Some Extended Results for Multivalued F-Contraction Mappings

Abstract

In this study, we introduce a new notion, so-called $KW$-type F-contraction mapping, inspired by the ideas of Wardowski and Klim–Wardowski. Then, we investigate the existence of a best proximity point for such mappings by considering a new family, which is larger than the family of functions that is often used in fixed-point results for multivalued mappings. To demonstrate the effectiveness of our result, we also give a comparative example to which similar results in the literature cannot be applied. Moreover, we present an application of our main result to homotopic mappings.

1. Introduction and Preliminaries

Fixed-point theory in metric spaces was initiated by the result which states that every self-mapping $\phi$ on a complete metric space $(\Im ,\rho )$, such that $\rho (\phi \check{s},\phi \breve{u})\le k\rho (\check{s},\breve{u})$, for all $\check{s},\breve{u}\in \Im$ where $k\in [0,1)$, has a unique fixed point in ℑ. The result is known as Banach’s contraction principle [1]. Banach’s contraction principle has been generalized in different ways, especially because of its applications in non-linear analysis [2,3]. In this sense, Wardowski [4] introduced the useful and interesting concept of F-contraction mappings and obtained a fixed-point result for new kinds of mappings. Thus, previously established results in the literature, including the famous Banach contraction principle, were extended [5]. Then, in 2015, Cosentino et al. [6] adapted Wardowski’s result to b-metric spaces. In 2016, Durmaz et al. [7] took ordered metric spaces and F-contractions into account together. In 2017, combining F-contractions and c-comparison functions, Aydi et al. [8] obtained a fixed-point theorem for such mappings. In 2019, Sahin et al. [9] investigated the existence of a fixed point for Feng–Liu-type multivalued mappings defined via F-contraction in M-metric spaces. A reminder of some notions and notations related to F-contraction is provided below.

Let $\mathcal{F}$ be a family function $\mathrm{\Psi }:(0,\infty )\to \mathbb{R}$ satisfying the following conditions:

(F${}_1$)

For all ${\lambda }_1,{\lambda }_2>0$ with ${\lambda }_1<{\lambda }_2$, we have $\mathrm{\Psi }\left({\lambda }_1\right)<\mathrm{\Psi }\left({\lambda }_2\right).$

(F${}_2$)

For every sequence $\left\{{\lambda }_t\right\}\subseteq (0,\infty )$, ${lim}_{t\to \infty }{\lambda }_t=0$ if ${lim}_{t\to \infty }\mathrm{\Psi }\left({\lambda }_t\right)=-\infty .$

(F${}_3$)

There is $k\in (0,1)$ satisfying ${lim}_{\lambda \to 0^{+}}{\lambda }^k\mathrm{\Psi }\left(\lambda \right)=0.$

Let $\phi :\Im \to \Im$ be a mapping and $(\Im ,\rho )$ be a metric space. If there is $\tau >0$ and $\mathrm{\Psi }\in \mathcal{F}$, such that

$$\tau +\mathrm{\Psi }\left(\rho (\phi \check{s},\phi \breve{u})\right)\le \mathrm{\Psi }\left(\rho (\check{s},\breve{u})\right)$$

for all $\check{s},\breve{u}\in \Im$ with $\rho (\phi \check{s},\phi \breve{u})>0$, then $\phi$ is called an F-contraction on ℑ.

Then, Wardowski proved the following result for the F-contraction mappings.

Theorem 1 

([4]). Let φ be an F-contraction on $(\Im ,\rho )$ where $(\Im ,\rho )$ is a complete metric space. Then, there exists a unique point ${\check{s}}^{*}$ in ℑ satisfying $\phi {\check{s}}^{*}={\check{s}}^{*}$ and, for each ${\check{s}}_0\in \Im$, the sequence $\left\{{\phi }^t{\check{s}}_0\right\}$ converges to ${\check{s}}^{*}$.

Recently, Secelean [10] presented the following conditions which are equivalent to the ($F_2$), but easy condition

($F_2^{\prime }$)

$inf\mathrm{\Psi }=-\infty$

or

($F_2^{″}$)

there exists the sequence $\left\{{\lambda }_t\right\}\subseteq (0,\infty )$, such that ${lim}_{t\to \infty }\mathrm{\Psi }\left({\lambda }_t\right)=-\infty .$

Then, Secelean [10] proved the following important lemma:

Lemma 1. 

Let$\mathrm{\Psi }:(0,\infty )\to \mathbb{R}$be an increasing function and$\left\{{\lambda }_t\right\}\subseteq (0,\infty )$be a sequence. Then, the following statements are true.

(i) 

if ${lim}_{t\to \infty }\mathrm{\Psi }\left({\lambda }_t\right)=-\infty ,$ then $li{m}_{t\to \infty }{\lambda }_t=0.$

(ii) 

if $inf\mathrm{\Psi }=-\infty$ and ${lim}_{t\to \infty }{\lambda }_t=0,$ then ${lim}_{t\to \infty }\mathrm{\Psi }\left({\lambda }_t\right)=-\infty .$

We will denote the set of all functions $\mathrm{\Psi }:(0,\infty )\to \mathbb{R}$ satisfying ($F_1$), ($F_2^{\prime }$) and ($F_3$) by ${\mathcal{F}}^{\prime }.$

Combining multivalued mappings and Lipschitz mappings, Nadler [11] presented one of the important generalizations of the Banach contraction principle for multivalued mappings as follows:

Theorem 2 

([11]). Let φ be a multivalued mapping on $(\Im ,\rho )$, where $(\Im ,\rho )$ is a complete metric space, such that $\phi \check{s}$ is a non-empty, closed and bounded subset of ℑ (denote by $CB(\Im )$) for any $\check{s}\in \Im$. If there exists $c\in (0,1)$, such that

$$H(\phi \check{s},\phi \breve{u})\le c\rho (\check{s},\breve{u})$$

for all $\check{s},\breve{u}\in \Im$, then the mapping φ has a fixed point in ℑ where

$$H(\mathrm{\Gamma },\mathrm{\Lambda })=max\{\underset{\check{s}\in \mathrm{\Gamma }}{sup}\rho (\check{s},\mathrm{\Lambda }),\underset{\breve{u}\in \mathrm{\Lambda }}{sup}\rho (\mathrm{\Gamma },\breve{u})\}$$

for all $\mathrm{\Gamma },\mathrm{\Lambda }\in CB(\Im ).$

Considering the family of all non-empty closed subsets of ℑ (denoted by $C(\Im )$) instead of $CB(\Im )$, an interesting and nice generalization of Nadler’s result, without using the Hausdorff metric, was obtained by Feng and Liu [12] as follows:

Theorem 3 

([12]). Let $\phi :\Im \to C(\Im )$ be a multivalued mapping on a complete metric space $(\Im ,\rho )$. Assume that there exists $\breve{u}\in I_b^{\check{s}}=\{\breve{u}\in \phi \check{s}:b\rho (\check{s},\breve{u})\le \rho (\check{s},\phi \check{s})\}$ for all $\check{s}\in \Im$ and for some $b,c\in [0,1)$ with $c<b$, such that

$$\rho (\breve{u},\phi \breve{u})\le c\rho (\check{s},\breve{u}).$$

Then, the mapping φ has a fixed point in ℑ provided that the mapping $\check{s}\to \rho (\check{s},\phi \check{s})$ is lower semi-continuous.

Then, Klim and Wardowski [13] introduced a new contraction for multivalued mappings by considering the ideas of Feng–Liu [12] and Mizoguchi–Takahashi [14]. Hence, they proved the following nice result for such contractive mappings:

Theorem 4 

([13]). Let $\phi :\Im \to C(\Im )$ be a multivalued mapping on a complete metric space $(\Im ,\rho )$ and a function $h:\Im \to \mathbb{R}$ defined by $h\left(\check{s}\right)=\rho (\check{s},\phi \check{s})$ be lower semi-continuous. If there exist $b\in (0,1)$ and $\phi :[0,\infty )⟶[0,b)$, such that

$$\underset{\ell \to s^{+}}{lim}sup\phi (\ell )<b$$

(1)

for all $s\ge 0$ and there exists $\breve{u}\in I_b^{\check{s}}$ for all $\check{s}\in \Im$ satisfying

$$\rho (\breve{u},\phi \breve{u})\le \phi \left(\rho (\check{s},\breve{u})\right)\rho (\check{s},\breve{u}),$$

then the mapping φ has a fixed point in ℑ.

Recently, taking into account the idea of F-contraction for multivalued mappings, multivalued F-contraction mappings were presented by Altun et al. [15].

Let $\phi :\Im \to CB(\Im )$ be a multivalued mapping on a metric space $(\Im ,\rho )$. If there is $\tau >0$ and $\mathrm{\Psi }\in \mathcal{F}$, such that

$$\tau +\mathrm{\Psi }\left(H(\phi \check{s},\phi \breve{u})\right)\le \mathrm{\Psi }\left(\rho (\check{s},\breve{u})\right)$$

for all $\check{s},\breve{u}\in \Im$ with $H(\phi \check{s},\phi \breve{u})>0$, then $\phi$ is called a multivalued F-contraction mapping.

Then, they proved that a compact valued ($K(\Im )$) multivalued F-contraction mapping $\phi$on a complete metric space $(\Im ,\rho )$ has a fixed point in ℑ. In addition, they investigated whether the same result is valid if the mapping $\phi$ is $CB(\Im )$ valued and proved a fixed-point result for the new type of mappings by assuming the following property in addition to (F${}_1$)–(F${}_3$):

(F${}_4$) For all $\mathrm{\Gamma }\subseteq (0,\infty )$ with $inf\mathrm{\Gamma }>0$, we have $\mathrm{\Psi }(inf\mathrm{\Gamma })=inf\mathrm{\Psi }\left(\mathrm{\Gamma }\right)$.

We denote the family of functions $\mathrm{\Psi }\in \mathcal{F}$ satisfying (F${}_4$) by ${\mathcal{F}}_{*}$ throughout this study.

On the other hand, if for a non-self mapping $\phi :\mathrm{\Gamma }\to \mathrm{\Lambda }$ on a metric space $(\Im ,\rho )$,we have $\rho (\mathrm{\Gamma },\mathrm{\Lambda })>0$, then the equation $\phi \check{s}=\check{s}$ may not have a solution. In this case, the question whether there is a point $\check{s}$, such that $\rho (\check{s},\phi \check{s})=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$, has become very popular in recent times. The point $\check{s}$ is said to be the best proximity point of $\phi$ [16]. It is clear that, if the mapping $\phi$ has a best proximity point, then the minimization problem $min\left\{\rho (\check{s},\phi \check{s}):\check{s}\in \mathrm{\Gamma }\right\}$ has a solution. Therefore, this topic has been studied by many authors [17,18,19,20,21,22,23,24]. Now, we recall some concepts and notations which are important for our results.

Let us consider the following subsets of a metric space $(\Im ,\rho )$:

$${\mathrm{\Gamma }}_0=\{\check{s}\in \mathrm{\Gamma }:\rho (\check{s},\breve{u})=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\mathrm{for}\mathrm{some}\breve{u}\in \mathrm{\Lambda }\}$$

and

$${\mathrm{\Lambda }}_0=\{\breve{u}\in \mathrm{\Lambda }:\rho (\check{s},\breve{u})=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\mathrm{for}\mathrm{some}\check{s}\in \mathrm{\Gamma }\}.$$

Definition 1 

([25]). Let Γ and Λ be non-empty subsets of a metric space $(\Im ,\rho )$. Then, the pair $(\mathrm{\Gamma },\mathrm{\Lambda })$ is said to have a modified P-property, if it is satisfied

$$\left.\begin{array}{c}\rho ({\check{s}}_1,{\breve{u}}_1)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\\ \rho ({\check{s}}_2,{\breve{u}}_2)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\end{array}\right\}\Rightarrow \rho ({\check{s}}_1,{\check{s}}_2)=\rho ({\breve{u}}_1,{\breve{u}}_2)$$

for all ${\check{s}}_1,{\check{s}}_2\in \mathrm{\Gamma }$ with ${\check{s}}_1\ne {\check{s}}_2$ and ${\breve{u}}_1,{\breve{u}}_2\in \mathrm{\Lambda }$ with ${\breve{u}}_1\ne {\breve{u}}_2$.

Very recently, Sahin [25] introduced generalized multivalued F-contraction mapping inspired by the ideas of Feng–Liu and Wardowski, and then proved the following result for such mappings.

Theorem 5 

([25]). Let $\varnothing \ne \mathrm{\Gamma },\mathrm{\Lambda }\subseteq \Im$ be closed, having a modified P-property, where $(\Im ,\rho )$ is a complete metric space and ${\mathrm{\Gamma }}_0\ne \varnothing$. Suppose that $\phi :\mathrm{\Gamma }\to C\left(\mathrm{\Lambda }\right)$ is a multivalued mapping and $f(\check{s},\breve{u})=\rho (\breve{u},\phi \check{s})$ is lower semi-continuous on $\mathrm{\Gamma }\times \mathrm{\Lambda }$. If there are $\tau >0$ and $\mathrm{\Psi }\in {\mathcal{F}}_{*}$ such that for all $\check{s}\in {\mathrm{\Gamma }}_0$ with $\rho (\check{s},\phi \check{s})>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$ and $\breve{u}\in \phi \check{s}$, there exists $z\in {\mathrm{\Gamma }}_0$ satisfying

$$\rho (\breve{u},z)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$$

and

$$\tau +\mathrm{\Psi }\left(\rho (\breve{u},\phi z)\right)\le \mathrm{\Psi }\left(\rho (\check{s},z)\right),$$

then, φ has a best proximity point in $\mathrm{\Gamma }.$

Thus, the main result for $CB(\Im )$ valued mappings in [15] has been generalized in an interesting way. Note that one needs the family ${\mathcal{F}}_{*}$ to show the existence of a fixed point for $CB(\Im )$ valued mappings. However, Aslantas [26] presented another approach by a new family, which includes ${\mathcal{F}}_{*}$. We combine the approaches of Secelean [10] and Aslatas [26]. Now, let $\phi :\mathrm{\Gamma }\to P\left(\mathrm{\Lambda }\right)$ (the family of all non-empty subsets of $\mathrm{\Lambda }$) be a mapping defined on a metric space $(\Im ,\rho )$, where $\varnothing \ne \mathrm{\Gamma },\mathrm{\Lambda }\subseteq \Im$. Define a family by

$${}_{\phi }^{\rho }{\mathcal{F}}_{**}=\{\mathrm{\Psi }\in {\mathcal{F}}^{\prime }:{\mathrm{\Psi }}_{\sigma ,\phi }^{\check{s},\breve{u}}\ne \varnothing \mathrm{for}\mathrm{all}\left(\check{s},\breve{u}\right)\in {\rho }_{\mathrm{\Lambda }}^{\mathrm{\Gamma }}\mathrm{and}\sigma \ge 0\},$$

where

$${\mathrm{\Psi }}_{\sigma ,\phi }^{\check{s},\breve{u}}=\{z\in \phi \check{s}:\mathrm{\Psi }\left(\rho (\breve{u},z)\right)\le \mathrm{\Psi }\left(\rho (\breve{u},\phi \check{s})\right)+\sigma \}$$

and

$${\rho }_{\mathrm{\Lambda }}^{\mathrm{\Gamma }}=\left\{\left(\check{s},\breve{u}\right)\in \mathrm{\Gamma }\times \mathrm{\Lambda }:\rho (\check{s},\breve{u})=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\mathrm{and}\rho (\breve{u},\phi \check{s})>0\right\}.$$

Then, it can be shown that ${\mathcal{F}}_{*}$ is a proper subset of ${}_{\phi }^{\rho }{\mathcal{F}}_{**}$.

In this study, we offer a novel idea known as $KW$-type F-contraction mapping, which is inspired by the theories of Klim–Wardowski and Wardowski. Then, we examine the existence of a best proximity point for such mappings by considering a new family ${}_{\phi }^{\rho }{\mathcal{F}}_{**}$, which is larger than the family of functions ${\mathcal{F}}_{*}$. In addition, we give a comparative example to which similar results in the literature cannot be applied to demonstrate the effectiveness of our result.

2. Main Results

Now, we introduce the following definition:

Definition 2. 

Let $(\Im ,\rho )$ be a metric space and $\varnothing \ne \mathrm{\Gamma },\mathrm{\Lambda }\subseteq \Im$. Assume that $\phi :\mathrm{\Gamma }\to C\left(\mathrm{\Lambda }\right)$ is a mapping and $\mathrm{\Psi }\in {\mathcal{F}}^{\prime }$. Then, we say that φ is $KW$-type F-contraction mapping if there are $\sigma \ge 0$ and $\tau :(0,\infty )\to (\sigma ,\infty )$, such that, for all $s>0$

$$\underset{\ell \to s^{+}}{lim}inf\tau (\ell )>\sigma ,$$

(2)

and there exists $z\in {\mathrm{\Gamma }}_0$ for all $\check{s}\in {\mathrm{\Gamma }}_0$ with $\rho (\check{s},\phi \check{s})>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$ and $\breve{u}\in \phi \check{s}$satisfying

$$\rho (z,\breve{u})=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$$

and

$$\tau \left(\rho (\check{s},z)\right)+\mathrm{\Psi }\left(\rho (\breve{u},\phi z)\right)\le \mathrm{\Psi }\left(\rho (\check{s},z)\right).$$

The following is the main result of this study:

Theorem 6. 

Let $(\Im ,\rho )$ be a complete metric space, $\phi :\mathrm{\Gamma }\to C\left(\mathrm{\Lambda }\right)$ be a $KW$-type F-contraction mapping, where $\mathrm{\Gamma },\mathrm{\Lambda }$ are closed non-empty subsets having a modified P-property, ${\mathrm{\Gamma }}_0\ne \varnothing$ and $\mathrm{\Psi }\in {}_{\phi }^{\rho }{\mathcal{F}}_{**}$. Then, φ has a best proximity point in Γ provided that $f(\check{s},\breve{u})=\rho (\breve{u},\phi \check{s})$ is lower semi-continuous on Δ, where

$$\Delta =\{(\check{s},\breve{u})\in \mathrm{\Gamma }\times \mathrm{\Lambda }:\rho (\check{s},\breve{u})=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\}.$$

Proof. 

Let ${\check{s}}_0$ be an arbitrary point in ${\mathrm{\Gamma }}_0$ and choose ${\breve{u}}_0\in \phi {\check{s}}_0.$ If $\rho ({\check{s}}_0,\phi {\check{s}}_0)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$, then ${\check{s}}_0$ is a best proximity point of $\phi$, and so the proof is completed. Now, assume $\rho ({\check{s}}_0,\phi {\check{s}}_0)>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$. Since $\phi$ is a $KW$-type F-contraction mapping, there is ${\check{s}}_1\in {\mathrm{\Gamma }}_0$ satisfying

$$\rho ({\check{s}}_1,{\breve{u}}_0)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$$

and

$$\tau \left(\rho ({\check{s}}_0,{\check{s}}_1)\right)+\mathrm{\Psi }\left(\rho ({\breve{u}}_0,\phi {\check{s}}_1)\right)\le \mathrm{\Psi }\left(\rho ({\check{s}}_0,{\check{s}}_1)\right).$$

(3)

Hence, we get ${\check{s}}_0\ne {\check{s}}_1$. Since $\mathrm{\Psi }\in {}_{\phi }^{\rho }{\mathcal{F}}_{**}$, we get ${\mathrm{\Psi }}_{\sigma ,\phi }^{{\check{s}}_1,{\breve{u}}_0}\ne \varnothing$. So, there exists ${\breve{u}}_1\in \phi {\check{s}}_1$ satisfying

$$\mathrm{\Psi }\left(\rho ({\breve{u}}_0,{\breve{u}}_1)\right)\le \mathrm{\Psi }\left(\rho ({\breve{u}}_0,\phi {\check{s}}_1)\right)+\sigma .$$

If $\rho ({\check{s}}_1,\phi {\check{s}}_1)=\rho (\mathrm{\Gamma },\mathrm{\Lambda }),$ then ${\check{s}}_1$ is a best proximity point of $\phi$, and so the proof is completed. Now, assume $\rho ({\check{s}}_1,\phi {\check{s}}_1)>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$. Then, there is ${\check{s}}_2\in {\mathrm{\Gamma }}_0$ satisfying

$$\rho ({\check{s}}_2,{\breve{u}}_1)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$$

and

$$\tau \left(\rho ({\check{s}}_1,{\check{s}}_2)\right)+\mathrm{\Psi }\left(\rho ({\breve{u}}_1,\phi {\check{s}}_2)\right)\le \mathrm{\Psi }\left(\rho ({\check{s}}_1,{\check{s}}_2)\right).$$

(4)

Hence, we get ${\check{s}}_1\ne {\check{s}}_2$. Since $\mathrm{\Psi }\in {}_{\phi }^{\rho }{\mathcal{F}}_{**}$, we get ${\mathrm{\Psi }}_{\sigma ,\phi }^{{\check{s}}_2,{\breve{u}}_1}\ne \varnothing$. So, there exists ${\breve{u}}_2\in \phi {\check{s}}_2$ satisfying

$$\mathrm{\Psi }\left(\rho ({\breve{u}}_1,{\breve{u}}_2)\right)\le \mathrm{\Psi }\left(\rho ({\breve{u}}_1,\phi {\check{s}}_2)\right)+\sigma .$$

By continuing, one can find two sequences $\left\{{\check{s}}_t\right\}$ in $\mathrm{\Gamma }$ and $\left\{{\breve{u}}_t\right\}$ in $\mathrm{\Lambda }$ with ${\check{s}}_t\ne {\check{s}}_{t+1}$ and ${\breve{u}}_t\ne {\breve{u}}_{t+1}$ satisfying

$$\rho ({\check{s}}_{t+1},{\breve{u}}_t)=\rho (\mathrm{\Gamma },\mathrm{\Lambda }),$$

(5)

$$\tau \left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)+\mathrm{\Psi }\left(\rho ({\breve{u}}_t,\phi {\check{s}}_{t+1})\right)\le \mathrm{\Psi }\left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right),$$

(6)

and

$$\mathrm{\Psi }\left(\rho ({\breve{u}}_t,{\breve{u}}_{t+1})\right)\le \mathrm{\Psi }\left(\rho ({\breve{u}}_t,\phi {\check{s}}_{t+1})\right)+\sigma .$$

(7)

for all $t\ge 0$. Further, since the pair $(\mathrm{\Gamma },\mathrm{\Lambda })$ has the modified P-property, then, from (5), we get

$$\rho ({\check{s}}_t,{\check{s}}_{t+1})=\rho ({\breve{u}}_{t-1},{\breve{u}}_t)$$

(8)

for all $t\ge 1$. Thus, using (6) and (7), we obtain

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)& =& \mathrm{\Psi }\left(\rho ({\breve{u}}_{t-1},{\breve{u}}_t)\right)\hfill \\ & \le & \mathrm{\Psi }\left(\rho ({\breve{u}}_{t-1},\phi {\check{s}}_t)\right)+\sigma \hfill \\ & \le & \mathrm{\Psi }\left(\rho ({\check{s}}_{t-1},{\check{s}}_t)\right)+\sigma -\tau \left(\rho ({\check{s}}_{t-1},{\check{s}}_t)\right)\hfill \\ & <& \mathrm{\Psi }\left(\rho ({\check{s}}_{t-1},{\check{s}}_t)\right)\hfill \end{array}$$

(9)

for all $t\ge 1$. Hence, the sequence $\left\{\rho ({\check{s}}_t,{\check{s}}_{t+1})\right\}$ is decreasing. Then, $\left\{\rho ({\check{s}}_t,{\check{s}}_{t+1})\right\}$ converges for some $u\in [0,\infty )$. Assume that $u>0$. Then, from the hypothesis, we obtain

$$\underset{t\to \infty }{lim}inf\tau \left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)>\sigma ,$$

and so there is $\eta \in \mathbb{R}$ with $\eta >\sigma$ and $t_0\in \mathbb{N}$ satisfying

$$\tau \left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)>\eta$$

for all $t\ge t_0$. Therefore, for all $t\ge t_0$, we have

$$\begin{array}{ccc}\hfill \mathrm{\Psi }\left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)& =& \mathrm{\Psi }\left(\rho ({\breve{u}}_{t-1},{\breve{u}}_t)\right)\hfill \\ & \le & \mathrm{\Psi }\left(\rho ({\check{s}}_{t-1},{\check{s}}_t)\right)+\sigma -\tau \left(\rho ({\check{s}}_{t-1},{\check{s}}_t)\right)\hfill \\ & <& \mathrm{\Psi }\left(\rho ({\check{s}}_{t-1},{\check{s}}_t)\right)+\left(\sigma -\eta \right)\hfill \\ & \le & \mathrm{\Psi }\left(\rho ({\check{s}}_{t-2},{\check{s}}_{t-1})\right)+\sigma -\tau \left(\rho ({\check{s}}_{t-2},{\check{s}}_{t-1})\right)+\left(\sigma -\eta \right)\hfill \\ & <& \mathrm{\Psi }\left(\rho ({\check{s}}_{t-2},{\check{s}}_{t-1})\right)+2\left(\sigma -\eta \right)\hfill \\ & & ⋮\hfill \\ & <& \mathrm{\Psi }\left(\rho ({\check{s}}_{t_0},{\check{s}}_{t_0+1})\right)+\left(t-t_0\right)\left(\sigma -\eta \right).\hfill \end{array}$$

(10)

Taking the limit as $t\to \infty$ in (10), we have

$$\underset{t\to \infty }{lim}\mathrm{\Psi }\left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)=-\infty ,$$

and so, from (F${}_2^{\prime }$) and Lemma 1, we get

$$u=\underset{t\to \infty }{lim}\rho ({\check{s}}_t,{\check{s}}_{t+1})=0$$

which is a contradiction. Therefore, we get

$$\underset{t\to \infty }{lim}\rho ({\check{s}}_t,{\check{s}}_{t+1})=0.$$

From the condition (F${}_3$), there exists $k\in (0,1)$ satisfying

$$\underset{t\to \infty }{lim}\rho {({\check{s}}_t,{\check{s}}_{t+1})}^k\mathrm{\Psi }\left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)=0.$$

From (10), we have

$$\left\{\begin{array}{c}\rho {({\check{s}}_t,{\check{s}}_{t+1})}^k\mathrm{\Psi }\left(\rho ({\check{s}}_t,{\check{s}}_{t+1})\right)-\\ \rho {({\check{s}}_t,{\check{s}}_{t+1})}^k\mathrm{\Psi }\left(\rho ({\check{s}}_{t_0},{\check{s}}_{t_0+1})\right)\end{array}\right\}\le \rho {({\check{s}}_t,{\check{s}}_{t+1})}^{k}t\left(\sigma -\eta \right)\le 0$$

for all $t\ge t_0$. Hence, we get

$$t{\left[\rho ({\check{s}}_t,{\check{s}}_{t+1})\right]}^k\to 0\mathrm{as}t\to \infty .$$

(11)

Therefore, there exists $t_1\in \mathbb{N}$ satisfying

$$t{\left[\rho ({\check{s}}_t,{\check{s}}_{t+1})\right]}^k\le 1$$

for all $t\ge t_1$, which implies that

$$\rho ({\check{s}}_t,{\check{s}}_{t+1})\le \frac{1}{t^{1/k}}$$

for all $t\ge t_1$. Now, let $t,m\in \mathbb{N}$ with $m>t\ge t_1$. Hence, we get

$$\begin{array}{ccc}\hfill \rho ({\check{s}}_t,{\check{s}}_m)& \le & \rho ({\check{s}}_t,{\check{s}}_{t+1})+\rho ({\check{s}}_{t+1},{\check{s}}_{t+2})+\cdots +\rho ({\check{s}}_{m-1},{\check{s}}_m)\hfill \\ & \le & \frac{1}{t^{1/k}}+\frac{1}{{(t+1)}^{1/k}}+\cdots +\frac{1}{{(m-1)}^{1/k}}\hfill \\ & \le & \sum _{i=t}^{m-1}\frac{1}{i^{1/k}}\hfill \\ & \le & \sum _{i=t}^{\infty }\frac{1}{i^{1/k}}.\hfill \end{array}$$

Since ${\displaystyle \sum _{i=1}^{\infty }}\frac{1}{i^{1/k}}<\infty$, $\left\{{\check{s}}_t\right\}$ is a Cauchy sequence in $\mathrm{\Gamma }$. From (8), $\left\{{\breve{u}}_t\right\}$ is also a Cauchy sequence in $\mathrm{\Lambda }$. Since $\mathrm{\Gamma }$ and $\mathrm{\Lambda }$ are closed subsets of the complete metric space $(\Im ,\rho )$, there exist ${\check{s}}^{*}\in \mathrm{\Gamma }$ and ${\breve{u}}^{*}\in \mathrm{\Lambda }$ satisfying ${\check{s}}_t\to {\check{s}}^{*}$ and ${\breve{u}}_t\to {\breve{u}}^{*}$ as $t\to \infty$. Taking the limit as $t\to \infty$ in (5), we have

$$\rho ({\check{s}}^{*},{\breve{u}}^{*})=\rho (\mathrm{\Gamma },\mathrm{\Lambda }),$$

(12)

and so, we get $({\check{s}}^{*},{\breve{u}}^{*})\in \Delta$. Moreover, from (9), (F${}_2^{\prime }$) and Lemma 1, we get

$$\underset{t\to \infty }{lim}\mathrm{\Psi }\left(\rho ({\breve{u}}_t,\phi {\check{s}}_{t+1})\right)=-\infty ,$$

and so

$$\underset{t\to \infty }{lim}\rho ({\breve{u}}_t,\phi {\check{s}}_{t+1})=0.$$

(13)

Now, since the function $f(\check{s},\breve{u})=\rho (\breve{u},\phi \check{s})$ is lower semi-continuous on $\Delta$, ${\check{s}}_{t+1}\to {\check{s}}^{*}$ and ${\breve{u}}_t\to {\breve{u}}^{*}$ as $t\to \infty$, from (13), we obtain

$$\rho ({\breve{u}}^{*},\phi {\check{s}}^{*})=f({\check{s}}^{*},{\breve{u}}^{*})\le \underset{t\to \infty }{lim}inff({\check{s}}_{t+1},{\breve{u}}_t)=\underset{t\to \infty }{lim}inf\rho ({\breve{u}}_t,\phi {\check{s}}_{t+1})=0,$$

and so, ${\breve{u}}^{*}\in \overline{\phi {\check{s}}^{*}}=\phi {\check{s}}^{*}$. Therefore, from (12), we have

$$\rho (\mathrm{\Gamma },\mathrm{\Lambda })\le \rho ({\check{s}}^{*},\phi {\check{s}}^{*})\le \rho ({\check{s}}^{*},{\breve{u}}^{*})=\rho (\mathrm{\Gamma },\mathrm{\Lambda }).$$

So, we get

$$\rho ({\check{s}}^{*},\phi {\check{s}}^{*})=\rho (\mathrm{\Gamma },\mathrm{\Lambda }),$$

that is, $\stackrel{ˇ\mathrm{*}}{s}$ is a best proximity point of $\phi$. □

We obtain the generalized result below by using Theorem 6. The result generalizes the main result of [25].

Corollary 1. 

Let $(\Im ,\rho )$ be a complete metric space and $\phi :\mathrm{\Gamma }\to C\left(\mathrm{\Lambda }\right)$ be a mapping, where $\mathrm{\Gamma },\mathrm{\Lambda }$ are closed non-empty subsets having a modified P-property. Assume that ${\mathrm{\Gamma }}_0\ne \varnothing$, $\tau >0$ and $\mathrm{\Psi }\in {}_{\phi }^{\rho }{\mathcal{F}}_{**}$. If, for all $\check{s}\in {\mathrm{\Gamma }}_0$ with $\rho (\check{s},\phi \check{s})>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$ and $\breve{u}\in \phi \check{s}$, there exists $z\in {\mathrm{\Gamma }}_0$, such that

$$\rho (z,\breve{u})=\rho (\mathrm{\Gamma },\mathrm{\Lambda }),$$

and

$$\tau +\mathrm{\Psi }\left(\rho (\breve{u},\phi z)\right)\le \mathrm{\Psi }\left(\rho (\check{s},z)\right)$$

then, φ has a best proximity point in Γ provided that $f(\check{s},\breve{u})=\rho (\breve{u},\phi \check{s})$ is lower semi-continuous on $\Delta .$

Proof. 

To show the existence by Theorem 6, it is enough to demonstrate that the contraction condition of Theorem 6 is satisfied. Now, let $\check{s}\in {\mathrm{\Gamma }}_0$ with $\rho (\check{s},\phi \check{s})>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$ and $\breve{u}\in \phi \check{s}$ be arbitrary points. Using Corollary 1, we say that there exists $z\in {\mathrm{\Gamma }}_0$, such that

$$\rho (\breve{u},z)=\rho (\mathrm{\Gamma },\mathrm{\Lambda }),$$

and

$$\tau +\mathrm{\Psi }\left(\rho (\breve{u},\phi z)\right)\le \mathrm{\Psi }\left(\rho (\check{s},z)\right).$$

(14)

On the other hand, since $\tau >0$, there exists a real number $\sigma$, such that $0<\sigma <\tau$. Now, if we define the mapping ${\tau }^{\prime }:(0,\infty )\to (\sigma ,\infty )$ by ${\tau }^{\prime }(\ell )=\tau$ for all $\ell >0$, then the inequality (2) is satisfied. In addition, from (14), we have

$${\tau }^{\prime }\left(\rho (\check{s},z)\right)+\mathrm{\Psi }\left(\rho (\breve{u},\phi z)\right)\le \mathrm{\Psi }\left(\rho (\check{s},z)\right),$$

and so, from Theorem 6, the mapping $\phi$ has a best proximity point in $\mathrm{\Gamma }.$ □

Theorem 6 is a proper generalization of Corollary 1, as demonstrated by the example that follows.

Example 1. 

Let $\Im ={\mathbb{R}}^2$ and the function $\rho :\Im \times \Im \to \mathbb{R}$ be the taxi-cab metric; that is, for all $\check{s}=({\check{s}}_1,{\check{s}}_2)$, $\breve{u}=({\breve{u}}_1,{\breve{u}}_2)$$\in \Im$, $\rho (\check{s},\breve{u})=|{\check{s}}_1-{\breve{u}}_1|+|{\check{s}}_2-{\breve{u}}_2|.$ If we take the subsets

$$\mathrm{\Gamma }=\left\{0,\frac{1}{\gamma }:\gamma \in \mathbb{N}\right\}\times \left\{0\right\}$$

and

$$\mathrm{\Lambda }=\left\{0,\frac{1}{\gamma }:\gamma \in \mathbb{N}\right\}\times \left\{1\right\}$$

then, Γ and Λ are closed subsets of the complete metric space $(\Im ,\rho )$. Moreover, $\rho (\mathrm{\Gamma },\mathrm{\Lambda })=1,{\mathrm{\Gamma }}_0=\mathrm{\Gamma }$, ${\mathrm{\Lambda }}_0=\mathrm{\Lambda }$ and the pair $(\mathrm{\Gamma },\mathrm{\Lambda })$ has the modified P-property. Consider the mappings $\mathrm{\Psi }:{\mathbb{R}}^{+}\to \mathbb{R}$ and $\phi :\mathrm{\Gamma }\to C\left(\mathrm{\Lambda }\right)$ as $\mathrm{\Psi }\left(\alpha \right)=ln\alpha$ and

$$\phi \check{s}=\left\{\begin{array}{ccc}\mathrm{\Lambda }& ,& \check{s}=(0,0)\\ & & \\ \left\{\left(\frac{1}{\gamma +1},1\right)\right\}& ,& \check{s}=\left(\frac{1}{\gamma },0\right),\gamma \in \mathbb{N}\end{array}\right..$$

Choose $\sigma =0$ and define the function $\tau :(0,\infty )\to (0,\infty )$ by

$$\tau (\ell )=\left\{\begin{array}{ccc}ln\left(\frac{\gamma +1}{\gamma }\right)& ,& \ell =\frac{1}{\gamma (\gamma +1)},\gamma \in \mathbb{N}\\ & & \\ 1& ,& \mathrm{otherwise}\end{array}\right..$$

Then, we have $\mathrm{\Psi }\in {}_{\phi }^{\rho }{\mathcal{F}}_{**}$ and ${lim}_{\ell \to s^{+}}inf\tau (\ell )=1>0=\sigma$ for all $s>0$. We also have $f(\check{s},\breve{u})=\rho (\breve{u},\phi \check{s})$ is lower semi-continuous on Δ. Now, we indicate that φ is a $KW$-type F-contraction mapping. Let $\check{s}=\left(\frac{1}{\gamma },0\right)$ for arbitrary $\gamma \in \mathbb{N}$ and choose $\breve{u}\in \left\{\left(\frac{1}{\gamma +1},1\right)\right\}=\phi \check{s}$. Then, it has to be $z=\left(\frac{1}{\gamma +1},0\right),$ and so $\phi z=\left\{\left(\frac{1}{\gamma +2},1\right)\right\}$. In this case, it is satisfied

$$\begin{array}{ccc}\hfill \tau \left(\frac{1}{\gamma (\gamma +1)}\right)+\mathrm{\Psi }\left(\frac{1}{(\gamma +1)(\gamma +2)}\right)& =& ln\left(\frac{\gamma +1}{\gamma }\right)+ln\left(\frac{1}{(\gamma +1)(\gamma +2)}\right)\hfill \\ & =& ln\left(\frac{1}{\gamma (\gamma +2)}\right)\hfill \\ & \le & ln\left(\frac{1}{\gamma (\gamma +1)}\right)\hfill \\ & =& \mathrm{\Psi }\left(\frac{1}{\gamma (\gamma +1)}\right)\hfill \end{array}$$

for all $\gamma \in \mathbb{N}$. Hence, all conditions of Theorem 6 hold; so, φ has a best proximity point in Γ. However, Theorem 1 is not applied to this example because the contraction condition of Corollary 1 is not satisfied. Assume the contrary, that is, there exists $\tau >0$, such that, for all $\check{s}\in {\mathrm{\Gamma }}_0$ with $\rho (\check{s},\phi \check{s})>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$ and $\breve{u}\in \phi \check{s}$, there exists $z\in {\mathrm{\Gamma }}_0$, such that

$$\rho (\breve{u},z)=\rho (\mathrm{\Gamma },\mathrm{\Lambda }),$$

and

$$\tau +\mathrm{\Psi }\left(\rho (\breve{u},\phi z)\right)\le \mathrm{\Psi }\left(\rho (\check{s},z)\right).$$

In this case, for $\check{s}=\left(\frac{1}{\gamma },0\right)$, $\gamma \in \mathbb{N}$ and choose $\breve{u}\in \left\{\left(\frac{1}{\gamma +1},1\right)\right\}=\phi \check{s}$. Then, it has to be $z=\left(\frac{1}{\gamma +1},0\right),$ and so $\phi z=\left\{\left(\frac{1}{\gamma +2},1\right)\right\}$. In this case, it is satisfied

$$\tau +\mathrm{\Psi }\left(\frac{1}{(\gamma +1)(\gamma +2)}\right)\le \mathrm{\Psi }\left(\frac{1}{\gamma (\gamma +1)}\right),$$

which implies that

$$\tau \le ln\left(\frac{\gamma +2}{\gamma }\right)$$

for all $\gamma \in \mathbb{N}$. Taking the limit as $\gamma \to \infty$, we have

$$\tau \le 0$$

which is a contradiction.

If we define the function $\phi :[0,\infty )⟶[0,1)$ by

$$\phi \left(\ell \right)=\left\{\begin{array}{ccc}0& ,& \ell =0\\ & & \\ e^{-\tau (\ell )}& ,& \ell >0\end{array}\right.$$

via the function $\tau$ in Theorem 6, then we have ${lim}_{\ell \to s^{+}}sup\phi (\ell )<1$ for all $s>0$. Taking into account the function $\mathrm{\Psi }:{\mathbb{R}}^{+}\to \mathbb{R}$ by $\mathrm{\Psi }\left(\alpha \right)=ln\alpha$ in Theorem 6, we also get, for all $\check{s}\in {\mathrm{\Gamma }}_0$ and $\breve{u}\in \phi \check{s}$, there exists $z\in {\mathrm{\Gamma }}_0$satisfying

$$\rho (\breve{u},z)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$$

and

$$\rho (\breve{u},\phi z)\le \phi \left(\rho (\check{s},z)\right)\rho (\check{s},z).$$

Therefore, we can obtain the following best proximity point version of the fixed-point result of Klim and Wardowski with the help of Theorem 6.

Corollary 2. 

Let $(\Im ,\rho )$ be a complete metric space, $\phi :\mathrm{\Gamma }\to C\left(\mathrm{\Lambda }\right)$ be a mapping where $\mathrm{\Gamma },\mathrm{\Lambda }$ are closed non-empty subsets having a modified P-property and ${\mathrm{\Gamma }}_0\ne \varnothing$. If there exist $\phi :[0,\infty )⟶[0,1)$, such that, for all $s>0$

$$\underset{\ell \to s^{+}}{lim}sup\phi (\ell )<1$$

(15)

and there exists $z\in {\mathrm{\Gamma }}_0$for all $\check{s}\in {\mathrm{\Gamma }}_0$ and $\breve{u}\in \phi \check{s}$ satisfying

$$\rho (\breve{u},z)=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$$

and

$$\rho (\breve{u},\phi z)\le \phi \left(\rho (\check{s},z)\right)\rho (\check{s},z),$$

and $f(\check{s},\breve{u})=\rho (\breve{u},\phi \check{s})$ is lower semi-continuous on Δ, then φ has a best proximity point in Γ.

When $\mathrm{\Gamma }=\mathrm{\Lambda }=\Im$ in Theorem 6, we conclude the following fixed-point theorem which is a generalization of the main result of [27].

Corollary 3. 

Let $\phi :\Im \to C(\Im )$ be a multivalued mapping on a complete metric space $(\Im ,\rho )$ and $\mathrm{\Psi }\in {}_{\phi }^{\rho }{\mathcal{F}}_{**}$. If there exist $\sigma >0$ and $\tau :(0,\infty )\to (\sigma ,\infty )$, such that, for all $s>0$

$$\underset{\ell \to s^{+}}{lim}inf\tau (\ell )>\sigma$$

and, for all $\check{s}\in \Im$ with $\rho (\check{s},\phi \check{s})>0$ and $\breve{u}\in \phi \check{s}$, it is satisfied

$$\tau \left(\rho (\check{s},\breve{u})\right)+\mathrm{\Psi }\left(\rho (\breve{u},\phi \breve{u})\right)\le \mathrm{\Psi }\left(\rho (\check{s},\breve{u})\right).$$

Then, φ has a fixed point in ℑ, provided that $f(\check{s},\check{s})=\rho (\check{s},\phi \check{s})$ is lower semi-continuous.

3. Homotopy Result

Recently, there has been a new trend towards interest in homotopy theory due to the close relation between several branches of mathematics and homotopy theory. In this sense, many mathematicians have produced applications of homotopy using their fixed-point results [28,29,30]. Taking into account this approach, we prove a best proximity point result for the homotopy. Now, we recall some basic concepts of this theory:

Definition 3. 

Let $({\Im }_1,{\tau }_1)$ and $({\Im }_2,{\tau }_2)$ be topological spaces, and ${\psi }_1,{\psi }_2:{\Im }_1\to {\Im }_2$ be continuous mappings. Then, the mapping ψ is called homotopy if there exists a continuous function $\psi :{\Im }_1\times [0,1]\to {\Im }_2$, such that $\psi (\check{s},0)={\psi }_1\check{s}$ and $\psi (\check{s},1)={\psi }_2\check{s}$, for all $\check{s}\in {\Im }_1$. Moreover, ${\psi }_1$ and ${\psi }_2$ are called homotopic mappings.

The authors give the following definition in [31].

Definition 4. 

Let Γ be a non-empty subset of a metric space $(\Im ,\rho )$ and $\psi :\mathrm{\Gamma }\times [0,1]\to C\left(\mathrm{\Gamma }\right)$ be a mapping. If $G\left(\psi \right)$ is a closed subset of $(\mathrm{\Gamma }\times [0,1]\times \mathrm{\Gamma },{\rho }^{*})$, then ψ is called a closed multivalued mapping, where

$$G\left(\psi \right)=\{(\check{s},\gamma ,\breve{u}):\check{s}\in \mathrm{\Gamma },\gamma \in [0,1]\mathrm{and}\breve{u}\in \psi (\check{s},\gamma )\}$$

and

$${\rho }^{*}(({\check{s}}_1,{\gamma }_1,{\breve{u}}_1),({\check{s}}_2,{\gamma }_2,{\breve{u}}_2))=\rho ({\check{s}}_1,{\check{s}}_2)+\left|{\gamma }_1-{\gamma }_2\right|+\rho ({\breve{u}}_1,{\breve{u}}_2)$$

for all $({\check{s}}_1,{\gamma }_1,{\breve{u}}_1),({\check{s}}_2,{\gamma }_2,{\breve{u}}_2)\in \mathrm{\Gamma }\times [0,1]\times \mathrm{\Gamma }$.

Considering the best proximity point theory and Definition 4, Sahin [25] gave the following definition:

Definition 5. 

Let $\varnothing \ne \mathrm{\Gamma },\mathrm{\Lambda }\subseteq \Im$, where $(\Im ,\rho )$ is a metric space, and $\psi :\mathrm{\Gamma }\times [0,1]\to C\left(\mathrm{\Lambda }\right)$ be a multivalued mapping. Then, ψ is said to be a ρ-closed multivalued mapping if

$$G_{\rho }\left(\psi \right)=\{(\check{s},\gamma ,\breve{u}):\check{s},\breve{u}\in Y\mathrm{and}\gamma \in [0,1]\mathrm{with}\rho (\breve{u},\psi (\check{s},\gamma )=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\}$$

is a closed subset of $(\mathrm{\Gamma }\times [0,1]\times \mathrm{\Gamma },{\rho }^{*})$.

Notice, if we take $\mathrm{\Gamma }=\mathrm{\Lambda }$ in Definition 5, then Definition 5 turns to Definition 4. The following result is the main result of this section:

Theorem 7. 

Let $(\Im ,\rho )$ be a complete metric space, $\varnothing \ne \mathrm{\Gamma },\mathrm{\Lambda }\subseteq \Im$, where $\mathrm{\Gamma },\mathrm{\Lambda }$ are closed, $\varnothing \ne U\subseteq \mathrm{\Gamma }$ and ${\mathrm{\Gamma }}_0\ne \varnothing .$ Suppose that the pair $(\mathrm{\Gamma },\mathrm{\Lambda })$ has the modified P-property, $\psi :\mathrm{\Gamma }\times [0,1]\to C\left(\mathrm{\Lambda }\right)$ is ρ-closed multivalued mapping and $\mathrm{\Psi }\in {}_{\psi (.,\gamma )}^{\rho }{\mathcal{F}}_{**}\mathrm{for}\mathrm{all}\gamma \in [0,1]$. Assume that the following conditions are satisfied:

(i) 

$\rho (\check{s},\psi (\check{s},\gamma ))>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$ for all $\check{s}\in \mathrm{\Gamma }\setminus U$ and $\gamma \in [0,1]$,

(ii) 

there exist $\sigma \ge 0$ and $\tau :(0,\infty )\to (\sigma ,\infty )$, such that, for all $s>0$

$$\underset{\ell \to s^{+}}{lim}inf\tau (\ell )>\sigma$$

(16)

and for all $\overline{\check{s}}\in {\mathrm{\Gamma }}_0$, $m,\gamma \in [0,1]$ and for all $\check{s}\in \overline{B}(\overline{\check{s}},m)=\{\check{s}\in \mathrm{\Gamma }:\rho (\check{s},\overline{\check{s}})\le m\}$ with $\rho (\check{s},\psi (\check{s},\gamma ))>\rho (\mathrm{\Gamma },\mathrm{\Lambda })$ and $\breve{u}\in \psi (\check{s},\gamma )$, there is $\zeta \in \overline{B}(\overline{\check{s}},m)$, such that

$$\rho (\breve{u},\zeta )=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$$

and

$$\tau +\mathrm{\Psi }\left(\rho (\breve{u},\psi (\zeta ,\gamma ))\right)\le \mathrm{\Psi }\left(\rho (\check{s},\zeta )\right).$$

(iii) 

for all $\gamma \in [0,1]$, $f(\check{s},\breve{u})=\rho (\breve{u},\psi (\check{s},\gamma ))$ is lower semi-continuous on Δ.

Then, $\psi (·,1)$ has a best proximity point in Γ, if $\psi (·,0)$ has a best proximity point in Γ.

Proof. 

Define the following set

$$H=\left\{(\gamma ,\check{s}):\rho (\check{s},\psi (\check{s},\gamma ))=\rho (\mathrm{\Gamma },\mathrm{\Lambda })\right\}\subseteq [0,1]\times U.$$

Since there is a point $\check{s}$ in U such that $\rho (\check{s},\psi (\check{s},0))=\rho (\mathrm{\Gamma },\mathrm{\Lambda })$, we get $H\ne \varnothing$. Now, consider the following partial order on H

$$(\gamma ,\check{s})⪯(\mu ,\breve{u})\iff \gamma \le \mu \mathrm{and}\rho (\check{s},\breve{u})\le \mu -\gamma .$$

Assume that K is a totally ordered subset of H and ${\gamma }^{*}=sup\{\gamma :(\gamma ,\check{s})\in K\}$. Consider $\left\{({\gamma }_n,{\check{s}}_n)\right\}\subseteq K$ with $({\gamma }_n,{\check{s}}_n)⪯$$({\gamma }_{n+1},{\check{s}}_{n+1})$ for all $n\in \mathbb{N}$ and ${\gamma }_n\to {\gamma }^{*}$ as $n\to \infty .$ So, we get

$$\rho ({\check{s}}_n,{\check{s}}_m)\le {\gamma }_m-{\gamma }_n,$$

for all $n,m\in \mathbb{N}$ with $m>n$. Therefore, we get $\left\{{\check{s}}_n\right\}\subseteq U$ is a Cauchy sequence. There exists ${\check{s}}^{*}\in \mathrm{\Gamma }$, such that ${\check{s}}_n\to {\check{s}}^{*}$ as $n\to \infty$, since $\mathrm{\Gamma }$ is a closed subset of complete metric space $(\mathrm{\Lambda },\rho )$. In addition, we have $({\check{s}}_n,{\gamma }_n,{\check{s}}_n)\in G_{\rho }\left(\psi \right)$ and $({\check{s}}_n,{\gamma }_n,{\check{s}}_n)\stackrel{{\rho }^{*}}{\to }({\check{s}}^{*},{\gamma }^{*},{\check{s}}^{*})$ as $n\to \infty$. Since $\psi$ is a closed multivalued mapping, we get

$$\rho ({\check{s}}^{*},\psi ({\check{s}}^{*},{\gamma }^{*}))=\rho (\mathrm{\Gamma },\mathrm{\Lambda }).$$

From (i), we have ${\check{s}}_n\in U,$ and so $({\gamma }^{*},{\check{s}}_n)\in H$. It is satisfied $(\gamma ,\check{s})⪯({\gamma }^{*},{\check{s}}_n)$ for all $(\gamma ,\check{s})\in K$, since K is totally ordered. Hence, K has an upper bound $({\gamma }^{*},{\check{s}}_n)$. Therefore, we conclude H has a maximal element $({\gamma }_0,{\check{s}}_0)\in H$ using the Zorn Lemma. We claim that ${\gamma }_0=1$. If we suppose the contrary, then there is $\gamma$ with ${\gamma }_0<\gamma <1$. Let $R=\gamma -{\gamma }_0>0$. Hence, considering (ii) and (iii), we say that $f:\overline{B}({\check{s}}_0,M)\times \mathrm{\Lambda }\to [0,\infty )$ defined by $f(\check{s},\breve{u})=\rho (\breve{u},\psi (\check{s},\gamma ))$ is lower semi-continuous on $\Delta$ and the mapping $\psi (·,\gamma ):\overline{B}({\check{s}}_0,M)\to C\left(\mathrm{\Lambda }\right)$ is a $KW$-type F-contraction mapping. Therefore, we conclude that $\psi (·,\gamma )$ has a best proximity point ${\check{s}}_{\gamma }$ in $\overline{B}({\check{s}}_0,M)$ using Theorem 6. From (i), ${\check{s}}_{\gamma }\in U,$ and so, $(\gamma ,{\check{s}}_{\gamma })\in H$, which contradicts that $({\gamma }_0,{\check{s}}_0)$ is a maximal element of H. So, ${\gamma }_0=1$ and $\psi (·,1)$ has a best proximity point ${\check{s}}_0$ in $\mathrm{\Gamma }$. □

4. Conclusions

In this paper, we sought to combine the approaches of Wardowski and Klim–Wardowski. Hence, we first introduced a new type of F-contraction, named $KW$-type F-contraction mapping. Then, we investigated the conditions of existence of a best proximity point for such mappings by considering a new family ${}_{\phi }^{\rho }{\mathcal{F}}_{**}$which was larger than the family of functions ${\mathcal{F}}_{*}$. Moreover, we provided an example where our results can be applied, but to which the results in the literature cannot be applied. Finally, considering a new trend towards homotopy theory, we present an application for homotopic mappings.