Volume 2011, Article ID 712706,22pages doi:10.1155/2011/712706
Research Article
Quasigauge Spaces with Generalized
Quasipseudodistances and Periodic Points of Dissipative Set-Valued Dynamic Systems
Kazimierz Włodarczyk and Robert Plebaniak
Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University ofŁ´od´z, Banacha 22, 90-238Ł´od´z, Poland
Correspondence should be addressed to Kazimierz Włodarczyk,wlkzxa@math.uni.lodz.pl
Received 13 September 2010; Revised 19 October 2010; Accepted 10 November 2010 Academic Editor: Jen Chih Yao
Copyrightq2011 K. Włodarczyk and R. Plebaniak. 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.
In quasigauge spaces, we introduce the families of generalized quasipseudodistances, and we define three kinds of dissipative set-valued dynamic systems with these families of generalized quasi-pseudodistances and with some families of not necessarily lower semicontinuous entropies and next, assuming that quasigauge spaces are leftKsequentially completebut not necessarily Hausdorff, we prove that for each starting point each dynamic process or generalized sequence of iterations of these dissipative set-valued dynamic systems left converges and we also show that if an iterate of these dissipative set-valued dynamic systems is left quasiclosed, then these limit points are periodic points. Examples illustrating ideas, methods, definitions, and results are constructed.
1. Introduction
The study of quasigauge spaces, initiated by Reilly 1, has a long history. These spaces generalize topological spaces, quasiuniform spaces, and quasimetric spaces. Studies of asymmetric structures in these spaces and their applications to problems in theoretical computer science are important. There exists an extensive literature concerning unsymmetric distances, topological properties, and fixed point theory in these spaces. Some researches tools for many problems in these spaces were provided by Reilly 1, 2, Reilly et al. 3, Kelly4, Subrahmanyam5, Alemany and Romaguera6, Romaguera7, Stoltenberg8, Wilson9, Gregori and Romaguera10, Lee et al.11, Frigon12, and Chis¸-Novac et al.
13. For quasiuniformities over the past 20 years, see also to the Fletcher and Lindgren book 14and to the K ¨unzi surveys15,16.
Recall that a set-valued dynamic systems is defined as a pairX, T, whereXis a certain space andT is a set-valued map T : X → 2X; in particular, a set-valued dynamic system
includes the usual dynamic system, whereT is a single-valued map. Here, 2X denotes the family of all nonempty subsets of a spaceX.
For eachx∈X, a sequencewm:m∈ {0} ∪Nsuch that
∀m∈{0}∪N{wm1∈Twm}, w0x, 1.1
is called a dynamic process or a trajectory starting atw0xof the systemX, T for details see Aubin and Siegel17, Aubin and Ekeland18, and Aubin and Frankowska19. For each x∈X, a sequencewm:m∈ {0} ∪N, such that
∀m∈{0}∪N
wm1∈Tm1x
, w0x, 1.2
Tm T ◦T ◦ · · · ◦T m-times,m ∈ N, is called a generalized sequence of iterations starting atw0 xof the system X, T for details see Yuan20, page 557, Tarafdar and Vyborny 21, and Tarafdar and Yuan22. Each dynamic process starting fromw0 is a generalized sequence of iterations starting fromw0, but the converse may not be true; the setTmw0is, in general, bigger thanTwm−1. IfX, Tis a single-valued, then, for eachx∈X, a sequence wm:m∈ {0} ∪Nsuch that
∀m∈{0}∪N
wm1Tm1x
, w0x, 1.3
is called a Picard iteration starting atw0 xof the systemX, T. IfX, Tis a single valued, then1.1–1.3are identical.
IfX, Tis a dynamic system, then by FixT, PerT, and EndT, we denote the sets of all fixed points, periodic points, and endpoints ofT, respectively, that is, FixT {w∈X:w∈ Tw}, PerT {w∈X:w∈Tqwfor someq∈N}, and EndT {w∈X:{w}Tw}.
Let X be a metric space with metric d, and let X, T be a single-valued dynamic system. Racall that if∃λ∈0,1∀x,y∈X{dTx, Tyλdx, y}, thenX, Tis called a Banach’s contraction Banach 23. X, T is called contractive if ∀x,y∈X{dTx, Ty < dx, y}. If
∃>0∀x,y∈X{0 < dx, y < ⇒ dTx, Ty < dx, y}, then X, T is called-contractive Edelstein 24. Contractive and -contractive maps are some modifications of Banach’s contractions. If
∀x∈X{dx, Txωx−ωTx} 1.4
for someω :X → 0,∞, thenTis called Caristi’s mapCaristi25, Caristi and Kirk26 andωis called entropy. Caristi’s maps generalize Banach’s contractionsfor details see Kirk and Saliga27, page 2766. Recall that Ekeland’s28variational principle concerning lower semicontinuous maps and Caristi’s fixed point theorem Caristi 25 when entropyω is lower semicontinuous are equivalent.
In metric spacesX, d, mapω :X → 0,∞is called a weak entropy or entropy of a set-valued dynamic systemX, Tif
∀x∈X∃y∈Tx d
x, y
ωx−ω
y
1.5
or
∀x∈X∀y∈Tx d
x, y
ωx−ω
y
, 1.6
respectively, andX, Tis called weak dissipative or dissipative if it has a weak entropy or an entropy, respectively; here,ω is not necessarily lower semicontinuous. These two kinds of dissipative maps were introduced and studied by Aubin and Siegel17. IfX, Tis a single valued, then1.4–1.6are identical.
Various results concerning the convergence of Picard iterations and the existence of periodic points, fixed points, and invariant sets for contractive and -contractive single- valued and set-valued dynamic systems in metric spaces have been established by Edelstein 24, Ding and Nadler Jr.29, and Nadler Jr.30. Periodic point theorem for special single- valued dynamic systems of Caristi’s type in quasimetric spaces haS been obtained by ´Ciri´c 31, Theorem 2.
Investigations concerning the existence of fixed points and endpoints and convergence of dynamic processes or generalized sequences of iterations to fixed points or endpoints of single-valued and set-valued dissipative dynamic systems of the types1.4–1.6when entropy ω is not necessarily lower semicontinuous have been conducted by a number of authors in different settings; for example, see Aubin and Siegel17, Kirk and Saliga27, Yuan20, Willems32, Zangwill33, Justman34, Maschler and Peleg35, and Petrus¸el and Sˆınt˘am˘arian36.
In this paper, in quasigauge spaces see Section 2, we introduce the families of generalized quasipseudodistances and define three new kinds of dissipative set-valued dynamic systems with these families of generalized quasipseudodistances and with some families of not necessarily lower semicontinuous entropiessee Section3and next, assuming that quasigauge spaces are leftKsequentially completebut not necessarily Hausdorff, we prove that for each starting point each dynamic process or generalized sequence of iterations of these dissipative set-valued dynamic systems left converges, and we also show that if some iterates of these dissipative set-valued dynamic systems are left quasiclosed, then these limit points are periodic pointssee Section4. Examples are includedsee Section5.
The presented methods and results are different from those given in the literature and are new even for single-valued and set-valued dynamic systems in topological, quasiuniform, and quasimetric spaces.
This paper is a continuation of37–41.
2. Quasigauge Spaces
The following terminologies will be much used.
Definition 2.1. LetXbe a nonempty set. A quasipseudometric onXis a mapp:X×X → 0,∞ such that
P1∀x∈X{px, x 0},
P2∀x,y,z∈X{px, zpx, y py, z}.
If, additionally,
P3∀x,y∈X{px, y 0⇒xy}, thenpis called quasimetric onX.
Definition 2.2. LetXbe a nonempty set.
iEach familyP{pα:α∈ A}of quasipseudometricspα:X×X → 0,∞,α∈ Ais called a quasigauge onX.
iiLet the familyP{pα:α∈ A}be a quasigauge onX. The topologyTPhaving as a subbase the familyBP {Bx, εα:x∈X, εα>0, α∈ A}of all ballsBx, εα
{y∈X:pαx, y< εα},x∈X,εα >0,α∈ Ais called the topology induced byPon X.
iii Dugundji 42, Reilly 1, 2 A topological space X,T such that there is a quasigaugePonX withT TPis called a quasigauge space and is denoted by X,P.
Theorem 2.3see Reilly1, Theorem 2.6. Any topological space is a quasigauge space.
Definition 2.4. LetXbe a nonempty set.
iA quasiuniformity onXis a filterUonX×Xsuch that U1∀U∈U{ΔXU},
U2∀U∈U∃V∈U{V2 U}.
Here,ΔX {x, x:x∈X}denotes the diagonal ofX×Xand, for eachM⊂X×X, M2{x, y∈X×X :∃z∈X{x, z∈M∧z, y∈M}}. The elements ofUare called entouragesor vicinities.
iiA subfamilyBofUis called a base of the quasiuniformityUonXif∀U∈U∃V∈B{V ⊂ U}.
iiiThe topology TU onX induced by the quasiuniformityU on X is {A ⊆ X :
∀x∈A∃U∈U{Ux A}}; hereUx {y ∈ X : x, y ∈ U}wheneverU ∈ Uand x∈X. A neighborhood base for each pointx∈Xis given by{Ux:U∈ U}.
ivIfUis a quasiuniformity onX, then the pairX,Uis called a quasiuniform space.
Theorem 2.5see Reilly1, Theorem 4.2. Any quasiuniform space is a quasigauge space.
Definition 2.6. LetX,Pbe a quasigauge space.
i Reilly et al.3, Definition 1vand page 129We say that a sequencewm:m∈N inXis left-P,K-Cauchy sequence inXif
∀α∈A∀ε>0∃k∈N∀m,n∈N;kmn
pαwm, wn< ε
. 2.1
ii Reilly et al.3, Definition 1iand page 129We say that a sequencewm:m∈N inXis leftP-Cauchy sequence inXif
∀α∈A∀ε>0∃w∈X∃k∈N∀m∈N;km
pαw, wm< ε
. 2.2
iiiWe say that a sequencewm:m∈NinXis left convergent inXif
∃w∈X∀α∈A∀ε>0∃k∈N∀m∈N;km
pαw, wm< ε
. 2.3
iv Reilly 1, Definition 5.3 and 2, Definition 4 If every left- P, K- Cauchy sequence in X is left convergent to some point inX, then X,P is called left K sequentially complete quasigauge space.
v Reilly1, Definition 5.3and2, Definition 4If every leftP-Cauchy sequence in Xis left convergent to some point inX, thenX,Pis called left sequentially complete quasigauge space.
Remark 2.7. LetX,Pbe a quasigauge space.
a Reilly 2, page 131Every left-P, K-Cauchy sequence inX is left P-Cauchy sequence inX.
b Reilly2, Example 1, Reilly et al.3, Example 2, and Kelly4, Example 5.8Every left convergent sequence inX is leftP-Cauchy sequence inX and the converse is false.
c Reilly et al. 3, Section 3Every left sequentially complete quasigauge space is leftKsequentially complete quasigauge space.
3. Three Kinds of Dissipative Set-Valued Dynamic Systems in Quasigauge Spaces with Generalized Quasipseudodistances
First, we introduce the concepts of JP-family of generalized quasipseudodistances in quasigauge space X,P and left- JP, K- Cauchy sequences in quasigauge space X,P withJP-family of generalized quasipseudodistances.
Definition 3.1. LetX,Pbe a quasigauge space.
iThe familyJ {Jα :α∈ A}of mapsJα :X×X → 0,∞, α∈ A, is said to be a JP-family onXif the following two conditions hold:
J1∀α∈A∀x,y,z∈X{Jαx, zJαx, y Jαy, z}, J2for any sequencewm:m∈NinXsuch that
∀α∈A∀ε>0∃k∈N∀m,n∈N;kmn{Jαwm, wn< ε}, 3.1
if there exists a sequencevm:m∈NinXsatisfying
∀α∈A∀ε>0∃k∈N∀m∈N;km{Jαwm, vm< ε}, 3.2
then
∀α∈A∀ε>0∃k∈N∀m∈N;km
pαwm, vm< ε
. 3.3
iiThe elements ofJP-family onXare called generalized quasipseudodistances onX. iiiLet the family J {Jα : α ∈ A} be a JP-family onX. We say that a sequence
wm:m∈NinXis left-JP,K-Cauchy sequence inXif3.1holds.
Remark 3.2. LetXbe a nonempty set.
aIf X,P is a quasigauge space, J {Jα : α ∈ A} is a JP-family on X and
∀α∈A∀x∈X{Jαx, x 0}, then, for eachα∈ A,Jαis quasipseudometric.
bEach quasigaugePonXisJP-family onXand the converse is falsesee Section5, e.g., in Example5.1II, ifx /∈E, then∀α∈A{Jαx, x cα>0}.
Now, we introduce the following three kinds of dissipative set-valued dynamic systems in quasigauge spaces with generalized quasipseudodistances.
Definition 3.3. Let X,P be a quasigauge space, and let X, T be a set-valued dynamic system. LetJ {Jα : α ∈ A},Jα : X×X → 0,∞,α ∈ Abe aJP-family onX, and let Γ {γα:α∈ A},γα:X → 0,∞,α∈ Abe a family of maps.
iWe say that a sequencewm:m∈ {0} ∪NinXisJ,Γadmissible if
∀α∈A∀m∈{0}∪N
Jαwm, wm1γαwm−γαwm1
. 3.4
iiIf the following two conditions hold:
C1∅/X0⊂X,
C2x ∈ X0 if and only if there exists aJ,Γ-admissible dynamic processwm : m ∈ {0} ∪Nstarting atw0 xof the systemX, T, then we say thatT is a weakJ,Γ;X0dissipative onX.
iiiWe say that T isJ,Γ-dissipative onX if, for eachx ∈ X, each dynamic process wm:m∈ {0} ∪Nstarting atw0xof the systemX, TisJ,Γ-admissible.
ivWe say thatTis a strictlyJ,Γ dissipative onXif, for eachx∈X, each generalized sequence of iterationswm :m∈ {0} ∪Nstarting atw0 xof the systemX, Tis J,Γadmissible.
If one from the conditionsii–ivholds, then we say thatX, Tis a dissipative set- valued dynamic system with respect toJ,Γ dissipative set-valued dynamic system, for shortand elements of the familyΓwe call entropies onX.
Remark 3.4. LetX,Pbe a quasigauge space, and letX, Tbe a set-valued dynamic system.
aIf a sequencewm :m∈ {0} ∪NinX isJ,Γadmissible, then, for eachk ∈N, a sequencewmk:m∈ {0} ∪NisJ,Γadmissible.
bBya, ifT is a weakJ,Γ;X0dissipative on X,x ∈ X0 andwm :m ∈ {0} ∪N is aJ,Γ-admissible dynamic process starting atw0 xof the systemX, T, then
∀m∈N{wm∈X0}.
cIfX, Tis a single-valued dynamic system, theniiiandivare identical.
Proposition 3.5. LetX,Pbe a quasigauge space, and letX, Tbe a set-valued dynamic system.
aIfT is a weakJ,Γ;X0dissipative onX, thenX0,KJ;Tis a set-valued dynamic system where, for eachx∈X0,
KJ;Tx
{{w0, w1, w2, . . .}:wm:m∈ {0} ∪N∈ KJT, x}, 3.5 KJT, x {wm:m∈ {0} ∪N:w0x
∧∀m∈{0}∪N
wm1∈Twm∧ ∀α∈A
Jαwm, wm1γαwm−γαwm1 . 3.6
bIfT isJ,Γdissipative onX, thenX,WJ;Tis a set-valued dynamic system where, for eachx∈X,
WJ;Tx
{{w0, w1, w2, . . .}:wm:m∈ {0} ∪N∈ WJT, x}, 3.7 WJT, x
wm:m∈ {0} ∪N:w0 x∧ ∀m∈{0}∪N{wm1∈Twm}
. 3.8
cIf T is a strictly J,Γ dissipative on X, thenX,SJ;Tis a set-valued dynamic system where, for eachx∈X,
SJ;Tx
{{w0, w1, w2, . . .}:wm:m∈ {0} ∪N∈ SJT, x}, 3.9 SJT, x
wm:m∈ {0} ∪N:w0x∧ ∀m∈{0}∪N
wm1∈Tm1w0
. 3.10
Proof. The fact thatKJ;T :X0 → 2X0,WJ;T :X → 2X, andSJ;T :X → 2X follows from1.1, 1.2, Definition3.3, Remark3.4, and3.5–3.10.
Remark 3.6. By Definition3.3and Proposition3.5, we obtain the following.
aIfTisJ,Γdissipative onX, thenTis a weakJ,Γ;X0dissipative onXforX0X and∀x∈X0{KJ;Tx WJ;Tx}.
bIf T is strictly J,Γ dissipative on X, then T is J,Γ dissipative on X and
∀x∈X{WJ;Tx⊂SJ;Tx}.
4. Convergence of Dynamic Processes and Generalized Sequences of Iterations and Periodic Points of Dissipative Set-Valued Dynamic Systems in Quasigauge Spaces with Generalized Quasipseudodistances
We first recall the definition of closed maps in topological spaces given in Berge43and Klein and Thompson44.
Definition 4.1. Let L be a topological vector space. The set-valued dynamic system X, T is called closed if whenever wm : m ∈ Nis a sequence in X converging to w ∈ X and vm:m∈Nis a sequence inXsatisfying the condition∀m∈N{vm ∈Twm}and converging tov∈X, thenv∈Tw.
By Definition2.6iii, we are able to revise the above definition, and we define left quasiclosed maps and left quasiclosed sets in quasigauge spaces as follows.
Definition 4.2. LetX,Pbe a leftKsequentially complete quasigauge space.
iThe set-valued dynamic systemX, T is called left quasiclosed if wheneverwm : m ∈ Nis a sequence in X left converging to each point of the set W ⊂ X and vm:m∈Nis a sequence inXsatisfying the condition∀m∈N{vm∈Twm}and left converging to each point of the setV ⊂X, then∃v∈V∀w∈W{v∈Tw}.
iiFor an arbitrary subsetEofX, the left quasi-closure ofE, denoted by clLE, is defined as the set
clLE
w∈X:∃wm:m∈N⊂E∀α∈A∀ε>0∃k∈N∀m∈N;km
pαw, wm< ε
. 4.1
iiiThe subsetEofXis said to be left quasiclosed subset inXif clLE E.
Remark 4.3. LetX,Pbe a leftK sequentially complete quasigauge space. For each subset E ofX, E ⊂ clLE. Indeed, by Definition 4.2iiand P1, for each w ∈ E, the sequence wm:m∈N, where∀m∈N{wmw}, is left convergent tow.
Now we are ready to prove the following main result of this paper.
Theorem 4.4. LetX,Pbe a leftKsequentially complete quasigauge space, and letX, Tbe a set- valued dynamic system. LetJ{Jα :α∈ A},Jα :X×X → 0,∞,α∈ Abe aJP-family onX and letΓ {γα:α∈ A},γα:X → 0,∞,α∈ Abe a family of maps. The following hold.
A A1IfT is weakJ,Γ;X0dissipative onX, then, for eachx∈X0and for each dynamic processwm :m∈ {0} ∪N∈ KJT, x, there exists a nonempty setW ⊂ clLX0such that, for eachw∈W,wm:m∈ {0} ∪Nis left convergent tow.
A2If, in addition, the mapTqis left quasiclosed inXfor someq∈N, then there exists w∈Wsuch thatw∈Tqw.
B B1If T isJ,Γdissipative onX, then, for eachx ∈ X and for each dynamic process wm:m∈ {0} ∪N∈ WJT, x, there exists a nonempty setW ⊂ Xsuch that, for each w∈W,wm:m∈ {0} ∪Nis left convergent tow.
B2If, in addition, the mapTqis left quasiclosed inX for someq∈N, then there exists w∈Wsuch thatw∈Tqw.
C C1IfT is strictlyJ,Γdissipative onX, then, for eachx∈Xand for each generalized sequence of iterationswm:m∈ {0} ∪N∈ SJT, x, there exists a nonempty setW⊂X such that, for eachw∈W,wm:m∈ {0} ∪Nis left convergent tow.
C2If, in addition, the mapTqis left quasiclosed in X for someq ∈ N, then, for each x∈X, there exists a generalized sequence of iterationswm:m∈ {0} ∪N∈ SJT, x, a nonempty setW⊂Xandw∈Wsuch thatwm:m∈ {0} ∪Nis left convergent to each points ofWandw∈Tqw.
Proof. The proof will be broken into five steps.
Step 1. Letix ∈ X0 andwm : m ∈ {0} ∪N ∈ KJT, xor ii x ∈ X and wm : m ∈ {0} ∪N ∈ WJT, x∪ SJT, x. We show thatwm : m ∈ {0} ∪Nis left-JP, K-Cauchy sequence inX, that is,
∀α∈A∀ε>0∃k∈N∀m,n∈N;kmn{Jαwm, wn< ε}. 4.2 Indeed, by 3.6, 3.8, 3.10, Definition 3.3iii and iv, and definition of J,
∀α∈A∀m∈{0}∪N{γαwm1 γαwm}. According to the fact that ∀α∈A∀x∈X{γαx 0}, we conclude that, for eachα∈ A, the sequenceγαwm:m∈ {0} ∪Nis bounded from below and nonincreasing. Hence, we have
∀α∈A∃uα0 lim
m→ ∞γαwm−uα0
. 4.3
Let nowα0 ∈ Aandε0>0 be arbitrary and fixed. By4.3,
∃n0∈N∀m;n0mγα0wm−uα0< ε0
2
. 4.4
Furthermore, for n0 m n, using J1 and 3.4, we obtain 0 Jα0wm, wn n−1
kmJα0wk, wk1 γα0wm−γα0wn and next, by 4.4, we have that Jα0wm, wn γα0wm−γα0wn |γα0wm−uα0 −γα0wn uα0| |γα0wm−uα0||γα0wn−uα0| <
ε0/2ε0/2ε0. Therefore,4.2holds.
Step 2. Let ix ∈ X0 andwm : m ∈ {0} ∪N ∈ KJT, x orii x ∈ X andwm : m ∈ {0} ∪N ∈ WJT, x∪ SJT, x. We show thatwm : m ∈ {0} ∪Nis left-P,K-Cauchy sequence inX, that is,
∀α∈A∀ε>0∃k∈N∀m,n∈N;kmn
pαwm, wn< ε
. 4.5
Indeed, by4.2,∀α∈A∀ε>0∃k∈N∀mk∀l∈{0}∪N{Jαwm, wlm< ε}. Hence, ifi0∈ {0} ∪Nis arbitrary and fixed and if we define a sequencevm :m∈Nasvm wi0mform∈N, then we obtain
∀α∈A∀ε>0∃k∈N∀mk{Jαwm, vm< ε}. 4.6
ByJ2,4.2, and4.6,
∀α∈A∀ε>0∃k∈N∀mk
pαwm, vm< ε
. 4.7
Consequence of4.7and the definition ofvm:m∈Nis
∀α∈A∀ε>0∃k∈N∀mk
pαwm, wi0m< ε
. 4.8
Now, letα0∈ A,ε0>0 be arbitrary and fixed. By4.2,
∃n1∈N∀mn1∀l∈{0}∪N{Jα0wm, wlm< ε0}. 4.9
From4.8, we get
∃n2∈N∀mn2∀i∈{0}∪N
pα0wm, wim< ε0
. 4.10
Letn0max{n1, n2}1. Hence, ifn0mn, thenni0mfor somei0∈ {0}∪N. Therefore, by4.9and4.10,pα0wm, wn pα0wm, wi0m< ε0. The proof of4.5is complete.
Step 3. Let ix ∈ X0 andwm : m ∈ {0} ∪N ∈ KJT, x oriix ∈ X and wm : m ∈ {0} ∪N∈ WJT, x∪ SJT, x. We show thatwm:m∈ {0} ∪Nis leftP-Cauchy sequence in X, that is,
∀α∈A∀ε>0∃w∈X∃k∈N∀m∈N;km
pαw, wm< ε
. 4.11
Indeed, by Remark2.7a, property4.11is a consequence of Step2.
Step 4. Assertions ofAandBhold.
Indeed, letix∈X0andwm :m∈ {0} ∪N∈ KJT, xoriix∈Xandwm:m∈ {0} ∪N∈ WJT, x.
Since∀m∈{0}∪N{wm ∈ KJ;Tx}or ∀m∈{0}∪N{wm ∈ WJ;Tx},X is leftK sequentially complete quasigauge space and 4.5holds; therefore, by Definition 2.6iv, we claim that there exists a nonempty set W ⊂ clLKJ;Tx or W ⊂ clLWJ;Tx, respectively, where KJ;Tx⊂X0, clLKJ;Tx⊂ clLX0,WJ;Tx ⊂X, and clLX X, such that the sequence wm:m∈ {0} ∪Nis left convergent to each pointwofW.
Now, we see that ifTq is left quasiclosed for someq ∈ N, then there exists a point w∈W such thatw∈Tqw. Indeed, by1.1, we conclude that
∀m∈N
wm∈Twm−1⊂T2wm−2⊂ · · · ⊂Tm−1w1⊂Tmw0
, 4.12
which gives
wmqk∈Tq
wm−1qk
fork1,2, . . . , q, m∈N. 4.13 It is clear that, for eachk 1,2, . . . , q, the sequenceswmqk :m ∈ {0} ∪Nandwm−1qk : m ∈ {0} ∪N, as subsequences of wm : m ∈ {0} ∪ N, also left converge to each point ofW. Further, sinceTq is left quasiclosed inX, by 4.13and Definition4.2i, we obtain
∃v∈VW∀w∈W{v∈Tqw}, which gives∃w∈W{w∈Tqw}.
Step 5. Assertion ofC1holds.
Indeed, let x ∈ X and wm : m ∈ {0} ∪N ∈ SJT, x be arbitrary and fixed. By Step2and Proposition3.5c, we claim that∀m∈{0}∪N{wm ∈ Tmx ⊂ SJ;Tx ⊂ X},wm :
m ∈ {0} ∪N is left P, K-Cauchy sequence in left K-sequentially complete quasigauge spaceX, and, by Definition2.6iv, there exists a nonempty setW ⊂ clLSJ;Txsuch that the sequence wm : m ∈ {0} ∪Nis left convergent to each pointw ofW. This gives that assertion ofC1holds.
Step 6. Assertion ofC2holds.
Initially, we will prove that if, for someq∈N,Tqis left quasiclosed inX, then, for each x∈X, we may construct a generalized sequence of iterationswm:m∈ {0} ∪N∈ SJT, x which converge to each pointwof some nonempty setW ⊂ clLSJ;Tx, and the following property holds:∃w∈W{w∈Tqw}.
Indeed, letx ∈ X andk ∈ {1,2, . . . , q}be arbitrary and fixed. First, we construct a sequenceumqk : m ∈ {0} ∪N as follows. Form 0, we define uk as an arbitrary and fixed point satisfyinguk∈Tkw0. Then, we have thatTquk⊂Tqkw0and, form1, we defineuqkas an arbitrary and fixed point satisfyinguqk ∈Tquk ⊂Tqkw0. Then, we have thatTquqk ⊂ T2qkw0and, for m 2, we defineu2qk as an arbitrary and fixed point satisfying u2qk ∈ Tquqk ⊂ T2qkw0. In general, for each m ∈ {0} ∪N, if we defineum−1qk satisfyingum−1qk ∈ Tqum−2qk ⊂ Tqm−1kw0, then we have thatTqum−1qk⊂Tmqkw0and defineumqkas an arbitrary and fixed point satisfying umqk∈Tqum−1qk⊂Tmqkw0.
Consequently, for arbitrary and fixed x ∈ X and k ∈ {1,2, . . . , q}, there exists a sequenceumqk:m∈ {0} ∪Nsatisfying
uk∈Tkw0, 4.14
umqk∈Tq
um−1qk
⊂Tmqkw0, m∈N. 4.15
Let nowwm:m∈ {0} ∪Nbe an arbitrary and fixed sequence satisfying1.2and the condition
wmqkumqk, m∈ {0} ∪N. 4.16
Obviously, by Step5,wm : m ∈ {0} ∪Nis left convergent to each point w of some set W⊂clLSJ;Tx. Moreover,umqk:m∈ {0} ∪Nandum−1qk:m∈N, as subsequences of wm:m∈ {0} ∪N, also converge to each pointwof some setW. Hence, using4.15,4.16, assumption inC2, and Definition4.2i, we get that∃v∈VW∀w∈W{v∈Tqw}, which gives
∃w∈W{w∈Tqw}.
Remark 4.5. IfX, Tis a single-valued dynamic system, then Theorems4.4Band4.4Care identical.
5. Examples
In this section, we present some examples illustrating the concepts introduced so far.
In Example5.1, we define twoJP-families in quasigauge spaces.
Example 5.1. Let the familyP {pα :α ∈ A}of quasipseudometricspα :X×X → 0,∞, α∈ Abe a quasigauge onXand letX,Pbe a quasigauge space.
IThe familyPis aJP-family onXsee Remark3.2b.
IILetXcontain at least two different points. Let the setE⊂Xcontaining at least two different points be arbitrary and fixed, and let {cα}α∈A satisfy∀α∈A{δαE < cα}, where∀α∈A{δαE sup{pαx, y : x, y ∈ E}}. Let the familyJ {Jα : α ∈ A}, Jα:X×X → 0,∞,α∈ Abe defined by the following formula:
Jα
x, y
⎧⎨
⎩ pα
x, y
if E∩ x, y
x, y cα if E∩
x, y /
x, y
, x, y∈X. 5.1
We show that the familyJis aJP-family onX.
Indeed, we see that the conditionJ1does not hold only if there exist some α∈ A andx, y, z∈X such thatJαx, y cα,Jαx, z pαx, z,Jαz, y pαz, y, andpαx, z pαz, y < cα. However, then we conclude that there exists v ∈ {x, y} such thatv /∈Eand x, y, z∈E, which is impossible. Therefore,∀α∈A∀x,y,z∈X{Jαx, yJαx, z Jαz, y}, that is, the conditionJ1holds.
For proving thatJ2holds, we assume that the sequences{wm}and{vm}inXsatisfy 3.1and3.2. Then, in particular,3.2yields
∀α∈A∀0<ε<cα∃m0m0α∈N∀mm0{Jαwm, vm< ε}. 5.2
By5.2and5.1, denotingmmin{m0α:α∈ A}, we conclude that
∀mm{E∩ {wm, vm}{wm, vm}}. 5.3
From5.3, definition ofJ, and5.2, we get
∀α∈A∀0<ε<cα∃m∈N∀mm
pαwm, vm Jαwm, vm< ε
. 5.4
The result is that the sequences {wm}and {vm}satisfy3.3. Therefore, the propertyJ2 holds.
The following example illustrates Theorem4.4Ain not Hausdorffquasigauge space.
Example 5.2. LetX 0,1/2∪ {3/4,1} ⊂R, and letT:X → 2Xbe of the form
Tx
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
{1} forx0 0,1/2x forx∈0,1/2 {0,1} forx3/4 {0,3/4} forx1.
5.5
Let the mapp:X×X → 0,∞be defined by the formula
p x, y
⎧⎨
⎩
0 ifxy, 1 ifx < y,
x, y
∈X×X, 5.6
and letP {p}; this mappis a modification of a mappdue to Reilly et al.3, Example 1.
For a setE 0,1/2, let the familyJ{J:X×X → 0,∞}be defined as follows
J x, y
⎧⎨
⎩ p
x, y
ifE∩ x, y
x, y
,
2 ifE∩
x, y /
x, y
, x, y∈X, 5.7
and letΓ {γ}, whereγ :X → 0,∞is of the formγx x,x∈X.
aThe mapp:X×X → 0,∞ is a quasipseudometric onX.
Indeed, we have thatpx, x 0 sincexx, and thusP1holds. AlsoP2is satisfied since we obtain the following. 1 If x z, thenpx, z 0 px, y py, z for each y ∈X.2Ifx < z, thenpx, z 1, and suppose thatpx, y py, z 0 for somey ∈X, thenx y andy zwhich impliesx z. This is absurd becausex < z. Consequently,
∀y∈X{px, y 1∨py, z 1}.
bQuasigauge spaceX,P is a left sequentially complete.
Indeed, letwm :m∈Nbe leftP-Cauchy sequence inX i.e., let2.2hold, and let η0, 0< η0<1, be arbitrary and fixed. Then, by2.2, we get
∃w0∈X∃k0∈N∀mk0
pw0, wm< η0 <1
. 5.8
Now, if ε0 > 0 is arbitrary and fixed, then, by 5.6 and 5.8, for each m k0, pw0, wm 0< ε0.
Hence, we conclude that∃w0∈X∀ε>0∃k∈N∀mk{pw0, wm< ε}. This gives thatwm:m∈ Nleft converges tow0.
cQuasigauge spaceX,Pis a leftKsequentially complete.
This follows fromband Remark2.7c.
dThe familyJ{J :X×X → 0,∞} is aJP-family onX.
This is the consequence of Example 5.1II. Let us observe, additionally, that clLE X.
eT is weakJ,Γ;X0dissipative onX, whereX0 0,1/2.
Indeed, letx∈0,1/2be arbitrary and fixed. We have thatT2x 0,1/4x∪ {1}
andTmx 0,1/2mx∪ {3/4,1}form 3. Thus, there exists a dynamic processwm : m ∈ {0} ∪Ngiven by the formulaw0 x,wm 1/2mx,m ∈ N, such thatJw0, w1
px,1/2x 0 x−x/2 γw0−γw1and∀m∈N{Jwm, wm1 pwm, wm1 0 γwm−γwm1}. This gives that the dynamic processwm:m∈ {0} ∪Nsatisfies1.1and 3.4. Consequently,x∈X0. Therefore, we proved that0,1/2⊂X0.
Now, we show thatX0⊂0,1/2. Suppose thatX0∩ {X\0,1/2}/∅then there exists x∈X0such thatx /∈0,1/2, and we consider the following three cases.
Case 1. Ifx0, then each dynamic processwm:m∈ {0}∪Nstarting atw00 of the system X, Tsatisfiesw1 1 Tw0and 0 < Jw0, w1 2 >−1 0−1 γw0−γw1, that is, 3.4does not hold.
Case 2. Letx 3/4 and letwm : m ∈ {0} ∪Nbe an arbitrary and fixed dynamic process starting at w0 3/4 of the systemX, T. Ifw1 1, then 0 < Jw0, w1 2 > 3/4−1 γw0−γw1, that is,3.4does not hold. Ifw1 0, thenw2 1 and 0 < Jw1, w2 2 >
0−1γw1−γw2, that is,3.4does not hold.
Case 3. Letx 1 and letwm : m ∈ {0} ∪N be an arbitrary and fixed dynamic process starting atw0 1 of the systemX, T. Ifw10, thenw21 and 0< Jw1, w2 2>0−1 γw1−γw2, that is,3.4does not hold. Ifw1 3/4, then 0 < Jw0, w1 2 >1−3/4 1/4γw0−γw1, that is,3.4does not hold.
Consequently,X0 0,1/2.
fWe have that clLX0 0,1/2∪ {3/4,1}.
First, we show thatX\ {0} ⊂clLX0, that is, by Definition4.2ii, for eachw∈X\ {0}, there exists a sequencewm:m∈N⊂X0which left converges tow. Indeed, ifw∈X\ {0}is arbitrary and fixed, then for the sequence defined byw1wandwmcmwform2, where 0 < c < 1/2 is arbitrary and fixed, we get that∃k∈N∀mk{wm < w}andwm :m ∈N⊂ X0. Consequently, for arbitrary and fixedε > 0, by 5.6, we have ∀mk{pw, wm 0 < ε}.
Therefore,wm:m∈Nleft converges tow. By Definition4.2ii,w∈clLX0. This gives that X\ {0} ⊂clLX0.
Now, we show that clLX0 ⊂ X \ {0}, that is, 0/∈clLX0. Otherwise, 0 ∈ clLX0, and thus there exists a sequence wm : m ∈ N ⊂ X0 which left converges to 0, that is,
∀ε>0∃k∈N∀m∈N;km{p0, wm< ε}. This is absurd because∀m∈N{0 < wm}which, by5.6, gives
∀m∈N{p0, wm 1}.
gTheorem4.41)holds.
Indeed, by the considerations ine, ifx ∈ 0,1/2 X0, thenKJT, x/∅, and if wm :m∈ {0} ∪N ∈ KJT, xis arbitrary and fixed, then, by Cases1–3ine,∀m∈N{wm ∈ 0,1/2mx}. Therefore, fromf,wm :m∈ {0} ∪Nleft converges to each pointw ∈W clLX0.
hThe mapT is not left quasiclosed inX.
Indeed, ifwm:m∈N⊂Xis such that∀m∈N{wm 0}, thenW 0,1/2∪ {3/4,1}.
Next, we see that if vm : m ∈ Nsatisfies∀m∈N{vm ∈ Twm}, then ∀m∈N{vm 1} which givesV {1}. Consequently,∃v∈V∀w∈W{v ∈Tw}does not hold becausew 1 ∈ Wand V∩T1 {1} ∩ {0,3/4}∅.
iThe mapT2 is not left quasiclosed inX.
Indeed, we have that
T2x
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
{0,3/4} forx0 0,1/4x∪ {1} forx∈0,1/2 {0,3/4,1} forx3/4 {0,1} forx1.
5.9
Thus, ifwm :m∈Nis such that∀m∈N{wm 0}, thenW 0,1/2∪ {3/4,1}, and ifvm : m ∈ Nsatisfying∀m∈N{vm ∈ T2wm}is of the form∀m∈N{vm 3/4}, thenV {3/4,1}.
Consequently,∃v∈V∀w∈W{v ∈ T2w}does not hold becauseIif v 3/4 ∈ V, then, for w1∈W, we have{3/4} ∩T21 {3/4} ∩ {0,1}∅,IIifv1∈V, then, forw0∈W, we have{1} ∩T20 {1} ∩ {0,3/4}∅.
jThe mapT3 is left quasiclosed inX.
Indeed, we have that
T3x
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
{0,1} forx0
0,1/8x∪ {3/4,1} forx∈0,1/2 {0,3/4,1} forx3/4 {0,3/4,1} forx1,
5.10
and ifwm :m∈Nis an arbitrary and fixed sequence inX,W is a set of all left limit points of wm : m ∈ N,vm : m ∈ N is an arbitrary and fixed sequence satisfying∀m∈N{vm ∈ T3wm}, andV is a set of all left limit points ofvm : m ∈N, then we see that∀x∈X{1 ∈ T3x}and 1∈V. Consequently,∃v1∈V∀w∈W{v∈T3w}.
kThe assumptions of Theorem4.4(A2) hold forq3.
This is the consequence of h–j. The assertion is that {w ∈ W ⊂ clLX0 : w ∈ Tqw}{3/4,1} ⊂FixT3, that is,{3/4,1} ⊂PerTforq3.
lThe mapT is notJ,Γdissipative onX and not strictlyJ,Γdissipative onX.
This is the consequence of Cases1–3ine.
mFor anyΓ,T is notJ,Γdissipative onX.
Indeed, suppose that there existsΓ {γ} such that γ : X → 0,∞and that T is J,Γdissipative onX. Then, for a dynamic processwm:m∈ {0} ∪Nstarting atw03/4 defined byw1 0 ∈ Tw0,w2 1 ∈ Tw1,w3m1 0 ∈ Tw3m,w3m2 1 ∈ Tw3m1, andw3m 3/4 ∈ Tw3m−1form ∈ N, we have 0 < Jw0, w1 2 γw0−γw1, 0 <
Jw1, w2 2 γw1−γw2, and 0 < Jw2, w3 2 γw2−γw3 γw2−γw0. Hence,γw0< γw2< γw1< γw0, which is impossible.
nFor anyΓ,T is not strictlyJ,Γdissipative onX.
This is the consequence ofmand Remark3.6b.
oQuasigauge spaceX,Pis not Hausdorff.
Indeed, forx0,y0 ∈Xsuch thatx0 > y0, there existsz0 ∈X such thatz0 y0. Then, for eachε,η >0, by5.6, we have thatpx0, z0 0< εandpy0, z0 0< η, which implies thatz0∈Bx0, ε∩By0, η.
The following example illustrates Theorems4.4Band4.4Cin quasimetric space.
Example 5.3. LetXN⊂R, letT :X → Xbe defined by
Tx
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
1 ifx1
x−1 ifx2k, k∈N x−2 ifx2k1, k∈N,
5.11
and letp:X×X → 0,∞be defined by the formulae
pm, n 0 ifmn, 5.12
pm, n m−1 if m < nandmis even andnis odd, 5.13
pm, n 1 otherwise; 5.14
this mappis a modification of a mappdue to Reilly et al.3, Example 5.
LetP {p}, letJ {J}whereJm, n pm, nform, n∈N, and letΓ {γ}where γ:X → 0,∞is of the form
γx x, x∈X. 5.15
a The mappis quasimetric onX.
This is the consequence of the following useful observations.
Case 1. Ifmn, thenpm, n 0pm, k pk, nfor eachk∈X.
Case 2. Ifm > nandk ∈X, thenpm, n pm, k pk, n. Indeed, by5.14,pm, n 1.
On the other hand,k nmeans thatk < m. By5.14,k < mimpliespm, k 1 andn < k impliespk, n 1. Hence, for eachk∈X, 1pm, k pk, n.
Case 3. Ifm < nandk < morn < k, thenpm, npm, k pk, n. Indeed, by5.12–5.14, pm, n1. Next, fork < morn < k, we have 1pm, k pk, nsince, by5.14,k < m impliespm, k 1 andn < kimpliespk, n 1.
Case 4. Ifm < nandm k n, thenpm, n pm, k pk, nsince the following five propertiesIV1–IV2are satisfied.
IV1Letkmorknthenpm, n pm, k pk, n.
IV2Letmbe even, let nbe odd and letm < k < nthen, by5.12,pm, n m−1. If k is odd, then, by 5.13,pm, k m−1 and, by 5.14,pk, n 1. Ifk is even, then, by 5.13,pk, n k−1 and, by5.14,pm, k 1. Consequently,pm, n <
pm, k pk, n.
IV3Letmandnbe even, and letm < k < n. Then, by5.14,pm, n 1. Ifkis odd, then, by5.13,pm, k m−1and, by5.14,pk, n 1. Ifkis even, then, by5.14, pk, n pm, k 1. Consequently,pm, n< pm, k pk, n.
IV4Letmandnbe odd, and letm < k < n. Then, by5.14,pm, n 1. Ifkis odd, then, by5.14,pm, k pk, n 1. Ifkis even, then, by5.13,pk, n k−1and, by5.14,pm, k 1. Consequently,pm, n< pm, k pk, n.
IV5Letmbe odd, letnbe even, and letm < k < n. Then, by5.14,pm, n pm, k pk, n 1.Consequently,pm, n< pm, k pk, n.
bThe familyJ is aJP-family.
This follows from Example5.1I.
c X,P is a left K sequentially complete quasimetric space.
Indeed, we see that only sequenceswm:m∈NinXsatisfying
∃k∈N∃w∈N∀mk{wmw} 5.16
are left-P,K-Cauchy sequences inX, that is, satisfy 2.1. Further, each sequence wm : m ∈ NinX satisfying5.16 is left convergent inX, that is, satisfies2.3. Consequently, X,Pis a leftKsequentially complete.
d X,Pis not a left sequentially complete quasimetric space.
Indeed, using5.13, we see that the sequencewm : m∈ NinX of the formwm 2m−1,m∈N, satisfies∀ε>0∃w2m0∈X∀m∈N{pw, wm 1/2m0< ε}, and thus2m−1 :m∈N is leftP-Cauchy sequence inXi.e., satisfies2.2.
Now, suppose that for this sequence the condition 2.3 holds, that is, that
∃w∈N∀ε>0∃k∈N∀m∈N;km{pw, wm< ε}. It is clear that then∃s∈N∀ms{w < wm}. Hence, since, for eachm∈N,wmis odd, using5.13and5.14, we obtain that
∃w∈N∀ε>0∀ms
ε > pw, wm
⎧⎨
⎩
1/w ifw is even
1 ifw is odd, 5.17
which is impossible.
eThe mapT is strictlyJ,Γ dissipative onX.
Indeed, letx∈Xbe arbitrary and fixed, and consider the following three cases.
Case 1. Ifx1, then there exists a unique generalized sequence of iterationswm:m∈ {0} ∪ Nstarting atw0xof the systemX, T, given by the formulawm1,m∈ {0}∪N. Hence, by 5.12and5.15, we have∀m∈{0}∪N{Jwm, wm1 pwm, wm1 0γwm−γwm1 0}.
Therefore, the sequence wm : m ∈ {0} ∪Nsatisfies1.2and 3.4 and left converges to w1.
Case 2. Ifx2k, wherek ∈N, then there exists a unique generalized sequence of iterations wm :m∈ {0} ∪Nstarting atw0 xof the systemX, Tdefined by the formulaw0 2k, wm 2k−2m−1 Tmw0form∈ {1,2,3, . . . , k−1}, andwm1Tmw0formk.
Hence, by5.14 and5.15, we get thatJw0, w1 p2k,2k−1 1 2k−2k−1 γw0−γw1and∀m∈{1,...,k−1}{Jwm, wm1 pwm, wm1 1 2wm−wm1 γwm− γwm1} and, by5.12and 5.15, we get that ∀mk{Jwm, wm1 pwm, wm1 0 γwm−γwm1}. Therefore, the sequencewm:m∈ {0} ∪Nsatisfies1.2and3.4and left converges tow1.
Case 3. Ifx2k1, wherek∈N, then there exists a unique generalized sequence of iterations wm :m∈ {0} ∪Nstarting atw0xof the systemX, Tdefined by the formulawm 2k− 2m−1 Tmw0form∈ {0,1,2,3, . . . , k−1}andwm1Tmw0formk. Hence, by
5.14and5.15, we get thatJw0, w1 p2k1,2k−1 12k1−2k−1 γw0−γw1 and∀m∈{1,...,k−1}{Jwm, wm1 pwm, wm1 12 wm−wm1 γwm−γwm1}and, by5.12and5.15, we get that∀mk{Jwm, wm1 pwm, wm1 0γwm−γwm1}.
Therefore, the sequence wm : m ∈ {0} ∪Nsatisfies1.2and 3.4 and left converges to w1.
The above implies thatTis strictlyJ,Γdissipative onX.
fThe assertion of Theorem4.4(C1) holds.
From considerations ine, it follows that, for eachx ∈X, a generalized sequence of iterationswm : m ∈ {0} ∪Nstarting atw0 xof the systemX, Tsatisfies{wm : m ∈ {0} ∪N}SJT, xand left converges tow∈W {1} ⊂X.
gThe mapT is left quasiclosed inX.
Indeed, in X only sequences wm : m ∈ N satisfying the condition
∃k∈N∃w∈X∀m;km{wm w}are left convergent to w and W {w}. Further, if ∀m∈N{vm Twm}, then a sequencevm:m∈Nleft converges tovTwandV {v}where
V
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
{1} ifw1
{w−1} ifw2k, k∈N {w−2} ifw2k1, k∈N.
5.18
By Definition4.2i, the mapT is left quasiclosed inX.
hThe assertion of Theorem4.4(C2) holds.
Indeed, bye–g, for each x ∈ X, there exists a generalized sequence of iterations wm : m ∈ {0} ∪Nstarting atw0 xof the systemX, Tsatisfying∃k∈N∀m;km{wm 1}
and{wm :m∈ {0} ∪N}SJT, x, left converging tow ∈W {1}, for which{vm :m∈ {0} ∪N}, where∀m∈N{vm Twm}satisfies∀m;km{vm 1}, left converges tov∈V {1}, andw1 is the fixed point ofT.
From a–h, it follows that Theorem 4.4C also Theorem 4.4B by Remark 4.5 holds.
The following example shows that in Theorem4.4the assumptions inA2,B2, and C2 i.e., the assumptions that the mapTq is left quasiclosed in X for someq ∈ Nare essential.
Example 5.4. LetX 0,1, and letP{p}wherepx, y |x−y|,x, y∈X. Then, the family J{J}, whereJ p, is aJP-family. LetT :X → 2Xbe of the form
Tx
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
{1} if x0 0,1/2x if x∈0,1 {1} if x1,
5.19
