On the Extensions of Krasnoselskii-Type Theorems to p-Cyclic Self-Mappings in Banach Spaces

A set of np ≥2 -cyclic and either continuous or contractive self-mappings, with at least one of them being contractive, which are defined on a set of subsets of a Banach space, are considered to build a composed self-mapping of interest. The existence and uniqueness of fixed points and the existence of best proximity points, in the case that the subsets do not intersect, of such composed mappings are investigated by stating and proving ad hoc extensions of several Krasnoselskii-type theorems.


Introduction
In the last years, important attention is being devoted to extend the fixed point theory by weakening the conditions on both the maps and the sets where those maps operate 1, 2 .For instance, every nonexpansive self-mappings on weakly compact subsets of a metric space has fixed points if the weak fixed point property holds 1 .It has also to be pointed out the relevance of fixed point theory in the stability of complex continuous-time and discrete-time dynamic systems 3-5 .On the other hand, Meir-Keeler self-mappings have received important attention in the context of fixed point theory perhaps due to the associated relaxing in the required conditions for the existence of fixed points compared with the usual contractive mappings 6-10 .Another interest of such maps is their usefulness as formal tool for the study of p-cyclic contractions even if the involved subsets of the metric space under study do not intersect 6 .The underlying idea is that the best proximity points are fixed points if such subsets intersect while they play a close role to fixed points otherwise.It has to be pointed out that there are close links between contractive self-mappings and Kannan self-mappings 2, 10-13 .
A rich research is being devoted to the existence and uniqueness of best proximity points of cyclic mappings under different assumptions on the vector space to which the subsets involved in the cyclic mapping belong.For instance, in 14 , the concept of 2-cyclic self-mappings is extended to p ≥2 -cyclic self-mappings and results about fixed points are derived in the case that the subsets of the considered complete metric space have a nonempty intersection.Some of the ideas in such a manuscript inspired the definition of p-cyclic self-mappings from X to X with X being the union of the subsets involved in the cyclic representation.On the other hand, the existence and uniqueness of best proximity points of p ≥2 -cyclic ϕ-contractive self-mappings are investigated in 15 borrowing the previous scenario investigated in 16, 17 for 2-cyclic ϕ-contractive self-mappings.Generally speaking, the so-called cyclic ϕ-contractive self-mappings, which are associated with some strictly increasing unbounded map, are based on a concept of weak contractiveness contrarily to the standard, and commonly used, strict contractive concept being inspired in the well-known Banach contraction principle.In those papers, the Banach space X under consideration is also assumed to be reflexive and strictly convex.These joint assumptions, which are less restrictive than the assumption that the space is uniformly convex since uniformly convex Banach spaces are reflexive and strictly convex but the converse is not true in general , are proven to keep intact the essential properties of existence and uniqueness of best proximity points for cyclic ϕ-contractive self-mappings if the involved subsets are nonempty, weakly closed, and convex.The above formalism is revisited in 18 for cyclic ϕ-contractive self-mappings in the framework of ordered metric spaces.In 19 , characterization of best proximity points is studied for non-self-mappings S, T : A → B, where A and B are nonempty subsets of a metric space.In general, best proximity points do not fulfil in this context the usual condition x Sx Tx.However, they jointly globally optimize the mappings from x to the distances d x, T x and d x, Sx .
In this manuscript, X, d is a complete metric space and is considered associated to a Banach space X endowed with translation-invariant and homogeneous metric d : X × X → R 0 and S X : {A i ⊆ X : A j p ≡ A j ; ∀i ∈ p, ∀j ∈ Z 0 } is a set of p subsets of X. T : i∈p A i → i∈p A i is a p-cyclic self-mapping if it satisfies T A i ⊆ A i 1 ; for all i ∈ p.A valid metric is the norm of the Banach space X.If the p-cyclic self-mapping T : i∈p A i → i∈p A i is nonexpansive resp., contractive-also referred to as a 2-cyclic contraction then there exists a real constant k ∈ 0, 1 resp., k ∈ 0, 1 such that The self-mapping This concept of 2-cyclic large contraction on intersecting subsets extends that of large contraction 20 , and both concepts extend to p-cyclic contractions.A p-cyclic large contraction satisfies also 1.2 by replacing A → A i , B → A i 1 ; for all i ∈ p provided that i∈p A i / ∅.In Section 2 of this paper, we consider a mapping T : 2 y; for all j ≥2 ∈ Z ; for all x, y ∈ A i , for all i ∈ 2 and then T x, y ∈ A i 1 where that such a map is not required to be 2-cyclic either.For the obtained results related to boundedness of distances between iterates through T , it is not required for the set of subsets of X to be either closed or convex.For the obtained results concerning fixed points and best proximity point, the sets A 1 and A 2 are required to be convex but they are not necessarily closed if the self-mapping T can be defined on the union of the closures of the sets A 1 and A 2 .Also, concerning best proximity points in the case that the subsets do not intersect, it is assumed that the space is restricted to be a uniformly convex Banach space 6-9 .It turns out that since uniformly convex Banach spaces are also strictly convex, the results also hold under this more general assumption used in several papers see, for instance 15-18 .In Section 3, the results of Section 2 are extended to mappings built in a close way via p ≥2 -cyclic self-mappings on a set of p subsets A i i ∈ p of X.

Notation
Superscript denotes vector or matrix transpose, Fix T is the set of fixed points of a selfmapping T on some nonempty convex closed subset A of a metric space X, d , cl A denotes the closure of a subset A of X, Dom T and Im T denote, respectively, the domain and image of the self-mapping T and 2 X is the family of subsets of X, dist A, B d AB denotes the distance between the sets A and B for a 2-cyclic self-mapping for all i ∈ p for distances between adjacent subsets of p-cyclic self-mappings T on p i 1 A i where A i i ∈ p are subsets of X. BP i T is the set of best proximity points on a subset A i of a metric space X, d of a p-cyclic self-mapping T on p i 1 A i , the union of a collection of nonempty subsets of X, d which do not intersect.

Results for Mappings Defined by 2-Cyclic Self-Mappings
The following result is concerned with the above-defined map T : Theorem 2.1.The following properties hold.

2.2
ii 2 are both 2-cyclic nonexpansive self-mappings and at least one of them is contractive then Property (i) holds and, furthermore, iii 2 are both 2-cyclic contractive self-mappings then Properties (i), (ii) hold and, furthermore, (i.e., A 1 and A 2 are closed and intersect or at least one of them is open while their boundaries intersect).
Proof.Take x, y ∈ A 1 and x , y ∈ A 2 .Direct calculation yields by taking into account that , and 1.1 with respective contraction constants k 1 and k 2 , and that the metric is translation-invariant and homogeneous:

2.8
If k 1 k 2 1 then one gets from 2.8 If Thus, Property i follows from 2.9 .If only one of them is nonexpansive and the other is contractive then max k 1 , k 2 1 and 0 ≤ min k 1 , k 2 < 1 so that 2.3 follow from 2.9 and Property ii is proven.For real constants for some real constants k i ∈ 0, 1 , T i : are both contractive and one gets from 2.9 what leads to 2.5a , 2.5b , and to 2.6 , 2.7 by taking j → ∞.Property iii has been proven.

Corollary 2.2. If T
that is, they are nonexpansive with time-varying real sequences {k ij } j∈Z in 0, 1 for i 1, 2 which converge asymptotically to two respective real numbers k 1 and k 2 in 0, 1 as j → ∞.Assume that the sequences is satisfied for some real K ∈ R .Then, 2.1 holds and, furthermore, the following properties hold: for any bounded positive decreasing real sequence ε j , with arbitrary prescribed upperbound ε, which converges asymptotically to zero as j → ∞ for some finite j 0 j 0 ε ∈ Z and some bounded nondecreasing positive real sequence K j of upper-bound K.
ii lim sup

2.13
Proof.Note that Since the terms in the sum the summation term of the right-hand-side of the above equation are all nonnegative, it follows that:

2.15
Note also that j for all j ≥ j 0 for any given real constant ε ∈ 0, 1 , an infinite subsequence according to 2.15 then if 2.11 holds, one gets from 2.11 that 2.12 holds for any given ε ∈ R and some positive decreasing real sequence {ε j } j∈Z and some nondecreasing positive real sequence {K j } j∈Z subject to K j ≤ K j 1 ≤ K, so that lim sup j → ∞ K j ≤ K, and ε j 1 ≤ ε j ≤ ε which converges then to zero as j → ∞ since it has an infinite monotone decreasing subsequence {ε j i } j i j ∈Z for each j ∈ Z , satisfying to 0 2.17

2.19
where ρ : min Proof.i First, note that the integer subset Z exists since the time-varying contraction sequences converge to real constants being less than unity.One gets from 2.5a for k : and 2.17 holds.Property i has been proven.
ii The proof of Property ii is close to that of Property i by using a close method to that of the proof of Corollary 2.2 ii : for all x, y ∈ A 1 , for all x , y ∈ A 2 , where ε ij : for all i ∈ Z 0 , and {K j, i } j >i ∈Z ; are sequences of positive finite real constants whose integer arguments j and i are related to the iterate of the left and right hand sides of 2.20 , respectively, defined by

2.22
Since 0 ≤ k 1j k 2j ≤ 1; for all j ∈ Z 0 then 0 ≤ 2k ε ij ≤ 1; for all i, j ∈ Z and since the limit k 1j k 2j → k : max k 1 , k 2 < 1/2 as j → ∞, {K j, i } j >i ∈Z is uniformly bounded and lim i → ∞ K j i, i 0; for all j ∈ Z .Thus, combining 2.20 and 2.21 , one gets d T j i x, y , T j i x , y

2.23
and then lim sup

2.24
Some Krasnoselskii-type fixed point results follow for the map T : defined trough 2-cyclic binary self-mappings for the case when A 1 and A 2 intersect.
Theorem 2.5.Assume that X is a Banach space which has an associate complete metric space X, d with the metric d : X × X → R 0 being translation-invariant and homogeneous.Assume that A 1 and A 2 are nonempty, convex, and closed subsets of X which intersect.Assume also that T i : are also the respective unique fixed points of T i : Proof.One gets from Theorem 2.1 iii , 2.9 Since X is a Banach space then X, d is a complete metric space, since T i : A 1 ∪A 2 → A 1 ∪A 2 i 1, 2 are 2-cyclic contractive self-mappings then satisfying 1.1 with contraction constants k i ∈ 0, 1 and dist A 1 , A 2 0 since A 1 ∩ A 2 / ∅ and subject T i A j ⊆ A for j, / j , i 1, 2, and since ∅, the unique limits below for Cauchy sequences exist

2.26
where for all i ∈ Z .Since z 1 and z 2 then z is unique so that Fix T {z}.
The following result follows from Corollary 2.2 and Theorem 2.5.
Theorem 2.7.Assume that X is a Banach space which has an associate complete metric space X, d with the metric d : X × X → R 0 being translation-invariant and homogeneous.Assume that A 1 and A 2 are nonempty, convex, and closed subsets of X, which intersect and have a convex union.Assume that T i : A 1 ∪A 2 → A 1 ∪A 2 (i 1, 2) are both 2-cyclic self-mappings, the first one being continuous, the second one being contractive, and, furthermore, the mapping T :

2.27
Then, there is a (in general, nonunique) fixed point which has the properties below: 28 Proof.Note that the following properties hold 1 By hypothesis, A 1 ∪A 2 is a nonempty convex set which is, furthermore, closed since A 1 and A 2 are both nonempty and closed.
2 A 1 ∩A 2 is nonempty, convex, and closed since A 1 and A 2 are nonempty, closed, and convex with nonempty intersection. 3 4 A 1 ∪ A 2 is a nonempty compact set which is trivially invariant under the 2-cyclic continuous self-mapping T 1 : 5 Thus, from the standard Krasnoselskii fixed point theorem 20, 21 , under the above Properties 1 -4 , it exists z ∈ Fix T ⊆ A 1 ∪ A 2 which satisfies T i 1 z T i 2 z z T z, z ; for all i ∈ Z .Also, one gets from Property 5 that lim j → ∞ T j 2 z T 2 y y ∈ Fix T 2 ≡ {y} ⊆ A 1 ∩ A 2 are limits of Cauchy sequences so that T j The following results follow from Krasnoselskii fixed point theorem extended to 2cyclic self-mappings since A 1 ∪ A 2 is a compact set 20 .

Corollary 2.10. Theorem 2.7 also follows with the replacement of the 2-cyclic contractive selfmapping T
2 with time-varying contraction sequences in 0, 1 with limits in 0, 1 .
Corollary 2.10 follows by using Corollary 2.2 instead of Theorem 2.1.Also, Theorems 2.8 and 2.9 can be extended for T 2 : A 1 ∪ A 2 → A 1 ∪ A 2 being asymptotically contractive closely to the extension Corollary 2.10 to Theorem 2.7.
On the other hand, note that if A 1 ∩ A 2 ∅ then the convergence through the mapping to a fixed point under the conditions of Theorem 2.5, or those of Corollary 2.6, cannot be achieved since fixed points, if any, are in A 1 ∩ A 2 since A 1 ∩ A 2 ∅.See Theorem 2.1, 2.6 , Corollary 2.2, 2.13 , Corollary 2.4, 2.18 , or 2.19 .It is not either guaranteed the convergence to best proximity points because the guaranteed upperbounds for the limit superiors of distances of the iterates exceed the distance between the adjacent subsets A 1 and A 2 since the lower upper-bounds of the above respective referred to limit superiors are of the form λ dist A 1 , A 2 for some real constant λ > 1 defined directly by inspection of 2.6 , 2.13 , or 2.18 , or 2.19 .Then, the following result follows instead of Theorem 2.5.
Theorem 2.11.Assume that X is a Banach space which has an associate complete metric space X, d with the metric d : X × X → R 0 being translation-invariant and homogeneous.Assume also that A 1 and A 2 are nonempty subsets of X, which do not intersect.Assume that T i :

2.29
Then, T j x, y is asymptotically permanent as j → ∞; for all x ∈ A 1 , for all y ∈ A 2 entering a compact subset A 1 ∪ A 2 of cl A 1 ∪ A 2 , where: used in Theorem 2.5 and in Corollary 2.6 is essential for the existence of the residual set where the iterates T j x, y enter asymptotically as j → ∞.It is now proven how Corollary 2.4 may be improved to obtain λ 1 in Theorem 2.11 so that the convergence of the iterates T j : to a best proximity point if A 1 and A 2 are nonempty, disjoint, convex, and closed.The part of Theorem 2.11 referred to the fulfilment of Corollary 2.4 i.e., either the sum of both contraction constants or the sum of their limits if they are time-varying sequences is less than unity is improved as follows.
Corollary 2.12.Assume that X is a Banach space which has an associate complete metric space X, d with the metric d : X × X → R 0 being translation-invariant and homogeneous.Assume that A 1 and A 2 are nonempty, disjoint, convex, and closed subsets of X. Assume also that T i :

) are both 2-cyclic either contractive under Corollary 2.4(i) with (or asymptotically contractive under Corollary 2.4(ii)
) self-mappings and, furthermore, (see 2.29 ).Then, any iterates T j x, y , respectively, T j 1 x, y converge to a best proximity point of either A 1 , respectively, A 2 , or conversely, as j → ∞ for any given Proof.Take some real constant k ∈ k 1 k 2 , 1 and note from 1.1 that 2.8 is modified as follows:

2.32
Thus, 2.33 so that the limit lim j → ∞ d T j x, y , T j x , y dist A 1 , A 2 exists modifying 2.18 in Corollary 2.4 i .Then, Theorem 2.11 holds with λ 1 and A i is a subset of the boundary of A i containing the best proximity points of A i to the set A i 1 i 1, 2 with A 3 A 1 .Thus, since {T j x, y } j∈Z is a Cauchy sequence since X, d is complete, then T j x, y converges to a best proximity point of A 2 resp.,A 1 if j → ∞ and is odd resp., even .A close proof leading to a similar modified limit follows by modifying 2.19 in Corollary 2.4 ii by using the limits k i of the time-varying asymptotically contractive sequences for i 1, 2, instead of the constants k i , satisfying Note that Corollary 2.4 and its referred to part of Corollary 2.12 also hold in a more general version for limits Some illustrative examples follow.
Example 2.13.Consider the scalar differential equations ẋi t a i x i t r i , x i 0 x i0 , i 1, 2, 2.34 with a i < 0 for i 1, 2. The solution trajectories are defined by contractive self-mappings T i : R → R for i 1, 2 as follows: Both solutions converge to respective unique fixed points z i r i /|a i | as t → ∞ which are also stable equilibrium points, for i 1, 2 since by using the Euclidean metric, it follows trivially for any two initial conditions x i0 , x i0 i 1, 2 and x i t, x i0 → z i with x i0 ∈ R as t → ∞ for i 1, 2. Note that the same property holds by redefining the self-mappings T ih : R → R for i 1, 2 so as to pick up by successive iterates starting from initial conditions the sequence of points of the solutions {x i kh, x i0 } k∈Z 0 for some h ∈ R as ; for all t ∈ R 0 .Then, the above results still hold with the individual and combined resulting mappings still being contractive leading to respective unique fixed points z 1 r 1 , z 2 r 2 /|a 2 |, and z r 1 r 2 /|a 2 | with r 1 : Example 2.15.Consider self-mappings T i : A ∪ B → A ∪ B i 1, 2 defined for any given h, ε ∈ R , a i ∈ R − , and r i ∈ R i 1, 2 by for all k ∈ Z 0 , k i ∈ 0, 1 being given contraction constants for i 1, 2 for disjoint sets for some given ε ∈ R .The above equations are interpreted as the solution of the discretized versions of the differential equations 2.34 discussed in Examples 2.13 and 2.14 subject to saturated values on the near boundaries of the real subintervals A and B. Note that T i A ⊆ B and T i B ⊆ A for i 1, 2 so that T i : A ∪ B → A ∪ B i 1, 2 are 2-cyclic self-mappings for which any succession of iterates converges to best proximity points −2ε ∈ A and ε ∈ B for any bounded initial conditions which restricts the iterates to be real successions on bounded convex disjoint subsets of R. The composed self-mapping T : by a domain restriction converges to best proximity points −ε ∈ A and ε ∈ B Corollary 2.12 .If now a 1 ε x 10 0 then A ∩ B {0} and z 0 is a fixed point of T : A ∪ B × A ∪ B → A ∪ B. Note that the mapping T 1 is continuous although it is noncontractive while some Krasnoselskii-type theorems still apply.
The above examples are easily extended for nonscalar cases by using the same tools.

Results for Mappings Defined by P ≥ 2 -Cyclic Self-Mappings
Define n : {1, 2, . . ., n} for any n ∈ Z .Then, the above results are easily extended to the case of ≥2 p ≥2 p-cyclic T : i∈p A i → i∈p A i ; for all ∈ n self-mappings which define a mapping T : i∈p A i × i∈p A i → X defined by T x 1 , x 2 , . . ., x n n i 1 T i x i , for any x i ∈ A j and any given j ∈ p T j : To facilitate the exposition, assume that where card n 2 ≥ 1, with contraction constants k ∈ 0, 1 ; for all ∈ n 2 .
That is, we allocate, with no loss of generality, the n-cyclic self-mappings which are not contractive, while they are typically nonexpansive or continuous, all in the first strictly ordered set of integers and those being contractive in the second strictly ordered one.The identities of sets A i p ≡ A i ; for all i ∈ p are assumed in all the necessary notations as associated with p-cyclic self-mappings.It is also used that any p-cyclic nonexpansive self-mappings have identical distances d : dist A i , A i 1 > 0; for all i ∈ p between disjoint adjacent sets 6 .The following three results can be proven in a similar way as those counterparts of Section 2 for p q 2. Theorem 3.1.Assume that T i : ∈p A i → ∈p A i for i ∈ n are p-cyclic nonexpansive for Part (i), with at least one of them is being contractive p − 1 ≥ p 2 ≥ 1 for Part (ii).Then, Theorem 2.1 holds with the subsequent replacements: and all further necessary replacements borrowed from Theorem 3.1.

Corollary 3.3. Assume that T i :
∈p A i → ∈p A i for i ∈ n are all p-cyclic contractive for Part (i) and p-cyclic nonexpansive and asymptotically contractive for Part (ii).Thus, Corollary 2.4 holds by replacing: and all further necessary replacements borrowed from Theorem 3.1 or Corollary 2.2.
The following results based on Krasnoselskii theorems follow for fixed and best proximity points in a Banach space endowed with a translation-invariant and homogeneous metric.The results are a direct extension from the results for mappings T : with mixed continuous and contractive 2-cyclic self-mappings of Section 2 to mappings T : i∈p A i × i∈p A i → i∈p A i built with p ≥2 -cyclic selfmappings, some of them being continuous while the remaining ones are contractive.The underlying ideas are a a mapping which is a sum of continuous mappings is continuous, b a set of contractive p-cyclic self-mappings for the same set of convex and closed subsets of X has each a unique fixed point if the p subsets involved in the p-cyclic contractive selfmappings intersect.The sum of all those contractive self-mappings has a unique fixed point which is the sum of the whole set of contractive self-mappings, c the combinations of both results yields the existence of a fixed point from Krasnoselskii fixed point theorems of T : i∈p A i × i∈p A i → i∈p A i if all the subsets of the p-cyclic structure intersect.If those sets do not intersect, there is a limiting best proximity point at any of the p subsets of X for any p-cyclic iterates for initial points in between adjacent subsets.Theorem 3.4.Assume that X is a Banach space which has an associate complete metric space X, d with the metric d : X × X → R 0 being translation-invariant and homogeneous.Assume also that A i ; for all i ∈ p are nonempty, convex, and closed subsets of X with i∈p A i / ∅.Assume also that T , where z ∈ Fix T ⊆ i∈p A i ; for all ∈ n are also the respective unique fixed points of T : i∈p A i → i∈p A i ; for all ∈ n.Theorem 3.4 above and Corollary 3.5 below extend, respectively, Theorem 2.5 and Corollary 2.6 to the existence of fixed points for a mapping built from a set of n contractive, respectively, asymptotically contractive, p ≥2 -cyclic self-mappings.Corollary 3.5.Theorem 3.4 also holds if T : i∈p A i → i∈p A i ; for all ∈ n are p-cyclic asymptotically contractive self-mappings with respective time-varying contraction sequences being in 0, 1 with limits in 0, 1 .
Theorem 2.7 extends directly as follows Theorem 3.6.Assume that X is a Banach space which has an associate complete metric space X, d with the metric d : X × X → R 0 being translation-invariant and homogeneous.Assume that A 1 and A 2 are nonempty, convex, and closed subsets of X, which intersect and have convex union.Assume that T : i∈p A i → i∈p A i ; for all ∈ n are p-cyclic self-mappings, of which n 1 are continuous and n 2 are contractive subject to 1 ≤ n 1 n − n 2 and n 2 ≥ 1. Assume, furthermore, that the mapping

3.4
Then, there exists a (in general, nonunique) fixed point z ∈ i∈p A i of T : i∈p A i × i∈p A i → X which has the properties below: One gets by extending Corollary 2.12 based on the ad hoc extensions of Theorem 2.20 and Corollary 2.4 for the composition of p-cyclic self-mappings.Theorem 3.9.Assume that X is a uniformly convex Banach space which has an associate complete metric space X, d with the metric d : X × X → R 0 being translation-invariant and homogeneous.Assume also that A i for i ∈ p are nonempty, disjoint, convex and closed subsets of X.Furthermore, assume that T : i∈p A i × i∈p A i → i∈p A i .Then, the following properties hold.
i Assume that T i : i∈p A i → i∈p A i are p-cyclic contractive self-mappings with timevarying real sequences {k ij } j∈Z ; for all i ∈ p such that the sum sequence { n i 1 k ij } j∈Z is in 0, 1 with k ≤ max j∈ Z n i 1 k ij ∈ 0, 1 and k : min j∈Z n i 1 k ij ≥ 0 where Z is some infinite subset of Z .Then, any iterates T j i x, y converges as j → ∞ to a best proximity point of A i ≡ A i −p for some integer p subject to 2p − i ≥ ≥ p − i − 1 for any given x ∈ A μ , y ∈ A μ 1 ; for all μ ∈ p.
ii Assume that T i : i∈p A i → i∈p A i are p-cyclic asymptotically contractive self-mappings, and, furthermore, with the sequences { n i 1 k ij } j∈Z in 0, 1 and having limits k i for i ∈ n satisfying k : max k i : i ∈ n < 1/2, k : min k i : i ∈ n ≥ 0.Then, any iterates T j i x, y converges as j → ∞ to a best proximity point of A i ≡ A i −p for some integer p subject to 2p − i ≥ ≥ p − i − 1 for any given x ∈ A μ , y ∈ A μ 1 ; for all μ ∈ p.
y are in such an interval and converge to the fixed point z for any initial conditions x, y ∈ A satisfying the constraint.Note that T i A , T i B ⊆ A ≡ B and T ih A , T ih B ⊆ A ≡ B are trivially 2-cyclic contractive self-mappings from A to A Theorem 2.5 .Example 2.14.Now, assume the replacement r 1 → r 1 t in Example 2.13 where it exists the integral t 0 e |a 1 |τ r 1 τ dτ O e |a 1 |t

1 T 3 . 7 .Theorem 3 . 8 .
i z − y, ∀i ∈ n 1 , y , y ∈ Fix T ≡ {y } ⊆ i∈p A i ; for all ∈ n 2 , for all i ∈ Z , is the unique fixed point of the p-cyclic self-mapping T : i∈p A i → i∈p A i ; for all ∈ n.Theorem 2.7 extends directly as followsTheorem Theorem 3.6 holds if T : i∈p A i →i∈p A i ; for all ∈ n 2 is a p-cyclic large contraction instead of being a contraction, for all ∈ n.Theorem 3.6 holds if the hypothesis T As a result, 2.13 follows, whose upper-bound is zero if dist A 1 , A 2 0. Note that a simple comparison between 2.6 in Theorem 2.1 and 2.12 in Corollary 2.2 concludes that K in 2.12 can be taken as small as 2 if the sequences {k ij } j∈Z in Corollary 2.2 are constant and equal to k i ∈ 0, 1 for i 1, 2. Therefore, a logic procedure of accomplishing with 2.11 is to check for the existence of valid constants K possessing a lower-bound 2.If the constants k 1 k 2 i 1,2 in Theorem 2.1, or the time-varying sequences {k 1j k 2j } j∈Z for i 1, 2 of Corollary 2.2, are less than unity then the following result holds.
A 2 and then A 1 ∩ A 2 are nonempty, convex, and closed.It follows that z z 1 z 2 by making .30 and λ > 1 is defined directly from 2.6 , under Theorem 2.1; from 2.13 , under Corollary 2.2, or from either 2.18 or 2.19 , under Corollary 2.4.Proof.It follows from either Theorem 2.1, Corollary 2.2 or Corollary 2.4 since 2.29 holds.

1
Corollary 3.2.Assume that T i :∈p A i → ∈p A i for i ∈ n are p-cyclic asymptotically contractive self-mappings.Then, Corollary 2.2 holds with the replacements: