Best proximity point of $\alpha$-$\eta$-type generalized $F$-proximal contractions in modular metric spaces

1.   Introduction

The first attempts to generalize the classical function spaces of the Lebesgue type $L_{p}$ were made by Birnhaum and Orlicz in 1931 ^[1]. The more abstract generalization was established by Nakano ^[2] in 1950 and refined and generalized by Musielak and Orlicz ^[3] in 1959 under the name of modular and modular spaces. Lately, Chistyakov ^[4,5] developed modular spaces and metric spaces by introducing modular metric spaces (or metric modular spaces). The main idea behind this new concept is physical interpretation of the modular metric spaces ^[6]. Here, we look at modular metric spaces as the nonlinear version of the classical modular spaces as introduced by Nakano ^[7], on the vector spaces and modular function spaces introduced by Musielack ^[8] and Orlicz ^[9]. It is worth noting that there is another similar generalization, i.e., $(q_{1}, q_{2})$-quasimetric spaces, which were recently introduced and studied by Arutyunov and Greshnov in ^[10,11,12].

Over the past hundred years, fixed-point theory, as one of the centers of mathematical analysis, has been used in many different fields of mathematics such as topology, analysis and operator theory; see ^{[13,14,15,16,17,18]}. Let $A$ be a non-empty subset of a metric space $(X, d)$ and $T:A \rightarrow A$ be a self-mapping. A point $x\in A$ is said to be a fixed point of $T$ if $Tx = x$. However, in many practical applications, $T$ does not satisfy the condition for a self-mapping. In other words, the mapping $T : A \rightarrow B$ $(A \cap B = \emptyset)$ does not have any fixed point. In this case, it is quite natural to investigate an element $x\in X$ such that $d(x, Tx)$ is minimized. In 2010, Basha ^[19] introduced the notion for a best proximity point of non-self mappings. Let $A, B$ be two non-empty subsets of a metric space $(X, d)$ and $T:A \rightarrow B$ be a non-self mapping. A point $x\in A$ satisfying that $d(x, Tx) = d(A, B)$ is called a best proximity point of the non-self mapping $T$. If $A \cap B \ne \emptyset$, then the best proximity point becomes a fixed point of $T$.

Recently, Beg et al. ^[20] introduced a new type of generalized $F$-proximal contractions and investigated the unique best proximity point of generalized $F$-proximal contractions on a complete metric space. Motivated by the recent results, in this paper, we investigate $\alpha$-$\eta$-type generalized $F$-proximal contractions of the first and second kind in the context of modular metric spaces by focusing on the uniform approximation property of the set. Then, we state several best proximity point theorems for some proximal contractions in modular metric spaces. Some examples are given to demonstrate our theoretical results. Moreover, we give an application of our main results to establish the existence of the solution of a non-linear integral equation.

2.   Preliminaries

Throughout this paper, $\mathbb{N}$ and $\mathbb{R}$ denote the sets of positive integers and real numbers respectively. We write ${\mathbb{N}_0} = \mathbb{N } \cup \{ 0\}$. First, we recall some prerequisites.

Let $X$ be a non-empty set. For any $x, y \in X$, we also write ${w_\lambda }(x, y): = w(\lambda, x, y)$ for all $\lambda > 0$ and $w = {\left\{ {{w_\lambda }} \right\}_{\lambda > 0}}$ for which ${w_\lambda }:X \times X \to [0, \infty ]$.

Definition 2.1. ^[4,5] Let $X$ be a non-empty set and $x, y, z \in X$. A function $w:(0, \infty) \times X \times X \to [0, \infty ]$ is said to be modular (metric) on $X$ if it satisfies the following conditions:

(i) ${w_\lambda }(x, y) = 0$ if and only if $x = y$ for all $\lambda > 0$;

(ii) ${w_\lambda }(x, y) = {w_\lambda }(y, x)$ for all $\lambda > 0$;

(iii) ${w_{\lambda + \mu }}(x, y) \le {w_\lambda }(x, z) + {w_\mu }(z, y)$ for all $\lambda, \mu > 0$.

If we utilize the condition

(${i_p}$) ${w_\lambda }(x, x) = 0$ for all $\lambda > 0$,

instead of (i), then $w$ is called pseudomodular on $X$. If $w$ satisfies ${i_p}$ and

(${i_s}$)${w_\lambda }(x, y) = 0$ if and only if $x = y$ for some $\lambda > 0$,

then $w$ is called strictly modular on $X$. If condition (ⅲ) is replaced by

(${i_c}$) ${w_{\lambda + \mu }}(x, y) \le \frac{\lambda }{{\lambda + \mu }}{w_\lambda }(x, z) + \frac{\mu }{{\lambda + \mu }}{w_\mu }(z, y)$, for all $\lambda, \mu > 0$,

then $w$ is called convexly modular on $X$.

Some examples of modular metrics are as follows.

Example 2.1. ^[4] If $(X, d)$ is a metric space for any $x, y \in X$ and $\lambda > 0$, then

is a modular metric and

is a convex modular.

Definition 2.2. ^[4] Let $w$ be pseudomodular on $X$ and ${x_0}$ be a fixed element of $X$. Then the sets

are called modular metric spaces (around ${x_0}$).

Obviously, ${X_w} \subset X_w^*$ holds. If $w$ is a modular metric on $X$, then the modular space $X_w$ can be equipped with a (nontrivial) metric $d_w$, was generated by $w$ and given by

If $w$ is a convex modular metric on $X$, then ${X_w} = X_w^*$ and this common set can be endowed with a metric $d_w^*$ given by

Given $\lambda, r > 0$ and $x \in X_w^*$, set

Definition 2.3. ^[21] A non-empty set $A \subset X$ is said to be $w$-open (or modular open) if, for every $x \in A$ and $\lambda > 0$ there is $r > 0$ (possibly depending on $x$ and $\lambda$) such that ${B_{\lambda, r}} \subset A$.

Denote by $\pi (w)$ the family of all $w$-open subsets of $X_w^*$. Clearly, $\pi (w)$ is a topology on $X_w^*$; see ^[21].

Definition 2.4. ^[6,21] Let ${X_w}$ and $X_w^*$ be modular metric spaces and $\left\{ {{x_n}} \right\}$ be in ${X_w}$ (or $X_w^*$); then,

(1) the sequence $\left\{ {{x_n}} \right\}$ is said to be $w$-convergent to $x \in X$ if and only if ${w_\lambda }({x_n}, x) \to 0$ as $n \to \infty$ for some $\lambda > 0$;

(2) the sequence $\left\{ {{x_n}} \right\}$ is said to be $w$-Cauchy if ${w_\lambda }({x_n}, {x_m}) \to 0$ as $m, n \to \infty$ for some $\lambda > 0$;

(3) a subset $A$ of ${X_w}$ (or $X_w^*$) is said to be $w$-complete if any $w$-Cauchy sequence in $A$ is a $w$-convergent sequence and its $w$-limit lies in $A$.

(4) a subset $A$ of ${X_w}$ (or $X_w^*$) is said to be $w$-closed if the $w$-limit of a $w$-convergent sequence of $A$ always belongs to $A$.

It is easy to see that if $w$ is strict, then we have uniqueness of the $w$-limit. Indeed, If ${x_n} \to x$ and ${x_n} \to y$, then ${w_\lambda }({x_n}, x) \to 0$ and ${w_\lambda }({x_n}, y) \to 0$ for some $\lambda > 0$. By axiom (iii), ${w_{2\lambda }}(x, y) \le {w_\lambda }(x, {x_n}) + {w_\lambda }({x_n}, y)$; thus, ${w_{2\lambda }}(x, y) = 0$. If $w$ is strict, then $x = y$.

Definition 2.5. ^[22] Let $w$ be a modular metric on $X$. We say that $w$ satisfies the Fatou property if

for some $\lambda > 0$ whenever $\left\{ {{x_n}} \right\}$ is $w$-convergent to $x \in X$ and $\left\{ {{y_n}} \right\}$ is $w$-convergent to $y \in X$.

Samet et al.^[23] introduced the notion of $\alpha$-admissible mappings as follows.

Definition 2.6. ^[23] Let $T$ be a self-mapping on $X$ and $\alpha: X\times X \rightarrow [0, \infty)$ be a function. We say that $T$ is an $\alpha$-admissible mapping if

for all $x, y \in X$.

Salimi et al. ^[24] modified the notion of $\alpha$-admissible mappings as follows.

Definition 2.7. ^[24] Let $T$ be a self-mapping on $X$ and $\alpha, \eta : X\times X \rightarrow [0, \infty)$ be two functions. We say that $T$ is an $\alpha$-admissible mapping with respect to $\eta$ if

for all $x, y \in X$.

In 2012, Wardowski ^[25] introduced the concept of an $F$-contraction.

Definition 2.8. ^[25] Let $F:(0, \infty) \to \mathbb{R}$ be a function such that

(F1) $F$ is strictly increasing;

(F2) for any sequence $\left\{ {{\xi _n}} \right\} \subseteq (0, \infty)$, then

(F3) there exists $k \in (0, 1)$ such that $\mathop {\lim }\limits_{\xi \to {0^ + }} {\xi ^k}F(\xi) = 0$ for any $\xi \in (0, \infty).$

We denote by $\mathfrak{F}$ the set of all functions $F:(0, \infty) \to \mathbb{R}$ satisfying (F1)–(F3).

Example 2.2. ^[25] Let $t > 0$. The following functions $F:(0, \infty) \to \mathbb{R}$ belong to $\mathfrak{F}$:

(1) $F(t) = \ln t$;

(2) $F(t) = t + \ln t$;

(3) $F(t) = - \frac{1}{{\sqrt t }}$;

(4) $F(t) = \ln ({t^2} + t)$.

Definition 2.9. ^[25] Let $(X, d)$ be a metric space and $T:X \to X$ a mapping. If there exist $F \in \mathfrak{F}$ and $\tau > 0$ such that

for all $x, y \in X$ with $d(Tx, Ty) > 0$, then $T$ is called an $F$-contraction.

Define

Now, we put forward the definition of a $P$-property.

Definition 2.10. ^[26] Let $(A, B)$ be a pair of non-empty subsets of a metric space $(X, d)$ with ${A_0} \ne \emptyset$. We say that the pair $(A, B)$ has the weak $P$-property if and only if the following holds:

for all ${x_1}, {x_2} \in A$ and ${y_1}, {y_2} \in B$.

Zhang et al. ^[27] introduced the notion of the weak $P$-property which is weaker than the $P$-property.

Definition 2.11. ^[27] Let $(A, B)$ be a pair of non-empty subsets of a metric space $(X, d)$ with ${A_0} \ne \emptyset$. We say that the pair $(A, B)$ has the weak $P$-property if and only if the following holds:

for all ${x_1}, {x_2} \in A$ and ${y_1}, {y_2} \in B$.

Recently, Basha ^[28] introduced the concept of the uniform approximation of a set.

Definition 2.12. ^[28] Let $(A, B)$ be a pair of non-empty subsets of a metric space $(X, d)$ with ${A_0} \ne \emptyset$. $A$ is said to have uniform approximation in $B$ if and only if, given $\varepsilon > 0$, there exists $\sigma > 0$ such that the following holds:

for all ${x_1}, {x_2} \in A$ and ${y_1}, {y_2} \in B$.

It is trivial to see that $A$ (or $B$) has uniform approximation in $B$ (or $A$) and the pairs ($A, B$) and ($B, A$) do not necessarily have the weak $P$-property (see ^[28] and Example 3.1).

3.   Main results

Let $A$ and $B$ be two non-empty subsets of a modular metric space $(X, w)$. For all $\lambda > 0$, we set

Definition 3.1. Let $(A, B)$ be a pair of non-empty subsets of a modular metric space $(X, w)$, and let $\alpha, \eta: X \times X\rightarrow [0, \infty)$ be two functions. We say that $T:A \rightarrow B$ is an $\alpha$-proximal admissible mapping with respect to $\eta$ if the following holds:

for all $x_{1}, x_{2}, u_{1}, u_{2} \in A$ and $\lambda > 0$.

Definition 3.2. Let $(A, B)$ be a pair of non-empty subsets of a modular metric space $(X, w)$; the subspace $A$ is said to have uniform approximation in the subspace $B$ if and only if, for given $\varepsilon > 0$, there exists $\delta > 0$ such that the following holds:

for all ${x_1}, {x_2} \in A$, ${y_1}, {y_2} \in B$ and $\lambda > 0$.

Here, we introduce the concept of an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first and second kind in modular metric spaces.

Definition 3.3. Let $(A, B)$ be a pair of non-empty subsets of a modular metric space $(X, w)$. A non-self mapping $T : A \rightarrow B$ is said to be an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first kind if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $a, b, c, e, \tau > 0$ with $a + b + c + 2e = 1$ such that the following holds:

for any ${x_1}, {x_2}, {u_1}, {u_2} \in A$ with ${x_1} \ne {x_2}$ and $0 < \lambda \le {\lambda _0}$.

Definition 3.4. Let $(A, B)$ be a pair of non-empty subsets of a modular metric space $(X, w)$. A non-self mapping $T : A \rightarrow B$ is said to be an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the second kind if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $a, b, c, e, \tau > 0$ with $a + b + c + 2e = 1$ such that the following holds:

for any ${x_1}, {x_2}, {u_1}, {u_2} \in A$ with $T{x_1} \ne T{x_2}$ and $0 < \lambda \le {\lambda _0}$.

Now we state and prove the main results of this section.

Theorem 3.1. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Suppose that $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$ such that $A$ has uniform approximation in $B$. Assume that $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first kind that satisfies the following assertions:

(1) $A_0^\lambda$ and $B_0^\lambda$ are non-empty sets and $T({A_0^\lambda}) \subseteq {B_0^\lambda}$ for all $0 < \lambda \le {\lambda _0}$;

(2) $T$ is an $\alpha$-admissible mapping with respect to $\eta$;

(3) there exist elements ${x_0}, {x_1} \in A_0^\lambda$ for all $0 < \lambda \le {\lambda _0}$ such that ${w_\lambda }({x_1}, T{x_0}) = {w_\lambda }(A, B)$ and $\alpha ({x_0}, {x_1}) \ge \eta ({x_0}, {x_1})$;

(4) if $\{ {x_n}\}$ is a sequence such that $\alpha ({x_n}, {x_{n + 1}}) \ge \eta({x_n}, {x_{n + 1}})$ for all $n \in {\mathbb{N}_0}$ and ${x_n} \to x$ as $n \to \infty$, then $\alpha ({x_n}, x) \ge \eta({x_n}, x)$ for all $n\in \mathbb{\mathbb{N}}_{0}$.

If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a best proximity point. If, in addition, for any $x, u \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(u, Tu) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, u) < \infty$ and $\alpha (x, u) \ge \eta (x, u)$, then the best proximity point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by

$w$-converges to the best proximity point.

Proof. Let ${x_0}, {x_1} \in A_0^\lambda$ such that

Given the fact that $T(A_0^\lambda) \subseteq B_0^\lambda$, there exists ${x_2}$ in $A_0^\lambda$ such that

Since $T$ is an $\alpha$-admissible mapping with respect to $\eta$, we have that $\alpha ({x_1}, {x_2}) \ge \eta ({x_1}, {x_2})$. Again, in view of the fact that $T(A_0^\lambda) \subseteq B_0^\lambda$, there exists ${x_3} \subseteq A_0^\lambda$ such that

Similarly, we have that $\alpha ({x_2}, {x_3}) \ge \eta({x_2}, {x_3})$. Continuing this process, we get:

for all $n\in \mathbb{N}$. If there exists ${n_0} \in {\mathbb{N}_0}$ such that ${x_{{n_0}}} = {x_{{n_0} + 1}}$, then ${w_\lambda }({x_{{n_0}}}, T{x_{{n_0}}}) = {w_\lambda }(A, B)$, which implies that ${x_{{n_0}}}$ is a best proximity point of $T$. Hence, we suppose that ${x_n} \ne {x_{n + 1}}$ for all $n \in {\mathbb{N}_0}$. Given the fact that $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first kind, we have

By the convexity of $w$, we get

which implies that

Similarly, we can obtain

Also,

Applying (3.2)–(3.5) in (3.1), we obtain

Since $F$ is strictly increasing, we derive

Thus,

Since $a + b + c + 2e = 1$ and $a, b, c, e > 0$, we have

for any $n \in {\mathbb{N}_0}$ and $0 < \lambda \le {\lambda _0}$. Thus, from (3.6), we have

Therefore,

Denote $\gamma _n^\lambda = {w_\lambda }({x_n}, {x_{n + 1}})$ for any $n \in {\mathbb{N}_0}$ and $0 < \lambda \le {\lambda _0}$. From (3.7), we deduce that $\mathop {\lim }\limits_{n \to \infty } F(\gamma _n^\lambda) = - \infty$. Using (F2), we get

Taking into account (F3), there exists $k \in (0, 1)$ such that

It follows from (3.7) that

Letting $n\rightarrow \infty$ in the above inequality, and combining it with (3.8) and (3.9) we get

So, there exists ${n_1} \in {\mathbb{N}_0}$ such that ${\left({\gamma _n^\lambda } \right)^k}n \le 1$ for all $n \ge {n_1}$. Consequently, we have

for all $n \ge {n_1}$ and $0 < \lambda \le {\lambda _0}$. Set ${\lambda _i} = {\left({\frac{1}{2}} \right)^i}{\lambda _0}$ for any $i\in \mathbb{N}$. It is easy to see that $0 < {\lambda _i} \le {\lambda _0}$. Thus, we have

for all $n \ge {n_1}$. For any positive integers $m, n$ with $1 < m < n$, we obtain

where ${\lambda _h} = {\lambda _m} + {\lambda _{m + 1}} +... + {\lambda _{n - 1}}$. Since ${\lambda _h} = \frac{1}{{{2^m}}}{\lambda _0} + \frac{1}{{{2^{m + 1}}}}{\lambda _0} +... + \frac{1}{{{2^{n + 1}}}}{\lambda _0} \le \frac{1}{{{2^{m - 1}}}}{\lambda _0} < {\lambda _0}$, the inequalities (3.10) and (3.11) imply that

Hence, $\mathop {\lim }\limits_{m, n \to \infty } {w_{{\lambda _h}}}({x_m}, {x_n}) = 0$. Given that $0 < {\lambda _h} < {\lambda _0}$, we have that $\mathop {\lim }\limits_{m, n \to \infty } {w_{{\lambda _0}}}({x_m}, {x_n}) = 0$, which implies that $\{ {x_n}\}$ is a $w$-Cauchy sequence. Because the space $A$ has uniform approximation in the space $B$, it follows that $\{ T{x_n}\}$ must be a $w$-Cauchy sequence in $B$. Since $X_w^*$ is a $w$-complete modular metric space and $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$, the sequence $\{ {x_n}\}$ $w$-converges to some element $x$ in $A$ and the sequence $\{ T{x_n}\}$ $w$-converges to some element $y$ in $B$. Noting that $w$ satisfies the Fatou property, then

thus, ${w_{{\lambda _0}}}(x, y) = {w_{{\lambda _0}}}(A, B)$, which implies that $x$ is a member in $A_{0}$. Given that $T({A_0}) \subseteq {B_0}$, we have that

for some element $p$ in $A$. If, for some $n\in \mathbb{N}_{0}$, we have that ${x_{n + 1}} = p$, so ${w_{{\lambda _0}}}({x_{n + 1}}, Tx) = {w_{{\lambda _0}}}(A, B)$; then,

which implies that ${w_{{\lambda _0}}}(x, Tx) = {w_{{\lambda _0}}}(A, B)$ and the conclusion is immediate. Therefore, we assume that ${x_n} \ne p$ for all $n\in \mathbb{N}_{0}$. Again, since $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first kind, we have

Since $F$ is strictly increasing, we obtain

Letting $n\rightarrow \infty$ in the above inequality, we get that $\mathop {\lim }\limits_{n \to \infty } {w_{{\lambda _0}}}(p, {x_{n + 1}}) \le (b + e){w_{{\lambda _0}}}(p, x)$; hence, ${w_{{\lambda _0}}}(p, x) \le (b + e){w_{{\lambda _0}}}(p, x)$, which implies that $p$ and $x$ should be identical. Thus, ${w_{{\lambda _0}}}(x, Tx) = {w_{{\lambda _0}}}(A, B)$ and $x$ is a best proximity point of $T$. To prove the uniqueness of the result, suppose that there is another best proximity point $u$ of $T$ such that ${w_{{\lambda _0}}}(u, Tu) = {w_{{\lambda _0}}}(A, B)$. Given that $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first kind, we have

Since $F$ is strictly increasing, we obtain

Since $a + 2e > 0$, it follows that ${w_{{\lambda _0}}}(x, u) = 0$, which implies that $x$ and $u$ are identical. This complete the proof.

Let $\alpha (x, y) = \eta (x, y) = 1$ for all $x, y\in X$ in Theorem 3.1; we can deduce the following best proximity point theorem in the setting of a modular metric space.

Corollary 3.1. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Suppose that $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$ such that $A$ has uniform approximation in $B$. Assume that $T:A \rightarrow B$ is a generalized $F$-proximal contraction of Reich type of the first kind, that is, for any ${x_1}, {x_2}, {u_1}, {u_2} \in A$ with ${x_1} \ne {x_2}$, $F\in \mathfrak{F}$ and $\tau > 0$, there exists $\lambda_{0} > 0$ such that the following holds:

for all $0 < \lambda \le {\lambda _0}$ and $a, b, c > 0$ with $a+b+c = 1$. Also,

(1) $A_0^\lambda$ and $B_0^\lambda$ are non-empty sets and $T({A_0^\lambda}) \subseteq {B_0^\lambda}$ for all $0 < \lambda \le {\lambda _0}$;

(2) there exist elements ${x_0}, {x_1} \in A_0^\lambda$ for all $0 < \lambda \le {\lambda _0}$ such that ${w_\lambda }({x_1}, T{x_0}) = {w_\lambda }(A, B)$.

If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a best proximity point. If in addition, for any $x, u \in X_w^*$, ${w_\lambda }(x, Tx) = {w_\lambda }(u, Tu) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, u) < \infty$, then the best proximity point of $T$ is unique. Further, for any fixed element ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by

$w$-converges to the best proximity point.

The following results without the convexity assumption of Corollary 3.1.

Corollary 3.2. Let $w$ be a strict modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Suppose that $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$ such that $A$ has uniform approximation in $B$. Assume that $T:A \rightarrow B$ is an $F$-proximal contraction of the first kind, that is, for any ${x_1}, {x_2}, {u_1}, {u_2} \in A$ with ${x_1} \ne {x_2}$, there exists $F\in \mathfrak{F}$, $\tau > 0$, and $\lambda_{0} > 0$ such that the following holds:

for all $0 < \lambda \le {\lambda _0}$. Also, suppose that $T$ satisfies the following assertions:

(1) $A_0^\lambda$ and $B_0^\lambda$ are non-empty sets and $T({A_0^\lambda}) \subseteq {B_0^\lambda}$ for all $0 < \lambda \le {\lambda _0}$;

(2) there exist elements ${x_0}, {x_1} \in A_0^\lambda$ for all $0 < \lambda \le {\lambda _0}$ such that ${w_\lambda }({x_1}, T{x_0}) = {w_\lambda }(A, B)$.

If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a best proximity point. If in addition, for any $x, u \in X_w^*$, ${w_\lambda }(x, Tx) = {w_\lambda }(u, Tu) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, u) < \infty$, then the best proximity point of $T$ is unique. Further, for any fixed element ${x_0} \in {A_0}$, sequence $\{ {x_n}\}$ defined by

$w$-converges to the best proximity point.

We present an illustrative example.

Example 3.1. Let us consider the subsets

in the space $X = \mathbb{R}^{2}$ with the modular metric ${w_\lambda }:(0, \infty) \times X \times X \to [0, \infty ]$ defined by

for any $\lambda > 0$. Then, we have that ${w_\lambda }(A, B) = 1$, ${A_0} = A$, and ${B_0} = B$. It is easy to see that $(X, w_{\lambda})$ is a $w$-complete modular metric space, $w$ satisfies the Fatou property, and $A$ has uniform approximation in $B$; however, the pair $(A, B)$ does not have the weak $P$-property. Let $F:(0, \infty) \to \mathbb{R}$ defined by $F(x) = \ln x$ for all $x > 0$. Thus, $F$ belongs to $\mathfrak{F}$. Let $T:A \to B$ be a mapping that satisfies the following for each $({x_1}, {x_2}) \in A$:

where $P{x_1} = \frac{{{x_1}}}{{2 + {x_1}}}$. We can observe that $T(A_0^\lambda) \subseteq B_0^\lambda$ for all $\lambda > 0$. Assume that ${u_1}, {u_2}, {u_3}, {u_4}$ are elements in $A$ such that ${w_\lambda }({u_1}, T{u_2}) = {w_\lambda }({u_3}, T{u_4}) = {w_\lambda }(A, B)$. Set ${u_2} = ({r_1}, \sqrt {4 - r_1^2})$ and ${u_4} = ({r_2}, \sqrt {4 - r_2^2})$ for some $1 \le {r_1}, {r_2} \le 2$. Then, $T{u_2} = (\frac{{{r_1}}}{{2 + {r_1}}}, \sqrt {1 - {{(P{r_1})}^2}})$ and $T{u_4} = (\frac{{{r_2}}}{{2 + {r_2}}}, \sqrt {1 - {{(P{r_2})}^2}})$. So ${u_1} = (\frac{{2{r_1}}}{{2 + {r_1}}}, 2\sqrt {1 - {{(P{r_1})}^2}})$ and ${u_3} = (\frac{{2{r_2}}}{{2 + {r_2}}}, 2\sqrt {1 - {{(P{r_2})}^2}})$. We obtain

When we set ${r_1} = 0$ and ${r_2} = 1$, we get that ${e^{ - \tau }} \geq \frac{{6 - 3\sqrt 3 }}{{8 - 2\sqrt 2 }}$ or $\tau \in (0, \ln \frac{{8 - 2\sqrt 2 }}{{6 - 3\sqrt 3 }})$. Consequently, $T$ is an $F$-proximal contraction of the first kind. Thus, all of the conditions of Corollary 3.2 are satisfied. Hence, $T$ has a unique best proximity point $(2, 0)$.

Next, we state and prove the best proximity point theorem for a $\alpha$-$\eta$-type generalized $F$-proximal contraction of the second kind in a modular metric space.

Theorem 3.2. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Suppose that $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$ such that $B$ has uniform approximation in $A$. Assume that $T$ is a continuous $\alpha$-$\eta$-type generalized $F$-proximal contraction of the second kind that satisfies the following assertions:

(1) $A_0^\lambda$ and $B_0^\lambda$ are non-void and $T({A_0^\lambda}) \subseteq {B_0^\lambda}$;

(2) $T$ is an $\alpha$-admissible mapping with respect to $\eta$;

(3) there exist elements ${x_0}, {x_1} \in A_0^\lambda$ for all $0 < \lambda \le {\lambda _0}$ such that ${w_\lambda }({x_1}, T{x_0}) = {w_\lambda }(A, B)$ and $\alpha ({x_0}, {x_1}) \ge \eta ({x_0}, {x_1})$.

If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a best proximity point. If in addition, for any $x, u \in X_w^*$ satisfying ${w_\lambda }(x, Tx) = {w_\lambda }(u, Tu) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, u) < \infty$ and $\alpha (x, u) \ge \eta (x, u)$, then the best proximity point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by

$w$-converges to the best proximity point.

Proof. Similar to Theorem 3.1, we can obtain that there is a sequence $\left\{ {{x_n}} \right\}$ in $A_0^\lambda$ such that

for all $n \in {\mathbb{N}_0}$ and $0 < \lambda \le {\lambda _0}$. Without loss of generality, we assume that ${x_{n + 1}} \ne {x_n}$ for all $n \in {\mathbb{N}_0}$. Given the fact that $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the second kind, we have

Since $F$ is strictly increasing, we obtain

and, thus,

We can obtain that $\mathop {\lim }\limits_{m, n \to \infty } {w_{{\lambda _0}}}(T{x_m}, T{x_n}) = 0$ and $\{ T{x_n}\}$ is a $w$-Cauchy sequence by using a similar technique as in Theorem 3.1. Because the space $B$ has uniform approximation in the space $A$, it follows that $\{ T{x_n}\}$ must be a $w$-Cauchy sequence in $A$. Since $X_w^*$ is a $w$-complete modular metric space and $A$ is a non-empty $w$-closed subset of $X_w^*$, the sequence $\{ {x_n}\}$ $w$-converges to some element $x$ in $A$. By virtue of the fact that $w$ satisfies the Fatou property and $T$ is a continuous mapping, we have

So, ${w_{{\lambda _0}}}(x, Tx) = {w_{{\lambda _0}}}(A, B)$, which implies that $x$ is a best proximity point of $T$. To prove the uniqueness of the result, suppose that there is another best proximity point $u$ of $T$ such that ${w_{{\lambda _0}}}(u, Tu) = {w_{{\lambda _0}}}(A, B)$. Given that $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the second kind, we have

Since $F$ is strictly increasing, we obtain

Since $a + 2e > 0$, we have that ${w_{{\lambda _0}}}(Tx, Tu) = 0$, which implies that $Tx = Tu$. This completes the proof.

Letting $\alpha (x, y) = \eta (x, y) = 1$ for all $x, y\in X$ in Theorem 3.2, we can deduce the following best proximity point theorem in the setting of a modular metric space.

Corollary 3.3. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Suppose that $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$ such that $B$ has uniform approximation in $A$. Assume that $T:A \rightarrow B$ is a generalized $F$-proximal contraction of Reich type of the second kind, that is, for any ${x_1}, {x_2}, {u_1}, {u_2} \in A$ with ${Tx_1} \ne {Tx_2}$, there exist $F\in \mathfrak{F}$, $\tau > 0$, and a $\lambda_{0} > 0$ such that the following holds:

for all $0 < \lambda \leq \lambda_{0}$ and $a, b, c > 0$ with $a+b+c = 1$. Also,

(1) $A_0^\lambda$ and $B_0^\lambda$ are non-empty sets and $T({A_0^\lambda}) \subseteq {B_0^\lambda}$ for all $0 < \lambda \le {\lambda _0}$;

(2) there exist elements ${x_0}, {x_1} \in A_0^\lambda$ for all $0 < \lambda \le {\lambda _0}$ such that ${w_\lambda }({x_1}, T{x_0}) = {w_\lambda }(A, B)$.

If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a best proximity point. If, in addition, for any $x, u \in X_w^*$, ${w_\lambda }(x, Tx) = {w_\lambda }(u, Tu) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, u) < \infty$, then the best proximity point of $T$ is unique. Further, for any fixed element ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by

$w$-converges to the best proximity point.

Corollary 3.4. Let $w$ be a strict modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Suppose that $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$ such that $B$ has uniform approximation in $A$. Assume that $T:A \rightarrow B$ is an $F$-proximal contraction of the second kind, that is, for any ${x_1}, {x_2}, {u_1}, {u_2} \in A$ with ${Tx_1} \ne {Tx_2}$, there exist $F\in \mathfrak{F}$, $\tau > 0$, and a $\lambda_{0} > 0$ such that the following holds:

for all $0 < \lambda \leq \lambda_{0}$. Also,

(1) $A_0^\lambda$ and $B_0^\lambda$ are non-empty sets and $T({A_0^\lambda}) \subseteq {B_0^\lambda}$ for all $0 < \lambda \le {\lambda _0}$;

(2) there exist elements ${x_0}, {x_1} \in A_0^\lambda$ for all $0 < \lambda \le {\lambda _0}$ such that ${w_\lambda }({x_1}, T{x_0}) = {w_\lambda }(A, B)$;

If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a best proximity point. If, in addition, for any $x, u \in X_w^*$, ${w_\lambda }(x, Tx) = {w_\lambda }(u, Tu) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, u) < \infty$, then the best proximity point of $T$ is unique. Further, for any fixed element ${x_0} \in {A_0}$, sequence $\{ {x_n}\}$ defined by

$w$-converges to the best proximity point.

We present an illustrative example.

Example 3.2. Let us consider the subsets

in the space $X = \mathbb{R}^{2}$ with the modular metric ${w_\lambda }:(0, \infty) \times X \times X \to [0, \infty ]$ defined by

for any $\lambda > 0$. Then, we have that ${w_\lambda }(A, B) = \sqrt 2$, ${A_0} = A$ and ${B_0} = B$. It is easy to see that $(X, w_{\lambda})$ is a $w$-complete modular metric space, $w$ satisfies the Fatou property and $B$ has uniform approximation in $A$. Let $F:(0, \infty) \to \mathbb{R}$ be defined by $F(x) = \ln x$ for all $x > 0$. Thus, $F$ belongs to $\mathfrak{F}$. Let $T:A \to B$ be a mapping that satisfies the following for each $({x_1}, {x_2}) \in A$:

where $P{x_2} = \frac{{{x_2}}}{{1 + {x_2}}}$. We can observe that $T(A_0^\lambda) \subseteq B_0^\lambda$ for all $\lambda > 0$. Assume that ${u_1}, {u_2}, {u_3}, {u_4}$ are elements in $A$ such that ${w_\lambda }({u_1}, T{u_2}) = {w_\lambda }({u_3}, T{u_4}) = {w_\lambda }(A, B)$. Set ${u_2} = (1, {i_1})$ and ${u_4} = (1, {i_2})$ for some $1 \le {i_1}, {i_2} \le 2$. Then, $T{u_2} = (0, P{i_1})$ and $T{u_4} = (0, P{i_2})$. So, ${u_1} = (1, P{i_1})$ and ${u_3} = (1, P{i_2})$. We obtain

When we set ${i_1} = 0$ and ${i_2} = 1$, we get that ${e^{ - \tau }} \geq \frac{2}{3}$ or $\tau \in (0, \ln \frac{3}{2})$. Consequently, $T$ is an $F$-proximal contraction of the second kind. Thus, all of the conditions of Corollary 3.4 are satisfied. Hence, $T$ has a unique best proximity point $(1, 0)$.

Our next result is obtained for $\alpha$-$\eta$-type generalized $F$-proximal contractions of the first kind, as well as $\alpha$-$\eta$-type generalized $F$-proximal contractions of the second kind without the assumption of uniform approximation of the domains or the co-domain of the mappings.

Theorem 3.3. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Suppose that $(A, B)$ is a pair of non-empty $w$-closed subsets of $X_w^*$. Moreover, assume the following:

(1) $A_0^\lambda$ and $B_0^\lambda$ are non-empty sets and $T({A_0^\lambda}) \subseteq {B_0^\lambda}$;

(2) $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first kind as well as an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the second kind;

(3) there exist elements ${x_0}, {x_1} \in A_0^\lambda$ for all $0 < \lambda \le {\lambda _0}$ such that ${w_\lambda }({x_1}, T{x_0}) = {w_\lambda }(A, B)$ and $\alpha ({x_0}, {x_1}) \ge \eta({x_0}, {x_1})$;

(4) if $\{ {x_n}\}$ is a sequence such that $\alpha ({x_n}, {x_{n + 1}}) \ge \eta ({x_n}, {x_{n + 1}})$ for all $n \in {\mathbb{N}_0}$ and ${x_n} \to x$ as $n \to \infty$, then $\alpha ({x_n}, x) \ge \eta ({x_n}, x)$ for all $n\in \mathbb{\mathbb{N}}_{0}$.

If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) = {w_\lambda }(A, B)$ implies that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a best proximity point. If, in addition, for any $x, y \in {X^*}$ such that ${w_\lambda }(x, Tx) = {w_\lambda }(u, Tu) = {w_\lambda }(A, B)$, we have that ${w_\lambda }(x, u) < \infty$ and $\alpha (x, u) \ge \eta (x, u)$, then the best proximity point of $T$ is unique. Further, for any fixed element ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by

$w$-converges to the best proximity point.

Proof. Similar to Theorem 3.1, we can obtain that there is a sequence $\left\{ {{x_n}} \right\}$ in $A_0^\lambda$ such that

for all $n \in {\mathbb{N}_0}$ and $0 < \lambda \le {\lambda _0}$. Proceeding as in Theorem 3.1, we obtain that the sequence $\left\{ {{x_n}} \right\}$ is a $w$-Cauchy sequence and $w$-converges to some element $x$ in $A$. Similar to Theorem 3.2, we obtain that the sequence $\left\{ {{Tx_n}} \right\}$ is a $w$-Cauchy sequence and $w$-converges to some element $y$ in $B$. Noting that $w$ satisfies the Fatou property, it follows that

thus, ${w_{{\lambda _0}}}(x, y) = {w_{{\lambda _0}}}(A, B)$, which implies that $x$ is a member in $A_0^\lambda$. Given that $T({A_0^\lambda }) \subseteq {B_0^\lambda }$, we have

for some element $p$ in $A$. If for some $n\in \mathbb{N}_{0}$ such that ${x_{n + 1}} = p$ we have that ${w_{{\lambda _0}}}({x_{n + 1}}, Tx) = {w_{{\lambda _0}}}(A, B)$, then

which implies that ${w_{{\lambda _0}}}(x, Tx) = {w_{{\lambda _0}}}(A, B)$ and the conclusion is immediate. Therefore, we assume that ${x_n} \ne p$ for all $n\in \mathbb{N}_{0}$. Again, since $T$ is an $\alpha$-$\eta$-type generalized $F$-proximal contraction of the first kind, we have

Since $F$ is strictly increasing, we obtain

Letting $n\rightarrow \infty$ in the above inequality, we get that $\mathop {\lim }\limits_{n \to \infty } {w_{{\lambda _0}}}(p, {x_{n + 1}}) \le (b + e){w_{{\lambda _0}}}(p, x)$; hence, ${w_{{\lambda _0}}}(p, x) \le (b + e){w_{{\lambda _0}}}(p, x)$, which implies that $p$ and $x$ should be identical. Thus, ${w_{{\lambda _0}}}(x, Tx) = {w_{{\lambda _0}}}(A, B)$ and $x$ is a best proximity point of $T$. As in the proof of Theorem 3.1, the uniqueness of the best proximity point of mapping $T$ follows.

4.   Applications

Let $(X, w_{\lambda})$ be a metric space and $A, B$ be two subsets of $X$. If $A \cap B \ne \emptyset$, then $d(A, B) = 0$. In this case, a best proximity result turns to a fixed point result.

Letting $\alpha (x, y) = \eta (x, y) = 1$ for all $x, y\in X$ in Theorem 3.1, we can obtain the following:

Corollary 4.1. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Assume that $T$ is a self mapping on $X$ if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $a, b, c, e, \tau > 0$ with $a + b + c + 2e = 1$ such that the following holds:

for all $x, y\in X$ and $0 < \lambda\leq \lambda_{0}$. If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a fixed point. If, in addition, for any $x, u \in X_w^*$ satisfying that ${w_\lambda }(x, u) < \infty$, then the fixed point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by $\left\{ {T{x_n}} \right\}$ $w$-converges to the fixed point.

Corollary 4.2. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Assume that $T$ is a self-mapping on $X$ if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $a, b, c, \tau > 0$ with $a + b + c = 1$ such that the following holds:

for all $x, y\in X$ and $0 < \lambda\leq \lambda_{0}$. If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a fixed point. If, in addition, for any $x, u \in X_w^*$ satisfying that ${w_\lambda }(x, u) < \infty$, then the fixed point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by $\left\{ {T{x_n}} \right\}$ $w$-converges to the fixed point.

Corollary 4.3. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Assume that $T$ be a self mapping on $X$, if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $\tau > 0$ such that

for all $x, y\in X$ and $0 < \lambda\leq \lambda_{0}$. If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a fixed point. If, in addition, for any $x, u \in X_w^*$ satisfying that ${w_\lambda }(x, u) < \infty$, then the fixed point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by $\left\{ {T{x_n}} \right\}$ $w$-converges to the fixed point.

Letting $\alpha (x, y) = \eta (x, y) = 1$ for all $x, y\in X$ in Theorem 3.2, we can obtain the following:

Corollary 4.4. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Assume that $T$ is a self-mapping on $X$, if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $a, b, c, e, \tau > 0$ with $a + b + c + 2e = 1$ such that the following holds:

for all $x, y\in X$ and $0 < \lambda\leq \lambda_{0}$. If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a fixed point. If, in addition, for any $x, u \in X_w^*$ satisfying that ${w_\lambda }(x, u) < \infty$, then the fixed point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by $\left\{ {T{x_n}} \right\}$ $w$-converges to the fixed point.

Corollary 4.5. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Assume that $T$ is a self-mapping on $X$, if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $a, b, c, \tau > 0$ with $a + b + c = 1$ such that the following holds:

for all $x, y\in X$ and $0 < \lambda\leq \lambda_{0}$. If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a fixed point. If, in addition, for any $x, u \in X_w^*$ satisfying that ${w_\lambda }(x, u) < \infty$, then the fixed point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by $\left\{ {T{x_n}} \right\}$ $w$-converges to the fixed point.

Corollary 4.6. Let $w$ be a strict convex modular metric with the Fatou property on $X$ and $X_w^*$ be a $w$-complete modular metric space induced by $w$. Assume that $T$ is a self-mapping on $X$, if there exist $F \in \mathfrak{F}$, ${\lambda _0} > 0$, and $\tau > 0$ such that the following holds:

for all $x, y\in X$ and $0 < \lambda\leq \lambda_{0}$. If, for every $0 < \lambda \leq \lambda_{0}$, there exists an $x \in X_w^*$ satisfying that ${w_\lambda }(x, Tx) < \infty$, then $T$ has a fixed point. If, in addition, for any $x, u \in X_w^*$ satisfying that ${w_\lambda }(x, u) < \infty$, then the fixed point of $T$ is unique. Further, for any ${x_0} \in {A_0}$, the sequence $\{ {x_n}\}$ defined by $\left\{ {T{x_n}} \right\}$ $w$-converges to the fixed point.

Taking into account Corollary 4.3, we give an existence and uniqueness result for a solution of the Fredholm linear integral equation:

for $t \in I = [0, 1]$, where $h:I \to X$, $\beta :I \to I$, $G:I \times I \to \mathbb{R}$, and $L:I \times I \to \mathbb{R}$ are continuous functions. Let $C(I, \mathbb{R})$ be the space of all continuous functions on $I$ with the norm $\left\| x \right\| = \mathop {\sup }\limits_{t \in [0, 1]} \left| {x(t)} \right|$, and the modular metric ${w_\lambda }(x, y) = \frac{{\left\| {x - y} \right\|}}{\lambda }$ for all $x, y \in C(I, \mathbb{R})$. We consider the following assumptions:

(A1) The function $G(t, \theta)$ is continuous and nonnegative on $I\times I$ with ${\left\| G \right\|_\infty } = \sup \left\{ {G(t, \theta):t, \theta \in I} \right\}$;

(A2) $\left| {L(\theta, x(\theta)) - L(\theta, y(\theta))} \right| \le \delta \left| {x(\theta) - y(\theta)} \right|$ for all $\theta \in I$.

Theorem 4.1. Assume that the hypotheses (A1) and (A2) hold. If ${\left\| G \right\|_\infty }\delta \le {e^{ - \frac{1}{\delta }}}$ for some $\delta > 0$, then (4.2) has a unique solution in $C(I, \mathbb{R})$.

Proof. Note that $(C(I, \mathbb{R}), {w_\lambda })$ is a $w$-complete modular metric space. Define a self-map $T$ on $C(I, \mathbb{R})$ by

By the hypotheses (H1) and (H2), and by using the Cauchy-Schwarz inequality, we get

This implies that ${w_\lambda }(Tx, Ty) \le {e^{ - \frac{1}{\delta }}}{w_\lambda }(x, y)$. Hence, (4.1) is satisfied for $F(\alpha) = \ln \alpha$, $\alpha > 0$ and $\tau = \frac{1}{\delta }$. Therefore, all conditions of Corollary 4.3 hold; thus, the integral given by (4.2) has a unique positive solution.

5.   Conclusions

The main motivation of the current paper is to show that the best proximity point results for $\alpha$-$\eta$-type generalized $F$-proximal contraction mappings in the framework of modular metric spaces. We have achieved some best proximity point theorems for modular metric spaces. Our new results extend and improve many recent results. We also gave some examples to show the validity of our results and an application to nonlinear integral inclusions. Finally, we plan on looking into two future directions: the first direction is proving the existence of the best proximity points for cyclic mappings in modular metric spaces and the second direction is applying the results of this paper in the settings of other spaces, such as fuzzy metric spaces ^[29] and $(q_{1}, q_{2})$-quasimetric spaces ^[11].

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported by the Natural Science Foundation of Heilongjiang Province of China (grant no. YQ2021C025).

Conflict of interest

The authors declare no conflict of interest.