Some Results on Fixed and Best Proximity Points of Multivalued Cyclic Self-Mappings with a Partial Order

Abstract

This paper is devoted to investigate the fixed points and best proximity points of multivalued cyclic self-mappings on a set of subsets of complete metric spaces endowed with a partial order under a generalized contractive condition involving a Hausdorff distance. The existence and uniqueness of fixed points of both the cyclic self-mapping and its associate composite self-mappings on each of the subsets are investigated, if the subsets in the cyclic disposal are nonempty, bounded and of nonempty convex intersection. The obtained results are extended to the existence of unique best proximity points in uniformly convex Banach spaces.

1. Introduction

Important attention is being devoted recently to the investigation of fixed points of self-mappings as well as to the investigation of associate relevant properties like, for instance, stability of the iterations [1–3] and existence and uniqueness of fixed points. On the other hand, the extension of those topics to the existence of either fixed points of multivalued self-mappings [1, 4–19], or common fixed points of several multivalued mappings or operators has received important attention; see, for example, [15–19] and references therein. This paper investigates some properties of fixed point and best proximity point results for multivalued cyclic self-mappings under a general contractive-type condition based on the Hausdorff metric between subsets of a metric space [4, 7–9] and which includes a particular case the contractive condition for contractive single-valued self-mappings, [1, 4–10] including the problems related to cyclic self-mappings, see for example, [7, 8, 11] and references therein. This includes strict contractive cyclic self-mappings and Meir-Keeler type cyclic contractions, [20, 21]. There is a rich background literature available on cyclic self-mappings and related fixed point and best proximity point results; see, for example, [22–30] and references therein. Some existing fixed point results on contractive single and multivalued self-mappings provided in [1, 4, 5, 9, 10, 31, 32] and references therein, under various types of contractive conditions, have been revisited and extended in [4]. There is also a wide sample of fixed point type results available on fixed points and asymptotic properties of the iterations for self-mappings satisfying a number of contractive-type conditions while being endowed with partial order conditions. See, for instance, [18, 19], and references therein. The main objective of this paper is the investigation of fixed point/best proximity point results for multivalued cyclic self-mappings in complete metric spaces, or uniformly convex Banach spaces. Such multivalued cyclic self-mappings satisfy a contractive-type condition, which is specified on the Hausdorff metric, for all pairs of elements in the union of the subsets defining the cyclic disposal which are subject to a partial order.

2. Properties of Distances and Fixed Points for Multivalued Cyclic Self-Mappings with a Partial Order

Assume that (𝑋,𝑑) is a metric space for a set X endowed with some metric: 𝑋×𝑋→𝐑₀₊ with 𝐑₀₊=𝐑₊∪{0}. Let 𝐶𝐿(𝑋) be the family of all nonempty and closed subsets of the set 𝑋. If 𝐴,𝐵∈𝐶𝐿(𝑋) then we can define (𝐶𝐿(𝑋),𝐻) being the generalized hyperspace of (𝑋,𝑑) equipped with the Hausdorff metric 𝐻∶𝐶𝐿(𝑋)→𝐑₀₊ induced by the metric 𝑑∶𝑋×𝑋→𝐑₀₊: 𝐻(𝐴,𝐵)=maxsup_𝑥∈𝐴𝑑(𝑥,𝐵),sup_𝑦∈𝐵𝑑𝑦,𝐴 (1) for two sets 𝐴⊆𝑋 and 𝐵⊆𝑋 which are finite if both sets are bounded and zero if they have the same closure. The distance between 𝐴⊆𝑋 and 𝐵⊆𝑋 is 𝐷=𝑑(𝐴,𝐵)=inf_{𝑥∈𝐴,𝑦∈𝐵}𝑑𝑥,𝑦=inf_𝑥∈𝐴𝑑(𝑥,𝐵)=inf_𝑦∈𝐵𝑑𝑦,𝐴. (2) Denote by 𝑃(𝑋), 𝐵(𝑋), and 𝐶𝐵(𝑋) the sets of nonempty, and nonempty, bounded and nonempty, and bounded and closed sets of 𝑋, respectively. The following relations hold: ()()()𝐷≤𝐻𝐴,𝐵≤𝛿𝐴,𝐵=𝛿𝐵,𝐴=sup_{𝑥∈𝐴,𝑦∈𝐵}𝑑()()()(𝑋),(𝑋()]⟹[()]𝑎,𝑏≤𝛿𝐴,𝐵+𝛿𝐵,𝐶;∀𝐴,𝐵,𝐶∈𝐵[(𝐴,𝐵∈𝐶𝐵))∧𝐻𝐴,𝐵<𝜀∃𝑏∈𝐵∶𝑑𝑎,𝑏<𝜀,∀𝑎∈𝐴 (3)(4) and 𝛿(𝐴,𝐵)=0 if and only if 𝐴=𝐵={𝑥}. Consider also a self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖, where 𝐴_𝑖 are nonempty closed sets of 𝑋; ∀𝑖∈𝑝={1,2,…,𝑝}, subject to the constraints 𝑇(𝐴_𝑖)⊆𝐴_𝑖+1 such that 𝐴_{𝑖𝑝+𝑗}≡𝐴_𝑗 for any integer numbers 𝑗∈[1,𝑝−1)∩𝐙 and 𝑖∈𝐙₀₊=𝐙₊∪{0} with 𝐑₀₊=𝐑₊∪{0}. If 𝑝≥2 then 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a 𝑝-cyclic self-mapping. If 𝑝=1 then 𝑇∶𝐴₁→𝐴₁ is, in particular, a self-mapping on 𝐴₁. We will also consider a partial order ⪯ on 𝑋 so that (𝑋,⪯) is a partially ordered space and will assume, in general, that 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a multivalued 𝑝-cyclic self-mapping so that 𝐴_𝑖∋𝑥→𝑇𝑥(≠Ø)⊂𝐴_𝑖+1; ∀𝑖∈𝑝, ∀𝑥∈⋃_𝑖∈𝑝𝐴_𝑖. The subsequent result does not assume a contractive condition for each iteration on adjacent subsets of the contractive mapping but a global contractive condition for the cyclic mapping for iterations on multiple strips of the 𝑝 subsets 𝐴_𝑖⊂𝑋; 𝑖∈𝑝. Therefore, the result that the distances between any two subsets being adjacent or not of [33] for nonexpansive self-mappings is not required.

If 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 is a multivalued 𝑝-cyclic self-mapping then the set 𝐵𝑃(𝐴_𝑖)⊂𝐴_𝑖 will be said to be the set of best proximity points between 𝐴_𝑖 to 𝐴_𝑖+1 if 𝑑(𝐴_𝑖,𝐴_𝑖+1)=𝐷_𝑖=𝑑(𝑧,𝑦) for all 𝑧∈𝐴_𝑖 and some 𝑦∈𝑇𝑧. This concept generalizes that of best proximity points of subsets of single valued cyclic self-mappings which is established as follows. If 𝑇∶𝐴₁∪𝐴₂→𝐴₁∪𝐴₂ is cyclic and single-valued then 𝑥∈𝐴₁ and 𝑇𝑥∈𝐴₂ are best proximity point if 𝑑(𝐴₁,𝐴₂)=𝑑(𝑥,𝑇𝑥), [33, 34]. The following result extends a previous one for the case of noncyclic multivalued self-mappings, [18, 19].

Theorem 1. Let (𝑋,⪯) be a partially ordered space and 𝑑∶𝑋×𝑋→𝐑₀₊ with (𝑋,𝑑) being a complete metric space. Let 𝐴_𝑖 be a set of 𝑝(≥2) nonempty, bounded, and closed subsets of 𝑋; ∀𝑖∈𝑝 (i.e., 𝐴_𝑖∈𝐶𝐵(𝑋); ∀𝑖∈𝑝) with 𝐷_𝑖=𝑑(𝐴_𝑖,𝐴_𝑖+1); ∀𝑖∈𝑝 and let 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 be a multivalued 𝑝-cyclic self-mapping on ⋃_𝑖∈𝑝𝐴_𝑖 satisfying.(1)There exist 𝑝 real constants 𝑘_𝑖∈𝐑₀₊ satisfying 𝑘=∏_𝑖∈𝑝[𝑘_𝑖]∈[0,1) such that the following condition holds: 𝐻𝑇𝑥,𝑇𝑦≤𝑘_𝑖𝑑𝑥,𝑦+1−𝑘_𝑖𝐷_𝑖 (5) for any given 𝑥∈𝐴_𝑖 and 𝑦∈𝐴_𝑖+1 which fulfil 𝑥⪯𝑦, ∀𝑖∈𝑝. (2)If 𝑑(𝑥,𝑦)<𝑑₀ for some given 𝑑₀∈𝐑₊, 𝑦∈𝑇𝑥, and any given 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, then 𝑥⪯𝑦 with 𝑦∈𝐴_𝑗+1 if 𝑥∈𝐴_𝑗 for any given 𝑗∈𝑝. (3)There are some 𝑖∈𝑝, some 𝑥=𝑥_𝑖∈𝐴_𝑖, and some 𝑥_𝑖+1∈𝑇𝑥_𝑖⊂𝐴_𝑖+1 such that 𝑑(𝑥_𝑖,𝑥_𝑖+1)<𝑑_0𝑖 for some 𝑑_0𝑖>𝐷_𝑖.(4)𝑑₀≥maxmax_𝑗∈𝑝𝑑_0𝑗,max_𝑗∈𝑝𝑘_𝑗𝑑_0𝑗−𝐷_𝑗+𝐷_𝑗. (6)Note that (6) implies that 𝑑_0𝑗≤min(𝑑₀,(𝑑₀−(1−𝑘_𝑗)𝐷_𝑗)/𝑘_𝑗); ∀𝑗∈𝑝. Then, the following properties hold.(i) There is a partially ordered subsequence 𝑆_𝑖={𝑥_{𝑖+𝑗+𝑛𝑘𝑝}}_{𝑛𝑘∈𝐙0+} of the partially ordered sequence 𝑆(𝑥_𝑖)={𝑥_𝑖+𝑗}_{𝑗∈𝐙0+}, both of them of the first element 𝑥_𝑖, with respect to the partial order (𝑋,⪯), such that 𝑥_{𝑖+𝑗+𝑛𝑘𝑝}∈𝑄_𝑖+𝑗 for 𝑗∈𝑝; ∀𝑘≥𝑘₀, 𝑛_𝑘∈𝐙₀₊ for some 𝑘₀∈𝐙₀₊ and the given 𝑖∈𝑝, where 𝑄_𝑖+𝑗⊆𝑇𝑥_{𝑖+𝑗−1}⊆𝑇^𝑗−1𝑥_𝑖⊆𝐴_𝑖+𝑗, for any 𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝, are 𝑝 closed “quasi-proximity” sets in-between each pair of adjacent subsets of the multivalued 𝑝-cyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 such that 𝐷_𝑖+𝑗≤𝑑𝑥_{𝑛𝑝+𝑖+𝑗+1},𝑥_{𝑛𝑝+𝑖+𝑗}≤𝑘_𝑖+𝑗𝐷+1−𝑘_𝑖+𝑗𝐷=𝐷,∀𝑗∈𝑝−1∪{0},∀𝑛∈𝐙₀₊, (7) where 𝐷=min_𝑗∈𝑝𝐷_𝑗 with 𝑥_{𝑛𝑝+𝑖+𝑗}∈𝑇𝑥_{𝑛𝑝+𝑖+𝑗−1}⊆𝐴_𝑖+𝑗; ∀𝑗∈𝑝−1∪{0},  ∀𝑛∈𝐙₀₊ for the given 𝑖∈𝑝. (ii) If 𝐷_𝑗=𝐷; ∀𝑗∈𝑝 then any partially ordered sequence 𝑆(𝑥_𝑖) of first element 𝑥=𝑥_𝑖∈𝐴_𝑖 fulfills ∃lim_𝑛→∞𝑑𝑥_{𝑛𝑝+𝑖+𝑗+1},𝑥_{𝑛𝑝+𝑖+𝑗}=𝐷; (8)∀𝑗∈𝑝 and the given 𝑖∈𝑝, and 𝑥_{𝑛𝑝+𝑗+1}∈𝑇𝑥_{𝑛𝑝+𝑗}⊆𝐴_𝑖+𝑗+1; ∀𝑗∈𝑝 (i.e., 𝑥_{𝑛𝑝+𝑗+1}∈𝐴_𝑖+𝑗+1 if 0≤𝑗≤𝑝−𝑖−1 and 𝑥_{𝑛𝑝+𝑗+1}∈𝐴_{𝑗−𝑝+𝑖+1} if 𝑝−𝑖<𝑗≤𝑝−1), ∀𝑛∈𝐙₀₊. Let 𝐵𝑃(𝐴_𝑗) be the set of best proximity points between 𝐴_𝑗 and 𝐴_𝑗+1; ∀𝑗∈𝑝. Then, there is a sequence {𝑧_𝑛^(𝑗)}⊂𝐵𝑃(𝐴_𝑗); ∀𝑗∈𝑝 such that the following limit exists: lim_𝑛→∞𝑑𝑥_{𝑛𝑝+𝑗+1},𝑧_𝑛^(𝑗)=𝐷;∀𝑗∈𝑝𝑤𝑖𝑡ℎ𝑥_{𝑛𝑝+𝑗+1}∈𝑇𝑥_{𝑛𝑝+𝑗};∀𝑛∈𝐙₀₊ (9)(iii) If assumption (3) is removed and (6) in assumption (4) is replaced by the stronger condition (5)𝑑₀>maxmax_𝑗∈𝑝𝐷_𝑗+diam𝐴_𝑗,max_𝑗∈𝑝𝑘_𝑗𝑑_0𝑗−𝐷_𝑗+𝐷_𝑗 (10)then, properties (i)-(ii) hold for any 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.

Proof. Let 𝑥_𝑖∈𝐴_𝑖 for the given 𝑖∈𝑝 which satisfy assumption (3). Then, from such an assumption, there is 𝑥_𝑖+1∈𝑇𝑥_𝑖, which is also in 𝐴_𝑖+1, since 𝑇𝑥∈𝐴_𝑖+1 for any 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, such that 𝑑(𝑥_𝑖,𝑥_𝑖+1)<𝑑_0𝑖≤𝑑₀. Thus, 𝑥_𝑖⪯𝑥_𝑖+1 from assumption (2), since 𝑑_0𝑖≤𝑑₀. From (6) and assumptions (1)-(2) by considering the distance between adjacent subsets, 𝐷_𝑖≤𝐻𝑇𝑥_𝑖,𝑇≤𝑘_𝑖𝑑𝑥_𝑖,𝑥_𝑖+1+1−𝑘_𝑖𝐷_𝑖<𝑘_𝑖𝑑_0𝑖+1−𝑘_𝑖𝐷_𝑖≤𝑑_0𝑖≤𝑑₀, (11) since 𝐷_𝑖<𝑑_0𝑖≤(1/𝑘_𝑖)[(𝑘_𝑖−1)𝐷_𝑖+𝑑₀] from assumptions (3)-(4). From assumption (2) and (11), there is 𝑥_𝑖+2∈𝑇𝑥_𝑖+1⊂𝐴_𝑖+2 such that 𝑥_𝑖+1⪯𝑥_𝑖+2, and then 𝑑(𝑥_𝑖+1,𝑥_𝑖+2)<𝑑_0,𝑖+1, and 𝑑(𝑥_𝑖+1,𝑥_𝑖+2)≤𝐻(𝑇𝑥_𝑖,𝑇𝑥_𝑖+1)<𝑑_0𝑖≤𝑑₀. Then, one gets from (11) and assumption (4): 𝐷_𝑖+1≤𝐻𝑇𝑥_𝑖+1,𝑇𝑥_𝑖+2≤𝑘_𝑖+1𝑑𝑥_𝑖+1,𝑥_𝑖+2+1−𝑘_𝑖+1𝐷_𝑖+1≤𝑘_𝑖+1𝑑_0,𝑖+1+1−𝑘_𝑖+1𝐷_𝑖+1<𝑘_𝑖+1𝑑_0,𝑖+1+1−𝑘_𝑖+1𝑑₀−𝑘_𝑖+1𝑑_0,𝑖+1−𝐷_𝑖+1≤𝑑₀+𝑘_𝑖+1𝑑₀−𝑘_𝑖+1𝑑₀+1−𝑘_𝑖+1𝑘_𝑖+1𝐷_𝑖+1−𝑑_0𝑖+1<𝑑₀,𝑑𝑥_𝑖+2,𝑥_𝑖+3≤𝐻𝑇𝑥_𝑖+1,𝑇𝑥_𝑖+2≤𝑘_𝑖+1𝑑𝑥_𝑖+1,𝑥_𝑖+2+1−𝑘_𝑖+1𝐷_𝑖+1≤𝑘_𝑖+1𝑘_𝑖𝑑𝑥_𝑖,𝑥_𝑖+1+1−𝑘_𝑖𝐷_𝑖+1−𝑘_𝑖+1𝐷_𝑖+1≤𝑘_𝑖+1𝑘_𝑖𝑑𝑥_𝑖,𝑥_𝑖+1+𝑘_𝑖+11−𝑘_𝑖+1−𝑘_𝑖+1max_𝑖∈𝑝𝐷_𝑖<𝑑₀. (12a)(12b)Again, from assumption (2), there is 𝑥_𝑖+3∈𝑇𝑥_𝑖+2 such that 𝑥_𝑖⪯𝑥_𝑖+1⪯𝑥_𝑖+2⪯𝑥_𝑖+3. Now, proceeding by complete induction with (12a) from 𝑗=0 to 𝑗=𝑝−1, it follows that the existence of a partially ordered space 𝑥_𝑖⪯𝑥_𝑖+1⪯𝑥_𝑖+2⪯𝑥_{𝑖+𝑝−2} implies, from assumption (2), the existence of the partially ordered space 𝑥_𝑖⪯𝑥_𝑖+1⪯𝑥_𝑖+2⪯⋯⪯𝑥_{𝑖+𝑝−1} satisfying 𝑥_𝑖+𝑗∈𝑇𝑥_{𝑖+𝑗−1}⊆𝐴_𝑖+𝑗; ∀𝑗∈𝑝 with 𝐴_{𝑘𝑝+𝑗}≡𝐴_𝑗; ∀𝑘∈𝑝. Also, proceeding recursively with (12b), one concludes, if 𝑘=∏_𝑖∈𝑝[𝑘_𝑖]∈[0,1) and 𝐷=max_𝑖∈𝑝𝐷_𝑖, that there is a partially ordered sequence 𝑥_𝑖⪯𝑥_𝑖+1⪯𝑥_𝑖+2⪯⋯⪯𝑥_{𝑖+𝑝−1}⪯𝑥_𝑖+𝑝 such that 𝑥_𝑖+𝑗∈𝑇𝑥_{𝑖+𝑗−1}⊆𝑇(𝐴_{𝑖+𝑗−1})⊆𝐴_𝑖+𝑗; ∀𝑗∈𝑝, and 𝑑𝑥_𝑖+𝑝+1,𝑥_𝑖+𝑝≤𝑘𝑑𝑥_𝑖,𝑥_𝑖+1+(1−𝑘)𝐷 (13) so that there is a partially ordered sequence 𝑥_𝑖⪯𝑥_𝑖+1⪯𝑥_𝑖+2⪯⋯⪯𝑥_{𝑖+𝑝−1}⪯𝑥_{𝑖+𝑛𝑝}⪯⋯ such that 𝑥_𝑖+𝑗∈𝑇𝑥_{𝑖+𝑛𝑝+𝑗−1}⊆𝑇(𝐴_𝑖+𝑗); ∀𝑗∈𝑝−𝑖: 𝐷_𝑖+1≤𝑑𝑥_{𝑛𝑝+𝑖+2},𝑥_{𝑛𝑝+𝑖+1}≤𝐻𝑇𝑥_{𝑖+𝑛𝑝+1},𝑇𝑥_{𝑛𝑝+𝑖}+1−𝑘_𝑖𝐷_𝑖≤limsup𝑘^𝑛𝑑𝑥_𝑖,𝑥_𝑖+1+1−𝑘^𝑛𝐷_𝑛→∞=𝐷;∀𝑛∈𝐙₊ (14)
so that 𝐷_𝑖+𝑗≤𝑑𝑥_{𝑛𝑝+𝑖+𝑗+1},𝑥_{𝑛𝑝+𝑖+𝑗}≤𝑘_𝑖+𝑗𝐷+1−𝑘_𝑖+𝑗𝐷=𝐷;∀𝑗∈𝑝−1∪{0},∀𝑛∈𝐙₀₊. (15) Then, from (15), there are 𝑝 closed “quasi-proximity” sets 𝑄_𝑖⊆𝐴_𝑖, 𝑄_𝑖+1⊆𝑇𝑥_𝑖⊆𝐴_𝑖+1, 𝑄_{𝑖+𝑝−1}⊆𝑇^𝑝−1𝑥_𝑖⊆𝐴_{𝑖+𝑝−1} between each pair of adjacent subsets of the cyclic self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 in view of (14), such that there is a partially ordered subsequence {𝑥_{𝑖+𝑗+𝑛𝑘𝑝}}_{𝑛𝑘∈𝐙+} of the partially ordered sequence 𝑥_𝑖⪯𝑥_𝑖+1⪯𝑥_𝑖+2⪯⋯⪯𝑥_{𝑖+𝑝−1}⪯𝑥_{𝑖+𝑛𝑝}⪯⋯, being subject to 𝑥_{𝑖+𝑗+𝑛𝑘𝑝}∈𝑄_𝑖+𝑗 for ∀𝑗∈𝑝−1∪{0}; ∀𝑘≥𝑘₀, some 𝑘₀∈𝐙₀₊. Thus, (7) holds and then the property (i) has been proven. The relation (8) of property (ii) for 𝑗=0 is a direct consequence of property (i). From (8), it is also proven that the sequence 𝑆_𝑖 of first element 𝑥_𝑖∈𝐴_𝑖 in property (i) satisfies the following property ∃lim_𝑛→∞𝑑(𝑥_{𝑗+𝑛𝑝+1},𝑥_{𝑗+𝑛𝑝})=𝐷; ∀𝑗∈𝑝. Assume not, then, it follows that 𝐷≤liminf_𝑛→∞𝑑𝑥_{𝑖+𝑗+𝑛𝑝+1},𝑥_{𝑖+𝑗+𝑛𝑝}≤limsup_𝑛→∞𝑑𝑥_{𝑖+𝑗+𝑛𝑝+1},𝑥_{𝑖+𝑗+𝑛𝑝}≤𝑘_𝑗lim_𝑛→∞𝑑𝑥_{𝑖+𝑗+𝑛𝑝},𝑥_{𝑖+𝑗+𝑛𝑝−1}+1−𝑘_𝑗𝐷≤𝑘_𝑗𝐷+1−𝑘_𝑗𝐷=𝐷; (16)∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝 so that, by using complete induction, ∃lim_𝑛→∞𝑑(𝑥_{𝑖+𝑗+𝑛𝑝+1},𝑥_{𝑖+𝑗+𝑛𝑝})=𝐷; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝 with {𝑥_𝑘}_𝑘≥𝑖 being partially ordered with respect to (X,⪯), that is, 𝑥_𝑖+𝑘(∈𝑇𝑥_{𝑖+𝑘−1}⊆𝐴_𝑖+𝑘)⪯𝑥_𝑖+𝑘+1(∈𝑇𝑥_𝑖+𝑘⊆𝐴_𝑖+𝑘+1); ∀𝑘∈𝐙₀₊ and we can then reformulate the above limits of the distances as ∃lim_𝑛→∞𝑑(𝑥_{𝑖+𝑗+𝑛𝑝+1},𝑥_{𝑖+𝑗+𝑛𝑝})=𝐷; ∀𝑗∈𝑝−1∪{0} for the given 𝑖∈𝑝.
The remaining proof of property (ii) follows by contradiction. Suppose that the limit (9) does not exist for some sequence {𝑧_𝑛^(𝑗)}⊂𝐵𝑃(𝐴_𝑗) for some 𝑗∈𝑝. Since {𝑧_𝑛^(𝑗)}⊂𝐵𝑃(𝐴_𝑗), liminf_𝑛→∞𝑑(𝑥_{𝑛𝑝+𝑗+1},𝑧_𝑛^(𝑗))<𝐷 is impossible in the case that liminf_𝑛→∞𝑑(𝑥_{𝑛𝑝+𝑖+1},𝑧_𝑛^(𝑗)) would not exist for some 𝑗∈𝑝. Then, liminf_𝑛→∞𝑑𝑥_{𝑛𝑝+𝑗+1},𝑧_𝑛^(𝑗)=𝐷≠liminf_𝑘→∞𝑑𝑥_{𝑛𝑘𝑝+𝑗+1},𝑧_𝑛^(𝑗)_𝑘𝐴>𝐷=𝑑_𝑗,𝐴_𝑗+1=𝑑𝑧,𝑦 (17) for some 𝑧∈𝐵𝑃(𝐴_𝑗), 𝑦∈𝑇𝑧⊂𝐵𝑃(𝐴_𝑗+1), since 𝐴_𝑗 and 𝐴_𝑗+1 are boundedly compact for all 𝑗∈𝑝 since they are bounded and closed and, then, compact, [7, 8]. This leads to a contradiction, since {𝑧_𝑛^(𝑗)}⊂𝐵𝑃(𝐴_𝑗) and 𝑥_{𝑛𝑝+𝑗+1}∈𝑇_{𝑛𝑝+𝑗}⊂𝐴_𝑗+1; ∀𝑗∈𝑝. The property (ii) has been proven.
If assumption (3) is removed, while 𝑑₀ satisfies the stronger constraint (10), then there are infinitely many sequences 𝑆(𝑥) for any arbitrary first element 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖, in the partial order ⪯, of an iterated sequence through 𝑇 for which property (i) and thus property (ii) both hold since 𝑑(𝑥,𝑦)=𝑑(𝑥,𝑇𝑥)<𝑑₀ from assumption (2). Hence, property (iii) follows so that the theorem has been fully proven.

Note that (5) is not guaranteed to be a cyclic contractive condition for each restricted map 𝑇∶(⋃_𝑗∈𝑝𝐴_𝑗)∣𝐴_𝑖→(⋃_𝑗∈𝑝𝐴_𝑗)∣𝐴_𝑖+1, since all the constants are not required to be less than one in (5), and furthermore, (5) and assumption (3) are fulfilled for some first element 𝑥_𝑖∈𝐴_𝑖, 𝑥_𝑖+1∈𝑇𝑥_𝑖⊆𝐴_𝑖+1and some given 𝑖∈𝑝 in the partial order (𝑋,𝑑). Note also that sequences fulfilling the partial order of Theorem 1 can always be built through iterations with the multivalued 𝑝-self-mapping for any arbitrarily chosen 𝐴_𝑖 for any 𝑖∈𝑝 from (6) characterizing assumption (4) of Theorem 1. The subsequent particular case of Theorem 1 applies when all the iterations between the cyclic disposal satisfy a cyclic contractive condition, that is, 𝑘_𝑖<1; ∀𝑖∈𝑝.

Note that Theorem 1(iii) also holds in the particular case that the partial order is a total order for all pairs in any Cartesian product 𝐴_𝑖×𝐴_𝑖+1 of adjacent subsets 𝐴_𝑖⊂𝑋; ∀𝑖∈𝑝, since both elements of any ordered pair (𝑥,𝑦)∈𝐴_𝑖×𝐴_𝑖+1, 𝑦∈𝑇𝑥; ∀𝑖∈𝑝 are comparable with respect to the partial order ⪯. Theorem 1(iii) establishes that any element in any subset 𝐴_𝑖⊂𝑋; ∀𝑖∈𝑝 is a first element of a nondecreasing (i.e., partially ordered) sequence with respect to the partial order ⪯ which fulfils properties (i)-(ii) of Theorem 1.

Theorem 2. In addition to assumptions (1)–(4) of Theorem 1, assume, furthermore, (6)𝐷_𝑗=𝐷=0; ∀𝑗∈𝑝 (i.e., ⋂_𝑗∈𝑝𝐴_𝑗≠Ø);(7) the limit 𝑥 of any converging nondecreasing sequence {𝑥_𝑛}_{𝑛∈𝐙0+}is comparable to each 𝑥_𝑛; ∀𝑛∈𝐙₀₊ in the partial order (𝑋,⪯), that is, 𝑥_𝑛⪯𝑥≠𝑥_𝑛𝑓𝑜𝑟𝑥∈𝐴_𝑗,𝑥_𝑛∈𝐴_𝑗,∀𝑗∈𝑝,∀𝑛∈𝐙₀₊⟹𝐻𝑇𝑥,𝑇𝑥_𝑛>𝑘_𝑖𝑑𝑥,𝑥_𝑛. (18)Then, there is a sequence {𝑥_{𝑛𝑝+𝑖+𝑗}}_{𝑛∈𝐙0+}satisfying 𝑥_{𝑛𝑝+𝑖+𝑗}∈𝑇^{𝑛𝑝+𝑗}𝑥_𝑖 for some given initial element 𝑥=𝑥_𝑖∈𝐴_𝑖 and some given 𝑖∈𝑝; ∀𝑗∈𝑝−1∪{0} which is non-decreasing and ordered with respect to the partial order (𝑋,⪯) and fulfils the following properties.(i)∃lim_𝑛→∞𝑑(𝑥_{𝑛𝑝+𝑖+𝑗+2},𝑥_{𝑛𝑝+𝑖+𝑗+1})=0; 𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝 with 𝑥_{𝑛𝑝+𝑗+2}∈𝑇𝑥_{𝑛𝑝+𝑗+1}; 𝑗∈𝑝−1∪{0}, ∀𝑛∈𝐙₀₊ and the sequence {𝑥_{𝑛𝑝+𝑖+𝑗}}_{𝑛∈𝐙0+}is a Cauchy sequence; 𝑗∈𝑝−1∪{0}.(ii)The sequence {𝑥_{𝑛𝑝+𝑖+𝑗}}_{𝑛∈𝐙0+} for any 𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝 converge to a limit 𝑥 in ⋂_𝑗∈𝑝𝐴_𝑗, which is a fixed point of the composite self-mapping 𝑇_𝑗∶𝐴_𝑗→𝐴_𝑗, where 𝑇_𝑗=𝑇^𝑝=𝑇∘𝑇∘⋯𝑇(𝑝𝑡𝑖𝑚𝑒𝑠)=𝑇^𝑝∣𝐴_𝑗 of domain 𝐴_𝑗; ∀𝑗∈𝑝 and also a fixed point of the self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖, that is, 𝑥∈𝑇_𝑗𝑥(⊆⋂_𝑗∈𝑝𝐴_𝑗) and 𝑥∈𝑇^𝑝𝑥(⊆⋂_𝑗∈𝑝𝐴_𝑗); ∀𝑗∈𝑝.(iii) If, in addition, (𝑋,𝑑) is a convex metric space, what holds, in particular, if 𝑋 is a Euclidean vector space and 𝑑∶𝑋×𝑋→𝐑₀₊ is the Euclidean metric, and ⋂_𝑗∈𝑝𝐴_𝑗 is convex, then 𝑥∈𝑇𝑥(⊆⋂_𝑗∈𝑝𝐴_𝑗) is the unique fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 and 𝑇_𝑗∶𝐴_𝑗→𝐴_𝑗; ∀𝑗∈𝑝 and also the unique fixed point of   𝑇^𝑝∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖.(iv)If assumption (4) of Theorem 1 is replaced by assumption (5) then properties (i)–(iii) hold for any 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.(v)If 𝑋 is a Euclidean vector space then property (iii) holds also if the condition of (𝑋,𝑑) being a convex metric space is removed.

Proof. The property (i) follows from Theorem 1 when 𝐷_𝑗=𝐷=0; ∀𝑗∈𝑝. To address the proof of property (ii), it is first proven that {𝑥_{𝑛𝑝+𝑗}}_{𝑖∈𝐙0+} is a Cauchy sequence in 𝑋; ∀𝑗∈𝑝. Take 𝑚∈𝐙₊ so that one gets from (10) that 𝑑𝑥_{(𝑛+𝑚+1)𝑝+𝑖+𝑗},𝑥_{(𝑛+𝑚)𝑝+𝑖+𝑗}≤𝑘𝑑𝑥_{(𝑛+𝑚)𝑝+𝑖+𝑗},𝑥_{(𝑛+𝑚−1)𝑝+𝑖+𝑗}≤⋯≤𝑘^𝑚𝑑𝑥_{𝑛𝑝+𝑖+𝑗},𝑥_{(𝑛+1)𝑝+𝑖+𝑗} (19) for some 𝑥=𝑥_𝑖∈𝐴_𝑖, ∀𝑗∈𝑝−1∪{0} for any given 𝑖∈𝑝 from assumption (1) of Theorem 1, where 𝑘=∏^𝑝_𝑖=1𝑘_𝑖. Then, 𝑑(𝑥_{(𝑛+𝑚+1)𝑝+𝑖+𝑗},𝑥_{(𝑛+𝑚)𝑝+𝑖+𝑗})→0 as 𝑚→∞; ∀𝑗∈𝑝−1∪{0}. Thus, {𝑥_{𝑛𝑝+𝑖+𝑗}} is a Cauchy sequence for the given 𝑖∈𝑝; ∀𝑗∈𝑝−1∪{0}. Then, {𝑥_{𝑛𝑝+𝑗}} is a Cauchy sequence; ∀𝑗∈[𝑖,𝑝+𝑖]. Then, such a sequence converges to some 𝑥_𝑖+𝑗∈𝑇𝑥_{𝑖+𝑗−1}∩𝐴_𝑖+𝑗; ∀𝑗∈𝑝−1∪{0} and any given 𝑖∈𝑝, such that 𝑥_{𝑛𝑝+𝑖+𝑗}⪯𝑥_𝑖+𝑗; ∀𝑛∈𝐙₀₊ from (18), since all the elements of the generated non-decreasing sequence in the partial order (𝑋,⪯) are comparable from (18) and (𝑋,𝑑) is complete. The property (i) has been proven.
To prove property (ii), assume that there are two distinct limits 𝑥_𝑖∈𝑇𝑥_𝑖−1∩𝐴_𝑖 and 𝑥_𝑗∈𝑇𝑥_𝑗−1∩𝐴_𝑗 for some distinct 𝑖,𝑗 in 𝑝. Since the restricted composite self-mapping 𝑇_𝑗(≡𝑇^𝑝∣𝐴_𝑗)∶𝐴_𝑗→𝐴_𝑗 is a cyclic contraction on the nonempty closed set 𝐴_𝑗 for any 𝑗∈𝑝, then we can built uniquely a restricted composite self-mapping 𝑇_𝑗∶𝐴_𝑖→𝐴_𝑖 defining the partially ordered sequence {𝑥_{𝑛𝑝+𝑖}}, with first element 𝑥=𝑥_𝑖∈𝐴_i, which converges to 𝑥_𝑖 as 𝑛→∞, since such a restricted composite self-mapping satisfies also assumptions (1)-(2) of Theorem 1. Then, we can proceed, in the same way, with 𝑇_𝑗∶𝐴_𝑗→𝐴_𝑗 generating {𝑥_{𝑛𝑝+𝑗}} converging to 𝑥_𝑗≠𝑥_𝑖 as 𝑛→∞ for any 𝑗∈𝑝. Both such composite self-mappings are Lipschitz-continuous, since they are contractive with the Lipschitz constant being the contractive constant 𝑘<1, so that the limit of the distance can be permuted with the distance of the limits. Then, since 𝑥_ℓ=𝑇_𝑗𝑥_ℓ for ℓ=𝑖,𝑗 and since 𝑥_ℓ is a fixed point of 𝑇_ℓ∶𝐴_ℓ→𝐴_ℓ for ℓ=𝑖,𝑗(≠𝑖)∈𝑝, the following contradiction holds to the existence of two distinct fixed points 𝑥_𝑖(∈𝑇^𝑛_𝑖𝑥_𝑖)≠𝑥_𝑗(∈𝑇^𝑛_𝑗𝑥_𝑗) for some ℓ=𝑖,𝑗(≠𝑖)∈𝑝; ∀𝑛∈𝐙₀₊: 0=𝐻lim_𝑛→∞𝑇^𝑛_𝑖𝑥_𝑖,lim_𝑛→∞𝑇^𝑛_𝑗𝑥_𝑗=𝐻lim_𝑛→∞𝑇^𝑛𝑝𝑥_𝑖,lim_𝑛→∞𝑇^𝑛𝑝𝑥_𝑗=lim_𝑛→∞𝐻𝑇^𝑛𝑝𝑥_𝑖,𝑇^𝑛𝑝𝑥_𝑗≤lim_𝑛→∞𝑘^𝑛𝑑𝑥_𝑖,𝑥_𝑗=0⟹𝑥_𝑖=𝑥_𝑗for𝑖,𝑗(≠𝑖)∈𝑝. (20) Since any existing fixed point in 𝐴_𝑖∩𝐴_𝑗 of 𝑇_ℓ(=𝑇^𝑝∣𝐴_ℓ)∶𝐴_ℓ→𝐴_ℓ for ℓ=𝑖,𝑗; ∀𝑖,𝑗(≠𝑖)∈𝑝 is comparable in the partial order (𝑋,⪯) to any element of 𝐴_𝑖∩𝐴_𝑗, diam(𝐴_𝑖∩𝐴_𝑗)<𝑑₀ and 𝑥_𝑖∈𝑇_𝑖𝑥_𝑖⊆𝐴_𝑖, 𝑥_𝑗∈𝑇_𝑗𝑥_𝑗⊆𝐴_𝑗; ∀𝑖,𝑗(≠𝑖)∈𝑝. Assuming, with no loss in generality, that 𝑥_𝑖⪯𝑥_𝑗, one can build, from the assumptions of Theorem 1 and the current comparability assumption (7), a nondecreasing, converging and partially ordered sequence: 𝑧₁=𝑥_𝑖⪯𝑧₂=𝑥_𝑗⪯⋯⪯𝑧_𝑛⋯⪯𝑧=lim_𝑛→∞𝑧_𝑛 (21) such that 𝑧_2𝑛+1∈𝑇^{(2𝑛+1)𝑝}⊆𝐴_𝑖,𝑧_2𝑛∈𝑇^{(2𝑛+𝑗−𝑖)𝑝}⊆𝐴_𝑗,∀𝑛∈𝐙₀₊. (22) Thus, lim_𝑛→∞𝑇^𝑛𝑝𝑥_𝑖(⊆𝐴_𝑖)=lim_𝑛→∞𝑇^𝑛𝑝𝑥_𝑗(⊆𝐴_𝑗) for any𝑖,𝑗(≠𝑖)∈𝑝. Since, 𝐴_𝑖 and 𝐴_𝑖 are closed and nonempty for any distinct 𝑖,𝑗 in 𝑝, then 𝑧∈𝐴_𝑖∩𝐴_𝑗. Since the pair (𝑖,𝑗) is arbitrary and the set ⋂_𝑖∈𝑝𝐴_𝑖 is nonempty and closed, then 𝑧∈𝑇_𝑗𝑧(⊆⋂_𝑖∈𝑝𝐴_𝑖) for any 𝑗∈𝑝. Then, 𝑧∈𝑇_𝑖𝑧 is a fixed point of 𝑇_𝑖∶𝐴_𝑖→𝐴_𝑖; ∀𝑖∈𝑝. But, since 𝑥_𝑖∈𝑇_𝑖𝑥_𝑖 is a fixed point of 𝑇_𝑖∶𝐴_𝑖→𝐴_𝑖, it cannot converge through an iterated sequence to another distinct fixed point of the same self-mapping or to be distinct of it. Thus, 𝑥=𝑥_𝑖=𝑥_𝑗 is a fixed point of the restricted composite self-mapping of 𝑇_𝑖∶𝐴_𝑖→𝐴_𝑖; ∀𝑖∈𝑝 which is in the closed nonempty set ⋂_𝑗∈𝑝𝐴_𝑗. Then, 𝑇^𝑝𝑛𝑥=𝑇^𝑝𝑥=Fix(𝑇^𝑝)⊆∈⋂_𝑖∈𝑝𝐴_𝑖; ∀𝑛∈𝐙₊. Also, note that, since 𝑥⪯𝑥, 𝑥∈𝑇𝑥 and 𝐷_𝑗=0, ∀𝑗∈𝑝, one gets from (10) that 0=𝑑𝑥,𝑇^𝑝+1𝑥=𝐻𝑇𝑥,𝑇^𝑝+1𝑥≤𝑘_𝑗𝑑(𝑥,𝑥)=𝑘_𝑗𝑑𝑥,𝑇^𝑝𝑥=0, (23) since 𝑥∈𝑇^𝑝+1𝑥 from 𝐻(𝑇𝑥,𝑇^𝑝+1𝑥)=0. Thus, Fix(𝑇^𝑝)=Fix(𝑇^𝑝_𝑖)⊆Fix(𝑇^𝑝+1)⊆𝑇𝑥.
It remains to be proven that 𝑥∈𝑇𝑥⊆⋂_𝑗∈𝑝𝐴_𝑗 is the unique fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖, since 𝑑₀>diam(𝐴_𝑖∩𝐴_𝑗)≥diam(⋂_𝑘∈𝑝𝐴_𝑘) and any existing fixed point in ⋂_𝑘∈𝑝𝐴_𝑘 is comparable, with respect to the partial order (𝑋,⪯), to any element of ⋂_𝑘∈𝑝𝐴_𝑘. This is a consequence of assumption (2) of Theorem 1, so that 𝑑(𝑥_𝑖,𝑇^{𝑛𝑝+𝑗}𝑥_𝑖+𝑗)→0 as 𝑛→∞, and 𝑑𝑥_𝑖,𝑇^{𝑛𝑝+𝑗}𝑥_𝑖+𝑗≤𝐻𝑇^𝑛𝑝𝑥_𝑖,𝑇^{𝑛𝑝+𝑗}𝑥_𝑖+𝑗≤𝑘^{𝑛𝑖+𝑗−1}_ℓ=𝑖𝑘_ℓ𝑑𝑥_𝑖,𝑇^𝑗−1𝑥_𝑖+𝑗<𝑑𝑥_𝑖,𝑥_𝑖+𝑗;∀𝑛∈𝐙₀₊, (24) if 𝑥_𝑖(≠𝑥_𝑖+𝑗)∉𝑇^𝑗−1𝑥_𝑖+𝑗, since 𝑘<1, and then 𝑘^𝑛(∏^{𝑖+𝑗−1}_ℓ=𝑖𝑘_ℓ)<1; ∀𝑗∈𝑝, ∀𝑛(≥𝑛₀) and some 𝑛₀∈𝐙₊. Thus, 𝑥_𝑖∈𝑇𝑥_𝑖⊆𝑇^𝑝𝑛𝑥_𝑖⊆𝐴_𝑖 and 𝑥_𝑖+𝑗∈𝑇𝑥_𝑖+𝑗⊆𝑇^{𝑝𝑛+𝑗}𝑥_𝑖+𝑗(=𝑇^𝑗𝑥_𝑖+𝑗)⊆𝐴_𝑖+𝑗.
Thus, 𝑥_𝑖∈𝑇𝑥_𝑖⇒𝑥_𝑖+𝑗∈𝑇^𝑗𝑥_𝑖+𝑗. It can be also proven in the same way by interchanging the roles of both fixed points that 𝑥_𝑖+𝑗∈𝑇^𝑗𝑥_𝑖+𝑗⇒𝑥_𝑖∈𝑇𝑥_𝑖. As a result, 𝑇^𝑗𝑥_𝑖+𝑗=𝑇𝑥_𝑖=Fix(𝑇)⊆⋂_𝑖∈𝑝𝐴_𝑖.
On the other hand, since any fixed point of the self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; ∀𝑖∈𝑝 has to be also a fixed point of each restricted composite self-mapping 𝑇_𝑗∶𝐴_𝑗→𝐴_𝑗, ∀𝑗∈𝑝, which is a unique 𝑥∈𝑇𝑥, ∀𝑖∈𝑝, then, Fix(𝑇)=Fix(𝑇^𝑝)=Fix(𝑇^𝑝_𝑗)⊆⋂_𝑖∈𝑝𝐴_𝑖; ∀𝑗∈𝑝. Hence, property (ii) is proven.
The property (iii) is proven as follows. Assume that there are 𝑥_𝑖∈𝑇𝑥₁(⊆⋂_𝑗∈𝑝𝐴_𝑗) for 𝑖=1,2 with 𝑥₁≠𝑥₂, since it has been proven that 𝑇𝑥₁=𝑇𝑥₂ if 𝑥_𝑖∈Fix(𝑇); 𝑖=1,2. Then, since (𝑋,𝑑) is a convex metric space (which is guaranteed, in particular, under a sufficiency-type condition, if the vector space 𝑋 is Euclidean and the metric is the Euclidean norm) and ⋂_𝑗∈𝑝𝐴_𝑗 is convex, there is a sequence {𝑥₁^(𝑛)}⊂[𝑥₁,𝑥₂]⊂⋂_𝑗∈𝑝𝐴_𝑗, for some 𝑥₁⁽¹⁾∈𝑇𝑥₁ being such that 𝑑(𝑥₁,𝑥₁⁽¹⁾)=𝑑(𝑥₂,𝑥₁⁽¹⁾)=𝑑(𝑥₁,𝑥₂)/2, fulfilling 0=𝑑𝑥₁^(𝑛+1),𝑥₂≤𝑑𝑥₁^(𝑛),𝑥₂≤2^𝑛𝑑𝑥₁^(𝑛),𝑥₂=2𝑑𝑥₁⁽¹⁾,𝑥₂≤𝐻𝑇𝑥₁,𝑇𝑥₂=𝐻𝑇^𝑝𝑥₁,𝑇^𝑝𝑥₂=0≤𝑘𝑑𝑥₁,𝑥₂<𝑑𝑥₁,𝑥₂<𝑑₀; (25)∀𝑛∈𝐙₊ such 𝑥₁⁽¹⁾∈𝑇𝑥₁ always exists, since 𝑥₁=𝑥₁⁽⁰⁾⪯𝑥₂, since all elements in ⋂_𝑗∈𝑝𝐴_𝑗 are pair wise comparable by hypothesis, 𝑇𝑥₁=𝑇𝑥₂, 𝑥_1,2∈𝑇𝑥₁, [𝑥₁,𝑥₂]⊂⋂_𝑗∈𝑝𝐴_𝑗(𝑋,𝑑) is a convex metric space, and ⋂_𝑗∈𝑝𝐴_𝑗 is nonempty and convex. Then, from the convexity of ⋂_𝑗∈𝑝𝐴_𝑗 and (25), one gets that [𝑥₁^(𝑛+1),𝑥₂]⊂[𝑥₁^(𝑛),𝑥₂]⊂⋂_𝑗∈𝑝𝐴_𝑗, 𝑥₁^(𝑛)=𝑥₂ and 𝑥₁^(𝑛)⪯𝑥₂; ∀𝑛∈𝐙₀₊ and then there are infinitely many fixed points of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 so that 𝑇𝑥₁ has infinite cardinal and 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 would not be a multivalued self-mapping, a contradiction. Thus, 𝑥₁=𝑥₂=𝑥is the unique fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 and also the unique fixed point of 𝑇^𝑝∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 using a similar argument to the above one to prove the uniqueness. Hence, property (iii) follows.
The proof of property (iv) follows directly from the above properties (i)–(iii) and property (iii) of Theorem 1; property (v) follows directly, since 𝐴_𝑖⊂𝑋; ∀𝑖∈𝑝 being closed implies that ⋂_𝑖∈𝑝𝐴_𝑖 is also closed, which together with the condition that ⋂_𝑖∈𝑝𝐴_𝑖 is convex, leads to the property that (⋂_𝑖∈𝑝𝐴_𝑖,𝑑) is a convex metric space if 𝑋 is an Euclidean vector space while the complementary to ⋂_𝑖∈𝑝𝐴_𝑖in 𝑋 is not invoked in the proof of the uniqueness of the fixed point so that if (⋂_𝑖∈𝑝𝐴_𝑖,𝑑) is a convex metric space the uniqueness proof follows as that of property (v).

Remarks 1. (1) Note that the restricted composite multivalued self-mapping 𝑇_𝑗∶𝐴_𝑗→𝐴_𝑗; ∀𝑗∈𝑝 can be extended in a natural way to the composite self-mapping 𝑇^𝑝∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 in the sense that 𝑇^𝑝∣𝐴_𝑗=𝑇_𝑗; ∀𝑗∈𝑝.
(2) The convexity of the subsets 𝐴_𝑖; 𝑖∈𝑝 is not required in Theorem 2(iii) but that of their intersection.
(3) Finally, note that a convex set in a Euclidean space is convex metric space under the Euclidean induced norm and that closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex. This property is used in the proof of property (v) of Theorem 2. Finally, note that Theorem 2(v) holds independently of the metric (not necessarily for a norm-induced metric) and that properties (iii)-(iv) do not require that the subsets 𝐴_𝑖⊂𝑋 for 𝑖∈𝑝 are convex but that their intersection is convex.

3. The Main Result on Best Proximity Points for Nonintersecting Subsets

An “ad hoc” version of Theorem 2 will be obtained in this section for the case of nonintersecting subsets by proving the convergence to unique best proximity points within each subset 𝐴_𝑖, which are also unique fixed points of each of the composed self-mappings 𝑇_𝑖∶𝐴_𝑖→𝐴_𝑖; ∀𝑖∈𝑝. It is assumed that (𝑋,‖‖) is a uniformly convex Banach space endowed with the partial order ⪯ and that the subsets 𝐴_𝑖; ∀𝑖∈𝑝 are nonempty, closed, and convex sets. The following remark describes conditions to characterize a class of Banach spaces from complete metric spaces provided that the norm is induced from a metric.

Remark 3. It is well known that a norm defines a metric. In this sense, a Banach space (𝑋,‖‖) can be considered also a complete metric space (𝑋,𝑑) under the norm-induced metric. To practical effects, the induced metric is identical to the norm. The contrary is not true in general since metrics are subject to less restrictive conditions than norms. However, under certain conditions, as for instance, if the metric is homogeneous and translation-invariant, then it can be considered as a norm in a natural way, say, a metric-induced norm. In this case, we can also consider the norm to be identical to the metric-induced norm. If (𝑋,𝑑) is a complete metric space and 𝑋 is a vector space and 𝑑∶𝑋×𝑋→𝐑₀₊ is a homogeneous and translation invariant metric, then (𝑋,‖‖) is also a Banach space under such a metric-induced norm 𝑑∶𝑋×𝑋→𝐑₀₊.

The next result is an “ad hoc” version for this paper of previous technical results. See Lemma 3.7, Lemma 3.8 and Theorem 3.10 in [33].

Lemma 4 (Lemma 3.7 and Lemma 3.8 of [33]). Let 𝐴_𝑖⊆𝑋 for 𝑖∈𝑝 be nonempty closed subsets of the vector space 𝑋 of a uniformly convex Banach space (𝑋,‖‖) with norm-induced metric 𝑑∶𝑋×𝑋→𝐑₀₊ and 𝑑(𝐴_𝑖,𝐴_𝑖+1)=𝐷_𝑖 and either 𝐴_𝑖 or 𝐴_𝑖+1 for 𝑖∈𝑝 are, furthermore, convex (i.e., at least one of each two adjacent subsets is, in addition, convex). Consider sequences {𝑥_{𝑛𝑝+𝑖}}_{𝑛∈𝐙0+}⊂𝐴_𝑖 and {𝑧_𝑛𝑝+i}_{𝑛∈𝐙0+}⊂𝐴_𝑖 and {𝑦_{𝑛𝑝+𝑖+1}}_{𝑛∈𝐙0+}⊂𝐴_𝑖+1 satisfying(1)‖𝑧_{𝑛𝑝+𝑖}−𝑦_{𝑛𝑝+𝑖}‖→𝐷_𝑖 as 𝑛→∞ for any given 𝑖∈𝑝.(2)For every 𝜀∈𝐑₊, there is 𝑛₀∈𝐙₀₊ such that ‖𝑥_{𝑚𝑝+𝑖}−𝑦_{𝑛𝑝+𝑖+1}‖≤𝐷+𝜀 for all 𝑚,𝑛∈𝐙₀₊ with 𝑚>𝑛≥𝑛₀ for any given 𝑖∈𝑝.
Then, the following properties hold:(i)For every 𝜀∈𝐑₊, there is 𝑛₁∈𝐙₀₊ such that ‖𝑥_{𝑚𝑝+𝑖}−𝑧_{𝑛𝑝+𝑖}‖≤𝜀 for all 𝑚,𝑛∈𝐙₀₊ with 𝑚>𝑛≥𝑛₁ for any given 𝑖∈𝑝. (ii)If ‖𝑧_{𝑛𝑝+𝑖}−𝑦_{𝑛𝑝+𝑖}‖→𝐷_𝑖 as 𝑛→∞ and ‖𝑥_{𝑛𝑝+𝑖}−y_{𝑛𝑝+𝑖}‖→𝐷_𝑖 as 𝑛→∞ then ‖𝑥_{𝑛𝑝+𝑖}−𝑧_{𝑛𝑝+𝑖}‖→0 as 𝑛→∞ for any given 𝑖∈𝑝. (iii){𝑥_{𝑛𝑝+𝑖+𝑗}}_{𝑛∈𝐙0+} and {𝑧_{𝑛𝑝+𝑖+𝑗}}_{𝑛∈𝐙0+}are Cauchy sequences; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝.

The proof of Lemma 4(i)-(ii) is supported by the nonemptiness, closeness, and convexity of the subsets 𝐴_𝑗⊆𝑋; 𝑗∈𝑝 and the uniform convexity of the Banach space (𝑋,‖‖) [33]. The following main result for multivalued 𝑝-cyclic self-mappings is obtained from Theorem 1 and Lemma 4 while taking into account Remark 3.

Theorem 5. Let 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 be a multivalued 𝑝(≥2)-cyclic self-mapping on ⋃_𝑖∈𝑝𝐴_𝑖 with 𝐴_𝑖∈𝐶𝐵(𝑋)⊆𝑋; ∀𝑖∈𝑝 being all nonempty and convex with 𝐷_𝑖=𝑑(𝐴_𝑖,𝐴_𝑖+1); ∀𝑖∈𝑝. Assume the following:(1)Let 𝑋 be a vector space and let (𝑋,𝑑) be a convex complete metric space with 𝑑∶𝑋×𝑋→𝐑₀₊ being a homogeneous translation-invariant metric which induces a norm ‖‖ on 𝑋 such that (𝑋,‖‖) is a Banach space.(2)(𝑋,‖‖) is a uniformly convex Banach space with metric convexity.(3)The complete metric space (𝑋,𝑑), equivalently, the Banach space (𝑋,‖‖), is endowed with a partial order ⪯ defined by (5) with 𝑥=𝑥_𝑖(∈𝐴_𝑖)⪯𝑦∈𝑇𝑥(⊆𝐴_𝑖+1) for any (𝑥,𝑦)∈𝐴_𝑖×𝐴_𝑖+1 and some given 𝑖∈𝑝 such that the resulting (𝑋,⪯) partially ordered space is subject to assumptions (1)–(4) of Theorem 1 and assumption (7) of Theorem 2.
Then, the following properties hold.(i)There are unique best proximity points 𝑥_𝑗+1∈𝑇𝑥_𝑗⊆𝐴_𝑗+1 with 𝑑(𝑥_𝑗,𝑥_𝑗+1)=𝑑(𝑥_𝑗,𝑇𝑥_𝑗+1)=𝐷_𝑗, for each 𝑗∈𝑝 which are also unique fixed points of each of the restricted composite self-mappings 𝑇_𝑗(≡𝑇^𝑝∣𝐴_𝑗)∶𝐴_𝑗→𝐴_𝑗; ∀𝑗∈𝑝.(ii)Take any 𝑥=𝑥_𝑖(∈𝐴_𝑖)⪯𝑦=𝑥_𝑖+1∈𝑇𝑥_𝑖 for any given 𝑖∈𝑝 (i.e., 𝑥 and 𝑦 are partially ordered with respect to the partial ordered set (𝑋,⪯) and consider the partially ordered sequences {𝑥_{𝑛𝑝+𝑗}}, being nondecreasing with respect to ⪯while satisfying 𝑥_{𝑛𝑝+𝑗+1}∈𝑇𝑥_{𝑛𝑝+𝑗}; ∀𝑗∈𝑝 of first element subject to 𝑥=𝑥_𝑖(∈𝐴_𝑖)⪯𝑦=𝑥_𝑖+1∈𝑇𝑥_𝑖 for any given 𝑖∈𝑝. Then, each of such sequences {𝑥_{𝑛𝑝+𝑗}} converges to the unique best proximity point 𝑥_𝑗 in 𝐴_𝑗; ∀𝑗∈𝑝 which is also the unique fixed point of each of the restricted composite self-mapping 𝑇_𝑗∶𝐴_𝑗→𝐴_𝑗. If ⋂_𝑖∈𝑝𝐴_𝑖≠Ø, then 𝑥=𝑥_𝑗∈⋂_𝑖∈𝑝𝐴_𝑖 is the unique fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖, 𝑇_𝑗(≡𝑇^𝑝∣𝐴_𝑗) and a fixed point of 𝑇^𝑝∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖; ∀𝑗∈𝑝.(iii) If assumption (4) of Theorem 1 is replaced by its assumption (5), then the convergence to the above unique best proximity points holds for partially ordered sequences of first element 𝑥∈⋃_𝑖∈𝑝𝐴_𝑖.

Proof. Note from the various hypothesis the uniformly convex Banach space (𝑋,‖‖) possesses the metric convexity property with respect to the norm metric ‖‖ while it is endowed with a partial order ⪯ under assumptions (1)–(4) of Theorem 1. From property (ii) of Theorem 1, (8), the nonemptiness and closeness of the subsets 𝐴_𝑖⊆𝑋; ∀𝑖∈𝑝, and Lemma 4(i)-(ii), it follows that ∃lim_𝑛→∞𝑑𝑥_{𝑛𝑝+𝑖+𝑗+1},𝑥_{𝑛𝑝+𝑖+𝑗}=𝐷_𝑖+𝑗;∃lim_𝑛→∞𝑑𝑥_{(𝑛+1)𝑝+𝑖+𝑗},𝑥_{𝑛𝑝+𝑖+𝑗}=0;∀𝑗∈𝑝−1∪{0}, (26) where 𝑥_{𝑛𝑝+𝑖+𝑗}∈𝑇𝑥_{𝑛𝑝+𝑖+𝑗−1}⊆𝐴_𝑖+𝑗, 𝑥_{(𝑛+1)𝑝+𝑖+𝑗}∈𝑇^𝑝𝑥_{𝑛𝑝+𝑖+𝑗−1}⊆𝑇𝑥_{(𝑛+1)𝑝+𝑖+𝑗−1}⊆𝐴_𝑖+𝑗; ∀𝑗∈𝑝−1∪{0}, ∀𝑛∈𝐙₀₊ for the given 𝑖∈𝑝 and the iterated sequences; {𝑥_{𝑛𝑝+𝑖+𝑗}}_{𝑛∈𝐙0+}; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝 are partially ordered with respect to the partial order ⪯, from Theorem 1, of first element 𝑥_𝑖+𝑗=𝑥_𝑗 generated from the iteration 𝑥_{𝑛𝑝+𝑖+𝑗}∈𝑇𝑥_{𝑛𝑝+𝑖+𝑗−1}; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝 are all Cauchy sequences. Since (𝑋,𝑑)≡(𝑋,‖‖) is complete, it follows that 𝑥_{𝑛𝑝+𝑖+𝑗}→𝑥_𝑖+𝑗(∈𝑇^𝑝𝑥_{(𝑛−1)𝑝+𝑖+𝑗}⊆𝑇𝑥_{𝑛𝑝+𝑖+𝑗−1}⊆𝐴_𝑖+𝑗) and 𝑥_{𝑛𝑝+𝑖+𝑗+1}→𝑥_𝑖+𝑗+1(∈𝑇^𝑝𝑥_{(𝑛−1)𝑝+𝑖+𝑗+1}⊆𝑇𝑥_{𝑛𝑝+𝑖+𝑗}⊆𝐴_𝑖+𝑗+1) as 𝑛→∞; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝 since 𝐴_𝑗⊆𝑋 is nonempty, bounded and closed; ∀𝑗∈𝑝and the given 𝑖∈𝑝. Thus, one gets from (26), since 𝐴_𝑗⊆𝑋 is nonempty, bounded and closed, and then boundedly compact, and also approximatively compact with respect to 𝐴_𝑗−1 [8, 35], that: 𝐷_𝑖+𝑗≤𝑑𝑥_{𝑛𝑝+𝑖+𝑗+1},𝑥_{𝑛𝑝+𝑖+𝑗}→𝑑𝑥_𝑖+𝑗,𝑥_𝑖+𝑗+1=𝐷_𝑖+𝑗=𝑑𝑥_𝑖+𝑗,𝑇𝑥_𝑖+𝑗as𝑛→∞; (27)∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝, where 𝑥_𝑖+𝑗+1∈𝑇^𝑝𝑥_{𝑖+𝑗+1−𝑝}⊆𝑇𝑥_𝑖+𝑗; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝. Since all the subsets 𝐴_𝑗⊂𝑋; ∀𝑗∈𝑝 are nonempty, closed, and boundedly compact; ∀𝑗∈𝑝 then 𝑥_𝑗∈𝐴_𝑗 is a best proximity point in 𝐴_𝑗 of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 and it is also a fixed point of the restricted composite self-mapping 𝑇_𝑗∶⋃_𝑖∈𝑝𝐴_𝑖∣𝐴_𝑗→⋃_𝑖∈𝑝𝐴_𝑖∣𝐴_𝑗; ∀𝑗∈𝑝. Thus, there are Cauchy, then convergent since (𝑋,𝑑) is complete, sequences {𝑥_{𝑛𝑝+𝑖+𝑗}}_{𝑛∈𝐙0+} with respective first elements 𝑥_𝑖+𝑗∈𝑇𝑥_{𝑖+𝑗−1}; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝, each being convergent to 𝑥_𝑖+𝑗∈𝐴_𝑖+𝑗, such that 𝑥=𝑥_𝑖 is the first element of {𝑥_{𝑛𝑝+𝑖}}_{𝑛∈𝐙0+}⊆𝐴_𝑖 which consists of partially ordered elements with respect to the partial order ⪯ such that 𝑥_𝑖+𝑗⪯⋯⪯𝑥_{(𝑛+1)𝑝+𝑖+𝑗}∈𝑇^𝑝𝑥_{𝑛𝑝+𝑖+𝑗}⊆𝑇𝑥_{(𝑛+1)𝑝+𝑖+𝑗−1}⪯𝑥_{(𝑛+1)𝑝+𝑖+𝑗}⪯⋯⪯𝑥_𝑖+𝑗=lim_𝑛→∞𝑥_{𝑛𝑝+𝑖+𝑗}, (28) with {𝑥_{ℓ𝑝+𝑖+𝑗}}⊆𝐴_𝑖+𝑗; 𝑗∈𝑝−1∪{0}, ∀ℓ∈𝐙₀₊ for the given 𝑖∈𝑝. But 𝑥_𝑖+𝑗∈𝐴_𝑖+𝑗; ∀𝑗∈𝑝−1∪{0} and the given 𝑖∈𝑝, is a fixed point of the restricted composite self-mapping 𝑇_𝑗∶⋃_𝑖∈𝑝𝐴_𝑖∣𝐴_𝑗→⋃_𝑖∈𝑝𝐴_𝑖∣𝐴_𝑗, ∀𝑗∈𝑝 and a fixed point of the composite self-mapping 𝑇^𝑝∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 from Lemma 4(iii) to which the partially ordered sequences of first element 𝑥=𝑥_𝑖∈𝐴_𝑖 converge. It is also a best proximity point in 𝐴_𝑖+𝑗 of the self-mapping 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 from Lemma 4(iii) and the second part of Lemma 4(ii). Then, 𝑥_𝑗∈𝑇𝑥_𝑗⊆𝑇^𝑝𝑥_𝑗; ∀𝑗∈𝑝. The uniqueness property of each of those 𝑝 best proximity points 𝑥_𝑗∈𝑇𝑥_𝑗 in each of the subsets 𝐴_𝑗⊆𝑋 follows from their uniqueness as fixed points of the restricted self-mappings 𝑇_𝑗∶⋃_𝑖∈𝑝𝐴_𝑖∣𝐴_𝑗→⋃_𝑖∈𝑝𝐴_𝑖∣𝐴_𝑗 from Theorem 2, since (𝑋,𝑑) is a convex metric space and the subsets 𝐴_𝑗⊆𝑋 are convex; ∀𝑗∈𝑝. On the other hand, it turns out that if all the subsets have nonempty intersection, such an intersection is convex so that the best proximity points are all identical and the unique fixed point of 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 and 𝑇^𝑝∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 from Theorem 2, this leads to the proofs of properties (i)–(iii).

Remarks 2. (1) Theorem 5 proves the uniqueness of the best proximity points for any partially ordered sequences with first elements in any of the subsets of the multivalued 𝑝-cyclic self-mapping on ⋃_𝑖∈𝑝𝐴_𝑖 satisfying assumptions (1)–(4) of Theorem 1 as it was commented, in Section 2 concerning such a theorem, the given 𝐴_𝑖⊂𝑋 for some 𝑖∈𝑝 to select the first two elements of the partial order can be chosen arbitrarily by construction from (6), namely, from assumption (4) of Theorem 1.
(2) The value of the individual contractive constants being less than, equal to, or larger than one for each pair of adjacent subsets is irrelevant in Theorem 5 provided that its product is less than one. Note also that Theorem 5 holds also if the distances between each pair of adjacent subsets are not necessarily identical.
(3) Note also that, for Euclidean metric, the convexity of 𝑋 is kept as hypothesis for the uniqueness of the best proximity points of the multivalued self-mapping, since although the subsets 𝐴_𝑖 of 𝑋, 𝑖∈𝑝 are convex, the existence of points belonging to such subsets guaranteeing the equality in the triangle inequality for the metric would not be otherwise guaranteed, since such sets are disjoint and pair-wise disjoint.
(4) It can be observed that the metric convexity of the space (𝑋,𝑑) cannot be relaxed to that of (⋂_𝑖∈𝑝𝐴_𝑖,𝑑), since the subsets 𝐴_𝑖⊂𝑋 do not necessarily intersect.
(5) Note that the results of Sections 2 and 3 obtained from the contractive condition (5) also hold for multivalued self-mappings 𝑇∶⋃_𝑖∈𝑝𝐴_𝑖→⋃_𝑖∈𝑝𝐴_𝑖 which are not cyclic; that is, 𝑇(𝐴_𝑖)∩𝐴_𝑖+1≠Ø for some 𝑖∈𝑝 but fulfil the condition 𝑇𝑥∩𝐴_𝑖+1≠Ø; ∀𝑥∈𝐴_𝑖, ∀𝑖∈𝑝.

4. Example

Consider two bounded and closed real subsets 𝐴₁=[𝜀,𝑀];  𝐴₂=[−𝑀,−𝜀]=−𝐴₁ for nonnegative positive real constants 𝜀,𝑀 with 𝜀≤𝑀 under the Euclidean metric so that 𝐷=𝑑(𝐴₁,𝐴₂)=2𝜀.

Consider also a scalar discrete dynamic system of state 𝑥_𝑘 operating at each state value under 𝑁 tentative feedback controls 𝑢^(𝑖𝑘)_𝑘=−𝐾^(𝑖𝑘)_𝑘𝑥^(𝑖𝑘)_𝑘; ∀𝑖_𝑘∈𝐼_𝑘, where the indexing set of tentative states at the (𝑘+1)-th sampling point is defined by 𝐼_𝑘+1=𝐼_𝑘×𝑁=𝐼₀×𝑁×⋯×𝑁={1}×𝑁^𝑘,∀𝑘∈𝐙₀₊, (29) where “×” stands for the Cartesian product of sets, 𝑥^(𝑖0)₀=𝑥₀⁽¹⁾=𝑥∈[−𝑀,−𝜀]∪[𝜀,𝑀] is the initial point of an iteration through a self-mapping 𝑇 from [−𝑀,−𝜀]∪[𝜀,𝑀] to itself and 𝐼₀={1}. Then the discrete state trajectory takes values in alternated points at 𝐴₁ and 𝐴₂ from the initial state condition 𝑥 such that 𝑥^(𝑖𝑘+1)_𝑘+1∈𝑇𝑥^(𝑖𝑘)_𝑘={𝑥^(𝑖𝑘,1)_𝑘+1,…,𝑥^{(𝑖𝑘,𝑁)}_𝑘+1}; ∀𝑘∈𝐙₀₊ obtained as follows: 𝑥^(𝑖𝑘+1)_𝑘+1=𝑥^(𝑖𝑘+1)_𝑘+1if𝑥^(𝑖𝑘+1)_𝑘+1∈[𝜀,𝑀],𝑀sgn𝑥^(𝑖𝑘+1)_𝑘+1if𝑥^(𝑖𝑘+1)_𝑘+1>𝑀,∀𝑖_𝑘∈𝐼_𝑘,∀𝑘∈𝐙₀₊,𝜀sgn𝑥^(𝑖𝑘+1)_𝑘+1if𝑥^(𝑖𝑘+1)_𝑘+1<𝜀, (30) where 𝑥^(𝑖𝑘+1)_𝑘+1=𝑎_𝑘𝑥^(𝑖𝑘)_𝑘+𝑏_𝑘𝑢^(𝑖𝑘)_𝑘=𝑎_𝑘−𝑏_𝑘𝐾^(𝑖𝑘)_𝑘𝑥^(𝑖𝑘)_𝑘;∀𝑖_𝑘∈𝐼_𝑘,∀𝑘∈𝐙₀₊ (31) with 𝑎_𝑘≠0 and 𝑏_𝑘>0 being nonzero real numbers under a sequence of controllers of gains 𝐾^(𝑖𝑘)_𝑘=𝜌^(𝑖𝑘)_𝑘+𝑎_𝑘𝑏_𝑘,𝜌^(𝑖𝑘)_𝑘∈0,𝜌;∀𝑖_𝑘∈𝐼_𝑘,∀𝑘∈𝐙₀₊ (32) for some nonnegative real constant 𝜌 so that, after replacing (32) into (31), this leads to the controlled closed-loop trajectory sequence given by (30) subject to 𝑥^(𝑖𝑘+1)_𝑘+1=−𝜌^(𝑖𝑘)_𝑘𝑥^(𝑖𝑘)_𝑘; ∀𝑖_𝑘∈𝐼_𝑘 with |𝑥^(𝑖𝑘+1)_𝑘+1|≤min(max(𝜌^(𝑖𝑘)_𝑘|𝑥^(𝑖𝑘)_𝑘|,𝜀),𝑀); ∀𝑖_𝑘∈𝐼_𝑘, ∀𝑘∈𝐙₀₊. Assume that (5) holds for 𝑥⪯𝑦 so that, for some 𝑦∈𝑇𝑥 and some 𝑧∈𝑇𝑦, one gets 𝑑𝑧,𝑦≤𝐻𝑇𝑥,𝑇𝑦≤𝐾𝑑𝑥,𝑦+2(1−𝐾)𝜀 (33) for some 𝐾∈(0,1). Note also from (30)-(31) that (5) is guaranteed under the set of constraints below: 2𝜀≤𝑑𝑥^(𝑖𝑘+1)_𝑘+2,𝑥^(𝑖𝑘)_𝑘+1≤𝑑𝑥^(𝑖𝑘+1)_𝑘+2,𝑥^(𝑖𝑘)_𝑘+1=𝑥^(𝑖𝑘+1)_𝑘+2−𝑥^(𝑖𝑘)_𝑘+1=𝑥^(𝑖𝑘+1)_𝑘+2+𝑥^(𝑖𝑘)_𝑘+1=𝑎_𝑘+1−𝑏_𝑘+1𝐾^(𝑖𝑘+1)_𝑘+1−1𝑥^(𝑖𝑘+1)_𝑘+1≤𝑎_𝑘+1−𝑏_𝑘+1𝐾^(𝑖𝑘+1)_𝑘+1−1min𝑥^(𝑖𝑘+1)_𝑘+1≤𝑎,𝑀_𝑘+1−𝑏_𝑘+1𝐾^(𝑖𝑘+1)_𝑘+1−1min𝜌^(𝑖𝑘)_𝑘𝑥^(𝑖𝑘)_𝑘𝑑,𝑀≤min₀,𝐾𝑑𝑥^(𝑖𝑘+1)_𝑘+1,𝑥^(𝑖𝑘)_𝑘()𝜀𝑑+21−𝐾=min₀,𝐾𝑎_𝑘−𝑏_𝑘𝐾^(𝑖𝑘)_𝑘−1𝑥^(𝑖𝑘)_𝑘()𝜀𝑑+21−𝐾≤min₀,𝐾𝑎_𝑘−𝑏_𝑘𝐾^(𝑖𝑘)_𝑘−1𝑀+2(1−𝐾)𝜀≤𝑑₀;forsome𝑖_𝑘∈𝐼_𝑘,∀𝑘∈𝐙₀₊ (34) which hold provided that there is at least a controller gain sequence {𝐾^(𝑖𝑘)_𝑘}_{𝑖𝑘∈𝐼𝑘} subject to the constraints:

𝑑₀>max2𝜀,𝑀𝑎_𝑘+𝑎_𝑘−1,0<𝜌^(𝑖𝑘)_𝑘≤𝑑min1,⁰𝑀−𝑎_𝑘−𝑎_𝑘,𝐾𝑎−1^𝑘+1−𝑏_𝑘𝐾^(𝑖𝑘)_𝑘−1+2𝜀(1−𝐾)/𝑥_𝑘𝑎_𝑘+2−𝑏_𝑘+1𝐾^(𝑖𝑘+1)_𝑘+1𝑑−1if𝜀>0min1,⁰𝑀−𝑎_𝑘−𝑎_𝑘,𝐾𝑎−1^𝑘+1−𝑏_𝑘𝐾^(𝑖𝑘)_𝑘−1𝑎_𝑘+2−𝑏_𝑘+1𝐾^(𝑖𝑘+1)_𝑘+1−1if𝜀=0 (35a)(35b)

for some 𝑖_𝑘∈𝐼_𝑘; ∀𝑘∈𝐙₀₊. The above constraints guarantee the generation of an iterated nondecreasing partially ordered sequence 𝑆(𝑥)={𝑥=𝑥₀⁽¹⁾,𝑥^(𝑖𝑖)₁,…,𝑥^(𝑖𝑛)_𝑛,…} by the self-mapping 𝑇 on [−𝑀,−𝜀]∪[𝜀,𝑀] with respect to the partial order “⪯” under a sequence of control gains satisfying 0<𝐾^(𝑖𝑘)_𝑘=(𝜌^(𝑖𝑘)_𝑘+𝑎_𝑘)/𝑏_𝑘≤(1/|𝑏_𝑘|)((𝑑₀/𝑀)−|𝑎_𝑘−1|) and |𝑎_𝑘−1|<2𝜀/𝑀≤𝑑₀/𝑀; ∀𝑘∈𝐙₀₊.

It follows from Theorem 5 that such a sequence has two subsequences, built by keeping the same order with alternated terms of 𝑆(𝑥), converging, respectively, to the unique best proximity points, 𝑥^∗₁=𝜀∈𝐴₁ and 𝑥^∗₂=−𝜀∈𝐴₂ of the self-mapping. If 𝜀=0 both such sequences and 𝑆(𝑥) converge to the resulting unique fixed point of the 2-cyclic composite self-mapping 𝑇 on [−𝑀,−𝜀]∪[𝜀,𝑀]. If the considered subsets are 𝐴₁=[−𝜀,𝑀];𝐴₂=[−𝑀,𝜀]=−𝐴₁, the first term of the chain of inequalities (34) is valid with the replacement 2𝜀→0 while the terms 2(1−𝐾)𝜀 are all zeroed in such a chain of inequalities. Thus, (34) holds if there is a sequence of controller gains satisfying 0<𝐾^(𝑖𝑘)_𝑘=𝜌^(𝑖𝑘)_𝑘+𝑎_𝑘𝑏_𝑘≤1𝑏_𝑘𝑑₀−𝑎𝐾𝑀^𝑘−1forsome𝑖_𝑘∈𝐼_𝑘;∀𝑘∈𝐙₀₊ (36) under sufficiency-type constraints 𝑑₀>𝐾𝑀 if |𝑎_𝑘|<𝑑₀/𝐾𝑀−1; ∀𝑘∈𝐙₀₊ or 𝑑₀<𝐾𝑀 and 𝑎_𝑘>1−𝑑₀/𝐾𝑀 if 𝑎_𝑘∈[0,1]; ∀𝑘∈𝐙₀₊. Thus, 𝑆(𝑥)={𝑥=𝑥₀⁽¹⁾,𝑥^(𝑖𝑖)₁,…,𝑥^(𝑖𝑛)_𝑛,…} converges to 𝑥^∗=0 which is the a unique fixed point of 𝑇 on [−𝑀,−𝜀]∪[𝜀,𝑀].

The following extension of the example is direct. Assume that (31) is replaced by the second-order uncoupled discrete dynamic system: 𝑥^(𝑖𝑘+1)_𝑘+1=𝑥^(𝑖𝑘+1)_1𝑘+1,𝑥^(𝑖𝑘+1)_2𝑘+1^𝑇=𝐴_𝑘𝑥^(𝑖𝑘)_𝑘+𝐵_𝑘𝑢^(𝑖𝑘)_𝑘=𝐴_𝑘−𝐵_𝑘𝐾^(𝑖𝑘)_𝑘^𝑇𝑥^(𝑖𝑘)_𝑘;∀𝑖_𝑘∈𝐼_𝑘,∀𝑘∈𝐙₀₊, (37) where the superscript 𝑇 stands for transposition and 𝑥^(𝑖𝑘+1)_𝑘+1=(𝑥^(𝑖𝑘+1)_1𝑘+1,𝑥^(𝑖𝑘+1)_2𝑘+1)^𝑇, subject to the tentative scalar feedback controls 𝑢^(𝑖𝑘)_𝑘=−𝐾^{(𝑖𝑘)𝑇}_𝑘𝑥^(𝑖𝑘)_𝑘, where 𝐴_𝑘=Diag(𝑎_1𝑘,𝑎_2𝑘) is a sequence of real 2×2 diagonal matrices, 𝐵_𝑘=(𝑏_1𝑘,𝑏_2𝑘)^𝑇 is a sequence of two dimensional real column vectors, 𝐾^(𝑖𝑘)_𝑘^𝑇are sequences of two-dimensional real row vectors. The set of inequalities (34) are replaced by parallel ones by using the maxima for norms/distances of the two state vector components. The resulting inequalities are obtained from the initial basic constraints: 2𝜀≤𝑑𝑥^(𝑖𝑘+1)_𝑘+2,𝑥^(𝑖𝑘)_𝑘+1≤𝑑𝑥^(𝑖𝑘+1)_𝑘+2,𝑥^(𝑖𝑘)_𝑘+1=max_1≤𝑗≤2𝑑𝑥^(𝑖𝑘+1)_𝑗𝑘+2,𝑥^(𝑖𝑘)_𝑗,𝑘+1 (38) for some 𝑖_𝑘∈𝐼_𝑘,∀𝑘∈𝐙₀₊. To keep the validity of (30) for each state component, the constraints (35a) and (35b) are replaced by

𝑑₀>max_1≤𝑗≤2max2𝜀,𝑀𝑎_𝑗𝑘+𝑎_𝑗𝑘−1,0<𝜌^(𝑖𝑘)_𝑘≤min1,min_1≤𝑗≤2𝑑₀𝑀−𝑎_𝑗𝑘−𝑎_𝑗𝑘−1,min_1≤𝑗≤2𝐾𝑎_𝑗,𝑘+1−𝑏_𝑗𝑘𝐾^(𝑖𝑘)_𝑗𝑘−1+2𝜀(1−𝐾)/𝑥_𝑗𝑘𝑎_𝑗,𝑘+2−𝑏_𝑗,𝑘+1𝐾^(𝑖𝑘+1)_𝑗,𝑘+1−1if𝜀>0min1,min_1≤𝑗≤2𝑑₀𝑀−𝑎_𝑗𝑘−𝑎_𝑗𝑘−1,min_1≤𝑗≤2𝐾𝑎_𝑗,𝑘+1−𝑏_𝑗𝑘𝐾^(𝑖𝑘)_𝑗𝑘−1𝑎_𝑗,𝑘+2−𝑏_𝑗,𝑘+1𝐾^(𝑖𝑘+1)_𝑗,𝑘+1−1if𝜀=0 (39a)(39b)

for some 𝑖_𝑘∈𝐼_𝑘; ∀𝑘∈𝐙₀₊.

Acknowledgments

The author is very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government by its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. He is also grateful to UPV/EHU for its financial support through UFI 2011/07 and to the referees for their useful comments.