JOPERJournal of Operators2314-50722314-5064Hindawi Publishing Corporation10.1155/2014/741818741818Research ArticleNew Results in the Startpoint Theory for Quasipseudometric SpacesGabaYaé UlrichJeribiAref^{1}Department of Mathematics and Applied MathematicsUniversity of Cape TownRondebosch, Cape Town 7701South Africauct.ac.za

Dedicated to Professor Guy A. Degla for his mentorship

We give two generalizations of Theorem 35 proved by Gaba (2014). More precisely, we change the structure of the contractive condition; namely, we introduce a function Φ instead of a simple constant c.

1. Introduction and Preliminaries

In [1], we introduced the concept of startpoint and endpoint for set-valued mappings defined on quasipseudometric spaces. As mentioned there, the purpose of this theory is to study fixed point like related properties. In the present, we give more results from the theory. More precisely, we generalize Theorem 35 of [1] by changing the structure of the contractive condition; namely, we introduce a function Φ instead of a simple constant c (as it appears in the original statement). This new condition is interesting in the sense that it allows us to have a condition involving a functional of the variables and not just the variables themselves. For the convenience of the reader, we will recall some necessary definitions but for a detailed exposé of the definition and examples, the interested reader is referred to [1].

Definition 1.

Let X be a nonempty set. A function d:X×X→[0,∞) is called quasipseudometric on X if

d(x,x)=0∀x∈X;

d(x,z)≤d(x,y)+d(y,z)∀x,y,z∈X.

Moreover, if d(x,y)=0=d(y,x)⇒x=y, then d is said to be a T0-quasipseudometric. The latter condition is referred to as the T0-condition.

Remark 2.

(i) Let d be quasipseudometric on X; then the map d-1 defined by d-1(x,y)=d(y,x) whenever x,y∈X is also a quasipseudometric on X, called the conjugate of d. In the literature, d-1 is also denoted as dt or d¯.

(ii) It is easy to verify that the function ds defined by ds:=d∨d-1, that is, ds(x,y)=max{d(x,y),d(y,x)}, defines a metric on X whenever d is a T0-quasipseudometric on X.

The quasipseudemetric d induces a topology τ(d) on X.

Definition 3.

Let (X,d) be a quasipseudometric space. The d-convergence of a sequence (xn) to a point x, also called left-convergence and denoted by xn→dx, is defined in the following way:
(1)xn→dx⟺dx,xn⟶0.

Similarly, we define the d-1-convergence of a sequence (xn) to a point x or right convergence and denote it by xn→d-1x, in the following way:
(2)xn→d-1x⟺d(xn,x)⟶0.

Finally, in a quasipseudometric space (X,d), we will say that a sequence (xn) ds-converges to x if it is both left and right convergent to x, and we denote it as xn→dsx or xn→x when there is no confusion. Hence
(3)xn→dsx⟺xn→dx,xn→d-1x.

Definition 4.

A sequence (xn) in quasipseudometric (X,d) is called

left d-Cauchy if, for every ϵ>0, there exist x∈X and n0∈N such that
(4)∀n≥n0d(x,xn)<ϵ;

left K-Cauchy if, for every ϵ>0, there exists n0∈N such that
(5)∀n,k:n0≤k≤nd(xk,xn)<ϵ;

ds-Cauchy if, for every ϵ>0, there exists n0∈N such that
(6)∀n,k≥n0d(xn,xk)<ϵ.

Dually, we define right d-Cauchy and right K-Cauchy sequences.

Definition 5.

A quasipseudometric space (X,d) is called

left K-complete provided that any left K-Cauchy sequence is d-convergent,

left Smyth sequentially complete if any left K-Cauchy sequence is ds-convergent.

Definition 6.

A T0-quasipseudometric space (X,d) is called bicomplete provided that the metric ds on X is complete.

As usual, a subset A of a quasipseudometric space (X,d) will be called bounded provided that there exists a positive real constant M such that d(x,y)<M whenever x,y∈A.

Let (X,d) be a quasipseudometric space. We set P0(X):=2X∖{∅} where 2X denotes the power set of X. For x∈X and A,B∈P0(X), we define
(7)dx,A=infdx,a,a∈A,d(A,x)=inf{d(a,x),a∈A},
and H(A,B) by
(8)H(A,B)=maxsupa∈Ad(a,B),supb∈Bd(A,b).

Then H is an extended quasipseudometric on P0(X). Moreover, we know from [2] that the restriction of H to Scl(X)={A⊆X:A=(clτ(d)A)∩(clτ(d-1)A)} is an extended T0-quasipseudometric. We will denote by CB(X) the collection of all nonempty bounded and ds-closed subsets of X.

Let (X,d) be a quasipseudometric space. For every fixed x∈X, the mapping y↦d(x,y) is τ(d) upper semicontinuous (τ(d)-usc in short) and τ(d-1) lower semicontinuous (τ(d-1)-lsc in short). For every fixed y∈X, the mapping x↦d(x,y) is τ(d)-lsc and τ(d-1)-usc.

2. Main Results

We commence this section with the main result of this paper.

Theorem 9.

Let (X,d) be a left K-complete quasipseudometric space. Let T:X→CB(X) be a set-valued map and define f:X→R as f(x)=H({x},Tx). Let Φ:[0,∞)→[0,1) be a function such that limsupr→t+Φ(r)<1 for each t∈[0,∞). Moreover, assume that for any x∈X there exists y∈Tx satisfying
(9)H({y},Ty)≤ΦHx,yHx,y,
and then T has a startpoint.

Proof.

First observe that, since Φ(H({x},{y}))<1 for any x,y∈X, it follows that 2-Φ(H({x},{y}))>1 for any x,y∈X, and hence
(10)H({x},{y})≤2-ΦHx,yH({x},Tx),
for any x∈X and y∈Tx.

For any initial x0∈X, there exists (for all actually) x1∈Tx0⊆X such that
(11)Hx0,x1≤2-ΦHx0,x1Hx0,Tx0.

From (9) we get
(12)Hx1,Tx1≤Φ(H({x0},{x1}))H({x0},{x1})≤Φ(H({x0},{x1}))×[2-Φ(H({x0},{x1}))]H({x0},Tx0)=Ψ(H({x0},{x1}))H({x0},Tx0),
where Ψ:[0,∞)→[0,1) is defined by
(13)Ψ(t)=Φ(t)(2-Φ(t)).

Now choosing x2∈Fx1⊆X, we have that
(14)H({x1},{x2})≤[2-Φ(H({x1},{x2}))]H({x1},Tx1),
and from (9) we get
(15)H({x2},Tx2)≤Ψ(H({x1},{x2}))H({x1},Tx1).

Continuing this process, we obtain a sequence (xn) where xn+1∈Txn⊆X, with
(16)Hxn,xn+1≤[2-Φ(H({xn},{xn+1}))]H({xn},Txn),(17)Hxn+1,Txn+1≤ΨHxn,xn+1Hxn,Txn,n=1,2,….

For simplicity, denote dn:=H({xn},{xn+1}) and Dn:=H({xn},Txn) for all n≥0. So from (17) we can write
(18)Dn+1≤ΨdnDn≤Dn,
for all n≥0. Hence (Dn) is a strictly decreasing sequence and hence there exists δ≥0 such that
(19)limn→∞Dn=δ.

From (16), it is easy to see that
(20)dn<2Dn.

Thus the sequence (dn) is bounded and so there is d≥0 such that limsupn→∞dn=d and hence a subsequence (dnk) of (dn) such that limk→∞dnk=d+. From (17) we have Dnk+1≤Ψ(dnk)Dnk and thus
(21)δ=limsupk→∞Dnk+1≤limsupk→∞Ψdnklimsupk→∞Dnk≤limsupdnk→d+Ψ(dnk)δ.

This together with the fact limsupr→t+Φ(r)<1 for each t∈[0,∞) implies that δ=0. Then from (19) and (20) we derive that limn→∞dn=0.

Claim 1. (xn) is a left K-Cauchy sequence.

Now let α:=limsupdn→0+Ψ(dn) and q such that α<q<1. This choice of q is always possible since α<1. Then there is n0 such that Ψ(dn)<q for all n≥n0. So from (18) we have Dn+1≤qDn for all n≥n0. Then by induction we get Dn≤qn-n0Dn0 for all n≥n0+1. Combining this and inequality (20) we get
(22)∑k=n0mdxk,xk+1≤2∑k=n0mqk-n0Dn0≤211-qDn0,
for all m>n≥n0+1. Hence (xn) is a left K-Cauchy sequence.

According to the left K-completeness of (X,d), there exists x*∈X such that xn→dx*.

Claim 2. x* is a startpoint of T.

Observe that the sequence Dn=(fxn)=(H({xn},Txn)) converges to 0. Since f is τd-lower semicontinuous (as supremum of τ(d)-lower semicontinuous functions), we have
(23)0≤f(x*)≤liminfn→∞f(xn)=0.

Hence f(x*)=0; that is, H({x*},Tx*)=0.

This completes the proof.

We give below an example to illustrate the theorem.

Example 10.

Let X=[0,1] and d:X×X→R be the mapping defined by d(a,b)=max{a-b,0}. Then d is a T0-quasipseudometric on X. Observe that any left K-Cauchy sequence in (X,d) is d-convergent to 0. Indeed, if (xn) is a left K-Cauchy sequence, for every ϵ>0, there exists n0∈N such that
(24)∀n,k:n0≤k≤nd(xk,xn)<ϵ.

This entails that ∀n:n0<n(25)d(0,xn)≤d(0,xn-1)+d(xn-1,xn)=0+d(xn-1,xn)<ϵ.

Hence d(0,xn)→0; that is, xn→d0. Therefore (X,d) is left K-complete. Let T:X→CB(X) be such that
(26)T(x)=12x2,forx∈0,1532∪1532,1,1796,14,forx=1532.

Let Φ:[0,∞)→[0,3/4)⊂[0,1) be defined by
(27)Φ(r)=32r,forr∈0,724∪724,12425768,forr=72412,forr∈12,∞.

An explicit computation of f(x)=H({x},Tx) gives
(28)f(x)=x-12x2,forx∈0,1532∪1532,1,2896,forx=1532.

Moreover, for each x∈0,15/32∪15/32,1, y=1/2x2 and we have
(29)H({y},Ty)≤Φ(H({x},{y}))H({x},{y}).

Of course inequality (29) also holds in the case of x=15/32 and y=17/96. Therefore, all assumptions of Theorem 9 are satisfied and the endpoint of T is x=0.

Remark 11.

In fact, every sequence in (X,d)d-converges to 0.

Corollary 12.

Let (X,d) be a right K-complete quasipseudometric space. Let T:X→CB(X) be a set-valued map and define f:X→R as f(x)=H(Tx,{x}). Let Φ:[0,∞)→[0,1) be a function such that limsupr→t+Φ(r)<1 for each t∈[0,∞). Moreover, assume that for any x∈X there exists y∈Tx satisfying
(30)H(Ty,{y})≤Φ(H({y},{x}))H({y},{x}),
and then T has an endpoint.

Corollary 13.

Let (X,d) be a bicomplete quasipseudometric space. Let T:X→CB(X) be a set-valued map and f:X→R defined by f(x)=Hs(Tx,{x})=max{H(Tx,{x}),H({x},Tx)}. If there exists c∈(0,1) such that for all x∈X there exists y∈Fx satisfying
(31)Hs({y},Fy)≤min{Φ(a)a,Φ(b)b},
where a=H({y},{x}) and b=H({x},{y}), then T has a fixed point.

Proof.

We give here the main idea of the proof. Observe that inequality (31) guarantees that the sequence (xn) constructed in the proof of Theorem 9 is a ds-Cauchy sequence and hence ds-converges to some x*. Using the fact that f is τ(ds)-lower semicontinuous (as supremum of τ(ds)-continuous functions), we have
(32)0≤f(x*)≤liminfn→∞f(xn)=0.

Hence f(x*)=0; that is, H({x*},Tx*)=0=H(Tx*,{x*}), and we are done.

The following theorem is the second generalization that we propose.

Theorem 14.

Let (X,d) be a left K-complete quasipseudometric space. Let T:X→CB(X) be a set-valued map and define f:X→R as f(x)=H({x},Tx). Let b:[0,∞)→[a,1),a>0 be a nondecreasing function. Let Φ:[0,∞)→[0,1) be a function such that Φ(t)<b(t) for each t∈[0,∞) and limsupr→t+Φ(r)<limsupr→t+b(r) for each t∈[0,∞). Moreover, assume that for any x∈X there exists y∈Tx satisfying
(33)H({y},Ty)≤Φ(H({x},{y}))H({x},{y}),
and then T has a startpoint.

Proof.

Observe that because b(H({x},{y}))<1 for all x,y∈X,
(34)bHx,yHx,y≤Hx,Tx,
for any y∈Tx. Let x0∈X be arbitrary. Then we can choose x1∈Tx0 such that
(35)bHx0,x1Hx0,x1≤Hx0,Tx0,(36)H({x1},Tx1)≤Φ(H({x0},{x1}))H({x0},{x1}).

Define the function Ψ:[0,∞)→[0,1) by Ψ(t)=Φ(t)/b(t). Hence (35) and (36) together sum to
(37)H({x1},Tx1)≤Ψ(H({x0},{x1}))H({x0},Tx0).

Now we choose x2∈Tx1 such that
(38)bHx1,x2Hx1,x2≤Hx1,Tx1,Hx2,Tx2≤ΦHx1,x2Hx1,x2,
which lead to
(39)Hx2,Tx2≤ΨHx1,x2Hx1,Tx1.

Continuing this process, we get an iterative sequence (xn) where xn+1∈Txn⊆X and, denoting dn=H({xn},{xn+1}) and Dn=H({xn},Txn) for all n≥0, we can write that
(40)bdndn≤Dn,Dn+1≤Φdndn,
for all n≥0. Hence
(41)Dn+1≤ΨdnDn.

If Dk=0 for some k≥0, then we trivially have limn→∞Dn=0 and the conclusion is immediate. So without loss of generality, we can assume that Dn>0 for all n≥0 and, from (41), we have Dn+1<Dn for all n≥0.

Observe also that if, for some n, dn≤dn+1, we are led to a contradiction. Indeed from (40) and using the fact that the function b is nondecreasing, we have
(42)dn≤dn+1≤Φ(dn)b(dn+1)dn≤Φ(dn)b(dn)dn=Ψ(dn)dn<dn.

Hence dn+1<dn for all n≥0. Thus there exist δ≥0 and d≥0 such that limn→∞Dn=δ and limn→∞dn=d. From (41), we get
(43)δ≤(limsupn→∞Ψ(dn))δ=limsupdn→d+Ψ(dn)δ,
and hence δ=0 (since limsupr→t+Φ(r)<1 for all t∈[0,∞)). Moreover, since
(44)adn≤bdndn≤Dn≤Ψdn-1Dn-1,
this forces d to be 0; that is, limn→∞dn=0.

Furthermore, setting α:=limsupdn→0+Ψ(dn) and letting q>0 be a positive number such that α<q<1, there is n0 such that Ψ(dn)<q for all n≥n0. Hence from (41) and (44), we get
(45)adn≤Ψdn-1Dn-1≤qDn-1≤⋯≤qn-n0Dn0,
for all n≥n0. So dn≤1/aqn-n0Dn0 for all n≥n0. Using a similar argument as the one used in the proof of Theorem 9, we conclude that (xn) is a left K-Cauchy sequence and that its limit point is a starpoint of T.

Example 15.

Let X=[0,1) and d:X×X→R be the mapping defined by d(a,b)=max{a-b,0}. Then d is a T0-quasipseudometric on X. We know that (X,d) is left K-complete. Let T:X→CB(X) be such that
(46)Tx=12x2.

Let Φ:[0,∞)→[0,3/4)⊂[0,1) be defined by
(47)Φ(r)=32r,forr∈0,120,forr∈12,∞.

Let b:[0,∞)→[2/3,1) be defined by
(48)b(t)=56t,fort∈0,1212,fort∈12,∞.

An explicit computation of f(x)=H({x},Tx) gives
(49)fx=x-12x2,forx∈0,1.

Moreover, for each x∈0,1, there exists y=1/2x2 and we have
(50)Hy,Ty≤32Hx,yHx,y=Φ(H({x},{y}))H({x},{y}).

Therefore, all assumptions of Theorem 14 are satisfied and the endpoint of T is x=0.

Corollary 16.

Let (X,d) be a right K-complete quasipseudometric space. Let T:X→CB(X) be a set-valued map and define f:X→R as f(x)=H-1({x},Tx). Let b:[0,∞)→[a,1),a>0 be a nondecreasing function. Let Φ:[0,∞)→[0,1) be a function such that Φ(t)<b(t) for each t∈[0,∞) and limsupr→t+Φ(r)<limsupr→t+b(r) for each t∈[0,∞). Moreover, assume that for any x∈X there exists y∈Tx satisfying
(51)H-1({y},Ty)≤Φ(H-1({x},{y}))H-1({x},{y}),Φ(t)<b(t), and limsupr→t+Φ(r)<limsupr→t+b(r) for each t∈[0,∞). Moreover, assume that for any x∈X there exists y∈Tx satisfying and then T has an endpoint.

Corollary 17.

Let (X,d) be a bicomplete quasipseudometric space. Let T:X→CB(X) be a set-valued map and define f:X→R as f(x)=Hs({x},Tx). Let b:[0,∞)→[a,1), a>0 be a nondecreasing function. Let Φ:[0,∞)→[0,1) be a function such that Φ(t)<b(t) and limsupr→t+Φ(r)<limsupr→t+b(r) for each t∈[0,∞). Moreover, assume that for any x∈X there exists y∈Tx satisfying
(52)Hsy,Ty≤minΦaa,Φbb,
where a=H({x},{y}) and b=H-1({x},{y}). Then T has a fixed point.

Remark 18.

All the results given remain true when we replace accordingly the bicomplete quasipseudometric space (X,d) by a left Smyth sequentially complete/left K-complete or a right Smyth sequentially complete/right K-complete space.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

GabaY. U.Startpoints and α,γ-contractions in quasi-pseudometric spacesKünziH.-P.RyserC.The Bourbaki quasi-uniformity