Theory and Modern Applications

# 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).

## 1 Introduction and preliminaries

### 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.

## 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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Each author equally contributed to this paper, read and approved the final manuscript.

### Corresponding author

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