Fixed Point and Best Proximity Theorems

Volume 2010, Article ID 510974,12pages doi:10.1155/2010/510974

Research Article

Fixed Point and Best Proximity Theorems

under Two Classes of Integral-Type Contractive Conditions in Uniform Metric Spaces

M. De la Sen

Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box. 644- Bilbao, 48080 Bilbao, Spain

Correspondence should be addressed to M. De la Sen,[email protected] Received 5 July 2010; Accepted 14 October 2010

Academic Editor: S. Reich

Copyrightq2010 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the existence of fixed points and best proximity points of p-cyclic self- maps on a set of subsets of a certain uniform space under integral-type contractive conditions. The parallel properties of the associated restricted composed maps from any of the subsets to itself are also investigated. The subsets of the uniform space are not assumed to intersect.

1. Introduction

Fixed point theory is of an intrinsic theoretical interest but also a useful tool in a wide class of practical problems. There is an exhaustive variety of results concerning fixed point theory in Banach spaces and metric spaces involving different types of contractive conditions including those associated with the so-called Kannan maps and with Meir-Keeler contractionssee, e.g., 1–6. There is also a rich background literature concerning the use of contractive conditions in integral form using altering distances, Lebesgue integrable test functions, and comparison functions,7–9. Also, the so-called reasonable expansive mappings have been investigated in10, and conditions for the existence of fixed points have been given. It has been used, for instance, for the study of the Lyapunov stability of delay-free dynamic systems and also for that of dynamic systems subject to delays and then described by functional differential equationssee, for instance,11,12concerning a related fixed point background for those systems and12–15concerning some related background for stability. On the other hand, it has also been useful for investigating the stability of hybrid systems consisting of coupled continuous-time and discrete-time or digital dynamic subsystems16. This paper considers p-cyclic self-maps in a uniform space X,Φ, whereX is a nonempty set equipped with a nonempty familyΦof subsets ofX×Xsatisfying certain uniformity properties. The familyΦ

is called the uniform structure ofX, and its elements are called entourages, neighbourhoods, or surroundings. The uniform spaceX,Φis assumed to be endowed with anA-distance or andE-distance. The existence of fixed points and best proximity points in restrictedp-cyclic self-mapsF :X |

i∈pAi → X |

i∈pAiofX,8, subject to the constraintFAi ⊆A_i1 for each pair of adjacent subsetsAi ofX, for all i∈p: {1,2, . . . , p},p≥2, under the cyclic conditionA_pi A_i; for alli ∈ Z, is investigated separately under two groups of integral- type contractive conditions. One of such groups involves a positive integrand test function while the other combines a positive integrand with a comparison function. Some properties of the composed restricted self-maps on each of the subsets are also investigated. The subsets of the uniform structure do not necessarily intersect. If the sets do not intersect, then it is proven thatg gi ≡ g_ii1 : distAi, A_i1 dzi, Fzi ifX, dis a metric space endowed with a distance map d:X×X → R₀, somez_i ∈A_i, for alli∈p, andz_iis said to be a best proximity point. Also, it follows thatg g_ij : distAi, A_j dzi, F^j−iz_i, for alli, j> i∈p for some zi ∈ Ai, for all i ∈ p. If the self-mapT ofX is nonexpanding, theng g_ii1 : distAi, A_i1>0, for alli∈p,8.

2. Basic Results about A-Distances, E-Distances, and V -Closeness

Define the nonempty familyΦof subsets ofX×X of the formΦ :

i,j∈pΦijwithΦij : A_i×A_j, for alli, j∈p: {1,2, . . . , p}. Note by construction that

V ∈Φ ⇒

⎡

⎣V ∈A_i×A_j; ∀i, j∈p

∨V ∈

⎛

⎝

i∈px

A_i

⎞

⎠×

⎛

⎝

i∈py

A_j

⎞

⎠

for some nonempty finite subsets of positive integers p_x, p_y ⊆p

⎤

⎦.

2.1

The following definitions ofV-closeness and anAandE-distances are used throughout the paper.

Definition 2.1see7,9. IfV ∈Φandx, y ∈ V andy, x ∈ V, thenxandyare said to beV-close. A sequence{xn}^∞₀ ⊂X is a Cauchy sequence forΦif for anyV ∈Φ, there exists N≥1 such thatx_nandx_mareV-close forn, m≥N.

Definition 2.2. A functiond:X×X → R0is said to be anA-distance if 1dx, y 0⇔x yfor allx, y∈X;

2for eachV ∈Φ, there existδ δV>0 such that

max

dz, x, d z, y

≤δ for somez∈X ⇒ x, y

∈V. 2.2

Definition 2.2 generalizes slightly that of 7 by admitting δ to depend on V since it is being used on distinct setsA_i, i ∈ p. Note that V ∈ Φis symmetrical, that is, V V⁻¹ {y, x :x, y∈ V}thenx, y∈ V ⇔ x, y∈ V so thatxandyareV-close under Definition 2.2.

Definition 2.3see7. A functiond:X×X → R0is said to be anE-distance if 1it is anA-distance;

2dx, y≤dx, z dz, y; for allx, y, z.

Assertion 1. Assume that anyZ ∈ Ψ ⊂ X×X is symmetrical, that is, Z Z⁻¹ {y, x | x, y∈Z}. Then,d:X×X → R₀is anA-distance if and only ifxandyareZ-close for all Z∈Ψprovided that maxdz, x, dz, y< δfor somez∈Xand someδ >0.

Proof. It follows from the symmetry of allZ∈ΨandDefinition 2.3by a simple contradiction argument. Take a pairx, y∈ZfromDefinition 2.3sinced:X×X → R₀is anA-distance fulfilling maxdz, x, dz, y< δfor somez∈Xand someδ >0. Such a pair always exists for anyZ∈Ψ. SinceVis symmetrical, theny, x∈Z. Sincex, y∈Zif and only ify, x∈Z thenxandyareV-close.

Assertion 2. 1 ΦijandΦ:

i,j∈pΦijare symmetrical.

2IfV ∈Φis of the formVX×VXthenV is symmetrical.

3IfV ⊆

i∈px,j∈pyΦij is nonempty andp_x, p_y ⊂ p with p_x∩p_y ∅, thenV is not symmetrical.

If, in-addition,Ai∩Aj ∅, for alli, j∈pthen there are nox, yinΦbeingV-close.

Proof. 1 x, y ∈ Φij A_i×A_j ⇔ x ∈ A_i, y ∈ A_j ⇔ y, x ∈ Φji A_j×A_i ⇔ Φij is symmetrical.x, y ∈ Φ⇔ x, y ∈

i∈pA_i ⇔ y, x∈ Φ⇔ Φis symmetrical.Assertion 21 has been proven.

2 x, y ∈ V ∈ Φ ⇒ x, y ∈

i∈pxA_i×

i∈pxA_i∩V ⇔ y, x ∈

i∈pxA_i×

i∈pxA_i∩V for somep_x⊆p ⇒Vis symmetrical.Assertion 22is proven . 3Proceed by contradiction:V symmetrical⇔x, y∈V ⇒x, y∈

i∈px,j∈pyΦij

i∈pxA_i×

j∈pyA_j≡

i∈pyA_i×

j∈pxA_jy, x⇐y, x∈V⇒p_x∩p_y/∅what contradicts p_x∩p_y ∅.

Assertion 2states that some, but not all, nonempty subsetsVof∅are symmetrical. For instance, ifV A1∪A2×A3∪A4, thenV is not symmetrical since there arex, y∈ V such thaty, xare not inV; that is, there are pairsx,ywhich are notV-close. If furthermore the setsA_·are disjoint, then there is no pair inΦbeingV-closeAssertion 23. Note that under symmetry ofV, the second property of anA-distance can be rewritten in an equivalent form by replacing x, y ∈ V with x, ybeing V-close. The subsequent result states that, contrarily to results in former studies related toAandE-distances7,9, the second property guaranteeing anA-distance necessarily involvesδ-values exceeding distances between the various subsetsA_i, i∈p.

Lemma 2.4. Assume thatd:X×X → R0is anA-distance andV ⊆Φij ∈Φfor somei, j∈p. If x, y∈V, then maxdz, x, dz, y< δijfor somez∈Xand someδij >distAi, Aj.

Proof. Assume that x, y ∈ V, so that x ∈ A_i and y ∈ A_j, and maxdz, x, dz, y <

distAi, A_j, for somez∈X. The following cases can occur.

1Ifz x∈Ai, and sincey∈Aj, then 0≤dist

Ai, Aj

≤d z, y

max

dx, x, d z, y

<dist Ai, Aj

, 2.3 which leads to the contradiction distAi, A_j<distAi, A_j.

2Ifz/x∈Ai, and sincey∈Aj, then

0≤dist A_i, A_j

≤max

dz, x,dist

A_i, A_j

≤max

dz, x, d z, y

<dist A_i, A_j

, 2.4

which leads to the same contradiction as in Case1.

3Ifz y ∈ Ajand ifz/y ∈Aj, the above contradiction of cases1and2, is also obtained by replacingA_i↔A_j.

4Ifz∈A_i∪A_j∩X, then

0≤dist Ai, Aj

≤max

dz, x,dist Ai, Aj

≤max

dz, x, d z, y

< dist Ai, Aj

2.5

which leads to the same contradiction as that of case1.

The following lemma is a direct consequence ofLemma 2.4:

Lemma 2.5. Assume thatd:X×X → R0is anA-distance andV ⊆

i∈px,j∈pyΦij forpx, py ⊂p.

Ifx, y∈V then maxdz, x, dz, y< δfor somez∈Xand someδ >max_{i∈px,j∈py}distAi, Aj.

3. Main Results about Fixed Points and Best Proximity Points

Consider p-cyclic self-maps F : X |

i∈pA_i → X |

i∈pA_i subject to FAi ⊆ A_i1; for alli ∈ p {1,2, . . . , p}. The objective is to first investigate if each of them has a fixed point and then if they have a common fixed point through contraction conditions on Lebesgue integrals and use of comparison functions. Without loss of generality, we discuss the fixed points of self-mapsF ofX. Consider a Lebesgue-integrable mapϕ : R0 → R0

which satisfies_ε

0ϕtdt >0, for allε∈Rsuch that for allx∈Ai, for ally∈A_i1. Define also the composed self-mapF^p : X | p

i 1A_i → X |p

i 1A_i asp

i 1A_i y F^px ∈ p

i 1A_i from the self-mapF : X | p

i 1A_i → X | p

i 1A_i whose restrictions toA_i, F^p:X|Ai → X |Ai; for all i∈p, are defined via the restrictionF^p|Ai: F_p−i|A_p−i◦ · · · ◦ F_i1 |A_i1◦F_i |A_i; for alli∈pbyy F^px∈A_i for eachx∈A_i; for alli∈p. Note that the domain of the self-mapF^pofXisp

i 1A_i while that ofF^p|A_iisA_i. The paper investigates, under two types of integral-type contractive conditions of self-mapsFofX, the existence of fixed points of such a self-map inp

i 1A_i, provided that the intersection is nonempty. In that case, the fixed points coincide with those of the self-mapF^p:X |p

i 1A_i → X|p

i 1A_i. It also investigated the existence of best proximity points between adjacent and nonadjacent subsets A_i; for alli∈pfor the case thatp

i 1A_i ∅. In such a case, the best proximity points at each pair of adjacent subsetsA_i, A_i1; for alli∈pare also fixed points of the composed self-maps F^p:X|Ai → X|Aifrom each subsetAito itself; for alli∈peven under weaker contractive integral-type conditions. A key basic result used in the mathematical proofs is that the distance between any pair ofadjacent or nonadjacentsubsets is identical for nonexpansive contractions.

It is first assumed that the integral-type contractive Condition1below holds.

Condition 1. One has _dFnpj1x,F^npj1y

0

ϕtdt≤ 1−α_jn

dF^npjx,F^npjy 0

ϕtdtα_jn

_distAj,Aj1

0

ϕtdt; ∀j∈p, 3.1 where{αjn}^∞_{n 1} are sequences of nonnegative real numbers subject to_∞

n 1 α_jn ∞; for all j ∈p, for alln∈Z. The self-mapFofX is said to be reasonably nonexpansive through this paper if

_dF^np_x,F^np_y

0

ϕtdt≤KM

_dx,y

0

ϕtdtK_M d

F^ipx, F^ipy

≤KMd x, y

K_M ∀x, y∈X; ∀n∈Z

3.2

for some nonnegative real constantsK_M andK_M. In particular,Fis reasonably nonexpansive ifF^pis nonexpansive. The following result follows from Condition1.

Theorem 3.1. The following properties hold under Condition1for anyA-distanced:X×X → R₀: iThe restricted self-mapsF^p |Ai; for alli∈psatisfying3.1are all nonexpansive, and so

it is the self-mapF:X|p

i 1A_i → X|p i 1A_i; ii∞>distAi, A_i1 g ≥0; for alli∈p;

iiilim sup_{n→ ∞dF}n1px, F^n1py

0 ϕtdt≤M <∞, for allx, y∈Ai×A_i1; for alli∈p, withM MgandM:R₀ → R₀being monotone increasing withM0 0;

ivIf distAi, A_i1 0; for alli ∈ p, that is,

i∈pA_i/∅, then there is a fixed point x ∈

i∈pA_i∩FixF^pof the self-mapF^pofXand of its restrictions to

i∈pA_i and

i∈pA_i defined through the natural set inclusions

i∈pAi⊆

i∈pAi⊂X. Also,x∈FixFfor the self-mapF:X|p

i 1A_i → X|p i 1A_i.

Proof. Consider someA-distanced:X×X → R0. Note that for each

V ⊆

⎛

⎝

i∈p

A_i

⎞

⎠×

⎛

⎝

i∈p

A_i

⎞

⎠∈Φ, ∃δ δV>max

i∈p distAi, A_i1≥0; ∀i∈p 3.3

such that

max

dz, x, d z, y

≤δ for somez∈

i∈p

Ai⊂X

⇒ x, y

∈V∧ y, x

∈V

⇐⇒x, yare V-close

3.4

If, in particular,V ⊆ Ai×A_i1; for alli∈p, thenx, y∈ V if maxdz, x, dz, y ≤δwith anyδ >distAi, A_i1≥0 andz∈A_i∪A_i1; for alli∈p, and ifV ⊆Ai∪A_i1×Ai∪A_i1

thenx, y∈V ∧y, x∈V⇔x, yareV-close. It is first proven that the self-mapF^pofX satisfying3.1is nonexpansive. Proceed by contradiction by assuming that it is expansive.

Then, one gets the following by defining a real sequence {αn}^∞_{n 1} with general termα_n : 1−_n

j 11−αjn∈0,1:

⎛

⎝^p

j 1

1−αjn

⎞

⎠_dF^np_x,F^np_y

0

ϕtdt p 1

p j 1

1−αjn

αn

_distA,A1

0

ϕtdt

≥αn

_dF^np_x,F^np_y

0

ϕtdtmin

j∈p1−αn

_distAj,Aj1

0

ϕtdt

≥

_dF^n1p_x,F^n1p_y

0

ϕtdt >

_dF^np_x,F^np_y

0

ϕtdt

⇒min

j∈p

_distAj,Aj1

0

ϕtdt >

_dF^np_x,F^np_y

0

ϕtdt

⇒min

j∈p dist

A_j, A_j1

> d

F^npx, F^npy

≡d x₁, y₁

3.5

for somex₁, y₁ ∈ p

i 1A_i which is a contradiction, and the self-mapF^p and then the self- mapFofXis nonexpansive and propertyiholds. Now, its is proven by contradiction that distAi, A_i1 g≥0, for all i∈p. Assume that there existi, j∈psatisfying 1≤j≤p−isuch that distAi, A_i1> distA_ij, A_ij1. Then there are best proximity pointsz_i∈A_i, ξ_ij∈A_ij and somez_i∈A_isuch that, sincep > jand the self-mapFofXis nonexpanding, one gets

dzi, Fzi distAi, A_i1> dist

A_ij, A_ij1 d

ξ_ij, Fξ_ij d

F^jzi, F^j1zi

≥d

F^pzi, F^p1zi

dzi, Fzi≤distAi, A_i1. 3.6

for somezi ∈Ai with the last inequality being strict unlesszi zi, what is a contradiction if

z_i/z_i. Now, assume thatz_i F^pz_i z_i, then the best proximity pointz_i z_i∈FixF^psince F^pz_i F^2pz_i F^pz_i z_i z_iand distAi, A_i1 0, that is,A_i∩Ai1/∅. This is a contradiction to the assumption distAi, A_i1> distAij, A_ij1. Then, distAi, A_i1 g≥0, for alli∈p and p

i 1A_i/∅ if and only if g 0. Since the self-map F^p of X restricted to p i 1A_i is nonexpansive, then the self-map F of X restricted to p

i 1A_i is reasonably nonexpansive.

It also follows by contradiction that g < ∞. Assume that g ∞. Then, the following contradiction follows from3.1:

0≤

_dF^n1p_x,F^n1p_y

0

ϕtdt≤

⎛

⎝^p

j 1

1−αjn

⎞

⎠_dF^np_x,F^np_y

0

ϕtdt

p 1

p j 1

1−αjn

αn

_distA,A1

0

ϕtdt <0

3.7

for some n ∈ Z, j ∈ p, unless _dF^np_{x, F}^np_y

0 ϕtdt ∞ and then dF^npx, F^npy

∞; for allx, y ∈p

i 1A_iprovided thatα_jn> 0. Such aα_jnalways exists since_∞

n 1α_jn ∞;

for allj∈p. Then,∞>distAi, A_i1 g≥0; for alli∈p, and Propertyiifollows.

Note that3.1yields directly via recursion _dFn1px,F^n1py

0

ϕtdt

≤

⎛

⎝^p

j 1

1−α_jn⎞

⎠_dF^np_x,F^np_y

0

ϕtdt p 1

p j 1

1−α_jn α_n

_distA,A1

0

ϕtdt

≤ⁿ⁻¹

1

⎛

⎝^p

j 1

1−α_j⎞

⎠_dF^p_x,F^p_y

0

ϕtdtⁿ⁻¹

i 1

⎛

⎝ⁿ⁻¹

m i1

⎛

⎝^p

j 1

1−α_jm⎞

⎠

⎞

⎠

×

⎛

⎝^p

1

p j 1

1−α_ji α_i

_distA,A1

0

ϕtdt

⎞

⎠; ∀n∈Z,∀x∈A_i,∀y∈A_i1. 3.8

Note thatρn: p

j 11−αjn∈0,1. DefineZ: Zs∪ZcwithZs: {n∈Z:ρn<1},Zc: {n∈Z:ρ_n 1}and

ρ: lim sup

m→ ∞

⎛

⎝^μZsm

n 1

ρn

⎞

⎠

1/m

Z_sm:

n∈Z:

ρ_n<1∧n≤m

⊂Z_s⊆Z.

3.9

Note also that the cardinalor discrete measureofZ_sisμZs χ₀i.e., infinity numerable, since otherwise,_∞

n 1αjn<∞for somej∈pcontrarily to one of the given hypothesisand μZc≤χ₀. Since_m

n 1α_jn ∞andμZs χ₀, so thatρ∈0,1, it follows that

mlim→ ∞

m n 1

ρ_n

n∈Zs

ρ_n

n∈Zc

ρ_n

n∈Zs

ρ_n

m→ ∞limρ^m 0. 3.10

Then, since the distance between any two adjacent setsAi, A_i1is a real constantg, one gets the following from3.8, and3.10:

lim sup

n→ ∞

_dF^n1p_x,F^n1p_y

0

ϕtdt

lim sup

n→ ∞ n−1

i 1

⎛

⎝ⁿ⁻¹

m i1

⎛

⎝^p

j 1

1−αjm

⎞

⎠

⎞

⎠

⎛

⎝^p

1

p j 1

1−αji

αi

_g

0

ϕtdt

⎞

⎠;

≤M: α 1−ρ

g 0

ϕtdt

<∞; ∀x∈Ai, ∀y∈A_i1, ∀i∈p,

3.11

whereα: sup_{j∈p, n∈Z}αjn≤1 since limn→ ∞_n−1

1p

j 11−αj lim_{n→ ∞}ρⁿ 0,

0≤ⁿ⁻¹

i 1

⎛

⎝ⁿ⁻¹

m i1

⎛

⎝^p

j 1

1−α_jm⎞

⎠

⎞

⎠

⎛

⎝^p

1

p j 1

1−α_ji α_i

_g

0

ϕtdt

⎞

⎠

<∞; ∀x∈Ai, ∀y∈A_i1.

3.12

Note thatM Mgis monotone increasing with it argumentgand thatM0 0. Property iiihas been proven. If g 0, then

i∈pA_i/∅ so that there is a fixed pointxofF₁^p : X |

i∈pAi → Xwhich is also a fixed point of its extensionsF₂^p: X|

i∈pAi → XandF^p:X → XsinceF^p |

i∈pAi F₂^p,F^p|

i∈pAi F₁^pandF₂^p|

i∈pAi F₁^p. It turns out thatxis also a fixed point ofF :X|p

i 1A_i → p

i 1A_i,17. Propertyivhas been proven.

Note that the proved boundedness property of theA-distance∞>distAi, A_i1also relies on the fact that this is a distance between best proximity points in adjacent sets. It is well known that a distance map in a metric space has always a uniform equivalent distance which is finite. The following two concluding results from3.11are direct since_ε

0ϕtdt >0; for all ε∈R.

Corollary 3.2. Assume thatg : distAi, A_i1>0; for alli∈pand that

∃ lim

n→ ∞

n−1 i 1

⎛

⎝ⁿ⁻¹

m i1

⎛

⎝^p

j 1

1−α_jm⎞

⎠

⎞

⎠

⎛

⎝^p

1

p j 1

1−α_ji α_i

_g

0

ϕtdt

⎞

⎠<∞. 3.13

Then, there is a setS: {xi∈A_i:i∈p}of cardS psuch that

dxi, x_i1 lim

n→ ∞d

F^npx, F^npy gα 1−ρ; ∀

x, y

∈

⎛

⎝

i∈p

Ai

⎞

⎠×

⎛

⎝

i∈p

Ai

⎞

⎠; ∀i∈p. 3.14

If α 1−ρ, then the points of the setSsatisfyg dxi, x_i1, for alli∈pso thatSis a set of best proximity points in

i∈pAi of the self-mapsFandF^pofX. Eachxi ∈Sis a fixed point ofF |Ai, a best proximity point ofF^p |Ai; for alli∈pand satisfiesFxi x_i1andxi ∈FixF^p |Aiso that F^px_i x_i; for alli∈p.

Corollary 3.3. Ifg 0 so that

i∈pAi/∅; for alli ∈ p, thenCorollary 3.2still holds with the set Sconsisting only of a set of identical pointsxin

i∈pAi such thatx ∈ FixF,x ∈ FixF^p, and x∈F^p|A_i; for alli∈p.

Since an E-distance is also an A-distance, the following conclusion is direct from Theorem 3.1andCorollary 3.2.

Corollary 3.4. Theorem 3.1andCorollary 3.2also hold ifd:X×X → R₀is anE-distance.

An important relaxation of Condition1allows the reformulation ofTheorem 3.1and Corollaries3.2–3.4except in the result thatx∈FixFwheng 0 as follows.

Corollary 3.5. Assume that Condition1is reformulated as thep-cyclic contractive Condition2below.

Condition 2. One has _dF^n1p_x,F^n1p_y

0

ϕtdt≤1−αn

_dF^np_x,F^np_y

0

ϕtdtαnmin

j∈p

_distAj,Aj1

0

ϕtdt 3.15

for a new real sequence{αn}^∞_{n 1}under the weaker constraints

αn: 1− p

j 1

1−αjn

∈0,1; ∀n∈Z, ∞ n 1

αn ∞ 3.16

and that the finite limit ofCorollary 3.2exists. Then, the following properties hold.

iTheorem 3.1 and Corollaries 3.2–3.4 still hold, except that in the case that the distance between adjacent setsgis zeroi.e., if all subsetsA_i, i∈phave a nonempty intersection, the propertyx ∈ FixFis not guaranteed, since the restricted self- mapsF:X|Ai → X |A_i1can be expansive for somei∈p.

iiIfg >0 then there exists a setS: {xi ∈A_i:i∈p}of cardS pof best proximity points of the self-mapF ofX such thatxi ∈ FixF^p | Ai; for alli ∈ p, and there are Cauchy sequences {xin}^∞_{n 1}; for alli ∈ pwhich satisfyx_in1 F^px_in Fx_i−1n1andx_in → x_i asn → ∞; for alli∈ p. The pointsx_i, x_i1 areAi∪ A_i1×Ai∪A_i1-close for eachi∈pvia some existing real constantδ > αg/1−ρ inDefinition 2.2. Also, the pairs of Cauchy sequences{xin}^∞_{n 1},{x_i1n}^∞_{n 1}have subsequences{xin}^∞_{n N},{xi1n}^∞_{n N}which areAi∪A_i1×Ai∪A_i1-close via a real constantδ0ε εδ > εαg/1−ρinDefinition 2.2for any givenε >0 and some integerN Nδ, ε.

Proof. First note thatTheorem 3.1i–iiiis independent of the above modification. Note also that now 1−αn _n

j 11 −αjn < 1 on a subset of Z infinite discrete measure so that 3.8–3.12 still hold except that x ∈ FixF is not guaranteed wheng 0 last part of Theorem 3.1iv, andCorollary 3.3, sinceα_jn ≡1 forj belonging to some proper nonempty subset ofp, for alln∈Z. It still holds thatx∈FixF^p. Propertyihas been proven. Now, note fromCorollary 3.2that fromTheorem 3.1there is a setSofppoints each being a fixed point of the restricted self-mapF^p|A_i, for alli∈punder the pairwise constraints

dxi, x_i1 lim

n→ ∞d

F^npx, F^npy gα 1−ρ; ∀

x, y

∈

⎛

⎝

i∈p

Ai

⎞

⎠×

⎛

⎝

i∈p

Ai

⎞

⎠; ∀i∈p 3.17

which are necessarily in disjoint adjacent sets since the distances between all the sets are a constantg >0 andFAi⊆A_i1; for alli∈p. Then theA-distancedx, yof any pairx, y∈ Ai×A_i1; for alli∈pconverges to a constant distancegα/1−ρ. Then, there is a convergent sequence{xin}^∞_{n 1} of points in A_i verifying x_in → x_i asn → ∞ since x_i ∈ FixF^p | A_ifor eachi ∈ p. Those sequences are Cauchy sequences since each convergent sequence in a metric space is a Cauchy sequence. Furthermore, xin1 F^pxin Fx_i−1n1 since x_in ∈ A_i implies that x_i ∈ A_i,x_in1 F^px_in ∈ A_i, andx_i1n Fx_in ∈

A_i1, for alln∈ Z, for alli∈p. The remaining parts of Propertyiiconcerning closeness according toDefinition 2.2follow the fact that the best proximity points of the self-mapFof X are also fixed points of restricted composed maps to which Cauchy sequences of points converge and whose distance isαg/1−ρ. Propertyiihas been proven.

Since the validity ofTheorem 3.1iiiis independent of the modification of Condition 1 to the weaker one Condition 2 implying the use of the sequence {αn}^∞_{n 1} see proof of Corollary 3.5, Condition2ofCorollary 3.5may be replaced with the following.

Condition 3. One has _dF^n1p_x,F^n1p_y

0

ϕtdt≤1−αn

_dF^np_x,F^np_y

0

ϕtdtαnmin

j∈p

_g

0

ϕtdt. 3.18

The above discussion may be discussed under any of the following replacements of Conditions1–3.

Condition 4. One has _dF^npj1_x,F^npj1_y

0

ϕtdt≤ψ

_dF^npj_x,F^npj_y

0

ϕtdt _g

0

ϕtdt

; ∀j∈p. 3.19

Condition 5. One has _dF^npj1_x,F^npj1_y

0

ϕtdt≤ψ1

_dF^npj_x,F^npj_y

0

ϕtdt

ψ2

g 0

ϕtdt

; ∀j∈p. 3.20

Condition 6. One has

_dF^n1p_x,F^n1p_y

0

ϕtdt≤ψ

_dF^np_x,F^np_y

0

ϕtdt _g

0

ϕtdt

. 3.21

Condition 7. One has _dFnpj1x, F^npj1y

0

ϕtdt≤ψ₁

_dF^npj_x,F^npj_y

0

ϕtdt

ψ_{2 g}

0

ϕtdt

, 3.22

where ψ, ψ₁, ψ₂ : R0 → R0 are comparison functions, namely, monotone increasing satisfying lim_{t→ ∞}ψⁿt ψ₁ⁿt ψ₂ⁿt 0, for all t∈R0.

Thus, ψ0 ψ₁0 ψ₂0 0 andψt ψ₁t ψ₂t < t; for allt ∈ R as a consequence of their above properties to be comparison functions. In addition,ψ:R0 → R0

satisfies the subadditive conditionψt1t2≤ψt1ψt2. As a result of the above properties, note that:

aConditions4and5imply that _dF^npj1_x,F^npj1_y

0

ϕtdt≤

_dF^npj_x,F^npj_y

0

ϕtdt _g

0

ϕtdt, 3.23

for allx, y∈

i∈pAi, for all j∈pwith the equality standing for somej ∈pand somex, y∈

i∈pAi if and only ifg dF^npjx, F^npjy 0, that is, the distance between relevant points in the upper-limits of the integral and between all the adjacent sets are zero.

bConditions6and7imply that _dF^n1p_x,F^n1p_y

0

ϕtdt≤

_dF^np_x,F^np_y

0

ϕtdt _g

0

ϕtdt; 3.24

for allx, y ∈

i∈pA_i with the equality standing for somex, y ∈

i∈pA_i if and only if g dF^npx, F^npy 0.

The following results follow.

Corollary 3.6. Theorem 3.1and Corollaries 3.2–3.4hold “mutatis-mutandis” under any of the p- cyclic contractive Conditions6and7.

Corollary 3.7. Theorem 3.1and Corollaries 3.2–3.4hold “mutatis-mutandis” under any of the p- cyclic contractive Conditions4and5except thatx∈FixFif the distance between adjacent sets g is zero (i.e., all setsA_i, i∈phave a nonempty intersection).

The proofs are direct as that ofTheorem 3.1see also that ofCorollary 3.5by using the properties3.24for that ofCorollary 3.6and3.23for that ofCorollary 3.7.

Acknowledgments

The author is grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI2009-07197. He is also grateful to the Basque Government by its support through Grants IT378-10, SAIOTEK S-PE08UN15, and SAIOTEK S-PE09UN12. He thanks the anonymous reviewers by their suggestions to improve the paper.

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