1 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, 18–20, 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, [2–4, 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

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

2 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.

3 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. □

4 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.