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Fixed point results for rational contraction in function weighted dislocated quasi-metric spaces with an application

Abstract

The objective of this article is to introduce function weighted L-R-complete dislocated quasi-metric spaces and to present fixed point results fulfilling generalized rational type F-contraction for a multivalued mapping in these spaces. A suitable example confirms our results. We also present an application for a generalized class of nonlinear integral equations. Our results generalize and extend the results of Karapınar et al. (IEEE Access 7:89026–89032, 2019).

Introduction and preliminaries

In functional analysis, fixed point theory plays a vital role in elaborating the problems. Fixed point results for the multivalued functions were first examined by Nadler [24]. The work of Nadler has been cited by many mathematicians and brings to the level of ultimate advancement, see [6, 25, 33]. Dislocated metric space [21] is one of the generalizations of metric spaces among several generalizations, and it has applications in logic programming semantics [10]. Hussain et al. [11] extended this concept to dislocated b-metric space and obtained results for weak contractions. On the other hand, Wilson [39] introduced the quasi-metric space by excluding the symmetric conditions in the definition of metric spaces. Several extensions of quasi-metric space have been made, and some fixed point theorems have been obtained, see [1, 9, 16, 1820, 28, 31]. Shoaib et al. [35] established results for multivalued functions in a dislocated quasi-metric space, see also [8, 37]. Rational type, Kannan type, and Reich type contractions on multivalued functions in double controlled quasi-metric type spaces [34, 36] have been introduced, and some fixed point theorems have been obtained. Another generalization of metric space, named function weighted metric space or F-metric space (see, [24, 22]), was defined by Jleli [13]. Recently, Panda et al. [29] defined extended F-metric space and discussed a solution for Atangana–Baleanu fractional and Lp-Fredholm integral equations. Karapınar et al. [17] gave the idea of a function weighted quasi-metric space and examined the presence of a fixed point of functions in function weighted bi-complete quasi-metric spaces. Different efforts have been made in the field of F-contraction mapping [38] to exhibit certain results on fixed points of multivalued mappings. Hussain et al. [12] introduced Suzuki–Wardowski type, Rasham et al. [30] established rational Ćirić type, and Sgroi et al. [32] defined Hardy–Roger type F-contraction mappings. Some applications were also discussed by them. For more results, see [5, 7, 14, 15, 23, 26, 27]. In this article, we introduce function weighted L-R-complete dislocated quasi-metric spaces and obtain fixed point results for multivalued mappings satisfying generalized rational type F-contraction in such spaces without the second condition (F2) and the third condition (F3) imposed on Wardowski’s function [38]. A suitable example and an application confirm our results. We start with some basic concepts.

Definition 1.1

([17])

A function \(h:(0,+\infty )\rightarrow \mathbb{R} \) is said to be

  1. (i)

    logarithmic-like, if:

    $$\begin{aligned}& \text{for each sequence }\{\tau _{m}\}\subset (0,+\infty ) \text{ satisfies} \\& \underset{m\rightarrow +\infty }{\lim }h(\tau _{m})=-\infty \quad \text{if and only if}\quad \underset{m\rightarrow +\infty }{\lim }\tau _{m}=0. \end{aligned}$$
  2. (ii)

    nondecreasing function, if:

    $$0< \sigma < \tau \quad \text{implies} \quad h(\sigma )< h(\tau ). $$

Let γ denote the set of all logarithmic-like nondecreasing functions.

Definition 1.2

([13])

For a mapping δ: \(M \times M\rightarrow {}[ 0,+\infty )\), if a pair \((h,C)\in \gamma \times {}[ 0,+\infty )\) exists for all \(u,v,w\in M\), we have

\((\Delta _{1})\):

\(\delta (u,w)=\delta (w,u)\);

\((\Delta _{2})\):

\(\delta (u,w)=0\) if and only if \(u=w\);

\((\Delta _{3})\):

For any \(j\in \mathbb{N} \), \(j\geq 2\), we have

$$ \delta (u,w)>0\quad \text{implies}\quad h \bigl(\delta (u,w) \bigr)\leq h ( \sum _{i=1}^{j-1}\delta (v_{i},v_{i+1} ) +C $$

for every \((v_{i})_{i=1}^{j}\subset M\) with \((v_{1},v_{j})=(u,w)\). Then δ is called an \(\mathcal{F}\)-metric or a function weighted metric [17] and \((M,\delta )\) is known as an \(\mathcal{F}\)-metric space or a function weighted metric space. If we exclude the condition \((\Delta _{1})\) from Definition 1.2, then \((M,\delta _{q})\) represents a function weighted quasi-metric space [17].

Definition 1.3

Let \((M,\delta _{q})\) be a function weighted quasi-metric space. If we replace \((\Delta _{2})\) with \(\delta _{q}(u,w)=0\) implies \(u=w\), that is, \(\delta _{q}(u,u)\) may not be equal to zero, then we say that \(\delta _{q}\) is a function weighted dislocated quasi-metric on M. We will denote this new metric by \(\delta _{dq}\). Furthermore, the couple \((M,\delta _{dq})\) is called a function weighted dislocated quasi-metric space. Note that any function weighted quasi-metric space is also a function weighted dislocated quasi-metric space but the converse is not true in general.

Definition 1.4

Let \((M,\delta _{dq})\) be a function weighted dislocated quasi-metric space. A sequence \(\{u_{t}\}\) in M is

  1. (i)

    left convergent to some \(u\in M\) if and only if \(\underset{m\rightarrow +\infty }{\lim }\delta _{dq}(u_{m},u)=0\) or, for every \(\varepsilon >0\), we have \(\delta _{dq}(u_{m},u)<\varepsilon \) for all \(m\geq t_{\varepsilon }\), where \(t_{\varepsilon }\) is some integer depending on ε.

  2. (ii)

    right convergent to some \(u\in M\) if and only if \(\underset{t\rightarrow +\infty }{\lim }\delta _{dq}(u,u_{t})=0\) or, for every \(\varepsilon >0\), we have \(\delta _{dq}(u,u_{t})<\varepsilon \) for all \(t\geq t_{\varepsilon }\), where \(t_{\varepsilon }\) is some integer depending on ε.

  3. (iii)

    The sequence \(\{u_{t}\}\) is L-R-convergent if and only if it is both left and right convergent.

  4. (iv)

    The sequence \(\{u_{t}\}\) is bi-convergent to some \(u\in M\) if and only if \(\underset{t\longrightarrow +\infty }{\lim }\delta _{dq}(u,u_{t})= \underset{t\longrightarrow +\infty }{\lim }\delta _{dq}(u_{t},u)=0\).

Lemma 1.5

Every L-R-convergent sequence in a function weighted dislocated quasi-metric space is bi-convergent.

Definition 1.6

Let \((M,\delta _{dq})\) be a function weighted dislocated quasi-metric space. A sequence \(\{u_{t}\}\) in M is

  1. (i)

    left Cauchy if and only if \(\underset{t>m}{\lim_{t,m\rightarrow +\infty }}\delta _{dq}(u_{m},u_{t})=0\) or, for every \(\varepsilon >0\), we have \(\delta _{dq}(u_{m},u_{t})<\varepsilon \) for all \(t>m\geq t_{\varepsilon }\), where \(t_{\varepsilon }\) is some integer depending on ε.

  2. (ii)

    right Cauchy if and only if \(\underset{m>t}{\lim_{t,m\rightarrow +\infty }}\delta _{dq}(u_{m},u_{t})=0\) or, for every \(\varepsilon >0\), we have \(\delta _{dq}(u_{m},u_{t})<\varepsilon \) for all \(m>t\geq t_{\varepsilon }\), where \(t_{\varepsilon }\) is some integer depending on ε.

  3. (iii)

    The sequence \(\{u_{t}\}\) is bi-Cauchy if and only if it is both left and right Cauchy.

Definition 1.7

Let \((M,\delta _{dq})\) be a function weighted dislocated quasi-metric space. Then \((M,\delta _{dq})\) is

  1. (i)

    right-complete if and only if each right-Cauchy sequence in M is bi-convergent to some \(u\in M\).

  2. (ii)

    left-complete if and only if each left-Cauchy sequence in M is bi-convergent to some \(u\in M\).

  3. (iii)

    bi-complete (or dual complete) if and only if it is both right- and left-complete.

  4. (iv)

    L-R-complete if and only if for every bi-Cauchy in M is L-R-convergent to some \(u\in M\).

Remark 1.8

Every right-complete, left-complete, and bi-complete function weighted dislocated quasi-metric space is L-R-complete, but the converse is not true in general, so it is better to prove results in L-R-complete function weighted dislocated quasi-metric space instead of right-complete or left-complete or bi-complete.

Definition 1.9

Let Q be a nonempty subset in a function weighted dislocated quasi-metric space \((M,\delta _{dq})\), and let \(u\in M\). An element \(w_{0}\in Q\) is called the best approximation in Q for u if

$$\begin{aligned} \delta _{dq}(u,Q) =&\delta _{dq}(u,w_{0}),\quad \text{where }\delta _{dq}(u,Q)= \underset{w\in Q}{\inf }\delta _{dq}(u,w), \\ \delta _{dq}(Q,u) =&\delta _{dq}(w_{0},u),\quad \text{where }\delta _{dq}(Q,u)= \underset{w\in Q}{\inf }\delta _{dq}(w,u). \end{aligned}$$

If each \(a\in M\) has at least one best approximation in Q, then Q is called a proximinal set. The set of all closed proximinal subsets of M is denoted by \(P(M)\).

Definition 1.10

The function \(H_{\delta _{dq}}:P(M)\times P(M)\rightarrow {}[ 0,+\infty )\), defined by

$$ H_{\delta _{dq}}(G,H)=\max \Bigl\{ \sup_{g\in G}\delta _{dq}(g,H), \sup_{h\in H}\delta _{dq}(G,h) \Bigr\} , $$

is called Hausdorff–Pompeiu function weighted dislocated quasi-metric on \(P(M)\).

Lemma 1.11

Suppose that \((M,\delta _{dq})\) is a function weighted dislocated quasi-metric. Let \((P(M),H_{\delta _{dq}})\) be a function weighted Hausdorff–Pompeiu quasi-metric space on \(P(M)\). Then, for all \(G,F\in P(M)\) and for each \(g\in G\), there exists \(f_{g}\in F\) that satisfies \(\delta _{dq}(g,F)=\delta _{dq}(g,f_{g})\), and then

$$ H_{\delta _{dq}}(G,F)\geq \delta _{dq}(g,f_{g}). $$

Main results

Let \((M,\delta _{dq})\) be an L-R-complete function weighted dislocated quasi-metric, \(a_{0}\in M\) and \(S:M\rightarrow P(M)\) be the multivalued mapping on M. Let \(a_{1}\in Sa_{0} \) such that \(\delta _{dq}(a_{0},Sa_{0})=\delta _{dq}(a_{0},a_{1})\) and \(\delta _{dq}(Sa_{0},a_{0})=\delta _{dq}(a_{1},a_{0})\). Now, for \(a_{1}\in M\), there exists \(a_{2}\in Sa_{1}\) such that \(\delta _{dq}(a_{1},Sa_{1})=\delta _{dq}(a_{1},a_{2})\) and \(\delta _{dq}(Sa_{1},a_{1})=\delta _{dq}(a_{2},a_{1})\). Continuing this process, we construct a sequence \(a_{n}\) of points in M such that \(a_{n+1}\in Sa_{n}\), and \(a_{n+2}\in Sa_{n+1}\) with \(\delta _{dq}(a_{n},Sa_{n})=\delta _{dq}(a_{n},a_{n+1})\), \(\delta _{dq}(Sa_{n},a_{n})=\delta _{dq}(a_{n+1},a_{n})\) and \(\delta _{dq}(a_{n+1},Sa_{n+1})=\delta _{dq}(a_{n+1},a_{n+2})\), \(\delta _{dq}(Sa_{n+1},a_{n+1})=\delta _{dq}(a_{n+2},a_{n+1})\). We denote this iterative sequence by \(\{MS(a_{n})\}\) and say that \(\{MS(a_{n})\}\) is a sequence in M generated by \(a_{0}\). Now, we announce our first new result in this paper.

Theorem 2.1

Suppose that \((M,\delta _{dq})\) is an L-R-complete function weighted dislocated quasi-metric with respect to \((h,C)\in \gamma \times {}[ 0,+\infty )\). Let \(S:M\rightarrow P(M)\) be a multivalued mapping, \(\mathcal{F}:(0,+\infty )\rightarrow \mathbb{R} \) be a strictly increasing mapping, \(\tau >0\), \(\mu _{1},\mu _{2},\mu _{3},\mu _{4}\geq 0\), \(\eta _{1}=\frac{\mu _{1}+\mu _{2}}{1-\mu _{3}-\mu _{4}}<1\) and \(\eta _{2}=\frac{\mu _{1}+\mu _{3}}{1-\mu _{2}-\mu _{4}}<1\) such that

$$\begin{aligned}& \tau +\max \bigl\{ \mathcal{F} \bigl(H_{\delta _{dq}}(Sg,Sw) \bigr), \mathcal{F} \bigl(H_{\delta _{dq}}(Sw,Sg) \bigr) \bigr\} \\& \quad \leq \min \biggl\{ \mathcal{F} \biggl( \mu _{1}\delta _{dq}(g,w)+ \mu _{2}\delta _{dq}(g,Sg)+\mu _{3}\delta _{dq}(w,Sw)+\mu _{4} \frac{\delta _{dq}(g,Sg).\delta _{dq}(w,Sw)}{1+\delta _{dq}(g,w)} \biggr) , \\& \qquad {} \mathcal{F} \biggl( \mu _{1}\delta _{dq}(w,g)+\mu _{2}\delta _{dq}(Sg,g)+ \mu _{3}\delta _{dq}(Sw,w)+\mu _{4} \frac{\delta _{dq}(Sg,g). \delta _{dq}(Sw,w)}{1+\delta _{dq}(w,g)} \biggr) \biggr\} , \end{aligned}$$
(2.1)

whenever \(\min \{ H_{\delta _{dq}}(Sg,Sw),H_{\delta _{dq}}(Sw,Sg) \} >0\), \(g,w\in \{MS(g_{t})\}\cup \{ z^{\ast } \} \), where \(\{ MS(g_{t}) \} \rightarrow z^{\ast }\). Then \(z^{\ast }\) is the fixed point of S.

Proof

Consider the sequence \(\{MS(g_{t})\}\). By using Lemma 1.11 and inequality (2.1), we have

$$\begin{aligned} \tau +\mathcal{F} \bigl(\delta _{dq}(g_{t+1},g_{t+2}) \bigr) \leq &\tau + \mathcal{F} \bigl(H_{\delta _{dq}}(Sg_{t},Sg_{t+1}) \bigr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t},g_{t+1} ) +\mu _{2}\delta _{dq} ( g_{t},Sg_{t} ) +\mu _{3} \delta _{dq}(g_{t+1},Sg_{t+1}) \\ &{} +\mu _{4} \frac{\delta _{dq} ( g_{t},Sg_{t} ) .\delta _{dq}(g_{t+1},Sg_{t+1})}{1+\delta _{dq} ( g_{t},g_{t+1} ) } \biggr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t},g_{t+1} ) +\mu _{2}\delta _{dq} ( g_{t},g_{t+1} ) +\mu _{3} \delta _{dq}(g_{t+1},g_{t+2}) \\ & {}+\mu _{4} \frac{\delta _{dq} ( g_{t},g_{t+1} ) .\delta _{dq}(g_{t+1},g_{t+2})}{1+\delta _{dq} ( g_{t},g_{t+1} ) } \biggr) \\ \leq &\mathcal{F} \bigl( (\mu _{1}+\mu _{2})\delta _{dq} ( g_{t},g_{t+1} ) +(\mu _{3}+\mu _{4})\delta _{dq}(g_{t+1},g_{t+2}) \bigr) . \end{aligned}$$

As \(\tau >0\), we have

$$ \mathcal{F} \bigl(\delta _{dq}(g_{t+1},g_{t+2}) \bigr)< \mathcal{F} \bigl( (\mu _{1}+ \mu _{2})\delta _{dq} ( g_{t},g_{t+1} ) +(\mu _{3}+\mu _{4}) \delta _{dq}(g_{t+1},g_{t+2}) \bigr) . $$

As \(\mathcal{F}\) is a strictly increasing mapping, we have

$$ \delta _{dq}(g_{t+1},g_{t+2})< (\mu _{1}+ \mu _{2})\delta _{dq} ( g_{t},g_{t+1} ) +( \mu _{3}+\mu _{4})\delta _{dq}(g_{t+1},g_{t+2}). $$

We get

$$\begin{aligned}& (1-\mu _{3}-\mu _{4})\delta _{dq}(g_{t+1},g_{t+2}) < (\mu _{1}+\mu _{2}) \delta _{dq} ( g_{t},g_{t+1} ), \\& \delta _{dq}(g_{t+1},g_{t+2}) < \biggl( \frac{\mu _{1}+\mu _{2}}{1-\mu _{3}-\mu _{4}} \biggr) \delta _{dq} ( g_{t},g_{t+1} ) . \end{aligned}$$

As \(\eta _{1}=\frac{\mu _{1}+\mu _{2}}{1-\mu _{3}-\mu _{4}}<1\), so

$$ \delta _{dq}(g_{t+1},g_{t+2})< \eta _{1} \delta _{dq} ( g_{t},g_{t+1} ) . $$

Let \(\eta =\max \{ \eta _{1},\eta _{2} \} <1\), hence

$$ \delta _{dq}(g_{t+1},g_{t+2})< \eta \delta _{dq} ( g_{t},g_{t+1} ) . $$
(2.2)

Now, by using Lemma 1.11 and inequality (2.1), we have

$$\begin{aligned} \tau +\mathcal{F} \bigl(\delta _{dq}(g_{t},g_{t+1}) \bigr) \leq &\tau + \mathcal{F} \bigl(H_{\delta _{dq}}(Sg_{t-1},Sg_{t}) \bigr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t-1},g_{t} ) +\mu _{2}\delta _{dq} ( g_{t},Sg_{t} ) +\mu _{3} \delta _{dq}(g_{t-1},Sg_{t-1}) \\ & {}+\mu _{4} \frac{\delta _{dq}(g_{t},Sg_{t}).\delta _{dq} ( g_{t-1},Sg_{t-1} ) }{1+\delta _{dq} ( g_{t-1},g_{t} ) } \biggr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t-1},g_{t} ) +\mu _{2}\delta _{dq} ( g_{t},g_{t+1} ) +\mu _{3} \delta _{dq}(g_{t-1},g_{t}) \\ & {}+\mu _{4} \frac{\delta _{dq} ( g_{t-1},g_{t} ) .\delta _{dq}(g_{t},g_{t+1})}{1+\delta _{dq} ( g_{t-1},g_{t} ) } \biggr) \\ \leq &\mathcal{F} \bigl( (\mu _{1}+\mu _{3})\delta _{dq} ( g_{t-1},g_{t} ) +(\mu _{2}+\mu _{4})\delta _{dq}(g_{t},g_{t+1}) \bigr) . \end{aligned}$$

This implies

$$ \mathcal{F} \bigl(\delta _{dq}(g_{t},g_{t+1}) \bigr)< \mathcal{F} \bigl( (\mu _{1}+ \mu _{3})\delta _{dq} ( g_{t-1},g_{t} ) +(\mu _{2}+\mu _{4}) \delta _{dq}(g_{t},g_{t+1}) \bigr) . $$

Since \(\mathcal{F}\) is a strictly increasing mapping, we have

$$ \delta _{dq}(g_{t},g_{t+1})< (\mu _{1}+ \mu _{3})\delta _{dq} ( g_{t-1},g_{t} ) +(\mu _{2}+\mu _{4})\delta _{dq}(g_{t},g_{t+1}). $$

We get

$$\begin{aligned}& (1-\mu _{2}-\mu _{4})\delta _{dq}(g_{t},g_{t+1}) < (\mu _{1}+\mu _{3}) \delta _{dq} ( g_{t-1},g_{t} ), \\& \delta _{dq}(g_{t},g_{t+1}) < \biggl( \frac{\mu _{1}+\mu _{3}}{1-\mu _{2}-\mu _{4}} \biggr) \delta _{dq} ( g_{t-1},g_{t} ) . \end{aligned}$$

As \(\eta _{2}=\frac{\mu _{1}+\mu _{3}}{1-\mu _{2}-\mu _{4}}<1\), so

$$ \delta _{dq}(g_{t},g_{t+1})< \eta _{2} \delta _{dq} ( g_{t-1},g_{t} ) < \eta \delta _{dq} ( g_{t-1},g_{t} ) . $$
(2.3)

By using (2.3) in (2.2), we have

$$ \delta _{dq}(g_{t+1},g_{t+2})< \eta ^{2} \delta _{dq} ( g_{t-1},g_{t} ) . $$

Continuing in this way, we have

$$ \delta _{dq}(g_{t+1},g_{t+2})< \eta ^{t+1} \delta _{dq} ( g_{0},g_{1} ) . $$
(2.4)

By using Lemma 1.11 and inequality (2.1), we have

$$\begin{aligned} \tau +\mathcal{F} \bigl(\delta _{dq}(g_{t+2},g_{t+1}) \bigr) \leq &\tau + \mathcal{F} \bigl(H_{\delta _{dq}}(Sg_{t+1},Sg_{t}) \bigr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t+1},g_{t} ) +\mu _{2}\delta _{dq} ( Sg_{t},g_{t} ) +\mu _{3} \delta _{dq}(Sg_{t+1},g_{t+1}) \\ & {}+\mu _{4} \frac{\delta _{dq} ( Sg_{t},g_{t} ) .\delta _{dq}(Sg_{t+1},g_{t+1})}{1+\delta _{dq} ( g_{t+1},g_{t} ) } \biggr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t+1},g_{t} ) +\mu _{2}\delta _{dq} ( g_{t+1},g_{t} ) +\mu _{3} \delta _{dq}(g_{t+2},g_{t+1}) \\ & {}+\mu _{4} \frac{\delta _{dq} ( g_{t+1},g_{t} ) .\delta _{dq}(g_{t+2},g_{t+1})}{1+\delta _{dq} ( g_{t+1},g_{t} ) } \biggr) \\ \leq &\mathcal{F} \bigl( (\mu _{1}+\mu _{2})\delta _{dq} ( g_{t+1},g_{t} ) +(\mu _{3}+\mu _{4})\delta _{dq}(g_{t+2},g_{t+1}) \bigr) . \end{aligned}$$

Again by doing similar steps to obtain (2.2) from (2.1), we have

$$ \delta _{dq}(g_{t+2},g_{t+1})< \eta _{1} \delta _{dq} ( g_{t+1},g_{t} ) < \eta \delta _{dq} ( g_{t+1},g_{t} ) . $$
(2.5)

By using Lemma 1.11 and inequality (2.1), we have

$$\begin{aligned} \tau +\mathcal{F} \bigl(\delta _{dq} ( g_{t+1},g_{t} ) \bigr) \leq & \tau + \mathcal{F} \bigl(H_{\delta _{dq}}(Sg_{t},Sg_{t-1}) \bigr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t},g_{t-1} ) +\mu _{2}\delta _{dq} ( Sg_{t},g_{t} ) +\mu _{3} \delta _{dq}(Sg_{t-1},g_{t-1}) \\ & {}+\mu _{4} \frac{\delta _{dq} ( Sg_{t},g_{t} ) .\delta _{dq}(Sg_{t-1},g_{t-1})}{1+\delta _{dq} ( g_{t},g_{t-1} ) } \biggr) \\ \leq &\mathcal{F} \biggl( \mu _{1}\delta _{dq} ( g_{t},g_{t-1} ) +\mu _{2}\delta _{dq} ( g_{t+1},g_{t} ) +\mu _{3} \delta _{dq}(g_{t},g_{t-1}) \\ & {}+\mu _{4} \frac{\delta _{dq}(g_{t+1},g_{t}).\delta _{dq} ( g_{t},g_{t-1} ) }{1+\delta _{dq} ( g_{t},g_{t-1} ) } \biggr) \\ \leq &\mathcal{F} \bigl( (\mu _{1}+\mu _{3})\delta _{dq} ( g_{t},g_{t-1} ) +(\mu _{2}+\mu _{4})\delta _{dq}(g_{t+1},g_{t}) \bigr) . \end{aligned}$$

Again by doing similar steps to obtain (2.3) from (2.1), we have

$$ \delta _{dq}(g_{t+1},g_{t})< \eta _{2} \delta _{dq} ( g_{t},g_{t-1} ) < \eta \delta _{dq} ( g_{t},g_{t-1} ) . $$
(2.6)

By using (2.6) in (2.5), we have

$$ \delta _{dq}(g_{t+2},g_{t+1})< \eta ^{2} \delta _{dq} ( g_{t},g_{t-1} ) . $$

Continuing in this way, we have

$$ \delta _{dq}(g_{t+2},g_{t+1})< \eta ^{t+1} \delta _{dq} ( g_{1},g_{0} ) . $$
(2.7)

As \((h,C)\in \gamma \times [ 0,+\infty ) \) satisfies \((\Delta _{3})\), then for fixed \(\epsilon >0\) there exists \(\delta >0\) such that

$$ 0< \sigma < \delta \quad \text{implies}\quad h(\sigma )< h(\epsilon )-C. $$
(2.8)

By using (2.4), we have

$$\begin{aligned}& \sum_{k=n}^{m-1}\delta _{dq}(g_{k,}g_{k+1})< \eta ^{n} \bigl(1+ \eta +\eta ^{2}\ldots \eta ^{m-n-1} \bigr)\delta _{dq} ( g_{0},g_{1} ) , \\& \sum_{k=n}^{m-1}\delta _{dq}(g_{k,}g_{k+1})< \frac{\eta ^{n}}{1-\eta } \delta _{dq} ( g_{0},g_{1} ) , \quad m>n. \end{aligned}$$
(2.9)

Since \(\underset{n\rightarrow +\infty }{\lim }\frac{\eta ^{n}}{1-\eta } \delta _{dq} ( g_{0},g_{1} ) =0\), then for \(\delta >0\) there exists some \(n_{0}\in \mathbb{N} \) such that \(0<\frac{\eta ^{n}}{1-\eta }\delta _{dq} ( g_{0},g_{1} ) < \delta \), \(n\geq n_{0}\). By (2.8) and (2.9), we write

$$\begin{aligned} h \Biggl( \sum_{k=n}^{m-1}\delta _{dq}(g_{k,}g_{k+1}) \Biggr) < &h \biggl( \frac{\eta ^{n}}{1-\eta }\delta _{dq} ( g_{0},g_{1} ) \biggr) \\ < &h(\epsilon )-C\quad \text{for all }m,n\geq n_{0}. \end{aligned}$$

Suppose that \(\delta _{dq}(g_{p},g_{dq})=0\) for some \(p,q\in \{ 0,1,2,3,\ldots \} \) with \(q>p\), then \(g_{p}=g_{dq}\)

$$\begin{aligned}& \delta _{dq} ( g_{p},g_{p+1} ) = \delta _{dq} ( g_{p},Sg_{p} ) =\delta _{dq} ( g_{dq},Sg_{dq} ) =\delta _{dq} ( g_{dq},g_{q+1} ) \leq \eta ^{q-p}\delta _{dq} ( g_{p},g_{p+1} ), \\& \bigl( 1-\eta ^{q-p} \bigr) \delta _{dq} ( g_{p},g_{p+1} ) \leq 0. \end{aligned}$$

So \(\delta _{dq} ( g_{p},g_{p+1} ) =0\) and \(g_{p}=g_{p+1}\). Now, \(g_{p+1}\in Sg_{p}\) implies that \(g_{p}\in Sg_{p}\). Hence \(g_{p}\) is the fixed point of S. Now suppose that \(\delta _{dq}(g_{m},g_{n})\neq 0\) for all \(m,n\in \{ 0,1,2,3,\ldots \} \) with \(m>n\). Using \((\Delta _{3})\) and the inequality, \(\delta _{dq}(g_{n,}g_{m})>0\) for all \(m,n\geq n_{0}\), we have

$$\begin{aligned}& h \bigl( \delta _{dq}(g_{n,}g_{m}) \bigr) < h \Biggl(\sum_{k=n}^{m-1} \delta _{dq}(g_{k,}g_{k+1}) \Biggr)+C < h(\epsilon ), \\& \delta _{dq}(g_{n,}g_{m}) < \epsilon \quad \text{for all }m,n\geq n_{0}. \end{aligned}$$

This proves that \(\{ g_{n} \} \) is a right-Cauchy sequence in M. Again by using (2.7), we have

$$\begin{aligned} \sum_{k=n}^{m-1}\delta _{dq}(g_{k+1,}g_{k}) \leq &\eta ^{n} \bigl(1+ \eta +\eta ^{2}\ldots \eta ^{m-n-1} \bigr)\delta _{dq} ( g_{1},g_{0} ) \\ \leq &\frac{\eta ^{n}}{1-\eta }\delta _{dq} ( g_{1},g_{0} ) ,\quad m>n. \end{aligned}$$

Since \(\underset{n\rightarrow +\infty }{\lim }\frac{\eta ^{n}}{1-\eta } \delta _{dq} ( g_{1},g_{0} ) =0\), for any \(\delta >0\) there exists some \(n_{1}\in \mathbb{N} \) such that \(0<\frac{\eta ^{n}}{1-\eta }\delta _{dq} ( g_{1},g_{0} ) < \delta \) for all \(n\geq n_{1}\). Furthermore, assume that \((h,C)\in \gamma \times [ 0,+\infty ) \) satisfies \((\Delta _{3})\), and let \(\epsilon >0\) be fixed, by using similar steps as above, we have

$$ \delta _{dq}(g_{m,}g_{n})< \epsilon \quad \text{for all }m,n\geq n_{1}. $$

This proves that \(\{ g_{n} \} \) is a left-Cauchy sequence in M. Hence, \(\{ g_{n} \} \) is a bi-Cauchy sequence in M. Since \((M,\delta _{dq})\) is L-R-complete, there will be some \(y^{\ast }\in M\) such that \(\{ g_{n} \} \) is L-R-convergent to \(y^{\ast }\). By Lemma 1.5, every L-R-convergent sequence is bi-convergent, that is,

$$ \underset{t\longrightarrow +\infty }{\lim }\delta _{dq} \bigl(z^{\ast },g_{t} \bigr)= \underset{t\longrightarrow +\infty }{\lim }\delta _{dq} \bigl(g_{t},z^{ \ast } \bigr)=0. $$

Suppose \(\delta _{dq}(z^{\ast },Sz^{\ast })>0\), we have

$$\begin{aligned} \tau +\mathcal{F} \bigl(\delta _{dq} \bigl(g_{t+1},Sz^{\ast } \bigr) \bigr) \leq &\tau + \mathcal{F} \bigl(H_{\delta _{dq}} \bigl(Sg_{t},Sz^{\ast } \bigr) \bigr) \\ \leq &\mathcal{F} \biggl(\mu _{1}\delta _{dq} \bigl( g_{t},z^{\ast } \bigr) +\mu _{2}\delta _{dq} ( g_{t},Sg_{t} ) +\mu _{3}\delta _{dq} \bigl(z^{ \ast },Sz^{\ast } \bigr) \\ &{}+\mu _{4} \frac{\delta _{dq} ( g_{t},Sg_{t} ) .\delta _{dq}(z^{\ast },Sz^{\ast })}{1+\delta _{dq}(g_{t},z^{\ast })} \biggr). \end{aligned}$$

This implies that

$$\begin{aligned} \delta _{dq} \bigl(g_{t+1},Sz^{\ast } \bigr) < &\mu _{1}\delta _{dq} \bigl( g_{t},z^{ \ast } \bigr) +\mu _{2}\delta _{dq} ( g_{t},Sg_{t} ) + \mu _{3}\delta _{dq} \bigl(z^{\ast },Sz^{\ast } \bigr) \\ &{}+\mu _{4} \frac{\delta _{dq} ( g_{t},Sg_{t} ) .\delta _{dq}(z^{\ast },Sz^{\ast })}{1+\delta _{dq}(g_{t},z^{\ast })}. \end{aligned}$$

Taking \(t\rightarrow +\infty \), we have

$$\begin{aligned}& \delta _{dq} \bigl(z^{\ast },Sz^{\ast } \bigr) < \mu _{3}\delta _{dq} \bigl(z^{\ast },Sz^{ \ast } \bigr), \\& (1-\mu _{3})\delta _{dq} \bigl(z^{\ast },Sz^{\ast } \bigr) < 0. \end{aligned}$$

This is a contradiction, so \(\delta _{dq}(z^{\ast },Sz^{\ast })=0\), so \(z^{\ast }\in Sz^{\ast }\). Hence \(z^{\ast }\) is a fixed point of S. □

Example 2.2

Let \(M= [ 0,+\infty ) \). Consider \(\delta _{dq}:M\times M\longrightarrow [ 0,+\infty ) \) to be an L-R-complete function weighted dislocated quasi-metric on M defined as

$$ \delta _{dq}(g,w)= ( 2g+3w ) ^{2}. $$

Obviously, \(\delta _{dq}\) satisfies axiom \((\Delta _{1})\). However, \(\delta _{dq} \) is not symmetric, as \(\delta _{dq}(1,2)=64\neq 49=\delta _{dq}(2,1)\). Define \(S:M\times M\longrightarrow P(M)\) as \(S(g)= [ \frac{3g}{10},\frac{2g}{3} ] \). Take \(\mu _{1}=\frac{1}{2}\), \(\mu _{2}=\frac{1}{4}\), \(\mu _{3}= \frac{1}{8}\), \(\mu _{4}=\frac{1}{10}\), then \(\mu _{1}+\mu _{2}+\mu _{3}+\mu _{4}<1\). Taking \(\tau =0.2\) and \(\mathcal{F}(g)=\ln g\), we have

$$\begin{aligned}& \tau +\max \bigl\{ \mathcal{F} \bigl(H_{\delta _{dq}}(Sg,Sw) \bigr), \mathcal{F} \bigl(H_{\delta _{dq}}(Sw,Sg) \bigr) \bigr\} \\& \quad \leq \min \biggl\{ \mathcal{F} \biggl( \mu _{1}\delta _{dq}(g,w)+ \mu _{2}\delta _{dq}(g,Sg)+\mu _{3}\delta _{dq}(w,Sw)+\mu _{4} \frac{\delta _{dq}(g,Sg).\delta _{dq}(w,Sw)}{1+\delta _{dq}(g,w)} \biggr) , \\& \qquad {} \mathcal{F} \biggl( \mu _{1}\delta _{dq}(w,g)+\mu _{2} \delta _{dq}(Sg,g)+\mu _{3}\delta _{dq}(Sw,w)+\mu _{4} \frac{\delta _{dq}(Sg,g).\delta _{dq}(Sw,w)}{1+\delta _{dq}(w,g)} \biggr) \biggr\} \\& \quad = \mathcal{F} \biggl( \mu _{1}\delta _{dq}(w,g)+\mu _{2}\delta _{dq}(Sg,g)+ \mu _{3}\delta _{dq}(Sw,w)+\mu _{4} \frac{\delta _{dq}(Sg,g).\delta _{dq}(Sw,w)}{1+\delta _{dq}(w,g)} \biggr) \\& \quad = \ln \biggl( \frac{1}{2} ( 2g+3w ) ^{2}+\frac{1}{4} \biggl( \frac{3g}{5}+3g \biggr) ^{2}+\frac{1}{8} \biggl( \frac{3w}{5}+3w \biggr) ^{2}+\frac{1}{10} \frac{ ( \frac{3g}{5}+3g ) ^{2}. ( \frac{3w}{5}+3w ) ^{2}}{1+ ( 2g+3w ) ^{2}} \biggr). \end{aligned}$$

Since all the conditions of Theorem 2.1 are fulfilled and 0 is a fixed point of S.

Corollary 2.3

Suppose that \((M,\delta _{dq})\) is an L-R-complete function weighted dislocated quasi-metric space with respect to \((h,C)\in \gamma \times {}[ 0,+\infty )\). Let \(S:M\rightarrow P(M)\) be a multivalued mapping, \(\mathcal{F}:(0,+\infty )\rightarrow \mathbb{R} \) be a strictly increasing mapping, \(\tau >0\), \(\mu _{1},\mu _{3},\mu _{4}\geq 0\), \(\eta _{1}=\frac{\mu _{1}}{1-\mu _{3}-\mu _{4}}<1\) and \(\eta _{2}=\frac{\mu _{1}+\mu _{3}}{1-\mu _{4}}<1\) such that

$$\begin{aligned}& \tau +\max \bigl\{ \mathcal{F} \bigl(H_{\delta _{dq}}(Sg,Sw) \bigr), \mathcal{F} \bigl(H_{\delta _{dq}}(Sw,Sg) \bigr) \bigr\} \\& \quad \leq \min \biggl\{ \mathcal{F} \biggl( \mu _{1}\delta _{dq}(g,w)+ \mu _{3}\delta _{dq}(w,Sw)+\mu _{4} \frac{\delta _{dq}(g,Sg).\delta _{dq}(w,Sw)}{1+\delta _{dq}(g,w)} \biggr) , \\& \qquad {} \mathcal{F} \biggl( \mu _{1}\delta _{dq}(w,g)+\mu _{3} \delta _{dq}(Sw,w)+\mu _{4} \frac{\delta _{dq}(Sg,g).\delta _{dq}(Sw,w)}{1+\delta _{dq}(w,g)} \biggr) \biggr\} \end{aligned}$$

whenever \(\min \{ H_{\delta _{dq}}(Sg,Sw),H_{\delta _{dq}}(Sw,Sg) \} >0\), \(g,w\in \{MS(g_{t})\}\cup \{ z^{\ast } \} \), where \(\{ MS(g_{t}) \} \rightarrow z^{\ast }\). Then \(z^{\ast }\) is the fixed point of S.

Corollary 2.4

Suppose that \((M,\delta _{dq})\) is an L-R-complete function weighted dislocated quasi-metric space with respect to \((h,C)\in \gamma \times {}[ 0,+\infty )\). Let \(S:M\rightarrow P(M)\) be a multivalued mapping, \(\mathcal{F}:(0,+\infty )\rightarrow \mathbb{R} \) be a strictly increasing mapping, \(\tau >0\), \(\mu _{1},\mu _{2},\mu _{4}\geq 0\), \(\eta _{1}=\frac{\mu _{1}+\mu _{2}}{1-\mu _{4}}<1\) and \(\eta _{2}=\frac{\mu _{1}}{1-\mu _{2}-\mu _{4}}<1\) such that

$$\begin{aligned}& \tau +\max \bigl\{ \mathcal{F} \bigl(H_{\delta _{dq}}(Sg,Sw) \bigr), \mathcal{F} \bigl(H_{\delta _{dq}}(Sw,Sg) \bigr) \bigr\} \\& \quad \leq \min \biggl\{ \mathcal{F} \biggl( \mu _{1}\delta _{dq}(g,w)+ \mu _{2}\delta _{dq}(g,Sg)+\mu _{4} \frac{\delta _{dq}(g,Sg).\delta _{dq}(w,Sw)}{1+\delta _{dq}(g,w)} \biggr) , \\& \qquad {} \mathcal{F} \biggl( \mu _{1}\delta _{dq}(w,g)+\mu _{2} \delta _{dq}(Sg,g)+\mu _{4} \frac{\delta _{dq}(Sg,g).\delta _{dq}(Sw,w)}{1+\delta _{dq}(w,g)} \biggr) \biggr\} \end{aligned}$$

whenever \(\min \{ H_{\delta _{dq}}(Sg,Sw),H_{\delta _{dq}}(Sw,Sg) \} >0\), \(g,w\in \{MS(g_{t})\}\cup \{ z^{\ast } \} \), where \(\{ MS(g_{t}) \} \rightarrow z^{\ast }\). Then \(z^{\ast }\) is the fixed point of S.

Corollary 2.5

Suppose that \((M,\delta _{dq})\) is an L-R-complete function weighted dislocated quasi-metric space with respect to \((h,C)\in \gamma \times {}[ 0,+\infty )\). Let \(S:M\rightarrow P(M)\) be a multivalued mapping, \(\mathcal{F}:(0,+\infty )\rightarrow \mathbb{R} \) be a strictly increasing mapping, \(\tau >0\), \(\mu _{1},\mu _{2},\mu _{3}\geq 0\), \(\eta _{1}=\frac{\mu _{1}+\mu _{2}}{1-\mu _{3}}<1\) and \(\eta _{2}=\frac{\mu _{1}+\mu _{3}}{1-\mu _{2}}<1\) such that

$$\begin{aligned}& \tau +\max \bigl\{ \mathcal{F} \bigl(H_{\delta _{dq}}(Sg,Sw) \bigr), \mathcal{F} \bigl(H_{\delta _{dq}}(Sw,Sg) \bigr) \bigr\} \\& \quad \leq \min \bigl\{ \mathcal{F} \bigl( \mu _{1}\delta _{dq}(g,w)+ \mu _{2}\delta _{dq}(g,Sg)+\mu _{3}\delta _{dq}(w,Sw) \bigr) , \\& \qquad {} \mathcal{F} \bigl( \mu _{1}\delta _{dq}(w,g)+\mu _{2} \delta _{dq}(Sg,g)+\mu _{3}\delta _{dq}(Sw,w) \bigr) \bigr\} \end{aligned}$$

whenever \(\min \{ H_{\delta _{dq}}(Sg,Sw),H_{\delta _{dq}}(Sw,Sg) \} >0\), \(g,w\in \{MS(g_{t})\}\cup \{ z^{\ast } \} \), where \(\{ MS(g_{t}) \} \rightarrow z^{\ast }\). Then \(z^{\ast }\) is the fixed point of S.

Application

In this section, we present our main result for single-valued mappings and investigate the uniqueness of the fixed point as well. An application is given to the obtained result.

Theorem 3.1

Suppose that \((M,\delta _{dq})\) is an L-R-complete function weighted dislocated quasi-metric space with respect to \((h,C)\in \gamma \times {}[ 0,+\infty )\). Let \(S:M\rightarrow M\) be a mapping, \(\mathcal{F}:(0,+\infty )\rightarrow \mathbb{R} \) be a strictly increasing mapping, \(\tau >0\), \(\mu _{1},\mu _{2},\mu _{3},\mu _{4}\geq 0\), \(\eta _{1}=\frac{\mu _{1}+\mu _{2}}{1-\mu _{3}-\mu _{4}}<1\) and \(\eta _{2}=\frac{\mu _{1}+\mu _{3}}{1-\mu _{2}-\mu _{4}}<1\) such that

$$\begin{aligned}& \tau +\max \bigl\{ \mathcal{F} \bigl( \delta _{dq}(Sg,Sw) \bigr) , \mathcal{F} \bigl( \delta _{dq}(Sw,Sg) \bigr) \bigr\} \\& \quad \leq \min \biggl\{ \mathcal{F} \biggl( \mu _{1}\delta _{dq}(g,w)+ \mu _{2}\delta _{dq}(g,Sg)+\mu _{3}\delta _{dq}(w,Sw)+\mu _{4} \frac{\delta _{dq}(g,Sg).\delta _{dq}(w,Sw)}{1+\delta _{dq}(g,w)} \biggr) , \\& \qquad {}\mathcal{F} \biggl( \mu _{1}\delta _{dq}(w,g)+\mu _{2}\delta _{dq}(Sg,g)+ \mu _{3}\delta _{dq}(Sw,w)+\mu _{4} \frac{\delta _{dq}(Sg,g).\delta _{dq}(Sw,w)}{1+\delta _{dq}(w,g)} \biggr) \biggr\} , \end{aligned}$$
(3.1)

where, \(g,w\in M\). Then there exists a unique fixed point of S.

Proof

The proof of Theorem 3.1 is similar to the proof of Theorem 2.1. Here we prove only uniqueness. Suppose that \(g^{\ast }\) and \(w^{\ast }\) are the two distinct fixed points of S, then \(\delta _{dq}(g^{\ast },w^{\ast })>0\). By inequality (3.1), we have

$$\begin{aligned}& \tau +\mathcal{F}(\delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr) \leq \tau + \max \bigl\{ \mathcal{F}(\delta _{dq} \bigl(Sg^{\ast },Sw^{\ast } \bigr), \mathcal{F}(\delta _{dq} \bigl(Sw^{\ast },Sg^{\ast } \bigr) \bigr\} \\& \hphantom{\tau +\mathcal{F}(\delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr) }\leq \mathcal{F} \biggl( \mu _{1}\delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr)+ \mu _{2}\delta _{dq} \bigl(g^{\ast },Sg^{\ast } \bigr)+\mu _{3} \delta _{dq} \bigl(w^{ \ast },Sw^{\ast } \bigr) \\& \hphantom{\tau +\mathcal{F}(\delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr) \leq} {}+\mu _{4} \frac{\delta _{dq}(g^{\ast },Sg^{\ast }).\delta _{dq}(w^{\ast },Sw^{\ast })}{1+\delta _{dq}(g^{\ast },w^{\ast })} \biggr), \\& \tau +\mathcal{F}(\delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr) \leq \mathcal{F} \bigl( \mu _{1}\delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr) \bigr), \\& \delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr) < \mu _{1}\delta _{dq} \bigl(g^{\ast },w^{ \ast } \bigr), \\& \delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr) < \delta _{dq} \bigl(g^{\ast },w^{\ast } \bigr). \end{aligned}$$

As \(\delta _{dq}(g^{\ast },w^{\ast })>0\), therefore a contradiction arises. So, we have \(g^{\ast }\in M\), a unique fixed point of S. □

Remark

By taking a bi-complete function weighted quasi-metric space, \(\mu _{2}=\mu _{3}=\mu _{4}=0\), \(\tau >0\), and \(\mathcal{F}(\alpha )=\ln (\alpha )\) in Theorem 3.1, we obtain the result of Karapınar et al. [17] as follows.

Corollary 3.2

Let \((M,\delta _{q})\) be a bi-complete function weighted quasi-metric space and S be a mapping from M to M. Suppose that there exists \(k=\mu _{1}e^{-\tau }\in (0,1)\) such that

$$ \delta _{q}(Sg,Sw)\leq k\delta _{q}(g,w),\quad g,w\in M. $$
(3.2)

Then S possesses a unique fixed point \(g\in M\).

Remark

By taking a bi-complete function weighted quasi-metric space, \(\mu _{1}=\mu _{4}=0\) and \(\mu _{2}=\mu _{3}\), \(\tau >0\) and \(\mathcal{F}(\alpha )=\ln (\alpha )\) in Theorem 3.1, we obtain the result of Karapınar et al. [17] as follows.

Corollary 3.3

Let \((M,\delta _{q})\) be a bi-complete function weighted quasi-metric space and S be a mapping from M to M. Suppose that there exists \({\mu }=\mu _{2}e^{-\tau }\in (0,1/2)\) such that

$$ \delta _{q}(Sg,Sw)\leq {\mu } \bigl[ \delta _{q}(g,Sg)+\delta _{q}(w,Sw) \bigr] ,\quad g,w\in M. $$
(3.3)

Then S possesses a unique fixed point \(g\in M\).

Now we discuss the solution of Volterra type integral equation which is an application of Theorem 3.1. Consider the equation

$$ m(r)= \int _{0}^{r}H \bigl(r,q,m(q) \bigr)\,dq $$
(3.4)

for all \(r,q\in {}[ 0,1]\). For solution of (3.4), we follow the following process.

Let M be a collection of all real-valued continuous functions on \([0,1]\) endowed with the L-R-complete function weighted dislocated quasi-metric space. Define the supremum norm as \(\Vert m\Vert _{\tau }=\sup_{r\in {}[ 0,1]}\{ \vert m(r) \vert e^{-\tau r}\}\) for \(m\in M\), where \(\tau >0\). Now, define

$$ \delta _{dq}^{\tau }(m,z)= \Bigl[ \sup_{r\in {}[ 0,1]} \bigl\{ \bigl\vert 2m(r)+3z(r) \bigr\vert e^{-\tau r} \bigr\} \Bigr] ^{2}= \Vert 2m+3z \Vert _{\tau }^{2} $$

for all \(m,z\in M\), with these settings, \((M,\delta _{dq}^{\tau })\) becomes an L-R-complete function weighted dislocated quasi-metric space.

Let us prove the theorem given as under to make sure the existence of solution of (3.4).

Theorem 3.4

Suppose that the following conditions are satisfied:

  1. (i)

    \(H:[0,1]\times {}[ 0,1]\times C([0,1],\mathbb{R} _{+})\rightarrow \mathbb{R} _{+}\);

  2. (ii)

    \(S:M\rightarrow M\) is defined by

    $$ Sm(r)= \int _{0}^{r}H \bigl(r,q,m(q) \bigr)\,dq. $$

Suppose that \(\tau >0\) exists, such that

$$ \max \bigl\{ 2H(r,q,m)+3H(r,q,z),2H(r,q,z)+3H(r,q,m) \bigr\} \leq \frac{\tau N(m,z)e^{\tau q}}{\tau N(m,z)+1} $$

for \(m,z\in C([0,1],\mathbb{R} _{+})\) and for all \(r,q\in {}[ 0,1]\), where

$$\begin{aligned} N(m,z) =&\mu _{1} \Vert 2m+3z \Vert ^{2}+\mu _{2} \Vert 2m+3Sm \Vert ^{2}+\mu _{3} \Vert 2z+3Sz \Vert ^{2} \\ &{}+\mu _{4} \frac{ \Vert 2m+3Sm \Vert ^{2}. \Vert 2z+3Sz \Vert ^{2}}{1+ \Vert 2m+3z \Vert ^{2}}, \end{aligned}$$

where \(\tau ,\mu _{1},\mu _{2},\mu _{3},\mu _{4}>0\) and \(\mu _{1}+\mu _{2}+\mu _{3}+\mu _{4}<1\). Then \(( 3.4 ) \) has a unique solution.

Proof

By supposition (ii)

$$\begin{aligned}& \bigl\vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \bigr\vert \\& \quad = \max \biggl\{ \int _{0}^{r} \bigl( 2H(r,q,m)+3H(r,q,z) \bigr)\,dq, \int _{0}^{r} \bigl( 2H(r,q,z)+3H(r,q,m) \bigr)\,dq \biggr\} \\& \quad < \int _{0}^{r}\frac{\tau N(m,z)}{\tau N(m,z)+1}e^{\tau q}\,dq \\& \quad = \frac{\tau N(m,z)}{\tau N(m,z)+1} \int _{0}^{r}e^{\tau q}\,dq, \\& \bigl\vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \bigr\vert < \frac{\tau N(m,z) ( e^{\tau r}-1 ) }{ ( \tau N(m,z)+1 ) \tau } \\& \hphantom{\bigl\vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \bigr\vert }< \frac{N(m,z)e^{\tau r}}{\tau N(m,z)+1}, \\& \bigl\vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \bigr\vert e^{- \tau r} < \frac{N(m,z)}{\tau N(m,z)+1}, \\& \bigl\Vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \bigr\Vert _{ \tau } < \frac{N(m,z)}{\tau N(m,z)+1}. \end{aligned}$$

This implies

$$ \frac{\tau N(m,z)+1}{N(m,z)}< \frac{1}{ \Vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \Vert _{\tau }}. $$

That is,

$$ \tau +\frac{1}{N(m,z)}< \frac{1}{ \Vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \Vert _{\tau }}. $$

This further implies

$$\begin{aligned}& \tau - \frac{1}{ \Vert \max \{ 2Sm+3Sz,2Sz+3Sm \} \Vert _{\tau }} < \frac{-1}{N(m,z)}, \\& \tau +\max \biggl\{ \frac{-1}{ \Vert 2Sm+3Sz \Vert }, \frac{-1}{ \Vert 2Sz+3Sm \Vert } \biggr\} < \frac{-1}{N(m,z)}. \end{aligned}$$

For \(\mathcal{F}(z)=\frac{-1}{\sqrt{z}}\); \(z >0\) and \(\delta _{dq}^{\tau }(m,z)=\Vert 2m+3z\Vert _{\tau }^{2}\), the conditions of Theorem 3.1 are fulfilled. Hence the Volterra integral equation given in (3.4) has a unique solution. □

Conclusion

The notion of a function weighted L-R-complete dislocated quasi-metric space has been introduced. The condition \(\delta _{dq}(g,g)=0\) from function weighted quasi-metric space has been excluded. The concept of bi-completeness has been generalized by introducing the concept of L-R-completeness. We have established fixed point results fulfilling generalized rational type F-contraction for a multivalued mapping in this new framework. We have presented results for single-valued mappings and have investigated the uniqueness of the fixed point as well. An application and an example have also been constructed.

Availability of data and materials

All the data utilized in this article have been included, and the sources where they were adopted were cited accordingly.

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Acknowledgements

The fourth author would like to thank Ministry of Education Malaysia and Universiti Kebangsaan Malaysia for their research support.

Funding

This work was supported by the Ministry of Education Malaysia through grant (FRGS/1/2019/STG06/UKM/01/3).

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Correspondence to Abdullah Shoaib.

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Shoaib, A., Mahmood, Q., Shahzad, A. et al. Fixed point results for rational contraction in function weighted dislocated quasi-metric spaces with an application. Adv Differ Equ 2021, 310 (2021). https://doi.org/10.1186/s13662-021-03458-x

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  • DOI: https://doi.org/10.1186/s13662-021-03458-x

MSC

  • 46S40
  • 47H10
  • 54H25

Keywords

  • Function weighted L-R-complete dislocated quasi-metric spaces
  • Fixed point
  • Set-valued mappings