Some applications of fixed point results for generalized two classes of Boyd–Wong’s Fcontraction in partial bmetric spaces
 293 Downloads
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.
References
 1.Bakhtin, I.A.: The contraction mapping principle in quasi metric spaces. Funct. Anal. Unianowsk Gos. Ped. Inst. 30, 26–37 (1989)MATHGoogle Scholar
 2.Czerwik, S.: Contraction mappings in \(b\)metric spaces. Acta Math. Inform. Univ. Ostrav. 1, 5–11 (1993)MathSciNetMATHGoogle Scholar
 3.Czerwik, S.: Nonlinear setvalued contraction mappings in \(b\)metric spaces. Atti Semin. Mat. Fis. Univ. Modena 46, 263–276 (1998)MathSciNetMATHGoogle Scholar
 4.Matthews, S.G.: Partial metric topology, in proceeding of the 8th summer conference on general topology and application. Ann. N. Y. Acad. Sci. 728, 183–197 (1994)CrossRefGoogle Scholar
 5.Shukla, S.: Partial \(b\)metric spaces and fixed point theorems. Mediterr. J. Math. 11, 703–711 (2014)MathSciNetCrossRefMATHGoogle Scholar
 6.Mustafa, Z., Roshan, J.R., Parvanesh, V., Kadelburg, Z.: Some common fixed point results in ordered partial \(b\)metric spaces. J. Ineq. Appl. 2013, 562 (2013)MathSciNetCrossRefMATHGoogle Scholar
 7.Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)MathSciNetCrossRefMATHGoogle Scholar
 8.Secelean, NA.: Iterated function systems consisting of Fcontractions. Fixed Point Theory Appl. 2013, 277 (2013)MathSciNetCrossRefMATHGoogle Scholar
 9.Piri, H., Kumam, P.: Some fixed point theorems concerning \(F\)contraction in complete metric spaces. Fixed Point Theory Appl. 2014, 210 (2014)MathSciNetCrossRefMATHGoogle Scholar
 10.Gopal, D., Abbas, M., Patel, D.K., Vetro, C.: Fixed points of \(alpha\)type \(F\)contractive mappings with an application to nonlinear fractional differential equation. Acta Math. Sci. 36(3), 957–970 (2016)MathSciNetCrossRefMATHGoogle Scholar
 11.Budhia, L.B., Kumam, P., Moreno, J.M., Gopal, D.: Extensions of almost\(F\) and \(F\)Suzuki contractions with graph and some applications to fractional calculus. Fixed Point Theory Appl. 2016, 2 (2016)MathSciNetCrossRefMATHGoogle Scholar
 12.Padcharoen, A., Gopal, D., Chaipuniya, P., Kumam, P.: Fixed point and periodic point results for \(\alpha \)type \(F\)contractions in modular metric spaces. Fixed Point Theory Appl. 2016, 39 (2016)MathSciNetCrossRefMATHGoogle Scholar
 13.Sumala, P., Kumam, P., Gopal, D.: Computational coupled fixed points for \(F\)contractive mappings in metric spaces endowed with a graph. J. Math. Comput. Sci. 16, 372–385 (2016)CrossRefGoogle Scholar
 14.Singh, D., Chauhan, V., Wangkeeree, R.: Geraghty type generalized \(F\) contractions and related applications in partial bmetric spaces. Int. J. Anal. 2017, 14 (2017)MathSciNetMATHGoogle Scholar
 15.Samet, B., Vetro, C., Vetro, P.: Fixed point theorem for \(\alpha \)\(\psi \)contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)MathSciNetCrossRefMATHGoogle Scholar
 16.Rosa, V.L., Vetro, P.: Common fixed points for \(\alpha \)\(\psi \)\(\phi \)contractions in generalized metric spaces. Model. Control 19(1), 43–54 (2014)MathSciNetGoogle Scholar
 17.Karapinar, E., Kumam, P., Salimi, P.: On \(\alpha \)\(\psi \)Meir–Keeler contractive mappings. Fixed Point Theory Appl. 2013, 94 (2013)MathSciNetCrossRefMATHGoogle Scholar
 18.Roshan, J.R., Parvaneh, V., Kadelburg, Z.: Common fixed point theorems for weakly isotone increasing mappings in ordered \(b\)metric spaces. J. Nonlinear Sci. Appl. 7, 229–245 (2014)MathSciNetCrossRefMATHGoogle Scholar
 19.Shukla, S., Radenovic, S.: Some common fixed point theorems for \(F\)contraction type mappings in 0complete partial metric spaces. J. Math. 2013, 7 (2013)MATHGoogle Scholar
 20.Shukla, S., Radenovic, S., Kadelburg, Z.: Some fixed point theorems for ordered \(F\)generalized contractions in 0forbitally complete partial metric spaces. Theory Appl. Math. Compt. Sci. 4(1), 87–98 (2014)MATHGoogle Scholar
 21.Alsulami, H.H., Karapinar, E., and Piri, H.: Fixed points for generalized FSuzuki type contraction in complete bmetric spaces. Discrete Dyn. Nat. Soc. 2015, 969726 (2015)MathSciNetCrossRefGoogle Scholar
 22.Singh, D., Chauhan, V., Kumam, P., Joshi, V.: Application of fixed point results for cyclic BoydWong type generalized \(F\)\(\psi \)contractions to dynamic programming. J. Math. Comput. Sci. 17, 200–215 (2017)CrossRefGoogle Scholar
 23.Huang, H., Vujakovic, J., Radenovic, S.: A note on common fixed point theorems for isotone increasing mappings in ordered \(b\)metric spaces. J. Nonlinear Sci. Appl. 8, 808–815 (2015)MathSciNetCrossRefMATHGoogle Scholar
 24.Romaguera, S.: Matkowski’s type theorems for generalized contractions on (ordered) partial metric spaces. Appl. Gen. Topol. 12(2), 213–220 (2011)MathSciNetMATHGoogle Scholar
 25.Arab, R., Haghighi, A.A.S.: Fixed pints of admissible almost contractive type mappings on \(b\)metric spaces with an application to quadratic integral equations. J. Inequal. Appl. 2015, 32 (2015)CrossRefMATHGoogle Scholar
 26.Latif, A., Roshan, J.R., Parvaneh, V., Hussain, N.: Fixed point results via \(\alpha \)admissible mappings and cyclic contractive mappings in partial \(b\)metric spaces. J. Inequal. Appl. 2014, 345 (2014)MathSciNetCrossRefMATHGoogle Scholar
 27.Shahkoohi, R.J., Razani, A.: Some fixed point theorems for rational geraghty contractive mappings in ordered \(b\)metric spaces. J. Inequal. Appl. 2014, 373 (2014)MathSciNetCrossRefMATHGoogle Scholar
 28.Harjani, J., Sadarangani, K.: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 71, 3403–3410 (2009)MathSciNetCrossRefMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.