Some Results for Ćirić–Prešić Type Contractions in F-Metric Spaces

Abstract

In this study, first, we introduce Ćirić–Prešić type contraction in F-metric spaces and prove a fixed point theorem for self mappings. We apply the fixed point results for a second-order differential equation. Therefore, we define Prešić type almost contraction and F-contraction, and we prove some fixed point theorems. In the last section, we prove the best proximity point theorems for Ćirić–Prešić type proximal contraction in F-metric spaces. Our results generalize the existing results in the literature.

1. Introduction and Preliminaries

Fixed point theory started with Brouwer’s fixed point theorem in 1912. Brouwer proved the existence of the fixed point of the continuous mappings defined from the closed unit ball of R${}^n$ to itself. Fixed point theory studies developed into three main branches, topological, metric, and separated spaces. Metric fixed point theory started with the Banach contraction principle in 1922. Banach’s fixed point theorem has recently become a fundamental and rapidly developing topic in nonlinear analysis. It has useful applications in fields such as science, economics, engineering sciences, as well as mathematics. In some cases where this principle is insufficient, some generalized metric spaces and generalized contraction principles were defined, and many fixed point theorems were proved.

It is established knowledge that for a self mapping $f$ on a nonempty set X, if $f\left(x\right)=x$ for any $x\in X$, then x is fixed point of f. In 1965, Preŝić [1] introduced this definition as $f\left(x,x,\dots ,x\right)=x$ for the mapping $f:X^k\to X.$ Additionally, he proved a fixed point theorem for a mapping satisfying generalized contraction. In 2007, Ćirić-Prešić [1,2] produced a new type contraction. Some researchers have generalized the Prešić type contraction in metric and generalized metric spaces [3,4,5,6,7,8,9,10,11].

Jleli and Samet [12] presented the concept of F-metric space and proved the Banach contraction principle. The authors defined the F-metric by using of the F-functions in [13]. Later on, Mitrovic et al. [14] obtained common fixed results for Banach, Jungck, Reich, and Berinde type contractions in F-metric spaces before applying the results to dynamic programming. Zhou et al. [15] proved best proximity results in F-metric spaces. Lateefa et al. [16] proved fixed point theorems for Dass–Gupta type contraction. Jahangir et al. [17] discovered nonlinear contraction principles in F-metric spaces. Faraji et al. [18] defined ($\alpha$-$\beta$)-admissible type contraction and proved fixed point theorems in F-metric spaces.

In this work, we define concept of Ćirić–Prešić type contractions in F-metric spaces. We prove fixed point theorems for almost contraction and F-contraction mapping. We apply the main fixed point result for a second-order differential equation. In the last section, we prove the best proximity point theorems for Ćirić-Prešić type contraction in F-metric spaces. Every metric space is F-metric space. However, the converse may not be true. So, our results generalize the existing results in the literature.

In this work, we denote $\mathsf{\Lambda }$ the family of all functions $F:\left(0,\infty \right)\to \mathbb{R}$ satisfying the following properties;

(F₁)

F is increasing, i.e., for all $s,u\in {\mathbb{R}}^{+}$ such that $s<u,$ $F\left(s\right)<F\left(u\right);$

(F₂)

For each sequence ${\left\{{\alpha }_s\right\}}_{s\in \mathbb{N}}$, ${lim}_{s\to \infty }$${\alpha }_s=0$ ⇔${lim}_{n\to \infty }F($${\alpha }_s)=-\infty ;$

(F₃)

There exists $k\in \left(0,1\right)$ such that ${\alpha }^k$${lim}_{n\to \infty }F($$\alpha )=0.$

We denote $\mathsf{\Omega }$ the family of all functions $F:\left(0,\infty \right)\to \mathbb{R}$ satisfying properties of ($F_1$) and ($F_2$).

Definition 1

([12]). Let the set $X\ne ⌀$ and $(F,\alpha )\in \mathsf{\Omega }\times [0,+\infty )$. d$:X^2\to \left[0,\infty \right)$ satisfies:

(d₁) 

$d\left(x,y\right)=0$ ⇔ $x=y$,

(d₂) 

$d\left(x,y\right)=d\left(y,x\right),$

(d₃) 

for $\forall N\ge 2$ ($N\in \mathbb{N}$) and $\forall \left\{t_n\right\}\subset X$ such that

$$d\left(t_1,t_s\right)>0⟹F\left(d\left(t_1,t_s\right)\right)\le F\left({\Sigma }_{i=1}^{s}d\left(t_i,t_{i+1}\right)\right)+\alpha$$

for all $x,y\in X.$ Then $\left(X,d\right)$ is called a F-metric space.

Let $\left(X,d\right)$ be a F-metric space. For each $x\in X$ and $\epsilon >0$,

$$O\left(x,l\right)=\{v\in X:d(x,v)<l\}$$

is called F-open ball centered at x with radius $l.$

Definition 2

([12]). Let $\left\{x_s\right\}\subset X$, then

i. 

$\left\{x_s\right\}$ is called F-convergent if there is $x\in X$ such that $d\left(x_s,x\right)\to 0$ as $s\to \infty .$

ii. 

$\left\{x_s\right\}$ is called F-Cauchy sequence if $d\left(x_s,x_r\right)\to 0$ as $s,r\to \infty .$

iii. 

$\left(X,d\right)$ is called F-complete if each F-Cauchy sequence is F-convergent.

2. Ćirić–Prešić Type Contractions

Definition 3.

Let $\left(X,d\right)$ be a F-metric space and $f:X^t\to X$ be a mapping, where t is a positive integer. Then, a mapping f is said to be Ćirić–Prešić type contraction if there exists a $\gamma \in (0,1)$ such that

$$d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)\le \gamma max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}$$

(1)

for all $x_i\in X$.

Theorem 1.

Let $\left(X,d\right)$ be a F-complete F-metric space and let $f:X^t\to X$ be a Ćirić–Prešić type contraction mapping where t is a positive integer. Then f has a unique fixed point in X.

Proof. 

Let $x_1,x_2\dots ,x_t$ $\in X$ be some arbitrary elements. We define a sequence $\left\{x_s\right\}$ in X by

$$x_{s+t}=f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right)$$

for all $s=1,2,\dots$. For simplicity, we say $d_s=d\left(x_s,x_{s+1}\right).$ We shall prove by mathematical induction that

$$d_s\le K{\mu }^s$$

(2)

for all $s\ge 1$, where $\mu ={\gamma }^{1/t}$ and $K=max\left\{\frac{d_i}{{\gamma }^i}:1\le i\le t\right\}$.

By (1), we obtain

$$\begin{array}{ccc}\hfill d\left(x_{s+t},x_{s+t+1}\right)& =& d\left(f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right),f\left(x_{s+1},\dots ,x_{s+t}\right)\right)\hfill \\ & \le & \gamma max\{d\left(x_i,x_{i+1}\right):s\le i\le s+t-1\}\hfill \\ & =& \gamma max\{d_s,d_{s+1},\dots ,d_{s+t-1}\}\hfill \\ & =& \gamma K{\mu }^s\hfill \\ & =& K{\mu }^{s+t}.\hfill \end{array}$$

Hence, we have

$$\underset{s\to \infty }{lim}K{\mu }^{s+t}=0.$$

Now, for any $\delta >0$ there exists $s_0\in \mathbb{N}$ such that

$$0<K{\mu }^{s+t}<\delta ,s\ge s_0.$$

(3)

Let $\epsilon >0$ be fixed. Since $\left(F,\alpha \right)\in \mathsf{\Omega }\times [0,\infty )$ and by (d${}_3$) and ($F_2$), we obtain

$$0<k<\delta \Rightarrow F\left(k\right)<F\left(\epsilon \right)-\alpha$$

(4)

By (3) and (4), we have

$$\begin{array}{ccc}\hfill F\left(d\left(x_s,x_{s+r}\right)\right)& \le & F\left({\Sigma }_{i=s}^{s+r-1}d\left(x_i,x_{i+1}\right)\right)+\alpha \hfill \\ & \le & F\left({\Sigma }_{i=s}^{s+r-1}K{\mu }^i\right)+\alpha \hfill \\ & =& F\left(K\frac{{\mu }^s}{1-\mu }\right)+\alpha \hfill \\ & <& F\left(\epsilon \right)\hfill \end{array}$$

which implies that

$$d\left(x_s,x_{s+r}\right)<\epsilon$$

for any $s,r\in \mathbb{N}$. Hence, $\left\{x_s\right\}$ is a F-Cauchy sequence. The F-completeness of $\left(X,d\right)$ confirms that the availability of a $y\in X$ such that ${lim}_{s\to \infty }$$d(x_s,y)=0.$ Now, we show that y is a fixed point of f. Assume $d\left(y,f\left(y,y,\dots ,y\right)\right)>0.$ Using the condition (F₃) and contractive condition (1),

$$\begin{array}{ccc}\hfill F\left(d\left(y,f\left(y,y,\dots ,y\right)\right)\right)& \le & F[d\left(y,x_{s+t}\right)+d\left(x_{s+t},f(y,y,\dots ,y)\right)]+\alpha \hfill \\ & \le & F[d\left(y,x_{s+t}\right)+d\left(f(x_s,\dots ,x_{s+t-1}),f(y,y,\dots ,y)\right)]+\alpha \hfill \\ & \le & F[d\left(y,x_{s+t}\right)+d\left(f(y,y,\dots y,x_s),f(y,y,\dots ,y)\right)\hfill \\ & & +d\left(f(y,y,\dots ,y,x_s),f(y,y,\dots ,x_s,x_{s+1})\right)\hfill \\ & & +\dots d\left(f(y,x_s,\dots ,x_{s+t-2}),f(x_s,x_{s+1},\dots ,x_{s+t-1})\right)]+\alpha \hfill \\ & \le & F[d\left(y,x_{s+t}\right)+d\left(y,x_s\right)+\gamma max\left\{d\left(y,x_s\right),d\left(x_s,x_{s+1}\right)\right\}\hfill \\ & & +\dots +\gamma max\left\{d\left(y,x_s\right),d\left(x_s,x_{s+1}\right),\dots ,d\left(x_{s+t-2},x_{s+t-1}\right)\right\}]+\alpha .\hfill \end{array}$$

Now, letting $s\to \infty$ in the above inequality and property of ($F_2$), we obtain

$$d\left(y,f\left(y,y,\dots ,y\right)\right)<0,$$

which is a contradiction. Hence, $y=f\left(y,y,\dots ,y\right)$.

Now, we show that uniqueness of fixed point. Let $y,z$ be the two fixed points of f. By (1)

$$d\left(f\left(y,y,\dots ,y\right),f\left(z,z,\dots ,z\right)\right)\le \gamma d\left(y,z\right),$$

which is a contradiction with $\gamma \in \left[0,1\right).$ Hence, f has unique fixed point. □

Example 1.

Let $X=[0,2].$ Define $d:X^2\to \mathbb{R}$, $d\left(x,y\right)=\left\{\begin{array}{cc}0,& \mathit{if}x=v\\ & \\ max\{x,v\}+1,& \mathit{if}x\ne v\end{array}\right|.$

Then d is a F-complete F- metric on X with $F\left(t\right)=\frac{-1}{t}$ and $\alpha =2$.

Let $f:X^2\to X,$$f\left(x,v\right)=\frac{x+v}{4}.$ Take $x,v,u\in X$ and any $\gamma \in \left(0,1\right).$

If $x>v>u,$ then

$$\begin{array}{ccc}\hfill d\left(f\left(x,v\right),f\left(v,u\right)\right)& =& d\left(\frac{x+v}{4},\frac{v+u}{4}\right)=\frac{x+v}{4}+1\hfill \\ & \le & \gamma (x+1)=\gamma max\{x+1,v+1\}\hfill \\ & =& \gamma max\{d(x,v),d\left(v,u\right)\}.\hfill \end{array}$$

If $x>u>v,$ then

$$\begin{array}{ccc}\hfill d\left(f\left(x,v\right),f\left(v,u\right)\right)& =& d\left(\frac{x+v}{4},\frac{v+u}{4}\right)=\frac{x+v}{4}+1\hfill \\ & \le & \gamma (x+1)=\gamma max\{x+1,u+1\}\hfill \\ & =& \gamma max\{d(x,v),d\left(v,u\right)\}.\hfill \end{array}$$

If $v>x>u,$ then

$$\begin{array}{ccc}\hfill d\left(f\left(x,v\right),f\left(v,u\right)\right)& =& d\left(\frac{x+v}{4},\frac{v+u}{4}\right)=\frac{x+v}{4}+1\hfill \\ & \le & \gamma (v+1)=\gamma max\{v+1,v+1\}\hfill \\ & =& \gamma max\{d(x,v),d\left(v,u\right)\}.\hfill \end{array}$$

If $v>u>x,$ then

$$\begin{array}{ccc}\hfill d\left(f\left(x,v\right),f\left(v,u\right)\right)& =& d\left(\frac{x+v}{4},\frac{v+u}{4}\right)=\frac{v+u}{4}+1\hfill \\ & \le & \gamma (v+1)=\gamma max\{v+1,v+1\}\hfill \\ & =& \gamma max\{d(x,v),d\left(v,u\right)\}.\hfill \end{array}$$

If $u>x>v$ of if $u>v>x$ proof is similar. All conditions of Theorem 1 are satisfied. Hence $x=0$ is unique fixed point of f.

Definition 4.

Let $\left(X,d\right)$ be a F-metric space and let $f:X^t\to X$ be a mapping, where t is a positive. Then, a mapping f is said to be Ćirić–Prešić type almost contraction if there exists a $\gamma \in (0,1)$ and for some $Y\ge 0$.

$$\begin{array}{ccc}\hfill d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)& \le & \gamma max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}\hfill \\ & & +Ymin\{d\left(x_1,f\left(x_1,x_1\dots ,x_1\right)\right),\hfill \\ & & d\left(x_{t+1},f\left(x_{t+1},x_{t+1},\dots ,x_{t+1}\right)\right),\hfill \\ & & d\left(x_{t+1},f\left(x_1,x_1\dots ,x_1\right)\right),d\left(x_1,f\left(x_{t+1},x_{t+1},\dots ,x_{t+1}\right)\right),\hfill \\ & & d\left(x_{t+1},f\left(x_1,x_2,\dots ,x_t\right)\right)\}\hfill \end{array}$$

(5)

for all $x_i\in X$.

Theorem 2.

Let $\left(X,d\right)$ be a F-complete F-metric space and let $f:X^t\to X$ be a Ćirić–Prešić almost contraction. Then f has a unique fixed point in X.

Proof. 

Let $x_1,x_2\dots ,x_t$ $\in X$ be arbitrary elements. We define a sequence $\left\{x_s\right\}$ by

$$x_{s+t}=f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right)$$

(6)

for all $s=1,2,\dots$. For simplicity, we say $d_s=d\left(x_s,x_{s+1}\right).$ We shall prove by mathematical induction that

$$d_s\le K{\mu }^s$$

(7)

for all $s\ge 1$, where $\mu ={\gamma }^{1/t}$ and $K=max\left\{\frac{d_i}{{\gamma }^i}:1\le i\le t\right\}$.

By (6) and (7), we have

$$\begin{array}{ccc}\hfill d\left(x_{s+t},x_{s+t+1}\right)& =& d\left(f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right),f\left(x_{s+1},\dots ,x_{s+t}\right)\right)\hfill \\ & \le & \gamma max\{d\left(x_i,x_{i+1}\right):s\le i\le s+t-1\}\hfill \\ & & +Ymin\{d\left(x_s,f\left(x_s,x_s,\dots ,x_s\right)\right),d\left(x_{s+t},f\left(x_{s+t},x_{s+t},\dots ,x_{s+t}\right)\right),\hfill \\ & & d\left(x_{s+t},f\left(x_s,x_s,\dots ,x_s\right)\right),\hfill \\ & & d\left(x_s,f\left(x_{s+t},x_{s+t},\dots ,x_{s+t}\right)\right),d\left(x_{s+t},f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right)\right)\}\hfill \\ & =& \gamma max\{d_s,d_{s+1},\dots ,d_{s+t-1}\}+Y.0\hfill \\ & =& \gamma K{\mu }^s\hfill \\ & =& K{\mu }^{s+t}.\hfill \end{array}$$

For any $\delta >0$ there exists $s_0\in \mathbb{N}$ such that

$$0<K{\mu }^{s+t}<\delta ,s\ge s_0.$$

(8)

Let $\epsilon >0$ be fixed. Since $\left(F,\alpha \right)\in \mathsf{\Omega }\times [0,\infty )$ and by (d${}_3$) and ($F_2$)

$$0<k<\delta \Rightarrow F\left(k\right)<F\left(\epsilon \right)-\alpha$$

(9)

By (8) and (9), for any $s,r\in \mathbb{N}$ and by definition of $F,$

$$\begin{array}{ccc}\hfill F\left(d\left(x_s,x_{s+r}\right)\right)& \le & F\left({\Sigma }_{i=s}^{s+r-1}d\left(x_i,x_{i+1}\right)\right)+\alpha \hfill \\ & \le & F\left({\Sigma }_{i=s}^{s+r-1}K{\mu }^i\right)+\alpha \hfill \\ & =& F\left(K\frac{{\mu }^s}{1-\mu }\right)+\alpha \hfill \\ & <& F\left(\epsilon \right).\hfill \end{array}$$

which implies that by ($F_1$),

$$d\left(x_s,x_{s+r}\right)<\epsilon .$$

Hence $\left\{x_s\right\}$ is a F-Cauchy sequence. Hence $\left\{x_s\right\}$ is a F-Cauchy sequence. Since $\left(X,d\right)$ is F-complete, there exists a $y\in X$ with ${lim}_{s\to \infty }$$d(x_s,y)=0.$

Now, we show that y is a fixed point of f. Assume $d\left(y,f\left(y,y,\dots ,y\right)\right)>0,$

$$\begin{array}{ccc}\hfill F\left(d\left(y,f\left(y,y,\dots ,y\right)\right)\right)& \le & F[d\left(y,x_{s+t}\right)+d\left(x_{s+t},f(y,y,\dots ,y)\right)]+\alpha \hfill \\ & \le & F[d\left(y,x_{s+t}\right)+d\left(f(x_s,\dots ,x_{s+t-1}),f(y,y,\dots ,y)\right)]+\alpha \hfill \\ & \le & F[d\left(y,x_{s+t}\right)+d\left(f(y,y,\dots y,x_s),f(y,y,\dots ,y)\right)\hfill \\ & & +d\left(f(y,y,\dots ,y,x_s),f(y,y,\dots ,x_s,x_{s+1})\right)\hfill \\ & & +\dots +d\left(f(y,x_s,\dots ,x_{s+t-2}),f(x_s,x_{s+1},\dots ,x_{s+t-1})\right)]+\alpha \hfill \\ & \le & F[d\left(y,x_{s+t}\right)+d\left(y,x_s\right)++Y.0+\gamma max\left\{d\left(y,x_s\right),d\left(x_s,x_{s+1}\right)\right\}\hfill \\ & & +Ymin\{d\left(y,f\left(y,y\dots ,y\right)\right),d\left(x_{s+1},f\left(x_{s+1},x_{s+1},\dots ,x_{s+1}\right)\right),\hfill \\ & & d\left(y,f\left(x_{s+1},x_{s+1},\dots ,x_{s+1}\right)\right),d\left(x_{s+1},f\left(y,y,\dots ,y\right)\right)\hfill \\ & & ,d\left(y,f\left(y,,,,x_s,x_{s+1}\right)\right)\}\hfill \\ & & +\dots +\gamma max\left\{d\left(y,x_s\right),d\left(x_s,x_{s+1}\right),\dots ,d\left(x_{s+t-2},x_{s+t-1}\right)\right\}\hfill \\ & & +Ymin\{d\left(y,f\left(y,y\dots ,y\right)\right),d\left(x_{s+t-1},f\left(x_{s+t-1},x_{s+t-1},\dots ,x_{s+t-1}\right)\right),\hfill \\ & & d\left(y,f\left(x_{s+t-1},x_{s+t-1},\dots ,x_{s+t-1}\right)\right),\hfill \\ & & d\left(x_{s+t-1},f\left(y,y,\dots ,y\right)\right),d\left(y,f\left(x_s,\dots ,x_{s+t-1}\right)\right)\left\}\right]+\alpha \hfill \end{array}$$

Now, taking limit as $s,t\to \infty$ in the above inequality, we obtain

$$d\left(y,f\left(y,y,\dots ,y\right)\right)<0$$

which is a contradiction. Hence, we have $y=f\left(y,y,\dots ,y\right)$. Now, we show that uniqueness of fixed point. Let $y,z$ be two different fixed points of f. Then, By (6),

$$\begin{array}{ccc}\hfill d\left(f\left(y,y,\dots ,y\right),f\left(z,z,\dots ,z\right)\right)& \le & \gamma d\left(y,z\right)+Ymin\{d\left(y,f\left(y,\dots ,y\right)\right),d\left(z,f\left(z,\dots ,z\right)\right),\hfill \\ & & d\left(z,f\left(y,\dots ,y\right)\right),d\left(y,f\left(z,\dots ,z\right)\right),d\left(y,f\left(z,z,\dots ,z\right)\right)\}\hfill \\ \hfill d\left(f\left(y,y,\dots ,y\right),f\left(z,z,\dots ,z\right)\right)& \le & \gamma d\left(y,z\right)\hfill \end{array}$$

which is a contradiction with $\gamma \in \left[0,1\right).$ Thus, f has unique fixed point. □

Example 2.

Let $X=\mathbb{N}$. Define $d:X^2\to \mathbb{R}$, $d\left(x,v\right)=\left\{\begin{array}{cc}{\left(x-v\right)}^2& \mathit{if}x,v\in \left\{1,2\right\}\\ & \\ 2\left|x-v\right|& other\end{array}\right|$.Then d is a F-complete F- metric on X with $F\left(t\right)=\frac{-1}{t}$ and $\alpha =4$. Therefore d is not a metric on $X.$ For $x=1,$ $v=2$, $u=4$, we have

$$d\left(1,4\right)=2.3=6\ge 1+4=d(1,2)+d(2,4).$$

Define $f:X^2\to X,$$f\left(x,v\right)=\left\{\begin{array}{cc}\frac{\left|x-v\right|}{2},& \mathit{if}x,v\mathit{even},x\ne v\mathit{and}x>v+4\\ \left|x-v\right|& \mathit{if}x,v\mathit{odd},x\ne v,\\ 1,& \mathit{other}\end{array}\right|$.

If $x,v,u\in \left\{1,2\right\}$, $u=v=u$ or $x=u\ne v$ or $x\ne v=u$ we have,

$$d\left(f\left(x,v\right),f\left(v,u\right)\right)=d(1,1)=0\le \gamma .$$

If $x,v,u$ are consecutive even or consecutive odd, we obtain

$$d\left(f\left(x,v\right),f\left(v,u\right)\right)=d(1,1)=0\le \gamma 0.$$

If $x,v,u$ are even and $x>$ $v+4>u+4,$ we obtain

$$\begin{array}{ccc}\hfill d\left(f\left(x,v\right),f\left(v,u\right)\right)& =& d(\frac{\left|x-v\right|}{2},\frac{\left|v-u\right|}{2})=\left|\left|x-v\right|-\left|v-u\right|\right|\hfill \\ & \le & \gamma max\{2\left|x-v\right|,2\left|v-u\right|\}\hfill \\ & & +Ymin\{2\left|x-1\right|,2\left|u-1\right|,2\left|u-\frac{\left|x-v\right|}{2}\right|\}\hfill \\ & =& \gamma max\left\{d\right(x,v),d(v,u\left)\right\}\hfill \\ & & +Ymin\left\{d\right(x,f(x,x)),d(u,f(u,u),d(u,f(x,x\left)\right),\hfill \\ & & d(x,f(u,u\left)\right),d(u,f(x,v\left)\right)\}.\hfill \end{array}$$

If $x,v,u$ are odd but not consecutive and $x\ne u\ne v,$ we obtain

$$\begin{array}{ccc}\hfill d\left(f\left(x,v\right),f\left(v,u\right)\right)& =& d(\left|x-v\right|,\left|v-u\right|)=2\left|\left|x-v\right|-\left|v-u\right|\right|\hfill \\ & \le & \gamma max\{2\left|x-v\right|,2\left|v-u\right|\}\hfill \\ & & +Ymin\{2\left|x-1\right|,2\left|u-1\right|,2\left|u-\frac{\left|x-v\right|}{2}\right|\}\hfill \\ & =& \gamma max\left\{d\right(x,v),d(v,u\left)\right\}\hfill \\ & & +Ymin\left\{d\right(x,f(x,x)),d(u,f(u,u),d(u,f(x,x\left)\right),\hfill \\ & & d(x,f(u,u\left)\right),d(u,f(x,v\left)\right)\}.\hfill \end{array}$$

If $x,v,u$ are odd but not consecutive and $x\ne u\ne v,$ we obtain

$$\begin{array}{ccc}\hfill d\left(f\left(x,v\right),f\left(v,u\right)\right)& =& d(\left|x-v\right|,\left|v-u\right|)=2\left|\left|x-v\right|-\left|v-u\right|\right|\hfill \\ & \le & \gamma max\{2\left|x-v\right|,2\left|v-u\right|\}+Ymin\{2\left|x-1\right|,2\left|u-1\right|,2\left|u-\frac{\left|x-v\right|}{2}\right|\}\hfill \\ & =& \gamma max\left\{d\right(x,v),d(v,u\left)\right\}\hfill \\ & & +Ymin\left\{d\right(x,f(x,x)),d(u,f(u,u),d(u,f(x,x\left)\right),\hfill \\ & & d(x,f(u,u\left)\right),d(u,f(x,v\left)\right)\}.\hfill \end{array}$$

Hence, Ćirić–Prešić type almost contraction principle is satisfied for $Y>0$ and any $\gamma \in \left(0,1\right)$. $x=1$ is fixed point of f.

Definition 5.

Let $\left(X,d\right)$ be a F-metric space and let $f:X^t\to X$ be a mapping where t is a positive integer. Then f is said to be a Ćirić–Prešić type $-F$-contraction if there exists a function $F\in \mathsf{\Lambda }$ such that

$$\tau +F\left(d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)\right)\le F\left(max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}\right)$$

(10)

for each $\tau >0$ and for all $x_i\in X.$

Theorem 3.

Let $\left(X,d\right)$ be a F-complete F-metric space and let $f:X^t\to X$ be a continuous Ćirić–Prešić type F-contraction. Then f has a unique fixed point.

Proof. 

Suppose $x_1,x_2\dots ,x_t$ $\in X$ arbitrary elements. Define a sequence $\left\{x_s\right\}$ by

$$x_{s+t}=f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right)$$

Using (10) we can write

$$\tau +F\left(d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)\right)\le F\left(max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}\right)$$

So,

$$\begin{array}{ccc}\hfill F\left(d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)\right)& \le & F\left(max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}\right)-\tau \hfill \\ & <& F\left(max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}\right).\hfill \end{array}$$

Using (F${}_1$), we obtain

$$d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)<max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}$$

For simplicity, say $d_s=d\left(x_s,x_{s+1}\right).$ For all $s\ge 1$,

By (10),

$$\begin{array}{ccc}\hfill F\left(d_{s+1}\right)& =& F\left(d\left(x_s,x_{s+1}\right)\right)\hfill \\ & =& F\left(d\left(f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right),f\left(x_{s+1},\dots ,x_{s+t}\right)\right)\right)\hfill \\ & \le & F\left(max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}\right)-\tau ,\hfill \end{array}$$

and similarly

$$F\left(d_{s+2}\right)\le F\left(max\{d\left(x_i,x_{i+1}\right):2\le i\le t+1\}\right)-2\tau .$$

By induction,

$$\begin{array}{ccc}\hfill F\left(d_{s+t}\right)& =& F\left(d\left(x_{s+t},x_{s+t+1}\right)\right)\hfill \\ & =& F\left(d\left(f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right),f\left(x_{s+1},\dots ,x_{s+t}\right)\right)\right)\hfill \\ & \le & F\left(max\{d\left(x_i,x_{i+1}\right):s\le i\le s+t-1\}\right)-n\tau (n\ge 1).\hfill \end{array}$$

(11)

Letting $n\to \infty$ in the above inequality, we obtain

$$\underset{n\to \infty }{lim}F\left(d_{s+t}\right)=-\infty$$

which implies that ${lim}_{n\to \infty }{d}_{s+t}=0.$

So, for any $\delta \in \left(0,1\right)$ we obtain

$$\underset{n\to \infty }{lim}{d}_{s+t}^{\delta }F\left(d_{s+t}\right)=0.$$

By (11), we have

$$d_{s+t}^{\delta }F\left(d_{s+t}\right)-d_{s+t}^{\delta }F\left(max\{d\left(x_i,x_{i+1}\right):s\le i\le s+t-1\}\right)\le -n\tau d_{s+t}^{\delta }\le 0.$$

Letting $n\to \infty$ in the above inequality, we obtain ${lim}_{n\to \infty }n{d}_{s+t}^{\delta }=0.$

Let $\epsilon >0$ be fixed. Since $\left(F,\alpha \right)\in \mathsf{\Omega }\times [0,\infty )$, we obtain

$$0<k<l\Rightarrow F\left(k\right)<F\left(\epsilon \right)-\alpha .$$

(12)

By (d${}_3$) and ($F_2$), for any $s,t\in \mathbb{N}$,

$$\begin{array}{ccc}\hfill F\left(d\left(x_s,x_{s+t}\right)\right)& \le & F\left({\Sigma }_{i=s}^{s+r-1}d\left(x_i,x_{i+1}\right)\right)+\alpha \hfill \\ & =& F\left(d_s+d_{s+1}+\dots +d_{s+r-1}\right)+\alpha \hfill \\ & <& F\left(\epsilon \right)\hfill \end{array}$$

which implies that by ($F_1$),

$$d\left(x_s,x_{s+r}\right)<\epsilon .$$

Hence $\left\{x_s\right\}$ is a F-Cauchy sequence. Since $\left(X,d\right)$ is F-complete, there exists a $y\in X$ with ${lim}_{s\to \infty }$$d(x_{s+r},y)=0.$

Now, we show that y is a fixed point of f. Assume $d\left(y,f\left(y,y,\dots ,y\right)\right)>0.$ By continuity of f,

$$y=\underset{n\to \infty }{lim}{x}_{s+r}=\underset{n\to \infty }{lim}f\left(x_s,x_{s+1},\dots ,x_{s+r-1}\right)=f\left(y,y,\dots ,y\right).$$

Hence, $y=f\left(y,y,\dots ,y\right)$.

Now, we show that uniqueness of fixed point. Suppose f has different fixed points $y,z$ with $y\ne z$. By (10),

$$\begin{array}{ccc}\hfill \tau +F\left(d\left(f\left(y,y,\dots ,y\right),f\left(z,z,\dots ,z\right)\right)\right)& =& \tau +F\left(d\left(y,z\right)\right)\hfill \\ & \le & F\left(d\left(y,z\right)\right)\hfill \end{array}$$

which is a contradiction with $\tau >0.$ Thus f has unique fixed point. □

Corollary 1.

Let $\left(X,d\right)$ be a F-complete F-metric space and $f:X^2\to X$ be a mapping satisfying

$$d\left(f\left(x_1,x_2\right),f\left(x_2,x_3\right)\right)\le \gamma max\{d\left(x_1,x_2\right),\left(x_2,x_3\right)\}$$

for all $x_1,x_2,x_3\in X,$ where $\gamma \in (0,1)$. Then, there exists a $x\in X$ such that $f\left(x,x\right)=x.$

Corollary 2.

Let $\left(X,d\right)$ be a F-complete F-metric space and $f:X^t\to X$ be a mapping where t is a positive integer. If f satisfies

$$d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)\le {\Sigma }_{i=1}^t{\gamma }_{i}d\left(x_i,x_{i+1}\right)$$

for all $x_i\in X,$ where $\gamma \in (0,1)$ and ${\gamma }_1+\dots {\gamma }_t<1.$ Then f has a unique fixed point.

3. Application

Let consider the second order differential equation for $x\in (C\left[0,f\right],\mathbb{R}),$ $s\ne 0\in \mathbb{R},$

$$\begin{array}{c}\frac{d^{2}x}{d{t}^2}+s^{2}x=q\left(t,x\left(t\right)\right)\\ x\left(0\right)=a,x^{\prime }\left(0\right)=b\end{array}.$$

(13)

Let $q,\sigma :\left[0,f\right]\times {\mathbb{R}}^{+}\to \mathbb{R}$ are continuous. Differential Equation (13) is equivalent to the integral equation

$$x\left(t\right)=acos\left(st\right)+bsin\left(st\right){\int }_0^{t}T\left(t,\xi \right)\left[q\left(\xi ,x\left(\xi \right)\right)\right]d\xi ,t\in \left[0,f\right].$$

(14)

Here $T\left(t,\xi \right)=\frac{1}{s}sin\left(s(t-\xi )\right)Z(t-\xi )$ is Green function where Z is the Heaviside unit function.

Theorem 4.

Consider the differential Equation (13) and suppose

(i) q,$\sigma :\left[0,f\right]\times {\mathbb{R}}^{+}\to \mathbb{R}$ is continuous,

(ii) for all $t\in \left[0,f\right]$ and $x,y\in \mathbb{R}$, $\gamma \in \left[0,1\right),$$s\in \mathbb{R}$ ($s\ne 0$) such that

$$\left|q\left(t,x\right)-q\left(t,v\right)\right|\le \gamma s^2\left|x-v\right|$$

Then, Equation (14) has a unique solution.

Proof. 

Consider the F-metric

$$d(x,v)={∥x-v∥}_{\infty }=\underset{t\in \left[0,f\right]}{max}\left|x\left(t\right)-v\left(t\right)\right|.$$

for all $t\in \left[0,f\right]$. Then, we define any mapping

$$f:X\times X\to X,f\left(x\left(t\right),v\left(t\right)\right)=acos\left(st\right)+bsin\left(st\right)+{\int }_0^{t}T\left(t,\xi \right)[q\left(\xi ,x\left(\xi \right)\right)+\sigma \left(\xi ,v\left(\xi \right)\right)]d\xi$$

Then, we obtain for all $x,v,z\in X$

$$\begin{array}{ccc}\hfill \left|f\left(x\left(t\right),y\left(t\right)\right)-f\left(y\left(t\right),z\left(t\right)\right)\right|& =& \left|\begin{array}{c}{\int }_0^{t}T\left(t,\xi \right)[q\left(\xi ,x\left(\xi \right)\right)+\sigma \left(\xi ,v\left(\xi \right)\right)d\xi \\ -{\int }_0^{t}T\left(t,\xi \right)[q\left(\xi ,v\left(\xi \right)\right)+\sigma \left(\xi ,z\left(\xi \right)\right)]d\xi \end{array}\right|\hfill \\ & \le & {\int }_0^{t}T\left(t,\xi \right)[q\left(\xi ,x\left(\xi \right)\right)+\sigma \xi ,v\left(\xi \right)\hfill \\ & & -q\left(\xi ,v\left(\xi \right)\right)-\sigma \left(\xi ,z\left(\xi \right)\right)]d\xi \hfill \\ & \le & {\int }_0^{t}T\left(t,\xi \right)max\left\{\left|x\left(\xi \right)-v\left(\xi \right)\right|,\left|v\left(\xi \right)-z\left(\xi \right)\right|\right\}d\xi \hfill \\ & \le & {\int }_0^{t}T\left(t,\xi \right)s^{2}max\left\{\left|x\left(\xi \right)-v\left(\xi \right)\right|,\left|v\left(\xi \right)-z\left(\xi \right)\right|\right\}d\xi \hfill \\ & =& \gamma s^2\underset{t\in \left[0,f\right]}{max}\left\{{∥x-v∥}_{\infty },{∥v-z∥}_{\infty }\right\}\underset{t\in \left[0,f\right]}{sup}{\int }_0^{t}T\left(t,\xi \right)d\xi \hfill \\ & \le & \gamma s^2\underset{t\in \left[0,f\right]}{max}\left\{{∥x-v∥}_{\infty },{∥v-z∥}_{\infty }\right\}\underset{t\in \left[0,f\right]}{sup}{\int }_0^t\frac{1}{s}sin\left(s(t-\xi )\right)d\xi \hfill \\ & \le & \gamma \underset{t\in \left[0,f\right]}{max}\left\{{∥x-v∥}_{\infty },{∥v-z∥}_{\infty }\right\}.\hfill \end{array}$$

Thus, we obtain

$${∥f\left(x,v\right)-f\left(v,z\right)∥}_{\infty }\le \gamma \underset{t\in \left[0,f\right]}{max}\left\{{∥x-v∥}_{\infty },{∥v-z∥}_{\infty }\right\}.$$

By Theorem 1, f has a unique fixed point. (14) has a unique solution. □

4. Ćirić–Prešić Type Proximal Contraction

Definition 6

([19]). Let $\left(X,d\right)$ be a F-metric space and P and V be two nonempty subsets of $X.$ Then

$$\begin{array}{ccc}\hfill d\left(P,R\right)& =& inf\{d\left(p,r\right):p\in P,v\in V\},\hfill \\ \hfill d\left(v,P\right)& =& inf\{d\left(p,v\right):p\in P\},\hfill \\ \hfill P_0& =& \{p\in P:d\left(p,v\right)=d\left(P,V\right)\mathit{for}\mathrm{some}v\in V\},\hfill \\ \hfill V_0& =& \{v\in V:d\left(p,v\right)=d\left(P,V\right)\mathit{for}\mathit{some}p\in P\}.\hfill \end{array}$$

An element $p\in P$ is called best proximity point of the non-self mapping $f:P\to V$ if $d(p,fp)=d(P,V)$. If f is a self mapping, then best proximity point is fixed point.

Now we will generalize this definition in a F-metric space $(X,d)$ as for nonempty subsets P and V if $d(p,f(p,p,\dots ,p\left)\right)=d(P,V)$, then u is a best proximity point of the mapping $f:P^t\to V.$

Definition 7.

Let $\left(P,V\right)$ be a pair of nonempty subsets of a F-metric space $\left(X,d\right)$ with $P\ne ⌀.$ Then, the pair $\left(P,V\right)$ is named to satisfy the weak P-property ⇔ $\forall p_1$,$p_2\in P_0$, $v_1,v_2\in V_0,$

$$d\left(p_1,v_1\right)=d\left(P,V\right)=d\left(p_2,v_2\right)\Rightarrow d\left(p_1,p_2\right)\le d\left(v_1,v_2\right).$$

Theorem 5.

Let $\left(P,V\right)$ be a pair of nonempty subsets of a F-complete F-metric space $\left(X,d\right)$, $U_0$ is a nonempty set and closed and $\left(P,V\right)$ has weak P-property. Let $f:P^t\to P$ be a one to one mapping satisfying

$$d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)\le \gamma max\{d\left(x_i,x_{i+1}\right):1\le i\le t\}$$

(15)

for all $x_i\in X,$ where $\gamma \in (0,1)$. If $f\left(P_0^t\right)\subseteq V_0,$ then f has a unique best proximity point.

Proof. 

Let $(x_1,x_2\dots ,x_t)\in P_0^t.$ Since $f\left(P_0^t\right)\subseteq V_0$, for $x_{t+1}\in P_0$ we have

$$d\left(x_{t+1},f(x_1,\dots ,x_t)\right)=d\left(P,V\right).$$

Again, for $(x_2,x_3\dots ,x_{t+1})\in P_0^t,$ there exists $x_{t+2}\in P_0$ such that

$$d\left(x_{t+2},f(x_2,\dots ,x_{t+1})\right)=d\left(P,V\right).$$

Continuing this process, we obtain a sequence $\left\{x_s\right\}$ in $P_0$ such that for all n

$$d\left(x_{s+t},f(x_s,x_{s+1},\dots ,x_{s+t-1})\right)=d\left(P,V\right).$$

$\left(P,V\right)$ has weak P-property, we have

$$d\left(x_{s+t},x_{s+t+1}\right)\le d\left(f(x_s,x_{s+1},\dots ,x_{s+t-1}),f(x_{s+1},x_{s+2},\dots ,x_{s+t})\right).$$

Say $d_s=d\left(x_s,x_{s+1}\right).$ By induction for all $s\ge 1$,

$$d_s\le K{\mu }^s$$

(16)

where $\mu ={\gamma }^{1/t}$ and $K=max\left\{\frac{d_i}{{\gamma }^i}:1\le i\le t\right\}.$

From (15)

$$\begin{array}{ccc}\hfill d\left(x_{s+t},x_{s+t+1}\right)& =& d\left(f\left(x_s,x_{s+1},\dots ,x_{s+t-1}\right),f\left(x_{s+1},\dots ,x_{s+t}\right)\right)\hfill \\ & \le & \gamma max\{d\left(x_i,x_{i+1}\right):s\le i\le s+t-1\}\hfill \\ & =& \gamma max\{d_s,d_{s+1},\dots ,d_{s+t-1}\}\hfill \\ & =& \gamma K{\mu }^s\hfill \\ & =& K{\mu }^{s+t}.\hfill \end{array}$$

Similarly, the proof of Theorem 1, we obtain $\left\{x_s\right\}$ is a F-Cauchy sequence. Since $\left(X,d\right)$ is F-complete and $P_0$ is closed subset, then there exists a $y\in P_0$ with ${lim}_{s\to \infty }$$d(x_s,y)=0.$

Now, we show that y is a best proximity point of f. For $\left(y,y,\dots ,y\right)\in P_0^k$, we know that f$\left(y,y,\dots ,y\right)\in f\left(P_0^k\right)\subseteq V_0.$ Then for any $x\in P_0$

$$d\left(x,f\left(y,y,\dots ,y\right)\right)=d\left(P,V\right).$$

By (15),

$$\begin{array}{ccc}\hfill d\left(f\left(x,x,\dots ,x\right),f\left(y,y,\dots ,y\right)\right))& \le & \gamma max\{d\left(x,x\right),d\left(y,y\right)\}\hfill \\ & =& 0.\hfill \end{array}$$

Since f is one two one, then we have $x=y.$ Hence

$$d\left(y,f\left(y,y,\dots ,y\right)\right)=d\left(P,V\right).$$

Now, we show the uniqueness of the best proximity point. Suppose f has different best proximity points $y,z$ with $y\ne z$. By (15),

$$d\left(y,z\right)\le d\left(f\left(y,y,\dots ,y\right),f\left(z,z,\dots ,z\right)\right)\le \gamma d\left(y,z\right)$$

which is a contradiction with $\gamma \in \left[0,1\right).$ Thus f has a unique best proximity point. □

Example 3.

Let $X=\mathbb{N}$ and $d:X\times X\to [0,\infty )$, $d(x,v)=\left\{\begin{array}{cc}{\left(x-v\right)}^2,& \mathit{if}x,v\in [0,3]\\ & \\ \frac{\left|x-v\right|}{2}& \mathit{other}\end{array}\right|.$ Then $(X,d)$ is F-complete F-metric space with $F\left(t\right)=lnt$. Suppose $P=\{2,3\}$ and $V=\{4,5,6\}.$ We have $P_0=\left\{2\right\}$ and $V_0=\left\{4\right\}.$ Obviously, $P_0$ is closed and P and V satisfy weak P-property. Define

$$f:P^2\to V\mathrm{as}f(x,v)=x+v.$$

Obviously, for any $\gamma \in \left(\frac{1}{2},1\right),$ all conditions of Theorem 5 and $x=2$ is best proximity point of f with $d(2,f(2,2\left)\right)=1=d(P,V).$

Corollary 3.

Let $\left(P,V\right)$ be a pair of nonempty subsets of a F-complete F-metric space $\left(X,d\right)$, $U_0$ is a nonempty and closed set and $\left(P,V\right)$ has weak P-property. Let $f:X^k\to X$ be a one to one mapping satisfying

$$d\left(f\left(x_1,x_2\dots ,x_t\right),f\left(x_2\dots ,x_{t+1}\right)\right)\le {\Sigma }_{i=1}^t{\gamma }_{i}d\left(x_i,x_{i+1}\right)$$

for all $x_i\in X,$ where $\gamma \in (0,1)$. If $f\left(P_0^k\right)\subseteq V_0,$ then f has a unique best proximity point.

5. Conclusions and Future Work

The classical Banach contraction principle and most of the generalizations give unique fixed-point results for self mappings. In this work, we have proven fixed point and best proximity point theorems in F-metric spaces which have nonnegativity, symmetry, and generalized triangular inequality. Every metric is a F-metric, but the converse is not true. Therefore, it is more general than many distance functions in the literature. Consequently, applications of fixed point results are useful in many sciences, and they can make solving problems easier.

In future studies, weak Prešić type contractive condition can be defined, and fixed point theorems, best proximity theorems, and common fixed point theorems can be proved in F-metric. In addition, applications can be given to Fredholm and Volterra integral equations and differential equations. There will be several useful applications, especially in mathematics and engineering.