Some Fixed-Point Theorems in Proximity Spaces with Applications

Abstract

Considering the $\omega$-distance function defined by Kostić in proximity space, we prove the Matkowski and Boyd–Wong fixed-point theorems in proximity space using $\omega$-distance, and provide some examples to explain the novelty of our work. Moreover, we characterize Edelstein-type fixed-point theorem in compact proximity space. Finally, we investigate an existence and uniqueness result for solution of a kind of second-order boundary value problem via obtained Matkowski-type fixed-point results under some suitable conditions.

1. Introduction and Preliminaries

The basic concepts of proximity spaces were initially developed by Frigyes Riesz [1] in 1908, and later on this theory was revived and axiomatized by Efremovich [2] in 1934, which was published in 1951. Over the years, a lot of studies on proximity spaces have been carried out [3,4,5,6]. Smirnov [5] established the link between the proximity relation and topological spaces. In addition, he was the first to introduce the relationship between proximities and uniformities.

Let $\delta$ be a relation on a set $2^{\Omega }$. Then, the pair $(\Omega ,\delta )$ is said to a proximity space if the following hold: for all $U,V,W\in 2^{\Omega }$, where $2^{\Omega }$ the power set of $\Omega$

(Pr1)

$U\delta V$ implies $V\delta U$;

(Pr2)

$U\delta V$ implies $U,V\ne \varnothing$;

(Pr3)

$U\delta (V\cup W)$ iff $U\delta V$ or $U\delta W$;

(Pr4)

$U\cap V\ne \varnothing$ implies $U\delta V$;

(Pr5)

For all $Y\subseteq \Omega$, $U\delta Y$ or $V\delta (\Omega -Y)$ implies $U\delta V$;

For all $\xi \in \Omega$ and $U\subseteq \Omega$, we will use the notation $\xi \delta U$ and $U\delta \xi$ instead of $\left\{\xi \right\}\delta U$ and $U\delta \left\{\xi \right\}$, respectively. A proximity space $(\Omega ,\delta )$ is called separated if $\xi \delta \eta$ implies $\xi =\eta$ for all $\xi ,\eta \in \Omega$. The properties of proximity spaces are generalizations of uniform properties and continuity properties of metric and topology, respectively. Every proximity relation on a nonempty set $\Omega$ induces a topology ${\tau }_{\delta }$ via the Kuratowski closure operator. The Kuratowski closure operator using proximity relation can be defined by $cl\left(U\right)=\{\xi \in \Omega$:$\xi \delta U\}$ for all $U\subseteq \Omega$. In this case, the topology ${\tau }_{\delta }$ is always completely regular, further, it is Tychonoff if $(\Omega ,\delta )$ is separated. If $(\Omega ,\tau )$ is a topological space and $\delta$ is a proximity on $\Omega$ such that ${\tau }_{\delta }=\tau$, then $\tau$ and $\delta$ are said to be compatible. Every completely regular topology on a nonempty set $\Omega$ has a compatible proximity. Additionally, if a sequence $\left\{{\xi }_n\right\}$ converges to a point $\xi \in \Omega$ with respect to induced topology ${\tau }_{\delta }$, then we obtain $\xi \delta \left\{{\xi }_n\right\}$. Furthermore, every uniform space $(\Omega ,\mathcal{U})$ has an associated proximity structure defined by for all $U,V\subseteq \Omega$, $U\delta V$ if $(U\times V)\cap W\ne \varnothing$ for all $W\in \mathcal{U}$. For further information, see [7,8].

Now we state some examples of proximity spaces.

Example 1.

Let $(\Omega ,\rho )$ be a metric space. Consider the relation δ on $2^{\Omega }$ as

$$U\delta V\iff \rho (U,V)=0,$$

where $\rho (U,V)=inf\left\{\rho \right(u,v):u\in U,v\in V\}.$ Then δ is a proximity on Ω. In addition, the metric topology ${\tau }_{\rho }$ and δ are compatible.

Example 2.

Let $(\Omega ,\tau )$ be a $T_4$ (both normal and $T_1$) topological space. Consider the relation δ on $2^{\Omega }$ as

$$U\delta V\iff \overline{U}\cap \overline{V}\ne \varnothing .$$

Then δ is a proximity on Ω. In addition, τ and δ are compatible.

Example 3.

Let $(\Omega ,\tau )$ be a completely regular topological space. Consider the relation δ on $2^{\Omega }$ as $U\delta V$ if and only if there does not exist a continuous function $f:\Omega \to [0,1]$ such that $f\left(U\right)=0$ and $f\left(V\right)=1.$ Then δ is a proximity on Ω. In addition, τ and δ are compatible.

Considering that fixed-point theory on proximity space will bring a new direction and interesting results, Kostić [9] defined the concept of $\omega$-distance and ${\omega }_0$-distance inspired by [10] (see [11] for more information about $\omega$-distance) in proximity space as follows and obtained the proximity space version of Banach fixed-point theorem.

Definition 1.

Let $(\Omega ,\delta )$ be proximity space and $\omega :\Omega \times \Omega ⟶[0,\infty )$ be a function. If ω satisfies

(w1) 

, $\omega (\zeta ,U)=0$ and $\omega (\zeta ,V)=0$ implies $U\delta V$ for all $\zeta \in \Omega$ and $U,V\subseteq \Omega$,

then ω is said to be a ω-distance on Ω, where

$$\omega (\zeta ,U)=inf\left\{\omega (\zeta ,\xi ):\xi \in U\right\}.$$

In addition, a ω-distance on a proximity space $(\Omega ,\delta )$ is called ${\omega }_0$-distance if the following axioms hold:

(w2) 

$\omega (\xi ,\eta )\le \omega (\xi ,\zeta )+\omega (\zeta ,\eta )$ for all $\xi ,\eta ,\zeta \in \Omega$,

(w3) 

ω is lower semicontinuous in both variables with respect to ${\tau }_{\delta }$, i.e., for all $\xi ,\eta \in \Omega$, we have

$$\omega (\xi ,\eta )\le \underset{{\xi }^{\prime }\to \xi }{lim\; inf}\omega ({\xi }^{\prime },\eta )=\underset{V\in {\mathcal{U}}_{\xi }}{sup}\underset{{\xi }^{\prime }\in V}{inf}\omega ({\xi }^{\prime },\eta )$$

and

$$\omega (\eta ,\xi )\le \underset{{\xi }^{\prime }\to \xi }{lim\; inf}\omega (\eta ,{\xi }^{\prime })=\underset{V\in {\mathcal{U}}_{\xi }}{sup}\underset{{\xi }^{\prime }\in V}{inf}\omega (\eta ,{\xi }^{\prime }),$$

where ${\mathcal{U}}_{\xi }$ is a base of neighborhoods of the point $\xi \in \Omega$.

Remark 1.

It is clear that if ω is lower semicontinuous in both variables with respect to ${\tau }_{\delta }$, then we have $\omega (\xi ,\eta )\le \underset{n\to \infty }{lim\; inf}\omega ({\xi }_n,\eta )$ and $\omega (\eta ,\xi )\le \underset{n\to \infty }{lim\; inf}\omega (\eta ,{\xi }_n)$ for every sequence $\left\{{\xi }_n\right\}$ converging to ξ with respect to ${\tau }_{\delta }$.

Example 4.

Let $\Omega =\mathbb{R}$ be endowed with the usual metric and the proximity δ defined in Example 1. Define ${\omega }_1,{\omega }_2:\Omega \times \Omega ⟶[0,\infty )$ by

$${\omega }_1(\xi ,\eta )=max\left\{\left|\xi \right|,\left|\eta \right|\right\}$$

and

$${\omega }_2(\xi ,\eta )=\frac{\left|\xi \right|+\left|\eta \right|}{2},$$

then both ${\omega }_1$ and ${\omega }_2$ are ${\omega }_0$-distance on $\Omega .$

Example 5.

Let $\Omega =[0,\infty )$ be endowed with the lower limit topology ${\tau }_l$ and the proximity δ defined in Example 2. It is well-known that $(\mathbb{R},{\tau }_l)$ is not metrizable but normal topological space. Define $\omega :\Omega \times \Omega ⟶[0,\infty )$ by

$$\omega (\xi ,\eta )=\eta ,$$

then ω is ${\omega }_0$-distance on $\Omega .$

Example 6.

Let $\Omega =C[0,2]$ be endowed with the metric ρ defined by

$$\rho (f,g)={\int }_0^2\left|f\left(\xi \right)-g\left(\xi \right)\right|d\xi$$

and the proximity δ defined in Example 1. Define $\omega :\Omega \times \Omega ⟶[0,\infty )$ by

$$\omega (f,g)={\int }_0^2\left|g\left(\xi \right)\right|d\xi ,$$

then ω is ${\omega }_0$-distance on $\Omega .$

Proof. 

Let $U,V\subseteq C[0,2]$, $h\in C[0,2]$ and

$$\omega (h,U)=0=\omega (h,V).$$

(1)

Let $\epsilon >0$. Then by (1) there exist $f\in U$ and $g\in V$ such that

$$\omega (h,f)={\int }_0^2\left|f\left(\xi \right)\right|d\xi <\frac{\epsilon }{2}\mathrm{and}\omega (h,g)={\int }_0^2\left|g\left(\xi \right)\right|d\xi <\frac{\epsilon }{2}.$$

Hence we have

$$\begin{array}{ccc}\hfill \rho (f,g)& =& {\int }_0^2\left|f\left(\xi \right)-g\left(\xi \right)\right|d\xi \hfill \\ & \le & \omega (h,f)+\omega (h,g)\hfill \\ & <& \epsilon .\hfill \end{array}$$

Thus we have

$$\rho (U,V)=inf\left\{\rho (f,g):f\in U,g\in V\right\}=0$$

and hence $U\delta V$. Therefore, (w1) holds, and so $\omega$ is a $\omega$-distance on $\Omega$. It is clear that (w2) holds. Now, let $f,g\in \Omega$, then we have

$$\omega (f,g)={\int }_0^2\left|g\left(\xi \right)\right|d\xi =\underset{f^{\prime }\to f}{lim\; inf}\omega (f^{\prime },g)=\underset{V\in {\mathcal{U}}_f}{sup}\underset{f^{\prime }\in V}{inf}\omega (f^{\prime },g),$$

that is, $\omega$ is lower semicontinuous in the first variable. On the other hand, let $V\in {\mathcal{U}}_f$ and $f^{\prime }\in V$, then there exists $\epsilon >0$ such that $\rho (f,f^{\prime })<\epsilon$, that is ${sup}_{V\in {\mathcal{U}}_f}{inf}_{f^{\prime }\in V}\rho (f,f^{\prime })=0$. Hence, we have

$$\begin{array}{ccc}\hfill \omega (g,f)& =& {\int }_0^2\left|f\left(\xi \right)\right|d\xi \hfill \\ & \le & {\int }_0^2\left|f\left(\xi \right)-f^{\prime }\left(\xi \right)\right|d\xi +{\int }_0^2\left|f^{\prime }\left(\xi \right)\right|d\xi \hfill \\ & \le & \epsilon +\omega (g,f^{\prime })\hfill \end{array}$$

and so we obtain

$$\omega (g,f)\le \underset{V\in {\mathcal{U}}_f}{sup}\underset{f^{\prime }\in V}{inf}\omega (g,f^{\prime }),$$

that is, $\omega$ is lower semicontinuous in second variable. □

Example 7.

Let $\Omega =C[0,1]$ be endowed with the metric ρ defined by

$$\rho (f,g)=sup\left\{\left|f\left(\xi \right)-g\left(\xi \right)\right|:\xi \in \left[0,1\right]\right\}$$

and the proximity δ defined in Example 1. Define ${\omega }_1,{\omega }_2:\Omega \times \Omega ⟶[0,\infty )$ by

$${\omega }_1(f,g)=sup\left\{\left|g\left(\xi \right)\right|:\xi \in \left[0,1\right]\right\}$$

and

$${\omega }_2(f,g)=sup\left\{\left|f\left(\xi \right)\right|+\left|g\left(\xi \right)\right|:\xi \in \left[0,1\right]\right\},$$

then ${\omega }_1$ and ${\omega }_2$ are ${\omega }_0$-distance on $\Omega .$

Lemma 1

([9,10]). Let $(\Omega ,\delta )$ be a proximity space with ω-distance ω. Then, the following properties hold:

(i) 

If $(\Omega ,\delta )$ is separated, then $\omega (\zeta ,\xi )=0$ and $\omega (\zeta ,\eta )=0$ implies $\xi =\eta$.

(ii) 

If $\omega (\zeta ,\xi )=0$ and $\omega (\zeta ,{\xi }_n)\to 0$ as $n\to \infty$, then $\left\{{\xi }_n\right\}$ subsequently converges to ξ with respect to ${\tau }_{\delta }$.

In [9], Kostić introduced the Banach contraction principle in proximity space using ${\omega }_0$-distance $\omega$ as follows:

Theorem 1.

Let $(\Omega ,\delta )$ be a separated proximity space with ${\omega }_0$-distance ω. Suppose that the mapping $T:\Omega ⟶\Omega$ satisfies the following:

(i) 

There exists $k\in (0,1)$ such that

$$\omega (T\xi ,T\eta )\le k\omega (\xi ,\eta )$$

for all $\xi ,\eta \in \Omega$;

(ii) 

For all $\xi \in \Omega$, any iterative sequence $\left\{T^n\xi \right\}$ has a convergent subsequence with respect to ${\tau }_{\delta }$.

Then, there exists $\zeta \in \Omega$, such that $\zeta =T\zeta$ and $\omega (\zeta ,\zeta )=0$.

In this study, we will use some auxiliary functions in the contraction inequality to generalize Kostić’s theorem. Thus, we will obtain equivalents in proximity space of the famous Boyd–Wong [12] and Matkowski [13] fixed-point theorems known in metric fixed-point theory. In addition, for the equivalent of Edelstein’s theorem [14], in this space, we will consider that the space is compact according to the topology induced by proximity.

Let $\varphi :[0,\infty )\to [0,\infty )$ be a function. Next, we will consider the below properties for $\varphi$:

(${\varphi }_1$)

$\varphi$ is non-decreasing;

(${\varphi }_2$)

For all $t\ge 0$, $\underset{n\to \infty }{lim}{\varphi }^n\left(t\right)=0$;

(${\varphi }_3$)

For all $t>0$, $\varphi \left(t\right)<t;$

(${\varphi }_4$)

$\varphi$ is upper semicontinuous from the right.

We will symbolize the set of all functions with properties (${\varphi }_1$–${\varphi }_2$) and (${\varphi }_3$–${\varphi }_4$), by $\mathsf{\Phi }$ and $\mathsf{\Psi }$ respectively. It is easy to see that both $\mathsf{\Phi }/\mathsf{\Psi }$ and $\mathsf{\Psi }/\mathsf{\Phi }$ are nonempty and also every function in $\mathsf{\Phi }$ satisfies the property (${\varphi }_3$). Some examples of the functions belonging both $\mathsf{\Phi }$ and $\mathsf{\Psi }$ are ${\varphi }_1\left(t\right)=Lt$, where $0\le L<1$ and ${\varphi }_2\left(t\right)=\frac{t}{1+t}$.

2. Main Result

Here, we present our main theorems.

Theorem 2.

Let $(\Omega ,\delta )$ be a separated proximity space with ${\omega }_0$-distance ω and $T:\Omega ⟶\Omega$ be a mapping satisfying the followings:

(i) 

There exists $\varphi \in \mathsf{\Phi }$, satisfying

$$\omega (T\xi ,T\eta )\le \varphi \left(\omega \right(\xi ,\eta \left)\right)$$

for all $\xi ,\eta \in \Omega$;

(ii) 

For all $\xi \in \Omega$, any iterative sequence $\left\{T^n\xi \right\}$ has a convergent subsequence with respect to ${\tau }_{\delta }$.

Then there exists $\zeta \in \Omega$, such that $\zeta =T\zeta$ and $\omega (\zeta ,\zeta )=0$.

Proof. 

Let ${\xi }_0\in \Omega$ be arbitrary. Consider the corresponding Picard sequence $\left\{{\xi }_n\right\}$ constructed by

$${\xi }_n=T^n{\xi }_0=T{\xi }_{n-1}.$$

Since $\varphi$ is non-decreasing, and by (i), we have

$$\begin{array}{ccc}\hfill \omega ({\xi }_n,{\xi }_{n+1})& =& \omega (T{\xi }_{n-1},T{\xi }_n))\hfill \\ & \le & \varphi \left(\omega ({\xi }_{n-1},{\xi }_n)\right)\hfill \\ & ⋮& \\ & \le & {\varphi }^n\left(\omega ({\xi }_0,{\xi }_1)\right).\hfill \end{array}$$

Similarly, we can obtain

$$\omega ({\xi }_{n+1},{\xi }_n)\le {\varphi }^n\left(\omega ({\xi }_1,{\xi }_0)\right).$$

By (${\varphi }_2$), we have

$$\underset{n\to \infty }{lim}\omega ({\xi }_n,{\xi }_{n+1})=0$$

(2)

and

$$\underset{n\to \infty }{lim}\omega ({\xi }_{n+1},{\xi }_n)=0.$$

(3)

On the other hand, since $\varphi \in \mathsf{\Phi }$, we have $\varphi \left(ϵ\right)<ϵ$ for all $ϵ>0$. Hence by (2), there exists a natural number N satisfying

$$\omega ({\xi }_N,{\xi }_{N+1})<ϵ-\varphi \left(ϵ\right)$$

for all $ϵ>0$.

Since $\omega$ is a ${\omega }_0$-distance and $\varphi$ is non-decreasing, then we have

$$\begin{array}{ccc}\hfill \omega ({\xi }_N,{\xi }_{N+2})& \le & \omega ({\xi }_N,{\xi }_{N+1})+\omega ({\xi }_{N+1},{\xi }_{N+2})\hfill \\ & <& ϵ-\varphi \left(ϵ\right)+\omega (T{\xi }_N,T{\xi }_{N+1})\hfill \\ & \le & ϵ-\varphi \left(ϵ\right)+\varphi \left(\omega ({\xi }_N,{\xi }_{N+1})\right)\hfill \\ & <& ϵ-\varphi \left(ϵ\right)+\varphi (ϵ-\varphi (ϵ\left)\right)\hfill \\ & \le & ϵ-\varphi \left(ϵ\right)+\varphi \left(ϵ\right)\hfill \\ & =& ϵ.\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill \omega ({\xi }_N,{\xi }_{N+3})& \le & \omega ({\xi }_N,{\xi }_{N+1})+\omega ({\xi }_{N+1},{\xi }_{N+3})\hfill \\ & <& ϵ-\varphi \left(ϵ\right)+\omega (T{\xi }_N,T{\xi }_{N+2})\hfill \\ & \le & ϵ-\varphi \left(ϵ\right)+\varphi \left(\omega ({\xi }_N,{\xi }_{N+2})\right)\hfill \\ & <& ϵ-\varphi \left(ϵ\right)+\varphi \left(ϵ\right)\hfill \\ & =& ϵ.\hfill \end{array}$$

Continuing this process, we obtain $\omega ({\xi }_N,{\xi }_{N+k})<ϵ$ for all $k=1,2,\cdots$ and similarly by (3) we obtain $\omega ({\xi }_{N+k},{\xi }_N)<ϵ$ for all $k=1,2,\cdots$. Thus, for all $m,n>N$, we have

$$\omega ({\xi }_n,{\xi }_m)\le \omega ({\xi }_n,{\xi }_N)+\omega ({\xi }_N,{\xi }_m)<2ϵ.$$

(4)

By (ii), there exists a subsequence $\left\{{\xi }_{n_k}\right\}$ of the sequence $\left\{{\xi }_n\right\}$ which is convergent with respect to ${\tau }_{\delta }$ to some $\zeta \in \Omega$, and by $\left(w3\right)$, Remark 1 and Inequality (4), we have

$$\omega (\zeta ,{\xi }_{n_k})\le \underset{l\to \infty }{lim}inf\omega ({\xi }_{n_l},{\xi }_{n_k})\le 2ϵ,$$

(5)

and symmetrically, we obtain

$$\omega ({\xi }_{n_k},\zeta )\le 2ϵ$$

(6)

for all $n_k>N$. By Inequality (6), we have

$$\omega (T{\xi }_{n_k},T\zeta )\le \varphi \left(\omega ({\xi }_{n_k},\zeta )\right)\le \varphi \left(2ϵ\right)<2ϵ$$

(7)

for all $n_k>N$. Finally, by Inequalities (5) and (7), we have

$$\begin{array}{ccc}\hfill \omega (\zeta ,T\zeta )& \le & \omega (\zeta ,{\xi }_{n_k})+\omega ({\xi }_{n_k},T{\xi }_{n_k})+\omega (T{\xi }_{n_k},T\zeta )\hfill \\ & <& 2ϵ+{\varphi }^{n_k}\left(\omega ({\xi }_0,{\xi }_1)\right)+2ϵ\hfill \\ & =& 4ϵ+{\varphi }^{n_k}\left(\omega ({\xi }_0,{\xi }_1)\right)\hfill \end{array}$$

for all $n_k>N$. Since, by (${\varphi }_2$), ${\varphi }^{n_k}\left(\omega ({\xi }_0,{\xi }_1)\right)\to 0$ as $k\to \infty$, it follows that $\omega (\zeta ,T\zeta )=0$. On the other hand, by $\left(w3\right)$ and Remark 1, we have

$$\omega (\zeta ,\zeta )\le \underset{l\to \infty }{lim}inf\omega (\zeta ,{\xi }_{n_l})\le 2ϵ,$$

and consequently, $\omega (\zeta ,\zeta )=0$. Thus, Lemma 1 (i), we have $\zeta =T\zeta$. To check the uniqueness, let $\varsigma \in \Omega$ be a different fixed point of T. Then, $\omega (\zeta ,\varsigma )>0$, because $\zeta \ne \varsigma$ and $\omega (\zeta ,\zeta )=0$. By (i), we obtain

$$0<\omega (\zeta ,\varsigma )=\omega (T\zeta ,T\varsigma )\le \varphi \left(\omega \right(\zeta ,\varsigma \left)\right)<\omega (\zeta ,\varsigma ),$$

a contradiction. Hence, the fixed point of T is unique. □

Remark 2.

Note that for $\varphi \left(t\right)=kt$, $0\le k<1$, Theorem 2 reduces to Theorem 1.

Example 8.

Let $(\Omega ,\delta )$ be the proximity space and ω be the ${\omega }_0$-distance given in Example 5. Clearly $(\Omega ,\delta )$ is separated proximity space. Define a mapping $T:\Omega \to \Omega$ by $T\xi =\frac{\xi }{1+\xi }$ and consider the function $\varphi \in \mathsf{\Phi }$ as $\varphi \left(t\right)=\frac{t}{1+t}$. In this case, we have

$$\omega (T\xi ,T\eta )=T\eta =\frac{\eta }{1+\eta }=\varphi \left(\eta \right)=\varphi \left(\omega (\xi ,\eta )\right)$$

for all $\xi ,\eta \in \Omega$. In addition, for all $\xi \in \Omega$, $T^n$$\xi =\frac{\xi }{1+n\xi }$ is convergent with respect to ${\tau }_{\delta }$. Hence, all conditions of Theorem 2 hold. Therefore, T has a unique fixed point. Note that, since

$$\underset{\eta \in \Omega }{sup}\frac{\omega (T\xi ,T\eta )}{\omega (\xi ,\eta )}=1,$$

we can not find a constant $k\in (0,1)$ satisfying

$$\omega (T\xi ,T\eta )\le k\omega (\xi ,\eta ).$$

Hence, Theorem 1 cannot be applied.

Theorem 3.

Let $(\Omega ,\delta )$ be a separated proximity space with ${\omega }_0$-distance ω and $T:\Omega ⟶\Omega$ be a mapping satisfying the followings:

(i) 

There exists $\varphi \in \mathsf{\Psi }$ such that $\varphi \left(0\right)=0$ and ϕ is right continuous at 0 satisfying $\omega (T\xi ,T\eta )\le \varphi \left(\omega \right(\xi ,\eta \left)\right)$ for all $\xi ,\eta \in \Omega$;

(ii) 

For all $\xi \in \Omega$, any iterative sequence $\left\{T^n\xi \right\}$ has a convergent subsequence with respect to ${\tau }_{\delta }$.

Then, there exists $\zeta \in \Omega$ such that $\zeta =T\zeta$ and $\omega (\zeta ,\zeta )=0$.

Proof. 

Let ${\xi }_0\in \Omega$ be arbitrary. Consider the corresponding Picard sequence $\left\{{\xi }_n\right\}$ constructed by

$${\xi }_n=T^n{\xi }_0=T{\xi }_{n-1}.$$

for all $n\in \mathbb{N}$. For simplicity, call ${\omega }_n=\omega ({\xi }_n,{\xi }_{n+1})$. If there exists $n_0\in \mathbb{N}$ such that ${\omega }_{n_0}={\omega }_{n_0+1}=0$, then by triangular inequality we have

$$\begin{array}{ccc}\hfill \omega ({\xi }_{n_0},{\xi }_{n_0+2})& \le & \omega ({\xi }_{n_0},{\xi }_{n_0+1})+\omega ({\xi }_{n_0+1},{\xi }_{n_0+2})\hfill \\ & =& {\omega }_{n_0}+{\omega }_{n_0+1}=0\hfill \end{array}$$

and so by Lemma 1 (i) we have ${\xi }_{n_0+1}={\xi }_{n_0+2}$. This shows that ${\xi }_{n_0+1}$ is a fixed point of T. Now assume that any of consecutive terms of the $\left\{{\omega }_n\right\}$ are not zero. In this case, there exists a subsequence $\left\{{\omega }_{n_k}\right\}$ of $\left\{{\omega }_n\right\}$ such that ${\omega }_{n_k}>0$ for all $k\in \mathbb{N}$. By (i), we have

$$\begin{array}{ccc}\hfill {\omega }_{n_k+1}& =& \omega ({\xi }_{n_k+1},{\xi }_{n_k+2})\hfill \\ & =& \omega (T{\xi }_{n_k},T{\xi }_{n_k+1})\hfill \\ & \le & \varphi \left(\omega ({\xi }_{n_k},{\xi }_{n_k+1})\right)\hfill \\ & =& \varphi \left({\omega }_{n_k}\right)<{\omega }_{n_k}.\hfill \end{array}$$

(8)

Since the index sequence $n_k$ is strictly increasing, we have $n_{k+1}>n_k$ and so $n_{k+1}\ge n_k+1$. Therefore, from (8), we have

$${\omega }_{n_{k+1}}\le {\omega }_{n_k+1}<{\omega }_{n_k}.$$

(9)

It follows that $\left\{{\omega }_{n_k}\right\}$ is a decreasing sequence which is also bounded below. Thus, there exists $\gamma \ge 0$ such that $\underset{k\to \infty }{lim}{\omega }_{n_k}=\gamma$. If $\gamma >0$, then by (9), $\left({\varphi }_3\right)$ and $\left({\varphi }_4\right)$, we have

$$0<\gamma =\underset{k\to \infty }{lim}{\omega }_{n_{k+1}}\le \underset{k\to \infty }{lim}{\omega }_{n_k+1}\le \underset{k\to \infty }{lim}\varphi \left({\omega }_{n_k}\right)\le \underset{k\to \infty }{lim}sup\varphi \left({\omega }_{n_k}\right)\le \varphi \left(\gamma \right)<\gamma ,$$

a contradiction. Therefore, $\gamma =0$, and consequently,

$$\underset{k\to \infty }{lim}{\omega }_{n_k}=0.$$

(10)

Similarly, it can be obtained that the limit of all subsequences of $\left\{{\omega }_n\right\}$ whose all terms are greater than zero, is zero. Hence we have

$$\underset{n\to \infty }{lim}{\omega }_n=\underset{n\to \infty }{lim}\omega ({\xi }_n,{\xi }_{n+1})=0.$$

Now, if we call ${\varpi }_n=\omega ({\xi }_{n+1},{\xi }_n)$, similarly we can obtain

$$\underset{n\to \infty }{lim}{\varpi }_n=\underset{n\to \infty }{lim}\omega ({\xi }_{n+1},{\xi }_n)=0.$$

(11)

Now, we claim that for every $ϵ>0$, there exists a natural number N satisfying

$$\omega ({\xi }_m,{\xi }_n)<ϵ$$

for all $m>n>N$. Assume the contrary, then there exist $ϵ>0$, $m_k,n_k\in \mathbb{N}$ with $m_k>n_k>k$, $k\in \mathbb{N}$ such that $\omega ({\xi }_{m_k},{\xi }_{n_k})\ge ϵ$. Here, we can choose $m_k$ as the smallest positive integer that satisfies $\omega ({\xi }_{m_k},{\xi }_{n_k})\ge ϵ$. Therefore, we obtain $\omega ({\xi }_{m_k-1},{\xi }_{n_k})<ϵ$, and since $\omega$ is a ${\omega }_0$-distance, we obtain

$$\begin{array}{ccc}\hfill ϵ& \le & \omega ({\xi }_{m_k},{\xi }_{n_k})\hfill \\ & \le & \omega ({\xi }_{m_k},{\xi }_{m_k-1})+\omega ({\xi }_{m_k-1},{\xi }_{n_k})\hfill \\ & \le & \omega ({\xi }_{m_k},{\xi }_{m_k-1})+ϵ\hfill \\ & =& {\varpi }_{m_k-1}+ϵ.\hfill \end{array}$$

By Equation (11), it follows that $\underset{k\to \infty }{lim}\omega ({\xi }_{m_k},{\xi }_{n_k})=ϵ$. On other hand,

$$\begin{array}{ccc}\hfill \omega ({\xi }_{m_k},{\xi }_{n_k})& \le & \omega ({\xi }_{m_k},{\xi }_{m_k+1})+\omega ({\xi }_{m_k+1},{\xi }_{n_k+1})+\omega ({\xi }_{n_k+1},{\xi }_{n_k})\hfill \\ & =& {\omega }_{m_k}+\omega (T{\xi }_{m_k},T{\xi }_{n_k})+{\varpi }_{n_k}\hfill \\ & \le & {\omega }_{m_k}+\varphi \left(\omega ({\xi }_{m_k},{\xi }_{n_k})\right)+{\varpi }_{n_k}.\hfill \end{array}$$

Taking limit $k\to \infty$ and by $\left({\varphi }_4\right)$, we have $ϵ\le \varphi \left(ϵ\right)$, a contradiction. Thus, for every $ϵ>0$, there exists $N\in \mathbb{N}$, such that

$$\omega ({\xi }_m,{\xi }_n)<ϵ$$

(12)

for all $m>n>N$. Similarly, we obtain

$$\omega ({\xi }_n,{\xi }_m)<ϵ$$

(13)

for all $m>n>N$.

By (ii), there exists a subsequence $\left\{{\xi }_{n_k}\right\}$ of sequence $\left\{{\xi }_n\right\}$ which is convergent with respect to ${\tau }_{\delta }$ to some $\zeta \in \Omega$, and by $\left({\omega }_3\right)$ and Inequality (12), we have

$$\omega (\zeta ,{\xi }_{n_k})\le \underset{l\to \infty }{lim}inf\omega ({\xi }_{n_l},{\xi }_{n_k})\le ϵ,$$

(14)

for all $n_k\in \mathbb{N}$ and symmetrically by (13), we obtain

$$\omega ({\xi }_{n_k},\zeta )\le ϵ$$

(15)

for all $n_k\in \mathbb{N}$. Then, we have

$$\omega (\zeta ,\zeta )\le \underset{k\to \infty }{lim}inf\omega (\zeta ,{\xi }_{n_k})\le ϵ,$$

and consequently, $\omega (\zeta ,\zeta )=0$. On the other hand, we have

$$\begin{array}{ccc}\hfill \omega (\zeta ,T\zeta )& \le & \omega (\zeta ,{\xi }_{n_k})+\omega ({\xi }_{n_k},T\zeta )\hfill \\ & =& \omega (\zeta ,{\xi }_{n_k})+\omega (T{\xi }_{n_k-1},T\zeta )\hfill \\ & <& ϵ+\varphi \left(\omega ({\xi }_{n_k-1},\zeta )\right).\hfill \end{array}$$

Since $\varphi \left(0\right)=0$ and $\varphi$ is right continuous at 0, we have $\omega (\zeta ,T\zeta )=0.$ Thus, by Lemma 1 (i), we have $\zeta =T\zeta$.

To check the uniqueness, let $v\in \Omega$ with $v\ne \zeta$ be another fixed point of mapping T. Since $\omega (\zeta ,\zeta )=0$ it must be $\omega (\zeta ,v)>0$. Hence by (i), we obtain

$$\omega (\zeta ,v)=\omega (T\zeta ,Tv)\le \varphi \left(\omega \right(\zeta ,v\left)\right)<\omega (\zeta ,v),$$

a contradiction. Thus, the fixed point of T is unique. □

Finally, in the theoretical part, we present a fixed-point result, taking into account the compactness of the space and a strict contraction inequality. Here, we will use the following well-known lemma.

Lemma 2

([15]). Let $f:\Omega \to \mathbb{R}$ a lower semicontinuous function, where Ω is a compact topological space. Then, there exists an element ${\xi }_0\in \Omega$ such that

$$f\left({\xi }_0\right)=inf\{f\left(\xi \right):\xi \in \Omega \}.$$

Definition 2.

Let $(\Omega ,\delta )$ be a proximity space with ω-distance ω and $T:\Omega ⟶\Omega$ be a mapping. Then T is said to have 0-property if $\omega (T\xi ,T\eta )=0$ whenever $\omega (\xi ,\eta )=0$ for all $\xi ,\eta \in \Omega$.

Example 9.

Let $\Omega =[0,\infty )$ be endowed with the lower limit topology ${\tau }_l$ and the proximity δ defined in Example 2. Consider the ω-distance on Ω defined by $\omega (\xi ,\eta )=\eta$. Then every self-mapping T of Ω has 0-property, provided that $T0=0$.

Theorem 4.

Let $(\Omega ,\delta )$ be a separated proximity space with ${\omega }_0$-distance ω such that Ω is compact with respect to ${\tau }_{\delta }$, and $T:\Omega ⟶\Omega$ be a continuous mapping with 0-property. Assume that

$$\omega (T\xi ,T\eta )<\omega (\xi ,\eta )$$

for all $\xi ,\eta \in \Omega$ with $\omega (\xi ,\eta )>0$. Then, there exists $\varsigma \in \Omega$ such that $\varsigma =T\varsigma$.

Proof. 

Define $f,g:\Omega \to \mathbb{R}$ by $f\left(\xi \right)=\omega (\xi ,T\xi )$ and $g\left(\xi \right)=\omega (T\xi ,\xi )$. Since $\omega$ is lower semicontinuous in both variables and T is continuous, then f and g are lower semicontinuous. Since $\Omega$ is compact with respect to ${\tau }_{\delta }$, then from Lemma 2 there exist $\zeta ,\varsigma \in \Omega$, such that

$$f\left(\zeta \right)=inf\left\{f\right(\xi ):\xi \in \Omega \}$$

and

$$g\left(\varsigma \right)=inf\left\{g\right(\xi ):\xi \in \Omega \}.$$

In this case, if $\omega (\zeta ,T\zeta )>0$, then we obtain

$$f\left(T\zeta \right)=\omega (T\zeta ,TT\zeta )<\omega (\zeta ,T\zeta )=f\left(\zeta \right),$$

which is a contradiction. Hence, we obtain $f\left(\zeta \right)=\omega (\zeta ,T\zeta )=0$. Similarly, we obtain $\omega (T\varsigma ,\varsigma )=0$.

Now, if $\omega (\zeta ,\varsigma )>0$, then we obtain

$$\begin{array}{ccc}\hfill \omega (\zeta ,\varsigma )& \le & \omega (\zeta ,T\zeta )+\omega (T\zeta ,T\varsigma )+\omega (T\varsigma ,\varsigma )\hfill \\ & =& \omega (T\zeta ,T\varsigma )\hfill \\ & <& \omega (\zeta ,\varsigma ),\hfill \end{array}$$

which is a contradiction. Hence, we have $\omega (\zeta ,\varsigma )=0$. Therefore, from Lemma 1 (i), we have $\varsigma =T\zeta$ and so $\omega (T\varsigma ,T\zeta )=0$. Now,

$$\omega (T\varsigma ,T\varsigma )\le \omega (T\varsigma ,T\zeta )+\omega (T\zeta ,T\varsigma )=0,$$

and so from Lemma 1 (i), we have $\varsigma =T\varsigma$. Additionally,

$$\omega (\varsigma ,\varsigma )=\omega (T\zeta ,T\varsigma )=0.$$

Now let v be another fixed point of T. Then, we have $v=Tv$ and $\omega (\varsigma ,v)>0$. Therefore, we obtain

$$\omega (\varsigma ,v)=\omega (T\varsigma ,Tv)<\omega (\varsigma ,v),$$

which is a contradiction. □

3. Application

Now, by considering Theorem 2, we give existence and uniqueness results about the solution of the second-order boundary value problem (BVP) as follows:

$$\left\{\begin{array}{c}-\frac{d^2\xi }{d{t}^2}=F(t,\xi \left(t\right)),t\in [0,1]\hfill \\ \xi \left(0\right)=\xi \left(1\right)=0\hfill \end{array}\right.,$$

(16)

where $F:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous function. By taking into account some certain conditions $F,$ some existence theorems were recently presented for problem (16) in the literature (see [16,17,18]). Here, we will consider some different conditions on F, and we provide a new theorem. We can see that the problem (16) is equivalent to the integral equation

$$\xi \left(t\right)={\int }_0^{1}G(t,s)F(s,\xi \left(s\right))ds,t\in [0,1],$$

(17)

where $G(t,s)$ is associated Green’s function defined as

$$G(t,s)=\left\{\begin{array}{ccc}t(1-s)& ,& 0\le t\le s\le 1\\ & & \\ s(1-t)& ,& 0\le s\le t\le 1\end{array}\right..$$

Therefore, $\xi \in C^2[0,1]$ is a solution of (16) if and only if it is a solution of (17). It is clear that

$${\int }_0^{1}G(t,s)ds=\frac{t(1-t)}{2}.$$

Let $(\Omega ,\delta )$ be the proximity space, where $\Omega =C[0,1]$ and $\delta$ is induced by the uniform metric ${\rho }_{\infty }$

$${\rho }_{\infty }(\xi ,\eta )=sup\{\left|\xi \left(t\right)-\eta \left(t\right)\right|:t\in \left[0,1\right]\}.$$

In this case, $(\Omega ,\delta )$ is separated proximity space. Consider the following ${\omega }_0$-distances ${\omega }_{\alpha }$ on $\Omega$ defined by

$${\omega }_{\alpha }(\xi ,\eta )=sup\{e^{-\alpha t}\left|\xi \left(t\right)-\eta \left(t\right)\right|:t\in \left[0,1\right]\},$$

where $\alpha >0$ is a constant.

Theorem 5.

The second-order BVP given by (16) has a unique solution under the following assumptions:

(a) There exists a non-decreasing function $\phi :[0,\infty )\to [0,\infty )$ such that

$$\left|F(s,\xi )-F(s,\eta )\right|\le \phi \left(e^{-\alpha s}\left|\xi -\eta \right|\right)$$

for all $s\in \left[0,1\right]$.

(b) ${lim}_{n\to \infty }{\varphi }^n\left(t\right)=0$ for all $t\ge 0$, where $\varphi =K_{\alpha }\phi$ and

$$K_{\alpha }=sup\left\{e^{-\alpha t}\frac{t(1-t)}{2}:t\in \left[0,1\right]\right\}.$$

Proof. 

Consider the operator $T:C[0,1]\times C[0,1]\to C[0,1]$ defined by

$$T\xi \left(t\right)={\int }_0^{1}G(t,s)F(s,\xi \left(s\right))ds$$

Then, for any $\xi ,\eta \in C[0,1]$ and $t\in \left[0,1\right]$, we have

$$\begin{array}{ccc}\hfill \left|T\xi \left(t\right)-T\eta \left(t\right)\right|& =& \left|{\int }_0^{1}G(t,s)F(s,\xi \left(s\right))ds-{\int }_0^{1}G(t,s)F(s,\eta \left(s\right))ds\right|\hfill \\ & \le & {\int }_0^{1}G(t,s)\left|F(s,\xi (s\left)\right)-F(s,\eta (s\left)\right)\right|ds\hfill \\ & \le & {\int }_0^{1}G(t,s)\phi \left(e^{-\alpha s}\left|\xi \left(s\right)-\eta \left(s\right)\right|\right)ds\hfill \\ & \le & \phi \left({\omega }_{\alpha }(\xi ,\eta )\right){\int }_0^{1}G(t,s)ds\hfill \\ & =& \phi \left({\omega }_{\alpha }(\xi ,\eta )\right)\frac{t(1-t)}{2}\hfill \end{array}$$

and then we obtain

$$e^{-\alpha t}\left|T\xi \left(t\right)-T\eta \left(t\right)\right|\le \phi \left({\omega }_{\alpha }(\xi ,\eta )\right)e^{-\alpha t}\frac{t(1-t)}{2}.$$

By taking supremum over $t\in \left[0,1\right]$, we have

$${\omega }_{\alpha }(T\xi ,T\eta )\le K_{\alpha }\phi \left({\omega }_{\alpha }(\xi ,\eta )\right)=\varphi \left({\omega }_{\alpha }(\xi ,\eta )\right).$$

(18)

Therefore, condition (i) of Theorem 2 is satisfied. Now let $\xi \in C[0,1]$ be an arbitrary function. Define a sequence of functions $\left\{{\xi }_n\right\}$ as ${\xi }_n=T^n\xi$. As in the proof of Theorem 2 and by (18), we have

$${\omega }_{\alpha }({\xi }_n,{\xi }_m)\to 0$$

as $m,n\to \infty$. On the other hand, since

$$e^{-\alpha }{\rho }_{\infty }(\xi ,\eta )\le {\omega }_{\alpha }(\xi ,\eta )\le {\rho }_{\infty }(\xi ,\eta )$$

for all $\xi ,\eta \in C\left[0,1\right]$, we have

$${\rho }_{\infty }({\xi }_n,{\xi }_m)\to 0$$

as $m,n\to \infty$. That is, the sequence $\left\{{\xi }_n\right\}$ is Cauchy and so has a convergent subsequence with respect to ${\rho }_{\infty }$ since $(\Omega ,{\rho }_{\infty })$ is complete. Therefore, condition (ii) of Theorem 2 is satisfied. Consequently, there exists unique $\zeta \in C\left[0,1\right]$ which is a fixed point of the operator T, moreover ${\omega }_{\alpha }(\zeta ,\zeta )=0$. Hence, the (16) has a unique solution. □

4. Conclusions

In this paper, following Kostić’s, idea we present some fixed-point theorems for single-valued mappings on proximity space by using $\omega$-distance and some auxiliary functions. Then, we support our theoretical results with some examples and an existence and uniqueness theorem.