Fredholm Type Integral Equation in Controlled Rectangular Metric-like Spaces

Abstract

In this article, we present an extension of the controlled rectangular b-metric spaces, so-called controlled rectangular metric-like spaces, where we keep the symmetry condition and we only change the condition $\left[D\right(s,r)=0\iff s=r]\mathrm{to}\left[D\right(s,r)=0\Rightarrow s=r]$, which means we may have a non-zero self distance; also, $D(s,s)$ is not necessarily less than $D(s,r).$ This new type of metric space is a generalization of controlled rectangular b-metric spaces and partial rectangular metric spaces.

1. Introduction

The uniqueness of a fixed-point theory for self-contractive mapping, which was introduced by Banach in 1922 [1], opened a new area of research in various fields. It has become an interesting domain and an exciting field of mathematical research see [2,3,4]; in fact, it has become an important tool now in many fields of mathematics, such as variational inequalities, approximation theory, linear inequalities nonlinear analysis, differential, and integral equations; for more details on these type of applications, see [5,6,7]. Its applications appear in mathematical sciences, super fractals, and more recently, in discrete dynamics. Kamran et al. [8] introduced extended b-metric spaces, which is a generalization of metric spaces and b-metric spaces [9]. Then, the generalization of these metrics appeared in the form of a controlled metric [10] and double controlled metric spaces [11]. Further, Branciari [12], in 2000, introduced rectangular metric spaces. Then, in 2015, George et al. in [13], generalized rectangular metric spaces to rectangular b-metric spaces. In 2020, Mlaiki et al., in [14], generalized the rectangular b-metric spaces by introducing the controlled rectangular metric spaces. Inspired by the work of Matthews in [15], where he introduced the notion of partial metric spaces, which is basically assuming that the self distance is not necessarily zero; however, we have $D(s,s)\le D(s,r).$ Shukla, in [16], introduced the concept of partial rectangular metric spaces, which is basically the exact same work as Matthews, except it is in rectangular metric spaces. In this article, we present a generalization of controlled rectangular b-metric spaces and partial rectangular metric spaces, so-called controlled rectangular metric-like spaces. In the next section, we present some preliminaries and concepts needed later; while in the next section, we prove our main result. In the last section, we present an application of our results.

2. Preliminaries

We present some preliminary definitions of rectangular b-metric spaces, and controlled rectangular metric spaces, before presenting our new notion of a controlled rectangular metric-like space.

Definition 1

([12]). (Branciari metric spaces) Let $X\ne \varphi$. A mapping $D:X^2\to [0,\infty )$ is called a rectangular metric on X if for any $x,y\in X$ and all $u,\ne v\in X\backslash \{x,y\}$, having the following conditions:

($R_1$) $x=y\leftrightarrow D(x,y)=0$;

($R_2$) $D(x,y)=D(y,x)$;

($R_3$) $D(x,y)\le D(x,u)+D(u,v)+D(v,y)$.

In this case, then $(X,D)$ is a rectangular metric space.

As a generalization of rectangular metric spaces, rectangular b-metric spaces were introduced in [13], where the triangle inequality has a constant $a>1.$

Definition 2

([13]). (Rectangular b-metric spaces)) Let $X\ne \varphi$. A mapping $D:X^2\to [0,\infty )$ is known as rectangular b-metric on X for a constant $a\ge 1$ such that any $xy\in X$ and points $u\ne ,v\in X\backslash \{x,y\}$, which has the following conditions:

($R_{b1}$) $x=y\leftrightarrow D(x,y)=0$;

($R_{b2}$) $D(x,y)=D(y,x)$;

($R_{b3}$) $D(x,y)\le a\left[D\right(x,u)+D(u,v)+D(v,y\left)\right]$.

In this case, the pair $(X,D)$ is called a rectangular b-metric space.

In 2020, a new extension to the rectangular metric spaces was defined as follows.

Definition 3

([14]). Let X be a non empty set, a function $\theta :X^4\to [1,\infty )$ and $D:X^2\to [0,\infty ).$ We say that $(X,D)$ is a controlled rectangular b-metric space if all distinct $x,y,u,v\in X$ we have:

• $D(x,y)=0$ if and only if $x=y;$
• $D(x,y)=D(y,x);$
• $D(x,y)\le \theta (x,y,u,v)\left[D\right(x,u)+D(u,v)+D(v,y\left)\right].$

In this manuscript, we define controlled rectangular metric-like spaces as follows;

Definition 4.

Let X be a non empty set, a function $\theta :X^4\to [1,\infty )$and $D:X^2\to [0,\infty ).$ We say that $(X,D)$ is a controlled rectangular metric-like space if $x\ne y\ne u\ne v\in X$ having the functions:

• $D(x,y)=0\Rightarrow x=y;$
• $D(x,y)=D(y,x);$ (symmetric condition)
• $D(x,y)\le \theta (x,y,u,v)\left[D\right(x,u)+D(u,v)+D(v,y\left)\right].$

Remark 1.

Note that, in Definition 4, we are assuming that the space is symmetric. However, in the case where the symmetric condition is not satisfied, we will have a different space with a totally different topology.

Next, we present two examples of controlled rectangular metric-like spaces that are not controlled rectangular b-metric spaces.

Example 1.

Let $X=[0,\infty )$ and $p:[0,\infty )\times [0,\infty )\to (1,\infty ).$ Define $D:X^2\to [0,\infty )$ by

$$D(x,y)={(x+y)}^{p(x,y)}\mathrm{for}\mathrm{all}x,y\in X$$

Note that $(X,D)$ is a controlled rectangular metric-like space with

$$\theta (x,y,u,v)=2^{p(max\{x,y\},max\{u,v\left\}\right)-1}.$$

For all $0<y<x$ we have

$$D(x,x)={(x+x)}^{p(x,x)}>{(x+y)}^{p(x,y)}=D(x,y).$$

Thus, $(X,D)$ is not a controlled rectangular b-metric space nor a partial rectangular metric space.

Example 2.

Let $X=Y\cup Z$ where $Y=\{\frac{1}{m}\mid m\mathit{is}a\mathit{natural}\mathit{number}\}$ and $Z\subset {\mathbb{R}}^{+}$. We define $D:X^2\to [0,\infty )$ by

$$D(x,y)=\left\{\begin{array}{cc}0,\hfill & \mathit{if}\mathit{and}\mathit{only}\mathit{if}x=y\hfill \\ 2\beta ,\hfill & \mathit{if}x,y\in Y\hfill \\ \frac{1}{2},\hfill & x=y=1\hfill \\ \frac{\beta }{2},\hfill & \mathit{otherwise},\hfill \end{array}\right.$$

where β is a constant bigger than $0.$ Now, define $\theta :X^4\to [1,\infty )$ by $\theta (x,y,u,v)=max\{x,y,u,v\}+2\beta .$ It is quite easy to check that $(X,D)$ is a controlled rectangular metric-like space. However, $(X,D)$ is not a controlled rectangular metric type space nor a partial rectangular metric space, for example $D(1,1)=\frac{1}{2}\ne 0.$

Remark 2.

Notice that by Example 1, not every controlled rectangular metric-like space is a controlled rectangular b-metric space. On the other hand, every controlled rectangular b-metric space and every partial rectangular metric space is a controlled rectangular metric-like space.

Next, we present the topology of controlled rectangular metric-like spaces.

Definition 5.

Let $(X,D)$ be controlled rectangular metric-like space,

• A sequence $\left\{x_l\right\}$ in a controlled rectangular metric-like space $(X,D)$ is called $D-$convergent, if there exists $x\in X$ such that ${lim}_{l\to \infty }D(x_l,\nu )=D(\nu ,\nu ).$
• A sequence $\left\{x_l\right\}$ is called $D-$Cauchy if and only if ${lim}_{l,m\to \infty }D(x_l,x_m)$ exists and finite.
• A controlled rectangular metric-like space $(X,D)$ is called D-complete if for every $D-$Cauchy sequence $\left\{x_n\right\}$ in X, if there exists $\nu \in X,$ such that

  $$\underset{l\to \infty }{lim}D(x_l,\nu )=\underset{l,m\to \infty }{lim}D(x_l,x_m)=D(\nu ,\nu ).$$
• For $a\in X$, an open ball in a controlled rectangular metric-like space $(X,D)$ define by

  $$B_D(a,\eta )=\{b\in X\mid |D(a,b)-D(a,a)|<\eta \}.$$

Next, we define continuity in controlled rectangular metric-like spaces.

Definition 6.

A self-mapping function ζ in F is said continuous at $x\in F$ if for all $\epsilon >0$, there exists $\delta >0$ such that $\zeta \left(B\right(s,\delta \left)\right)\subseteq B\left(\zeta \right(s,\epsilon \left)\right)$, that is ${lim}_{n\to \infty }\zeta \left(x_n\right)=\zeta \left({lim}_{n\to \infty }{x}_n\right).$

In the next section, we present our main results by proving the existence of a fixed point for mappings that satisfies different types of contractions in controlled rectangular metric-like spaces.

3. Main Results

Theorem 1.

Let $(X,D)$ be a complete controlled rectangular metric-like space, and T is continuous and maps to itself on $X.$ If there exists $0<k<1,$ such that $D(Tx,Ty)\le kD(x,y)$ and

$$\underset{m>1}{sup}\underset{l\to \infty }{lim}\theta (x_l,x_{l+1},x_{l+2},x_m)\le \frac{1}{k},$$

then in X there is a unique fixed point of T.

Proof. 

Let $x_0\in X$ and it is a sequence $\left\{x_l\right\}$ as follows $x_1=T{x}_0,x_2=T^2{x}_0,\cdots ,x_l=T^l{x}_0,\cdots$ Now, by the hypothesis of the theorem, we have

$$D(x_l,x_{l+1})\le kD(x_{l-1},x_l)\le k^{2}D(x_{l-2},x_{l-1})\le \cdots \le k^{l}D(x_0,x_1).$$

Note that taking the limit of the above inequality as $n\to \infty$ we deduce that $D(x_l,x_{l+1})\to 0\mathrm{as}l\to \infty$. Denote by $D_i=D(x_{l+i},x_{l+i+1})$. For all $l\ge 1,$ we have two cases.

Case 1: 

Let $x_l=x_m$ for some integers $l\ne m$. Therefore, if for $m>l$ we have $T^{m-l}\left(x_l\right)=x_l$. Choose $y=x_l$ and $p=m-l$. Then $T^{p}y=y,$ and that is, y is a periodic point of T. Thus, $D(y,Ty)=D(T^{p}y,T^{p+1}y)\le k^{p}D(y,Ty).$ Since $k\in (0,1)$, we obtain $D(y,Ty)=0$, so $y=Ty$, therefore, T has a fixed point y.

Case 2: 

Suppose $T^{l}x\ne T^{m}x$ for all integers $l\ne m$. Let $l<m\in N,$ and to show that $\left\{x_l\right\}$ is a $D-$Cauchy sequence, we considered two subcases:

Subcase 1: Assume that $m=l+2p+1.$ By property $\left(3\right)$ of the controlled rectangular-like metric spaces, we have,

$$\begin{array}{cc}\hfill D(x_l,x_{l+2p+1})& \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})[D(x_l,x_{l+1})+D(x_{l+1},x_{l+2})+D(x_{l+2},x_{l+2p+1})]\hfill \\ \hfill & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D(x_l,x_{l+1})+\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D(x_{l+1},x_{l+2})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})[D(x_{l+2},x_{l+3})\hfill \\ & +D(x_{l+3},x_{l+4})+D(x_{l+4},x_{l+2p+1})]\hfill \\ & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D(x_l,x_{l+1})+\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D(x_{l+1},x_{l+2})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})D(x_{l+2},x_{l+3})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})D(x_{l+3},x_{l+4})\hfill \\ \hfill & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})D(x_{l+4},x_{l+2p+1})\hfill \\ & \le \cdots \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D(x_l,x_{l+1})+\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D(x_{l+1},x_{l+2})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})D(x_{l+2},x_{l+3})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})D(x_{l+3},x_{l+4})\hfill \\ \hfill & +\cdots +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})\hfill \\ & \cdots \theta (x_{l+2p-2},x_{l+2p-1},x_{l+2p},x_{l+2p+1})D(x_{l+2p},x_{l+2p+1})\hfill \\ \hfill & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D_0+\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})D_1\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})D_2\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})D_3\hfill \\ \hfill & +\cdots \hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})\times \cdots \hfill \\ \hfill & \times \cdots \theta (x_{l+2p-2},x_{l+2p-1},x_{l+2p},x_{l+2p+1})D_{2p}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})[D_0+D_1]\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})[D_2+D_3]\hfill \\ \hfill & +\cdots +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})\times \cdots \hfill \\ \hfill & \times \cdots \theta (x_{l+2p-2},x_{l+2p-1},x_{l+2p},x_{l+2p+1})[D_{2p-1}+D_{2p}]\hfill \\ \hfill & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\left[(k^l+k^{l+1})D(x_0,x_1)\right]\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})\left[(k^{l+2}+k^{l+3})D(x_0,x_1)\right]\hfill \\ \hfill & +\cdots +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})\times \cdots \hfill \\ \hfill & \times \cdots \theta (x_{l+2p-2},x_{l+2p-1},x_{l+2p},x_{l+2p+1})\left[(k^{l+2p-2}+k^{l+2p-1})D(x_0,x_1)\right]\hfill \\ & \le [\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})(k^l+k^{l+1})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})(k^{l+2}+k^{l+3})+\hfill \\ & \cdots +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p+1})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p+1})\times \cdots \hfill \\ & \times \cdots \theta (x_{l+2p-2},x_{l+2p-1},x_{l+2p},x_{l+2p+1})(k^{l+2p-2}+k^{l+2p-1})]D(x_0,x_1)\hfill \\ \hfill & =\sum _{r=0}^{p-1}\prod _{i=0}^r\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+2},x_{l+2p+1})[k^{l+2l}+k^{l+2l+1}]D(x_0,x_1)\hfill \\ \hfill & =\sum _{r=0}^{p-1}\prod _{i=0}^r\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+2},x_{l+2p+1})[1+k]k^{l+2r}D(x_0,x_1)\hfill \end{array}$$

As $k<1$, the above inequalities imply the following:

$$D(x_l,x_{l+2p+1})<\sum _{r=0}^{p-1}\prod _{i=0}^r\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+2},x_{l+2p+1})2k^{l+2r}D(x_0,x_1).$$

Since ${sup}_{m>1}{lim}_{l\to \infty }\theta (x_l,x_{l+1},x_{l+2},x_m)\le \frac{1}{k}$ we deduce that,

$$\begin{array}{cc}\hfill \underset{l,p\to \infty }{lim}D(x_l,x_{l+2p+1})& <\sum _{r=0}^{\infty }\prod _{i=0}^r\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+2},x_{l+2p+1})2k^{l+2r}D(x_0,x_1)\hfill \\ \hfill & \le \sum _{r=0}^{\infty }\frac{1}{k^{r+1}}2k^{l+2r}D(x_0,x_1)\hfill \\ \hfill & \le \sum _{r=0}^{\infty }2k^{l+r-1}D(x_0,x_1).\hfill \end{array}$$

The series ${\sum }_{r=0}^{\infty }2k^{l+r-1}D(x_0,x_1)$ is convergent by the ratio test, which implies that $D(x_l,x_{l+2p+1})$ converges as $l,p\to \infty .$

Subcase 2: $m=l+2p$ Fist of all, note that

$$D(x_l,x_{l+2})\le kD(x_{l-1},x_{l+1})\le k^{2}D(x_{l-2},x_l)\le \cdots \le k^{l}D(x_0,x_2),$$

which leads us to conclude that $D(x_l,x_{l+2})\to 0\mathrm{as}l\to \infty .$ Similarly to Subcase 1, we have:

$$\begin{array}{cc}\hfill D(x_l,x_{l+2p})& \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p})[D(x_l,x_{l+1})+D(x_{l+1},x_{l+2})+D(x_{l+2},x_{l+2p})]\hfill \\ \hfill & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p})D(x_l,x_{l+1})+\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})D(x_{l+1},x_{l+2})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p})[D(x_{l+2},x_{l+3})\hfill \\ & +D(x_{l+3},x_{l+4})+D(x_{l+4},x_{l+2p})]\hfill \\ & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p})D(x_l,x_{l+1})+\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})D(x_{l+1},x_{l+2})\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p})D(x_{l+2},x_{l+3})\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p})D(x_{l+3},x_{l+4})\hfill \\ \hfill & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p})D(x_{l+4},x_{l+2p})\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \theta (x_l,x_{l+1},x_{l+2},x_{l+2p})D_0+\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})D_1\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p})D_2\hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p})D_3\hfill \\ \hfill & +\cdots \hfill \\ & +\theta (x_l,x_{l+1},x_{l+2},x_{l+2p})\theta (x_{l+2},x_{l+3},x_{l+4},x_{l+2p})\times \cdots \hfill \\ \hfill & \times \cdots \theta (x_{l+2p-3},x_{l+2p-2},x_{l+2p-1},x_{l+2p})D_{2p}\hfill \\ & +\prod _{i=0}^{2p-2}\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+1},x_{l+2p})D(x_{l+2p-2},x_{l+2p})\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{r=0}^{p-1}\prod _{i=0}^r\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+2},x_{l+2p+1})[k^{l+2r}+k^{l+2r+1}]D(x_0,x_1)\hfill \\ \hfill & +\prod _{i=0}^{2p-2}\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+1},x_{l+2p})D(x_{l+2p-2},x_{l+2p})\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\sum _{r=0}^{p-1}\prod _{i=0}^r\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+2},x_{l+2p+1})[1+k]k^{l+2r}D(x_0,x_1)\hfill \\ \hfill & +\prod _{i=0}^{2p-2}\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+1},x_{l+2p})D(x_{l+2p-2},x_{l+2p})\hfill \\ \hfill & \le \sum _{r=0}^{p-1}\prod _{i=0}^r\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+2},x_{l+2p+1})[1+k]k^{l+2r}D(x_0,x_1)\hfill \\ \hfill & +\prod _{i=0}^{2p-2}\theta (x_{l+2i},x_{l+2i+1},x_{l+2i+1},x_{l+2p})k^{l+2p-2}D(x_0,x_2)\hfill \end{array}$$

Since ${sup}_{m>1}{lim}_{l\to \infty }\theta (x_l,x_{l+1},x_{l+2},x_m)\le \frac{1}{k}$ we deduce that,

$$\begin{array}{cc}\hfill \underset{l,p\to \infty }{lim}D(x_l,x_{l+2p})& \le \underset{l,p\to \infty }{lim}\sum _{r=0}^{p-1}\frac{1}{k^{l+1}}[1+k]k^{l+2r}D(x_0,x_1)+k^{2p-1}{k}^{l+2p-2}D(x_0,x_2)\hfill \\ \hfill & =\underset{l,p\to \infty }{lim}\sum _{r=0}^{p-1}[1+k]k^{l+r-1}D(x_0,x_1)+k^{l-1}D(x_0,x_2)\hfill \\ \hfill & \le \sum _{m=0}^{\infty }[1+k]k^{m}D(x_0,x_1)+k^{m}D(x_0,x_2)\hfill \end{array}$$

From the ratio test, it is easy to show that the series

$$\sum _{m=0}^{\infty }[1+k]k^{m}D(x_0,x_1)+k^{m}D(x_0,x_2)$$

converges. Hence, $D(x_l,x_{l+2p})$ converges as $l,p\mathrm{going}\mathrm{toward}\infty .$ Thus, by subcase 1 and subcase 2, it is proved that the sequence $\left\{x_l\right\}$ is a $D-$Cauchy sequence. Since $(X,D)$ is a D-complete extended rectangular metric-like space, we deduce that $\left\{x_l\right\}$ converges to some $\nu \in X.$ Now, we show that $\nu$ is fixed point of $T.$

$$D(x_l,x_{l+1})=D(T{x}_{l-1},T{x}_l)\le kD(x_{l-1},x_l)=kD(x_{l-1},T{x}_{l-1})<D(x_{l-1},T{x}_{l-1})$$

Now, taking the limit $l\to \infty$, and as T is continuous, we deduce that

$$D(\nu ,T\nu )<D(\nu ,T\nu ),$$

which leads us to a contradiction. Hence, $D(\nu ,T\nu )=0$ and that is $T\nu =\nu$ and $\nu$ is a fixed point of $T.$

Finally, for uniqueness, let us assume two fixed points of T say $\nu$ and $\mu$ such that $\nu \ne \mu .$ By the contractive property of T we have:

$$D(\nu ,\mu )=D(T\nu ,T\mu )\le kD(\nu ,\mu )<D(\nu ,\mu )$$

which leads us to contradiction. Thus, T has a unique fixed point as required. □

Theorem 2.

Let $(X,D)$ be a complete controlled rectangular metric-like space, and T a continuous self-mapping on X satisfying the following condition; for all $x,y\in X$ there exists $0<k<\frac{1}{2}$ such that

$$D(Tx,Ty)\le k\left[D\right(x,Tx)+D(y,Ty\left)\right]$$

Furthermore, if

$$\underset{m>1}{sup}\underset{l\to \infty }{lim}\theta (x_l,x_{l+1},x_{l+2},x_m)\le \frac{1}{k},$$

and for all $u,v\in X$, we have:

$$\underset{l\to \infty }{lim}\theta (u,v,x_l,x_{l+1})\le 1,$$

then T has a fixed point in $X.$ Moreover, if for every fixed point ν of T we have $D(\nu ,\nu )=0,$ then the fixed point of T is unique.

Proof. 

Let $x_0\in X$ and define the sequence $\left\{x_l\right\}$ as follows

$$x_1=T{x}_0,x_2=T{x}_1=T^2{x}_0,\cdots ,x_l=T{x}_{l-1}=T^n{x}_0,\cdots$$

First of all, note that for all $l\ge 1$, we have

$$\begin{array}{cc}\hfill & D(x_l,x_{l+1})\le k[D(x_{l-1},x_l)+D(x_l,x_{l+1})]\hfill \\ & \Rightarrow (1-k)D(x_l,x_{l+1})\le kD(x_{l-1},x_l)\hfill \\ & \Rightarrow D(x_l,x_{l+1})\le \frac{k}{1-k}D(x_{l-1},x_l).\hfill \end{array}$$

Since $0<k<\frac{1}{2}$, one can easily deduce that $0<\frac{k}{1-k}<1.$ Therefore, let $\mu =\frac{k}{1-k}.$

Hence,

$$\begin{array}{cc}\hfill D(x_l,x_{l+1})& \le \mu D(x_{l-1},x_l)\hfill \\ \hfill & \le {\mu }^{2}D(x_{l-2},x_{l-1})\hfill \\ & \le \cdots \hfill \\ & \le {\mu }^{l}D(x_0,x_1).\hfill \end{array}$$

Therefore,

$$D(x_l,x_{l+1})\to 0\mathrm{as}l\to \infty .$$

Furthermore, for all $l\ge 1$, we have

$$D(x_l,x_{l+2})\le k[D(x_{l-1},x_l)+D(x_{l+1},x_{l+2})]$$

Thus, by using the fact that $D(x_l,x_{l+1})\to 0\mathrm{as}l\to \infty ,$ we deduce that

$$D(x_l,x_{l+2})\to 0\mathrm{as}l\to \infty .$$

Now, similarly to the proof of case 1 and case 2 of Theorem 1, we deduce that the sequence $\left\{x_l\right\}$ is a $D-$Cauchy sequence. Since $(X,D)$ is a D-complete controlled rectangular metric-like space, we conclude that $\left\{x_l\right\}$ converges to some $\nu \in X.$ Now, we show that $\nu$ is a fixed point of $T.$

$$D(x_l,x_{l+1})=D(T{x}_{l-1},T{x}_l)\le k[D(x_{l-1},T{x}_{l-1})+D(x_l,T{x}_l)]$$

Now, taking the limit as $l\to \infty$ and using the fact that T is continuous, we deduce that

$$D(\nu ,T\nu )<k[D(\nu ,T\nu )+D(\nu ,T\nu )]=2kD(\nu ,T\nu )<D(\nu ,T\nu )\mathrm{since}k<\frac{1}{2},$$

which leads us to a contradiction. Hence, $D(\nu ,T\nu )=0$, and that is, $T\nu =\nu$, and $\nu$ is a fixed point of $T.$ To show uniqueness, we assume two fixed points of T, say $\nu$ and $\mu$ such that $\nu \ne \mu .$ By the contractive property of T, we have:

$$D(\nu ,\mu )=D(T\nu ,T\mu )\le k\left[D\right(\nu ,T\nu )+D(\mu ,T\mu \left)\right]=k\left[D\right(\nu ,\nu )+D(\mu ,T\mu \left)\right]=0.$$

Thus, $D(\nu ,\mu )=0$ and, that is, $\nu =\mu .$ Therefore, T has a unique fixed point as required. □

In the next section, we present an application of our result.

4. Application

Let X be the set $C\left(\right[0,1],\mathbb{R})$ and consider the following Fredholm type integral equation:

$${\zeta }^{\prime }\left(t\right)={\int }_0^{1}F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))ds,\mathrm{for}\psi ,\omega \in [0,1]$$

(1)

where $F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))$ is continuous from ${[0,1]}^2\to \mathbb{R}.$ Next, let

$$\begin{array}{cc}\hfill D:X& \times X⟶\mathbb{R}\hfill \\ & (\zeta ,\varrho )\mapsto \underset{t\in [0,1]}{sup}(\frac{|{\zeta }^{\prime }\left(t\right)|+|\varrho \left(t\right)|}{2})\hfill \end{array}$$

Notice that $(X,D)$ is a complete controlled rectangular metric-like space, where

$$\theta (x,y,u,v)=max\{x,y,u,v\}.$$

Theorem 3.

If $\zeta ,\varrho \in X$ satisfies the following conditions

$\left(1\right)$$|F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))|+|F(\psi ,\omega ,\varrho \left(t\right))|\le k(|{\zeta }^{\prime }\left(t\right)|+|\varrho \left(t\right)\left|\right),$ for some $k\in (0,1);$

$\left(2\right)$$F(\psi ,\omega ,{\int }_0^{1}F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))ds)<F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))$ for all $\psi ,\omega ,$

then Equation (1) has a unique solution.

Proof. 

Let $T:X⟶X$ be defined by $T{\zeta }^{\prime }\left(t\right)={\int }_0^{1}F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))ds,$ then

$D(T\zeta ,T\varrho )={sup}_{t\in [0,1]}\left(\frac{|T{\zeta }^{\prime }\left(t\right)|+|T\varrho \left(t\right)|}{2}\right).$ Hence,

$$\begin{array}{cc}\hfill \frac{|T{\zeta }^{\prime }\left(t\right)|+|T\varrho \left(t\right)|}{2}& =\frac{|{\int }_0^{1}F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))ds|+|{\int }_0^{1}F(\psi ,\omega ,\varrho \left(t\right))ds|}{2}\hfill \\ & \le \frac{{\int }_0^1|F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))|ds+{\int }_0^1\left|F(\psi ,\omega ,\varrho \left(t\right))\right|ds}{2}\hfill \\ & =\frac{{\int }_0^1\left(\right|F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))|+|F(\psi ,\omega ,\varrho \left(t\right))\left|\right)ds}{2}\hfill \\ & \le \frac{{\int }_0^{1}k\left(\right|{\zeta }^{\prime }\left(t\right)|+|\varrho \left(t\right)\left|\right)ds}{2}\hfill \\ & \le kD({\zeta }^{\prime },\varrho ).\hfill \end{array}$$

Thus, $D(T\zeta ,T\varrho )\le kD(\zeta ,\varrho ).$ Now, let $n\in {\mathbb{N}}^{☆}$ and $\zeta \in X;$

$$\begin{array}{cc}\hfill \left(T^n\zeta \right)\left(t\right)=T\left(T^{n-1}{\zeta }^{\prime }\left(t\right)\right)& ={\int }_0^{1}F(\psi ,\omega ,T^{n-1}{\zeta }^{\prime }\left(t\right))ds\hfill \\ & ={\int }_0^{1}F(\psi ,\omega ,T\left(T^{n-2}\zeta \right)\left(t\right))ds\hfill \\ & ={\int }_0^{1}F(\psi ,\omega ,{\int }_0^{1}F(\psi ,\omega ,(T^{n-2}{\zeta }^{\prime }(t))))ds\hfill \\ & <{\int }_0^{1}F(\psi ,\omega ,\left(T^{n-2}{\zeta }^{\prime }\left(t\right)\right))ds=\left(T^{n-1}{\zeta }^{\prime }\left(t\right)\right)\hfill \end{array}$$

Therefore, for all $t\in [0,1]$ we have ${\left(T^n{\zeta }^{\prime }\left(t\right)\right)}_n$, which is a strictly decreasing and bounded-below sequence and, hence, converges to some l. Since ${\left(T_n\right)}_n$ is a monotone sequence, it follows from Dini Theorem that ${sup}_t\left|T^n{\zeta }^{\prime }\left(t\right)\right|$ converges to some $l^{\prime }\le {sup}_{\psi ,\omega }\left|F(\psi ,\omega ,{\zeta }^{\prime }\left(t\right))\right|.$ Now, it is not difficult to see that all the hypotheses of Theorem 1 are satisfied, and therefore, Equation (1) has a unique solution as required. □

5. Conclusions

In this manuscript, we have introduced a new type of metric space, which is a generalization of rectangular metric spaces, rectangular b-metric spaces, and controlled rectangular metric spaces. We have proved the existence and uniqueness of a fixed point for self-mapping on controlled rectangular metric-like spaces. Our results are a generalization of many theorems in the literature. Finally, we gave an application of our result to the Fredholm-type integral equation.

In closing, we would like to present the following two questions;

Question 1.

Let $(X,D)$ be a controlled rectangular metric like space, and T a map from $X\to X.$ Assume that for all $\zeta ,\eta ,T\zeta ,T\eta \in X$ there exists $k\in (0,1)$, where

$$D(T\zeta ,T\eta )\le k\theta (\zeta ,\eta ,T\zeta ,T\eta )D(\zeta ,\eta )$$

under what other condition(s) does T have a unique fixed point in $X?$

Question 2.

Let$(X,D)$ be a controlled rectangular metric-like space, and T a map from$X\to X.$Assume that for all$\zeta ,\eta ,T\zeta ,T\eta \in X$there exists$k\in (0,1)$, where

$$D(T\zeta ,T\eta )\le \theta (\zeta ,\eta ,T\zeta ,T\eta )\left[D\right(\zeta ,T\zeta )+D(\eta ,T\eta \left)\right]$$

under what other condition(s) does T have a unique fixed point in $X?$