Some applications of fixed point results for generalized two classes of Boyd–Wong’s F-contraction in partial b-metric spaces

In this paper, we will present some fixed point results for two classes of generalized contractions of Boyd–Wong type in partial b-metric spaces. More precisely, the structure of the paper is the following. In section one, we present some useful notions and results. The aim of section two is to introduce the concepts of Boyd–Wong F-contractions of type A and of type B and establish some new common fixed point results in partial b-metric spaces. We show the validity and superiority of our main results by suitable examples which are visualized by corresponding surfaces and related graphs. In section three, we correct some slip-ups in some recent papers. Finally, in section four, two applications to integral equation and periodic boundary value problem are included which make effective the new concepts and results.


Introduction and preliminaries
There are lots of extensions and generalizations of metric space. In 1989, Bakhtin [1] introduced the notion of b-metric space, and in 1993, Czerwik [2,3] extensively used the concept of b-metric space. On the other hand, the concept of partial metric space was introduced by Mathews [4]. In recent times, Shukla [5] generalized both the concept of b-metric and partial metric space by introducing the partial b-metric space. After that, in [6], Mustafa et al. introduced a modified version of partial b-metric space. On the other hand, in 2012, Wardowski [7] introduced a new contraction called F-contraction and proved a fixed point result as a generalization of the Banach contraction principle. Very recently, Piri et al. [9] improve the result of Wardowski [7] by launching the concept of an F-Suzuki contraction and proved some curious fixed point results. The results of Wardowski [7] were generalized by several authors (see, e.g., [10][11][12][13][14] ).
The purpose of this article is to extend the concept of Fcontraction by introducing Boyd-Wong type A and type B F-contraction in partial b-metric space, motivated and inspired by the ideas of Wardowski [7] and Mustafa et al. [6]. Our results substantially generalize and extend the corresponding results contained in Shukla et al. [19,20], Alsulami et al. [21], Singh et al. [22], and many others. We also point out some slip-ups of recent papers present in the literature. Some examples and applications are presented to highlight the realized improvement.
In the sequel, R, N, and N Ã will represent the set of all real numbers, natural numbers, and positive integers, respectively. Some elementary definitions and fundamental results, which will be used in the sequel, are described here.
A partial metric space is a pair (X, p), such that X is a nonempty set and p is a partial metric on X. Definition 1.3 [5] Let X be a nonempty set and s ! 1 be a given real number. A function p b : X Â X ! ½0; 1Þ is called a partial b-metric if for all x; y; z 2 X the following conditions are satisfied: ðp b 1 Þ x ¼ y iff p b ðx; xÞ ¼ p b ðx; yÞ ¼ p b ðy; yÞ; ðp b 2 Þ p b ðx; xÞ p b ðx; yÞ; ðp b 3 Þ p b ðx; yÞ ¼ p b ðy; xÞ; ðp b 4 Þ p b ðx; yÞ s½p b ðx; zÞ þ p b ðz; yÞ À p b ðz; zÞ.
The pair ðX; p b Þ is called a partial b-metric space. The number s ! 1 is called the coefficient of ðX; p b Þ.
In the following definition, Mustafa et al. [6] modified the Definition 1.3 to find that each partial b-metric p b generates a b-metric d p b . Definition 1.4 [6] Let X be a nonempty set and s ! 1 be a given real number. A function p b : X Â X ! ½0; 1Þ is called a partial b-metric if for all x; y; z 2 X the following conditions are satisfied: ðp b 1 Þ x ¼ y iff p b ðx; xÞ ¼ p b ðx; yÞ ¼ p b ðy; yÞ; ðp b 2 Þ p b ðx; xÞ p b ðx; yÞ; ðp b 3 Þ p b ðx; yÞ ¼ p b ðy; xÞ; ðp b 4 Þ p b ðx; yÞ sðp b ðx; zÞ þ p b ðz; yÞ À p b ðz; zÞÞþ ð 1Às 2 Þðp b ðx; xÞ þ p b ðy; yÞÞ. The pair ðX; p b Þ is called a partial b-metric space. The number s ! 1 is called the coefficient of ðX; p b Þ.
Example 1.1 [5] Let X ¼ R þ , q [ 1 be a constant and p b : X Â X ! R þ be defined by p b ðx; yÞ ¼ ½maxfx; yg q þ jx À yj q ; for all x; y 2 X: Then, ðX; p b Þ is a partial b-metric space with the coefficient s ¼ 2 qÀ1 [ 1; but it is neither a bmetric nor a partial metric space.
Remark 1.1 The class of partial b-metric space ðX; p b Þ is effectively larger that the class of partial metric space, since a partial metric space is a special case of a partial bmetric space ðX; p b Þ when s ¼ 1. In addition, the class of partial b-metric space ðX; p b Þ is effectively larger that the class of b-metric space, since a b-metric space is a special case of a partial b-metric space ðX; p b Þ when the self distance pðx; xÞ ¼ 0.
Proposition 1.1 [5] Let X be a nonempty set, and let p be a partial metric and d be a b-metric with the coefficient s ! 1 on X. Then, the function p b : X Â X ! ½0; 1Þ defined by p b ðx; yÞ ¼ pðx; yÞ þ dðx; yÞ for all x; y 2 X is a partial bmetric on X with the coefficient s.
yÞ À p b ðx; xÞ À p b ðy; yÞ for all x; y 2 X: For p b -convergent, p b -Cauchy sequence, and p b -complete, we refer [6]. Lemma 1.1 [6] Let ðX; p b Þ be a partial b-metric space. Then Definition 1.5 [15] Let f be a self-mapping on X and a : X Â X ! ½0; 1Þ be a function. We say that f is an a-admissible mapping if x; y 2 X; aðx; yÞ ! 1 ¼) aðfx; fyÞ ! 1: Let f ; g : X ! X and a : X Â X ! ½0; 1Þ. The mapping f is g-a-admissible if for all x; y 2 X, such that aðgx; gyÞ ! 1; we have aðfx; fyÞ ! 1: If g is identity mapping, then f is called a-admissible.
Definition 1.8 [18] Let ðX; "Þ be a partially ordered set and let f, g be two maps on X. Then (i) the pair (f, g) is said to be weakly increasing if fx " gfx and gx " fgx for all x 2 X; (ii) f is said to be g-weakly isotone increasing if fx " gfx " fgfx for all x 2 X.
Note that if f ; g : X ! X are weakly increasing, then f is a g-weakly isotone increasing.
Lemma 1.2 [17] Let f be a triangular a-admissible mapping. Assume that there exists x 0 2 X, such that aðx 0 ; fx 0 Þ ! 1: Define the sequence fx n g by x n ¼ f n x 0 : Then aðx m ; x n Þ ! 1 for all m; n 2 N with m\n: Let U be the set of functions / : ½0; 1Þ ! ½0; 1Þ, such that

/ is continuous and /ðtÞ\t for each t [ 0:
Let W denote the set of all decreasing functions w : ð0; 1Þ ! ð0; 1Þ.
On the other hand, Wardowski [7] introduced the Fcontraction as follows: Definition 1.9 Let F : R þ ! R be a mapping satisfying: (F1) F is strictly increasing, that is, for a; b 2 R þ , such that a\b implies FðaÞ\FðbÞ. (F2) For each sequence fa n g of positive numbers lim n!1 a n ¼ 0 if and only if lim n!1 Fða n Þ ¼ À1.
We denote the set of all functions satisfying (F1)-(F3) by F . On the other hand, Secelean [8] proved the following lemma.

Lemma 1.3 [8]
Let F : R þ ! R be an increasing mapping and fa n g 1 n¼1 be a sequence of positive real numbers. Then the following assertions hold: Fða n Þ ¼ À1.
or, also by ðF2 0 0 Þ, there exists a sequence fa n g 1 n¼1 of positive real numbers, such that lim n!1 Fða n Þ ¼ À1.
We denote the set of all functions satisfying (F1), ðF2 0 Þ, and ðF3 0 Þ by D F .

Main results
Common fixed point results for Boyd-Wong type A F-contraction In this section, we present our essential results. For this, we introduce the following definition.
Definition 2.1 Let ðX; p b Þ be a partial b-metric space and f ; g : X ! X two mappings. In addition, suppose that a : X Â X ! ½0; 1Þ be a function, where a is defined as in Definition 1.7. We say that the pair (f, g) is a Boyd-Wong type A F-contraction on a partial b-metric space X if there exists F 2 D F ; w 2 W and / 2 U, such that for all x; y 2 X and s and [ 1 is a constant. If aðx; yÞ ¼ 1 for all x; y 2 X, then the pair (f, g) is called Boyd-Wong type A Ã Fcontraction.
It needs mentioning that the following lemma will be useful in proving our main results. Lemma 2.1 Let ðX; p b Þ be a complete partial b-metric space. Let f and g are self-mappings on X, such that (f, g) is a Boyd-Wong typeA F-contraction on ðX; p b Þ. If f or g has a fixed point u in X, then u is a unique common fixed point of f and g and p b ðu; uÞ ¼ 0: Proof Let u 2 X be a fixed point of f, i.e., fu ¼ u. We will prove that u is also a fixed point of g on the contrary assume that p b ðfu; guÞ [ 0: From inequality (1), we have a contradiction. Hence, we conclude that p b ðu; vÞ ¼ 0; that is, u ¼ v. Thus, the common fixed point of f and g is unique.
On the other hand, if u is the fixed point of g, then by following the same procedure one can show that u is the unique common fixed point of f and g.
One of our main result of this paper is the following one.
Theorem 2.1 Let ðX; p b Þ be a complete partial b-metric space. Let f and g are self-mappings on X satisfying the following conditions: There exists x 0 2 X, such that aðx 0 ; fx 0 Þ ! 1.
3. (f, g) is a Boyd-Wong type A F-contraction on ðX; p b Þ.
Then, f and g have a unique common fixed point u 2 X with p b ðu; uÞ ¼ 0.
Proof Let x 0 2 X, such that aðx 0 ; fx 0 Þ ! 1. Define a sequence fx n g in the following way: Since f is a-admissible, therefore, aðx 0 ; x 1 Þ ¼ aðx 0 ; fx 0 Þ ! 1 ¼) aðx 1 ; x 2 Þ ¼ aðfx 0 ; fx 1 Þ ! 1: By induction, we get aðx n ; x nþ1 Þ ! 1 for all n 2 N Ã : If x mþ1 ¼ x m for any m 2 N, without loss of generality let x 2nþ1 ¼ x 2n and p b ðx 2n ; x 2nþ1 Þ [ 0. Therefore, from ðD ðF1Þ Þ, property of / and in account of inequality (1), we arrive at where Putting the value of M s ðx 2n ; x 2nþ1 Þ in (5) and using the property of / and w, we have Consequently, we get Hence, x 2n is a common fixed point of f and g.
Assume that x m 6 ¼ x mþ1 for all m 2 N, that is, Taking (6) into account, we arrive at By the property of /; w and from (F1), one can easily get We get a contradiction. Thus, M s ðx 2n ; x 2nÀ1 Þ ¼ /ðp b ðx 2n ; x 2nÀ1 ÞÞ. Therefore, from (6) and taking the property of F; / and w in account of, we have Þg is a decreasing sequence of positive real numbers. Using the property of / and repeated use of (7) gives Since w is an decreasing function, therefore, it follows from the above inequality that from the successive application, we arrive at Similarly As F 2 D F , letting the limit as n ! 1 in (8) and (9), we get lim n!1 Fðp b ðx n ; x nþ1 ÞÞ ¼ À1. It follows from ðF2 0 Þ and Moreover, from ðp b 2 Þ, we have the following: To prove that fx n g is a p b -Cauchy sequence in X, it is sufficient to show that fx 2n g is a p b -Cauchy sequence in X. From Lemma 1.1, we need to prove that fx 2n g is a b-Cauchy sequence in the b-metric space ðX; d p b Þ. Suppose to the contrary that, there exists d [ 0, such that for an integer k, there exist integer nðkÞ [ mðkÞ ! k, such that For every integer k, let m(k) is the least positive integer exceeding n(k) satisfying (12), such that Then, from above inequality along with (14), we comes at Analogously, we derive As Fðp b ðfx 2mðkÞ ; gx 2nðkÞÀ1 ÞÞ ¼Fðp b ðx 2mðkÞþ1 ; x 2nðkÞ ÞÞ [ 0; therefore, the contractive condition (1) and property of w imply that Using the definition of M s ðx; yÞ and keeping the inequalities (16)- (19) in mind, it is easy to see that Indeed Furthermore, from (18), (20), and (21)

À Á
; which is impossible. This contradiction proves that fx n g is a b-Cauchy sequence in the b-metric space ðX; d p b Þ, then from Lemma 1.1, fx n g is a p b -Cauchy sequence in the partial b-metric space ðX; p b Þ: Since, ðX; p b Þ is complete then by Lemma 1.1 b-metric space ðX; d p b Þ is b-complete. Therefore, the sequence fx n g converges to some point u 2 X, that is, lim n!1 d p b ðx n ; uÞ ¼ 0: Next, we will show that u is a common fixed point of f and g. Since F 2 D F , therefore, from ðD F3 Þ, F is continuous. Then, we have to consider the following two cases: Case (i): For each n 2 N, there exists j n 2 N, such that p b ðx j n þ1 ; fuÞ ¼ 0, i.e., x j n þ1 ¼ fu and j n [ j nÀ1 , where j 0 ¼ 1. Then, we get This yields that u is a fixed point of f. Case (ii): There exists n 2 2 N, such that p b ðx nþ1 ; fuÞ 6 ¼ 0 for all n ! n 2 , which means p b ðfx n ; fuÞ [ 0 for all n ! n 2 . It follows from the inequality (1) that From (23) and (25), we conclude that for all n ! fn 2 ; n 3 g. Since F is continuous, taking the limit as n ! 1 in above inequality, we have Fðp b ðfu; uÞÞ\Fðp b ðfu; uÞÞ; a contradiction, which yields that our assumption is wrong, which means that u is the fixed point of f. From Lemma 2.1, u is an unique common fixed point of f and g with p b ðu; uÞ ¼ 0.

&
We also define w : ð0; 1Þ ! ð0; 1Þ by wðtÞ ¼ 1 50ðtþ1Þ and / : ½0; 1Þ ! ½0; 1Þ be given by /ðtÞ ¼ 10tþ1 12 . Let FðtÞ ¼ log t for all t 2 R þ . Now, we will show that f is a-admissible. Let x; y 2 X, such that aðx; yÞ ! 1. By the definition of f and a, we have aðfx; fyÞ ! 1, for all x; y 2 ½0; 30. Hence, f is an aadmissible. On the other hand, there exists x 0 ¼ 0 2 X, such that að0; f 0Þ ¼ að0; 0Þ ¼ 1 ! 1: Without loss of generality, we may take x; y 2 X, such that x [ y. To check the contractive condition (1) of Theorem 2.1, we have to consider the following cases: Case I. If x; y 2 ½0; 30, then from (1), we get L:H:S: ¼ aðx; yÞFðs p b ðfx; gyÞÞ By repeating the same procedure as in Case II, one can find that Case III holds for all y 2 ½0; 30 and x 2 ð30; 1.
Hence, we conclude that inequality (1) of Theorem 2.1 holds for all x; y 2 X. Notice that, in the foregoing example, all the conditions of Theorem 2.1 are fulfilled and x ¼ 0 is a unique common fixed point of the mappings f and g (see Fig. 5).
Example 2.2 Let X ¼ f0; 1; 2g. Inspired by [18], let we define a partial b-metric p b : X Â X ! ½0:1Þ by p b ðx; xÞ ¼ 0 for all x 2 X, p b ð0; 1Þ ¼ p b ð1; 0Þ ¼ p b ð1; 2Þ ¼ It is easy to obtain that ðX; p b Þ is a complete partial bmetric space with s ¼ 9=8: Define self maps f and g by Clearly, the mappings f and g are continuous. Let aðx; yÞ ¼ 1 for all x; y 2 X. Taking Let /ðtÞ ¼ 19tþ3 23 and wðtÞ ¼ 1 50ðtþ1Þ . It is easy to see that the contractive condition (1) of Theorem 2.1 is satisfied for the points x ¼ 1; y ¼ 2 and x ¼ 0; y ¼ 2 with 1\\5 and FðtÞ ¼ log t. However, it is not holding for the point x ¼ 0; y ¼ 1. Thus, x ¼ 1 is not the unique common fixed point of the mappings f and g. Example 2.3 Let X ¼ ½0; 23 100 . Clearly, X is a complete partial b-metric space with partial b-metric p b : X Â X ! ½0; 1Þ defined by p b ðx; yÞ ¼ ½maxfx; yg 2 ; 8x; y 2 X; where s ¼ 2. Let mappings f ; g : X ! X are defined by fx ¼ . By repeating the same process as in case I, one can easily say that (1) is satisfied for all x; y 2 ð 1 10 ; 23 100 . From all cases, we conclude that (f, g) is a Boyd-Wong type A F-contraction on X. Notice that, all the conditions of Theorem 2.1 are satisfied and x ¼ 2 10 is the unique common fixed point of the mappings f and g. The following result is an immediate consequence of Theorem 2.1 using g ¼ f for all x 2 X, wðtÞ ¼ s [ 0, aðx; yÞ ¼ 1 for all x; y 2 X, and /ðtÞ ¼ t for all t 2 ½0; 1Þ. Then, f has a unique fixed point and p b ðu; uÞ ¼ 0.
Common fixed point results for Boyd-Wong type A * Fcontraction Theorem 2.2 Let ðX; p b ; "Þ be a complete partially ordered partial b-metric space. Let f and g are self-mappings on X satisfying the following conditions: 1.
The pair (f, g) is weakly increasing.
2. For every two comparable elements x; y 2 X, (f, g) is a Boyd-Wong type A Ã F-contraction on ðX; p b Þ.
Then, f and g have a unique common fixed point u 2 X with p b ðu; uÞ ¼ 0.
Proof Let x 0 2 X be arbitrary. Define a sequence fx n g in the following way: fx 2n ¼ x 2nþ1 and gx 2nþ1 ¼ x 2nþ2 for all n 2 N Ã : Since f and g are weakly increasing with respect to ", therefore By induction, we obtain that x n " x nþ1 for all n ! 1. The rest of the proof run on the lines of the proof of Theorem 2.1. This conclude the proof. h

Common fixed point results for Boyd-Wong type B F-contraction
In this section, we launch the following definition: Definition 2.2 Let ðX; p b Þ be a partial b-metric space and f ; g : X ! X two mappings. In addition, suppose that a : X Â X ! ½0; 1Þ be a function, where a is defined as in Definition 1.8. We say that f is a Boyd-Wong type B Fcontraction with respect to g, on a partial b-metric space X, if there exists F 2 D F ; w 2 W and / 2 U, such that for all and [ 1 is a constant.
Theorem 2.3 Let ðX; p b Þ be a complete partial b-metric space. Let f and g are self-mappings on X, such that fX gX. Suppose that gX is closed and the following conditions hold: 1. f is g-a-admissible and triangular a-admissible.
3. f is a Boyd-Wong type B F-contraction with respect to g on ðX; p b Þ. 4. Either f or g is continuous. Then, f and g have a coincidence point in X. Moreover, f and g have a unique common fixed point if the following conditions hold: 5. The pair ff ; gg is weakly compatible. 6. Either aðu; vÞ ! 1 or aðv; uÞ ! 1 whenever fu ¼ gu and fv ¼ gv Proof By assumption (2), there exists a point x 0 2 X, such that aðgx 0 ; fx 0 Þ ! 1. As fX gX, we can find a point x 1 2 X, such that fx 0 ¼ gx 1 . By induction, we construct a sequence fx n g 2 X, such that Since f is g-a-admissible and we have aðgx 0 ; fx 0 Þ ! 1, therefore Similar to above, we get aðgx n ; gx nþ1 Þ ! 1; n ¼ 0; 1; 2; . . .: By assumption (1), f is triangular a-admissible; therefore, from Lemma 1.2, we obtain aðgx m ; gx n Þ ! 1 with m\n: If for some n 0 2 N; we have fx n 0 ¼ fx n 0 þ1 , then from (31), we have gx n 0 þ1 ¼ fx n 0 ¼ fx n 0 þ1 : Then, f and g have a coincidence point at x ¼ x n 0 þ1 . Therefore, in what follows, we assume that for each n ! 0, fx n 6 ¼ fx nþ1 holds. On using inequality (29) and property ðp b 4 Þ [5], we have Fðp b ðgx n ; gx nþ1 ÞÞ Fðs p b ðfx nÀ1 ; fx n ÞÞ aðx nÀ1 ; x n ÞFðs p b ðfx nÀ1 ; fx n ÞÞ Fð/ðM s ðx nÀ1 ; x n ÞÞÞ À wðp b ðx nÀ1 ; x n ÞÞ; in which If for some n 2 N Ã , M s ðx nÀ1 ; x n Þ ¼ p b ðgx n ; gx nþ1 Þ: Then, from (33), we get Fðp b ðgx n ; gx nþ1 ÞÞ Fð/ðp b ðgx n ; gx nþ1 ÞÞÞ À wðp b ðgx nÀ1 ; gx n ÞÞ: Utilizing the property of the function w, / and from ðD F1 Þ, we arrive at Fðp b ðgx n ; gx nþ1 ÞÞ\Fðp b ðgx n ; gx nþ1 ÞÞ; which gives a contradiction. Thus, M s ðx nÀ1 ; x n Þ ¼ p b ðgx nÀ1 ; x n Þ. Therefore, from (6) and by the property of F; / and w, we acquire Fðp b ðgx n ; gx nþ1 ÞÞ Fð/ðp b ðgx nÀ1 ; gx n ÞÞ À wðp b ðx nÀ1 ; x n ÞÞ; ð34Þ which yields p b ðgx n ; gx nþ1 Þ\p b ðgx nÀ1 ; gx n Þ: Thus, fp b ðgx n ; gx nþ1 Þg is a decreasing sequence of positive real numbers. Using the property of / and repeated use of (34), we deduce that As w is an decreasing function. Therefore, from the above inequality, we arrive at Fðp b ðgx n ; gx nþ1 ÞÞ\Fðp b ðgx nÀ2 ; gx nÀ1 ÞÞ À 2wðp b ðx nÀ2 ; x nÀ1 ÞÞ: It follows from the successive application that Since F 2 D F , making the limit as n ! 1 in (35) Since Fðp b ðfx mðkÞ ; fx nðkÞÀ1 ÞÞ ¼ Fðp b ðgx mðkÞþ1 ; gx nðkÞ ÞÞ [ 0; therefore, by inequality (29) and using the property of w, we have Hence, fgx n g is a b-Cauchy sequence in the b-metric space ðX; d p b Þ, then form Lemma 1.1, fgx n g is a p b -Cauchy sequence in the partial b-metric space ðX; p b Þ: Since gX is closed, there exists u 2 X, such that lim n!1 gx n ¼ gu: From Lemma 1.1, we have Mathematical Sciences (2018) 12:111-127 121 lim n!1 p b ðgx n ; guÞ ¼ lim n;m!1 p b ðgx n ; gx m Þ ¼ pðgu; guÞ ¼ 0: ð45Þ Now, assume that, g is continuous. Next, we will show that u is a coincidence point of f and g. On the contrary, suppose that, that is not true then p b ðfu; guÞ [ 0: From (29), we get Passing to the limit when n ! 1 in (46) and using (47), we get that a contradiction. Therefore, p b ðfu; guÞ ¼ 0, and hence, fu ¼ gu, i.e., u is a coincidence point of f and g. Similarly, one can show that u is a coincidence point of f and g when f is continuous.
Next, we will prove that gu ¼ gv. From hypotheses (5) and (6), let fu ¼ gu and fv ¼ gv whenever aðu; vÞ ! 1: Thus, by the routine calculation and from (29), we have a contradiction. Hence, we conclude that gu ¼ gv. Similarly, for aðv; uÞ ! 1, one can easily show that gu ¼ gv. By hypotheses, the pair ff ; gg is weakly compatible, i.e., fw ¼ f ðguÞ ¼ gðfuÞ ¼ gw; whenever fu ¼ gu ¼ w: Hence, w is a coincidence point of f and g, which yields Thus, w is a common fixed point of f and g. Furthermore, in a similar way as in the proof of Theorem 2.1, one can show the uniqueness of the common fixed point.
Thus, x ¼ 0 is the coincidence point of mappings f and g. It is easy to observe that the pair ff ; gg is weakly compatible and condition (6) holds whenever f 0 ¼ g0. Hence, all the conditions of Theorem 2.3 are satisfied and the subsequent figure (see Fig. 9) show that x ¼ 0 is a unique common fixed point f and g.
In view of Remark 1.1, the following observations are worth noticing in the perspective of Theorems 2.1, 2.2, and 2.3.
Remark 2.1 Theorem 10 and Corollary 13 of Shukla et al. [19] are particular case of Theorem 2.1 by taking  [21] for partial b-metric space.
In [7], author stated that the F-contraction is the modified version of Banach contraction principle. Wardowski deduced that the Banach contractions are particular case of F-contractions and the author supported his finding by presenting some F-contractions which are not Banach contractions.
In view of aforesaid, we generalized and extend the following results present in the literature: wðdðgx n ; gx nþ1 ÞÞ aðgx nÀ1 ; gx n Þwðs 3 dðTx nÀ1 ; Tx n ÞÞ ðsee inequality ð2:4ÞÞ; which is worthless, one need to replace aðgx nÀ1 ; gx n Þ by aðx nÀ1 ; x n Þ. The authors committed the same mistakes on the page no. 7 and 8. On the other hand, authors wrote wðs 3 dðTu; TvÞÞ aðu; vÞwðs 3 dðTu; TvÞÞ ðsee page no: 9Þ: Therefore, there was a dispute regarding to condition (2.1) in the whole paper [25]. 3. In [26], authors committed a blunder. Notice that, in the context of Theorem 2.6, authors defined M s ðx; yÞ in terms of d(x, y), and in whole proof of Theorem 2.6, they used M s ðx; yÞ in the form of p b ðx; yÞ, which is unsound as for this to hold one needs to supplant d(x, y) by p b ðx; yÞ in the statement of Theorem 2.6. 4. In [19] from inequality (7), authors got a contradiction and concluded that Fðpðx 2n ; x 2nþ2 ÞÞ ¼ 0 that is x 2n ¼ x 2nþ2 . Now, by the property of partial metric space x 2n ¼ x 2nþ2 when pðx 2n ; x 2nþ2 Þ ¼ 0, which is incorrect because the function F is not defined at the point 0. 5. Note that, in application section of [27], authors established equivalency between 2 pÀ1 jSxðtÞ À SyðtÞj ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lnðMðx; yÞ þ 1Þ p p and 2 pÀ1 jSxðtÞ À SyðtÞj p lnðMðx; yÞ þ 1Þ, which is not possible. For this, we suggested rectification in our application part.

Application to solutions of integral equations
In this section, we obtain the solution of the following integral equation for an unknown function u: where f : ½a; b Â R ! R, G : ½a; b Â ½a; b ! ½0; 1Þ, v : ½a; b ! R are given continuous functions. Let X be the set C[a, b] of real continuous functions on [a, b] and let p b : X Â X ! ½0; 1Þ be given by where q [ 1 and r ! 0. One can easily see that ðX; p b Þ is a complete partial b-metric space. Let the mapping T : X ! X is defined by then u is a solution of (50) if and only it is a fixed point of T. Now, we prove the subsequent theorem to show the existence of solution of integral equation.
Proof Utilizing conditions (1)-(2) and in account of inequality (50), we have for FðtÞ ¼ log t; t [ 0. Thus, all the conditions of Corollary 2.1 are satisfied. Hence, we conclude that T has a unique fixed point u Ã in X, which yields that integral equation (50) has a unique solution which belongs to X ¼ C½a; b: h Application to solutions of ordinary differential equations: Consider the following first-order periodic boundary value problem: where T [ 0 and k : I Â R ! R is a continuous function. Consider the space X ¼ CðI; RÞ of all real continuous functions on I ¼ ½0; T: Let p b : X Â X ! R þ be given by for all u; v 2 X, where q [ 1 and r ! 0. Obviously, the space ðX; p b Þ is a complete partial b-metric space with parameter s ¼ 2 qÀ1 . The space X ¼ CðI; RÞ can also equipped with the following order relation: x; y 2 CðI; RÞ x y () xðtÞ yðtÞ for all t 2 I: Consequently, by passing to logarithms, above inequality deduce to logð2 ðqÀ1Þ p b ðfu; fvÞÞ logðM s ðu; vÞÞ þ log e Às : This turns into s þ Fð2 ðqÀ1Þ p b ðfu; fvÞÞ FðM s ðu; vÞÞ: for FðtÞ ¼ log t; t [ 0. Finally, let a be a lower solution for (53). Then from [28] we have a f ðaÞ. Thus, the hypotheses of Corollary 2.1 are satisfied and f has a unique fixed point in X, i.e., the system of first-order periodic boundary value problem has a unique solution. h

Open problems
• In Theorems 2.1 and 2.3 can Boyd-Wong type A and type B F-contraction be improved by cyclic contraction.
• Can Theorem 2.1 be extended and generalized replacing a-admissible by twisted ða; bÞ-admissible. • Can Theorem 2.3 be extended and generalized replacing g-a-admissible by generalized-a-admissible.