Some applications of fixed point results for generalized two classes of Boyd–Wong’s Fcontraction in partial bmetric spaces
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Abstract
In this paper, we will present some fixed point results for two classes of generalized contractions of Boyd–Wong type in partial bmetric 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 Fcontractions of type A and of type B and establish some new common fixed point results in partial bmetric 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 slipups 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.
Keywords
Common fixed point Partial metric spaces bmetric spaces Fcontraction \(\alpha \)AdmissibleMathematics subject classification
47H10 54H25Introduction and preliminaries
There are lots of extensions and generalizations of metric space. In 1989, Bakhtin [1] introduced the notion of bmetric space, and in 1993, Czerwik [2, 3] extensively used the concept of bmetric 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 bmetric and partial metric space by introducing the partial bmetric space. After that, in [6], Mustafa et al. introduced a modified version of partial bmetric space. On the other hand, in 2012, Wardowski [7] introduced a new contraction called Fcontraction 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 FSuzuki 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 Fcontraction in partial bmetric 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 slipups of recent papers present in the literature. Some examples and applications are presented to highlight the realized improvement.
In the sequel, \(\mathbb {R}\), \(\mathbb {N}\), and \(\mathbb {N^{*}}\) 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.
Definition 1.1

\((b_{1})\) \(d(x,y)=0\) iff \(x=y\);

\((b_{2})\) \(d(x,y)=d(y,x)\);

\((b_{3})\) \(d(x,y)\le s[d(x,z)+d(z,y)].\)
Definition 1.2

\((p_1)\) \(x=y\) iff \(p(x,x)=p(x,y)=p(y,y)\);

\((p_2)\) \(p(x,x)\le p(x,y)\);

\((p_3)\) \(p(x,y)=p(y,x)\);

\((p_4)\) \(p(x,y)\le p(x,z)+p(z,y)p(z,z)\).
Definition 1.3

\((p_{b_1})\) \(x=y\) iff \(p_{b}(x,x)=p_{b}(x,y)=p_{b}(y,y)\);

\((p_{b_2})\) \(p_{b}(x,x)\le p_{b}(x,y)\);

\((p_{b_3})\) \(p_{b}(x,y)=p_{b}(y,x)\);

\((p_{b_4})\) \(p_{b}(x,y)\le s[p_{b}(x,z)+p_{b}(z,y)]p_{b}(z,z)\).
In the following definition, Mustafa et al. [6] modified the Definition 1.3 to find that each partial bmetric \(p_b\) generates a bmetric \(d_{p_{b}}\).
Definition 1.4

\((p_{b_1})\) \(x=y\) iff \(p_{b}(x,x)=p_{b}(x,y)=p_{b}(y,y)\);

\((p_{b_2})\) \(p_{b}(x,x)\le p_{b}(x,y)\);

\((p_{b_3})\) \(p_{b}(x,y)=p_{b}(y,x)\);

\((p_{b_4})\) \(p_{b}(x,y)\le s(p_{b}(x,z)+p_{b}(z,y)p_{b}(z,z))+(\frac{1s}{2})(p_{b}(x,x)+p_{b}(y,y))\).
Example 1.1
Remark 1.1
The class of partial bmetric space \((X,p_b)\) is effectively larger that the class of partial metric space, since a partial metric space is a special case of a partial bmetric space \((X,p_b)\) when \(s=1\). In addition, the class of partial bmetric space \((X,p_b)\) is effectively larger that the class of bmetric space, since a bmetric space is a special case of a partial bmetric space \((X,p_b)\) when the self distance \(p(x,x)=0\).
Proposition 1.1
[5] Let X be a nonempty set, and let p be a partial metric and d be a bmetric with the coefficient \(s\ge 1\) on X. Then, the function \(p_{b} : X \times X \rightarrow [0, \infty )\) defined by \(p_b(x,y)=p(x,y)+d(x,y)\) for all \(x,y\in X\) is a partial bmetric on X with the coefficient s.
Proposition 1.2
[6] Every partial bmetric \(p_b\) defines a bmetric \(d_{p_{b}}\), where
\(d_{p_{b}}(x,y)=2p_b(x,y)p_b(x,x)p_b(y,y)\,\,\,\,\) for all \(\,\,\,x,y\in X.\)
For \(p_b\)convergent, \(p_b\)Cauchy sequence, and \(p_b\)complete, we refer [6].
Lemma 1.1
 1.
A sequence \(\{x_n\}\) is a \(p_b\)Cauchy sequence in \((X,p_b)\) if and only if it is a bCauchy sequence in the bmetric space \((X,d_{p_b})\);
 2.
\((X,p_b)\) is \(p_b\)complete if and only if the bmetric space \((X,d_{p_b})\) is complete. Moreover, \(\mathop {\lim }\limits _{n\rightarrow \infty } d_{p_b}(x_n,x)=0\) if and only if \(p_b(x,x)= \mathop {\lim }\limits _{n\rightarrow \infty } p_b(x_n,x)= \mathop {\lim }\limits _{n,m\rightarrow \infty } p_b(x_n,x_m).\)
Definition 1.5
Definition 1.6
[16] Let \(f,g:X\rightarrow X\) and \(\alpha : X\times X\rightarrow [0,\infty )\). The mapping f is g\(\alpha \)admissible if for all \(x,y\in X\), such that \(\alpha (gx,gy)\ge 1,\) we have \(\alpha (fx,fy)\ge 1.\)
If g is identity mapping, then f is called \(\alpha \)admissible.
Definition 1.7
[17] An \(\alpha \)admissible map f is said to be triangular \(\alpha \)admissible if \(x,y,z\in X,\) \(\alpha (x,z)\ge 1\) and \(\alpha (z,y)\ge 1\) \(\Longrightarrow \) \(\alpha (x,y)\ge 1\).
Definition 1.8
 (i)
the pair (f, g) is said to be weakly increasing if \(fx\preceq gfx\) and \(gx\preceq fgx\) for all \(x\in X\);
 (ii)
f is said to be gweakly isotone increasing if \(fx\preceq gfx\preceq fgfx\) for all \(x\in X\).
Lemma 1.2
 1.
\(\phi \) is monotonic increasing, i.e., \(t_1\le t_2 \Longrightarrow \phi (t_1)\le \phi (t_2)\).
 2.
\(\phi \) is continuous and \(\phi (t)<t\) for each \(t>0.\)
On the other hand, Wardowski [7] introduced the Fcontraction as follows:
Definition 1.9
 (F1)
F is strictly increasing, that is, for \(\alpha , \beta \in \mathbb {R^+}\), such that \(\alpha <\beta \) implies \(F(\alpha )<F(\beta )\).
 (F2)
For each sequence \(\{\alpha _n\}\) of positive numbers \(\mathop {\lim }\limits _{n\rightarrow \infty } \alpha _n=0\) if and only if \(\mathop {\lim }\limits _{n\rightarrow \infty } F(\alpha _n)=\infty \).
 (F3)
There exists \(k\in (0,1)\), such that \(\mathop {\lim }\limits _{\alpha \rightarrow 0^+} \alpha ^k F(\alpha )=0\).
We denote the set of all functions satisfying (F1)–(F3) by \(\digamma \). On the other hand, Secelean [8] proved the following lemma.
Lemma 1.3
 (a)
If \(\mathop {\lim }\limits _{n\rightarrow \infty } F(\alpha _n)=\infty \), then \(\mathop {\lim }\limits _{n\rightarrow \infty } \alpha _n=0\).
 (b)
If \(\inf F=\infty \) and \(\mathop {\lim }\limits _{n\rightarrow \infty } \alpha _n=0\), then \(\mathop {\lim }\limits _{n\rightarrow \infty } F(\alpha _n)=\infty \).
Secelean [8] reintegrated the condition (F2) by more elementary condition \((F2^{'})\).
\((F2^{'})\) \(\inf F=\infty \),
or, also by
\((F2^{'^{'}})\), there exists a sequence \(\{\alpha _n\}_{n=1}^\infty \) of positive real numbers, such that \(\mathop {\lim }\limits _{n\rightarrow \infty } F(\alpha _n)=\infty \).
Most recently, Piri et al. [9] used the following condition \((F3^{'})\) instead of (F3).
\((F3^{'})\,\,\) F is continuous on \((0, \infty )\).
We denote the set of all functions satisfying (F1), \((F2^{'})\), and \((F3^{'})\) by \(\Delta _F\).
Main results
Common fixed point results for Boyd–Wong type A Fcontraction
In this section, we present our essential results. For this, we introduce the following definition.
Definition 2.1
It needs mentioning that the following lemma will be useful in proving our main results.
Lemma 2.1
Let \((X,p_b)\) be a complete partial bmetric space. Let f and g are selfmappings on X, such that (f, g) is a Boyd–Wong \(\mathbf type A \) Fcontraction on \((X,p_b)\). If f or g has a fixed point u in X, then u is a unique common fixed point of f and g and \(p_b(u,u)=0.\)
Proof
One of our main result of this paper is the following one.
Theorem 2.1
 1.
f is \(\alpha \)admissible.
 2.
There exists \(x_0\in X\), such that \(\alpha (x_0, fx_0)\ge 1\).
 3.
(f, g) is a Boyd–Wong type A Fcontraction on \((X,p_b)\).
Proof
The following examples show the superiority of our assertions. \(\square \)
Example 2.1
Now, we will show that f is \(\alpha \)admissible. Let \(x,y\in X\), such that \(\alpha (x,y)\ge 1\). By the definition of f and \(\alpha \), we have \(\alpha (fx,fy)\ge 1\), for all \(x,y\in [0,30]\). Hence, f is an \(\alpha \)admissible. On the other hand, there exists \(x_0=0\in X\), such that \(\alpha (0,f0)=\alpha (0,0)=1\ge 1.\)
Without loss of generality, we may take \(x,y\in X\), such that \(x>y\). To check the contractive condition (1) of Theorem 2.1, we have to consider the following cases:
From Figs. 1 and 2, we obtain that inequality (1) holds for all \(x,y\in [0,30]\) with \(\epsilon \in (1,4.5].\)
From Figs. 3 and 4, it is easy to verify that inequality (32) holds for all \(x,y\in (30,\infty ]\).
Example 2.2
Let \(X=\{0,1,2\}\). Inspired by [18], let we define a partial bmetric \(p_b:X\times X\rightarrow [0.\infty )\) by \(p_b(x,x)=0\) for all \(x\in X\), \(p_b(0,1)=p_b(1,0)=p_b(1,2)=p_b(2,1)=1\), \(p_b(0,2)=p_b(2,0)=9/4\) with the partialorder relation \(x\preceq y \,\,\, \Longleftrightarrow \,\,\, x<y\)
It is easy to obtain that \((X,p_b)\) is a complete partial bmetric space with \(s=9/8.\) Define self maps f and g by \(f0=1\); \(f1=1\); \(f2=0\) and \(g0=g2=0\); \(g1=1.\) Clearly, the mappings f and g are continuous. Let \(\alpha (x,y)=1\) for all \(x,y\in X\). Taking \(x_0=2\), we have \(\alpha (2,f2)=\alpha (2,0)=1\ge 1.\)
Let \(\phi (t)=\frac{19t+3}{23}\) and \(\psi (t)=\frac{1}{50(t+1)}\).
It is easy to see that the contractive condition (1) of Theorem 2.1 is satisfied for the points \(x=1, y=2\) and \(x=0, y=2\) with \(1<\epsilon < 5\) and \(F(t)=\log t\). However, it is not holding for the point \(x=0, y=1\). Thus, \(x=1\) is not the unique common fixed point of the mappings f and g.
Example 2.3
Case II: If \(x,y\in (\frac{1}{10}, \frac{23}{100}]\), then \(\alpha (x,y)=\frac{\log 1.35 (e^5e^4)}{e^3e^2}\). By repeating the same process as in case I, one can easily say that (1) is satisfied for all \(x,y\in (\frac{1}{10}, \frac{23}{100}]\).
From all cases, we conclude that (f, g) is a Boyd–Wong type A Fcontraction on X. Notice that, all the conditions of Theorem 2.1 are satisfied and \(x=\frac{2}{10}\) is the unique common fixed point of the mappings f and g.
The following result is an immediate consequence of Theorem 2.1 using \(g=f\) for all \(x\in X\), \(\psi (t)=\tau >0\), \(\alpha (x,y)=1\) for all \(x,y\in X\), and \(\phi (t)=t\) for all \(t\in [0,\infty )\).
Corollary 2.1
Common fixed point results for Boyd–Wong type \(A^{*}\) Fcontraction
Theorem 2.2
 1.
The pair (f, g) is weakly increasing.
 2.
For every two comparable elements \(x,y\in X\), (f, g) is a Boyd–Wong type \(A^{*}\) Fcontraction on \((X,p_b)\).
Proof
The rest of the proof run on the lines of the proof of Theorem 2.1. This conclude the proof. \(\square \)
Common fixed point results for Boyd–Wong type B Fcontraction
In this section, we launch the following definition:
Definition 2.2
Theorem 2.3
 1.
f is g\(\alpha \)admissible and triangular \(\alpha \)admissible.
 2.
There exists \(x_0\in X\), such that \(\alpha (gx_0, fx_0)\ge 1\).
 3.
f is a Boyd–Wong type B Fcontraction with respect to g on \((X,p_b)\).
 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 \(\{f,g\}\) is weakly compatible.
 6.
Either \(\alpha (u,v)\ge 1\) or \(\alpha (v,u)\ge 1\) whenever \(fu=gu\) and \(fv=gv\)
Proof
The following example demonstrates the usability of Theorem 2.3. \(\square \)
Example 2.4
Clearly, \(f,\,g\) are continuous mappings and \(fX\subseteq gX\). To prove that f is g\(\alpha \)admissible mapping, let \(x,y\in X\), such that \(\alpha (gx,gy)\ge 1\), then by the definition of \(\alpha \) and \(fX\subseteq gX\), we have \(\alpha (fx,fy)\ge 1\). Thus, we conclude that f is g\(\alpha \)admissible mapping. Taking \(x_0=0\in X\), we have \(\alpha (gx_0,fx_0)=\alpha (g0,f0)=\alpha (0,0)=1\ge 1.\) Let \(x,y,z\in X\), such that \(\alpha (x,z)\ge 1\) and \(\alpha (z,y)\ge 1\), from the definition of \(\alpha \), we have \(\alpha (x,y)\ge 1\), i.e., f is triangular \(\alpha \)admissible. To verify the inequality (29) of Theorem 2.3, we have to consider the following cases:
From Figs. 6 and 7, we have inequality (29) holds for all \(x,y\in [0,2.5]\) with \(\epsilon \in (1,1.9].\)
Case III. If \(y\in [0,2.5]\) and \(x\in (2.5,5]\), then Case III is analogous to Case II that is why we omit the details.
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 \(s=1\), \(\psi (p_b(x,y))=\tau >0\), \(\phi (t)=t\), \(\alpha (x,y)=1\) and \(s=1\), \(f=g\), \(\psi (p_b(x,y))=\tau >0\), \(\phi (t)=t\), \(\alpha (x,y)=1\), respectively.
Remark 2.2
If we take \(s=1\), \(\alpha (x,y)=1\) \(\psi (p_b(x,y))=\tau >0\), \(\phi (t)=t\) and \(f=g\) in Theorem 2.1, then we obtain Theorem 3.2 [20] of Radenovic and Kadelburg along with Shukla.
Remark 2.3
We generalize the Theorem 17 of Alsulami et al. in [21] for partial bmetric space.
Remark 2.4
Theorem 2.1 of Singh et al. [22] is particular case of Theorem 2.3 by taking \(\alpha (x,y)=1\), \(gx=x\), and \(s=1\).
In [7], author stated that the Fcontraction is the modified version of Banach contraction principle. Wardowski deduced that the Banach contractions are particular case of Fcontractions and the author supported his finding by presenting some Fcontractions which are not Banach contractions.
In view of aforesaid, we generalized and extend the following results present in the literature:
Remark 2.5
By introducing Theorems 2.1 and 2.3, we generalized the results of Satish Shukla [5] and obtained the Fcontraction version of [5] in partial bmetric spaces.
Remark 2.6
Taking \(\epsilon =1\), \(f=g\) and \(\phi (t)=kt\), where \(k\in [0,1)\) in Theorem 2.2 is akin to Corollary 1 of Mustafa [6] in the sense of Fcontraction. On the other hand, to be specific taking \(\alpha (x,y)=1\), \(\epsilon =2\) and \(\phi (t)=kt\), where \(k\in [0,1)\) in Theorem 2.2 reduces to Corollary 3 of Mustafa [6] for Fcontraction.
Remark 2.7
If we take \(\epsilon =1\) and \(f=g\) in Theorem 2.1, then Theorem 2.6 in [26] due to Latif et al. is attained.
Remark 2.8
Theorem 2.1 of Huang et al.[23] is particular case of Theorem 2.2 for Fcontraction by taking \(\alpha (x,y)=1\) and \(\phi (t)=t\)
Remark 2.9
In Theorem 2.3, if we put \(s=1\), \(\alpha (x,y)=1\), and \(gx=x\), then we obtain Theorem 2.3 for Fcontraction by S. Romaguera in [24].
Slipups in some recent papers and their remedies
 1.
In [25], we point out that how \(\mathbb {R^{+}}\) can be extended to hold the definition of \(\alpha \)admissible map.
 2.In Definition 2.1, authors [25] defined the almost generalized \((\alpha \)–\(\psi \)–\(\phi \)–\(\theta )\)contraction. On using this authors reported thatwhich is worthless, one need to replace \(\alpha (gx_{n1},gx_{n})\) by \(\alpha (x_{n1},x_{n})\). The authors committed the same mistakes on the page no. 7 and 8. On the other hand, authors wrote$$\begin{aligned} \psi (d(gx_n,gx_{n+1}))\le \alpha (gx_{n1},gx_{n})\psi (s^{3}d(Tx_{n1},Tx_n))\,\,\,\,\,\,(\mathrm{{see \,\,\,inequality}}\,\, (2.4)), \end{aligned}$$Therefore, there was a dispute regarding to condition (2.1) in the whole paper [25].$$\begin{aligned} \psi (s^{3}d(Tu,Tv))\le \alpha (u,v)\psi (s^{3}d(Tu,Tv))\,\,\,\,\,(\mathrm{{see \,\,\,page\, no.}}\,\, 9). \end{aligned}$$
 3.
In [26], authors committed a blunder. Notice that, in the context of Theorem 2.6, authors defined \(M_s(x,y)\) in terms of d(x, y), and in whole proof of Theorem 2.6, they used \(M_s(x,y)\) in the form of \(p_b(x,y)\), which is unsound as for this to hold one needs to supplant d(x, y) by \(p_b(x,y)\) in the statement of Theorem 2.6.
 4.
In [19] from inequality (7), authors got a contradiction and concluded that \(F(p(x_{2n},x_{2n+2}))=0\) that is \(x_{2n}=x_{2n+2}\). Now, by the property of partial metric space \(x_{2n}=x_{2n+2}\) when \(p(x_{2n},x_{2n+2})=0\), 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 \(2^{p1}Sx(t)Sy(t)\le \root p \of {ln (M(x,y)+1)}\) and \(2^{p1}Sx(t)Sy(t)^{p}\le ln(M(x,y)+1)\), which is not possible. For this, we suggested rectification in our application part.
Applications
Application to solutions of integral equations
Theorem 4.1
 (1)$$\begin{aligned} \max _{a\le t \le b} \int _{a}^{b}G(t,z)^{q} \mathrm{d}z\le \frac{2}{ba}. \end{aligned}$$
 (2)For all \(x,y\in \mathbb {R},\) the following inequality holds:Then, the integral equation (50) has a solution.$$\begin{aligned} f(z,x)f(z,y)^{q}\le \frac{1}{2s^{\epsilon }}([xy^{q}+r] e^{\tau }s^{\epsilon }r). \end{aligned}$$
Proof
Application to solutions of ordinary differential equations:
Definition 4.1
Theorem 4.2
Proof
\(\square \)

In Theorems 2.1 and 2.3 can Boyd–Wong type A and type B Fcontraction be improved by cyclic contraction.

Can Theorem 2.1 be extended and generalized replacing \(\alpha \)admissible by twisted \((\alpha , \beta )\)admissible.

Can Theorem 2.3 be extended and generalized replacing g\(\alpha \)admissible by generalized\(\alpha \)admissible.
Notes
Acknowledgements
The authors acknowledge the financial support provided by King Mongkut’s University of Technology Thonburi through the “KMUTT 55th Anniversary Commemorative Fund”. This project was supported by the Theoretical and Computational Science (TaCS) Center under Computational and Applied Science for Smart Innovation Cluster (CLASSIC), Faculty of Science, KMUTT.
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