# Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations

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

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## Abstract

This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.


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## Body

## 1. Introduction

Fixed point theory has an increasing interest in research in the last years especially because of its high richness in bringing together several fields of Mathematics including classical and functional analysis, topology, and geometry [1–8]. There are many fields for the potential application of this rich theory in Physics, Chemistry, and Engineering, for instance, because of its usefulness for the study of existence, uniqueness, and stability of the equilibrium points and for the study of the convergence of state-solution trajectories of differential/difference equations and continuous, discrete, hybrid, and fuzzy dynamic systems as well as the study of the convergence of iterates associated to the solutions. A basic key point in this context is that fixed points are equilibrium points of solutions of most of many of the above problems. Fixed point theory has also been investigated in the context of the so-called cyclic self-mappings [8–20] and multivalued mappings [21–32]. One of the relevant problems under study in fixed point theory is that associated with p-cyclic mappings which are defined on the union of a number of nonempty subsets Ai⊂X; ∀i∈p-={1,2,…,p} of metric (X,d) or Banach spaces (X,∥∥). There is an exhaustive background literature concerning nonexpansive, nonspreading, and contractive p-cyclic self-mappings T:⋃i∈p-Ai→⋃i∈p-Ai, for example, [8–20], including rational contractive-type conditions and [20, 33], and references therein, and for various kinds of multivalued mappings. See, for instance [21–32] and references therein. A key point in the study of contractive cyclic self-mappings is that if the subsets Ai for i∈p- are disjoint then the convergence of the sequence of iterates xn+1=Txn; ∀n∈Z0+(Z0+=Z+∪{0}), x0∈⋃i∈p-Ai, is only possible to best proximity points. The existence of such fixed points, its uniqueness and associated properties are studied rigorously in [11–13] in the framework of uniformly convex metric spaces, in [14–17], and in [12, 19] for Meir-Keeler type contractive cyclic self-mappings. In this paper, we introduce the notions of nonexpansive and contractive p-semicyclic impulsive self-mappings and investigate the best proximity and fixed points of those maps. The properties of boundedness and convergence of distances are studied in metric spaces, while those of the iterated sequences  xn+1=Txn;  ∀n∈Z0+, x0∈⋃i∈p-Ai, are studied in uniformly convex Banach spaces. It is also seen through examples that the above combined constraint for distances is relevant for the description of the solutions of impulsive differential equations and discrete impulsive equations and for associate dynamic systems. The boundedness of the sequences of distances between consecutive iterates is guaranteed for nonexpansive p-semicyclic self-mappings while its convergence is proved for  asymptotically contractive p-semicyclic self-mappings. In this case, the existence of a limit set for such sequences is proved. Such a limit set contains best proximity points if the asymptotically contractive p-semicyclic self-mapping is asymptotically p-cyclic, (X,d) is a complete metric space which is also a uniformly convex Banach space (X,∥∥), and the subsets Ai⊂X; ∀i∈p- are nonempty, closed, and convex. It has to be pointed out that the standard nonexpansive and contractive cyclic self-mappings may be viewed as a particular case of those proposed in this paper since it suffices to define the map so that any point of a subset is mapped in one of the adjacent subsets in the cyclic disposal and to define the second self-mapping of the composite impulsive one as identity.


## 2. Nonexpansive and Contractivep-Semicyclic and p-Cyclic Impulsive Self-Mappings

Consider a metric space(X,d) and a composite self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai of the form T=T+T-, where Ai, i∈p- are p(≥2) nonempty closed subsets of X with Anp+i≡Ai; ∀i∈p-, ∀n∈Z0+ (in particular, Ap+1≡A1) having a distance Di=d(Ai,Ai+1)≥0 between any two adjacent subsets Ai and Ai+1 of X; ∀i∈p-. In order to facilitate the reading of the subsequent formal results obtained in the paper, it is assumed that D=Di; ∀i∈p-. Some useful types of such composite self-mappings for applications together with some of their properties in metric spaces are studied in this paper according to the following definition and its subsequent extensions.Definition 1.
The composite self-mappingT(≡T+T-):⋃i∈p-Ai→⋃i∈p-Ai is said to be a p-semicyclic impulsive self-mappingif the following conditions hold:(1)
T-:⋃i∈p-Ai→⋃i∈p-Ai is such that T-Ai⊆Ai∪Ai+1; ∀i∈p- satisfies the constraint d(T-x,T-y)≤Kd(x,y)+(1-K)D; ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p-  for some real constant K∈R.0+(2)
T+:⋃i∈p-Ai→⋃i∈p-Ai is such that  T+T-(Ai∪Ai+1)⊆Ai∪Ai+1; ∀i∈p- satisfies the constraint d(T+(T-x),T+(T-y))≤m(T-x,T-y)d(T-x,T-y) for some given bounded function m:(⋃i∈p-Ai)×(⋃i∈p-Ai)→R0+.Note thatp-semicyclic impulsive self-mappings satisfy the subsequent combined constraint as follows:
(1)d(Tx,Ty)≤m(T-x,T-y)[Kd(x,y)+(1-K)D],d(Tx,Ty)≤mv∀x∈Ai,∀y∈Ai+1,∀i∈p-;
then T:⋃i∈p-Ai→⋃i∈p-Ai which follows after combining the two ones given in Definition 1.The following specializations of thep-semicyclic impulsive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai of Definition 1 are of interest.(a)
It is said to benonexpansive (resp., contractive)p-semicyclic impulsive if, in addition, K∈[0,1] (resp., if K∈[0,1)) and m(T-x,T-y)≤1.(b)
It is said to bep-cyclic impulsiveif  TAi⊆Ai+1, ∀i∈p-. It is said to be a nonexpansive (resp., contractive)p-cyclic impulsive if, in addition, K∈[0,1] (resp., if K∈[0,1)) and m(T-x,T-y)≤1.(c)
It is said to bestrictlyp-semicyclic impulsive self-mapping if it satisfies the more stringent constraint
(2)d(Tx,Ty)≤Km(T-x,T-y)d(x,y)d(Tx,Ty)≤+(1-Km(T-x,T-y))D,d(T,T)∀x∈Ai,∀y∈Ai+1,∀i∈p-.A motivation for such a concept is direct sinceT:⋃i∈p-Ai→⋃i∈p-Ai is nonexpansive (resp., contractive) if Km(T-x,T-y)≤1 (resp., if Km(T-x,T-y)<1), ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p-. This motivates, as a result, the concepts of nonexpansive and contractive strictlyp-semicyclic impulsive self-mappings and the parallel ones of nonexpansive and contractive strictlyp-cyclic impulsive self-mappings for the particular case that Ai⊆Ai+1, ∀i∈p-.Remark 2.
Note that ifm(T-x,T-y)≤1, ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p-, then m(T-x,T-y)(1-K)D≤(1-Km(T-x,T-y))D, ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p-, and this holds if D=0 (i.e., ⋂i∈p-Ai≠∅) irrespective of the value of m(T-x,T-y), ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p-.The subsequent result follows directly from Remark2.Proposition 3.
Assume that any of the two conditions below holds:(1)
⋂i∈p-Ai≠∅;(2)
⋂i∈p-Ai=∅and0≤m(T-x,T-y)≤1,∀x∈Ai,∀y∈Ai+1, and∀i∈p-.
Then, the self-mappingT:⋃i∈p-Ai→⋃i∈p-Ai is(i)
strictlyp-semicyclic if it isp-semicyclic;(ii)
strictly nonexpansive (resp., contractive)p-semicyclic if it is nonexpansive (resp., contractive)p-semicyclic;(iii)
strictlyp-cyclic if it isp-cyclic;(iv)
strictly nonexpansive (resp., contractive)p-cyclic if it is nonexpansive (resp., contractive)p-cyclic.It is of interest the study of weaker properties than the above ones in an asymptotic context to be then able to investigate the asymptotic properties of distances for sequences{xn}n∈Z0+of iterates built through T:⋃i∈p-Ai→⋃i∈p-Ai according to xn+1=Txn for all n∈Z0+ and some x0∈⋃i∈p-Ai as well as the existence and uniqueness of fixed and best proximity points.Lemma 4.
Consider thep-semicyclic impulsive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai with K∈[0,1], and define
(3)m′(T-x,T-y)=m(T-x,T-y)-1,δk(x)=m′(T(k+1)-x,Tk-x)×(Kd(Tkx,Tk-1x)+(1-K)D),
for x and y in adjacent subsets Ai and Ai+1 of X for any i∈p-. Then, the following properties hold.
(i) The sequence{d(Tk+np+jx,Tk+np+j-1x)}k∈Z0+ is bounded for all k∈Z0+, and ∀n∈Z+, ∀j∈p-1¯∪{0} if
(4)-d(Tk+1x,Tkx)≤∑i∈S+(k,n,j)δk+j+np-i(x)-d(Tk+1xxx,Tkx)-∑i∈S-(k,n,j)δk+j+np-i(x)<∞∀k∈Z0+,∀n∈Z+,∀j∈p-1¯∪{0},
where
(5)δk(x)=m′(T(k+1)-x,Tk-x)×(Kd(Tkx,Tk-1x)+(1-K)D),∀k∈Z0+,(6)S+(k,n,j)={(m′(T(k+np+j-i+1)-x,T(k+np+j-i)-x))>0i∈Z+:(i≤np+j)∧(m′(T(k+np+j-i+1)-x,T(k+np+j-i)-x))>0(m′(T(k+np+j-i+1)-x,T(k+np+j-i)-x))},S-(k,n,j)={(-1≤m′(T(k+np+j-i+1)-x,T(k+np+j-i)-x))<0i∈Z+:(i≤np+j)∧(-1≤m′(T(k+np+j-i+1)-x,T(k+np+j-i)-x))<0},∀k∈Z0+,∀n∈Z+,∀j∈p-1¯∪{0}.
If, furthermore,T:⋃i∈p-Ai→⋃i∈p-Ai is p-cyclic then the lower-bound in (4) is replaced with D-d(Tk+1x,Tkx). If  T:⋃i∈p-Ai→⋃i∈p-Ai is a nonexpansive p-semicyclic impulsive self-mapping (in particular, p-cyclic), then {d(Tk+np+jx,Tk+np+j-1x)}k∈Z0+ is bounded, ∀k∈Z0+, and ∀n∈Z+, ∀j∈p-1¯∪{0}.
(ii) If, furthermore,K∈[0,1), then
(7)0≤limsupn→∞d(Tk+np+jx,Tk+np+j-1x)≤D+limsupn→∞∑i=1np+jδk+j+np-i(x)≤D+limsupn→∞(∑i∈S+(k,n,j)δk+j+np-i(x)-∑i∈S-(k,n,j)δk+j+np-i(x))<∞,∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0}.
If, in addition,T:⋃i∈p-Ai→⋃i∈p-Ai is p-cyclic, then the lower-bound in (7) is replaced with D.
IfT:⋃i∈p-Ai→⋃i∈p-Ai is contractive p-semicyclic, then
(8)0≤limsupn→∞d(Tk+np+jx,Tk+np+j-1x)≤D,∀k∈Z0+,∀j∈p-1¯∪{0},∀x∈⋃i∈p-Ai.
IfT:⋃i∈p-Ai→⋃i∈p-Ai is contractive p-cyclic, then there exists limn→∞d(Tk+np+1x,Tk+npx)=D, ∀x∈⋃i∈p-Ai.Proof.
Build a sequence of iterates{Tkx}k∈Z according to TTk-1x=T+T-Tk-1x with T0-x=x, T0x=T0+T0-x=x, for any given x∈Ai and any i∈p- that is,  T=T0+=T0-=id so that
(9)d(Tk+1x,Tkx)≤(1+m′(T(k+1)-x,Tk-x))×d(T(k+1)-x,Tk-x)≤(1+m′(T(k+1)-x,Tk-x))×(Kd(Tkx,Tk-1x)+(1-K)D)=Kd(Tkx,Tk-1x)+(1-K)D+δk(x),=Kd(Tkx,Tk-1x)+=Kd,222∀k∈Z0+.
Through a recursive calculation with (4), one get:
(10)0≤d(Tk+np+jx,Tk+np+j-1x)≤Kd(Tk+np+j-1x,Tk+np+j-2x)+(1-K)D+δk+np+j-1(x)≤K2d(Tk+np+j-2x,Tk+np+j-3x)+K[(1-K)D+δk+np+j-2(x)]+(1-K)D+δk+np+j-1(x)≤⋯≤Knp+j-1d(Tk+1x,Tkx)+(1-Knp+j-1)D+∑i=1np+jKiδk+np+j-i(x),+2∀k∈Z0+,∀n∈Z+,∀j∈p-1¯∪{0}.
IfK=1, then
(11)0≤d(Tk+np+jx,Tk+np+j-1x)≤d(Tk+1x,Tkx)+∑i∈S+(k,n,j)δk+np+j-i(x)-∑i∈S-(k,n,j)δk+np+j-i(x),∀k∈Z0+,∀n∈Z+,∀j∈p-1¯∪{0}.
Take anyk∈Z0+, any n∈Z+, and any x∈⋃i∈p-Ai. Since d(Tk+1x,Tkx) is finite and (4) holds, it follows that 0≤d(Tk+np+j+1x,Tk+np+jx)<∞. If, in addition, T:⋃i∈p-Ai→⋃i∈p-Ai is p-cyclic, then the zero lower-bound of (7) is replaced with D. If T:⋃i∈p-Ai→⋃i∈p-Ai is p-semicyclic (in particular, p-cyclic) nonexpansive, then (4) always holds since m(T(k+np+j+i)-x,T(k+np+j+i-1)-x)≤1, -1≤m′(T(k+j+np-i+1)-x,T(k+j+np-i)-x)≤0 so that
(12)∑i∈S+(k,n,j)δk+j+np-i(x)-∑i∈S-(k,n,j)δk+j+np-i(x)=∑i∈S-(k,n,j)δk+j+np-i(x)≤0,
if m(T(k+j+np-i+1)-x,T(k+j+np-i)-x)=1 and {d(Tk+np+j+1x,Tk+np+jx)}k∈Z0+ is always bounded; ∀k∈Z0+, ∀n∈Z+, and ∀j∈p-1¯∪{0}. Property (i) has been proven. If K∈[0,1), then
(13)0≤d(Tk+np+jx,Tk+np+j-1x)≤Knp+j-1d(Tk+1x,Tkx)+(1-Knp+j-1)D+∑i∈S+(k,n,j)δk+j+np-i(x)-∑i∈S-(k,n,j)δk+j+np-i(x)(14)0≤limsupn→∞d(Tk+np+jx,Tk+np+j-1x)≤D+limsupn→∞∑i=1np+jδk+j+np-i(x).
If, in addition,T:⋃i∈p-Ai→⋃i∈p-Ai is p-cyclic, then the zero lower-bound of (13)-(14) is replaced with D.
IfT:⋃i∈p-Ai→⋃i∈p-Ai is contractive p-semicyclic, then (14) becomes 0≤limsupn→∞d(Tk+np+1x,Tk+npx)≤D from (12). If, in addition, T:⋃i∈p-Ai→⋃i∈p-Ai is contractive p-cyclic, then D≤limsupn→∞d(Tk+np+1x,Tk+npx)≤D, ∀x∈⋃i∈p-Ai so that there is limn→∞d(Tk+np+1x,Tk+npx)=D, ∀x∈⋃i∈p-Ai. Property (ii) has been proven.The following result establishes an asymptotic property of the limits superiors of distances of consecutive points of the iterated sequences which implies thatT:⋃i∈p-Ai→⋃i∈p-Ai is asymptotically contractive, and the limit limn→∞(∑k=0np+j-2(∏ℓ=knp+j-2[Kℓ+i])(m(T(k+1)-x,Tk-x)-1))=0, ∀x∈⋃i∈p-Ai, ∀j∈p-1¯∪{0} exists. In particular, it is not required that m(x,y)≤1 for any x∈Ai, y∈Ai+1, and ∀i∈p- as in contractive and, in general, nonexpansive p-semicyclic impulsive self-mappings.Theorem 5.
Consider the following generalization of condition 3 of Definition1:
(15)D≤d(T2-x,T-y)≤Kid(Tx,x)+(1-Ki)D,
for any given x∈Ai,   ∀i∈p-, and define K-=∏i=1p-1[Ki]. Define
(16)K^=K-supx∈⋃i∈p-Aimaxn∈Z0+(∏i=np+1(n+1)p-1[m(T(i+1)-x,Ti-x)])=∏i=1p-1[Ki]supx∈⋃k∈p-Akmaxn∈Z0+(∏i=np+1(n+1)p-1[m×(T(i+1)-x,Ti-x)]),
such that K^∈[0,1). Then, the following properties hold.
(i)(17)D0≤limsupn→∞d(Tnp+jx,Tnp+j-1x)≤(∏ℓ=0j-1[Ki+ℓ]1+11-K^(∏ℓ=0j-1[Ki+ℓ])×supx∈⋃k∈p-Akmaxℓ∈Z0+|m′(T(ℓ+1)-x,Tℓ-x)|∏ℓ=0j-1[Ki+ℓ])D,∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0},D0≤d(Tnp+jx,Tnp+j-1x)≤(∏ℓ=0j-1[Ki+ℓ])K^nd(Tx,x)+[supx∈⋃k∈p-Akmaxℓ∈Z0+|m′(T(ℓ+1)-x,Tℓ-x)|(1-(∏ℓ=0j-1[Ki+ℓ])K^n)+1-K^n1-K^(∏ℓ=0j-1[Ki+ℓ])×supx∈⋃k∈p-Akmaxℓ∈Z0+|m′(T(ℓ+1)-x,Tℓ-x)|]D,supx∈⋃k∈p-Ak∀x∈Ai,∀i∈p-,∀j∈p-1¯∪{0},
where D0=0 if T:⋃i∈p-Ai→⋃i∈p-Ai is p-semicyclic and D0=D if T:⋃i∈p-Ai→⋃i∈p-Ai is  p-cyclic.
(ii) If, furthermore, there is a real constantε0≥-1 such that
(18)limsupn→∞(∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i])×(m(T(k+1)-x,Tk-x)-1)∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i]))≤ε0,2222222222222∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0},
then
(19)D0≤limsupn→∞d(Tnp+jx,Tnp+j-1x)≤D(1+ε0),∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0}.Proof.
SinceK^∈[0,1), one has through iterative calculation via (15)
(20)d(T2x,Tx)≤m(T2-x,T-x)(Kid(Tx,x)+(1-K)D)=(m(T2-x,T-x)Ki)d(Tx,x)+(1-m(T2-x,T-x)Ki)D+m′(T2-x,T-x)D,d(Tpx,Tp-1x)≤(∏i=1p-1[m(T(i+1)-x,Ti-x)])K-d(Tx,x)+(1-(∏i=1p-1[m(T(i+1)-x,Ti-x)])K-)D+(∑k=0p-2(∏ℓ=k-1p-2[Kℓ+i])m′(T(k+1)-x,Tk-x))Dd(Tnp+jx,Tnp+j-1x)≤(∏i=1np+j-1[m(T(i+1)-x,Ti-x)])(∏ℓ=0j-1[Ki+ℓ])×K-nd(Tx,x)+(1-(∏i=1np+j-1[m(T(i+1)-x,Ti-x)])×(∏ℓ=0j-1[Ki+ℓ])K-n)D+(∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i])m′(T(k+1)-x,Tk-x))D,∑k=0np+j-2(23456789)∀x∈Ai,∀i∈p-,∀j∈p-1¯∪{0},
with the convention (∏ℓ=0-1[Ki+ℓ])=1, ∀i∈p-. Then, one gets (17), and Property (i) has been proven. To prove Property (ii), use the indicator sets (6) and, since m′(T2-x,T-x)≥-1, ∀x∈⋃i∈p-Ai, one also gets from (15)-(16)
(21)D0+[liminfn→∞(∑k∈S-(k,n,j-2)(∏ℓ=knp+j-2[Kℓ+i])×|m′(T(k+1)-x,Tk-x)|∑k∈S-(k,n,j-2)(∏ℓ=knp+j-2[Kℓ+i]))liminfn→∞(∑k∈S-(k,n,j-2)(∏ℓ=knp+j-2[Kℓ+i])]D≤limsupn→∞d(Tnp+jx,Tnp+j-1x)+[liminfn→∞(∑k∈S-(k,n,j-2)(∏ℓ=k-1np+j-2[Kℓ+i])×|m′(T(k+1)-x,Tk-x)|∑k∈S-(k,n,j-2)(∏ℓ=k-1np+j-2[Kℓ+i]))liminfn→∞(∑k∈S-(k,n,j-2)(∏ℓ=k-1np+j-2[Kℓ+i])]D≤D[m′(T(k+1)-x,Tk-x)(∏ℓ=k-1np+j-2[Kℓ+i]))(∑k∈S+(k,n,j-2)(∏ℓ=k-1np+j-2[Kℓ+i]))];(∏ℓ=k-1np+j-2[Kℓ+i])limsupn→∞(1+(∑k∈S+(k,n,j-2)(∏ℓ=k-1np+j-2[Kℓ+i])×m′(T(k+1)-x,Tk-x)∑k∈S+(k,n,j-2)(∏ℓ=k-1np+j-2[Kℓ+i]))(∑k∈S+(k,n,j-2)(∏ℓ=k-1np+j-2[Kℓ+i]))],21222222222222222222∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0},and (19), and then Property (ii), follows from (18).Note from (19) in Theorem 5 that if D0=D=0, that is, ⋂i∈p-Ai≠∅, and ε0∈[-1,∞), then ∃limn→∞d(Tnp+jx,Tnp+j-1x)=0, ∀x∈⋃i∈p-Ai, ∀j∈p-1¯∪{0} from (19) since K^∈[0,1). In this case, T:⋃i∈p-Ai→⋃i∈p-Ai is an asymptotically contractive p-cyclic (and also p-semicyclic since D=0) self-mapping on the union on intersecting closed subsets of X. A close property follows if D0=D≠0, and ε0=0 implying from (19) that
(22)limsupn→∞(∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i])×(m(T(k+1)-x,Tk-x)-1)∑k=0p+j-2(∏ℓ=k-1np+j-2[Kℓ+i]))=limn→∞(∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i])×(m(T(k+1)-x,Tk-x)-1)∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i]))=0,222m(2)2∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0}
and leading to ∃limn→∞d(Tnp+jx,Tnp+j-1x)=D such that T:⋃i∈p-Ai→⋃i∈p-Ai is a contractive p-cyclic self-mapping on the union on disjoint closed subsets of X. The above discussion is summarized in the subsequent result.Corollary 6.
Assume that (15) holds with K^ defined in (16) being in [0,1), and assume also that
(23)∞>ε0≥max((∏ℓ=k-1np+j-2[Kℓ+i])limsupn→∞(∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i])×(m(T(k+1)-x,Tk-x)-1),-1∏ℓ=k-1np+j-2[Kℓ+i]s)),222121212121222222222222∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0}.
Then, the following properties hold
(i) If⋂i∈p-Ai≠∅, then T:⋃i∈p-Ai→⋃i∈p-Ai is an asymptotically contractive p-cyclic impulsive self-mapping so that there is the limit
(24)limn→∞d(Tnp+jx,Tnp+j-1x)=0,∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0}.
(ii) If⋂i∈p-Ai=∅, d(Tx,Ty)≥D, ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p- and the following limit exists:
(25)limn→∞(∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i])×(m(T(k+1)-x,Tk-x)-1)∑k=0np+j-2(∏ℓ=k-1np+j-2[Kℓ+i]))=0;222222222∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0}
then T:⋃i∈p-Ai→⋃i∈p-Ai is an asymptotically contractive p-cyclic impulsive self-mapping so that the limit
(26)limn→∞d(Tnp+jx,Tnp+j-1x)=D,∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0}exists.A particular result got from Theorem5 follows for contractive p-semicyclic and p-cyclic impulsive self-mappings T:⋃i∈p-Ai→⋃i∈p-Ai.Corollary 7.
Theorem5 holds with D0=0 if T:⋃i∈p-Ai→⋃i∈p-Ai is contractive p-semicyclic and with D0=D if the impulsive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is contractive p-cyclic provided that K-=∏i=1p-1[Ki]∈[0,1).Proof.
It is a direct consequence of Theorem5 since K-=∏i=1p-1[Ki]∈[0,1) implies that K^∈[0,1) since  m(T-x,T-y)≤1, ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p-.Remark 8.
Note that ifT:⋃i∈p-Ai→⋃i∈p-Ai is a nonexpansive p-cyclic impulsive self-mapping, the following constraints hold:
(27)m(Tx-,Ty-)≤1,D≤m(Tx-,Ty-)(Kd(x,y)-D)+m(Tx-,Ty-)D,D≤m(Tx-,T0)∀x∈Ai,∀y∈Ai+1,∀i∈p,-
and equivalently,
(28)1≥m(Tx-,Ty-)≥DKd(x,y)+(1-K)D=DD+K(d(x,y)-D),
implying that(a)
1≥m(Tx-,Ty-)≥0, ∀x∈Ai, ∀y∈Ai+1, and ∀i∈p- if D=0; that is, if the sets Ai intersect ∀i∈p-.(b)
m(Tx-,Ty-)=1 if d(x,y)=D; that is, for best proximity points associated with any two adjacent disjoint subsets Ai, y∈Ai+1 for i∈p-.
On the other hand, note that Corollary6 (ii) implies the asymptotic convergence of distances in-between consecutive points of the iterated sequences generated via T:⋃i∈p-Ai→⋃i∈p-Ai to the distance D between adjacent sets. This property does not imply 1≥m(Tx-,Ty-), ∀x∈Ai, and ∀y∈Ai+1, ∀i∈p- as required for nonexpansive (and, in particular, for contractive) p-cyclic impulsive self-mappings. However, it implies m(T(n+1)-x,Tn-x)→1 as n→∞ from (25), since the sequence defining its left-hand-side sequence has to converge asymptotically to zero.Define recursively global functions to evaluate the nonexpansive and contractive properties of the impulsive self-mappingT:⋃i∈p-Ai→⋃i∈p-Ai which take into account the most general case that the constant K in Definition 1 (1) can be generalized to be set dependent and point-dependent leading to a combined extended constraint as follows:
(29)d(T2x,Tx)≤Ki(x,Tx)m(T2-x,T-x)d(x,Tx)+m(T2-x,T-x)(1-Ki(x,Tx))D,22222+m(T2-x,T-x)∀x∈⋃i∈p-Ai,
so that
(30)K^(j)(x,Tx)=(∏i=1p-1[m(T(i+jp+1)-x,T(i+jp)-x)×Ki(Ti+jpx,Ti+jp-1x)]∏i=1p-1(∏i=1p-1[m(T(i+jp+1)-x,T(i+jp)-x)[m(T(i+jp+1)-x,T(i+jp)-x))K^(j-1)(x)(∏i=1p-1[m(T(i+jp2)-x,)∀x∈⋃i∈p-Ai,∀j∈Z+,
with  x=T0x and initial, in general, point-dependent value
(31)K^(0)(x,Tx)=∏i=1p-1[m(T(i+1)-x,T(i)-x)×Ki(Tix,Ti-1x)],222∀x∈⋃i∈p-Ai,∀j∈Z+,
for each iterated sequence constructed through the impulsive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai. The following related result follows.Theorem 9.
Consider thep-semicyclic impulsive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai under the constraint (29) subject to (30)-(31). If limn→∞K^(n)(x,Tx)=0, ∀x∈⋃i∈p-Ai, then the following properties hold.
(i) If⋂i∈p-Ai≠∅ then
(32)limn→∞d(T(n+1)px,T(n+1)p-1x)=0;∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0},∀n∈Z0+
so that T:⋃i∈p-Ai→⋃i∈p-Ai is asymptotically contractive p-semicyclic cyclic in the sense that, given x∈Ai, there is a sufficiently large n0=n0(x)∈Z0+ such that, together with (32), Tnpx∈Ai, T(n+1)px∈Ai∪Ai+1 for n≥n0.
(ii) If⋂i∈p-Ai=∅ and the limit below exists:
(33)limn→∞∑j=0n-1K^(n-j)(x,Tx)×(∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)])×(m(T(k+1)-x,Tk-x)-1)∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)]))=0,∀x∈⋃i∈p-Ai,
then
(34)limn→∞d(T(n+1)px,T(n+1)p-1x)=D,∀x∈⋃i∈p-Ai,∀j∈p-1¯∪{0},∀n∈Z0+,
and T:⋃i∈p-Ai→⋃i∈p-Ai is asymptotically contractive p-cyclic in the sense that, given x∈Ai, together with (34), there is a sufficiently large n0=n0(x)∈Z0+ such that, together with (34), Tnpx∈Ai, T(n+1)px∈Ai+1 for n≥n0.(iii)
The limit (33) exists and then (34) holds if m:(⋃i∈p-Ai)×(⋃i∈p-Ai)→R0+ satisfies the identity(35)
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One gets from (20), (29)–(31) that
(36)d(Tpx,Tp-1x)≤K^(0)(x,Tx)d(Tx,x)+(1-K^(0)(x,Tx))D+(∑k=0p-2(∏ℓ=k+i-1p-2[Kℓ(T(k-1)x,Tkx)])×m′(T(k+1)-x,Tk-x)∑k=0p-2(∏ℓ=k+i-1p-2[Kℓ(T(k-1)x,Tkx)]))D,(37)d(T(n+1)px,T(n+1)p-1x)≤K^(n)(x,Tx)d(Tx,x)+(1-K^(n)(x,Tx))D+∑j=0n-1K^(n-j)(x,Tx)×(∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)])×(m(T(k+1)-x,Tk-x)-1)∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)]))D,∀x∈Ai,∀i∈p-,∀j∈p-1¯∪{0},∀n∈Z0+,
where m′(T(n+1)-x,Tn-x)=m(T(n+1)-x,Tn-x)-1. If limn→∞K^(n)(x,Tx)=0, ∀x∈⋃i∈p-Ai and (33) holds, then d(T(n+1)px,T(n+1)p-1x)→D as n→∞, ∀x∈⋃i∈p-Ai, ∀j∈p-1¯∪{0}, and ∀n∈Z0+. This leads directly to Property (i) for D=0 if ⋂i∈p-Ai≠∅ (without the constraint (33) being needed) and to Property (ii) for D≠0 if ⋂i∈p-Ai=∅.
Consider that(38)∑j=0nK^(n+1-j)(x,Tx)×(∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)])×(m(T(k+1)-x,Tk-x)-1)∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)]))=K^(1)(x,Tx)×(∑k=np(n+1)p-2(∏ℓ=(n+1)p+i-3(n+1)p-2[Kℓ(T((n+1)p-3)x,T(n+1)p-2x)])×(m(T((n+1)p-1)-x,T((n+1)p-2)-x)-1)∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)]))+K^(1)(x,Tx)×(∑k=np(n+1)p-3(∏ℓ=k+i-1(n+1)p-2[Kℓ(T(k-1)x,Tkx)])×(m(T(k+1)-x,Tk-x)-1)∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)]))+∑j=0n-1K^(n-j)(x,Tx)×(∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)])×(m(T(k+1)-x,Tk-x)-1)∑k=jp(j+1)p-2(∏ℓ=k+i-1(j+1)p-2[Kℓ(T(k-1)x,Tkx)]))
converges to zero as n→∞ if for some real sequence {εn}n∈Z0+ which converges to zero, the function m:(⋃i∈p-Ai)×(⋃i∈p-Ai)→R0+ satisfies (35). This proves Property (iii).Theorem9 has a counterpart in terms of asymptotically strict p-semicyclic and cyclic versions established as follows.Corollary 10.
Assume that the following strict-type contractive condition holds:(39)d(T2x,Tx)≤Ki(x,Tx)m(T2-x,T-x)d(x,Tx)+(1-m(T2-x,T-x)Ki(x,Tx))D,(1-m(T2-x,T-x)222)∀x∈⋃i∈p-Ai,
subject to the constraints (30) and (31). If  limn→∞K^(n)(x,Tx)=0, ∀x∈⋃i∈p-Ai, then (34) holds, and T:⋃i∈p-Ai→⋃i∈p-Ai is a strictly asymptotically contractive p-cyclic impulsive self-mapping in the sense that, given any x∈Ai, there is a sufficiently large n0=n0(x)∈Z0+ such that, together with (34), Tnpx∈Ai,T(n+1)px∈Ai+1 for all n≥n0 if ⋂i∈p-Ai=∅.
If⋂i∈p-Ai≠∅, then T:⋃i∈p-Ai→⋃i∈p-Ai is (at least) strictly asymptotically contractive p-semicyclic in the sense that there is a sufficiently large n0=n0(x)∈Z0+ such that, together with (32), Tnpx∈Ai, T(n+1)px∈Ai∪Ai+1 for n≥n0 for any given x∈Ai.Proof (outline of proof).
It follows directly by replacing (37) with
(40)d(T(n+1)px,T(n+1)p-1x)≤K^(n)(x,Tx)d(Tx,x)+(1-K^(n)(x,Tx))D,≤K^(n)(x,1)∀x∈Ai,∀i∈p-,∀j∈p-1¯∪{0},
so that there is the limit limn→∞d(T(n+1)px,T(n+1)p-1x)=0;∀x∈Ai, ∀i∈p-, and ∀j∈p-1¯∪{0}.


## 3. Convergence of the Iterations to Best Proximity Points and Fixed Points

Important results about convergence of iterated sequences of 2-cyclic self-mappings to unique best proximity points were firstly stated and proven in [11] and then widely used in the literature. Some of them are quoted here to be then used in the context of this paper. Consider a metric space (X,d) with nonempty subsets A, B⊂X such that D=d(A,B)≥0. The following basic results have been proven in the existing background literature.Result 1 (see [11]).
Let(X,d) be a metric space, and let A and B be subsets of  X. Then, if A is compact and B is approximatively compact with respect to A (i.e., d(y,xn)→d(y,B) as n→∞ for each sequence {xn}n∈Z0+⊂B for some y∈A), then Ao={x∈A:d(x,y′)=Dforsomey′∈B} and Bo={y∈B:d(x′,y)=Dforsomex′∈A} are nonempty.It is known that ifA and B are both compact, then A(resp.,B) is approximatively compact which respect to B(resp.,A).Result 2 (see [11]).
Let(X,∥∥) be a reflexive Banach space, let A be a nonempty, closed, bounded, and convex subset of X and let B be a nonempty, closed and convex subset of X. Then, the sets of best proximity points Ao and Bo are nonempty.Result 3 (see [11]).
Let(X,d) be a metric space, let A and B be nonempty closed subsets of X, and let T:A∪B→A∪B be a 2-cyclic contraction. If either A is boundedly compact (i.e., if any bounded sequence {xn}n∈Z0+⊂A has a subsequence converging to a point of A) or B is boundedly compact, then there is x∈A∪B such that d(x,Tx)=D.Remark 11.
It is known that ifA⊂X is boundedly compact, then it is approximatively compact. Also, a closed set A of a normed space is boundedly compact if it is locally compact (the inverse is not true in separable Hilbert spaces [34]); equivalently, if and only if the closure of each bounded subset C⊂A is compact and contained in A. If (X,d) is a linear metric space, a closed subset A⊂X is boundedly compact if each bounded C⊂A is relatively compact. It turns out that if A⊂X is closed and bounded then it is relatively compact [35]. It also turns out that if (X,d) is a complete metric space and the metric is homogeneous and translation-invariant, then (X,d) is a linear metric space and (X,∥∥) is also a Banach space with ∥∥ being the norm induced by the metric d. Note that, since the metric is homogeneous and translation-invariant and since (X,d) is a linear metric space, such a metric induces a norm. In such a Banach space, if A⊂X is bounded and closed, then A is boundedly compact and thus approximatively compact.Result 4 (see [11]).
Let(X,∥∥) be a uniformly convex Banach space, let A be a nonempty closed and convex subset of X, and let B be a nonempty closed subset of X. Let sequences {xn}n∈Z0+⊂A, {zn}n∈Z0+⊂A and {yn}n∈Z0+⊂B satisfy ∥xn-yn∥→D and ∥zn-yn∥→D as n→∞. Then ∥zn-xn∥→0 as n→∞.It is known that a uniformly convex Banach space(X,∥∥) is reflexive and that a Banach space is a complete metric space (X,d) with respect to the norm-induced distance.Result 5 (see [11]).
If(X,d) is a complete metric space, T:A∪B→A∪B is a 2-cyclic contraction, where A and B are nonempty closed subsets of X, and the sequence {xn}n∈Z0+ generated as xn+1=Txn, ∀n∈Z+ for a given x0∈A has a convergent subsequence {x2nk}nk∈Z0+⊂{x2n}n∈Z0+⊂{xn}n∈Z0+ in A, then there is x∈A∪B  such that d(x,Tx)=D.Sufficiency-type results follow below concerning the convergence of iterated sequences being generated by contractive and strictly contractivep-semicyclic self-mappings, which are asymptotically p-cyclic, to best proximity or fixed points.Theorem 12.
Assume that(X,∥∥) is a uniformly convex Banach space so that (X,d) is a complete metric space if d:X×X→R0+ is the norm-induced metric. Assume, in addition, that T:⋃i∈p-Ai→⋃i∈p-Ai is a p-semicyclic impulsive self-mapping, whereAi⊂X, ∀i∈p- are nonempty, closed, and convex subsets of X, and assume also that(1)
either the constraint (29), or the constraint (39) holds subject to (30) and (31) provided that the limit limn→∞K^(n)(x,Tx)=0, ∀x∈⋃i∈p-Ai exists and m:(⋃i∈p-Ai)×(⋃i∈p-Ai)→R0+ satisfies (35);(2)
for each givenx∈Ai for any i∈p-, there is a finite ki=ki(x)∈Z0+ such that liminfn→∞Tnp+ki(x)∈Ai+1 (i.e., the p-semicyclic impulsive self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is also an asymptotically p-cyclic one).
Then,T:⋃i∈p-Ai→⋃i∈p-Ai is either an asymptotically contractive or a strictly contractive p-semicyclic impulsive self-mapping, and, furthermore, the following properties hold.
(i) The limits below exist:(41)limn⟶∞d(T(n+1)px,T(n+j)p+jx)=D,∀x∈Ai,∀j∈k-i,∀i∈p,-(42)limn⟶∞d(T(n+1)p+k-i+1x,T(n+j)p+k-ix)=0,∀x∈Ai,∀i∈p-,
where k-i=supx∈Aiki(x), ∀i∈p-. Furthermore, {Tnpx}n∈Z+→zi, {Tnp+jx}n∈Z+→Tzi(j) for any given x∈Ai with {Tnp+jx}n∈Z+⊂Ai∪Ai+1,   ∀j∈k-i¯, limn→∞Tnp+k-ix⊂Ai+1, zi∈Ai, zi(j)∈Ai; ∀j∈k-i¯-1, and  zi+1=Tzi(ki)∈Ai+1, ∀i∈p-. The points zi and zi+1 are unique best proximity points in Ai and Ai+1, ∀i∈p- of  T:⋃i∈p-Ai→⋃i∈p-Ai, and there is a unique limiting set
(43)(z1,z1(1)=Tz1,…,z2=z1(k-1)=Tk-1z1,…,zp,zp(1)=Tzp,…,zp(k-p-1)=Tk-p-1zp)⊂A1k-1×⋯×A1k-p.
If ⋂i∈p-Ai≠∅, then the p best proximity points zi=z∈⋂j∈p-Aj, ∀i∈p- become a unique fixed point z of T:⋃i∈p-Ai→⋃i∈p-Ai.
(ii) Assume that the constraint (15) holds, subject to either (25), or (29), with K-=∏i=1p-1[Ki] and K^∈[0,1) defined in (16). Assume, in addition, that for each x∈Ai for any i∈p-, it exists a finite ki=ki(x)∈Z0+ such that liminfn→∞Tnp+ki(x)∈Ai+1 with k-i=supx∈Aiki(x), ∀i∈p-. Then, Property (i) still holds.Proof.
The existence of the limits (41) and (42) follows from (34) in Theorem 9 and the above background Result 4 [11] since, for each x∈Ai for any i∈p-, there is a finite ki=ki(x)∈Z0+ such that  liminfn→∞Tnp+ki(x)∈Ai+1 with k-i=supx∈Aiki(x), ∀i∈p- so that the limits (41) exist (note that k-i=1, ∀i∈p- if T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic impulsive self-mapping). The limit (42) exists from the background Results 1 and 5 of [11] with zi∈Ai and zi+1=Tzi(ki)∈Ai+1, ∀i∈p- being unique best proximity points of T:⋃i∈p-Ai→⋃i∈p-Ai in Ai and Ai+1; ∀i∈p- since (X,d) is also a (X,∥∥) uniformly convex Banach space for the norm-induced metric and the subsets  Ai of X, ∀i∈p- are nonempty, closed and convex. The limiting set (zi,zi(1)=Tzi,…,zi+1=Tk-izi) is unique with zi(j)∈Ai; ∀j∈k-i¯-1 since zi and zi+1; ∀i∈p- are unique best proximity points and T:⋃i∈p-Ai→⋃i∈p-Ai is single-valued. Property (i) has been proved. The same conclusions arise from (25) in Corollary 6 and from (39) in Corollary 10 leading to Property (ii).Remarks 13.
(1) Note that if the self-mappingT:⋃i∈p-Ai→⋃i∈p-Ai is an asymptotic p-cyclic impulsive one, then the limiting set (43) of Theorem 12 can only contain points which are not best proximity points in bounded subsets Ai of X whose diameter is not smaller than D.
(2) Under the conditions of Theorem12, if T:⋃i∈p-Ai→⋃i∈p-Ai is, in particular, a contractive or strictly contractive p-cyclic impulsive self-mapping, then the limiting set (43) only contains best proximity points; that is, it is of the form (z1,z2,…,zp). If ⋂i∈p-Ai≠∅, then such a set reduces to a unique best proximity point z∈⋂i∈p-Ai.
(3) Note that Theorem12 can be formulated also for a complete metric space (X,d) with a homogeneous translation-invariant metric d:X×X→R0+ being equivalent to a Banach space (X,∥∥), where ∥∥ is the metric-induced norm, which is uniformly convex so that it is also a complete. Note that such a statement is well-posed since a norm-induced metric exists if such a metric is homogeneous and translation invariant.
It turns out that Theorem12 and Remarks 13 also hold if T:⋃i∈p-Ai→⋃i∈p-Ai is either a contractive or a strictly contractive p-semicyclic impulsive self-mapping as stated in the subsequent result.Corollary 14.
Theorem12 holds, in particular, if T:⋃i∈p-Ai→⋃i∈p-Ai is a contractive or strictly contractive p-semicyclic impulsive self-mapping with Ki=K∈[0,1), ∀i∈p- being a constant in (29) or (39) subject to (35) and m:(⋃i∈p-Ai)×(⋃i∈p-Ai)→R0+ being not larger than unity.
Theorem12 also holds if T:⋃i∈p-Ai→⋃i∈p-Ai is, in particular, a contractive or strictly contractive p-cyclic impulsive self-mapping with Ki=K∈[0,1); ∀i∈p- being constant in (29) or (39) subject to (35) and m:(⋃i∈p-Ai)×(⋃i∈p-Ai)→R0+ being not larger than unity. In this case, the limiting set (43) only contains best proximity points; that is, it is of the form (z1,z2,…,zp).


## 4. Application Examples to Impulsive Differential and Difference Equations

Recent results about best proximity points concerning psi-Geraghty contractions and on cyclic orbital contractions are obtained in [36, 37], respectively. On the other hand, it turns out that fixed point theory is a useful tool to study the stability of differential and difference equations and dynamic systems [38–42]. Some worked examples are given in the sequel concerning the global feedback stabilization and the stability of the equilibrium points [43–46], linked with fixed points and best proximity points of impulsive and time-delayed differential equations. The subsequent examples rely on the properties of iterated sequences xn+1=Txn, ∀n∈Z0+ for any x0∈⋃i∈p-Ai being generated from nonexpansive or contractive p-semicyclic, impulsive self-mappings TAi→Ai∪Ai+1, where Ai⊂X, ∀i∈p- and (X,d) is a metric space, subject to theb following: (1)
T-:⋃i∈p-Ai→⋃i∈p-Ai is a nonexpansive, or a contractive, p-semicyclic self-mapping so that T-Ai⊆Ai∪Ai+1, ∀i∈p- subject to the cyclic nonexpansive/contractive constraint d(T-x,T-y)≤Kd(x,y)+(1-K)D, ∀i∈p- for K∈[0,1], where D is the distance between any two adjacent subsets. This self-mapping describes in the given examples the discretized impulsive-free solution of an ordinary differential equation;(2)
T+:⋃i∈p-Ai→⋃i∈p-Ai satisfies a distance discontinuity condition of the type d(T+(T-x),T+(T-y))≤m(T-x,T-y)d(T-x,T-y) for some given bounded function m:(⋃i∈p-Ai)×(⋃i∈p-Ai)→R0+.  This self-mapping describes the bounded steps in the solution due to eventual forcing impulses at certain impulsive time instants.Example 15.
Consider the real impulsive differential equation(44)x˙(t)=α(t)x(t)+β(t)u(t)+∑tk∈SIγ(tk)δ(t-tk),x(0-)=x0,x(0)=x(0+)=x0+γ(0),
where α,β∈B(R0+,R)∪PC(R0+,R) (i.e., bounded and piece-wise continuous real functions on R0+), {γ(tk)}tk∈SI is a bounded sequence, δ(t) is the Dirac distribution, u∈PC(R0+,R) is the nonimpulsive control, x:R0+→R is the unique solution of (44) which is continuous and time-differentiable on [0,t1)⋃(⋃k∈I[tk,tk+1)), tk∈SI⊂R0+ is a set of impulsive sampling instants with k∈I⊆Z+, and the indicator set I of SI has a finite or an infinite cardinal. Note that uimp(t)=∑tk∈SIγ(tk)δ(t-tk) is an impulsive control. Assume a linear-feedback control of the form u(t)=g(t)x(t) with g∈PC(R0+,R). The solution of (44) is
(45)x(t)=e∫tktα(τ)dτx(tk)+∫tkte∫τtα(σ)dσβ(τ)u(τ)dτ=e∫tkt(α(τ)+β(τ)g(τ))dτx(tk),∀t∈([0,t1)⋃(⋃tk∈SI[tk,tk+1))),(46)x(tk+1)∶=x(tk+1+)=x(tk+1-)+γ(tk)=e∫tkt(α(τ)+β(τ)g(τ))dτx(tk)+γ(tk+1)=e∫tktk+1α(τ)dτx(tk)+∫tktk+1e∫τtk+1α(σ)dσβ(τ)u(τ)dτ=e∫tktk+1(α(τ)+β(τ)g(τ))dτx(tk)+γ(tk+1);∀tk∈SI.
Then, the following results hold.Proposition 16.
Assume thatγ(0)=M(0), γ(tk+1)=M(tk+1)-e∫tktk+1(α(τ)+β(τ)g(τ))dτx(tk) for tk∈SI and {M(tk)}tk∈SI is some bounded real sequence, then(i)
{x(tk)}tk∈SIis bounded so that (44) is globally stable. If, in addition,{M(tk)}tk∈SIconverges to zero astk(∈SI)→∞(ifc= cardIis finite thenM(tc)=0), then{x(tk)}tk∈SIconverges to zero ask→∞.(ii)
ifIhas an infinite cardinal, so thatI=Z+, and|tk+1-tk|≤T-<∞,∀tk∈SI,then|x(t)|is bounded for allt∈R0+. If, in addition,{M(tk)}tk∈SIconverges to the stable zero equilibrium point ask→∞thenx(t)→0ast→∞so that (44) is globally asymptotically stable.Proof.
Property (i) follows from its statement and (46). Since {x(tk)}tk∈SI is bounded, the continuous function x(t) on (tk,tk+1) cannot be unbounded on the finite interval [tk,tk+1) if 0≤T-0≤|tk+1-tk|≤T-<∞, ∀tk∈SI. Since clR0+=cl([0,t1)⋃(⋃k∈I[tk,tk+1))) if the indicator set I of impulses is of infinite cardinal, it becomes obvious that x(t) is bounded on its definition domain R0+. If, in addition, {M(tk)}tk∈SI converges to zero as k→∞ then {x(tk)}tk∈SI converges to zero as k→∞ from Property (i) so that {γ(tk)}tk∈SI converges to zero as k→∞. Then, x(t)→0 on t∈[tk,tk+1) as tk(∈SI)→∞ from (45). Hence, Property (ii) is proven.Proposition 17.
Assume thatγ(tk)=λ(tk)x(tk-). Then, the following properties hold.
(i)Assume thatcardSI=χ0 (i.e., the infinity cardinal of a numerable set) fulfilling |tk+1-tk|≤T-<∞, ∀tk∈SI,  and define the self-mapping T:R→R generating the solution sequence {x(tk)}tk∈SI of (44) at the set of impulsive time instants SI={tk}k∈I. Assume that such a set has infinite cardinal. Then, T:R→R is asymptotically contractive and has a unique fixed point x-=0 if(47a)0≤limsupSI∋tk→∞|1+λ(tk+1)|e∫tktk+1(α(τ)+β(τ)g(τ))dτ<1.
IfcardI=c<χ0 then (47a) is replaced with
(47b)0≤limsupt→∞e∫tct(α(τ)+β(τ)g(τ))dτ<1⟺limsupt→∞∫tct(α(τ)+β(τ)g(τ))dτ<0.
Furthermore,{x(tk)}tk∈SI is bounded, x:R0+→R is bounded and limt→∞x(t)=0 so that (44) is globally asymptotically stable.
(ii) Property (i) still holds ifcardSI=χ0, and there is a nondecreasing sequence {Nk}k∈Z0+⊆I with |Nk+1-Nk|≤N-<∞ such that
(48)limsupI∋Nk→∞|∏i=NkNk+1[(1+λ(ti))e∫titi+1(α(τ)+β(τ)g(τ))dτ]|<1.
(iii) Property (i) also holds withT:R→R being contractive if cardSI=χ0 and for some positive real sequence {εk}k∈Z0+(49)λ(tk+1)=e-∫tktk+1(α(τ)+β(τ)g(τ))dτ-1-εk,∀k∈Z0+.Proof.
Assume thatcardSI=χ0. It follows that
(50)x(tk+1)=(1+λ(tk+1))x(tk+1-)=(1+λ(tk+1))e∫tktk+1(α(τ)+β(τ)g(τ))dτx(tk),∀tk∈SI,
so that
(51)|x(tk+1)-x(tk)|≤suptk∈SI(|1+λ(tk+1)|e∫tktk+1(α(τ)+β(τ)g(τ))dτ)×|x(tk)-x(tk-1)|,∀tk∈SI
and one gets that {|x(tk+1)-x(tk)|}tk∈SI converges to zero and {x(tk)}tk∈SI converges to a unique fixed point x-∈R as SI∋tk→∞ from Theorem 12, supported by Theorem 9, with the complete metric space and Banach space (R0+,d)≡(R0+,∥∥) the metric being the Euclidean distance. Also, since the sequence {|x(tk+1)-x(tk)|}tk∈SI converges to zero as SI∋tk→∞ yields that x-=0 is the unique fixed point of T:R→R since, otherwise, (50) would contradict (47a) for x-=x(tk)=x(tk+1)≠0. The facts that {x(tk)}tk∈SI is bounded, x:R0+→R is bounded, and limt→∞x(t)=0 follow under the same reasoning as in Proposition 16. Hence, Property (i) follows for the case that cardSI=χ0. If such a cardinal is finite, we can remove a finite number of impulsive time instants from the discussion, and the property also holds under (47b). The proof of Property (ii) is similar leading to the convergence to zero of the sequence {δx^(tNk)}tNk∈SI¯⊆SI as tNk→∞ where δx^(tNk)=x(tNk+1)-x(tNk). Thus, x(tNk)→x^ as I-∋Nk→∞ where I-⊆I. As above, it turns out that x^=x-=0 under a similar contradiction argument to the above one. Hence, Property (ii) follows. Property (iii) follows directly since (49) leads to (47a).Note that Proposition17 states global properties for the solution so that the contractive condition is achievable with mixed conditions on the nonimpulsive and impulsive parts of the differential equation. For instance, it is clear from (49) that a certain condition on the impulsive controls can stabilize the system even if the nonimpulsive part is unstable; that is, if liminfSI∋tk→∞∫tktk+1(α(τ)+β(τ)g(τ))dτ>0. It is easy to deduce from a slightly extended Proposition 17 that T:R→R is asymptotically nonexpansive if the inequalities in (47a)–(48) are not strict. Note also that (49) can be checked in terms of the values of intervals in-between consecutive impulsive time instants and impulsive control gains with the following test if SI has infinite cardinal:
(52)ln|1+λ(tk+1)|+∫0tk+1-tk(α(tk+τ)+β(tk+τ)g(tk+τ))dτ<0,∫0tk+1-tk(α(tk+τ)222+β(tk+τ)g(tkτ))∀tk∈SI.Then, the mapping constructing the solution which iterates at the impulsive time instants is contractive. The above closed-loop stability condition (52) is guaranteed if |tk+1-tk|≤T-<∞, ∀tk∈SI and
(53)suptk∈SI[supτ∈[0,T-](α(tk+τ)+β(tk+τ)g(tk+τ))T-]ln|1+λ(tk+1)|+supτ∈[0,T-](α(tk+τ)+β(tk+τ)g(tk+τ))T-]<0.Related close conditions to (52) and (53) would follow being equivalent to (47a) and (48) to guarantee that the mapping building the solution sequence at impulsive time instants from any initial condition is asymptotically contractive. In particular, a close test can be jointly performed for finite sets of consecutive impulsive time instants defined bounded time intervals. Closed-loop global asymptotic stability of the feedback equation and the convergence to the unique equilibrium point x-=0 is also guaranteed by the subsequent result.Proposition 18.
Assume thatcardSI=χ0 and that there is a real sequence {q(tk)}k∈Z0+ fulfilling 0≤q(tk)<1;  ∀tk∈SI such that {γ(tk)}k∈Z0+ in (44) is defined by:
(54)γ(tk+1)=-(q(tk)+e∫tktk+1(α(τ)+β(τ)g(τ))dτ)x(tk),∀k∈SI.
Then,x(tk)→0 as tk→∞ and x(t)→x-=0 (the unique fixed point of T:R→R) as t→∞.Proof.
It follows from (45) by noting that (54) is equivalent to
(55)x(tk+1)=e∫tktk+1(α(τ)+β(τ)g(τ))dτx(tk)+γ(tk+1)=-q(tk)x(tk).Example 19.
Consider the differential equation (44) and the sets A={z∈R:z≤-D/2} and B={z∈R:z≥D/2} for some real D∈R0+. Define the self-mapping T:R→R for the solution sequence at impulsive time instants as follows for each tk∈SI assuming that cardSI=χ0 and that there are prefixed finite T->0 and T-0>0 with 0<T-0≤|tk+1-tk|≤T-.
(a)λ(tk+1) and tk+1≤tk+T- are chosen so that(56a)λ(tk+1)=-(1+e∫tktk+1(α(τ)+β(τ)g(τ))dτ)if x(tk)≤-D/2   (i.e., if x(tk)∈A) leading to x(tk+1)≥D/2 (i.e., x(tk+1)∈B) or if min(x(tk-1),x(tk))≥D/2 (i.e., if x(tk-1),x(tk)∈B) leading to x(tk+1)≤-D/2 (i.e., x(tk+1)∈A); and
(b)λ(tk+2) and tk+2≤tk+1+T- are chosen so that for some given positive real constant μ:
(56b)ln|1+λ(tk+2)|+supτ∈[0,tk+2-tk+1](α(tk+1+τ)+β(tk+1+τ)g(tk+1+τ))supτ∈[0,tk+2-tk+1]×(tk+2-tk+1)=-μk+1≤-μ<0 leading to x(tk+2)≥D/2 if x(tk)≤-D/2 and x(tk+1)≥D/2 (i.e., if x(tk)∈A and x(tk+1)∈B) and leading to x(tk+2)≤-D/2 if x(tk)≥D/2 and x(tk+1)≤-D/2 (i.e., if x(tk)∈B and x(tk+1)∈A).
Note that (56a) implies that γ(tk+1)=(1+e∫tktk+1(α(τ)+β(τ)g(τ))dτ)|x(tk)| and x(tk+1)=-x(tk). Note that (56b) leads to the strict contraction (39) of Corollary 10 in the particular form
(57)x(tk+2)-D=(1+λ(tk+2))e∫tk+1tk+2(α(τ)+β(τ)g(τ))dτ(x(tk+1)-D).
Note that (56a) and (56b) imply that the sequence of iterates is formed with consecutive sets of two consecutive points in B and one in A. Thus, the sequence of impulsive gains (56a) and (56b) implies that the self-mapping T:A∪B→A∪B, which generates the sequence {xtk}tk∈SI, which is bounded, is p-semicyclic nonexpansive, while the composite self-mapping T2:A∪B→A∪B, which generates the bounded subsequences {xt2k}t2k∈SI and {xt2k+1}t2k+1∈SI, is 2-semicyclic contractive. As a result, {xt2k+1}t2k∈SI and {xt2k+1}t2k+1∈SI converge each to one of the unique best proximity points  ±D/2, in particular, to the unique fixed point x-=0 if D=0.Example 20.
The differential equation (44) is now replaced by the functional impulsive differential equation with delay h>0 as follows:
(58)x˙(t)=α(t)x(t)+α0(t)x(t-h)+β(t)u(t)+∑tk∈SIγ(tk)δ(t-tk)
with α0∈B(R0+,R)∪PC(R0+,R) and φ:[-h,0]→R being any absolutely continuous of initial conditions of (58) with eventual bounded discontinuities on a subset of [-h,0] of zero measure with φ(0-)=x(0-)=x0 and φ(0+)=x(0+)=φ(0)=x0+γ(0) so that x(t)=φ(t),  t∈[-h,0]. Thus, the solution of (58) is unique and continuous and differentiable in [0,t1)  ⋃(⋃k∈I[tk,tk+1)). The unique solution of (58) is
(59)x(t)=Ψ(t,0)x(0)+∫0hΨ(t,τ)φ(τ-h)dτ+∫0tΨ(t,τ)u(τ)dτ+∑tk∈SIΨ(t,tk)γ(tk)1(t-tk),∀t∈R0+,
where 1(t) is the unit step (Heaviside) function, where the evolution operator satisfies
(60)Ψ˙(t,τ)=α(t)Ψ(t,τ)+α0(t)Ψ(t-h,τ),
for t≥τ with initial conditions Ψ(0)=1, Ψ(t)=0 for t<0. Thus, (60) has the unique continuously differentiable solution
(61)Ψ(t,0)=e∫0tα(τ)dτ+∫hte∫τtα(σ)dσα0(τ)Ψ(τ-h,0)dτ,∀t∈R0+.
Ifti∈SI then (60) has the unique solution at  t=ti as follows:
(62)x(ti-)=Ψ(ti,0)x(0)+∫0hΨ(ti,τ)φ(τ-h)dτ+∫0tiΨ(ti,τ)u(τ)dτ+∑tk(<ti)∈SIΨ(ti,tk)γ(tk)1(ti-tk),x(ti)=Ψ(ti,0)x(0)+∫0hΨ(ti,τ)φ(τ-h)dτ+∫0tiΨ(ti,τ)u(τ)dτ+∑tk(≤ti)∈SIΨ(ti,tk)γ(tk)1(ti-tk)=x(ti-)+γ(ti).
Equation (59) can also describe the interimpulses evolution of the solution under the expressions
(63)x(t)=Ψ(t,tk)x(tk)+∫0hΨ(t,tk+τ)x(tk+τ-h)dτ+∫0t-tkΨ(t,tk+τ)β(tk+τ)u(tk+τ)dτ,∀t∈[tk,tk+1),x(tk+1)=Ψ(tk+1,tk)x(tk)+∫0hΨ(tk+1,tk+τ)x(tk+τ-h)dτ+∫0tk+1-tkΨ(tk+1,tk+τ)β(tk+τ)u(tk+τ)dτ+γ(tk+1).
Now, assume for the sake of simplicity that the setSI of impulsive time instants is subject to the constraint h≤|tk+1-tk|≤T-; ∀tk∈SI and that the subsequent mixed piece-wise continuous impulsive-free and impulsive control law is used as follows:
(64)u(t)=g(tk)g-(t)x(tk)+g0(tk)g-0(t-h)x(t-h),∀t∈[tk,tk+h),∀tk∈SI,u(t)=g(tk)g-(t)x(tk),∀t∈[tk+h,tk+1),∀tk∈SI,γ(tk+1)=λ(tk+1)x(tk+1-),∀tk∈SI,
so that g-(·) and g-0 are piecewise continuous on R0+ and R0+∪[-h,0], respectively. Then, the solution of the controlled differential equation at the impulsive time instants is(65a)x(tk+1)=(1+λ(tk+1))×[∫0tk+1-tkΨ(tk+1,tk+τ)(∫0tk+1-tkΨ(tk+1,tk+τ)Ψ(tk+1,tk)+g(tk)×∫0tk+1-tkΨ(tk+1,tk+τ)×β(tk+τ)g-(tk+τ)dτ∫0tk+1-tkΨ(tk+1,tk+τ))x(tk)+∫0hΨ(tk+1,tk+τ)×(1+g0(tk)g-0(tk+τ-h)β(tk+τ-h))×x(tk+τ-h)dτ∫0tk+1-tkΨ(tk+1,tk+τ)](65b)=(1+λ(tk+1))ν(tk,tk+1)x(tk),∀tk∈SI,provided that sequences {g(tk)}tk∈SI and {g0(tk)}tk∈SI of the control law (64) are parameterized as follows:
(66)g(tk)=(∫0tk+1-tkΨ(tk+1,tk+τ)ν(tk,tk+1)-Ψ(tk+1,tk))×(∫0tk+1-tkΨ(tk+1,tk+τ)×β(tk+τ)g-(tk+τ)dτ∫0tk+1-tkΨ(tk+1,tk+τ))-1,g0(tk)=-(∫0hΨ(tk+1,tk+τ)x(tk+τ-h)dτ)×(∫0hΨ(tk+1,tk+τ)×(g-0(tk+τ-h)β(tk+τ-h))×x(tk+τ-h)dτ∫0hΨ(tk+1,tk+τ)(g-0(tk+τ-h)β(tk+τ-h))).-1
Thus, Propositions17 and 18 related to Example 15 can be applied to the sequence (65a) and (65b) and the self-mapping which generates it by replacing e∫tktk+1(α(τ)+β(τ)g(τ))dτ→ν(tk,tk+1). For instance, self-mapping which generates the solution sequence at impulsive time instants is asymptotically contractive if limsuptk∈SI(1+λ(tk+1))ν(tk,tk+1)<1, asymptotically nonexpansive if limsuptk∈SI(1+λ(tk+1))ν(tk,tk+1)≤1, and contractive and nonexpansive if the above conditions are replaced with similar ones at each tk∈SI. Similar extensions also apply to the extensions of Example 19 for a 2-semicyclic self-mapping of the delayed impulsive equation.



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