Higherorder Lipschitz mappings
 Jeffery Ezearn^{1}Email author
https://doi.org/10.1186/s1366301503341
© Ezearn 2015
Received: 10 March 2015
Accepted: 18 May 2015
Published: 16 June 2015
Abstract
We study selfmappings on complete metric spaces, which we refer to as higherorder Lipschitz mappings. These mappings generalise Lipschitz mappings, the latter which are equivalent to firstorder Lipschitz mappings studied in this paper. The main result of this paper is to extend the Banach fixed point theorem (and an oftencited generalisation) to higherorder contraction mappings. We also present results on the problem of local Lipschitzity of these higherorder Lipschitz mappings.
Keywords
MSC
1 Introduction
Let \((\mathcal{X},d) \) be a complete metric space and let \(T:\mathcal{X}\to\mathcal{X} \) be a Lipschitz mapping, that is, \(d(Ty,Tx)\leq cd(y,x)\) for all \(x,y \in\mathcal{X}\) where \(c\ge0 \). When \(0\le c<1 \), then T is referred to as a contraction mapping and when \(c=1 \), then T is referred to as a nonexpansive mapping. In this paper, we consider the following generalisation of Lipschitz mappings:
Definition 1.1
(Higherorder Lipschitz mapping)

T is an rthorder contraction mapping if the polynomial \(p(z):=z^{r}\sum_{k=0}^{r1}c_{k}z^{k}\) is stable, that is, \(\lambda<1\) if \(p(\lambda)=0 \).

T is an rthorder nonexpansive mapping if the polynomial \(p(z):=z^{r}\sum_{k=0}^{r1}c_{k}z^{k}\) is tamely unstable, that is, there exists at least a magnitudewise dominating root \(\lambda\in\mathbb{C} \) such that \(p(\lambda)=0 \) and \(\lambda=1\).

T is an rthorder expansive (Lipschitz) mapping if the polynomial \(p(z):=z^{r}\sum_{k=0}^{r1}c_{k}z^{k}\) is wildly unstable, that is, there exists \(\lambda\in\mathbb{C} \) such that \(\lambda>1\) and \(p(\lambda)=0 \).
In the following subsections, we review the pertinent results on the fixed point theory of Lipschitz mappings and show the relationship with the fixed point theory of the higherorder counterparts as introduced above.
1.1 Fixed point theory of contraction mappings in metric spaces
The basic result of metric fixed point theory is the Banach [3] fixed point theorem (or the contraction mapping theorem).
Theorem 1.2
(Banach fixed point)
Let \((\mathcal{X},d) \) be a complete metric space and let \(T:\mathcal{X}\to\mathcal{X} \) be a contraction mapping. Then T has a unique fixed point given by the limit of Picard iterates \(x_{n+1}:=Tx_{n} \).
Theorem 1.2 is particularly useful in the demonstration of existence and uniqueness of solutions to certain problems in analysis and economics (see [4–6]). A survey of various extensions of Theorem 1.2 can be found in [7]; we highlight the important results related to those demonstrated in this paper. First, the higherorder contraction case when \(r>1 \) and \(c_{k}=0 \) for all \(k\geq1 \) is an oftencited generalisation in many texts on Theorem 1.2; this is the case when \(T^{r} \), but not \(T^{k} \) for all \(k< r \), is a contraction mapping; that is:
Theorem 1.3
Let \((\mathcal{X},d) \) be a complete metric space and \(T:\mathcal{X}\to\mathcal{X} \) a mapping such that \(T^{r} \) is a contraction for some \(r>1 \). Then T has a unique fixed point given by the limit of Picard iterates \(x_{n+1}:=Tx_{n} \).
In Section 3, we demonstrate that the conclusions of Theorems 1.2 and 1.3 extend to all higherorder contraction mappings. Both (firstorder) contraction mappings and the rthorder contraction mappings defined in Theorem 1.3 are special cases of the nowproven generalised Banach contraction conjecture (see Jachymski [8], MerryfieldStein [9] and Arvanitakis [10]).
Theorem 1.4
(Generalised Banach contraction theorem)
Now an early continuous mapping generalisation of the Banach fixed point theorem is the following result due to Caccioppoli [11]:
Theorem 1.5
1.2 Fixed point theory of noncontraction mappings in Banach spaces
Now, for noncontraction mappings, complete metric spaces are in general not sufficient to guarantee the existence or uniqueness fixed points; in this regard, usually, compactness and/or convexity of subsets of normed linear spaces is required. Some noteworthy results are as follows.
Theorem 1.6
(Edelstein [12])
Let T be a contractive mapping on a compact metric space, that is, \(d(Ty,Tx)\le d(y,x) \) with equality only if \(x=y \). Then T has a unique fixed point given by the limit of Picard iteration \(x_{n+1}:=Tx_{n} \).
Theorem 1.7
(Kirk [13])
Let T be a nonexpansive selfmapping on a weakly compact convex subset \(\mathcal{C} \) of a Banach space with normal structure  that is, for any bounded nonempty convex subset \(\mathcal{K}\subset\mathcal{C} \) there exists a point \(x_{0}\in\mathcal {K} \) such that \(\sup_{x\in\mathcal{K}}\x_{0}x\<\operatorname{diam}(\mathcal {K}):=\sup_{x,y\in\mathcal{K}}\xy\ \). Then T has a fixed point.
Theorem 1.8
(Schauder^{1} [14])
Let T be a Lipschitz selfmapping on a compact convex subset of a Banach space. Then T has a fixed point.
In the present paper, we do not investigate the fixed point theory of higherorder Lipschitz mappings under the hypotheses in Theorems 1.6, 1.7 and 1.8 above. These are deferred to a sequel to this paper.
2 Preliminaries
First of all, we recall the following definitions:
A nowhere dense subset of a topological space is a set whose closure has an empty interior in the topological space; that is, it contains no open neighbourhood of its elements in the topological space.
A real matrix is nonnegative if all its entries are nonnegative real numbers; it is positive if all the entries are positive real numbers.
A nonnegative matrix A is irreducible if for every pair of indices i, j, there exists a natural number n such that \((\mathbf{A}^{n})_{ij}>0 \).
A real nonnegative matrix A is primitive if there exists an integer \(n\ge1 \) such that \(\mathbf{A}^{n} \) is positive; thus, a primitive matrix is irreducible.
A polynomial \(f(z) \) is nondegenerate if whenever \(\alpha\neq \beta\) but \(f(\alpha)=f(\beta)=0\), then \(\alpha\neq\zeta\beta\), where ζ is a root of unity.
An rthorder linear recurrence sequence \(S_{n} \), satisfying the recursive equation \(S_{n+r}=\sum_{k=0}^{r1}c_{k}S_{n+k} \), is nondegenerate if the associated characteristic polynomial \(p(z)=z^{r}\sum_{k=0}^{r1}c_{k}z^{k} \) is nondegenerate.
The following useful results are necessary for the proof of our main results. As used below and elsewhere, we employ the Kronecker delta symbol \(\delta_{jk} \), which equals 1 when \(j=k \) and equals 0 if otherwise.
Theorem 2.1
Proof
See for instance [15], Sections 1.1.4 through 1.1.6. □
The following corollary, which follows straightforwardly from the above theorem, is what is most useful for our purposes here.
Corollary 2.2
Using the notation of Theorem 2.1, define \(\lambda:=\max \lambda_{i} \) and \(\mu:=\max\mu_{i} \). Then \(S_{n}\leq K\lambda^{n}n^{\mu 1} \) for some absolute constant \(K>0 \) independent of n, λ, μ; moreover, if \(\lambda<1 \), then \(\lim_{n\to\infty}S_{n} =0\).
Theorem 2.3
The set of zeroes of a linear recurrence sequence \(S_{n} \) over a field of characteristic zero comprises a finite set together with a finite number of arithmetic progressions. If \(S_{n} \) is nondegenerate, then the set of zeroes is finite.
Remark
The set referred to is the set of indices n for which \(S_{n} = 0 \) and in either case it may be empty.
Theorem 2.4
(Baire category [20])
Let \(\mathcal{X} \) be a complete metric space. Then \(\mathcal{X} \) is not the countable union of nowhere dense closed sets.
Theorem 2.5
 1.
\(\rho(\mathbf{A}) \) is an eigenvalue of A and it is uniquely dominating if A is primitive;
 2.
\(\min_{i}\sum_{j}\mathbf{A}_{ij}\leq\rho(\mathbf{A})\leq\max_{i}\sum_{j}\mathbf{A}_{ij} \);
 3.
CollatzWielandt formula: Let \(\mathbf{N}:=\{\mathbf{v}=\{v_{j}\geq0\} _{j=1}^{r}:\exists i,v_{i}\neq0 \} \). Then \(\rho(\mathbf{A})=\max_{\mathbf{v}\in\mathbf{N}}\min_{1\leq i\leq r,v_{i}\neq0}\frac{1}{v_{i}}(\mathbf{A}\mathbf{v})_{i} \).
Remark
The PerronFrobenius theorem is more general than this but this suffices for our purposes here. By a uniquely dominating eigenvalue, we imply one which is a unique maximum in absolute value.
Theorem 2.6
(KeilsonStyan inequality [23])
Let A be a nonnegative \(r\times r \) matrix with spectral radius \(\rho(\mathbf{A})\). Then \(\det(tI\mathbf{A})\le t^{r}\rho (\mathbf{A})^{r} \) for all \(t\ge\rho(\mathbf{A}) \).
Theorem 2.7
(Rouché)
Let \(g(z)=z^{r} \) and \(h(z)=\sum_{k=0}^{r1}a_{k}z^{k} \) be complexvalued polynomials such that \(g(R)>\sum_{k=0}^{r1}a_{k}R^{k}\) for a real number \(R >0\), then the polynomial \(f(z):=g(z)h(z) \) has all its roots lying strictly inside the circle \(z= R \).
Proof
This follows immediately from a more general theorem of Rouché, a proof of which can be found in Titchmarsh [24]. □
Theorem 2.8
(Bolzano’s intermediate value [25])
If a continuous real function defined on an interval is sometimes positive and sometimes negative, then it must be 0 at some point in the interval.
Theorem 2.9
(Descartes’ rule of signs)
Let \(f(z):=\sum_{k=0}^{r}a_{k}z^{k} \) be an rth degree polynomial over the real numbers \(a_{k} \). Then the number of positive real roots of f is bounded above by the number of sign changes of the coefficients \(a_{k} \) as one proceeds from \(k=0 \) to \(k=r \) (ignoring zero coefficients).
Proof
See for instance [26]. □
Proposition 2.10
 (i)
If p is stable, then \(0< p(1)\leq1 \) and there exists \(\lambda\in [0,1) \), which is unique and positive if \(c_{0}\neq0 \), such that \(p(\lambda)=0 \).
 (ii)
If p is tamely unstable, then \(p(1)=0\) and 1 is the only positive root of p.
 (iii)
If p is wildly unstable, then \(p(1)< 0 \) and there exists a unique positive \(\lambda>1 \) such that \(p(\lambda)=0 \).
Proof
First of all, that \(p(1)\leq1 \) follows since \(c_{k}\geq0 \).
For (i): Suppose to the contrary that \(p(1)\leq0 \). Now we note that for real numbers t, we have \(\lim_{t\to\infty}p(t)=\infty\) and as such there exists \(t_{1}\geq1 \) such that \(p(t_{1})\geq0 \). Given that p is a continuous function (on the whole of the real line), then by Bolzano’s intermediate value theorem (Theorem 2.8), there exists \(t_{0}\in[1,t_{1}] \) such that \(p(t_{0})=0 \), contradicting the fact that by assumption we should rather have \(t_{0}<1 \). Finally since \(p(0)=c_{0}\leq0 \) and \(p(1)>0 \) then (by Bolzano’s intermediate value theorem again) there exists \(\lambda\in[0,1) \) such that \(p(\lambda )=0 \). The uniqueness of λ when \(c_{0}\neq0 \), in which case λ cannot be 0, follows from Descartes’ rule of signs (Theorem 2.9).
For (ii): If \(1p(1)=\sum_{k=0}^{r1}c_{k} < 1 \), then from Rouché’s theorem (Theorem 2.7) all roots of \(p(z) \) would be strictly less than 1 in absolute value; hence \(p(1)\leq0 \). Now suppose to the contrary that \(p(1) < 0 \); hence there exists \(t_{1}> 1\) such that \(p(t_{1})\geq0 \) and so by Bolzano’s intermediate value theorem there exists \(t_{0}\in(1,t_{1}] \) such that \(p(t_{0})=0 \), contradicting the fact that by assumption we should rather have \(t_{0}\leq1 \); thus \(p(1)=0 \). That 1 is the only positive root follows from Descartes’ rule of signs.
For (iii): Supposing to the contrary that \(p(1)\in[0,1] \), then we would have \(1p(1)\in[0,1] \). But that would imply from Rouché’s theorem that all roots of \(p(z) \) are at most 1 in absolute value, which is a contradiction. Finally, since \(p(1)<0 \) and \(\lim_{t\to \infty}p(t)=\infty\), by Bolzano’s intermediate value theorem, there exists \(\lambda>1 \) such that \(p(\lambda)=0 \), the uniqueness of which follows from Descartes’ rule of signs. □
Corollary 2.11
Proof
By Proposition 2.10, one root of the polynomial \(p(z)=z^{2}c_{1}zc_{0} \) is real and nonnegative, λ say; hence given that \(c_{0}\geq0 \) then the other root must be real and nonpositive, \(\lambda' \) say, where \(\lambda'\geq0 \). We can therefore factor \(p(z) \) as \(p(z)=(z\lambda)(z+\lambda')=z^{2}(\lambda \lambda')z\lambda\lambda' \) and given that \(c_{1}=\lambda\lambda'\geq 0 \), the conclusion follows. □
Lemma 2.12
 1.
λ dominates all other roots of \(p(z) \); furthermore, λ is a uniquely dominating root if \(p(z)\) is nondegenerate.
 2.
\(\lambda\in[1p(1),(1p(1))^{1/r}]\), \(\lambda=1 \) and \(\lambda\in (1,1p(1)] \) if p is stable, tamely unstable and wildly unstable, respectively.
Proof
Now from Proposition 2.10, λ is the unique positive root of \(p(z) \); consequently it follows from the first part of the PerronFrobenius theorem (Theorem 2.5) that \(\rho(\mathbf {C})=\lambda\) and the conclusion of the first part of the lemma follows immediately. Finally we observe that \(\min_{i}\sum_{j}\mathbf {C}_{ij}=\min\{1,\sum_{k=0}^{r1}c_{k}\}=\min\{1,1p(1)\}\) and similarly we have \(\max_{i}\sum_{j}\mathbf{C}_{ij}=\max\{1,\sum_{k=0}^{r1}c_{k}\}=\max \{1,1p(1)\} \); but by Proposition 2.10, we have \(p(1)>0 \), \(p(1)=0 \) and \(p(1)<0 \) if T is rthorder contraction, nonexpansive and expansive, respectively; hence given that \(\rho (\mathbf{C})=\lambda\), then from the second part of the PerronFrobenius theorem plus the fact that \(\lambda^{r}\le1p(1) \) when \(\lambda\le1 \) (the KeilsonStyan inequality, Theorem 2.6 with \(t=1 \)), the second part of the lemma follows. □
Proposition 2.13
Proof
Proposition 2.14
Let \(T:\mathcal{X}\to\mathcal{Y} \) be a mapping between metric spaces \((\mathcal{X},d_{\mathcal{X}}) \) and \((\mathcal {Y},d_{\mathcal{Y}}) \). Then T is continuous at \(x^{*}\in\mathcal{X} \) if and only if for every sequence \(\{x_{n}\}_{n\geq0} \) converging to \(x^{*} \) we find that the sequence \(\{Tx_{n}\}_{n\geq0} \) is Cauchy.
Proof
3 Main result
We now prove our main results. We begin with a direct proof of the fixed point theorem for higherorder contraction mappings; thereafter, we provide a remetrisation argument that relates higherorder Lipschitz mappings to (firstorder) Lipschitz mappings. This remetrisation of the original metric space does not necessarily result in a complete metric space even if the former is complete; consequently, we shall need a completion of the remetrised space and an extension of the higherorder Lipschitz mapping into the complete remetrised space.
3.1 Higherorder contraction mappings
The main result of this subsection is as follows, which extends the conclusion of the Banach fixed point theorem (Theorem 1.2) to higherorder contraction mappings:
Theorem 3.1
(Higherorder contraction mapping theorem)
Let \((\mathcal{X},d) \) be a complete metric space and let \(T:\mathcal{X}\to\mathcal{X} \) be an rthorder contraction mapping. Then T has a unique fixed point and \(\lim_{n\to\infty}T^{n}x \) converges to this fixed point for arbitrary \(x\in\mathcal{X} \).
We accomplish the proof below, but we require the following auxiliary lemma.
Lemma 3.2
The sequence \(\{T^{n}x\}_{n\geq0} \) is Cauchy for all \(x\in\mathcal{X} \); moreover, \(\lim_{n\to\infty}T^{n}y=\lim_{n\to\infty}T^{n}x \) for all \(x,y\in\mathcal{X} \).
Proof
Proof of Theorem 3.1
3.2 The general case
Now we consider the general case of higherorder Lipschitz mappings. First of all, it is worthy of note the theorem of Bessaga [27] states that whenever \(\mathcal{X} \) is an arbitrary set with a selfmap T satisfying the property that each iterate \(T^{n} \) has a unique fixed point, then for each \(c\in(0,1) \), there exists a metric \(d_{c} \) on \(\mathcal{X} \) such that \((\mathcal{X},d_{c}) \) is a complete metric space and T is a contraction mapping on \((\mathcal {X},d_{c}) \). Thus in light of Theorem 3.1 demonstrated in the previous subsection, we are motivated to consider a remetrisation of the space \((\mathcal{X},d) \) over which a higherorder Lipschitz mapping is defined.
Now we have the following lemma.
Lemma 3.3
First, we prove the following recurrence relation for the constants \(b_{k} \) in (8).
Proposition 3.4
Proof
Proof of Lemma 3.3
Theorem 3.5
Proof
4 Local Lipschitzity
If \(r>1 \), a natural question that arises is whether there are pairs \(x_{0},y_{0}\in\mathcal{X} \) and a constant \(c>0 \) such that \(d(T^{n}y_{0},T^{n}x_{0})\leq c^{n}d(y_{0},x_{0}) \) for all \(n\geq0 \); such a pair is therefore Lipschitzian under T, in the sense that this would be the case if T were a Lipschitz mapping with Lipschitz constant c. In this section, motivated by the existence of the unique positive real root λ of the polynomial \(p(z) \) (in Definition 1.1) when \(p(0)\neq0 \), we show that nearlocal Lipschitzity of open subsets exists with respect to an arbitrary point: that is, given \(x_{0}\in\mathcal{X} \) there exists an open subset \(\mathcal{S}\subset\mathcal{X} \) and positive real number \(m_{0}\geq1 \) such that \(d(T^{n}x,T^{n}x_{0})\leq m_{0}\lambda^{n}d(x,x_{0}) \) for all \(x\in\mathcal{S} \). We also prove the closely related result that there exists an open subset \(\mathcal{S} \) of \(\mathcal{X}^{2} \) and real number \(m_{0}\geq1 \) such that \(d(T^{n}y,T^{n}x)\leq m_{0}\lambda ^{n}d(y,x) \) for all \((x,y)\in\mathcal{S} \). Though plausible, we are not able to determine whether or when local Lipschitzity can occur (nontrivially) in either case, that is, whether or when \(m_{0} \) can take the value 1.
Unless otherwise mentioned, we assume \(p(0)\neq0 \) throughout the remaining part of this subsection.
Proposition 4.1
Proof
Remark
Obviously, \(M(y,x)\geq1 \) for every \(x,y\in\mathcal {X} \) so if \(M(y,x) \) can be 1 for some pair x, y then T is essentially locally Lipschitz on the pair x, y. We are not able to establish whether or when local Lipschitzity can occur, but the next results give near misses.
Theorem 4.2
Proof
Remark
We note that if the inequality in Theorem 4.2 is uniform for all \(1\leq i\leq r \) for some \(x,y\in \mathcal{X} \), then the maximum bound \(M(x,y) \) as defined in Proposition 4.1 would be exactly equal to 1.
Theorem 4.3
Proof
Theorem 4.4
Proof
Now we have the following problem.
Local Lipschitzity problem
Suppose T is a nontrivial rthorder Lipschitz mappings (that is, T is not of lower order on either \(T(\mathcal{X})\) or \(\mathcal{X}\)). Can the constants M and \(m_{0} \) appearing in Proposition 4.1 and Theorems 4.3 and 4.4 be exactly equal to 1?
The determination of whether or when this can be answered in the affirmative would demonstrate that there are subsets of elements in a complete metric space from which the Picard iterations are sharply convergent to the fixed point of a higherorder contraction mapping on the metric space in question.
5 Conclusion
Definition 5.1
When \(T_{k}:=T^{k} \), then (essentially) we achieve an rthorder Lipschitz mapping. One can then similarly inquire as regards the fixed point theory of such Lipschitz system of mappings.
Schauder’s theorem is more general than this and relates to continuous selfmappings on compact convex subsets of Banach spaces.
Declarations
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Authors’ Affiliations
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