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# Best proximity and fixed point results for cyclic multivalued mappings under a generalized contractive condition

- Manuel De la Sen
^{1}Email author, - Shyam Lal Singh
^{2}, - Madjid Eshaghi Gordji
^{3}, - Asier Ibeas
^{4}and - Ravi P Agarwal
^{5, 6}

**2013**:324

https://doi.org/10.1186/1687-1812-2013-324

© Sen et al.; licensee Springer. 2013

**Received: **26 July 2013

**Accepted: **5 November 2013

**Published: **27 November 2013

## Abstract

This paper is devoted to investigating the existence of fixed points and best proximity points of multivalued cyclic self-mappings in metric spaces under a generalized contractive condition involving Hausdorff distances. Some background results for cyclic self-mappings or for multivalued self-mappings in metric fixed point theory are extended to cyclic multivalued self-mappings. An example concerned with the global stability of a time-varying discrete-time system is also discussed by applying some of the results obtained in this paper. Such an example includes the analysis with numerical simulations of two particular cases which are focused on switched discrete-time control and integrate the associate theory in the context of multivalued mappings.

**MSC:**47H10, 55M20, 54H25.

## Keywords

## 1 Introduction

Important attention is being devoted to investigation of fixed point theory for single-valued and multivalued mappings concerning some relevant properties like, for instance, stability of the iterations, fixed points of contractive and nonexpansive self-mappings and the existence of either common or coupled fixed points of several multivalued mappings or operators. See, for instance, [1–24] and references therein. Related problems concerning the computational aspects of iterative calculations and best approximations based on fixed point theory have been also investigated. See, for instance, [21–23, 25, 26] and some references therein. On the other hand, a fixed point result for partial metric spaces and partially ordered metric spaces can be found in [27–30] and [4, 15, 31, 32], respectively, and references therein.

*et al.*and Ćirić’s results for single-valued self-mappings in some background literature papers. See, for instance, [2, 3, 33, 34] and references therein. Through this paper, we consider a metric space $(X,d)$ and a multivalued 2-cyclic self-mapping $T:A\cup B\to A\cup B$ (being simply referred to as a multivalued cyclic self-mapping in the sequel), where

*A*and

*B*are nonempty closed subsets of

*X*, so that $T(A)\subseteq B$ and $T(B)\subseteq A$ and $D=dist(A,B)\ge 0$. Let us consider the subset of the set of real numbers ${\mathbf{R}}_{0+}={\mathbf{R}}_{+}\cup \{0\}=\{z\ge 0:z\in \mathbf{R}\}$, ${\mathbf{R}}_{+}=\{z>0:z\in \mathbf{R}\}$, let the symbols ‘∨’ and ‘∧’ denote the logic disjunction (‘or’) and conjunction (‘and’), and define the functions $M:(A\cup B)\times (B\cup A)\times [0,1)\times \Delta \to {\mathbf{R}}_{0+}$ and $\phi :(A\cup B)\times (B\cup A)\times [0,1)\times \Delta \to (0,1]$ as follows:

Note that $\phi :(A\cup B)\times (B\cup A)\times [0,1)\times \Delta \to (0,1]$ is non-increasing since all its partial derivatives with respect to *K*, *α*, *β* exist and are non-positive; $\mathrm{\forall}x,y\in (A\cup B)\times (B\cup A)$ and note also that *Δ* is the union of the subsets ${\Delta}_{i}\subset \Delta $; $i=1,2,3,4$.

*X*. If $A,B\in \mathit{CL}(X)$, then we can define $(\mathit{CL}(X),H)$ as the generalized hyperspace of $(X,d)$ equipped with the Hausdorff metric $H:\mathit{CL}(X)\to {\mathbf{R}}_{0+}$

*A*and

*B*is defined by

where $T:A\cup B\to A\cup B$ is a multivalued cyclic self-mapping on the subset $A\cup B$ of *X*, that is, $T(A)\subseteq B$ and $T(B)\subseteq A$, where $(X,d)$ is a complete metric space including the case that $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$ is a Banach space with a norm-induced metric $d:X\times X\to {\mathbf{R}}_{0+}$, so that $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )\equiv (X,d)$ is a complete metric space, is used, subject to (1.1)-(1.4), in the main result Theorem 2.1 below. In this context, *Tx* is the image set through *T* of any $x\in A\cup B$ which is in *B*, that is, $Tx\subset T(A)\subseteq B$ (respectively, $Tx\subset T(B)\subseteq A$) if $x\in A$ (respectively, if $x\in B$). It is inspired by that proposed in [34] for single-valued self-mappings while it generalizes that proposed and discussed in [1] for multivalued self-mappings which is based on the Hausdorff generalized metric.

*Δ*and ${\Delta}_{i}\subset \Delta $ for $i=1,2,3,4$ and some of their relevant subsets in the contractive condition subject to (1.1)-(1.4).

Note that the proposed contractive condition, in fact, considers the worst case, given by the maximum of (1.1), of such a contractive condition of [1], reflected in (1.2a), with one based on a Kannan-type contractive condition associated with the choice of possible distinct values for the constants *α* and *β*, which is reflected in (1.2b) subject to (1.3)-(1.4). In particular, the choice $\alpha =\beta \in [0,1/2)$ gives a Kannan-type contractive condition in (1.2b). Note the importance of Kannan-type contractions for single-valued mappings in the sense that a metric space is complete if and only if each Kannan contraction has a unique fixed point [27, 38, 39]. The incorporation of (1.2b), (1.3)-(1.4) to (1.1) to build the general contractive condition allows an obvious direct generalization of the usual contractive condition, based on the Banach principle combined with a Kannan-type constraint, since both of them do not imply each other. In this context, note, for instance, that the simple scalar single-valued sequence ${x}_{n+1}=a{x}_{n}$; $\mathrm{\forall}n\in {\mathbf{N}}_{0}=\mathbf{N}\cup \{0\}$, with initial condition ${x}_{0}\in \mathbf{R}$, is a strict contraction if $|a|<1$. However, it is not a Kannan contraction for all $|a|<1$. This is easily seen as follows. Check the Kannan condition $d(Tx,Ty)\le \alpha (d(x,Tx)+d(y,Ty))$ for the self-mapping *T* on **R** defining the sequence solution and $\alpha \in [0,1/2)$, for instance, for points $x={x}_{n}$, ${x}_{n+1}=Tx$, ${x}_{n+2}={T}^{2}x=y$ and ${x}_{n+3}={T}^{3}x=Ty$ for any $n\in {\mathbf{N}}_{0}$. Then the Kannan contractive test is subject to $\frac{1}{2}>\alpha \ge \frac{|a|}{1+{a}^{2}}$, which is not fulfilled for given nonzero sufficiently small values of $1>|a|>0$ and any real $\alpha \in [0,1/2)$. It is possible also to check in a similar way a failure of the generalized Kannan-extended contractive condition $d(Tx,Ty)\le \alpha d(x,Tx)+\beta d(y,Ty)$ with $0\le \beta <1-\alpha $, $\alpha \in [0,1)$ for given nonzero sufficiently small values of $1>|a|>0$.

In the current approach, a combination of distinct contractive conditions for the $(\alpha ,\beta )$ pairs of values belonging to some relevant sets constructed from the subsets ${\Delta}_{i}$; $i=1,2,3,4$ of *Δ* is itself combined with the two point-to-point possibilities of combinations of the comparisons ${M}_{2}(x,y,\alpha ,\beta )>(\text{or}\le )\phantom{\rule{0.25em}{0ex}}{M}_{1}(x,y,K)$ for each $(x,y)\in A\times B\cup B\times A$. The various constraints are used to prove the convergence of the iterated sequences constructed with cyclic self-mappings $T:A\cup B\to A\cup B$ to best proximity points. On the other hand, the use of *ωD* in the contractive condition, instead of the distance in-between subsets, allows via the choice of some real constant $\omega >1$ to deal with problems where the achievement of limits of sequences at best proximity points is not of particular interest but just their limits superior belonging to certain subsets of the relevant sets ${A}_{i}\subset X$; $i\in \overline{p}$ containing the best proximity points. In this case, the permanence of the relevant sequences after a finite time in subsets of the sets ${A}_{i}\subset X$; $i\in \overline{p}$ after a finite number of steps is guaranteed. The standard analysis can be used for the particular case $\omega =1$. The performed study in the manuscript seems to be also promising for its extension to the study of single-valued and multivalued proximal contraction mappings in-between subsets of metric spaces because of the close formal relation between cyclic self-mappings and proximal mappings. See, for instance, [40] and references therein.

## 2 Main results

The first main result follows.

**Theorem 2.1**

*Let*$(X,d)$

*be a complete metric space*,

*and let*$T:A\cup B\to A\cup B$

*be*,

*in general*,

*a multivalued cyclic self*-

*mapping*,

*where*$A,B\subset X$

*are nonempty*,

*closed and subject to the contractive constraint*

*subject to*(1.1)-(1.4),

*for some*$\omega \in {\mathbf{R}}_{+}$, $K\in [0,1)$

*and*$(\alpha ,\beta )\in \Delta $; $\mathrm{\forall}(x,y)\in (A\cup B)\times (B\cup A)$.

*Assume also that*

*Then the following properties hold*:

*There is a sequence*$\{{x}_{n}\}$

*in*$A\cup B$

*satisfying*${x}_{i+1}\in T{x}_{i}$, $i\in \mathbf{N}$

*such that*

*If* *A* *and* *B* *are bounded sets which intersect*, *then* ${\sum}_{n=1}^{\mathrm{\infty}}d({x}_{n+1},{x}_{n})<\mathrm{\infty}$ *and* $\{{x}_{n}\}$ *is a Cauchy sequence having its limit in* $A\cap B$, *with* ${x}_{n+1}\in T{x}_{n}$; $n\in \mathbf{N}$ *for any given* ${x}_{1}\in A\cup B$.

*If* *A* *and* *B* *are not bounded*, *then the above property still holds if* $d({x}_{1},{x}_{2})<\mathrm{\infty}$. *Furthermore*, ${lim}_{n\to \mathrm{\infty}}d({x}_{n+1},{x}_{n})=D$ *exists if* $\omega =\frac{1-{K}_{1}}{{K}_{2}}$ *for any given* ${x}_{1}\in A\cup B$ *with the sequence* $\{{x}_{n}\}$ *being constructed in such a way that* ${x}_{n+1}\in T{x}_{n}$.

*If* ${x}_{1}\in A$, *then the sequence of sets* $\{T{x}_{2n+1}\}\subset T(B)\subseteq A$ *converges to a subset* $\{{z}_{A}\}$ *of best proximity points in* *A* (*in the sense that any point* ${z}_{2n+1}\phantom{\rule{0.25em}{0ex}}(\in \{T{x}_{2n}\})\to {z}^{0}\in \{{z}_{A}\}$ *as* $n\to \mathrm{\infty}$) *and the sequence of sets* $\{T{x}_{2n}\}\subset T(A)\subseteq B$ *converges to a subset* $\{{z}_{B}\}\subset T\{{z}_{A}\}\subset T(A)\subset B$ *of best proximity points in* *B* *with* $\{{z}_{A}\}\subset T\{{z}_{B}\}\subset T(B)\subset A$.

*If* $D=0$, *i*.*e*., *if* $A\cap B\ne \mathrm{\varnothing}$, *then* ${lim}_{n\to \mathrm{\infty}}{sup}_{m>n}d({x}_{m},{x}_{n})={lim}_{n\to \mathrm{\infty}}d({x}_{n+1},{x}_{n})=0$, *and any sequence* $\{{x}_{n}\}$ *being iteratively generated as* ${x}_{n+1}\in T{x}_{n}$, *for any* ${x}_{1}=x\in A\cup B$, *is a Cauchy sequence which converges to a fixed point* $z\in Tz\cap (A\cap B)$ *of* $T:A\cup B\to A\cup B$.

(ii) *Assume that* $A\cap B\ne \mathrm{\varnothing}$, *that* *A* *and* *B* *are convex*, *and that* ${z}_{i}\in T{z}_{i}$; $i\in \overline{N}=\{1,2,\dots ,N\}$ *are fixed points of* $T:A\cup B\to A\cup B$. *Then* ${z}_{i}={z}_{j}\subset A\cap B$ *and* $T{z}_{i}\equiv T{z}_{j}\subset A\cap B$; $\mathrm{\forall}i,j\phantom{\rule{0.25em}{0ex}}(\ne i)\in \overline{N}=\{1,2,\dots ,N\}$, *that is*, *the image sets of any fixed points are identical*.

(iii) *Consider a uniformly convex Banach space* $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$, *so that* $(X,d)$ *is a complete metric space for the norm*-*induced metric* $d:X\times X\to {\mathbf{R}}_{0+}$, *and let* *A* *and* *B* *be nonempty*, *disjoint*, *convex and closed subsets of* *X* *with* $T:A\cup B\to A\cup B$ *satisfying the contractive conditions* (2.1)-(2.2) *with* $\omega =\frac{1-{K}_{1}}{{K}_{2}}$.

*Then a sequence* $\{{x}_{2n}\}$ *built so that* ${x}_{2n}\in T{x}_{2n-1}$ *with* ${x}_{2n-1}\in T{x}_{2n-2}$ *is a Cauchy sequence in* *A* *if* ${x}_{1}\in A$ *and a Cauchy sequence in* *B* *if* ${x}_{1}\in B$ *so that* ${lim}_{n\to \mathrm{\infty}}d({x}_{2n+3},{x}_{2n+1})={lim}_{n\to \mathrm{\infty}}d({x}_{2n+2},{x}_{2n})=0$; $\mathrm{\forall}{x}_{1}\in A\cup B$, *and* ${lim}_{n\to \mathrm{\infty}}d({x}_{2n+2},{x}_{2n+1})={lim}_{n\to \mathrm{\infty}}d({x}_{2n+1},{x}_{2n})=D$; $\mathrm{\forall}{x}_{1}\in A\cup B$. *If* ${x}_{1}\in A$ *and* ${x}_{2}\in T{x}_{1}\subset T(A)\subset B$, *then the sequences of sets* $\{{T}^{2n}{x}_{1}\}\equiv \{T({T}^{2n-1}{x}_{1})\}$ *and* $\{{T}^{2n+1}{x}_{1}\}$ *converge to unique best proximity points* ${z}_{A}\in T{z}_{B}$ *and* ${z}_{B}\in T{z}_{A}$ *in* *A* *and* *B*, *respectively*.

*Proof* The proof is organized by firstly splitting it into two parts, namely, the situations: (a) ${M}_{2}$ defined in (1.2a), or (b) ${M}_{1}$, defined in (1.2b), gives the maximum for *M*, defined in (1.1); and then in five distinct cases concerning (1.3), subject to (1.4), as follows.

since $d(x,Tx)\le d(x,y)$; $\mathrm{\forall}y\in Tx$.

*A*and

*B*, one also gets by proceeding in a similar way:

$\mathrm{\forall}(x,y)\in (A\cup B)\times [T(A\cup B)]$, where ${K}_{1}=max(K,\frac{\beta}{1-\alpha},\frac{\alpha}{1-\beta})\in [0,1)$ and ${K}_{2}=max(\frac{1}{1-\alpha},\frac{1}{1-\beta})$. Note that since $T:A\cup B\to A\cup B$ is cyclic, then $y,Tx\in B$ if $x\in A$ and conversely.

*i.e.*, if

*A*and

*B*intersect provided that they are bounded or simply if $d({x}_{2},{x}_{1})<\mathrm{\infty}$) since ${lim}_{n\to \mathrm{\infty}}{sup}_{m>n}d({x}_{m},{x}_{n})={lim}_{n\to \mathrm{\infty}}d({x}_{n+1},{x}_{n})=0$, which has a limit

*z*in

*X*, since $(X,d)$ is complete, which is also in $A\cap B$ which is nonempty and closed since

*A*and

*B*are both nonempty and closed since $T(A)\subseteq B$ and $T(B)\subseteq A$. On the other hand, for any distance $D\ge 0$ between

*A*and

*B*,

*A*and

*B*are bounded. If $\omega =\frac{1-{K}_{1}}{{K}_{2}}$, then one gets from the above relations that

where ${x}_{2n+1}\in T{x}_{2n}\subset A$, ${x}_{2n+2}\in T{x}_{2n+1}\subset B$ and ${x}_{2n+3}\in T{x}_{2n+2}\subset A$. Thus, any sequences of sets $\{{x}_{2n+1}\}$ and $\{{x}_{2n}\}$ contain the best proximity points of *A* and *B*, respectively, if ${x}_{1}\in A$ and, conversely, of *B* and *A* if ${x}_{1}\in B$ and converge to them. This follows by contradiction since, if not, for each $k\in \mathbf{N}$, there is some $\epsilon =\epsilon (k)\in {\mathbf{R}}_{+}$, some subsequence ${\{{n}_{kj}\}}_{j\in \mathbf{N}}$ of natural numbers with ${n}_{km}>{n}_{kj}>k$ for $m>j$, and some related subsequences of real numbers $\{{x}_{2{n}_{kj}+1}\}$ and $\{{x}_{2{n}_{kj}}\}$ such that $d({x}_{2{n}_{kj}+2},{x}_{2{n}_{kj}+1})\ge D+\epsilon $ so that $d({x}_{2{n}_{k}+2},{x}_{2{n}_{k}+1})\to D$ as ${n}_{k}\to \mathrm{\infty}$ is impossible.

Now, assume $D=0$ and consider separately the various cases in (1.3)-(1.4), by using the contractive condition (2.1), subject to (1.1)-(1.4), to prove that there is $z\in Tz$ in $A\cap B$ to which all sequences converge by using $D=0\Rightarrow {lim}_{n\to \mathrm{\infty}}{sup}_{m>n}d({x}_{m},{x}_{n})={lim}_{n\to \mathrm{\infty}}d({x}_{n+1},{x}_{n})=0\Rightarrow \{{x}_{n}\}\to z\in A\cap B$ with $\{{x}_{n}\}$ being a Cauchy sequence since $(X,d)$ is complete and *A* and *B* are nonempty and closed.

*Case* 1: $\phi (x,y,K,\alpha ,\beta )=1$, $([(\alpha ,\beta )\in {\Delta}_{1}]\vee [(\alpha ,\beta )\in {\Delta}_{2}])\wedge (M(z,z,K,\alpha ,\beta )={M}_{2}(z,z,\alpha ,\beta )>{M}_{1}(z,z,K))$.

Then $d(z,Tz)=\phi (z,z,K,\alpha ,\beta )d(z,Tz)\le (\alpha +\beta )d(z,Tz)\le (1-{\alpha}^{2})d(z,Tz)$ if $(\alpha ,\beta )\in {\Delta}_{1}$. Thus, the contradiction $d(z,Tz)<d(z,Tz)$ holds if $(\alpha ,\beta )\in {\Delta}_{1}$, $\alpha \ne 0$ and $z\notin Tz$. Hence, $z\in Tz$ if $(\alpha ,\beta )\in {\Delta}_{1}$ with $\alpha \ne 0$ since *Tz* is closed. If $\alpha =0$, then $0\le \beta <1$ so that $d(z,Tz)\le \beta d(z,Tz)<d(z,Tz)$ if $z\notin Tz$. Hence, $z\in Tz$ if $\alpha =0$ and $(0,\beta )\in {\Delta}_{1}$. The proof that $z\in Tz$ if $(\alpha ,\beta )\in {\Delta}_{2}$ is similar since $(\alpha ,\beta )\in {\Delta}_{2}\iff (\beta ,\alpha )\in {\Delta}_{1}$ from the definitions of the sets ${\Delta}_{1}$ and ${\Delta}_{2}$, and the fact that distances have the symmetry property.

*Case* 2: $\phi (z,z,K,\alpha ,\beta )=1-\beta $, $([(\alpha ,\beta )\in {\Delta}_{3}])\wedge (M(z,z,K,\alpha ,\beta )={M}_{2}(z,z,\alpha ,\beta )>{M}_{1}(z,z,K))$.

Then the contractive condition becomes $(1-\beta )d(z,Tz)=\phi (x,y,K,\alpha ,\beta )d(z,Tz)\le (\alpha +\beta )d(z,Tz)$. Then either $z\in Tz$ or $z\notin Tz$ and $1<\alpha +2\beta $ with $(\alpha ,\beta )\in {\Delta}_{3}$. But the second possibility is impossible since ${\Delta}_{3}=\{(\alpha ,\beta )\in \Delta :\alpha \in (0,1/2),\frac{1-\alpha}{2}\ge \beta \}$ so that $1\ge \alpha +2\beta $. Hence, $z\in Tz$ since *Tz* is closed.

*Case* 3: $\phi (z,z,K,\alpha ,\beta )=\frac{1-\beta}{1-\beta +\alpha}$, $((\alpha ,\beta )\in {\Delta}_{4})\wedge (M(z,z,K,\alpha ,\beta )={M}_{2}(z,z,\alpha ,\beta )>{M}_{1}(z,z,K))$.

Then $\frac{1-\beta}{1-\beta +\alpha}d(z,Tz)=\phi (z,z,K,\alpha ,\beta )d(z,Tz)\le (\alpha +\beta )d(z,Tz)$ if $(\alpha ,\beta )\in {\Delta}_{4}$, which implies for $z\notin Tz$ if $(\alpha ,\beta )\in {\Delta}_{4}$ that $\frac{1-\beta}{1-\beta +\alpha}>\alpha +\beta $, equivalently, $1>\alpha (1+\alpha )+\beta (2-\beta )$. Since ${\Delta}_{4}=\{(\alpha ,\beta )\in \Delta :\alpha (1+\alpha )+\beta (2-\beta )\ge 1\}$, $z\notin Tz$ with $(\alpha ,\beta )\in {\Delta}_{4}$ is impossible. Hence, $z\in Tz$ since *Tz* is closed.

*Case* 4: $\phi (x,y,K,\alpha ,\beta )=1$, $(0\le K<1/2)\wedge ({M}_{2}(z,z,\alpha ,\beta )\le M(z,z,K,\alpha ,\beta )={M}_{1}(z,z,K))$.

Then $d(z,Tz)=\phi (z,z,K,\alpha ,\beta )d(z,Tz)\le Kmax\{d(z,z),d(z,Tz),\frac{d(z,Tz)}{2}\}=Kd(z,Tz)<d(z,Tz)$, which is a contradiction for any $z\notin Tz$. Hence, $z\in Tz$ since *Tz* is closed.

*Case* 5: $\phi (x,y,K,\alpha ,\beta )=1-K$, $(1/2\le K<1)\wedge ({M}_{2}(z,z,\alpha ,\beta )\le M(z,z,K,\alpha ,\beta )={M}_{1}(z,z,K))$.

*Tz*is closed. A combined result of Cases 1-5 is that $D=0\Rightarrow \{{x}_{n}\}\to z\in Tz\cap (A\cap B)$ for any ${x}_{1}\in A\cup B$. Now, assume again that $A\cap B\ne \mathrm{\varnothing}$ and that there are two distinct fixed points ${z}_{x}\phantom{\rule{0.25em}{0ex}}(\ne {z}_{y}\in T{z}_{y})\in T{z}_{x}$ necessary located in $A\cap B$ to which the sequences $\{{x}_{n}\}$ and $\{{y}_{n}\}$ converge to $z\in A\cap B$ and $q\phantom{\rule{0.25em}{0ex}}(\ne z)\in A\cap B$, respectively, where ${x}_{n+1}\in T{x}_{n}$, ${y}_{n+1}\in T{y}_{n}$ for $n\in \mathbf{N}$, where ${x}_{1},{y}_{1}\phantom{\rule{0.25em}{0ex}}(\ne {x}_{1})\in A\cup B$. Assume also that $Tz\ne Tq$. One gets from the contractive condition (2.1), subject to (1.1)-(1.4), that

Thus, construct sequences ${z}_{n+1}\in T{z}_{n}$, ${q}_{n+1}\in T{q}_{n}$ with ${z}_{1}=z$ and ${q}_{1}=q$ such that $d(z,{q}_{n+1})<d(z,{q}_{n})$ and $d(q,{z}_{n+1})<d(q,{z}_{n})$ for $n\in \mathbf{N}$. Since $z,q\in A\cap B$ which is nonempty, closed and convex, for any given $\epsilon \in {\mathbf{R}}_{+}$, there is ${n}_{0}={n}_{0}(\epsilon )$ such that ${x}_{n}$ and ${q}_{n}$ are in $A\cap B$ for $n\ge {n}_{0}$. Then ${q}_{n}\to \stackrel{\u02c6}{z}$ ($\in Tz$) and ${z}_{n}\to \stackrel{\u02c6}{q}$ ($\in Tq$) as $n\to \mathrm{\infty}$ with $z\in Tz\cap A\cap B$ and $q\in Tq\cap A\cap B$. Hence, $Tz\equiv Tq$ in $A\cap B$ contradicting the hypothesis that such sets are distinct. Properties (i)-(ii) have been proven.

*A*and

*B*are nonempty and disjoint closed subsets of

*X*and

*A*is convex. Note that Lemma 3.8 of [35] and its given proof remain fully valid for multivalued cyclic self-maps since only metric properties were used in its proof. It turns out that $\{{x}_{2n+1}\}$ is a Cauchy sequence, then bounded, with a limit ${z}_{A}$ in

*A*, which is also a best proximity point of $T:A\cup B\to A\cup B$ in

*A*since

and then $\{{x}_{2n}\}$ converges to some point ${z}_{B}\in T{z}_{A}\subset B$, which is also a best proximity point in *B* (then ${z}_{B}\in T{z}_{A}$ and $T{z}_{A}\subset B$), since $(X,d)$ is a uniformly convex Banach space and *A* and *B* are nonempty closed and convex subsets of *X*. In the same way, ${z}_{A}\in T{z}_{B}\subset A$. Also, $\{{x}_{2n}\}$ and $\{{x}_{2n+1}\}$ are bounded sequences since $\{{x}_{2n}\}$ is bounded and $D<\mathrm{\infty}$. Also, if ${x}_{1}\in B$ and *B* is convex, then the above result holds with ${x}_{2n+1}\in T{x}_{2n}\subset B$, ${x}_{2n+2}\in T{x}_{2n+1}\subset A$ and ${x}_{2n+3}\in T{x}_{2n+2}\subset B$. Now, for $D>0$, the reformulated five cases in the proof of Property (i) would lead to contradictions $D=d({z}_{A},{z}_{B})<D\ne 0$ if ${z}_{A}\notin T{z}_{B}$ or if ${z}_{B}\notin T{z}_{A}$. From Proposition 3.2 of [35], there are ${z}_{A}\in T{z}_{B}$ and ${z}_{B}\in T{z}_{A}$ such that $D=d({z}_{A},T{z}_{A})=d({z}_{B},T{z}_{B})$ since $T:A\cup B\to A\cup B$ is cyclic satisfying the contractive conditions (2.1)-(2.2), where *A* and *B* are nonempty and closed subsets of a complete metric space $(X,d)$, with convergent subsequences $\{{x}_{2n+1}\}$ and $\{{x}_{2n}\}$ in both *A* and *B*, respectively, for any $x={x}_{1}\in A$ and in *B* and *A*, respectively, for any given $x={y}_{1}\in B$. Assume that some given sequence $\{{x}_{2n+1}\}$ in *A* is generated from some given ${x}_{1}\in A$ with ${x}_{2n+1}\in T{x}_{2n}$, which converges to the best proximity point ${z}_{A}\in A\cap T{z}_{B}$ in *A* of $T:A\cup B\to A\cup B$. Assume also that there is some sequence $\{{y}_{2n}\}$, distinct from $\{{x}_{2n}\}$, in *A* generated from ${y}_{1}\phantom{\rule{0.25em}{0ex}}(\ne {x}_{1})\in A$ with ${y}_{2n+1}\in T{y}_{2n}$ which converges to ${z}_{A1}\in A$, where ${z}_{B}\in B\cap T{z}_{A}$ is a best proximity point in *B* of $T:A\cup B\to A\cup B$. Consider the complete metric space $(X,d)$ obtained by using the norm-induced metric in the Banach space $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$ so that both spaces can be mutually identified to each other. Since $d(x,y)\ge D$ for any $x\in A$ and $y\in B$, it follows that $D=d({z}_{B},{z}_{A})=d({z}_{A},T{z}_{A})=d({z}_{B},T{z}_{B})<d({z}_{A1},T{z}_{A})$ if ${z}_{A1}\in A\cap \overline{T{z}_{B}}$, where ${z}_{A}$ and ${z}_{B}$ are best proximity points of $T:A\cup B\to A\cup B$ in *A* and *B* and $\overline{T{z}_{B}}$ is the closure of $T{z}_{B}$. Hence, ${z}_{A},{z}_{A1}\in T{z}_{B}$ and ${z}_{B}\in T{z}_{A}$ and then any sequence converges to best proximity points.

*A*and

*B*are unique. Assume that $x,y\in A$ are two distinct best proximity points of $T:A\cup B\to A\cup B$ in

*A*. Then there are ${z}_{x}\in (Tx\cap B)\subset B$, ${z}_{x}\in (Ty\cap B)\subset B$, ${z}_{x}^{\prime}\in (T{z}_{x}\cap A)\subset ({T}^{2}x\cap A)\subset A$ and ${z}_{y}^{\prime}\in (T{z}_{y}\cap A)\subset ({T}^{2}y\cap A)\subset A$ so that, one gets

which leads to the contradiction $D>\parallel y-{z}_{x}\parallel =\parallel {z}_{x}^{\prime}-{z}_{x}\parallel =D$, and then $x=y={z}_{A}\in A$. Hence Property (iii) has been proven. □

A special case of Theorem 2.1 is stated and proven in the subsequent result.

**Corollary 2.2** *Assume that* $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$ *is a uniform Banach space with associate norm*-*induced metric* $d:X\times X\to {\mathbf{R}}_{0+}$, *and let* *A* *and* *B* *be nonempty closed and convex subsets of* *X*. *Assume also that* $K=0$, ${K}_{1}=max(\frac{\beta}{1-\alpha},\frac{\alpha}{1-\beta})\in [0,1)$, ${K}_{2}=max(\frac{1}{1-\alpha},\frac{1}{1-\beta})\in [1,\mathrm{\infty})$ *and* $\omega =\frac{1-{K}_{1}}{{K}_{2}}$ *in the contractive condition* (2.1). *If* $max(\alpha ,\beta )<1/2$, *then there are* ${z}_{1}\in A$ *and* ${z}_{2}\in B$ *such that* ${z}_{1}\in ({T}^{2}{z}_{1}\cap T{z}_{2})$, ${z}_{2}\in ({T}^{2}{z}_{2}\cap T{z}_{1})$, *i*.*e*., ${z}_{1}$ *and* ${z}_{2}$ *are*, *respectively*, *best proximity points of* $T:A\cup B\to A\cup B$ *in* *A* *and* *B*, *respectively*, *and simultaneously*, *fixed points of* ${T}^{2}:A\cup B\to A\cup B$, *respectively*. *In addition*, *if* $A\cap B\ne \mathrm{\varnothing}$, *then* $\mathrm{\exists}z\in Tz$ *is a fixed point of* $T:A\cup B\to A\cup B$. *The result also holds if* $max(\alpha ,\beta )<1$ (*and*, *in particular*, *if* $min(\alpha ,\beta )=0$).

*Proof* Assume, with no loss in generality, that $0\le \beta \le \alpha \in [0,1/2)$. Take ${u}_{1}\in A$ and ${u}_{2}\in T{u}_{1}$ by noting that ${u}_{2}\in T{u}_{1}\subset T(A)\subset B$ since $T:A\cup B\to A\cup B$ a multivalued cyclic self-mapping. □

**Remark 2.3** Note that the particular case $M(Tx,Ty,K,\alpha ,\beta )={M}_{2}(Tx,Ty,\alpha ,\beta )$ in the contractive condition (2.1) is useful to investigate multivalued cyclic Kannan self-mappings which are contractive with $\alpha =\beta \in [0,1/2)$ and some of their generalizations [33, 34].

The following result follows directly from Theorem 2.1 and Corollary 2.2 without proof.

**Corollary 2.4** *Assume that* $T:A\cup B\to A\cup B$ *is a single*-*valued cyclic self*-*mapping where* *A* *and* *B* *are nonempty closed subsets of* *X* *where* $(X,d)$ *is a complete metric space*. *Then Theorem * 2.1 *and Corollary * 2.2 *still hold mutatis*-*mutandis for a fixed point* $z=Tz\in A\cap B$ *if* *A* *and* *B* *are convex and intersect and best proximity points are* ${z}_{A}\in A$, ${z}_{B}\in B$ *with* ${z}_{A}=T{z}_{B}={T}^{2}{z}_{A}$, *if*, *in addition* $(X,\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$ *is a uniformly convex Banach space*.

**Remark 2.5** The results of this section can be extended *mutatis*-*mutandis* to multivalued $s\phantom{\rule{0.25em}{0ex}}(\ge 2)$-cyclic self-maps $T:{\bigcup}_{i\in \overline{s}}{A}_{i}\to {\bigcup}_{i\in \overline{s}}{A}_{i}$, where $\overline{s}=\{1,2,\dots ,s\}$, ${A}_{i}\phantom{\rule{0.25em}{0ex}}(\ne \mathrm{\varnothing})\subset X$, $T({A}_{i})\subseteq {A}_{i+1}$ and ${A}_{s+1}\equiv {A}_{1}$ with $(X,d)$ being a complete metric space. See [2, 3, 36, 37] and references therein for some background results for single-valued cyclic *s*-self-mappings.

## 3 Example of application to time-varying discrete-time dynamic systems

### 3.1 Multi-control discrete-time linear dynamic system

*T*’ denotes matrix transposition. Note that if ${j}_{n}=p$ for $n\in {\mathbf{N}}_{0}$, then $C(n,p)$ is the $p\times qp$ controllability matrix of (3.1) on the sequence of samples ${\{j\}}_{n}^{n+p-1}$. Any prefixed state is reachable in any given prefixed number of samples from a null initial condition by some linear time-invariant state-feedback control in at most

*p*samples if and only if (3.1) is reachable, that is, if

for any sequence of integers $\{{j}_{n}\}$ with ${j}_{n}\ge p$, $n\in {\mathbf{N}}_{0}$ with ${j}_{0}=0$ such that $\{{j}_{n+1}-{j}_{n}\}$ is uniformly bounded. It is controllable to the origin if and only if it is reachable, that is, (3.6) holds and, furthermore, ${A}_{n}$ are all non-singular for $n\in {\mathbf{N}}_{0}$. It is well known that if the dynamic system (3.1) is controllable to the origin, then it is also stabilizable in the sense that some linear time-varying state-feedback control sequence $\{{u}_{k}\}$ is such that ${x}_{n}\to 0$ as $n\to \mathrm{\infty}$ for any ${x}_{0}\in {\mathbf{R}}^{n}$. The controllability assumption can be weakened while keeping the stabilizability property as follows.

**Proposition 3.1** *Assume that* (3.1) *is stabilizable* (*which is guaranteed if it is controllable to the origin*).

*Then the following properties hold*:

(i) *There is a sequence of control gain matrices* $\{{K}_{n}\}$ *such that all the matrices in the subsequence* $\{{\prod}_{i=0}^{{j}_{n}-1}[{A}_{{j}_{n}+i}+{B}_{{j}_{n}+i}{K}_{{j}_{n}+i}]\}$ *are convergent matrices with* ${j}_{n}\in \mathbf{N}$ *with* ${j}_{0}=0$ *being some existing sequence with* $\{{j}_{n+1}-{j}_{n}\}$ *being a uniformly bounded sequence for any* $n\in {\mathbf{N}}_{0}$.

(ii) *The subsequence* $\{{x}_{{\sum}_{i=0}^{n}{j}_{i}}\}$ *of states of the closed*-*loop system* (3.2) *converges to zero as* $n\to \mathrm{\infty}$. *As a result*, *the sequence* $\{{x}_{n}\}$ *of states of the closed*-*loop system also converges to zero as* $n\to \mathrm{\infty}$.

*Proof*One gets from (3.2) that if such a sequence $\{{j}_{n}\}$ of finite natural numbers exists for $n\in {\mathbf{N}}_{0}$ with ${j}_{0}=0$, then

*i.e.*, with all their eigenvalues being of modulus less than one. Note that, since system (3.1) is stabilizable, then such a sequence of nonnegative integers $\{{j}_{n}\}$ always exists since it exists with ${j}_{n}=1$ for all $n\in {\mathbf{N}}_{0}$. Now, it follows from (3.7) for any vector-induced matrix norm that

since the sequence $\{{j}_{n+1}-{j}_{n}\}$ is uniformly bounded. Thus, $\{{x}_{n}\}$ converges to zero for any given ${x}_{0}\in {\mathbf{R}}^{n}$. □

Proposition 3.1 is linked to Theorem 2.1 of Section 2 in the subsequent result.

**Theorem 3.2** *The following properties hold*:

(i) *Assume that system* (3.1) *is stabilizable and a linear time*-*varying feedback control* ${u}_{n}={K}_{n}{x}_{n}\in {\mathbf{R}}^{q}$ *is used where* ${K}_{n}\in {S}_{Kn}=\{{K}_{n}^{1},{K}_{n}^{2},\dots ,{K}_{n}^{{J}_{n}}\}$ *for any* $n\in {\mathbf{N}}_{0}$. *Assume also that for each* $j\in [{\sum}_{k=0}^{n}{j}_{k},{\sum}_{k=0}^{n+1}{j}_{k})$ *for some sequence of nonnegative integers* $\{{j}_{n}\}$, *such that* $\{{j}_{n+1}-{j}_{n}\}$ *is uniformly bounded*, *for any* $n\in {\mathbf{N}}_{0}$ *with* ${j}_{0}=0$, *there is a controller gain* ${K}_{j}^{k}\in {S}_{K{J}_{n}}=\{{K}_{{\sum}_{k=0}^{n}{j}_{k}}^{1},{K}_{{\sum}_{k=0}^{n}{j}_{k}}^{2},\dots ,{K}_{{\sum}_{k=0}^{n}{j}_{k}}^{{J}_{{j}_{n}}}\}$ *for some integer* $k\in [1,{J}_{{j}_{n}}]$ *with* ${J}_{n}\le J<\mathrm{\infty}$ *for all* $n\in {\mathbf{N}}_{0}$ *such that any matrix in the subsequence of matrices* $\{{\prod}_{i=0}^{{j}_{n+1}-1}[{A}_{{\sum}_{k=0}^{n}{j}_{k}+i}+{B}_{{\sum}_{k=0}^{n}{j}_{k}+i}{K}_{{\sum}_{k=0}^{n}{j}_{k}+i}^{k}]\}$ *is convergent for each* $k\in [1,{J}_{{j}_{n}}]$ *for some uniformly bounded sequence of samples* $\{{j}_{n+1}-{j}_{n}\}$ *and some set of upper*-*bounded positive integer numbers* ${J}_{n}$ *for all* $n\in {\mathbf{N}}_{0}$.

(ii) *If*, *in addition*, *the elements of the subsequence of pairs* $\{({A}_{{j}_{n}},{B}_{{j}_{n}})\}$ *are all controllable for some sequence of nonnegative integers* $\{{j}_{n}\}$, *with the sequence of natural numbers* $\{{j}_{n+1}-{j}_{n}\}$ *being uniformly bounded*, *then the closed*-*loop system can be exponentially stabilized via time*-*varying linear control with prescribed stability degree*.

*Outline of proof*Property (i) follows directly from Proposition 3.1. Property (ii) follows since all the pairs $({A}_{{j}_{n}},{B}_{{j}_{n}})$ being controllable implies that the matrices of the closed-loop dynamics satisfy at the subsequence $\{{j}_{n}\}$ of samples the following matrix relation:

where the superscript ‘$(i)$’ stands for the *i* th row vector of matrix, ‘≈’ stands for matrix similarity, ${I}_{p-1}$ denotes the $(p-1)$ identity matrix and ${g}_{{j}_{n}}^{T}$ denotes some prefixed *p*-real row vector by the appropriate choice of the real controller matrix ${K}_{{j}_{n}}$, since $({A}_{{j}_{n}},{B}_{{j}_{n}})$ is controllable, towards the achievement of a suitable closed-loop stability degree. Note that the closed-loop matrix of dynamics at the ${j}_{n}$-sample is similar by a similarity transformation to its companion block partitioned form in (3.10). Thus, both matrices have the same characteristic monic polynomial, thus the same characteristic roots which are also the prefixed eigenvalues of the closed-loop dynamics given by (3.10), which can be arbitrarily fixed via ${K}_{{j}_{n}}$ such that its non-leading real coefficients are the components of the real row vector ${g}_{{j}_{n}}^{T}$. Thus, the sequence of closed-loop matrices can be chosen with the sequence ${M}_{{j}_{n}}$ having a stability degree ${\rho}_{{j}_{n}}\in (0,1)$ such that the stability degree of $({\prod}_{i=0}^{{j}_{n+1}-1}[{M}_{{j}_{n}+i}]){M}_{{j}_{n}}\le \rho \in (0,1)$. This follows since $\parallel {\prod}_{i=0}^{{j}_{n+1}-1}[{M}_{{j}_{n}+i}]\parallel \le P<\mathrm{\infty}$ since $\{{j}_{n+1}-{j}_{n}\}$ is uniformly bounded. Thus, the time-varying closed-loop system is exponentially stable with prescribed stability degree *ρ* and $\parallel {x}_{j}\parallel \le P{\rho}^{n}\parallel {x}_{0}\parallel $ for any integer $j\in [{j}_{n},{j}_{n+1})$ and $n\in {\mathbf{N}}_{0}$. □

The stability degree is defined by the modulus of the dominant eigenvalue of the matrix of dynamics if the dominant eigenvalue is simple and such a number is a strict upper-bound of the stability degree, otherwise. At samples which are not in the subsequence $\{{j}_{n}\}$, the controller gains may be chosen arbitrarily. The exponential stabilization of the closed-loop system is now related to Theorem 2.1 as follows. Assume that the sequence $\{{S}_{K{j}_{n}}\}$ of sets of matrices ${S}_{K{j}_{n}}=\{{K}_{{j}_{n}}^{1},{K}_{{j}_{n}}^{2},\dots ,{K}_{{j}_{n}}^{{J}_{{j}_{n}}}\}$ contains at least a stabilizing matrix such that Theorem 3.2(ii) holds via stabilization with such stabilizing matrices.

*A*as the states ${x}_{{j}_{0}}\equiv {x}_{0}\phantom{\rule{0.25em}{0ex}}(\in {A}_{0})\to {x}_{1}\in A$ (the replacement is made following the notation of Theorem 2.1), ${x}_{{j}_{1}}\equiv {x}_{{j}_{1}}\phantom{\rule{0.25em}{0ex}}(\in A)\to {x}_{2}\in A,{x}_{{j}_{2}}\equiv {x}_{{j}_{2}}\phantom{\rule{0.25em}{0ex}}(\in A)\to {x}_{3}\in A,\dots $ . If Theorem 3.2(ii) holds, then there is some bounded $A\supset {A}_{0}$ such that $T:A\to A$ defined by $(T{x}_{n}={x}_{n+1})\to (({\prod}_{i=0}^{{j}_{n+1}-1}[{M}_{{j}_{n}+i}]){M}_{{j}_{n}}{x}_{{j}_{n}}={x}_{{j}_{n+1}})$ for $n\in \mathbf{N}$ is a contractive mapping which defines the state trajectory solution at the points of the sampling subsequence $\{{j}_{n}\}$. Take ${K}_{1}=K=\rho <1$ and $\alpha =\beta =0$ in Theorem 2.1. Note that if the stabilizing matrix is chosen within the sequence of matrices, then $T:A\to A$ is single-valued. If all the matrices in the sequence $K{J}_{n}=\{{K}_{{\sum}_{k=0}^{n}{j}_{k}}^{1},{K}_{{\sum}_{k=0}^{n}{j}_{k}}^{2},\dots ,{K}_{{\sum}_{k=0}^{n}{j}_{k}}^{{J}_{{j}_{n}}}\}$ are tested, then the multivalued map $T:A\to A$ is defined as $T{x}_{n}=\{{x}_{n+1}^{1},\dots ,{x}_{n+1}^{{J}_{n}}\}$ such that ${x}_{n},{x}_{n+1}\in T{x}_{n}$ satisfies the Hausdorff particular contractive condition of Theorem 2.1. Also, one of the points of the sequence of sets $\{T{x}_{n}\}$ satisfies the point-to-point contractive particular condition of Theorem 2.1, by virtue of such a theorem, according to the constraints

obtained from the stabilizing control matrix sequence, and, furthermore, $\parallel {x}_{{j}_{n}+j}\parallel \le P{K}_{1}\parallel {x}_{{j}_{n}}\parallel $ for all samples given by the integers $j\in [1,{j}_{n+1})$. Note that the ${\ell}_{2}$-matrix norm of any real matrix *M* of any order satisfies ${\lambda}_{\mathrm{min}}^{1/2}({M}^{T}M)\le {\parallel M\parallel}_{2}={\lambda}_{\mathrm{max}}^{1/2}({M}^{T}M)$, where ${\lambda}_{\mathrm{max}}(\cdot )$ and ${\lambda}_{\mathrm{min}}(\cdot )$ stand, respectively, for the maximum and minimum eigenvalues of the $(\cdot )$-matrix with all its eigenvalues being real. A weak result is obtained with the particular case $K=\beta =0$ and ${K}_{1}=\frac{\alpha}{1-\alpha}=\frac{\rho}{1-\rho}<1$ for $0<\rho <1/2$ in the contractive condition of Theorem 2.1. In this case, we have a multivalued contractive Kannan self-mapping. In both cases, $0\in {\mathbf{R}}^{p}$ is a fixed point of $T:A\to A$ for any $A\subset X$ which is also a stable equilibrium point of the closed-loop dynamic system. Now assume $p=q=1$, that is, the uncontrolled system (3.1) is scalar subject to a scalar control with $A=\{z\ge \epsilon :z\in \mathbf{R}\}\subseteq {\mathbf{R}}_{0+}$ and $B=-A$ for some given $\epsilon \in {\mathbf{R}}_{0+}$. Take ${x}_{0}\in A\cup B$, then $|{x}_{0}|\ge \epsilon $, and note that $D=dist(A,B)=2\epsilon $. The tentative controller gains used are ${K}_{n}^{i}=-{\delta}_{n}^{i}sgn{x}_{n}$ if ${x}_{n}\ne 0$ and ${K}_{n}^{i}=0$ if ${x}_{n}=0$ for the bounded sets of integers $i\in [1,{J}_{n}]$ for $n\in {\mathbf{N}}_{0}$, where the nonnegative real sequences of sets $\{{\delta}_{n}^{i}:1\le i\le {J}_{n}\}$ are uniformly bounded and contain a strictly decreasing positive real sequence $\{{\delta}_{{j}_{n}}^{{i}_{n}}\}$ with ${i}_{n}\in [1,{J}_{{j}_{n}}]$ and some existing difference sequence of integers $\{{j}_{n+1}-{j}_{n}\}$ being uniformly upper-bounded for $n\in {\mathbf{N}}_{0}$.

The formalism of Section 2 is applicable to bounded sets ${A}_{0}\subset A$ and ${B}_{0}\subset B$ with $max(diam{A}_{0},diam{B}_{0})=||{x}_{0}|-\epsilon |$. If $\epsilon =0$, then a particular case of the above result follows for $p=1$. If $\epsilon >0$, then the closed-loop state-trajectory solutions $\{{x}_{2n+1}\}$ and $\{{x}_{2n+1}\}$ converge to the best proximity points *ε* and −*ε*, respectively, if the initial condition is in *A* and, conversely, if it is in *B* under the sequence of stabilizing matrices $\{{\delta}_{{j}_{n}}^{{i}_{n}}\}$.

### 3.2 Numerical example: a vector-valued discrete-time dynamic system with multiple parameterizations

*n*. This dynamic system is a simplified version of an automobile roll dynamics enhancement control system given in [15]. The switching function is assumed, for simulation purposes, to be the 1-sample periodic (cyclic) sequence $1\to 2\to 3\to 1\to 2\to \cdots $ . The following state-feedback gains are considered:

*n*, one considers the multivalued map ${x}_{n+1}=T{x}_{n}={A}_{\sigma (n)}{x}_{n}+{B}_{\sigma (n)}{K}_{n}^{(i)}{x}_{n}=({A}_{\sigma (n)}+{B}_{\sigma (n)}{K}_{n}^{(i)}){x}_{n}$ for $i=1,2,3$ for each sample $n\in {\mathbf{N}}_{0}$ from ${\mathbf{R}}^{2}$ to ${\mathbf{R}}^{2}$. The Banach space $({\mathbf{R}}^{2},\parallel \phantom{\rule{0.25em}{0ex}}\parallel )$ can be identified with the metric space $({\mathbf{R}}^{2},d)$ by taking the distance $d:X\times X\to {\mathbf{R}}_{0+}$ to be the Euclidean norm. Thus, $d({x}_{n},0)=\parallel {x}_{n}\parallel $ and $d({x}_{n},{x}_{n+1})x=\parallel {x}_{n+1}-{x}_{n}\parallel $ so that it is direct to apply the formalism and results of Section 2. The multivalued composite map represents the set of reachable states starting from ${x}_{n}$ for all potential feedback gains $\{{K}_{n}^{(1)},{K}_{n}^{(2)},{K}_{n}^{(3)}\}$ at each sample. Figure 2 displays graphically this concept. The starting point is depicted with a circle. The application of the multivalued map

*T*to this point produces the three points (each one corresponding to one of the feedback matrices $\{{K}_{n}^{(1)},{K}_{n}^{(2)},{K}_{n}^{(3)}\}$), labeled as first iteration in Figure 2. A second application of

*T*generates three more points from each previous one, providing nine new points, which are depicted in Figure 2 as the second iteration. This procedure can be continued to generate the complete set of reachable states from ${x}_{n}$. The ‘plus’ symbols are used to represent the image for ${K}_{n}^{(1)}$, dots are used for ${K}_{n}^{(2)}$, while squares are used to represent the image for ${K}_{n}^{(3)}$.

*T*and then chooses the gain ${K}_{n}$ in such a way that the null vector, $z=0$, is a fixed point of the multivalued map

*T*. In this example, the choice ${K}_{n}={K}_{n}^{(1)}$ for all $n\ge 0$ allows stabilizing asymptotically the system. Then, according to Proposition 3.1, all the states are bounded and the norm of the state converges to zero asymptotically as Figures 3 and 4 show.

while the eigenvalues of the matrix product ${\prod}_{k=1}^{N}[{A}_{\sigma (k)}+{B}_{\sigma (k)}{K}_{1}]$, which describes the evolution of the discrete dynamics, converge asymptotically to zero as $N\to \mathrm{\infty}$.

### 3.3 Numerical example: a scalar discrete-time dynamic system with multiple parameterization

- (i)
${\{{K}_{n}\}}_{n=0}^{\mathrm{\infty}}$ contains ${K}_{n}^{(2)}$ at least once,

- (ii)
${\{{K}_{n}\}}_{n=0}^{\mathrm{\infty}}$ contains ${K}_{n}^{(1)}$ an infinite (countable) number of times.

## Declarations

### Acknowledgements

The first and fourth authors (M De la Sen and A Ibeas) are grateful to the Spanish Government for its support of this research through Grant DPI2012-30651, and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. The second author (SL Singh) acknowledges the support by the UGC New Delhi under Emeritus Fellowship. The authors are also grateful to the referees for their valuable comments which helped to improve the manuscript.

## Authors’ Affiliations

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