# Fixed Point Theorems for Contractions of Rational Type with PPF Dependence in Banach Spaces

Authors: J. Rocha; B. Rzepka; K. Sadarangani
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We prove the existence of the PPF dependent fixed point in the Razumikhin class for contractions of rational type in Banach spaces, by using a general class of pairs of functions. Our result has as particular cases a great number of interesting consequences which extend and generalize some results appearing in the literature.


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## Body

## 1. Introduction

Banach’s contraction principle is one of the pivotal results of analysis. Its significance lies in its vast applicability to a great number of branches of mathematics and other sciences, for example, theory of existence of solutions for nonlinear differential, integral, and functional equations, variational inequalities, and optimization and approximation theory.Generalizations of the contractive mapping theorem have been a heavily investigated branch of research. In particular, this principle was extended in [1], where the domain of the nonlinear operator involved is C([a,b],E) and the range is E, where E is a Banach space. This result is known as the contraction theorem for operator with PPF (past, present, and future) dependence. The PPF fixed point theorems are useful for proving the existence of solutions for nonlinear functional-differential and integral equations which may depend upon the past history, present data, and future considerations. Some papers about fixed point theorems with PPF dependence have appeared in the literature (see, e.g., [1–5]).On the other hand, Dass and Gupta in [6] and Jaggi in [7] were the pioneers in proving fixed point theorems using contractive conditions involving rational expressions. In [4], the authors present a fixed point theorem for contractions of rational type with PPF dependence.The purpose of this paper is to present a fixed point theorem for generalized contractions of rational type with PPF dependence which has, as particular cases, interesting consequences. Particularly, our result extends the one appearing in [4].


## 2. Preliminaries

Throughout this paper,E will denote a Banach space with norm ∥·∥E and E0=C([a,b],E) will denote the space of the continuous E-valued functions defined on [a,b] and equipped with the norm ∥·∥E0 given by
(1)∥ϕ∥E0=sup⁡t∈[a,b]∥ϕ(t)∥Eforϕ∈E0.LetT:E0→E be a mapping. A point ϕ∈E0 is said to be a PPF dependence fixed point of  T or a fixed point with PPF dependence of  T if  Tϕ=ϕ(c), for some c∈[a,b].For a fixedc∈[a,b], the Razumikhin class Rc is defined by
(2)Rc={ϕ∈E0:∥ϕ∥E0=∥ϕ(c)∥E}.Remark 1.
Notice that, forx∈E fixed, the function ϕx, defined by
(3)ϕx(t)=xfort∈[a,b],
satisfies ϕx∈E0, ∥ϕx∥E0=∥ϕx(c)∥E=∥x∥ for any c∈[a,b] and, therefore, ϕx∈Rc for any c∈[a,b]. Consequently, Rc≠∅ for any c∈[a,b].We say that the classRc is algebraically closed with respect to difference if for any  ϕ,ξ∈Rc we have  ϕ-ξ∈Rc. Similarly, we say that the class Rc is topologically closed if it is closed with respect to the topology on E0 induced by the norm ∥·∥E0.The Razumikhin class plays an important role in the existence of PPF fixed point.The first result about the existence of PPF fixed point appears in [1] and it is presented in the following theorem.Theorem 2 (see [1]).
Suppose thatT:E0→E is a mapping such that there exists α∈[0,1) satisfying
(4)∥Tϕ-Tξ∥E≤α∥ϕ-ξ∥E0
for any  ϕ,ξ∈E0. If  Rc is topologically closed and algebraically closed with respect to difference for some c∈[a,b], then T has a unique PPF dependent fixed point in Rc.Recently, in [4] the authors proved the following PPF dependent fixed point theorem for rational type contraction mappings.Theorem 3 (see [4]).
LetT:E0→E be a mapping satisfying
(5)∥Tϕ-Tξ∥E≤α∥ϕ-ξ∥E0+β∥ϕ(c)-Tϕ∥E∥ξ(c)-Tξ∥E1+∥Tϕ-Tξ∥E
for any ϕ,ξ∈E0 and where α,β∈[0,1) with α+β<1 and c∈[a,b]. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.The main purpose of this paper is, by using a class of pairs of functions satisfying certain assumptions, to present new PPF dependent fixed point theorems for contractions of rational type. Particularly, our result generalizes the main result of  [4] (Theorem 3).


## 3. Main Results

We start this section presenting the following class of pairs of functionsF. A pair of functions (φ,ϕ) is said to belong to the class F if they satisfy the following conditions:(i)
φ,ϕ:[0,∞)→[0,∞);(ii)
fort,s∈[0,∞), if φ(t)≤ϕ(s), then t≤s;(iii)
for(tn) and (sn) sequences in [0,∞) such that
(6)lim⁡n→∞tn=lim⁡n→∞sn=a,ifφ(tn)≤ϕ(sn) for any n∈N, then a=0.Remark 4.
Notice that if(φ,ϕ)∈F and φ(t)≤ϕ(t), then t=0, since we can take tn=sn=t for any n∈N and by (iii) we deduce t=0.In the sequel, we present some interesting examples of pairs of functions belonging to the classF which will be very important in our study.Example 5.
Letφ:[0,∞)→[0,∞) be a continuous and increasing function such that φ(t)=0 if and only if t=0 (these functions are known in the literature as altering distance functions).
Letϕ:[0,∞)→[0,∞) be a nondecreasing function such that ϕ(t)=0 if and only if  t=0 and suppose that ϕ≤φ.
Then the pair(φ,φ-ϕ)∈F.
In fact, it is clear that(φ,φ-ϕ) satisfy (i).
To prove (ii), suppose thatt,s∈[0,∞) and φ(t)≤(φ-ϕ)(s). Then, from
(7)φ(t)≤φ(s)-ϕ(s)≤φ(s)
and taking into account the increasing character of φ, we can deduce that t≤s.
In order to prove (iii), we suppose that(8)φ(tn)≤φ(sn)-ϕ(sn)≤φ(sn)foranyn∈N,
where tn,sn∈[0,∞) and
(9)lim⁡n→∞tn=lim⁡n→∞sn=a.
Taking n→∞ in (8), we infer that lim⁡n→∞ϕ(sn)=0.
Let us suppose thata>0. Since lim⁡n→∞sn=a>0, we can find ε>0 and a subsequence (snk) of (sn) such that snk>ε for any k∈N. As ϕ is nondecreasing, we have ϕ(snk)>ϕ(ε) for any k∈N and, consequently, lim⁡k→∞ϕ(snk)≥ϕ(ε). This contradicts the fact that lim⁡k→∞ϕ(snk)=0. Therefore, a=0.
This proves that(φ,φ-ϕ)∈F.An interesting particular case is whenφ is the identity mapping, φ=1[0,∞), and ϕ:[0,∞)→[0,∞) is a nondecreasing function such that ϕ(t)=0 if and only if  t=0 and ϕ(t)≤t for any t∈[0,∞).Example 6.
LetS be the class of functions defined by
(10)S={α:[0,∞)⟶[0,1):{α(tn)⟶1⟹tn⟶0}}.
Let us consider the pairs of functions (1[0,∞),α1[0,∞)), where α∈S and α1[0,∞) is defined by (α1[0,∞))(t)=α(t)t, for t∈[0,∞).
Then(1[0,∞),α1[0,∞))∈F.
It is clear that the pairs(1[0,∞),α1[0,∞)) with α∈S satisfy (i).
To prove (ii), from(11)1[0,∞)(t)≤(α1[0,∞)A11)(s)fort,s∈[0,∞)
we infer, since α:[0,∞)→[0,1), that
(12)t≤α(s)s<s
and, consequently, (1[0,∞),α1[0,∞)) satisfies (ii).
In order to prove (iii), we suppose that(13)1[0,∞)(tn)=tn≤(α1[0,∞)A11)(sn)=α(sn)snforanyn∈N,
where tn,sn∈[0,∞) and lim⁡n→∞tn=lim⁡n→∞sn=a.
Let us suppose thata>0.
Sincelim⁡n→∞sn=a>0, we can find a subsequence (snk) such that snk>0 for any k∈N. Now, as
(14)tn≤α(sn)sn≤snforanyn∈N,
in particular, we have
(15)tnk≤α(snk)snk≤snkforanyk∈N,
and, since snk>0 for any k∈N,
(16)tnksnk≤α(snk)≤1.
Taking k→∞ in the last inequality, we obtain
(17)lim⁡k→∞α(snk)=1.
Finally, since α∈S, we infer that lim⁡k→∞snk=0 and this contradicts the fact that lim⁡n→∞sn=a>0.
Therefore,a=0.
This proves that(1[0,∞),α1[0,∞))∈F for α∈S.Remark 7.
Suppose thatg:[0,∞)→[0,∞) is an increasing function and (φ,ϕ)∈F. Then it is easily seen that the pair (g∘φ,g∘ϕ)∈F.Now, we are ready to present our main result.Theorem 8.
LetT:E0→E be a mapping such that there exists a pair of functions (φ,ϕ)∈F and c∈[a,b] such that
(18)φ(A11∥Tξ-Tη∥E)≤max⁡{(∥ξ(c)-Tξ∥E·∥η(c)-Tη∥E1+∥Tξ-Tη∥E)ϕ(∥ξ-η∥E0),lllllllllllllllllϕ(∥ξ(c)-Tξ∥E·∥η(c)-Tη∥E1+∥Tξ-Tη∥E)}
for any ξ,η∈E0. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.Proof.
Letξ0 be an arbitrary function in Rc (whose existence is guaranteed by Remark 1). Since ξ0∈Rc⊂E0, put x1=Tξ0∈E.
Again, by Remark1, we can find ξ1∈Rc such that
(19)Tξ0=x1=ξ1(c).
Since ξ1∈Rc⊂E0, put x2=Tξ1∈E. Using the same argument, we can find ξ2∈Rc such that
(20)Tξ1=x2=ξ2(c).
Repeating this process, we can obtain a sequence (ξn) in Rc such that
(21)Tξn-1=ξn(c)foranyn∈N.
Since Rc is algebraically closed with respect to difference, we have
(22)∥ξp-ξq∥E0=∥ξp(c)-ξq(c)∥E
for any p,q∈N.
First, we will prove thatlim⁡n→∞∥ξn+1-ξn∥E0=0. In fact, taking into account (21) and (22), we get
(23)∥ξn+1-ξn∥E0=∥ξn+1(c)-ξn(c)∥E=∥Tξn-Tξn-1∥Eforanyn∈N
and, therefore, applying the contractive condition, we have
(24)φ(∥ξn+1-ξn∥E0)=φ(∥Tξn-Tξn-1∥EA11)≤max⁡{(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E)ϕ(∥ξn-ξn-1∥E0),llllllllllllllllllϕ(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E)}.
Let us suppose that there existsn0∈N such that ∥ξn0+1-ξn0∥E0=0.
In this case,ξn0+1=ξn0 and, consequently, ξn0+1(c)=ξn0(c).
By (21), we have
(25)Tξn0=ξn0+1(c)=ξn0(c)
and ξn0 would be the PPF dependent fixed point.
In the sequel we suppose that∥ξn+1-ξn∥E0≠0 for any n∈N.
Now, we can distinguish two cases.
Case  1. Consider
(26)max⁡{(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E)ϕ(∥ξn-ξn-1∥E0),lllllllllϕ(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E)}=ϕ(∥ξn-ξn-1∥E0).
In this case, from (24), we infer
(27)φ(∥ξn+1-ξn∥E0)≤ϕ(∥ξn-ξn-1∥E0),
and, since (φ,ϕ)∈F, we deduce
(28)∥ξn+1-ξn∥E0≤∥ξn-ξn-1∥E0.Case  2. Consider
(29)max⁡{(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E)ϕ(∥ξn-ξn-1∥E0),lllllllllϕ(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E)}=ϕ(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E).
In this case, from (24) and since (φ,ϕ)∈F, we infer
(30)∥ξn+1-ξn∥E0≤∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E.
By (21) and (22), we have
(31)∥ξn+1-ξn∥E0≤∥ξn(c)-ξn+1(c)∥E·∥ξn-1(c)-ξn(c)∥E1+∥ξn+1(c)-ξn(c)∥E=∥ξn+1-ξn∥E0·∥ξn-ξn-1∥E01+∥ξn+1-ξn∥E0.
Since ∥ξn+1-ξn∥E0≠0, from the last inequality, it follows that
(32)1≤∥ξn-ξn-1∥E01+∥ξn+1-ξn∥E0
and, therefore,
(33)∥ξn+1-ξn∥E0<1+∥ξn+1-ξn∥E0≤∥ξn-ξn-1∥E0.
In both cases, we obtain that inequality (28) is satisfied and, consequently, the sequence (∥ξn+1-ξn∥E0) is a decreasing sequence of nonnegative real numbers.
Putr=lim⁡n→∞∥ξn+1-ξn∥E0, where r≥0, and denote
(34)A={n∈N:nsatisfies(26)},B={n∈N:nsatisfies(29)}.
We remark the following.(1)
IfCardA=∞, then from (24) we can find infinitely many natural numbers n satisfying inequality (27) and, since lim⁡n→∞∥ξn+1-ξn∥E0=lim⁡n→∞∥ξn-ξn-1∥E0=r and (φ,ϕ)∈F, we deduce that r=0.(2)
IfCardB=∞, then from (24) we can find infinitely many n∈N such that
(35)φ(∥ξn+1-ξn∥E0)≤ϕ(∥ξn(c)-Tξn∥E·∥ξn-1(c)-Tξn-1∥E1+∥Tξn-Tξn-1∥E).Since(φ,ϕ)∈F and using a similar argument to the one used in Case 2, we obtain
(36)∥ξn+1-ξn∥E0≤∥ξn+1-ξn∥E0·∥ξn-ξn-1∥E01+∥ξn+1-ξn∥E0for infinitely manyn∈N. Taking n→∞ in the last inequality and taking into account that lim⁡n→∞∥ξn+1-ξn∥E0=r we deduce
(37)r≤r21+r,and, consequently,r≤0. Since r≥0, we obtain r=0.
Therefore,(38)lim⁡n→∞∥ξn+1-ξn∥E0=0.
Next, we will prove that(ξn) is a Cauchy sequence in E0. In contrary case, since lim⁡n→∞∥ξn+1-ξn∥E0=0, by Lemma 2.1 of  [8], we can find ε>0 and subsequences (ξn(k)) and (ξm(k)) of (ξn) satisfying
(39)(i)n(k)>m(k)≥kfork>0,(ii)ε≤∥ξn(k)-ξm(k)∥E0,∥ξn(k)-1-ξm(k)∥E0<εlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllfork>0,(iii)lim⁡k→∞∥ξn(k)-ξm(k)∥E0=lim⁡k→∞∥ξn(k)-ξm(k)+1∥E0(iii)lim⁡k→∞∥ξn(k)-ξm(k)∥E0=lim⁡k→∞∥ξn(k)+1-ξm(k)∥E0(iii)lim⁡k→∞∥ξn(k)-ξm(k)∥E0=lim⁡k→∞∥ξn(k)+1-ξm(k)+1∥E0=ε.
Since ξn(k)+1,ξm(k)+1∈Rc for any k∈N, from (21) and (22), we have
(40)∥ξn(k)+1-ξm(k)+1∥E0=∥ξn(k)+1(c)-ξm(k)+1(c)∥E=∥Tξn(k)-Tξm(k)∥E
for any k∈N.
Using the contractive condition and (21) and (22), we obtain
(41)φ(∥ξn(k)+1-ξm(k)+1∥E0)=φ(∥Tξn(k)-Tξm(k)∥E)≤max⁡{(∥ξn(k)(c)-Tξn(k)∥E·∥ξm(k)(c)-Tξm(k)∥E1+∥Tξn(k)-Tξm(k)∥E)ϕ(∥ξn(k)-ξm(k)∥E0),lllllllllllllllllϕ(∥ξn(k)(c)-Tξn(k)∥E·∥ξm(k)(c)-Tξm(k)∥E1+∥Tξn(k)-Tξm(k)∥E)}=max⁡{×(1+∥ξn(k)+1(c)-ξm(k)+1(c)∥E)-1)ϕ(∥ξn(k)-ξm(k)∥E0),lllllllllllllllllϕ((1+∥ξn(k)+1(c)-ξm(k)+1(c)∥E)-1(∥ξn(k)(c)-ξn(k)+1(c)∥Elllllllllllllllllllllll·∥ξm(k)(c)-ξm(k)+1(c)∥E)llllllllllllllllllll×(1+∥ξn(k)+1(c)-ξm(k)+1(c)∥E)-1)}=max⁡{(1+∥ξn(k)+1-ξm(k)+1∥E0)-1)ϕ(∥ξn(k)-ξm(k)∥E0),lllllllllllllllllϕ((1+∥ξn(k)+1-ξm(k)+1∥E0)-1(∥ξn(k)+1-ξn(k)∥E0·∥ξm(k)+1-ξm(k)∥E0)lllllllllllllllllllll×(1+∥ξn(k)+1-ξm(k)+1∥E0)-1)}
for any k∈N.
Let us put(42)C={(∥ξn(k)-ξm(k)∥E0)k∈N:lllllφ(∥ξn(k)+1-ξm(k)+1∥E0)≤ϕ(∥ξn(k)-ξm(k)∥E0)},D={(∥ξn(k)+1-ξn(k)∥E0·∥ξm(k)+1-ξm(k)∥E01+∥ξn(k)+1-ξm(k)+1∥E0)k∈N:lllllllφ(∥ξn(k)+1-ξm(k)+1∥E0)llllll≤ϕ(∥ξn(k)+1-ξn(k)∥E0·∥ξm(k)+1-ξm(k)∥E01+∥ξn(k)+1-ξm(k)+1∥E0)}.
By (41) we have  CardC=∞ or  CardD=∞.
Let us suppose thatCardC=∞. Then there exist infinitely many k∈N such that
(43)φ(∥ξn(k)+1-ξm(k)+1∥E0)≤ϕ(∥ξn(k)-ξm(k)∥E0)
and since (φ,ϕ)∈F and
(44)lim⁡k→∞∥ξn(k)+1-ξm(k)+1∥E0=lim⁡k→∞∥ξn(k)-ξm(k)∥E0=ε
we infer from (39) that ε=0. This is a contradiction.
Let us suppose thatCardD=∞. In this case, we can find infinitely many k∈N such that
(45)φ(∥ξn(k)+1-ξm(k)+1∥E0)≤ϕ(∥ξn(k)+1-ξn(k)∥E0·∥ξm(k)+1-ξm(k)∥E01+∥ξn(k)+1-ξm(k)+1∥E0)
and since (φ,ϕ)∈F, we infer
(46)∥ξn(k)+1-ξm(k)+1∥E0≤∥ξn(k)+1-ξn(k)∥E0·∥ξm(k)+1-ξm(k)∥E01+∥ξn(k)+1-ξm(k)+1∥E0.
Taking  k→∞ and in view of (38) and (39), it follows that ε≤0 and this is a contradiction.
Therefore, since in both possibilitiesCardC=∞  and  CardD=∞ we obtain a contradiction, we deduce that (ξn) is a Cauchy sequence in E0.
SinceE0 is a Banach space, we can find ξ*∈E0 such that lim⁡n→∞ξn=ξ*. As ξn∈Rc and Rc is topologically closed, we have ξ*∈Rc.
Next, we will prove thatξ* is a PPF dependent fixed point of T. In fact, by the contractive condition, we obtain
(47)φ(∥Tξ*-Tξn∥E)≤max⁡{((∥ξ*(c)-Tξ*∥E)·(∥ξn(c)-Tξn∥E)1+(∥Tξ*-Tξn∥E))ϕ(∥ξ*-ξn∥E0),llllllllllllllllllϕ(∥ξ*(c)-Tξ*∥E·∥ξn(c)-Tξn∥E1+∥Tξ*-Tξn∥E)}
for any n∈N.
We can distinguish two cases again.(1)
There exist infinitely manyn∈N such that
(48)φ(∥Tξ*-Tξn∥E)≤ϕ(∥ξ*-ξn∥E0).In this case, since(φ,ϕ)∈F, we obtain
(49)∥Tξ*-Tξn∥E≤∥ξ*-ξn∥E0for infinitely manyn∈N. Since lim⁡n→∞ξn=ξ*, taking n→∞ in the last inequality, we obtain
(50)lim⁡n→∞Tξn=Tξ*,where, to simplify our considerations, we will denote the subsequence by the same symbol(Tξn). By (21),
(51)Tξ*=lim⁡n→∞Tξn=lim⁡n→∞ξn+1(c).ξn→ξ* in E0; this means that
(52)sup⁡t∈[a,b]∥ξn(t)-ξ*(t)∥E⟶0and, consequently,lim⁡n→∞ξn+1(c)=ξ*(c). From this last result and from (51) we deduce that
(53)Tξ*=ξ*(c)and, therefore,ξ* is a PPF dependent fixed point of T in Rc.(2)
There exist infinitely manyn∈N such that
(54)φ(∥Tξ*-Tξn∥E)≤ϕ(∥ξ*(c)-Tξ*∥E·∥ξn(c)-Tξn∥E1+∥Tξ*-Tξn∥E).To simplify our considerations, we will denote the subsequence by the same symbol(Tξn). Since (φ,ϕ)∈F, we infer
(55)∥Tξ*-Tξn∥E≤∥ξ*(c)-Tξ*∥E·∥ξn(c)-Tξn∥E1+∥Tξ*-Tξn∥Efor anyn∈N. Using (21), we have that Tξn=ξn+1(c) and, therefore,
(56)∥Tξ*-Tξn∥E≤∥ξ*(c)-Tξ*∥E·∥ξn(c)-ξn+1(c)∥E1+∥Tξ*-Tξn∥Efor anyn∈N. Taking n→∞ and by (38) since lim⁡n→∞∥ξn(c)-ξn+1(c)∥E=0, we infer (50). From the above case, we deduce that ξ* is a PPF dependent fixed point of T in Rc. Therefore, we have proved that in both cases ξ* is a PPF dependent fixed point of T in Rc.
Finally, we will prove the uniqueness of PPF dependent fixed point ofT in Rc.
Suppose thatϕ* is another PPF dependent fixed point of T in Rc. Then, since ξ*, ϕ*∈Rc and Rc is algebraically closed, we obtain
(57)∥ξ*-ϕ*∥E0=∥ξ*(c)-ϕ*(c)∥E.
As ξ*(c)=Tξ* and ϕ*(c)=Tϕ* because ξ* and ϕ* are PPF dependent fixed points of T, we infer
(58)∥ξ*-ϕ*∥E0=∥ξ*(c)-ϕ*(c)∥E=∥Tξ*-Tϕ*∥E.
Consequently, using the contractive condition, we get
(59)φ(∥ξ*-ϕ*∥E0)=φ(A11∥Tξ*-Tϕ*∥E)≤max⁡{(∥ξ*(c)-Tξ*∥E·∥ϕ*(c)-Tϕ*∥E1+∥Tξ*-Tϕ*∥E)ϕ(∥ξ*-ϕ*∥E0),lllllllllllllllllϕ(∥ξ*(c)-Tξ*∥E·∥ϕ*(c)-Tϕ*∥E1+∥Tξ*-Tϕ*∥E)}=max⁡{ϕ(∥ξ*-ϕ*∥E0),ϕ(0)}.
We can distinguish two cases.(i)
Considermax⁡{ϕ(∥ξ*-ϕ*∥E0),ϕ(0)}=ϕ(∥ξ*-ϕ*∥E0). In this case, from (59) we have
(60)φ(∥ξ*-ϕ*∥E0)≤ϕ(∥ξ*-ϕ*∥E0).Now, since(φ,ϕ)∈F and using Remark 4, we get ∥ξ*-ϕ*∥E0=0 and, therefore, ξ*=ϕ*.(ii)
Considermax⁡{ϕ(∥ξ*-ϕ*∥E0),ϕ(0)}=ϕ(0). From (59) we obtain
(61)φ(∥ξ*-ϕ*∥E0)≤ϕ(0),and, since(φ,ϕ)∈F, we infer that ∥ξ*-ϕ*∥E0≤0. Therefore, ∥ξ*-ϕ*∥E0=0 or, equivalently, ξ*=ϕ*.
This result finishes the proof.By Theorem8, we obtain the following corollaries.Corollary 9.
LetT:E0→E be a mapping such that there exists a pair of functions (φ,ϕ)∈F and c∈[a,b] satisfying
(62)φ(∥Tξ-Tη∥E)≤ϕ(∥ξ-η∥E0)
for any ξ,η∈E0. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.Corollary 10.
LetT:E0→E be a mapping such that there exists a pair of functions (φ,ϕ)∈F and c∈[a,b] satisfying
(63)φ(∥Tξ-Tη∥E)≤ϕ(∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E)
for any ξ,η∈E0. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.The main result of [4] is Theorem 3. Notice that the contractive condition appearing in this theorem
(64)∥Tξ-Tη∥E≤α∥ξ-η∥E0+β∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E
for any ξ,η∈E0, where α,β∈[0,1) with α+β<1 and c∈[a,b], implies that
(65)∥Tξ-Tη∥E≤(α+β)max⁡{∥ξ-η∥E0,∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E}≤max⁡{∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E(α+β)∥ξ-η∥E0,lllllllllllllllll(α+β)∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E}
for any ξ,η∈E0. This condition is a particular case of the contractive condition appearing in Theorem 8 with the pair of functions (φ,ϕ)∈F given by φ=1[0,∞) and ϕ=(α+β)1[0,∞). Therefore, Theorem 3 is a particular case of the following corollary.Corollary 11.
LetT:E0→E be a mapping such that there exist real numbers α,β∈[0,1) with α+β<1 and c∈[a,b] satisfying
(66)∥Tξ-Tη∥E≤max⁡{∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E(α+β)∥ξ-η∥E0,llllllllllllll(α+β)∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E}
for any ξ,η∈E0. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.Taking into account Example5, we have the following corollary.Corollary 12.
LetT:E0→E be a mapping such that there exist two functions φ,ϕ:[0,∞)→[0,∞) and c∈[a,b] such that
(67)φ(∥Tξ-Tη∥E)≤max⁡{(∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E)φ(∥ξ-η∥E0)-ϕ(∥ξ-η∥E0),llllllllllllllφ(∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E)llllllllllll-ϕ(∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E)}
for any ξ,η∈E0, where φ is a continuous and increasing function satisfying φ(t)=0 if and only if t=0, and ϕ is a nondecreasing function such that ϕ(t)=0 if and only if t=0, and ϕ≤φ.
IfRc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.Corollary12 has the following consequences.Corollary 13.
LetT:E0→E be a mapping such that there exist two functions φ,ϕ:[0,∞)→[0,∞) and c∈[a,b] such that
(68)φ(∥Tξ-Tη∥E)≤φ(∥ξ-η∥E0)-ϕ(∥ξ-η∥E0)
for any ξ,η∈E0, where φ is an increasing function and ϕ is a nondecreasing function and they satisfy φ(t)=ϕ(t)=0 if and only if t=0, and φ is continuous with ϕ≤φ. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.Corollary13 can be considered as the version, in the context of PPF dependent fixed point theorems, of the following result about fixed point theorems which appears in [9].Theorem 14 (see [9]).
Let(X,d) be a complete metric space and T:X→X a mapping satisfying
(69)φ(d(Tx,Ty))≤φ(d(x,y))-ϕ(d(x,y))
for x,y∈X, where φ and ϕ satisfy the same conditions as in Corollary 13. Then T has a unique fixed point.Corollary 15.
LetT:E0→E be a mapping such that there exist two functions φ,ϕ:[0,∞)→[0,∞) satisfying the same conditions as in Corollary 13 and c∈[a,b] such that
(70)φ(∥Tξ-Tη∥E)≤φ(∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E)-ϕ(∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E)
for any ξ,η∈E0. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.Taking into account Example6, we have the following corollary.Corollary 16.
LetT:E0→E be a mapping such that there exist α∈S (see Example 6) and c∈[a,b] satisfying
(71)∥Tξ-Tη∥E≤max⁡{∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥Eα(∥ξ-η∥E0)∥ξ-η∥E0,llllllllllllllα(∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E)lllllllllllll×∥ξ(c)-Tξ∥E∥η(c)-Tη∥E1+∥Tξ-Tη∥E}
for any ξ,η∈E0. If Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.A consequence of Corollary16 is the following result.Corollary 17.
LetT:E0→E be a mapping such that there exists α∈S satisfying
(72)∥Tξ-Tη∥E≤α(∥ξ-η∥E)∥ξ-η∥E0
for any ξ,η∈E0.
Ifc∈[a,b] such that Rc is topologically closed and algebraically closed with respect to difference, then T has a unique PPF dependent fixed point in Rc.Corollary17 is the version, in the context of PPF dependent fixed point theorems, of the following result about fixed point theorems appearing in [10].Theorem 18.
Let(X,d) be a complete metric space and T:X→X a mapping satisfying
(73)d(Tx,Ty)≤α(d(x,y))d(x,y)
for any x,y∈X, where α∈S. Then T has a unique fixed point.



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*Source: 416187-2014-06-09.xml*