On Jungck–Branciari–Wardowski Type Fixed Point Results

Abstract

The terms of $F-$integral contraction as well as $(\varpi ,\tilde{\zeta },F,i)-$integral contraction are introduced. Fixed point and common fixed point theorems are established. For the mapping F we use only the supposition that it is strictly increasing. As a consequence of the main theorems we obtain Jungck–Wardowski, Branciari–Wardowski and Jungck–Branciari type results. Consequently, the results presented in the article enhance and complement some known results in literature.

1. Introduction and Preliminaries

In 1976, Jungck [1] generalized the principle proposed by Banach [2] as follows:

Theorem 1.

Let $h,i:\mathsf{\Omega }\to \mathsf{\Omega }$, $ih(a)=hi(a), a\in \mathsf{\Omega }$ where $(\mathsf{\Omega },w)$ is a complete metric space, and

$$w(ha,hb)\le \lambda w(ia,ib), a,b\in \mathsf{\Omega }, \lambda \in (0,1).$$

(1)

If $h(\mathsf{\Omega })\subset i(\mathsf{\Omega })$ and i is continuous then there exists a unique $u\in \mathsf{\Omega }$ so that $hu=iu=u$.

Wardowski [3] proposed a new contractive condition that generalizes [2].

Definition 1.

Let $\left(\mathsf{\Omega },w\right)$ be a metric space and $\mathcal{F}$ be a set of mappings $F: \left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ satisfying the next three conditions:

(F1) For all $l_1,l_2\in \left(0,+\infty \right),l_1<l_2$ yields $F \left(l_1\right) <F \left(l_2\right);$

(F2) If ${\left\{a_n\right\}}_{n\in \mathbb{N}}\subset (0,+\infty )$ and $\underset{n\to +\infty }{lim}{a}_n=0,$ then $\underset{n\to +\infty }{lim}F(a_n)=-\infty$ and vice versa.

(F3) $\underset{a\to 0^{+}}{lim}{a}^{ \mu }F\left(a\right)=0$ for some $\mu \in (0,1).$

A mapping $h:\mathsf{\Omega }\to \mathsf{\Omega }$ is $F-$contraction (in the sense of D. Wardowski) on $\left(\mathsf{\Omega },w\right)$ if there exists $\varpi >0$ such that for all $a,b\in \mathsf{\Omega },$$w \left(ha,hb\right)>0$ yields

$$\varpi +F \left(w\left(ha,hb\right)\right)\le F \left(w\left(a,b\right)\right).$$

(2)

Theorem 2.

Let $(\mathsf{\Omega },w)$ be a complete metric space, and let $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be a $F-$contraction. Then, there is a $b\in \mathsf{\Omega }$, $b=hb$ and it is unique.

Remark 1.

Based on $F\left(l-0\right)\le F\left(l\right)\le F\left(l+0\right),$ $l\in \left(0,+\infty \right),$ and (F1) we conclude that there are $\underset{a\to b^{-}}{lim}F\left(a\right)=F\left(b-0\right)$ and $\underset{a\to b^{+}}{lim}F\left(a\right)=F\left(b+0\right).$ For all particulars see [4,5]. More details of the property (F2) can be found in [6,7]. Likewise, if $F:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ is a strictly increasing function, then either $F\left(0+0\right)=\underset{a\to 0^{+}}{lim}F\left(a\right)=m,m\in \mathbb{R}$ or $F\left(0+0\right)=\underset{x\to 0^{+}}{lim}F\left(a\right)=-\infty .$

In the proofs of our results in the follow-up we will use the following known lemmas from ([8,9]).

Lemma 1.

[10] Suppose that ${\left\{a_n\right\}}_{n\in \mathbb{N}}$ which belongs to a metric space $(\mathsf{\Omega },w)$ and satisfies $\underset{n\to +\infty }{lim}w(a_n,a_{n+1})=0$ is not a Cauchy sequence. Therefore, there exists $\epsilon >0$ and sequences of positive integers $\left\{n_k\right\}, \left\{m_k\right\}, n_k>m_k>k$ such that the sequences

$$\{w\left(a_{n_k},a_{m_k}\right),w\left(a_{n_k+1},a_{m_k}\right),w\left(a_{n_k},a_{m_k-1}\right),w\left(a_{n_k+1},a_{m_k-1}\right),w\left(a_{n_k+1},a_{m_k+1}\right)\},$$

tend to ${\epsilon }^{+}$ when $k\to +\infty .$

The second significant Banach contraction principle generalization is established in 2002 by Branciari [11]. Firstly, we recall some necessary notions.

Let $\mathsf{\Psi }$ be the class of all functions $\tilde{\zeta }:[0,+\infty )\to [0,+\infty )$ which is Lebesgue integrable, summable on every compact set on $[0,+\infty )$ and $\underset{0}{\overset{ \epsilon }{\int }}\tilde{\zeta }\left(t\right)dt>0$ for all $\epsilon >0.$

The following lemmas are useful for our main results. We shall also suppose that $\tilde{\zeta }\in \mathsf{\Psi }.$

Lemma 2.

[12] Let ${\left\{l_n\right\}}_{n\in \mathbb{N}}$ be a non-negative sequence of real numbers so that $\underset{n\to +\infty }{lim}{l}_n=l.$ Then $\underset{n\to +\infty }{lim}\underset{0}{\overset{l_n}{\int }}\tilde{\zeta }\left(t\right)dt=\underset{0}{\overset{l}{\int }}\tilde{\zeta }\left(t\right)dt.$

Lemma 3.

[12] Let ${\left\{l_n\right\}}_{n\in \mathbb{N}}$ be a non-negative sequence of real numbers. Then $\underset{n\to +\infty }{lim}\underset{0}{\overset{l_n}{\int }}\tilde{\zeta }\left(t\right)dt=0$ if and only if $\underset{n\to +\infty }{lim}{l}_n=0.$

Here is the Branciaris theorem [11]:

Theorem 3.

Let h be a mapping from a complete metric space $\left(\mathsf{\Omega },w\right)$ into itself satisfying

$$\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt\le \lambda \underset{0}{\overset{w\left(a,b\right)}{\int }}\tilde{\zeta }\left(t\right)dt,$$

for all $a,b\in \mathsf{\Omega },$ where $\lambda \in \left(0,1\right)$ is a constant and $\tilde{\zeta }\in \mathsf{\Psi }.$ Then h has a unique fixed point $b\in \mathsf{\Omega }$ such that $\underset{n\to +\infty }{lim}{h}^{n}a=b$ for each $a\in \mathsf{\Omega }.$

For the further results, it is necessary to define the following terms, see Jungck [13,14], also see Abbas and Jungck [15] (Definition 1.3.).

Let $\mathsf{\Omega }\ne \varnothing$ and $h,i:\mathsf{\Omega }\to \mathsf{\Omega }$. If for some $a\in \mathsf{\Omega },$$b=ha=ia$ then a is a coincidence point and b is a point of coincidence of h and $i.$ A pair $\left(h,i\right)$ is compatible in $\left(\mathsf{\Omega },w\right)$ if $\underset{n\to +\infty }{lim}w \left(hi(a_n),ih(a_n)\right)=0,$ for every sequence $\left\{a_n\right\}$ in $\mathsf{\Omega }$ such that $\underset{n\to +\infty }{lim}h(a_n)=\underset{n\to +\infty }{lim}i(a_n)=t,$ for some $t\in \mathsf{\Omega }.$ In addition, a pair $(h,i)$ is weakly compatible if $ha=ia$ implies $hi(a)=ih(a), a\in \mathsf{\Omega }.$ A sequence $\left\{a_n\right\}$ in $\mathsf{\Omega }$ is a Picard–Jungck sequence of the pair $\left(h,i\right)$ (based on $a_0$) if $b_n=h{a}_n=i{a}_{n+1}$ for all $n\in \mathbb{N}\cup \left\{0\right\}.$

Proposition 1.

[15] If weakly compatible mappings $h,i:\mathsf{\Omega }\to \mathsf{\Omega }$ have a point of coincidence which is unique $b=ha=ia,$ then b is a unique common fixed point of h and i.

2. Main Result

In this section we shall combine Jungck’s, Braniciari’s and Wardowski’s results for obtaining common and usual fixed points of some self-mappings on metric space $\left(\mathsf{\Omega },w\right)$. Our results merge, generalize and refine several recent results in the literature. We commence with the following definition.

Definition 2.

Let $(\mathsf{\Omega },w)$ be a metric space and $\mathcal{F}$ be a family of mappings $F:(0,+\infty )\to (-\infty ,+\infty )$ which satisfy condition (F1). A mapping $h:\mathsf{\Omega }\to \mathsf{\Omega }$ is said to be an integral $F-$contraction on $(\mathsf{\Omega },w)$ if there exists $\varpi >0$ such that for all $a,b\in \mathsf{\Omega },$$w(ha,hb)>0$ we have

$$\varpi +F(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt), \tilde{\zeta }\in \mathsf{\Psi }.$$

(3)

Remark 2.

If $\tilde{\zeta }(t)\equiv 1$ then we have a $F-$contraction.

Theorem 4.

If $F(a)=lna$ then the notion of Branciari contraction and integral $F-$contraction are equivalent.

Proof. 

At first, we suppose that the mapping h is Branciari contraction. Then

$$-ln\lambda +ln(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)\le ln(\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt)$$

and accordingly we get integral $F-$contraction for $\varpi =-ln\lambda >0.$

If h is integral $F-$contraction then we have the following:

$$\varpi +ln(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)\le ln(\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt).$$

Let $\varpi =ln{\varpi }_1.$ Then ${\varpi }_1>1$ and $ln(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)\le ln(\frac{1}{{\varpi }_1}{\int }_0^{w(a,b)}\tilde{\zeta }(t)dt).$ Then h is Branciari contraction for $\lambda =\frac{1}{{\varpi }_1}<1.$ □

Our first new result on integral $F-$contraction is the following one:

Theorem 5.

Let $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be an integral $F-$contraction with property (F1) in $(\mathsf{\Omega },w)$, where $(\mathsf{\Omega },w)$ is a metric space which is completed. Then there exists a unique $a\in \mathsf{\Omega }$, $a=ha.$

Proof. 

We will initially show that fixed point is unique, under the assumption that such a point exists. We presume opposite i.e., there exist $u,v$, $u\ne v$ and $u=hu$ and $v=hv.$ This assumption is obviously false since $\varpi >0,$$F({\int }_0^{w(u,v)}\tilde{\zeta }(t)dt)\in \mathbb{R}.$ By (3) it follows:

$$\varpi +F(\underset{0}{\overset{w(u,v)}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{w(u,v)}{\int }}\tilde{\zeta }(t)dt).$$

Let $a_0\in \mathsf{\Omega }$ and $h{a}_n=a_{n+1}$. If $a_k=a_{k+1}$ for some $k\in \mathbb{N}\cup \left\{0\right\},$ then $a_k$ is a unique fixed point. So, $a_k\ne a_{k+1}$ for every $k\in \mathbb{N}\cup \left\{0\right\}$. Then,

$$F(\underset{0}{\overset{w(a_{n+1},a_n)}{\int }}\tilde{\zeta }(t)dt)<\varpi +F(\underset{0}{\overset{w(a_{n+1},a_n)}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{w(a_n,a_{n-1})}{\int }}\tilde{\zeta }(t)dt)$$

By (F1) we have that

$$\underset{0}{\overset{w(a_{n+1},a_n)}{\int }}\tilde{\zeta }(t)dt<\underset{0}{\overset{w(a_n,a_{n-1})}{\int }}\tilde{\zeta }(t)dt$$

and thence $w(a_{n+1},a_n)<w(a_n,a_{n-1})$ for all $n\in \mathbb{N}.$ Sequence $\left\{w(a_{n+1},a_n)\right\}$ is monotone decreasing, bounded from bellow and so there exists $\tilde{\rho }$ such that

$$\underset{n\to +\infty }{lim}w(a_n,a_{n+1})=\tilde{\rho }\ge 0.$$

In addition, $w(a_n,a_{n+1})>\tilde{\rho }$ for all $n\in \mathbb{N}\cup \left\{0\right\}.$ Suppose that $\tilde{\rho }>0$, then

$$\varpi +F(\underset{0}{\overset{\tilde{\rho }+0}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{\tilde{\rho }+0}{\int }}\tilde{\zeta }(t)dt),$$

so we have contradiction and thus $\tilde{\rho }=0.$ Therefrom we have that $\underset{n\to +\infty }{lim}w(a_n,a_{n+1})=0.$

It remains to prove that $\left\{a_n\right\}$ is a Cauchy sequence. Suppose the contrary. If we put $a=a_{n_k}$ and $b=a_{m_k}$ in contractive condition (3), we obtain

$$F(\underset{0}{\overset{w(a_{n_k+1},a_{m_k+1})}{\int }}\tilde{\zeta }(t)dt)<\varpi +F(\underset{0}{\overset{w(a_{n_k+1},a_{m_k+1})}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{w(a_{n_k},a_{m_k})}{\int }}\tilde{\zeta }(t)dt).$$

By Lemma 1 $w(a_{n_k+1},a_{m_k+1})\to {\epsilon }^{+}$ and $w(a_{n_k},a_{m_k})\to {\epsilon }^{+}$ as $k\to +\infty$ so we get that

$$F(\underset{0}{\overset{\epsilon +0}{\int }}\tilde{\zeta }(t)dt)<\varpi +F(\underset{0}{\overset{\epsilon +0}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{\epsilon +0}{\int }}\tilde{\zeta }(t)dt),$$

i.e., consequently, the sequence $\left\{a_n\right\}$ is a Cauchy sequence and there exists $a\in \mathsf{\Omega }$ such that $\underset{n\to +\infty }{lim}{a}_n=a$.

Using (3) we have that $w(ha,hb)<w(a,b)$ and therefore h must be continuous. Then $ha=h(\underset{n\to +\infty }{lim}{a}_n)=\underset{n\to +\infty }{lim}{a}_{n+1}=a.$ □

Example 1.

Let $\mathsf{\Omega }=[0,1]$ and $w(a,b)=|a-b|.$ Then metric space $(\mathsf{\Omega },w)$ is complete. Let $h(a)=\frac{a}{2}$, $\tilde{\zeta }(t)=2t$ and $F(a)=-\frac{1}{a}$. Then

$$F(\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt)-F(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)=-\frac{1}{{(a-b)}^2}+\frac{4}{{(a-b)}^2}=\frac{3}{{(a-b)}^2}\ge 3.$$

Therefore, all requirements of Theorem 5 are satisfied for $\varpi \in (0,3]$ and obviously is $h(0)=0.$

Corollary 1.

Let $(\mathsf{\Omega },w)$ be a complete metric space, $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be a function such that there exists $K_i>0,$$i=\overline{1,5}$ and for all $a,b\in \mathsf{\Omega }$ with $w(ha,hb)>0$, any of the following contractive conditions hold:

$$\begin{array}{c}{K}_1+\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt\le \underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt;\hfill \\ K_2-\frac{1}{\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt}\le -\frac{1}{\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt};\hfill \\ K_3-\frac{1}{\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt}+\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt\le -\frac{1}{\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt}+\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt;\hfill \\ K_4+\frac{1}{1-exp(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)}\le \frac{1}{1-exp(\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt)};\hfill \\ K_5+\frac{1}{exp(-\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)-exp(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)}\le \frac{1}{exp(-\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt)-exp(\underset{0}{\overset{w(a,b)}{\int }}\tilde{\zeta }(t)dt)};\hfill \end{array}$$

then in every case h has a fixed point which is unique.

Proof. 

Proof follows directly from Theorem 5. Indeed, since each of the functions $F(l)=l,$$F(l)=-\frac{1}{l}$, $F(l)=-\frac{1}{l}+l$, $F(l)=\frac{1}{1-exp(r)}$, $F(l)=\frac{1}{exp(-l)-exp(l)}$ is strictly increasing on $(0,+\infty )$ the result follows. □

Remark 3.

If in Theorem 5 instead of the contractive condition (3) we assume the following condition for all $a,b\in \mathsf{\Omega }$ and $w(ha,hb)>0,$

$$\varpi +F(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{\mathbf{L}(a,b)}{\int }}\tilde{\zeta }(t)dt), \tilde{\zeta }\in \mathsf{\Psi }.$$

(4)

where

$$\begin{array}{ccc}\hfill \mathbf{L}(a,b)& =& max\{w(a,b),w(a,ha),w(b,hb)\},\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill \mathbf{L}(a,b)& =& max\{w(a,b),w(a,ha),w(b,hb),\frac{w(a,hb)+w(ha,b)}{2}\},\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill \mathbf{L}(a,b)& =& max\{w(a,b),\frac{w(a,ha)+w(b,hb)}{2},\frac{w(a,hb)+w(b,ha)}{2}\}.\hfill \end{array}$$

(7)

then there exists a unique fixed point of the mapping h with the addition that one of the mappings h or F is continuous.

In the next definition the notion of $\left(\varpi ,\tilde{\zeta },F,i\right)-$integral contraction is introduced.

Definition 3.

Let $h,i:\mathsf{\Omega }\to \mathsf{\Omega }$ where $\left(\mathsf{\Omega },w\right)$ is a metric space. A mapping h is a $\left(\varpi ,\tilde{\zeta },F,i\right)-$ integral contraction if there exists a function $\varpi :\left(0,+\infty \right)\to \left(0,+\infty \right)$ satisfying

$$lim\underset{s\to t^{+}}{inf}\varpi \left(s\right)>0, for all t>0,$$

(8)

$\tilde{\zeta }\in \mathsf{\Psi }$ function $F:\left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ with property (F1) such that for all $a,b\in \mathsf{\Omega }$ with $ha\ne hb$ and $ia\ne ib$ one has

$$\varpi (\underset{0}{\overset{w(ia,ib)}{\int }}\tilde{\zeta }(t)dt)+F(\underset{0}{\overset{w(ha,hb)}{\int }}\tilde{\zeta }(t)dt)\le F(\underset{0}{\overset{w(ia,ib)}{\int }}\tilde{\zeta }(t)dt).$$

(9)

We now state a new result for the term $\left(\varpi ,\tilde{\zeta },F,i\right)-$ integral contraction. We succeed in generalizing results from several manuscripts in existing literature, for instance ([11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]).

Theorem 6.

Let $h,i:\mathsf{\Omega }\to \mathsf{\Omega }$, h is a $\left(\varpi ,\tilde{\zeta },F,i\right)-$integral contraction where $\left(\mathsf{\Omega },w\right)$ is a metric space. Presume that there exists a Picard sequence ${\left\{b_n\right\}}_{n\in \mathbb{N}\cup \left\{0\right\}}$ of $(h,i).$ Further, suppose that (i) or (ii) holds:

(i)

$\left(i\mathsf{\Omega },w\right)$ is complete,

(ii)

$\left(\mathsf{\Omega },w\right)$ is complete, i is continuous and $\left(h,i\right)$ is compatible.

Then h and i have a unique point of coincidence.

Proof. 

We initially prove that there is a unique point of coincidence of h and i, assuming that such a point exists. Let $b_1\ne b_2$ be points of coincidence for h and $i.$ Using that, we conclude that there exist $a_1$ and $a_2$ ($a_1\ne a_2$) so that $h{a}_1=i{a}_1=b_1$ and $h{a}_2=i{a}_2=b_2.$ The condition (9) yields that

$$\varpi \left(\underset{0}{\overset{w\left(i{a}_1,i{a}_2\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{w\left(h{a}_1,h{a}_2\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{w\left(i{a}_1,i{a}_2\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

(10)

i.e.,

$$\varpi \left(\underset{0}{\overset{w\left(b_1,b_2\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{w\left(b_1,b_2\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{w\left(b_1,b_2\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

(11)

which is a contradiction, because $\varpi \left(\underset{0}{\overset{w\left(b_1,b_2\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)>0.$

Suppose now that there is a Picard–Jungck sequence $$ such that $b_n=h{a}_n=i{a}_{n+1},$ where $n\in \mathbb{N}\cup \left\{0\right\}.$ If $b_p=b_{p+1}$ for some $p\in \mathbb{N}\cup \left\{0\right\},$ then $i{b}_{p+1}=b_p=h{b}_{p+1},$ and h and i have a unique point of coincidence. Accordingly, suppose that $b_n\ne b_{n+1}$ for every $n\in \mathbb{N}\cup \left\{0\right\}.$ By replacing $a=a_n$ and $b=a_{n+1}$ into (9), we get

$$\varpi \left(\underset{0}{\overset{w\left(i{a}_n,i{a}_{n+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{w\left(h{a}_n,h{a}_{n+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{w\left(i{a}_n,i{a}_{n+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right).$$

(12)

Guided by the properties of the $\varpi ,\tilde{\zeta }$ and $F,$ we get that $w\left(b_n,b_{n+1}\right)<w\left(b_{n-1},b_n\right),$ for all $n\in \mathbb{N}.$ Therefore, there exists $\overline{\delta }\ge 0$ so that $\underset{n\to +\infty }{lim}w\left(b_n,b_{n+1}\right)=\overline{\delta }.$ Suppose that $\overline{\delta }>0.$ Based on the condition of the function $\varpi$ we know that there exist ${\varpi }_0>0$ and $n_1\in \mathbb{N}$ such that for all $n\ge n_1$ we have

$${\varpi }_0+F\left(\underset{0}{\overset{w\left(b_n,b_{n+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)<$$

$$\varpi \left(\underset{0}{\overset{w\left(b_{n-1},b_n\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{w\left(b_n,b_{n+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{w\left(b_{n-1},b_n\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

(13)

that is,

$${\varpi }_0+F\left(\underset{0}{\overset{w\left(b_n,b_{n+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)<F\left(\underset{0}{\overset{w\left(b_{n-1},b_n\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

(14)

for all $n\ge n_1.$ Based on the conditions (F1), the last relation yields

$${\varpi }_0+F\left(\underset{0}{\overset{\overline{\delta }+0}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{\overline{\delta }+0}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

and it is a contradiction. Hence, $\underset{n\to +\infty }{lim}w\left(b_n,b_{n+1}\right)=0.$

Moreover, it remains to be shown $b_n\ne b_m$ whenever $n\ne m.$ We will assume the opposite, i.e., $b_n=b_m$ for some $n>m.$ Based on the definition of the Picard–Jungck sequence $\left\{b_n\right\}$ we can choose $b_{n+1}=b_{m+1}.$ Using the previous arguments, we have

$$w\left(b_n,b_{n+1}\right)=w\left(b_m,b_{m+1}\right)<w\left(b_{m-1},b_m\right)<...<w\left(b_{n+1},b_{n+2}\right)<w\left(b_n,b_{n+1}\right)$$

which is a contradiction.

Further we need to show that the sequence $$ is a Cauchy sequence. We will show this by the method of contradiction. Including $a=a_{n_k+1}$ and $b=a_{m_k+1}$ in (9), we obtain

$$\varpi \left(\underset{0}{\overset{w\left(i{a}_{n_k+1},i{a}_{m_k+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{w\left(h{a}_{n_k+1},h{a}_{m_k+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{w\left(i{a}_{n_k+1},i{a}_{m_k+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

i.e.,

$$\varpi \left(\underset{0}{\overset{w\left(b_{n_k},b_{m_k}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{w\left(b_{n_k+1},b_{m_k+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{w\left(b_{n_k},b_{m_k}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right).$$

(15)

Using Lemma 1, $w\left(b_{n_k+1},b_{m_k+1}\right)$ and $w\left(b_{n_k},b_{m_k}\right)$ tend to ${\epsilon }^{+}$ as $k\to +\infty ,$ and accordingly we obtain

$$lim\underset{w\left(b_{n_k},b_{m_k}\right)\to {\epsilon }^{+}}{inf}\varpi \left(\underset{0}{\overset{w\left(b_{n_k},b_{m_k}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+lim\underset{w\left(b_{n_k},b_{m_k}\right)\to {\epsilon }^{+}}{inf}F\left(\underset{0}{\overset{w\left(b_{n_k+1},b_{m_k+1}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)$$

$$\le lim\underset{w\left(b_{n_k},b_{m_k}\right)\to {\epsilon }^{+}}{inf}F\left(\underset{0}{\overset{w\left(b_{n_k},b_{m_k}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

that is,

$$lim\underset{w\left(b_{n_k},b_{m_k}\right)\to {\epsilon }^{+}}{inf}\varpi \left(\underset{0}{\overset{w\left(b_{n_k},b_{m_k}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{{\epsilon }^{+}+0}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{{\epsilon }^{+}+0}{\int }}\tilde{\zeta }\left(t\right)dt\right),$$

(16)

which is a contradiction with

$$lim\underset{w\left(b_{n_k},b_{m_k}\right)\to {\epsilon }^{+}}{inf}\varpi \left(\underset{0}{\overset{w\left(b_{n_k},b_{m_k}\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)>0.$$

So, we showed that the sequence $$ is a Cauchy sequence.

Now let (i) hold. Then, there exists $z\in X$ so that $b_n=i{a}_n\to iz$ as $n\to +\infty .$ We shall prove that $hz=iz.$ Since $b_n\ne b_m$ whenever $n\ne m$, we can suppose that $hz,iz\notin \left\{b_n:n\in \mathbb{N}\cup \left\{0\right\}\right\}.$ Therefore, by (9) we have

$$F\left(\underset{0}{\overset{w\left(b_n,hz\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)<\varpi \left(\underset{0}{\overset{w\left(b_{n-1},iz\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)+F\left(\underset{0}{\overset{w\left(b_n,hz\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le F\left(\underset{0}{\overset{w\left(b_{n-1},iz\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right).$$

(17)

Based on the properties of the function $F,$ we get that $w\left(b_n,hz\right)<w\left(b_{n-1},iz\right)\to 0$ as $n\to +\infty .$ Hence $hz=iz$ and z is unique.

At the end, let (ii) hold. From completeness of $$ it follows that there exists $v\in X$ such that $h{a}_n\to v,$ when $n\to +\infty .$ As i is continuous, $ih{a}_n\to iv$ when $n\to +\infty$. By (9) and the continuity of i we conclude that h must also be continuous. Therefore, $hi{a}_n\to hv$ as $n\to +\infty .$ As h and i are compatible, we have

$$w\left(hv,iv\right)\le w\left(hv,hi{a}_n\right)+w\left(hi{a}_n,ih{a}_n\right)+w\left(ih{a}_n,iv\right)\to 0+0+0=0.$$

(18)

Thus, our result is proved in both cases, and we realize that the mappings h and i have a unique point of coincidence. □

Remark 4.

(1) If (i) is satisfied and $\left(h,i\right)$ are weakly compatible, using Proposition 1, we conclude that h and i have a common fixed point. Moreover, the common fixed point is unique.

(2) Assuming that (ii) holds, h and i also have a unique common fixed point using Proposition 1. We conclude this based on the fact that every compatible pair $(h,i)$ is weakly compatible.

In the following corollary the mapping $F: \left(0,+\infty \right)\to \left(-\infty ,+\infty \right)$ is only strictly increasing one. Therefore our new Theorem 6 generalizes, improves, complements, unifies and enriches several results from $F-$contraction type in existing literature.

Corollary 2.

Putting in Theorem 6 condition $\tilde{\zeta }\left(t\right)\equiv 1$ for all $t\in [0,+\infty )$ we get a Jungck–Wardowski type result, i.e., Theorem 8 from [21]. Further if $\tilde{\zeta }\left(t\right)\equiv 1$ for all $t\in [0,+\infty )$ and $i=I_{\mathsf{\Omega }}$ the identity mapping on Ω then we obtain Theorem 2.1 from Wardowski [34]. If $\tilde{\zeta }\left(t\right)\equiv 1$ for all $t\in [0,+\infty ),\varpi \left(t\right)=\varpi =$constant from $\left(0,+\infty \right),$ and $i=I_{\mathsf{\Omega }}$ the identity mapping on X we have Wardowski’s Theorem 2.1. from [3]. Putting in Theorem 6 $i=I_X$ the identity mapping on Ω we get a Branciari–Wardowski type fixed point result in the sense of [34]. While for $i=I_{\mathsf{\Omega }}$ the identity mapping on Ω and $\varpi \left(t\right)=\varpi =$constant from $\left(0,+\infty \right)$ our Theorem 6 gives a Branciari–Wardowski type fixed point result in the sense of [3].

The direct consequences of the Theorem 6 are new contraction conditions that complement results from [18,28].

Corollary 3.

Let $\left(\mathsf{\Omega },w\right)$ be a metric space, $h,i:\mathsf{\Omega }\to \mathsf{\Omega }$ be a self-mapping and h be an $\left({\varpi }_i,\tilde{\zeta },F,i\right)-$ contraction, where $C_i>0,i=\overline{1,6}$ such that for all $a,b\in \mathsf{\Omega }$ with $w\left(ha,hb\right)>0$ and $w\left(ia,ib\right)>0$ any of the following inequalities hold true

$$\begin{array}{c}{C}_1+\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt\le \underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt,\hfill \\ C_2+exp\left(\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)\le exp\left(\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right),\hfill \\ C_3-\frac{1}{\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt}\le -\frac{1}{\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt},\hfill \\ C_4-\frac{1}{\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt}+\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt\le -\frac{1}{\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt}+\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt,\hfill \\ C_5+\frac{1}{1-exp\left(\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)}\le \frac{1}{1-exp\left(\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)},\hfill \\ C_6+\frac{1}{exp\left(-\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)-exp\left(\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)}\le \frac{1}{exp\left(-\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)-exp\left(\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt\right)},\hfill \\ C_7+{(\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt)}^k\le {(\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt)}^k, k>0\hfill \\ C_8+\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt·exp(\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt)\le \underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt·exp(\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt),\hfill \\ C_9+exp(\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt)·ln(\underset{0}{\overset{w\left(ha,hb\right)}{\int }}\tilde{\zeta }\left(t\right)dt)\le exp(\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt)·ln(\underset{0}{\overset{w\left(ia,ib\right)}{\int }}\tilde{\zeta }\left(t\right)dt).\hfill \end{array}$$

Suppose that there exists a Picard–Jungck sequence ${\left\{b_n\right\}}_{n\in \mathbb{N}\cup \left\{0\right\}}$ of $\left(h,i\right)$ and assume that at least one of the following two conditions holds true:

(i)

$\left(i\mathsf{\Omega },w\right)$ is a complete metric space;

(ii)

$\left(\mathsf{\Omega },w\right)$ is complete metric space, i is continuous and $\left(h,i\right)$ is compatible pair of self-mappings on $\left(X,w\right).$

Then, in each of these cases, h and i have a unique point of coincidence.

Proof. 

First of all, put ${\varpi }_i\left(t\right)=C_i\in \left(0,+\infty \right),i=\overline{1,9}$ for all $t\in \left(0,+\infty \right)$ and $F\left(l\right)=l,F\left(l\right)=exp\left(l\right),F\left(l\right)=-\frac{1}{l},F\left(l\right)=-\frac{1}{l}+l,F\left(l\right)=$$\frac{1}{1-exp\left(l\right)},F\left(l\right)=\frac{1}{exp\left(-l\right)-exp\left(l\right)},$$F(l)=l^k, k>0$, $F(l)=l·exp(l)$ and $F(l)=exp(l)·ln(l),$ respectively. Because every of the functions $l\mapsto F\left(l\right)$ is strictly increasing on $$ then the result follows by Theorem 6. □

Example 2.

Let Ω, $\omega ,$h, $\tilde{\zeta }$ and F be the same as in Example 1. Let $i(a)=\frac{2}{3}a$. Then all conditions of Corollary 3 are satisfied for $C_3\in (0,\frac{7}{4}]$ and obviously 0 is a unique point of coincidence for the mappings h and i.

3. Conclusions

In this paper, the new term of $F-$integral contraction is introduced. Fixed point and common fixed point theorems are established, and as a consequence of the main results we obtain Jungck–Wardowski, Branciari–Wardowski and Jungck–Branciari type results. The results presented in the article enhance and complement some of known results in literature.