# Some Results on Fixed and Best Proximity Points of Multivalued

Authors: M. De la Sen
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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+:
(1)H(A,B)=max{supx∈Ad(x,B),supy∈Bd(y,A)}
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
(2)D=d(A,B)=infx∈A,y∈Bd(x,y)=infx∈Ad(x,B)=infy∈Bd(y,A).
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:
(3)D≤H(A,B)≤δ(A,B)=δ(B,A)=supx∈A,y∈Bd(a,b)≤δ(A,B)+δ(B,C);∀A,B,C∈B(X),(4)[(A,B∈CB(X))∧H(A,B)<ε]⟹[∃b∈B:d(a,b)<ε,∀a∈A]
and δ(A,B)=0 if and only if A=B={x}. Consider also a self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai, where Ai are nonempty closed sets of X; ∀i∈p-={1,2,…,p}, subject to the constraints T(Ai)⊆Ai+1 such that Aip+j≡Aj for any integer numbers j∈[1,p-1)∩Z and i∈Z0+=Z+∪{0} with R0+=R+∪{0}. If p≥2 then T:⋃i∈p-Ai→⋃i∈p-Ai 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∈p-Ai→⋃i∈p-Ai is a multivalued p-cyclic self-mapping so that Ai∋x→Tx(≠⌀)⊂Ai+1; ∀i∈p-, ∀x∈⋃i∈p-Ai. 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 Ai⊂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.IfT:⋃i∈p-Ai→⋃i∈p-Ai is a multivalued p-cyclic self-mapping then the set BP(Ai)⊂Ai will be said to be the set of best proximity points between Ai to Ai+1 if d(Ai,Ai+1)=Di=d(z,y) for all z∈Ai 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 Ai be a set of p(≥2) nonempty, bounded, and closed subsets of X; ∀i∈p- (i.e., Ai∈CB(X); ∀i∈p-) with Di=d(Ai,Ai+1); ∀i∈p- and let T:⋃i∈p-Ai→⋃i∈p-Ai be a multivalued p-cyclic self-mapping on ⋃i∈p-Ai satisfying.(1)
There existp real constants ki∈R0+ satisfying k=∏i∈p-[ki]∈[0,1) such that the following condition holds:
(5)H(Tx,Ty)≤kid(x,y)+(1-ki)Di
for any given x∈Ai and y∈Ai+1 which fulfil x⪯y, ∀i∈p-.(2)
Ifd(x,y)<d0 for some given d0∈R+, y∈Tx, and any given x∈⋃i∈p-Ai, then x⪯y with y∈Aj+1 if  x∈Aj for any given j∈p-.(3)
There are somei∈p-, some x=xi∈Ai, and some xi+1∈Txi⊂Ai+1 such that d(xi,xi+1)<d0i for some d0i>Di.(4)
(6)d0≥max(maxj∈p-d0j,maxj∈p-(kj(d0j-Dj)+Dj)).Note that (6) implies that d0j≤min(d0,(d0-(1-kj)Dj)/kj); ∀j∈p-. Then, the following properties hold.(i)
There is a partially ordered subsequenceS^i={xi+j+nkp}nk∈Z0+ of the partially ordered sequence S(xi)={xi+j}j∈Z0+, both of them of the first element xi, with respect to the partial order (X,⪯), such that xi+j+nkp∈Qi+j for j∈p¯; ∀k≥k0, nk∈Z0+ for some k0∈Z0+ and the given i∈p-, where Qi+j⊆Txi+j-1⊆Tj-1xi⊆Ai+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∈p-Ai→⋃i∈p-Ai such that
(7)Di+j≤d(xnp+i+j+1,xnp+i+j)≤ki+jD+(1-ki+j)D=D,∀j∈p-1¯∪{0},∀n∈Z0+,
where D=minj∈p-Dj with xnp+i+j∈Txnp+i+j-1⊆Ai+j; ∀j∈p-1¯∪{0},  ∀n∈Z0+ for the given i∈p-.(ii)
IfDj=D; ∀j∈p- then any partially ordered sequence S(xi) of first element x=xi∈Ai fulfills
(8)∃limn→∞d(xnp+i+j+1,xnp+i+j)=D;∀j∈p- and the given i∈p-, and xnp+j+1∈Txnp+j⊆Ai+j+1; ∀j∈p- (i.e., xnp+j+1∈Ai+j+1 if 0≤j≤p-i-1 and xnp+j+1∈Aj-p+i+1 if p-i<j≤p-1), ∀n∈Z0+. Let BP(Aj) be the set of best proximity points between Aj and Aj+1; ∀j∈p-. Then, there is a sequence {zn(j)}⊂BP(Aj); ∀j∈p- such that the following limit exists:
(9)limn→∞d(xnp+j+1,zn(j))=D;∀j∈p-withxnp+j+1∈Txnp+j;∀n∈Z0+(iii)
If assumption (3) is removed and (6) in assumption (4) is replaced by the stronger condition(5)
(10)d0>max(maxj∈p-(Dj+diam(Aj)),maxj∈p-(kj(d0j-Dj)+Dj))then, properties (i)-(ii) hold for any x∈⋃i∈p-Ai.Proof.
Letxi∈Ai for the given i∈p- which satisfy assumption (3). Then, from such an assumption, there is xi+1∈Txi, which is also in Ai+1, since Tx∈Ai+1 for any x∈⋃i∈p-Ai, such that d(xi,xi+1)<d0i≤d0. Thus, xi⪯xi+1 from assumption (2), since d0i≤d0. From (6) and assumptions (1)-(2) by considering the distance between adjacent subsets,
(11)Di≤H(Txi,T)≤kid(xi,xi+1)+(1-ki)Di<kid0i+(1-ki)Di≤d0i≤d0,
since Di<d0i≤(1/ki)[(ki-1)Di+d0] from assumptions (3)-(4). From assumption (2) and (11), there is xi+2∈Txi+1⊂Ai+2 such that xi+1⪯xi+2, and then d(xi+1,xi+2)<d0,i+1, and d(xi+1,xi+2)≤H(Txi,Txi+1)<d0i≤d0. Then, one gets from (11) and assumption (4): (12a)Di+1≤H(Txi+1,Txi+2)≤ki+1d(xi+1,xi+2)+(1-ki+1)Di+1≤ki+1d0,i+1+(1-ki+1)Di+1<ki+1d0,i+1+(1-ki+1)[d0-ki+1(d0,i+1-Di+1)]≤d0+ki+1d0-ki+1d0+(1-ki+1)ki+1(Di+1-d0i+1)<d0,(12b)d(xi+2,xi+3)≤H(Txi+1,Txi+2)≤ki+1d(xi+1,xi+2)+(1-ki+1)Di+1≤ki+1[kid(xi,xi+1)+(1-ki)Di]+(1-ki+1)Di+1≤ki+1kid(xi,xi+1)+[ki+1(1-ki)+(1-ki+1)]maxi∈p-Di<d0.Again, from assumption (2), there is xi+3∈Txi+2 such that xi⪯xi+1⪯xi+2⪯xi+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 xi⪯xi+1⪯xi+2⪯xi+p-2 implies, from assumption (2), the existence of the partially ordered space xi⪯xi+1⪯xi+2⪯⋯⪯xi+p-1 satisfying xi+j∈Txi+j-1⊆Ai+j; ∀j∈p- with Akp+j≡Aj; ∀k∈p-. Also, proceeding recursively with (12b), one concludes, if k=∏i∈p-[ki]∈[0,1) and D=maxi∈p-Di, that there is a partially ordered sequence xi⪯xi+1⪯xi+2⪯⋯⪯xi+p-1⪯xi+p such that xi+j∈Txi+j-1⊆T(Ai+j-1)⊆Ai+j; ∀j∈p¯, and
(13)d(xi+p+1,xi+p)≤kd(xi,xi+1)+(1-k)D
so that there is a partially ordered sequence xi⪯xi+1⪯xi+2⪯⋯⪯xi+p-1⪯xi+np⪯⋯ such that xi+j∈Txi+np+j-1⊆T(Ai+j); ∀j∈p-i¯:
(14)Di+1≤d(xnp+i+2,xnp+i+1)≤H(Txi+np+1,Txnp+i)+(1-ki)Di≤limsup[knd(xi,xi+1)+(1-kn)D]n→∞=D;∀n∈Z+
so that(15)Di+j≤d(xnp+i+j+1,xnp+i+j)≤ki+jD+(1-ki+j)D=D;∀j∈p-1¯∪{0},∀n∈Z0+.
Then, from (15), there are p closed “quasi-proximity” sets Qi⊆Ai, Qi+1⊆Txi⊆Ai+1,  Qi+p-1⊆Tp-1xi⊆Ai+p-1 between each pair of adjacent subsets of the cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai in view of (14), such that there is a partially ordered subsequence {xi+j+nkp}nk∈Z+ of the partially ordered sequence xi⪯xi+1⪯xi+2⪯⋯⪯xi+p-1⪯xi+np⪯⋯, being subject to xi+j+nkp∈Qi+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 Si of first element xi∈Ai in property (i) satisfies the following property ∃limn→∞d(xj+np+1,xj+np)=D; ∀j∈p-. Assume not, then, it follows that
(16)D≤liminfn→∞d(xi+j+np+1,xi+j+np)≤limsupn→∞d(xi+j+np+1,xi+j+np)≤kjlimn→∞d(xi+j+np,xi+j+np-1)+(1-kj)D≤kjD+(1-kj)D=D;∀j∈p-1¯∪{0} and the given i∈p- so that, by using complete induction, ∃limn→∞d(xi+j+np+1,xi+j+np)=D; ∀j∈p-1¯∪{0} and the given i∈p- with {xk}k≥i being partially ordered with respect to (X,⪯), that is, xi+k(∈Txi+k-1⊆Ai+k)⪯xi+k+1(∈Txi+k⊆Ai+k+1); ∀k∈Z0+ and we can then reformulate the above limits of the distances as ∃limn→∞d(xi+j+np+1,xi+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 {zn(j)}⊂BP(Aj) for some j∈p-. Since {zn(j)}⊂BP(Aj), liminfn→∞d(xnp+j+1,zn(j))<D is impossible in the case that liminfn→∞d(xnp+i+1,zn(j)) would not exist for some j∈p-. Then,
(17)liminfn→∞d(xnp+j+1,zn(j))=D≠liminfk→∞d(xnkp+j+1,znk(j))>D=d(Aj,Aj+1)=d(z,y)
for some z∈BP(Aj), y∈Tz⊂BP(Aj+1), since Aj and Aj+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 {zn(j)}⊂BP(Aj) and xnp+j+1∈Tnp+j⊂Aj+1; ∀j∈p-. The property (ii) has been proven.
If assumption (3) is removed, whiled0 satisfies the stronger constraint (10), then there are infinitely many sequences S(x) for any arbitrary first element x∈⋃i∈p-Ai, 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∈p-Aj)∣Ai→(⋃j∈p-Aj)∣Ai+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 xi∈Ai, xi+1∈Txi⊆Ai+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 Ai 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, ki<1;  ∀i∈p-.Note that Theorem1(iii) also holds in the particular case that the partial order is a total order for all pairs in any Cartesian product Ai×Ai+1 of adjacent subsets Ai⊂X; ∀i∈p-, since both elements of any ordered pair (x,y)∈Ai×Ai+1, y∈Tx; ∀i∈p- are comparable with respect to the partial order ⪯. Theorem 1(iii) establishes that any element in any subset Ai⊂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 Theorem1, assume, furthermore, (6)
Dj=D=0; ∀j∈p- (i.e., ⋂j∈p-Aj≠⌀);(7)
the limitx of any converging nondecreasing sequence {xn}n∈Z0+is comparable to each xn; ∀n∈Z0+ in the partial order (X,⪯), that is,
(18)[xn⪯x(≠xn)forx∈Aj,xn∈Aj,∀j∈p-,∀n∈Z0+]⟹H(Tx,Txn)>kid(x,xn).Then, there is a sequence {xnp+i+j}n∈Z0+satisfying xnp+i+j∈Tnp+jxi for some given initial element x=xi∈Ai 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)
∃limn→∞d(xnp+i+j+2,xnp+i+j+1)=0; j∈p-1¯∪{0} and the given i∈p- with xnp+j+2∈Txnp+j+1; j∈p-1¯∪{0}, ∀n∈Z0+ and the sequence {xnp+i+j}n∈Z0+is a Cauchy sequence; j∈p-1¯∪{0}.(ii)
The sequence{xnp+i+j}n∈Z0+ for any j∈p-1¯∪{0} and the given i∈p- converge to a limit x- in ⋂j∈p-Aj, which is a fixed point of the composite self-mapping T^j:Aj→Aj, where T^j=Tp=T∘T∘⋯T(ptimes)=Tp∣Aj of domain Aj; ∀j∈p- and also a fixed point of the self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai, that is, x-∈T^jx-(⊆⋂j∈p-Aj) and x-∈Tpx-(⊆⋂j∈p-Aj); ∀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∈p-Aj is convex, then x-∈Tx-(⊆⋂j∈p-Aj) is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai and T^j:Aj→Aj; ∀j∈p- and also the unique fixed point of   Tp:⋃i∈p-Ai→⋃i∈p-Ai.(iv)
If assumption (4) of Theorem1 is replaced by assumption (5) then properties (i)–(iii) hold for any x∈⋃i∈p-Ai.(v)
IfX 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 Theorem1 when Dj=D=0; ∀j∈p-. To address the proof of property (ii), it is first proven that {xnp+j}i∈Z0+ is a Cauchy sequence in X; ∀j∈p-. Take m∈Z+ so that one gets from (10) that
(19)d(x(n+m+1)p+i+j,x(n+m)p+i+j)≤kd(x(n+m)p+i+j,x(n+m-1)p+i+j)≤⋯≤kmd(xnp+i+j,x(n+1)p+i+j)
for some x=xi∈Ai, ∀j∈p-1¯∪{0} for any given i∈p- from assumption (1) of Theorem 1, where k=∏i=1pki. Then, d(x(n+m+1)p+i+j,x(n+m)p+i+j)→0 as m→∞; ∀j∈p-1¯∪{0}. Thus, {xnp+i+j} is a Cauchy sequence for the given i∈p-; ∀j∈p-1¯∪{0}. Then, {xnp+j} is a Cauchy sequence; ∀j∈[i,p+i]. Then, such a sequence converges to some x-i+j∈Tx-i+j-1∩Ai+j; ∀j∈p-1¯∪{0} and any given i∈p-, such that xnp+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 limitsx-i∈Tx-i-1∩Ai and x-j∈Tx-j-1∩Aj for some distinct i,j in p¯. Since the restricted composite self-mapping T^j(≡Tp∣Aj):Aj→Aj is a cyclic contraction on the nonempty closed set Aj for any j∈p-, then we can built uniquely a restricted composite self-mapping T^j:Ai→Ai defining the partially ordered sequence {xnp+i}, with first element x=xi∈Ai, 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:Aj→Aj generating {xnp+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:Aℓ→Aℓ for ℓ=i,j(≠i)∈p-, the following contradiction holds to the existence of two distinct fixed points x-i(∈T^inx-i)≠x-j(∈T^jnx-j) for some ℓ=i,j(≠i)∈p-; ∀n∈Z0+:
(20)0=H(limn→∞T^inx-i,limn→∞T^jnx-j)=H(limn→∞Tnpx-i,limn→∞Tnpx-j)=limn→∞H(Tnpx-i,Tnpx-j)≤limn→∞(kn)d(x-i,x-j)=0⟹x-i=x-jfori,j(≠i)∈p-.
Since any existing fixed point in Ai∩Aj of T^ℓ(=Tp∣Aℓ):Aℓ→Aℓ for ℓ=i,j; ∀i,j(≠i)∈p- is comparable in the partial order (X,⪯) to any element of Ai∩Aj, diam(Ai∩Aj)<d0 and x-i∈T^ix-i⊆Ai, x-j∈T^jx-j⊆Aj; ∀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:
(21)z1=x-i⪯z2=x-j⪯⋯⪯zn⋯⪯z=limn→∞zn
such that
(22)z2n+1∈T(2n+1)p⊆Ai,z2n∈T(2n+j-i)p⊆Aj,∀n∈Z0+.
Thus, limn→∞Tnpx-i(⊆Ai)=limn→∞Tnpx-j(⊆Aj) for anyi,j(≠i)∈p-. Since, Ai and Ai are closed and nonempty for any distinct i,j in p-, then z∈Ai∩Aj. Since the pair (i,j) is arbitrary and the set ⋂i∈p-Ai is nonempty and closed, then z∈T^jz(⊆⋂i∈p-Ai) for any j∈p-. Then, z∈T^iz is a fixed point of T^i:Ai→Ai; ∀i∈p-. But, since x-i∈T^ix-i is a fixed point of T^i:Ai→Ai, 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:Ai→Ai; ∀i∈p- which is in the closed nonempty set ⋂j∈p-Aj. Then, Tpnx-=Tpx-=Fix(Tp)⊆∈⋂i∈p-Ai; ∀n∈Z+. Also, note that, since x-⪯x-,  x-∈Tx- and Dj=0, ∀j∈p-, one gets from (10) that
(23)0=d(x-,Tp+1x-)=H(Tx-,Tp+1x-)≤kjd(x-,x-)=kjd(x-,Tpx-)=0,
since x-∈Tp+1x- from H(Tx-,Tp+1x-)=0. Thus, Fix(Tp)=Fix(T^ip)⊆Fix(Tp+1)⊆Tx-.
It remains to be proven thatx-∈Tx-⊆⋂j∈p-Aj is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai, since d0>diam(Ai∩Aj)≥diam(⋂k∈p-Ak) and any existing fixed point in ⋂k∈p-Ak is comparable, with respect to the partial order (X,⪯), to any element of ⋂k∈p-Ak. This is a consequence of assumption (2) of Theorem 1, so that d(x-i,Tnp+jx-i+j)→0 as n→∞, and
(24)d(x-i,Tnp+jx-i+j)≤H(Tnpx-i,Tnp+jx-i+j)≤kn(∏ℓ=ii+j-1[kℓ])d(x-i,Tj-1x-i+j)<d(x-i,x-i+j);∀n∈Z0+,
if x-i(≠xi+j)∉Tj-1x-i+j, since k<1, and then kn(∏ℓ=ii+j-1kℓ)<1; ∀j∈p-, ∀n(≥n0) and some n0∈Z+. Thus, x-i∈Tx-i⊆Tpnx-i⊆Ai and x-i+j∈Tx-i+j⊆Tpn+jx-i+j(=Tjx-i+j)⊆Ai+j.
Thus,x-i∈Tx-i⇒x-i+j∈Tjx-i+j. It can be also proven in the same way by interchanging the roles of both fixed points that x-i+j∈Tjx-i+j⇒x-i∈Tx-i. As a result, Tjx-i+j=Tx-i=Fix(T)⊆⋂i∈p-Ai.
On the other hand, since any fixed point of the self-mappingT:⋃i∈p-Ai→⋃i∈p-Ai; ∀i∈p- has to be also a fixed point of each restricted composite self-mapping Tj:Aj→Aj, ∀j∈p-, which is a unique x-∈Tx-, ∀i∈p-, then, Fix(T)=Fix(Tp)=Fix(Tjp)⊆⋂i∈p-Ai; ∀j∈p-. Hence, property (ii) is proven.
The property (iii) is proven as follows. Assume that there arex-i∈Tx-1(⊆⋂j∈p-Aj) for i=1,2 with x-1≠x-2, since it has been proven that Tx-1=Tx-2 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∈p-Aj is convex, there is a sequence {x-1(n)}⊂[x-1,x-2]⊂⋂j∈p-Aj, for some x-1(1)∈Tx-1 being such that d(x-1,x-1(1))=d(x-2,x-1(1))=d(x-1,x-2)/2, fulfilling
(25)0=d(x-1(n+1),x-2)≤d(x-1(n),x-2)≤2nd(x-1(n),x-2)=2d(x-1(1),x-2)≤H(Tx-1,Tx-2)=H(Tpx-1,Tpx-2)=0≤kd(x-1,x-2)<d(x-1,x-2)<d0;∀n∈Z+ such x-1(1)∈Tx-1 always exists, since x-1=x-1(0)⪯x-2, since all elements in ⋂j∈p-Aj are pair wise comparable by hypothesis, Tx-1=Tx-2, x-1,2∈Tx-1, [x-1,x-2]⊂⋂j∈p-Aj(X,d) is a convex metric space, and ⋂j∈p-Aj is nonempty and convex. Then, from the convexity of ⋂j∈p-Aj and (25), one gets that [x-1(n+1),x-2]⊂[x-1(n),x-2]⊂⋂j∈p-Aj, x-1(n)=x-2 and x-1(n)⪯x-2; ∀n∈Z0+ and then there are infinitely many fixed points of T:⋃i∈p-Ai→⋃i∈p-Ai so that Tx-1 has infinite cardinal and T:⋃i∈p-Ai→⋃i∈p-Ai would not be a multivalued self-mapping, a contradiction. Thus, x-1=x-2=x-is the unique fixed point  of T:⋃i∈p-Ai→⋃i∈p-Ai and also the unique fixed point of Tp:⋃i∈p-Ai→⋃i∈p-Ai 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 Theorem1; property (v) follows directly, since Ai⊂X; ∀i∈p- being closed implies that ⋂i∈p-Ai is also closed, which together with the condition that ⋂i∈p-Ai is convex, leads to the property that (⋂i∈p-Ai,d) is a convex metric space if X is an Euclidean vector space while the complementary to ⋂i∈p-Aiin X is not invoked in the proof of the uniqueness of the fixed point so that if (⋂i∈p-Ai,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-mappingT^j:Aj→Aj; ∀j∈p- can be extended in a natural way to the composite self-mapping Tp:⋃i∈pAi→⋃i∈pAi in the sense that Tp∣Aj=T^j; ∀j∈p-.
(2) The convexity of the subsetsAi; 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 Theorem2. 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 Ai⊂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 Theorem2 will be obtained in this section for the case of nonintersecting subsets by proving the convergence to unique best proximity points within each subset Ai, which are also unique fixed points of each of the composed self-mappings T^i:Ai→Ai; ∀i∈p-. It is assumed that (X,∥∥) is a uniformly convex Banach space endowed with the partial order ⪯ and that the subsets Ai; ∀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 Lemma3.7, Lemma 3.8 and Theorem 3.10 in [33].Lemma 4 (Lemma3.7 and Lemma 3.8 of [33]).
LetAi⊆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(Ai,Ai+1)=Di and either Ai  or Ai+1 for i∈p- are, furthermore, convex (i.e., at least one of each two adjacent subsets is, in addition, convex). Consider sequences {xnp+i}n∈Z0+⊂Ai and {znp+i}n∈Z0+⊂Ai and {ynp+i+1}n∈Z0+⊂Ai+1 satisfying(1)
∥znp+i-ynp+i∥→Di as n→∞ for any given i∈p-.(2)
For everyε∈R+, there is n0∈Z0+ such that ∥xmp+i-ynp+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 everyε∈R+, there is n1∈Z0+ such that ∥xmp+i-znp+i∥≤ε for all m,n∈Z0+ with m>n≥n1 for any given i∈p-.(ii)
If∥znp+i-ynp+i∥→Di as n→∞ and ∥xnp+i-ynp+i∥→Di as n→∞ then ∥xnp+i-znp+i∥→0 as n→∞ for any given i∈p-.(iii)
{xnp+i+j}n∈Z0+ and {znp+i+j}n∈Z0+are Cauchy sequences; ∀j∈p-1¯∪{0} and the given i∈p-.The proof of Lemma4(i)-(ii) is supported by the nonemptiness, closeness, and convexity of the subsets Aj⊆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.
LetT:⋃i∈p-Ai→⋃i∈p-Ai be a multivalued p(≥2)-cyclic self-mapping on ⋃i∈p-Ai with Ai∈CB(X)⊆X; ∀i∈p- being all nonempty and convex with Di=d(Ai,Ai+1); ∀i∈p-. Assume the following:(1)
LetX 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=xi(∈Ai)⪯y∈Tx(⊆Ai+1) for any (x,y)∈Ai×Ai+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 pointsx-j+1∈Tx-j⊆Aj+1 with d(x-j,x-j+1)=d(x-j,Tx-j+1)=Dj, for each j∈p- which are also unique fixed points of each of the restricted composite self-mappings T^j(≡Tp∣Aj):Aj→Aj; ∀j∈p-.(ii)
Take anyx=xi(∈Ai)⪯y=xi+1∈Txi 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 {xnp+j}, being nondecreasing with respect to ⪯while satisfying xnp+j+1∈Txnp+j; ∀j∈p¯ of first element subject to x=xi(∈Ai)⪯y=xi+1∈Txi for any given i∈p-. Then, each of such sequences {xnp+j} converges to the unique best proximity point x-j in Aj; ∀j∈p¯ which is also the unique fixed point of each of the restricted composite self-mapping T^j:Aj→Aj. If ⋂i∈p-Ai≠⌀, then x-=x-j∈⋂i∈p-Ai is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai, T^j(≡Tp∣Aj) and a fixed point of Tp:⋃i∈p-Ai→⋃i∈p-Ai; ∀j∈p-.(iii)
If assumption (4) of Theorem1 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∈p-Ai.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 Ai⊆X; ∀i∈p-, and Lemma 4(i)-(ii), it follows that
(26)∃limn→∞d(xnp+i+j+1,xnp+i+j)=Di+j;∃limn→∞d(x(n+1)p+i+j,xnp+i+j)=0;∀j∈p-1¯∪{0},
where xnp+i+j∈Txnp+i+j-1⊆Ai+j, x(n+1)p+i+j∈Tpxnp+i+j-1⊆Tx(n+1)p+i+j-1⊆Ai+j; ∀j∈p-1¯∪{0}, ∀n∈Z0+ for the given i∈p- and the iterated sequences; {xnp+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 xi+j=xj generated from the iteration xnp+i+j∈Txnp+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 xnp+i+j→x-i+j(∈Tpx(n-1)p+i+j⊆Txnp+i+j-1⊆Ai+j) and xnp+i+j+1→x-i+j+1(∈Tpx(n-1)p+i+j+1⊆Txnp+i+j⊆Ai+j+1) as n→∞; ∀j∈p-1¯∪{0} and the given i∈p- since Aj⊆X is nonempty, bounded and closed; ∀j∈p¯and the given i∈p-. Thus, one gets from (26), since Aj⊆X is nonempty, bounded and closed, and then boundedly compact, and also approximatively compact with respect to Aj-1 [8, 35], that:
(27)Di+j≤d(xnp+i+j+1,xnp+i+j)→d(x-i+j,x-i+j+1)=Di+j=d(x-i+j,Tx-i+j)asn→∞;∀j∈p-1¯∪{0} and the given i∈p-, where x-i+j+1∈Tpx-i+j+1-p⊆Tx-i+j; ∀j∈p-1¯∪{0} and the given i∈p-. Since all the subsets Aj⊂X; ∀j∈p- are nonempty, closed, and boundedly compact; ∀j∈p- then x-j∈Aj is a best proximity point in Aj of T:⋃i∈p-Ai→⋃i∈p-Ai and it is also a fixed point of the restricted composite self-mapping T^j:⋃i∈p-Ai∣Aj→⋃i∈p-Ai∣Aj; ∀j∈p-. Thus, there are Cauchy, then convergent since (X,d) is complete, sequences {xnp+i+j}n∈Z0+ with respective first elements xi+j∈Txi+j-1; ∀j∈p-1¯∪{0} and the given i∈p-, each being convergent to x-i+j∈Ai+j, such that x=xi is the first element of {xnp+i}n∈Z0+⊆Ai which consists of partially ordered elements with respect to the partial order ⪯ such that
(28)xi+j⪯⋯⪯x(n+1)p+i+j(∈Tpxnp+i+j⊆Tx(n+1)p+i+j-1)⪯x(n+1)p+i+j⪯⋯⪯x-i+j=limn→∞xnp+i+j,
with {xℓp+i+j}⊆Ai+j; j∈p-1¯∪{0}, ∀ℓ∈Z0+ for the given i∈p-. But x-i+j∈Ai+j; ∀j∈p-1¯∪{0} and the given i∈p-, is a fixed point of the restricted composite self-mapping T^j:⋃i∈p-Ai∣Aj→⋃i∈p-Ai∣Aj, ∀j∈p- and a fixed point of the composite self-mapping Tp:⋃i∈p-Ai→⋃i∈p-Ai from Lemma 4(iii) to which the partially ordered sequences of first element x=xi∈Ai converge. It is also a best proximity point in Ai+j of the self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai from Lemma 4(iii) and the second part of Lemma 4(ii). Then, x-j∈Tx-j⊆Tpx-j; ∀j∈p-. The uniqueness property of each of those p best proximity points x-j∈Tx-j in each of the subsets Aj⊆X follows from their uniqueness as fixed points of the restricted self-mappings T^j:⋃i∈p-Ai∣Aj→⋃i∈p-Ai∣Aj from Theorem 2, since (X,d) is a convex metric space and the subsets Aj⊆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∈p-Ai→⋃i∈p-Ai and Tp:⋃i∈p-Ai→⋃i∈p-Ai from Theorem 2, this leads to the proofs of properties (i)–(iii).Remarks 2.
(1) Theorem5 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∈p-Ai satisfying assumptions (1)–(4) of Theorem 1 as it was commented, in Section 2 concerning such a theorem, the given Ai⊂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 Theorem5 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 ofX is kept as hypothesis for the uniqueness of the best proximity points of the multivalued self-mapping, since although the subsets Ai 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∈p-Ai,d), since the subsets Ai⊂X do not necessarily intersect.
(5) Note that the results of Sections2 and 3 obtained from the contractive condition (5) also hold for multivalued self-mappings T:⋃i∈p-Ai→⋃i∈p-Ai which are not cyclic; that is, T(Ai)∩A-i+1≠⌀ for some i∈p- but fulfil the condition Tx∩Ai+1≠⌀; ∀x∈Ai, ∀i∈p-.


## 4. Example

Consider two bounded and closed real subsetsA1=[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 statexk operating at each state value under N tentative feedback controls uk(ik)=-Kk(ik)xk(ik); ∀ik∈Ik, where the indexing set of tentative states at the (k+1)-th sampling point is defined by
(29)Ik+1=Ik×N-=I0×N-×⋯×N-={1}×N-k,∀k∈Z0+,
where “×” stands for the Cartesian product of sets, x0(i0)=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 xk+1(ik+1)∈Txk(ik)={xk+1(ik,1),…,xk+1(ik,N)}; ∀k∈Z0+ obtained as follows:
(30)xk+1(ik+1)={x-k+1(ik+1)if|x-k+1(ik+1)|∈[ε,M],Msgnx-k+1(ik+1)if|x-k+1(ik+1)|>M,∀ik∈Ik,∀k∈Z0+,εsgnx-k+1(ik+1)if|x-k+1(ik+1)|<ε,
where
(31)x-k+1(ik+1)=akxk(ik)+bkuk(ik)=(ak-bkKk(ik))xk(ik);∀ik∈Ik,∀k∈Z0+
with ak≠0 and bk>0 being nonzero real numbers under a sequence of controllers of gains
(32)Kk(ik)=ρk(ik)+akbk,ρk(ik)∈[0,ρ];∀ik∈Ik,∀k∈Z0+
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-k+1(ik+1)=-ρk(ik)xk(ik); ∀ik∈Ik with |xk+1(ik+1)|≤min(max(ρk(ik)|xk(ik)|,ε),M); ∀ik∈Ik, ∀k∈Z0+. Assume that (5) holds for x⪯y so that, for some y∈Tx and some z∈Ty, one gets
(33)d(z,y)≤H(Tx,Ty)≤Kd(x,y)+2(1-K)ε
for some K∈(0,1). Note also from (30)-(31) that (5) is guaranteed under the set of constraints below:
(34)2ε≤d(xk+2(ik+1),xk+1(ik))≤d(x-k+2(ik+1),xk+1(ik))=|x-k+2(ik+1)-xk+1(ik)|=|x-k+2(ik+1)|+|xk+1(ik)|=|ak+1-bk+1Kk+1(ik+1)-1||xk+1(ik+1)|≤|ak+1-bk+1Kk+1(ik+1)-1|min(|x-k+1(ik+1)|,M)≤|ak+1-bk+1Kk+1(ik+1)-1|min(ρk(ik)|xk(ik)|,M)≤min(d0,Kd(xk+1(ik+1),xk(ik))+2(1-K)ε)=min(d0,K|ak-bkKk(ik)-1||xk(ik)|+2(1-K)ε)≤min(d0,K|ak-bkKk(ik)-1|M+2(1-K)ε)≤d0;forsomeik∈Ik,∀k∈Z0+
which hold provided that there is at least a controller gain sequence {Kk(ik)}ik∈Ik subject to the constraints:(35a)
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0for someik∈Ik; ∀k∈Z0+. The above constraints guarantee the generation of an iterated nondecreasing partially ordered sequence S(x)={x=x0(1),x1(ii),…,xn(in),…} by the self-mapping T on [-M,-ε]∪[ε,M] with respect to the partial order “⪯” under a sequence of control gains satisfying 0<Kk(ik)=(ρk(ik)+ak)/bk≤(1/|bk|)((d0/M)-|ak-1|) and |ak-1|<2ε/M≤d0/M; ∀k∈Z0+.It follows from Theorem5 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, x1*=ε∈A1 and x2*=-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
(36)0<Kk(ik)=ρk(ik)+akbk≤1|bk|(d0KM-|ak-1|)forsomeik∈Ik;∀k∈Z0+
under sufficiency-type constraints d0>KM if |ak|<d0/KM-1; ∀k∈Z0+ or d0<KM and ak>1-d0/KM if ak∈[0,1]; ∀k∈Z0+. Thus, S(x)={x=x0(1),x1(ii),…,xn(in),…} 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:
(37)x-k+1(ik+1)=(x-1k+1(ik+1),x-2k+1(ik+1))T=Akxk(ik)+Bkuk(ik)=(Ak-BkKk(ik)T)xk(ik);∀ik∈Ik,∀k∈Z0+,
where the superscript T stands for transposition and xk+1(ik+1)=(x1k+1(ik+1),x2k+1(ik+1))T, subject to the tentative scalar feedback controls uk(ik)=-Kk(ik)Txk(ik), where Ak=Diag(a1k,a2k) is a sequence of real 2×2 diagonal matrices, Bk=(b1k,b2k)T is a sequence of two dimensional real column vectors, Kk(ik)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:
(38)2ε≤d(xk+2(ik+1),xk+1(ik))≤d(x-k+2(ik+1),xk+1(ik))=max1≤j≤2d(x-jk+2(ik+1),xj,k+1(ik))
for some ik∈Ik,∀k∈Z0+.  To keep the validity of (30) for each state component, the constraints (35a) and (35b) are replaced by(39a)
d
0
>
max
1
≤
j
≤
2
max
(
2
ε
,
M
(
a
j
k
+
|
a
j
k
-
1
|
)
)
,
(39b)
0
<
ρ
k
(
i
k
)
≤
{
min
(
1
,
min
1
≤
j
≤
2
(
d
0
M
-
a
j
k
-
|
a
j
k
-
1
|
)
,
min
1
≤
j
≤
2
K
|
a
j
,
k
+
1
-
b
j
k
K
j
k
(
i
k
)
-
1
|
+
(
2
ε
(
1
-
K
)
/
|
x
j
k
|
)
|
a
j
,
k
+
2
-
b
j
,
k
+
1
K
j
,
k
+
1
(
i
k
+
1
)
-
1
|
)
if
ε
>
0
min
(
1
,
min
1
≤
j
≤
2
(
d
0
M
-
a
j
k
-
|
a
j
k
-
1
|
)
,
min
1
≤
j
≤
2
K
|
a
j
,
k
+
1
-
b
j
k
K
j
k
(
i
k
)
-
1
|
|
a
j
,
k
+
2
-
b
j
,
k
+
1
K
j
,
k
+
1
(
i
k
+
1
)
-
1
|
)
if
ε
=
0for someik∈Ik; ∀k∈Z0+.



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*Source: 968492-2013-05-09.xml*