Best Proximity Points for Geraghty-Type Non-Self Mappings

Abstract

We study Geraghty-type non-self mappings within the framework of best proximity point theory. By introducing auxiliary functions with subsequential convergence, we establish general conditions ensuring the existence and uniqueness of best proximity points. Our results extend and unify earlier work on proximal and Kannan-type contractions under a Geraghty setting, and provide counterexamples showing that the auxiliary assumptions are essential. As an illustration, we construct an explicit non-self alignment mapping on subsets of ${\mathbb{R}}^2$ for which all hypotheses can be verified and the unique best proximity point, as well as the convergence of the associated proximal iteration, can be computed in closed form.

1. Introduction

Fixed point theory plays a fundamental role in nonlinear analysis, with applications ranging from differential equations to optimization problems. Classical results such as the Banach contraction principle provide powerful tools for self-mappings, but in many practical situations one encounters non-self mappings, where the appropriate substitute is the notion of best proximity points.

The development of the field has proceeded through several key milestones. In 1968, Kannan introduced a celebrated generalization of Banach’s contraction principle by proving that for a complete metric space $(X,d)$, any mapping $T:X\to X$ satisfying

$$d(Tx,Ty)\le \alpha \left[d(x,Tx)+d(y,Ty)\right],\forall x,y\in X,$$

with $\alpha \in [0,\frac{1}{2})$, admits a unique fixed point. A remarkable feature of Kannan’s theorem is that no continuity assumption on T is required; related notions of weakly contractive maps were subsequently investigated in [1]. Later, Ariza-Ruiz and Jiménez-Melado [2] extended this result by introducing weakly Kannan maps, in which the contractive coefficient depends on the points $x,y$. More precisely, there exists $\alpha :X\times X\to [0,1)$ such that

$$d(Tx,Ty)\le \alpha (x,y)\left[d\right(x,Tx)+d(y,Ty\left)\right],\forall x,y\in X,$$

with the uniform bound $sup\left\{\alpha \right(x,y):a\le d(x,y)\le b\}<1$ for all $0<a\le b$ [2]. This flexibility allows one to capture the behavior of a wider class of nonlinear operators.

The move from fixed point results to best proximity point results was initiated by Sadiq Basha [3], who introduced the concept of proximal contractions and established best proximity point theorems for non-self mappings. The key idea is to characterize conditions under which, even in the absence of fixed points, one can still guarantee the existence of a point $x^{*}$ such that the distance between $x^{*}$ and its image $T{x}^{*}$ is minimal. Best proximity point results for Geraghty-type proximal contraction mappings were later developed in [4]. In that work, the classical Geraghty condition $d(Tx,Ty)\le \beta \left(d\right(x,y\left)\right)d(x,y)$ was transferred to the proximal setting: one requires that whenever $d(u_1,T{x}_1)=d(u_2,T{x}_2)=d(A,B)$, the inequality $d(u_1,u_2)\le \beta \left(d(x_1,x_2)\right)d(x_1,x_2)$ holds with $\beta \in \mathrm{\Gamma }$. This framework, which we refer to as Geraghty-type non-self mappings, allows the contractive coefficient to vary with the distance between the points rather than being a fixed constant, thereby capturing a richer class of non-self operators than the standard proximal contractions of Sadiq Basha [3]. However, all existing Geraghty-type proximal results assume that T acts directly on the metric, without the mediation of an auxiliary function. In particular, there is no systematic theory addressing the role of subsequential convergence of auxiliary functions when Geraghty-type conditions are combined with non-self mappings—a gap that limits the applicability of the existing framework to problems where the contractive structure is naturally expressed through a mediating operator. The present paper fills this gap by incorporating an auxiliary mapping S into both the Geraghty and the Kannan–Geraghty proximal frameworks. We note that the Geraghty proximal setting continues to attract active interest: recent contributions include extensions to graph-endowed metric spaces [5], interpolative Kannan-type proximal contractions in fuzzy metric spaces [6], and generalized F-proximal contractions [7].

Subsequent work by Moradi provided a different perspective by introducing the use of an auxiliary function. Specifically, for mappings $T,S:X\to X$ in a metric space $(X,d)$, it was shown in [8] that if S is one-to-one, continuous, and sequentially convergent, and if

$$d(STx,STy)\le kd(Sx,Sy),\forall x,y\in X$$

for some constant $k\in (0,1)$, then T possesses a best proximity point.

Building on these foundations, the present work introduces two novel classes of mappings: proximal Geraghty and proximal Kannan–Geraghty mappings of the first kind. We establish existence and uniqueness theorems for these mappings and illustrate, via examples, that the auxiliary conditions we impose are both natural and necessary. Before turning to our main results, we recall some fundamental concepts from the best proximity theory.

2. Preliminaries

Definition 1

([9]). Let $A,B$ be nonempty subsets of a metric space $(X,d)$. The best proximity sets $A_0$ and $B_0$ are defined as

$$\begin{array}{cc}\hfill A_0& :=\{x\in A:d(x,y)=d(A,B)forsomey\in B\},\hfill \\ \hfill B_0& :=\{y\in B:d(x,y)=d(A,B)forsomex\in A\},\hfill \end{array}$$

where

$$d(A,B):=inf\left\{d\right(x,y):x\in A,y\in B\}$$

denotes the minimal distance between A and B.

When A and B are closed subsets of a normed linear space with $d(A,B)>0$, it is known that both $A_0$ and $B_0$ lie on the boundaries of their respective sets [3].

Definition 2

([9]). Given a mapping $T:A\to B$, a point $x^{*}\in A$ is called a best proximity point of T if

$$d(x^{*},T{x}^{*})=d(A,B).$$

Definition 3

([3]). A mapping $S:A\to B$ is called a proximal contraction of the first kind if there exists $\alpha \in [0,1)$ such that for all $x_1,x_2,u_1,u_2\in A$,

$$\left[d(u_1,S{x}_1)=d(A,B),d(u_2,S{x}_2)=d(A,B)\right]⟹d(u_1,u_2)\le \alpha d(x_1,x_2).$$

For self-maps, this reduces exactly to the Banach contraction principle. For non-self maps, proximal contractions need not be contractions in the usual sense.

Corollary 1

([10]). Let $A,B$ be nonempty closed subsets of a complete metric space such that $A_0$ and $B_0$ are nonempty. Suppose $T:A\to B$ and $g:A\to A$ satisfy the following:

(1) 

g is one-to-one and continuous, and $g^{-1}:g\left(A\right)\to A$ is uniformly continuous;

(2) 

T is a proximal contraction of the first kind with $T\left(A_0\right)\subseteq B_0$.

Then there exists a unique $x^{*}\in A$ such that $d(g{x}^{*},T{x}^{*})=d(A,B)$. Moreover, for any $x_0\in A_0$, the sequence defined by $d(g{x}_{n+1},T{x}_n)=d(A,B)$ converges to $x^{*}$.

When $g={\mathrm{Id}}_A$, we simply say that $T:A\to B$ is a proximal contraction if condition (2) holds.

Definition 4

([10]). Let $(A,B)$ be a nonempty pair of subsets of a metric space $(X,d)$. A mapping $T:A\to B$ is called a proximal Kannan non-self mapping if there exists $\alpha \in [0,\frac{1}{2})$ such that for all $u,v,x,y\in A$ with

$$d(u,Tx)=d(A,B),d(v,Ty)=d(A,B),$$

we have

$$d(u,v)\le \alpha \left[d^{*}(x,Tx)+d^{*}(y,Ty)\right],$$

where

$$d^{*}(x,Tx):=d(x,Tx)-d(A,B)\ge 0.$$

The class of proximal Kannan non-self mappings properly contains the class of Kannan non-self mappings.

Definition 5

(Weak Proximal Kannan Non-Self Mapping [10]). Let $(X,d)$ be a metric space, and let $A,B\subseteq X$. A mapping $T:A\to B$ is called a weak proximal Kannan non-self mapping if there exists $\alpha \in (0,\frac{1}{2})$ such that for all $u,v,x,y\in A$ with

$$d(u,Tx)=d(A,B),d(v,Ty)=d(A,B),$$

the implication

$${\displaystyle \frac{1}{r}}{d}^{*}(x,Tx)\le d(x,y)⟹d(u,v)\le \alpha [d^{*}(x,Tx)+d^{*}(y,Ty)]$$

holds, where $r={\displaystyle \frac{\alpha }{1-\alpha }}$.

Theorem 1

([10]). Let $(A,B)$ be a nonempty pair of subsets of a complete metric space $(X,d)$ such that $A_0$ is nonempty and closed. If $T:A\to B$ is a weak proximal Kannan non-self mapping with $T\left(A_0\right)\subseteq B_0$, then there exists a unique $x^{*}\in A$ such that $d(x^{*},T{x}^{*})=d(A,B)$. Moreover, if $\left\{x_n\right\}\subseteq A$ satisfies $d(x_{n+1},T{x}_n)=d(A,B)$, then $x_n\to x^{*}$.

Definition 6.

([11]). Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is called a Geraghty contraction if there exists $\beta \in \mathrm{\Gamma }$ such that

$$d(Tx,Ty)\le \beta \left(d\right(x,y\left)\right)d(x,y),\forall x,y\in X,$$

where Γ denotes the class of functions $\beta :[0,\infty )\to [0,1)$ with the property that

$$\beta \left(t_n\right)\to 1\Rightarrow t_n\to 0.$$

Theorem 2

([11]). Let $(X,d)$ be a complete metric space and $T:X\to X$ a Geraghty contraction. Then T has a unique fixed point.

According to [12], let $(X,d)$ be a metric space. A mapping $f:X\to X$ is said to be a Kannan–Geraghty self-mapping if there exists a function $\beta \in \mathrm{\Gamma }$ such that for all $x,y\in X$,

$$d(f\left(x\right),f\left(y\right))\le \beta \left(d(x,y)\right)\frac{1}{2}[d(x,f\left(x\right))+d(y,f\left(y\right))].$$

This definition combines the features of Kannan mappings with the flexibility of Geraghty-type control functions.

Definition 7

([8]). Let $(X,d)$ be a metric space. A mapping $S:X\to X$ is called subsequentially convergent if for every sequence $\left\{x_n\right\}$ in X such that $\left\{S{x}_n\right\}$ is convergent, the sequence $\left\{x_n\right\}$ has a convergent subsequence.

This notion, introduced by Moradi [8], plays a key role in recovering the convergence of the original sequence from the convergence of its image under the auxiliary mapping S. Motivated by the above developments, we now develop the non-self, proximal analogues.

3. Main Results

In this section, we introduce S-proximal contraction non-self mappings and S-proximal Kannan non-self mappings, and establish sufficient conditions for the existence and uniqueness of best proximity points in complete metric spaces. Before stating the main theorems, we recall the notion of an auxiliary function.

Definition 8.

Let $(X,d)$ be a metric space, and let $A,B\subseteq X$. A mapping $S:A\cup B\to A\cup B$ is called an auxiliary function if $S\left(A\right)\subseteq A$ and $S\left(B\right)\subseteq B$.

This definition is introduced here for the first time. It formalizes the structural requirement on the auxiliary mapping S that preserves the partition of the ambient space into the two sets A and B. The terminology is motivated by the role of S in Moradi’s framework [8,13], where a similar partition-preserving property is used implicitly; here we make it explicit and extend it to the non-self, proximal setting.

Theorem 3

(Extended Proximal Geraghty of the First Kind). Let $(A,B)$ be a pair of nonempty subsets of a complete metric space $(X,d)$ such that $A_0$ and $B_0$ are nonempty and closed. Let S be an auxiliary function that is continuous on A and B, one-to-one, subsequentially convergent, and satisfies $S\left(A_0\right)\subseteq A_0$ and $S\left(B_0\right)\subseteq B_0$.

Suppose $T:A\to B$ is a mapping such that $T\left(A_0\right)\subseteq B_0$, and assume that for all $u,v,x,y\in A$,

$$d(Su,STx)=d(Sv,STy)=d(A,B)⟹d(Su,Sv)\le \beta \left(d\right(Sx,Sy\left)\right)d(Sx,Sy),$$

where $\beta \in \mathrm{\Gamma }$ (that is, T is an S-proximal Geraghty contraction).

Then, there exists a unique $x^{*}\in A$ such that

$$d(S{x}^{*},ST{x}^{*})=d(A,B).$$

Moreover, if $\left\{x_n\right\}\subseteq A$ is a sequence satisfying

$$d(S{x}_{n+1},ST{x}_n)=d(A,B),\forall n\in \mathbb{N},$$

then $x_n\to x^{*}$.

Proof. 

We first introduce the images of the proximity sets under S

$$S\left(A_0\right):=\{Sx\mid d(Sx,Sy)=d(A,B)\mathrm{for}\mathrm{some}y\in B\},$$

$$S\left(B_0\right):=\{Sy\mid d(Sx,Sy)=d(A,B)\mathrm{for}\mathrm{some}x\in A\}.$$

Let $x_0\in A_0$. Since $T\left(A_0\right)\subseteq B_0$, we have $T{x}_0\in B_0$ and hence $ST{x}_0\in S\left(B_0\right)$. Thus, there exists $x_1\in A_0$ with

$$d(S{x}_1,ST{x}_0)=d(A,B).$$

Iterating this construction, we obtain a sequence $\left\{x_n\right\}\subseteq A_0$ such that

$$d(S{x}_{n+1},ST{x}_n)=d(A,B),\forall n\in \mathbb{N}.$$

(1)

Set

$$d_n:=d(S{x}_n,S{x}_{n-1}),n\ge 1.$$

For each $n\ge 1$, apply the S-proximal Geraghty condition with

$$u=x_{n+1},x=x_n,v=x_n,y=x_{n-1}.$$

Then, by (1),

$$d(Su,STx)=d(S{x}_{n+1},ST{x}_n)=d(A,B),d(Sv,STy)=d(S{x}_n,ST{x}_{n-1})=d(A,B),$$

and so, the hypothesis is satisfied, and we obtain

$$d(S{x}_{n+1},S{x}_n)\le \beta \left(d(S{x}_n,S{x}_{n-1})\right)d(S{x}_n,S{x}_{n-1}),$$

that is,

$$d_{n+1}\le \beta \left(d_n\right)d_n,n\ge 1.$$

(2)

By the standard argument in Geraghty’s fixed point theorem (see [11]), Equation (2) implies that $d_n\to 0$ and that $\left\{S{x}_n\right\}$ is a Cauchy sequence in X. Since X is complete and $A_0$ is closed, there exists $v\in A_0$ such that

$$\underset{n\to \infty }{lim}S{x}_n=v.$$

(3)

Because S is subsequentially convergent, $\left\{x_n\right\}$ has a subsequence $\left\{x_{n\left(k\right)}\right\}$ converging to some $u\in A_0$. By continuity of S,

$$\underset{k\to \infty }{lim}S{x}_{n\left(k\right)}=Su.$$

Comparing with (3), we conclude $Su=v$.

We claim that u is the unique best proximity point of T. Since $u\in A_0$ and $T\left(A_0\right)\subseteq B_0$, there exists $y^{*}\in A_0$ with $d(S{y}^{*},STu)=d(A,B)$. Combining this with (1) for the subsequence $\left\{x_{n\left(k\right)}\right\}$, we have

$$d(S{y}^{*},STu)=d(A,B),d(S{x}_{n\left(k\right)+1},ST{x}_{n\left(k\right)})=d(A,B).$$

Applying the contractive condition with

$$w=x_{n\left(k\right)+1},x=x_{n\left(k\right)},v=y^{*},y=u,$$

(where w plays the role of the variable u in the contractive hypothesis, distinct from the limit point u), we obtain

$$d(S{x}_{n\left(k\right)+1},S{y}^{*})\le \beta \left(d(S{x}_{n\left(k\right)},Su)\right)d(S{x}_{n\left(k\right)},Su).$$

Letting $k\to \infty$, we obtain $d(Su,S{y}^{*})=0$. Hence, $Su=S{y}^{*}$. Since S is one-to-one, $u=y^{*}$. Thus, u is a best proximity point of T.

Finally, for uniqueness, suppose $x_1,x_2\in A$ are two distinct best proximity points, so that $d(S{x}_i,ST{x}_i)=d(A,B)$ for $i=1,2$. Applying the contractive condition with $u=x_1$, $x=x_1$, $v=x_2$, $y=x_2$ gives

$$d(S{x}_1,S{x}_2)\le \beta \left(d(S{x}_1,S{x}_2)\right)d(S{x}_1,S{x}_2),$$

which is impossible unless $d(S{x}_1,S{x}_2)=0$. Hence $x_1=x_2$. □

Remark 1.

A fully worked example illustrating Theorem 3 is given in Section 4. There, the sets A and B are parallel line segments in ${\mathbb{R}}^2$, the auxiliary function is $S=\mathrm{Id}$, and the non-self mapping $T(0,t)=(\delta ,\kappa t)$ with $\kappa \in (0,1)$ is shown to be an S-proximal Geraghty contraction with constant $\beta \equiv \kappa$. The unique best proximity point $(0,0)$ and the geometric convergence of the proximal iteration $t_n={\kappa }^n{t}_0\to 0$ are computed in closed form.

We emphasize that the assumption of subsequential convergence of S in Theorem 3 cannot be omitted. This is demonstrated by an example (see Example 1 below).

In the next result, we extend the notion of proximal Kannan non-self mappings introduced by Gabeleh [10] to a Geraghty-type setting.

Theorem 4

(Extended Proximal Kannan–Geraghty of the First Kind). Let $(A,B)$ be a pair of nonempty subsets of a complete metric space $(X,d)$ such that $A_0$ and $B_0$ are nonempty and closed. Let S be an auxiliary function that is continuous on A and B, one-to-one, and subsequentially convergent, with $S\left(A_0\right)\subseteq A_0$ and $S\left(B_0\right)\subseteq B_0$. Suppose $T:A\to B$ is a mapping with $T\left(A_0\right)\subseteq B_0$, and assume that for all $u,v,x,y\in A$,

$$\begin{array}{cc}\hfill & d(Su,STx)=d(Sv,STy)=d(A,B)\hfill \\ \hfill & ⟹d(Su,Sv)\le \beta \left(d(Sx,Sy)\right)\left[d^{*}(Sx,STx)+d^{*}(Sy,STy)\right],\hfill \end{array}$$

(4)

where $\beta \in \mathrm{\Gamma }$ and

$$d^{*}(Sx,STx):=d(Sx,STx)-d(A,B)\ge 0.$$

Then there exists a unique point $x^{*}\in A$ such that

$$d(S{x}^{*},ST{x}^{*})=d(A,B).$$

Moreover, if $\left\{x_n\right\}\subseteq A$ is a sequence satisfying

$$d(S{x}_{n+1},ST{x}_n)=d(A,B),\forall n\in \mathbb{N},$$

then $x_n\to x^{*}$.

Proof. 

As before, define

$$\begin{array}{cc}\hfill S\left(A_0\right)& :=\{Sx\mid d(Sx,Sy)=d(A,B)\mathrm{for}\mathrm{some}y\in B\},\hfill \\ \hfill S\left(B_0\right)& :=\{Sy\mid d(Sx,Sy)=d(A,B)\mathrm{for}\mathrm{some}x\in A\}.\hfill \end{array}$$

Let $x_0\in A_0$. Since $T\left(A_0\right)\subseteq B_0$, we have $T{x}_0\in B_0$. Hence $ST{x}_0\in S\left(B_0\right)$. Thus, there exists $x_1\in A_0$ such that

$$d(S{x}_1,ST{x}_0)=d(A,B).$$

Proceeding inductively, we obtain a sequence $\left\{x_n\right\}\subseteq A_0$ with

$$d(S{x}_{n+1},ST{x}_n)=d(A,B),\forall n\in \mathbb{N}.$$

(5)

Set again $d_n:=d(S{x}_n,S{x}_{n-1})$ for $n\ge 1$. For each $n\ge 1$, apply (4) with

$$u=x_{n+1},x=x_n,v=x_n,y=x_{n-1}.$$

Using (5), we obtain

$$d(S{x}_n,S{x}_{n+1})\le \beta \left(d(S{x}_n,S{x}_{n-1})\right)\left[d^{*}(S{x}_n,ST{x}_n)+d^{*}(S{x}_{n-1},ST{x}_{n-1})\right].$$

By the triangle inequality and (5)

$$\begin{array}{cc}\hfill d^{*}(S{x}_n,ST{x}_n)& =d(S{x}_n,ST{x}_n)-d(A,B)\hfill \\ & \le d(S{x}_n,S{x}_{n+1})+d(S{x}_{n+1},ST{x}_n)-d(A,B)=d(S{x}_n,S{x}_{n+1})=d_{n+1},\hfill \end{array}$$

and similarly $d^{*}(S{x}_{n-1},ST{x}_{n-1})\le d(S{x}_{n-1},S{x}_n)=d_n$. Hence

$$d_{n+1}\le \beta \left(d_n\right)\left(d_{n+1}+d_n\right).$$

Rearranging

$$d_{n+1}\left(1-\beta \left(d_n\right)\right)\le \beta \left(d_n\right)d_n,$$

we have

$$d_{n+1}\le {\displaystyle \frac{\beta \left(d_n\right)}{1-\beta \left(d_n\right)}}{d}_n,n\ge 1.$$

(6)

Since $0\le \beta \left(t\right)<1$ for all t, the right-hand side is finite. We now show that $d_n\to 0$ and that $\left\{S{x}_n\right\}$ is Cauchy.

• Step 1: $d_n\to 0$. If $d_N=0$ for some N, then $d_n=0$ for all $n\ge N$, and we are done. So assume $d_n>0$ for all n. Write (6) as $d_{n+1}\le {\phi }_n{d}_n$ where ${\phi }_n:={\displaystyle \frac{\beta \left(d_n\right)}{1-\beta \left(d_n\right)}}$. We first show that $\left\{d_n\right\}$ is bounded. If $\left\{d_n\right\}$ were unbounded, there would exist a subsequence $d_{n_k}\to \infty$. Since $\beta \left(d_{n_k}\right)<1$, either $\beta \left(d_{n_k}\right)$ has a subsequence converging to some $c<1$, in which case ${\phi }_{n_k}\to {\displaystyle \frac{c}{1-c}}<\infty$ and $d_{n_k+1}\le {\displaystyle \frac{c}{1-c}}{d}_{n_k}$ eventually bounds consecutive terms by a constant multiple, preventing divergence to $+\infty$, or $\beta \left(d_{n_k}\right)\to 1$, but then $\beta \in \mathrm{\Gamma }$ forces $d_{n_k}\to 0$, contradicting $d_{n_k}\to \infty$. Hence $\left\{d_n\right\}$ is bounded.

Now suppose $d_n\nrightarrow 0$. Then ${lim\; sup}_{n\to \infty }{d}_n=:D>0$. Choose a subsequence $d_{n_k}\to D$. By telescoping, $d_{n+1}\le d_1{\prod }_{j=1}^n{\phi }_j$. Since $d_n\nrightarrow 0$, the product ${\prod }_{j=1}^{\infty }{\phi }_j$ does not converge to zero, so ${\sum }_{j=1}^{\infty }{(1-{\phi }_j)}^{+}<\infty$ and in particular ${\phi }_j$ cannot stay bounded below 1 permanently. More precisely, if $lim\; inf{\phi }_n<1$ permanently, then $d_n\to 0$ by comparison with a geometric series, contradicting $D>0$. So $lim\; sup{\phi }_n\ge 1$, meaning $lim\; sup\beta \left(d_n\right)\ge \frac{1}{2}$.

For the contradiction, consider two cases. If $\beta \left(d_{n_k}\right)\to 1$ along some subsequence with $d_{n_k}\to \ell >0$, then $\beta \in \mathrm{\Gamma }$ gives $d_{n_k}\to 0$, contradicting $\ell >0$. If $\beta \left(d_{n_k}\right)\to c<1$ along every convergent subsequence, then ${\phi }_{n_k}\to {\displaystyle \frac{c}{1-c}}$. For the product $\prod {\phi }_j$ not to vanish, we need ${\phi }_j\to 1$, i.e., $\beta \left(d_j\right)\to 1$, but this contradicts $c<1$. In either case we reach a contradiction, so $d_n\to 0$.

• Step 2: $\left\{S{x}_n\right\}$ is Cauchy. Suppose not. Then there exist $\epsilon >0$ and subsequences $m\left(k\right)<n\left(k\right)$ with $d(S{x}_{m\left(k\right)},S{x}_{n\left(k\right)})\ge \epsilon$ and $d(S{x}_{m\left(k\right)},S{x}_{n\left(k\right)-1})<\epsilon$. By the triangle inequality,

  $$\epsilon \le d(S{x}_{m\left(k\right)},S{x}_{n\left(k\right)})\le d(S{x}_{m\left(k\right)},S{x}_{n\left(k\right)-1})+d_{n\left(k\right)}<\epsilon +d_{n\left(k\right)},$$

  so $d(S{x}_{m\left(k\right)},S{x}_{n\left(k\right)})\to \epsilon$. Now applying (4) with the indices $m\left(k\right)$ and $n\left(k\right)$, and using the estimates $d^{*}(S{x}_n,ST{x}_n)\le d_{n+1}\to 0$ established above, the right-hand side of the contractive inequality tends to $\beta \left({\epsilon }_k\right)·0=0$, while the left-hand side tends to $\epsilon >0$, i.e., a contradiction. Because X is complete and $A_0$ is closed, there exists $v\in A_0$ such that

  $$\underset{n\to \infty }{lim}S{x}_n=v.$$

  (7)

Since S is subsequentially convergent, $\left\{x_n\right\}$ has a subsequence $\left\{x_{n\left(k\right)}\right\}$ converging to some $u\in A_0$. By continuity of S

$$\underset{k\to \infty }{lim}S{x}_{n\left(k\right)}=Su.$$

Comparing with (7), we conclude $Su=v$.

We claim u is the unique best proximity point of T. Since $u\in A_0$ and $T\left(A_0\right)\subseteq B_0$, there exists $y^{*}\in A_0$ with $d(S{y}^{*},STu)=d(A,B)$. Combining this with (5) for the subsequence $\left\{x_{n\left(k\right)}\right\},$ we have

$$d(S{y}^{*},STu)=d(A,B),d(S{x}_{n\left(k\right)+1},ST{x}_{n\left(k\right)})=d(A,B).$$

Applying the contractive condition (4) with

$$w=x_{n\left(k\right)+1},x=x_{n\left(k\right)},v=y^{*},y=u,$$

(where w plays the role of the variable u in (4), distinct from the limit point u), we obtain

$$d(S{x}_{n\left(k\right)+1},S{y}^{*})\le \beta \left(d(S{x}_{n\left(k\right)},Su)\right)\left[d^{*}(S{x}_{n\left(k\right)},ST{x}_{n\left(k\right)})+d^{*}(Su,STu)\right].$$

Letting $k\to \infty$ and using the convergence of $\left\{S{x}_n\right\}$, we obtain $d(Su,S{y}^{*})=0$. Hence $Su=S{y}^{*}$. Since S is one-to-one, $u=y^{*}$. Thus u is a best proximity point of T.

Uniqueness follows similarly: if $x_1,x_2\in A$ are both best proximity points, then, using (4) with $u=x_1$, $x=x_1$, $v=x_2$, $y=x_2$, we get

$$d(S{x}_1,S{x}_2)\le \beta \left(d(S{x}_1,S{x}_2)\right)\left[d^{*}(S{x}_1,ST{x}_1)+d^{*}(S{x}_2,ST{x}_2)\right],$$

but both defect terms vanish (since $d(S{x}_i,ST{x}_i)=d(A,B)$), so the right-hand side is zero. Hence $S{x}_1=S{x}_2$, and injectivity of S yields $x_1=x_2$. □

Example 1.

Consider the metric space

$$X:=\{0,1\}\times [0,\infty )$$

with the Euclidean metric. Define

$$A:=\left\{\right(0,x):x\in [0,\infty \left)\right\},B:=\left\{\right(1,y):y\in [0,\infty \left)\right\}.$$

For $(0,x)\in A$ and $(1,y)\in B$

$$d{\left((0,x),(1,y)\right)}^2=1+{(x-y)}^2,$$

and so

$$d(A,B)=1,A_0=A,B_0=B.$$

Let $T:A\to B$ be given by

$$T(0,x)=(1,2x+1),$$

and let $S:X\to X$ be defined by

$$S(x,y)=(x,e^{-y}).$$

Clearly, S is one-to-one, $S\left(A\right)\subseteq A$ and $S\left(B\right)\subseteq B$. Moreover, $T\left(A_0\right)\subseteq B_0$.

We first note that T has no best proximity point. Indeed, a best proximity point $x^{*}=(0,t)$ would have to satisfy

$$d\left((0,t),T(0,t)\right)=d(A,B)=1.$$

However,

$$d{\left((0,t),(1,2t+1)\right)}^2=1+{(t-(2t+1))}^2=1+{(t+1)}^2>1$$

for all $t\ge 0$, so no such t exists.

On the other hand, consider the sequence ${\left\{(0,n)\right\}}_{n\ge 1}\subseteq A$. We have

$$S(0,n)=(0,e^{-n})⟶(0,0)\in A.$$

So $\left\{S\right(0,n\left)\right\}$ converges in X, but $\left\{\right(0,n\left)\right\}$ has no convergent subsequence in A itself (the second coordinate diverges to $+\infty$). Hence S fails to be subsequentially convergent in the sense required by Theorems 3 and 4.

In this example, the S-proximal Geraghty (or Kannan–Geraghty) condition is satisfied only vacuously, since there are no quadruples $u,v,x,y\in A$ with

$$d(Su,STx)=d(Sv,STy)=d(A,B),$$

as one checks directly from the form of S and T. Nevertheless, the lack of subsequential convergence of S is accompanied by the failure of T to admit a best proximity point, showing that the subsequential convergence assumption on S is essential.

4. Application

In image registration and alignment problems, one seeks a transformation that maps feature points detected in one image to their corresponding positions in a second image [14,15]. When the two feature sets lie in disjoint coordinate frames (e.g., a source frame A and a target frame B), the matching transformation is inherently a non-self mapping $T:A\to B$, and exact alignment is generally impossible due to noise, occlusion, or model mismatch. The goal then reduces to finding a point $x^{*}\in A$ whose image $T{x}^{*}$ is as close as possible to the target frame—precisely the best proximity point problem.

As a concrete illustration, consider the task of aligning a set of equally spaced fiducial markers along a vertical line in a reference image (A) with a corresponding set of markers in a target image (B) that has been displaced horizontally by $\delta$ and scaled vertically by a factor $\kappa \in (0,1)$. The sets A and B model the marker positions in their respective coordinate frames, the mapping T encodes the displacement-plus-scaling transformation, and the best proximity point identifies the unique marker position that minimizes the residual alignment error.

We now formalize this scenario and verify that all hypotheses of our main results are satisfied.

4.1. Geometric Setting

Fix $\delta >0$ and consider the subsets

$$A=\left\{\right(0,t):t\in [0,1\left]\right\},B=\left\{\right(\delta ,s):s\in [0,1\left]\right\}$$

of ${\mathbb{R}}^2$ endowed with the Euclidean metric d. Since $A\cup B$ is compact, $(A\cup B,d)$ is a complete metric space.

For $(0,t)\in A$ and $(\delta ,s)\in B$

$$d{\left((0,t),(\delta ,s)\right)}^2={\delta }^2+{(t-s)}^2,$$

and this distance equals $d(A,B)$ if and only if $s=t$. Hence

$$d(A,B)=\delta ,A_0=A,B_0=B,$$

and both $A_0$ and $B_0$ are nonempty and closed.

Define $S:A\cup B\to A\cup B$ by

$$S={\mathrm{Id}}_{A\cup B}.$$

Then S is continuous, injective, $S\left(A_0\right)\subseteq A_0$, $S\left(B_0\right)\subseteq B_0$, and trivially subsequentially convergent.

4.2. Definition of the Non-Self Mapping

Let $\kappa \in (0,1)$ and define $T:A\to B$ by

$$T(0,t)=(\delta ,\kappa t).$$

Since $\kappa t\in [0,1]$ for all $t\in [0,1]$, it follows that $T\left(A_0\right)\subseteq B_0$.

4.3. Verification of the S-Proximal Geraghty Condition

Let $u=(0,u_2),v=(0,v_2),x=(0,x_2),y=(0,y_2)\in A$. Assume

$$d(Su,STx)=d(A,B)=\delta ,d(Sv,STy)=d(A,B)=\delta .$$

Using $S=\mathrm{Id}$ and $T(0,t)=(\delta ,\kappa t)$, these equalities become

$$d\left((0,u_2),(\delta ,\kappa x_2)\right)=\delta ,d\left((0,v_2),(\delta ,\kappa y_2)\right)=\delta .$$

Since

$$d{\left((0,a),(\delta ,b)\right)}^2={\delta }^2+{(a-b)}^2,$$

each equality holds if and only if $u_2=\kappa x_2$ and $v_2=\kappa y_2$. Thus

$$Su=(0,\kappa x_2),Sv=(0,\kappa y_2),$$

and hence

$$d(Su,Sv)=\kappa |x_2-y_2|=\kappa d(Sx,Sy).$$

Define $\beta :[0,\infty )\to [0,1)$ by $\beta \left(t\right)\equiv \kappa$. Since $\kappa <1$, we have $\beta \in \mathrm{\Gamma }$, and therefore

$$d(Su,Sv)\le \beta \left(d(Sx,Sy)\right)d(Sx,Sy)$$

holds for all $u,v,x,y\in A$, satisfying the proximal hypotheses. Hence T is an S-proximal Geraghty contraction.

4.4. Best Proximity Point

By Theorem 3, there exists a unique $x^{*}\in A$ such that

$$d(S{x}^{*},ST{x}^{*})=d(A,B)=\delta .$$

Writing $x^{*}=(0,t)$, we compute

$$d{\left((0,t),(\delta ,\kappa t)\right)}^2={\delta }^2+{(t-\kappa t)}^2={\delta }^2+{(1-\kappa )}^2{t}^2.$$

This equals ${\delta }^2$ if and only if $t=0$. Therefore,

$$x^{*}=(0,0),T{x}^{*}=(\delta ,0),$$

is the unique best proximity pair.

4.5. Convergence of the Proximal Sequence

Let $x_0=(0,t_0)\in A_0$ be arbitrary and define $\left\{x_n\right\}\subset A_0$ by

$$d\left(S{x}_{n+1},ST{x}_n\right)=d(A,B)=\delta ,n\ge 0.$$

With $x_n=(0,t_n)$, this becomes

$$d\left((0,t_{n+1}),(\delta ,\kappa t_n)\right)=\delta ,$$

which implies $t_{n+1}=\kappa t_n$. Thus,

$$t_n={\kappa }^n{t}_0⟶0,x_n⟶x^{*}=(0,0).$$

4.6. Kannan–Geraghty Variant

For the setting of Theorem 4, one may take

$$T(0,t)=(\delta ,0),t\in [0,1].$$

The proximal conditions again force $u=v=(0,0)$, and the defining inequality of the proximal Kannan–Geraghty condition holds trivially. The unique best proximity point is again $(0,0)$.

5. Conclusions

In this paper, we introduced two new classes of non-self mappings—S-proximal Geraghty contractions and S-proximal Kannan–Geraghty contractions of the first kind—and established existence and uniqueness theorems for their best proximity points in complete metric spaces (Theorems 3 and 4). These results unify and extend several earlier frameworks: they recover the classical proximal contractions of Sadiq Basha [3] when $S=\mathrm{Id}$ and $\beta$ is constant, the Geraghty-type proximal results of Mongkolkeha et al. [4] when the Kannan-type defect terms are absent, and the auxiliary-function approach of Moradi [8] when $\beta$ is constant. A counterexample (Example 1) demonstrated that the subsequential convergence assumption on the auxiliary function S is essential and cannot be omitted.

As an application, we modeled a simplified image-alignment problem in ${\mathbb{R}}^2$ and verified all hypotheses of our main theorems explicitly, computing the unique best proximity point and the geometric convergence rate of the proximal iteration in closed form.

To clarify the relationship between our results and the existing literature, we note the following specific comparisons. First, the S-proximal Geraghty framework (Theorem 3) strictly generalizes the Geraghty proximal results of Mongkolkeha et al. [4]: setting $S=\mathrm{Id}$ in our contractive condition precisely recovers their inequality $d(u_1,u_2)\le \beta \left(d(x_1,x_2)\right)d(x_1,x_2)$. Second, Theorem 4 extends the proximal Kannan results of Gabeleh [10] by replacing the fixed Kannan coefficient $\alpha$ with a Geraghty-type function $\beta \in \mathrm{\Gamma }$ and introducing the auxiliary mapping S; when $S=\mathrm{Id}$ and $\beta \equiv \alpha$, the Kannan–Geraghty condition (4) reduces to Gabeleh’s proximal Kannan inequality. Third, our results are consistent with, and do not contradict, the auxiliary-function framework of Moradi [8]: the Moradi setting corresponds to the special case $\beta \equiv k$ (constant) in Theorem 3. Finally, Example 1 shows that subsequential convergence of S cannot be dropped, a constraint not required in the non-auxiliary settings of [4] or [10], confirming that the auxiliary function introduces a genuinely new structural requirement.

Several directions for future work remain open. First, it would be interesting to extend the present results to partially ordered metric spaces or to b-metric spaces, where the triangle inequality constant may exceed one. Second, one could investigate whether the auxiliary function S can be relaxed from subsequential convergence to weaker compactness-type conditions while preserving the conclusions. Finally, the development of quantitative convergence rates—beyond the geometric rate obtained in our application—for general Geraghty-type control functions $\beta \in \mathrm{\Gamma }$ is an open problem of both theoretical and practical interest.