We develop fixed point and periodic point results for self-mappings on \(n\)-G-metric spaces. The definition is stated in a form compatible with the classical case \(n=3\), while the paper specializes to the genuinely higher-order case \(n\geq4\). The first result proves a sharp bound: a mapping satisfying an \(n\)-tuple contraction on pairwise distinct points has at most \(n-1\) periodic points, and hence at most \(n-1\) fixed points. This formulation makes clear the role of short periodic orbits, which are the natural obstruction to applying a distinct-tuple contraction along Picard blocks. Under \(d_G\)-completeness, orbital continuity and an \(n\)-orbit-separation condition, every Picard orbit either reaches a fixed point or converges to one. We also state direct \(n\)-ary Kannan- and Reich-type analogues and separate them from the standard consequences obtained from the associated metric. Finally, the total pairwise polygonal-length functional on normed spaces provides a concrete geometric model of the theory. To make the model computationally explicit, we add a Picard algorithm in \(n\)-G-metric spaces, pseudocode, convergence and error tables for several values of \(n\), \(\lambda\) and \(\rho\), and both linear and nonlinear discrete polygonal systems illustrating the comparison between the associated metric \(d_G\) and the pairwise polygonal-length dynamics.
Fixed point theory in generalized distance spaces is a classical part of nonlinear analysis. The Banach contraction principle [1], its Kannan-type and Reich-type variants [2,3], and the corresponding treatments in metric and distance spaces [4–6] provide the background for many modern extensions. Among these extensions, Mustafa and Sims introduced G-metric spaces through a three-variable distance function [7]. Subsequent works developed fixed point theory in this setting [8]; at the same time, the relation between G-metric fixed point theorems and the associated ordinary metric was clarified in [9]. Recent work of Jleli, Pacurar and Samet considered contractions acting on triples of distinct points and applications to mappings contracting triangle perimeters [10]. Related compact generalized metric models also appear in [11].
The present paper studies an \(n\)-variable version of this distinct-tuple phenomenon. Throughout the paper \(n\geq4\). The point is not merely to replace triples by \(n\)-tuples. If a contractive hypothesis is imposed only on pairwise distinct \(n\)-tuples, then it cannot be applied blindly to a Picard orbit: a short periodic repetition may appear before a block of length \(n\) is available. Thus the natural obstruction is not only the absence of a fixed point, but the possible presence of periodic points of length less than \(n\). This observation leads to a periodic-point formulation of the main result.
We use the term \(n\)-G-metric for the axiom system in Definition 1 because axiom (G3) was chosen so that the monotonicity feature of the original G-metric is preserved when one passes from three variables to \(n\) variables. The definition itself is meaningful for all \(n\geq3\); however, the present paper specializes to \(n\geq4\), and the case \(n=3\) is mentioned only to clarify compatibility with the Mustafa–Sims framework.
The main contributions are summarized as follows.
| Feature | Three-variable G-metric case | \(n\)-G-metric case |
|---|---|---|
| Number of variables | 3 | \(n\geq4\) |
| Contractive tuples | triples of distinct points | pairwise distinct \(n\)-tuples |
| Periodic obstruction | cycles of length \(<3\) | cycles of length \(<n\) |
| Periodic/fixed point bound | at most 2 periodic points | at most \(n-1\) periodic points |
| Orbit control | consecutive triples | blocks of length \(n\) |
| Geometric model | triangle perimeter | total pairwise polygonal-length functional |
The paper is organized as follows. §2 introduces \(n\)-G-metric spaces and their associated metric. §3 proves the sharp periodic and fixed point bounds and the main Picard convergence theorem. §4 establishes direct \(n\)-ary Kannan- and Reich-type results and then records the standard associated-metric corollaries. §5 presents examples and the pairwise polygonal-length dynamics. §6 gives a computational Picard scheme, pseudocode and numerical tests for discrete polygonal systems. §7 concludes the paper.
Throughout the paper, \(X\) denotes a nonempty set. Definition 1 is stated for every integer \(n\geq3\), in order to make explicit its compatibility with the classical three-variable G-metric. After the definition, all genuinely new results are formulated for the case \(n\geq4\).
Definition 1. A function
is called an \(n\)-G-metric on \(X\), where \(n\geq3\), if, for all \(\omega_1,\ldots,\omega_n,\theta,\upsilon,\phi_3,\ldots,\phi_n,\eta\in X\), the following conditions hold:
The pair \((X,G_n)\) is called an \(n\)-G-metric space.
Remark 1 (Terminology). The adjective “corrected”, used in an earlier version of this manuscript, has been removed. The present definition is called simply an \(n\)-G-metric. The reason for the form of axiom (G3) is that it preserves the monotonicity axiom of G-metric spaces introduced by Mustafa and Sims [7] when one passes from three variables to \(n\) variables.
Remark 2. When \(n=3\), axiom (G3) reduces to the usual condition
Thus the \(n\)-G-framework is compatible with the standard three-variable setting.
Remark 3 (On the exclusion in axiom (G3)). The restriction \((\phi_3,\ldots,\phi_n)\neq(\upsilon,\ldots,\upsilon)\) is retained in order to parallel the classical Mustafa–Sims monotonicity axiom, where the comparison tuple must not collapse to the repeated two-point configuration. In many concrete examples, including the diameter model and the pairwise polygonal-length model, the same inequality remains true even when \((\phi_3,\ldots,\phi_n)=(\upsilon,\ldots,\upsilon)\). Thus the exclusion is not needed for those examples; it is part of the abstract definition chosen to preserve the traditional G-metric pattern.
Example 1. Let \((X,d)\) be a metric space. Define
Then \(G_n\) is an \(n\)-G-metric on \(X\).
Proof. Conditions (G1), (G2) and (G4) are immediate. For (G3), both \(\theta\) and \(\upsilon\) appear among the entries of \((\theta,\upsilon,\phi_3,\ldots,\phi_n)\), and therefore
For (G5), the triangle inequality gives the usual diameter estimate:
and taking the maximum over \(i,j\) yields the desired inequality. \(\square\)
Definition 2. For an \(n\)-G-metric space \((X,G_n)\), define
where in each term the repeated variable occurs \(n-1\) times.
Proposition 1. The function \(d_G\) is a metric on \(X\).
Proof. Nonnegativity and symmetry are immediate. If \(d_G(\theta,\upsilon)=0\), then both terms in the definition vanish, and axiom (G1) gives \(\theta=\upsilon\). Conversely, \(d_G(\theta,\theta)=0\).
Let \(\theta,\upsilon,\eta\in X\). By axiom (G5) and symmetry,
and similarly,
Adding the two inequalities yields
Hence \(d_G\) is a metric. \(\square\)
Definition 3. An \(n\)-G-metric space \((X,G_n)\) is called \(d_G\)-complete, or complete with respect to the associated metric \(d_G\), if the associated metric space \((X,d_G)\) is complete. In the sequel, the phrase “complete \(n\)-G-metric space” always means \(d_G\)-complete; no independent \(G_n\)-Cauchy structure is being introduced.
Definition 4. A self-map \(T:X\to X\) is called orbitally continuous if, whenever a subsequence of iterates \(T^{m_k}\theta\) converges to some \(p\in X\), one has
Remark 4. The orbital continuity assumption is used only at the final step of the Picard argument, namely to identify the limit of a Cauchy orbit with a fixed point. For ordinary metric contractions of Banach, Kannan or Reich type this step is automatic. For contractions imposed only on pairwise distinct \(n\)-tuples, the contraction may no longer be applicable when the limiting tuple contains repeated entries. Therefore we keep orbital continuity as an explicit hypothesis in the direct \(n\)-tuple theorems. It could be replaced, without changing the proof, by any condition implying the following implication along Picard orbits:
For instance, it is enough to assume \(d_G\)-continuity of \(T\) at the orbit limit, closedness of the graph of \(T\) along Picard orbits, or an additional contractive condition valid also for tuples with repeated entries. In the associated-metric corollaries below, no such extra continuity is needed.
We begin with the fixed point cardinality bound.
Proposition 2. Let \((X,G_n)\) be an \(n\)-G-metric space, and let \(T:X\to X\) be a map such that there exists \(\lambda\in[0,1)\) with
for all pairwise distinct \(\omega_1,\ldots,\omega_n\in X\). Then
Proof. Assume, to the contrary, that there exist \(n\) pairwise distinct fixed points \(p_1,\ldots,p_n\in X\). Then \(Tp_i=p_i\) for \(i=1,\ldots,n\), and the contractive assumption gives
Thus \((1-\lambda)G_n(p_1,\ldots,p_n)\leq0\). Since \(\lambda<1\), it follows that \(G_n(p_1,\ldots,p_n)=0\), which contradicts axiom (G1) because the points are pairwise distinct. Therefore \(T\) cannot have \(n\) distinct fixed points, and \(|\operatorname{Fix}(T)|\leq n-1\). \(\square\)
Remark 5. The preceding proposition is the fixed point part of a stronger periodic-point statement. This is the form suggested by the distinct-tuple nature of the contraction: short cycles are the obstruction to applying the contraction along Picard blocks.
Definition 5. A point \(x\in X\) is called periodic for \(T\) if there exists an integer \(q\geq1\) such that \(T^q x=x\). The least such \(q\) is the period of \(x\). The set of all periodic points of \(T\) is denoted by \(\operatorname{Per}(T)\).
Proposition 3 (Periodic point bound). Let \((X,G_n)\) be an \(n\)-G-metric space, and let \(T:X\to X\) satisfy
for all pairwise distinct \(\omega_1,\ldots,\omega_n\in X\). Then the following assertions hold.
Proof. Assume first that \(x\) has least period \(q\geq n\). For \(j=0,1,\ldots,q\), define
Because the period of \(x\) is \(q\geq n\), every block
contains \(n\) pairwise distinct points. Indeed, if \(T^{j+r}x=T^{j+s}x\) for some \(0\leq r<s\leq n-1\), then \(T^{s-r}(T^{j+r}x)=T^{j+r}x\), so the orbit has period dividing \(s-r<n\leq q\), contrary to the minimality of \(q\). Hence the contractive assumption applies to each of these blocks and gives
Since \(T^q x=x\), periodicity gives \(A_q=A_0\). Therefore
As \(0\leq\lambda<1\), this implies \(A_0=0\), contradicting axiom (G1), because the entries \(x,Tx,\ldots,T^{n-1}x\) are distinct. Thus \(T\) has no periodic orbit of length at least \(n\).
Now suppose that \(p_1,\ldots,p_n\) are \(n\) distinct periodic points. Let \(Q\) be a common multiple of their periods; hence \(T^Qp_i=p_i\) for every \(i\). We first prove that, for every \(j=0,1,\ldots,Q\), the tuple
is pairwise distinct. If, for some \(j\) and some \(i\neq \ell\), one had \(T^j p_i=T^j p_\ell\), then
where the exponents are understood modulo the common period if necessary. This contradicts the assumed distinctness of \(p_i\) and \(p_\ell\). Thus all iterated tuples remain pairwise distinct, and the contraction can be applied repeatedly:
Consequently,
which forces \(G_n(p_1,\ldots,p_n)=0\), again contradicting (G1). Hence no \(n\) distinct periodic points can exist. Since fixed points are periodic points of period one, the fixed point bound follows. \(\square\)
Remark 6. For \(n=3\), this recovers the pattern of the three-variable result in which a contraction on distinct triples can allow at most two periodic points. The statement above makes explicit that the higher-order conclusion concerns periodic points, not only fixed points.
The existence part requires a condition ensuring that the contractive hypothesis can actually be applied along an orbit.
Definition 6. A self-map \(T:X\to X\) is called \(n\)-orbit-separated if, for every non-fixed point \(\theta\in X\), the points
are pairwise distinct.
Remark 7. For \(n=3\), the condition \(T\theta\neq\theta\Rightarrow T^2\theta\neq\theta\) is enough to guarantee that \(\theta,T\theta,T^2\theta\) are pairwise distinct whenever \(\theta\) is not a fixed point. For \(n\geq4\), the contraction acts on \(n\)-tuples, so the stronger orbit-separation condition in the preceding definition is a convenient sufficient hypothesis for a complete Picard proof. It is not a necessary dynamical requirement. If an orbit reaches a fixed point after finitely many steps, convergence is already trivial; if it enters a short cycle of length \(<n\), the distinct-tuple contraction cannot be applied indefinitely; and if the orbit remains separated in blocks of length \(n\), the contraction argument below applies.
Lemma 1. Let \((X,G_n)\) be an \(n\)-G-metric space, and let \((\omega_m)_{m\geq0}\) be a sequence in \(X\) such that, for some \(m\), the points
are pairwise distinct. Then
Proof. By axiom (G3),
By symmetry, the same estimate holds with \(\omega_m\) and \(\omega_{m+1}\) interchanged:
Adding the two inequalities proves the claim. \(\square\)
The next theorem is the \(n\)-ary counterpart of the Picard convergence theorem for contractions on distinct triples considered in [10]. The additional \(n\)-orbit-separation condition prevents repetitions in the first \(n\) consecutive Picard iterates, while orbital continuity identifies the limit as a fixed point.
Theorem 1. Let \((X,G_n)\) be a complete \(n\)-G-metric space, and let \(T:X\to X\) satisfy the following assumptions:
for all pairwise distinct \(\omega_1,\ldots,\omega_n\in X\).
Then, for every \(\omega_0\in X\), the Picard sequence \(\omega_m:=T^m\omega_0\) either reaches a fixed point after finitely many steps or converges in the metric \(d_G\) to a fixed point of \(T\). Consequently,
Proof. Fix \(\omega_0\in X\) and define \(\omega_m:=T^m\omega_0\) for \(m\geq0\). If some \(\omega_m\) is a fixed point, the proof is complete. We may therefore assume that no \(\omega_m\) is fixed. By \(n\)-orbit separation, each block
is pairwise distinct.
Set
Applying the contractive condition to the pairwise distinct tuple \((\omega_m,\omega_{m+1},\ldots,\omega_{m+n-1})\), we obtain
Hence, by induction, \(a_m\leq \lambda^m a_0\) for \(m\geq0\). By the block estimate lemma above,
Let \(p\geq1\). By the triangle inequality for \(d_G\),
This tends to \(0\) as \(m\to\infty\), uniformly in \(p\). Thus \((\omega_m)\) is a Cauchy sequence in the complete metric space \((X,d_G)\), so there exists \(u\in X\) such that \(\omega_m\to u\).
Since \(T\) is orbitally continuous, the convergence of the orbit implies
But \((\omega_{m+1})\) is a tail of \((\omega_m)\), so also \(\omega_{m+1}\to u\). By uniqueness of limits in the metric space \((X,d_G)\), we obtain \(Tu=u\). Hence \(u\in\operatorname{Fix}(T)\).
Finally, Proposition 2 yields \(|\operatorname{Fix}(T)|\leq n-1\). Since we have just shown that \(\operatorname{Fix}(T)\neq\varnothing\), it follows that
\(\square\)
Corollary 1. Under the assumptions of the Picard convergence theorem above, if \(T\) has \(n-1\) pairwise distinct fixed points, then they exhaust the whole fixed point set.
Proof. This follows immediately from Proposition 2. \(\square\)
Corollary 2 (Periodic alternative). Let \((X,G_n)\) be complete, let \(T:X\to X\) be orbitally continuous, and assume that \(T\) satisfies the \(n\)-tuple contraction from the Picard convergence theorem above. If \(T\) is not \(n\)-orbit-separated, then \(T\) has a periodic point of period strictly smaller than \(n\). If \(T\) is \(n\)-orbit-separated, then every Picard orbit converges to a fixed point.
Proof. If \(T\) is not \(n\)-orbit-separated, then there exists a non-fixed point \(\theta\) for which two elements among \(\theta,T\theta,\ldots,T^{n-1}\theta\) coincide. Hence \(z=T^i\theta\) satisfies \(T^qz=z\) for some \(1\leq q<n\). Thus \(z\) is a periodic point of period less than \(n\). If \(T\) is \(n\)-orbit-separated, the Picard convergence theorem above applies directly. \(\square\)
We begin with direct \(n\)-G-metric analogues of Kannan- and Reich-type contraction principles, where the contractive assumptions are imposed on pairwise distinct \(n\)-tuples.
Theorem 2 (\(n\)-G Kannan analogue). Let \((X,G_n)\) be a complete \(n\)-G-metric space, and let \(T:X\to X\) be orbitally continuous and \(n\)-orbit-separated. Assume that there exists \(\lambda\in[0,1/n)\) such that
for all pairwise distinct \(\omega_1,\ldots,\omega_n\in X\), where \(T\omega_i\) is repeated \(n-1\) times in each term of the sum. Then
Proof. Fix \(\omega_0\in X\) and define the Picard sequence \(\omega_m:=T^m\omega_0\) for \(m\geq0\). If \(\omega_m=\omega_{m+1}\) for some \(m\), then \(\omega_m\in\operatorname{Fix}(T)\) and the cardinality conclusion follows from the final part of the proof below. Assume therefore that \(\omega_m\neq\omega_{m+1}\) for all \(m\geq0\). Since \(T\) is \(n\)-orbit-separated, each block \(\omega_m,\omega_{m+1},\ldots,\omega_{m+n-1}\) is pairwise distinct. Set
Applying the contractive assumption to \((\omega_m,\ldots,\omega_{m+n-1})\), we obtain
For \(0\leq i\leq n-2\), the block \((\omega_m,\ldots,\omega_{m+n-1})\) is pairwise distinct and contains both \(\omega_{m+i}\) and \(\omega_{m+i+1}\). By symmetry, \(A_m\) can be written with these two entries in the first two positions and with the remaining \(n-2\) distinct entries afterwards. Axiom (G3) then gives
For \(i=n-1\), the same argument is applied to the next block \((\omega_{m+1},\ldots,\omega_{m+n})\), and yields
Consequently,
that is,
The condition \(\lambda<1/n\) is exactly what gives \(0\leq q<1\). By induction, \(A_m\leq q^mA_0\) for \(m\geq0\). The block estimate lemma gives
Thus \((\omega_m)\) is Cauchy in the complete metric space \((X,d_G)\), so there exists \(u\in X\) such that \(\omega_m\to u\). By orbital continuity,
Since limits in \((X,d_G)\) are unique, we get \(Tu=u\). Hence \(\operatorname{Fix}(T)\neq\varnothing\).
Finally, let \(u_1,\ldots,u_n\in\operatorname{Fix}(T)\) be pairwise distinct. The Kannan-type inequality gives
contrary to axiom (G1). Therefore \(|\operatorname{Fix}(T)|\leq n-1\). \(\square\)
Theorem 3 (\(n\)-G Reich analogue). Let \((X,G_n)\) be a complete \(n\)-G-metric space, and let \(T:X\to X\) be orbitally continuous and \(n\)-orbit-separated. Assume that there exist nonnegative constants \(a_1,\ldots,a_n,a_{n+1}\) such that
and
for all pairwise distinct \(\omega_1,\ldots,\omega_n\in X\), where \(T\omega_i\) is repeated \(n-1\) times in each term. Then
Proof. Fix \(\omega_0\in X\) and define \(\omega_m:=T^m\omega_0\). If \(\omega_m=\omega_{m+1}\) for some \(m\), then \(\omega_m\) is a fixed point. Assume that \(\omega_m\neq\omega_{m+1}\) for all \(m\geq0\). Since \(T\) is \(n\)-orbit-separated, each block \(\omega_m,\omega_{m+1},\ldots,\omega_{m+n-1}\) is pairwise distinct. Set
Applying the Reich-type assumption to \((\omega_m,\ldots,\omega_{m+n-1})\), we get
For \(i=1,\ldots,n-1\), the entries \(\omega_{m+i-1}\) and \(\omega_{m+i}\) both occur in the separated block defining \(A_m\). After permuting the variables by symmetry, axiom (G3) gives
For \(i=n\), the corresponding two entries occur in the next separated block defining \(A_{m+1}\), so the same symmetry–(G3) argument gives
Therefore,
Equivalently,
Since \(a_1+\cdots+a_n+a_{n+1}<1\), we have \(a_n<1\) and \(a_1+\cdots+a_{n-1}+a_{n+1}<1-a_n\), hence \(0\leq q<1\). Thus \(A_m\leq q^mA_0\) for \(m\geq0\). By the block estimate lemma above,
so \((\omega_m)\) is Cauchy in \((X,d_G)\). By completeness, there exists \(u\in X\) such that \(\omega_m\to u\), and orbital continuity yields \(\omega_{m+1}\to Tu\). Therefore \(Tu=u\), so \(\operatorname{Fix}(T)\neq\varnothing\).
Now let \(u_1,\ldots,u_n\in\operatorname{Fix}(T)\) be pairwise distinct. Then
so \((1-a_{n+1})G_n(u_1,\ldots,u_n)\leq0\). Since \(a_{n+1}<1\), this contradicts axiom (G1). Hence \(|\operatorname{Fix}(T)|\leq n-1\). \(\square\)
Remark 8. For \(n=3\), Theorems 2 and 3 recover the pattern of the corresponding three-variable fixed point results in G-metric spaces. In the higher-order \(n\)-G setting, the explicit \(n\)-orbit-separation hypothesis is the natural substitute that allows the Picard blocks of length \(n\) to remain pairwise distinct.
We next record two associated-metric consequences. Since \(d_G\) is canonically generated by the \(n\)-G-metric, they remain intrinsic to the present framework while giving the familiar uniqueness conclusions.
Corollary 3 (Kannan type via the associated metric). Let \((X,G_n)\) be a complete \(n\)-G-metric space, and let \(T:X\to X\) satisfy
for all \(x,y\in X\), where \(0\leq\kappa<1/2\). Then \(T\) has a unique fixed point \(u\in X\). Moreover, for every \(\omega_0\in X\), the Picard sequence \(\omega_m:=T^m\omega_0\) converges to \(u\) in the metric \(d_G\).
Proof. This is exactly Kannan’s fixed point theorem [2] applied to the complete metric space \((X,d_G)\). No additional \(n\)-tuple argument is involved. \(\square\)
Remark 9. The Kannan condition in Corollary 3 may be written explicitly in terms of \(G_n\) as
Corollary 4 (Reich type via the associated metric). Let \((X,G_n)\) be a complete \(n\)-G-metric space, and let \(T:X\to X\) satisfy
for all \(x,y\in X\), where \(\alpha,\beta,\gamma\geq0\) and
Then \(T\) has a unique fixed point \(u\in X\), and for every \(\omega_0\in X\), the Picard sequence \(T^m\omega_0\) converges to \(u\) in the metric \(d_G\).
Proof. This follows from the standard Reich fixed point theorem in the complete metric space \((X,d_G)\); see [3] and the monograph treatments [4–6]. \(\square\)
Remark 10. The Reich condition becomes the Banach condition when \(\beta=\gamma=0\), and it becomes a mixed Kannan-Reich inequality when \(\alpha=0\).
We now collect examples that complement the preceding theory. They show the sharpness of the fixed point bound, the non-vacuity of the direct \(n\)-tuple assumptions, and the usefulness of the associated-metric viewpoint.
Example 2 (Sharpness of the bound: vacuous model). Let \(X=\{1,\ldots,n-1\}\), and let \(d\) be the discrete metric on \(X\). Define
By the metric-diameter example above, \((X,G_n)\) is an \(n\)-G-metric space. Since \(X\) contains only \(n-1\) points, there are no \(n\) pairwise distinct elements of \(X\), so the distinct-tuple contractive hypothesis in Proposition 2 is vacuous. If \(T=\operatorname{id}_X\), then
Thus the estimate in Proposition 2 is sharp.
Example 3 (Sharpness of the bound: non-vacuous model). Fix \(\lambda\in(0,1)\) and choose numbers \(\varepsilon,L>0\) such that
Let
with \(L>(n-2)\varepsilon\), and endow \(X\) with the diameter \(n\)-G-metric
Define \(T:X\to X\) by
Then \(T\) has exactly the \(n-1\) fixed points
Moreover, \(X\) has exactly \(n\) elements, so every pairwise distinct \(n\)-tuple is a permutation of all elements of \(X\). For such a tuple,
whereas the image tuple consists of the cluster points \(0,\varepsilon,\ldots,(n-2)\varepsilon\) with one repetition of \(0\), hence
Therefore the distinct-tuple contraction has genuine content and the bound \(|\operatorname{Fix}(T)|\leq n-1\) is still attained.
Example 4 (A direct \(n\)-G Kannan model). Let \(X=[0,1]\), let
and choose \(\rho\in(0,1/(n+1))\). Define \(T(x)=\rho x\) for \(x\in X\). Then \((X,G_n)\) is complete and \(T\) is orbitally continuous. Also, if \(x\neq0\), then
are pairwise distinct; hence \(T\) is \(n\)-orbit-separated.
For pairwise distinct \(\omega_1,\ldots,\omega_n\in X\), put
Then
Moreover,
and, for every pair \(i,j\),
Thus
Consequently,
Since \(\rho<1/(n+1)\), one has \(\rho/(1-\rho)<1/n\). Therefore Theorem 2 applies with any
and the unique fixed point is \(0\).
Example 5 (A direct \(n\)-G Reich model). Let \(X=[0,1]\) and let \(G_n\) be the same \(n\)-G-metric as in Example 3. Fix \(p\in[0,1]\) and \(\rho\in(0,1)\). Define
Then, for every \(m\geq0\) and every \(x\in X\),
so \(T\) is orbitally continuous and every orbit converges to \(p\). If \(x\neq p\), then the first \(n\) iterates of \(x\) are pairwise distinct, so \(T\) is \(n\)-orbit-separated. Moreover, for pairwise distinct \(\omega_1,\ldots,\omega_n\in X\),
Thus the Reich-type inequality in Theorem 3 holds with \(a_1=\cdots=a_n=0\), \(a_{n+1}=\rho<1\). Consequently,
Example 6 (A Kannan contraction via the associated metric). Let \(X=[0,1]\) with the usual metric, and define
Then \((X,G_n)\) is complete, and the associated metric is
Fix \(c\in[0,1]\) and define the constant map \(T(x)=c\) for all \(x\in X\). Then, for all \(x,y\in X\),
for every \(\kappa\in[0,1/2)\). Hence Corollary 3 applies. The unique fixed point is \(c\), and every Picard orbit reaches it after one step.
Example 7 (A Reich contraction via the associated metric). Let \(X=\mathbb{R}\) and define
Fix \(\mu\in(0,1)\) and define \(T(x)=\mu x\). Then
Therefore the Reich condition in Corollary 4 holds with \((\alpha,\beta,\gamma)=(\mu,0,0)\). The unique fixed point is
and \(T^m x=\mu^m x\to0\) for every \(x\in\mathbb{R}\).
We now introduce a concrete \(n\)-G-metric generated by polygonal configurations.
Proposition 4. Let \(E\) be a normed linear space. For \(\omega_1,\ldots,\omega_n\in E\), define
Then \(\Pi_n\) is an \(n\)-G-metric on \(E\).
Proof. If all points are equal, then each term in the sum vanishes, so \(\Pi_n(\omega_1,\ldots,\omega_n)=0\). Conversely, if \(\Pi_n(\omega_1,\ldots,\omega_n)=0\), then every term \(\|\omega_i-\omega_j\|\) is zero, hence all \(\omega_i\) are equal. Thus (G1) holds.
If \(\theta\neq\upsilon\), then
which proves (G2).
To verify (G3), let \((\phi_3,\ldots,\phi_n)\neq(\upsilon,\ldots,\upsilon)\). Then
By the triangle inequality,
Therefore
Symmetry is immediate, so (G4) holds.
Finally, for (G5), write
Using the triangle inequality,
and hence
Thus (G5) holds, and \(\Pi_n\) is an \(n\)-G-metric. \(\square\)
Remark 11. The associated metric of this model is explicit:
Consequently, \(d_G\)-completeness in the pairwise polygonal-length model is exactly norm completeness, up to the constant factor \(2(n-1)\).
Remark 12. For \(n=3\), the quantity \(\Pi_3(\omega_1,\omega_2,\omega_3)\) is the usual perimeter of the triangle with vertices \(\omega_1,\omega_2,\omega_3\). For \(n\geq4\), \(\Pi_n\) is not the ordinary cyclic perimeter of an \(n\)-gon; rather, it is the total pairwise edge length, that is, the sum of all distances between pairs of vertices. It is symmetric in all variables and therefore fits naturally into the \(n\)-G-metric framework.
Example 8 (Polygon-perimeter dynamics). Let \(E=\mathbb{R}^m\), let \(p\in E\), and let \(K\subset E\) be a nonempty closed convex set such that \(p\in K\). On \(K\), consider the \(n\)-G-metric \(\Pi_n\) from Proposition 4. Fix \(\rho\in(0,1)\) and define
Then \(T(K)\subset K\) and, for all \(\omega_1,\ldots,\omega_n\in K\),
Thus the total total pairwise polygonal length contracts by the factor \(\rho\) at each iteration. Moreover, the associated metric is
Hence
so Corollary 4 applies with \((\alpha,\beta,\gamma)=(\rho,0,0)\). Consequently, \(T\) has the unique fixed point \(p\), and for every initial vertex \(x\in K\),
Geometrically, if one starts with an \(n\)-vertex configuration \((\omega_1,\ldots,\omega_n)\), then the iterated configuration \((T^m\omega_1,\ldots,T^m\omega_n)\) has total pairwise polygonal length
so the whole polygon collapses toward the equilibrium vertex \(p\) while preserving the affine directions of the edges.
This section records an explicit computational form of the Picard procedure used in the preceding results. The aim is not to replace the theoretical assumptions, but to show how the \(n\)-tuple estimates can be monitored numerically. The quantities used below are the block residual
the one-step associated-metric error
and, in the polygonal model, the total perimeter
The estimate proved in Lemma 1 gives \(S_k\leq2A_k\), while the Banach-type \(n\)-tuple contraction gives \(A_{k+1}\leq \lambda A_k\) whenever the consecutive block is pairwise distinct.
Let \((X,G_n)\) be a complete \(n\)-G-metric space and let \(T:X\to X\) be a self-map satisfying a direct \(n\)-tuple contraction on pairwise distinct tuples. Starting from \(x_0\in X\), the numerical Picard procedure is the following.
Algorithm 1: Picard iteration in an \(n\)-G-metric space
Input: \(x_0\in X\), map \(T\), order \(n\geq4\), tolerance \(\varepsilon>0\), maximum number of iterations \(N_{\max}\).
For \(k=0,1,\ldots,N_{\max}\):
Output: approximate fixed point, residual history \((A_k)\), associated-metric step history \((S_k)\).
The convergence proof of Theorem 1 gives the a posteriori tail estimate
whenever the hypotheses of that theorem hold and \(u\) is the fixed point reached by the Picard orbit.
For implementation in a numerical language, the previous procedure can be written as follows.
Pseudocode
We now give a computational model in \(\mathbb R^2\). Fix \(p=(0,0)\) and let
be a planar rotation. For a contraction factor \(\rho\in(0,1)\) define
For an \(n\)-vertex configuration \(V^{(k)}=(v_1^{(k)},\ldots,v_n^{(k)})\) the discrete polygonal dynamics is
Since rotations preserve Euclidean distances, the pairwise polygonal-length \(n\)-G-metric satisfies
Thus the \(n\)-tuple contraction constant is
Moreover, the associated metric of \(\Pi_n\) is
and hence
This gives a direct comparison between the polygonal dynamics and the associated metric: both decay with the same factor \(\rho\).
For the numerical tests, the initial polygon is chosen as
and we use \(\alpha=\pi/8\). The quantities reported are
and the residual
For this model, \(P_k/P_0=E_k/E_0=\rho^k\) and \(R_k=(1+\rho^2-2\rho\cos\alpha)^{1/2}\rho^k\).
The first table records the convergence history for \(n=6\), \(\lambda=\rho=0.55\) and \(\alpha=\pi/8\). Since this linear model has the exact identity \(P_k/P_0=E_k/E_0=\rho^k\), the table should be read as an illustrative verification of the theoretical formula rather than as independent computational validation.
| \(k\) | \(P_k\) | \(P_k/P_0\) | \(E_k/E_0\) | \(R_k\) |
|---|---|---|---|---|
| 0 | 22.3923 | 1.000000 | 1.000000 | 0.4814 |
| 1 | 12.3158 | 0.550000 | 0.550000 | 0.2648 |
| 2 | 6.7737 | 0.302500 | 0.302500 | 0.1456 |
| 4 | 2.0490 | 0.091506 | 0.091506 | 0.0441 |
| 8 | 0.1875 | 0.008373 | 0.008373 | 0.0040 |
| 12 | 0.0172 | 0.000766 | 0.000766 | 0.0004 |
The second table compares several values of \(n\), \(\lambda\) and \(\rho\). Here \(\lambda=\rho\), because the mapping is an exact pairwise polygonal-length contraction. The stopping index \(k_{10^{-6}}\) is the first integer \(k\) for which \(E_k/E_0\leq10^{-6}\).
| \(n\) | \(\rho\) | \(\lambda\) | \(P_0\) | \(P_{10}/P_0\) | \(k_{10^{-6}}\) |
|---|---|---|---|---|---|
| 4 | 0.25 | 0.25 | 9.6569 | \(9.5367\cdot10^{-7}\) | 10 |
| 4 | 0.50 | 0.50 | 9.6569 | \(9.7656\cdot10^{-4}\) | 20 |
| 4 | 0.75 | 0.75 | 9.6569 | \(5.6314\cdot10^{-2}\) | 49 |
| 6 | 0.25 | 0.25 | 22.3923 | \(9.5367\cdot10^{-7}\) | 10 |
| 6 | 0.50 | 0.50 | 22.3923 | \(9.7656\cdot10^{-4}\) | 20 |
| 6 | 0.75 | 0.75 | 22.3923 | \(5.6314\cdot10^{-2}\) | 49 |
| 8 | 0.25 | 0.25 | 41.0060 | \(9.5367\cdot10^{-7}\) | 10 |
| 8 | 0.50 | 0.50 | 41.0060 | \(9.7656\cdot10^{-4}\) | 20 |
| 8 | 0.75 | 0.75 | 41.0060 | \(5.6314\cdot10^{-2}\) | 49 |
The table shows two complementary effects. The decay rate is controlled by \(\rho=\lambda\), independently of \(n\). By contrast, the initial pairwise polygonal-length \(P_0\) increases with \(n\) because more pairwise edges are included in \(\Pi_n\). This is why the relative ratios \(P_k/P_0\) and \(E_k/E_0\) are the natural quantities for comparing different values of \(n\).
To include a less rigid example, consider
where \(0<\rho<1\). The radial map \(x\mapsto x/(1+\|x\|)\) is nonexpansive in the Euclidean norm, and the rotation is an isometry; hence
Therefore the pairwise polygonal-length functional satisfies
but the residual ratios are no longer forced to be exactly \(\rho^k\). For \(n=6\), \(\rho=0.55\), \(\alpha=\pi/8\) and the same initial regular hexagon, one obtains the following illustrative values.
| \(k\) | \(P_k\) | \(P_k/P_0\) | \(E_k/E_0\) | \(R_k\) |
|---|---|---|---|---|
| 0 | 22.3923 | 1.000000 | 1.000000 | 0.753320 |
| 1 | 6.1579 | 0.275000 | 0.275000 | 0.171519 |
| 2 | 2.6563 | 0.118627 | 0.118627 | 0.068481 |
| 4 | 0.6787 | 0.030311 | 0.030311 | 0.016563 |
| 8 | 0.0585 | 0.002614 | 0.002614 | 0.001401 |
| 12 | 0.0053 | 0.000238 | 0.000238 | 0.000127 |
This nonlinear test has the same qualitative conclusion as the theory: the associated-metric error and the pairwise polygonal length both tend to zero, while the decay history is not merely a restatement of a closed-form identity.
For the pairwise polygonal-length \(n\)-G-metric \(\Pi_n\), the associated metric measures the distance between two individual vertices, while \(\Pi_n\) measures the dispersion of the whole configuration. In the present model,
whereas
Thus both quantities have the same asymptotic convergence factor, but they encode different information: \(E_k\) measures how far the farthest vertex is from the equilibrium point, while \(P_k\) measures how much spread remains inside the polygon. Numerically, this gives a simple diagnostic: convergence in \(d_G\) confirms collapse toward the fixed point, and convergence of \(P_k\) confirms collapse of the entire polygonal configuration.
This discrete model can be interpreted as a damped rotating polygonal system. At each step every vertex is rotated by the same angle and multiplied by \(\rho\). The rotation preserves the shape angles, while the factor \(\rho\) reduces all pairwise distances. Hence the system is a genuine computational example of the theory: the fixed point is \(0\), every vertex converges to \(0\), and the total \(n\)-G pairwise polygonal-length decays exactly according to the predicted contraction law.
We revised the fixed point theory for self-maps on \(n\)-G-metric spaces in a form that emphasizes the role of periodic points. A contraction acting on pairwise distinct \(n\)-tuples cannot have \(n\) distinct periodic points; in particular, it cannot have a periodic orbit of length at least \(n\), and it has at most \(n-1\) fixed points. This periodic formulation clarifies the precise obstruction created by distinct-tuple hypotheses and distinguishes the present results from ordinary metric fixed point principles.
Existence of fixed points is obtained through Picard iteration under an \(n\)-orbit-separation condition and orbital continuity. The latter assumption is explicitly retained because the direct \(n\)-tuple contraction gives estimates only on separated blocks and does not by itself control tuples with repeated limiting entries. Direct \(n\)-ary Kannan and Reich analogues were proved in the same framework. By contrast, results formulated only through the associated metric \(d_G\) were stated as standard metric corollaries and not as new independent \(n\)-G principles.
The examples show sharpness, direct \(n\)-tuple models and metric consequences. The total pairwise polygonal-length functional \(\Pi_n\) gives a concrete geometric realization of the theory: under affine contractions, an \(n\)-vertex configuration collapses toward an equilibrium while its total pairwise perimeter decays geometrically. The computational section makes this mechanism explicit by giving a Picard algorithm, pseudocode, convergence tables and a damped rotating polygonal system. The numerical tests confirm that the associated-metric error and the pairwise polygonal-length residual decay with the predicted factor \(\lambda=\rho\). Possible further directions include Chatterjea-type contractions, common periodic point theorems, multivalued analogues, cyclic mappings, best-proximity versions and larger-scale computational schemes in \(n\)-G-metric spaces.
Conflicts of Interest: The author declares that there are no competing interests of a financial, personal or professional nature that could have influenced the work reported in this paper.
Funding Information: The author received no financial support for the research, authorship, or publication of this article.
Acknowledgments: The author sincerely thanks the editor and the anonymous reviewers for their valuable comments and suggestions, which improved the quality of this manuscript.
The author gratefully acknowledges Dr. Salem Rebhi for his valuable support, encouragement, and helpful advice during the preparation of this work.
