fu n g be defined as T n u 0 ¼ u n : Suppose that u n 6 ¼ u nþ1 for all n. Using the inequality (15), the condition (S2) and 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 Theorem 2.12 and so u ¼ 75 is a unique fixed point of T. Notice that T does not satisfy the inequality (2)  Á Á Á y n ðuÞ ¼ e þ ke ln u 1 þ k 2 þ k 2 4 þ Á Á Á þ k n 2 n ! e þ 2k 2 À k e ln u: Consequently, this is a solution of the Fredholm integral equation (18) for k j j\ 1 eÀ1 \1.
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