# PPF Dependent Fixed Point Results for Triangularαc-Admissible Mappings

Authors: Ljubomir Ćirić; Saud M. Alsulami; Peyman Salimi; Pasquale Vetro
Journal: The Scientific World Journal
 (2014)

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## Abstract

We introduce the concept of triangularαc-admissible
mappings (pair of mappings) with respect to ηc nonself-mappings and
establish the existence of PPF dependent fixed (coincidence) point theorems
for contraction mappings involving triangular αc-admissible mappings
(pair of mappings) with respect to ηc nonself-mappings in Razumikhin
class. Several interesting consequences of our theorems are also given.


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## Body

## 1. Introduction and Preliminaries

The applications of fixed point theory are very important and useful in diverse disciplines of mathematics. In fact, fixed point theory can be applied for solving equilibrium problems, variational inequalities, and optimization problems. In particular, a very powerful tool is the Banach fixed point theorem, which was generalized and extended in various directions: modifying Banach’s contractive condition, changing the space, or extending single-valued mapping to multivalued mapping (see [1–8] and references therein). In 1997, Bernfeld et al. [9] introduced the concept of fixed point for mappings that have different domains and ranges, which is called PPF dependent fixed point or the fixed point with PPF dependence. Furthermore, they gave the notion of Banach type contraction for nonself-mapping and also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings (also see [10]). The PPF dependent fixed point theorems are useful for proving the solutions of nonlinear functional differential and integral equations which may depend upon the past history, present data, and future consideration. On the other hand, Samet et al. [11] first introduced the concept of α-admissible self-mappings and proved the existence of fixed point results using contractive conditions involving α-admissible mappings in complete metric spaces. They also gave some examples and applications of the obtained results to ordinary differential equations. In this paper, we will introduce the concept of triangular αc-admissible mappings (pair of mappings) with respect to ηc nonself-mappings and establish the existence of PPF dependent fixed point theorems for contraction mappings involving triangular αc-admissible mappings (pair of mappings) with respect to ηc nonself-mappings in Razumikhin class.Throughout this paper, we assume that(E,∥·∥E) is a Banach space, I denotes a closed interval [a,b] in ℝ, and E0=(I,E) denotes the sets of all continuous E-valued functions on I equipped with the supremum norm ∥·∥E0 defined by
(1)∥ϕ∥E0=supt∈I∥ϕ(t)∥E.For a fixed elementc∈I, the Razumikhin or minimal class of functions in E0 is defined by
(2)ℛc={ϕ∈E0:∥ϕ∥E0=∥ϕ(c)∥E}.
Clearly, every constant function from I to E belongs to ℛc.Definition 1.
Letℛc be the Razumikhin class; then(i)
the classℛc is algebraically closed with respect to difference, if ϕ-ξ∈ℛc when ϕ,ξ∈ℛc;(ii)
the classℛc is topologically closed if it is closed with respect to the topology on E0 generated by the norm ∥·∥E0.Definition 2 (see [9]).
A mappingϕ∈E0 is said to be a PPF dependent fixed point or a fixed point with PPF dependence of mapping T:E0→E if Tϕ=ϕ(c) for some c∈I.Definition 3 (see [10]).
LetS:E0→E0 and T:E0→E. A point ϕ∈E0 is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of S and T if Tϕ=(Sϕ)(c) for some c∈I.Definition 4 (see [9]).
The mappingT:E0→E is called a Banach type contraction if there exists k∈[0,1) such that
(3)∥Tϕ-Tξ∥E≤k∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0.In 2012, Samet et al. [11] introduced the concepts of α-ψ-contractive and α-admissible mappings and established various fixed point theorems for such mappings in complete metric spaces. Afterwards, Karapinar and Samet [12] generalized these notions to obtain fixed point results. More recently, Salimi et al. [13] modified the notions of α-ψ-contractive and α-admissible mappings and established fixed point theorems which are proper generalizations of the recent results in [11, 12].Samet et al. [11] defined the notion of α-admissible mappings as follows.Definition 5.
LetT be a self-mapping on X and let α:X×X→[0,+∞) be a function. We say that T is an α-admissible mapping if
(4)x,y∈X,α(x,y)≥1⟹α(Tx,Ty)≥1.In [11] the authors consider the family Ψ of nondecreasing functions ψ:[0,+∞)→[0,+∞) such that ∑n=1+∞‍ψn(t)<+∞ for each t>0, where ψn is the nth iterate of ψ and give the following theorem.Theorem 6.
Let(X,d) be a complete metric space and let T be an α-admissible mapping. Assume that
(5)α(x,y)d(Tx,Ty)≤ψ(d(x,y))
for all x,y∈X, where ψ∈Ψ. Also, suppose that the following assertions hold:(i)
there existsx0∈X such that α(x0,Tx0)≥1,(ii)
eitherT is continuous or for any sequence {xn} in X with α(xn,xn+1)≥1 for all n∈ℕ∪{0} and xn→x as n→+∞, one has α(xn,x)≥1 for all n∈ℕ∪{0}.Then T has a fixed point.Salimi et al. [13] modified and generalized the notions of α-ψ-contractive mappings and α-admissible mappings by the following ways.Definition 7 (see [13]).
LetT be a self-mapping on X and α,η:X×X→[0,+∞) two functions. We say that T is an α-admissible mapping with respect to η if
(6)x,y∈X,α(x,y)≥η(x,y)⟹α(Tx,Ty)≥η(Tx,Ty).
Note that if we take η(x,y)=1, then this definition reduces to Definition 5. Also, if we take α(x,y)=1, then we say that T is an η-subadmissible mapping.The following result was proved by Salimi et al. [13].Theorem 8 (see [13]).
Let(X,d) be a complete metric space and let T be an α-admissible mapping. Assume that
(7)x,y∈X,α(x,y)≥1⟹d(Tx,Ty)≤ψ(M(x,y)),
where ψ∈Ψ and
(8)M(x,y)=max{d(x,y),d(x,Tx)+d(y,Ty)2,d(x,y),d(x,Tx)+d(y,Ty)2d(x,Ty)+d(y,Tx)2}.
Also, suppose that the following assertions hold:(i)
there existsx0∈X such that α(x0,Tx0)≥1,(ii)
eitherT is continuous or for any sequence {xn} in X with α(xn,xn+1)≥1 for all n∈ℕ∪{0} and xn→x as n→+∞, one has α(xn,x)≥1 for all n∈ℕ∪{0}.Then T has a fixed point.Recently Karapinar et al. [14] introduced the notion of triangular α-admissible mapping as follows.Definition 9 (see [14]).
LetT:X→X and α:X×X→(-∞,+∞). We say that T is a triangular α-admissible mapping if(T1)α(x,y)≥1impliesα(Tx,Ty)≥1,x,y∈X,(T2)α(x,z)≥1α(z,y)≥1implyα(x,y)≥1.For more details and applications of this line of research, we refer the reader to some related papers [15–21].Now, motivated by Salimi et al. [13] and Karapinar et al. [14] (see also [15–21]), we introduce the following notion.Definition 10.
Letc∈I and T:E0→E, α,η:E×E→[0,+∞). We say that T is a triangular αc-admissible mapping with respect to ηc if, for ϕ,ξ,π∈E0,
(TC1)α(ϕ(c),ξ(c))≥η(ϕ(c),ξ(c))⟹α(Tϕ,Tξ)≥η(Tϕ,Tξ),(TC2)α(ϕ(c),π(c))≥η(ϕ(c),π(c)),α(π(c),ξ(c))≥η(π(c),ξ(c))implyα(ϕ(c),ξ(c))≥η(ϕ(c),ξ(c)).
Note that if we takeη(x,y)=1 for all x,y∈E, then we say that T is a triangular αc-admissible mapping. Also, if we take α(x,y)=1 for all x,y∈E, then we say that T is a triangular ηc-subadmissible mapping.Example 11.
LetE=ℝ be a real Banach space with usual norm and let I=[0,1]. Define T:E0→E by Tϕ=2ϕ(1) for all ϕ∈E0 and α,η:E×E→[0,+∞) by
(9)α(x,y)={x2+y2+|x||y|+1,ifx≥y,0,otherwise;η(x,y)=x2+y2+|x||y|+1/2. Then T is a triangular αc-admissible mapping with respect to ηc. Indeed, if α(ϕ(1),ξ(1))≥η(ϕ(1),ξ(1)), then ϕ(1)≥ξ(1) and so 2ϕ(1)≥2ξ(1). That is, Tϕ≥Tξ which implies α(Tϕ,Tξ)≥η(Tϕ,Tξ). Also, if
(10)α(ϕ(c),π(c))≥η(ϕ(c),π(c)),α(π(c),ξ(c))≥η(π(c),ξ(c)),
then ϕ(c)≥π(c) and π(c)≥ξ(c) and so ϕ(c)≥ξ(c). That is, α(ϕ(c),ξ(c))≥η(ϕ(c),ξ(c)).The following lemma is necessary later on.Lemma 12.
LetT be a triangular αc-admissible mapping with respect to ηc. Define the sequence {ϕn} by the following way:
(11)Tϕn-1=ϕn(c);
for all n∈ℕ, where ϕ0∈ℛc is such that α(ϕ0(c),Tϕ0)≥η(ϕ0(c),Tϕ0). Then
(12)α(ϕm(c),ϕn(c))≥η(ϕm(c),ϕn(c)),(ϕm(c),ϕn.)∀m,n∈ℕwithm<n.Proof.
SinceT is a triangular αc-admissible mapping with respect to ηc,
(13)α(ϕ0(c),ϕ1(c))=α(ϕ0(c),Tϕ0)≥η(ϕ0(c),Tϕ0)=η(ϕ0(c),ϕ1(c))
and so
(14)α(ϕ1(c),Tϕ1)≥η(ϕ1(c),Tϕ1).
By continuing this process we get,
(15)α(ϕn(c),ϕn+1(c))≥η(ϕn(c),ϕn+1(c)),∀n∈ℕ.
Since
(16)α(ϕm(c),ϕm+1(c))≥η(ϕm(c),ϕm+1(c)),α(ϕm+1(c),ϕm+2(c))≥η(ϕm+1(c),ϕm+2(c)),
then by (TC2) we get α(ϕm(c),ϕm+2(c))≥η(ϕm(c),ϕm+2(c)). By continuing this process, we get
(17)α(ϕm(c),ϕn(c))≥η(ϕm(c),ϕn(c)),ϕm(c),ϕn(c)∀m,n∈ℕwithm<n.


## 2. Main Results

One of our main theorems is a result of Geraghty type [22] obtained by a modification of the approach in [13]. Let ℱ denote the class of all functions β:[0,+∞)→[0,1) satisfying the following condition:
(18)β(tn)⟶1impliestn⟶0,asn⟶+∞.Theorem 13.
LetT:E0→E, α,η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping with respect to ηc;(iii)
there existsβ∈ℱ such that
(19)α(ϕ(c),ξ(c))≥η(ϕ(c),ξ(c))⟹∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥η(ϕn(c),ϕn+1(c)) for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥η(ϕn(c),ϕ(c)) for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥η(ϕ0(c),Tϕ0).Then, T has a PPF dependent fixed point ϕ*∈ℛc.Proof.
Letϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥η(ϕ0(c),Tϕ0). Since Tϕ0∈E, there exists x1∈E such that Tϕ0=x1. Choose ϕ1∈ℛc such that
(20)x1=ϕ1(c).
By continuing this process, by induction, we can build a sequence {ϕn} in ℛc⊆E such that,
(21)Tϕn-1=ϕn(c),∀n∈ℕ.
Hence, from Lemma 12, we have
(22)α(ϕm(c),ϕn(c))≥η(ϕm(c),ϕn(c)),(ϕm(c),ϕn5)∀m,n∈ℕwithm<n.
Since ℛc is algebraically closed with respect to difference, it follows that
(23)∥ϕn-1-ϕn∥E0=∥ϕn-1(c)-ϕn(c)∥E,∀n∈ℕ.
Then, by (iii), we get
(24)∥ϕn-ϕn+1∥E0=∥ϕn(c)-ϕn+1(c)∥E=∥Tϕn-1-Tϕn∥E∥ϕn-ϕn+1∥E0≤β(∥ϕn-1-ϕn∥E0)∥ϕn-1-ϕn∥E0,
and so
(25)∥ϕn-ϕn+1∥E0≤β(∥ϕn-1-ϕn∥E0)∥ϕn-1-ϕn∥E0∥ϕn-ϕn+1∥E0<∥ϕn-1-ϕn∥E0,
for all n∈ℕ. This implies that the sequence {∥ϕn-ϕn+1∥E0} is decreasing in ℝ+. Then, there exists r≥0 such that limn→+∞∥ϕn-ϕn+1∥E0=r. Assume r>0. Now, by taking limit as n→+∞ in (24), we get
(26)r≤limn→+∞β(∥ϕn-1-ϕn∥E0)r,
which implies 1≤limn→+∞β(∥ϕn-1-ϕn∥E0). That is,
(27)limn→+∞β(∥ϕn-1-ϕn∥E0)=1,
and since β∈ℱ, limn→+∞∥ϕn-1-ϕn∥E0=0 which is a contradiction. Hence, r=0. That is,
(28)limn→+∞∥ϕn-1-ϕn∥E0=0.
Now, we prove that the sequence{ϕn} is Cauchy in ℛc. Assume the contrary; then there exist ɛ>0 and two sequences {mk} and {nk} with k≤mk<nk such that
(29)∥ϕmk-ϕnk∥E0≥ɛ,∥ϕmk-ϕnk-1∥E0<ɛ.
From
(30)ɛ≤∥ϕmk-ϕnk∥E0ɛ≤∥ϕmk-ϕnk-1∥E0+∥ϕnk-1-ϕnk∥E0ɛ<ɛ+∥ϕnk-1-ϕnk∥E0,
letting k→+∞, we get
(31)limn→+∞∥ϕmk-ϕnk∥E0=ɛ.
By triangle inequality, we have
(32)∥ϕmk-ϕnk∥E0≤∥ϕmk-ϕmk+1∥E0+∥ϕmk+1-ϕnk+1∥E0∥ϕmk-ϕnk∥E0≤+∥ϕnk-ϕnk+1∥E0.
On the other hand, by (iii) and (21), we have
(33)∥ϕmk+1-ϕnk+1∥E0≤β(∥ϕmk-ϕnk∥E0)∥ϕmk-ϕnk∥E0.
Therefore, we get(34)∥ϕmk-ϕnk∥E0≤∥ϕmk-ϕmk+1∥E0∥ϕmk-ϕnk∥E0≤+β(∥ϕmk-ϕnk∥E0)∥ϕmk-ϕnk∥E0∥ϕmk-ϕnk∥E0≤+∥ϕnk-ϕnk+1∥E0,
which implies
(35)(1-β(∥ϕmk-ϕnk∥E0))∥ϕmk-ϕnk∥E0≤∥ϕmk-ϕmk+1∥E0+∥ϕnk-ϕnk+1∥E0.
Taking limit as k→+∞ in the above inequality and applying (28) and (31), we get
(36)limk→+∞(1-β(∥ϕmk-ϕnk∥E0))=0,
which implies limk→+∞β(∥ϕmk-ϕnk∥E0)=1 and since β∈ℱ, we deduce
(37)limk→+∞∥ϕmk-ϕnk∥E0=0,
which is a contradiction. Consequently
(38)limm,n→+∞∥ϕn-ϕm∥E0=0,
and hence {ϕn} is a Cauchy sequence in ℛc⊆E0. By the completeness of E0 we get that {ϕn} converges to a point ϕ*∈E0; that is, ϕn→ϕ* as n→+∞. Since ℛc is topologically closed, we deduce ϕ*∈ℛc. From (iv) we have α(ϕn(c),ϕ*(c))≥η(ϕn(c),ϕ*(c)) for all n∈ℕ∪0. Then, from (iii) we get
(39)∥Tϕ*-ϕ*(c)∥E≤∥Tϕ*-Tϕn∥E+∥Tϕn-ϕ*(c)∥E∥Tϕ*-ϕ*(c)∥E=∥Tϕ*-Tϕn∥E+∥ϕn+1(c)-ϕ*(c)∥E∥Tϕ*-ϕ*(c)∥E≤β(∥ϕ*-ϕn∥E0)∥ϕ*-ϕn∥E0∥Tϕ*-ϕ*(c)∥E≤+∥ϕn+1(c)-ϕ*(c)∥E,
for all n∈ℕ. Taking limit as n→+∞ in the above inequality, we get
(40)∥Tϕ*-ϕ*(c)∥E=0;
that is,
(41)Tϕ*=ϕ*(c),
which implies that ϕ* is a PPF dependent fixed point of T in ℛc.If in Theorem13 we take η(ϕ(c),ξ(c))=1 for all ϕ,ξ∈E0, then we deduce the following corollary.Corollary 14.
LetT:E0→E and α:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping;(iii)
there existsβ∈ℱ such that
(42)α(ϕ(c),ξ(c))≥1⟹∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥1 for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.If in Theorem13 we take α(ϕ(c),ξ(c))=1 for all ϕ,ξ∈E0, then we deduce the following corollary.Corollary 15.
LetT:E0→E and η:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular ηc-subadmissible mapping;(iii)
there existsβ∈ℱ such that
(43)η(ϕ(c),ξ(c))≤1⟹∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and η(ϕn(c),ϕn+1(c))≤1 for all n∈ℕ∪0, then η(ϕn(c),ϕ)≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η(ϕ0(c),Tϕ0)≤1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Definition 16.
Letc∈I and S:E0→E0, T:E0→E, α,η:E×E→[0,+∞). We say that (S,T) is a triangular αc-admissible pair with respect to ηc if, for ϕ,ξ,π∈E0,
(ST1)α((Sϕ)(c),(Sξ)(c))≥η((Sϕ)(c),(Sξ)(c))⟹α(Tϕ,Tξ)≥η(Tϕ,Tξ),(44)α((Sϕ)(c),(Sξ)(c))≥η((Sϕ)(c),(Sξ)(c)),α((Sξ)(c),(Sπ)(c))≥η((Sϕ)(c),(Sξ)(c))⟹α((Sϕ)(c),(Sπ)(c))≥η((Sϕ)(c),(Sπ)(c)).
Note that if we take η(ϕ(c),ξ(c))=1, then, we say that (S,T) is a triangular αc-admissible pair. Also, if we take α(ϕ(c),ξ(c))=1, then we say that (S,T) is a triangular ηc-subadmissible pair.The following theorem gives a result of existence ofPPF dependent coincidence points.Theorem 17.
LetS:E0→E0, T:E0→E, and α,η:E×E→[0,+∞) be four mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair with respect to ηc;(iii)
there existsβ∈ℱ such that
(45)α((Sϕ)(c),(Sξ)(c))≥η((Sϕ)(c),(Sξ)(c))⟹∥Tϕ-Tξ∥E≤β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0,
for all ϕ,ξ∈ℛc;(iv)
if{Sϕn} is a sequence in ℛc such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥η((Sϕn)(c),(Sϕn+1)(c)) for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥η((Sϕn)(c),(Sϕ)(c)) for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥η((Sϕ0)(c),Tϕ0);(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, there exists ϕ*∈ℛc such that Sϕ*∈ℛc is a PPF dependent fixed point of T and hence ϕ* is a PPF dependent coincidence point of S and T.Proof.
Letϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥η((Sϕ0)(c),Tϕ0). By condition (vii), there exists ϕ1∈ℛc such that
(46)Tϕ0=(Sϕ1)(c).
By continuing this process, by induction, we can build a sequence {ϕn} in ℛc such that
(47)Tϕn-1=(Sϕn)(c),∀n∈ℕ.
Hence, from Lemma 12, we have
(48)α((Sϕm)(c),(Sϕn)(c))≥η((Sϕm)(c),(Sϕn)(c)),((Sϕm)(c),(Sϕn)(c)555)∀m,n∈ℕwithm<n.
Since S(ℛc) is algebraically closed with respect to difference, it follows that
(49)∥Sϕn-1-Sϕn∥E0=∥(Sϕn-1)(c)-(Sϕn)(c)∥E,∀n∈ℕ.
Then, by (iii), we get
(50)∥Sϕn-Sϕn+1∥E0=∥(Sϕn)(c)-(Sϕn+1)(c)∥E∥Sϕn-Sϕn+1∥E0=∥Tϕn-1-Tϕn∥E∥Sϕn-Sϕn+1∥E0≤β(∥Sϕn-1-Sϕn∥E0)∥Sϕn-1-Sϕn∥E0,
and so
(51)∥Sϕn-Sϕn+1∥E0≤β(∥Sϕn-1-Sϕn∥E0)∥Sϕn-1-Sϕn∥E0∥Sϕn-Sϕn+1∥E0<∥Sϕn-1-Sϕn∥E0,
for all n∈ℕ. This implies that the sequence {∥Sϕn-Sϕn+1∥E0} is decreasing in ℝ+. Then, there exists r≥0 such that limn→+∞∥Sϕn-Sϕn+1∥E0=r. Assume r>0. Now by taking limit as n→+∞ in (50) we get
(52)r≤limn→+∞β(∥Sϕn-1-Sϕn∥E0)r,
which implies 1≤limn→+∞β(∥Sϕn-1-Sϕn∥E0). That is,
(53)limn→+∞β(∥Sϕn-1-Sϕn∥E0)=1,
and since β∈ℱ, limn→+∞∥Sϕn-1-Sϕn∥E0=0 which is a contradiction. Hence, r=0. That is,
(54)limn→+∞∥Sϕn-1-Sϕn∥E0=0.
Now, we prove that the sequence{Sϕn} is Cauchy in S(ℛc). Assume the contrary; then there exist ɛ>0 and two sequences {mk} and {nk} with k≤mk<nk such that
(55)∥Sϕmk-Sϕnk∥E0≥ɛ,∥Sϕmk-Sϕnk-1∥E0<ɛ.
From
(56)ɛ≤∥Sϕmk-Sϕnk∥E0ɛ≤∥Sϕmk-Sϕnk-1∥E0+∥Sϕnk-1-Sϕnk∥E0ɛ<ɛ+∥Sϕnk-1-Sϕnk∥E0,
letting k→+∞, we get
(57)limn→+∞∥Sϕmk-Sϕnk∥E0=ɛ.
By triangle inequality, we have
(58)∥Sϕmk-Sϕnk∥E0≤∥Sϕmk-Sϕmk+1∥E0+∥Sϕmk+1-Sϕnk+1∥E0∥Sϕmk-Sϕnk∥E0≤+∥Sϕnk-Sϕnk+1∥E0.
On the other hand, by (iii) and (47), we have
(59)∥Sϕmk+1-Sϕnk+1∥E0≤β(∥Sϕmk-Sϕnk∥E0)∥Sϕmk-Sϕnk∥E0.
Therefore, we get(60)∥Sϕmk-Sϕnk∥E0≤∥Sϕmk-Sϕmk+1∥E0∥Sϕmk-Sϕnk∥E0≤+β(∥Sϕmk-Sϕnk∥E0)∥Sϕmk-Sϕnk∥E0∥Sϕmk-Sϕnk∥E0≤+∥Sϕnk-Sϕnk+1∥E0,
which implies
(61)(1-β(∥Sϕmk-Sϕnk∥E0))∥Sϕmk-Sϕnk∥E0≤∥Sϕmk-Sϕmk+1∥E0+∥Sϕnk-Sϕnk+1∥E0.
Taking limit as k→+∞ in the above inequality and applying (54) and (57), we get
(62)limk→+∞(1-β(∥Sϕmk-Sϕnk∥E0))=0,
which implies limk→+∞β(∥Sϕmk-Sϕnk∥E0)=1 and since β∈ℱ, we deduce
(63)limk→+∞∥Sϕmk-Sϕnk∥E0=0,
which is a contradiction. Consequently
(64)limm,n→+∞∥Sϕn-Sϕm∥E0=0,
and hence {Sϕn} is a Cauchy sequence in S(ℛc)⊂ℛc. By the completeness of S(ℛc), there exists ϕ*∈ℛc such that Sϕn→Sϕ* as n→+∞. From (iv), we have α((Sϕn)(c),(Sϕ*)(c))≥η((Sϕn)(c),(Sϕ*)(c)) for all n∈ℕ∪0. Then from (iii) we get
(65)∥Tϕ*-(Sϕ*)(c)∥E≤∥Tϕ*-Tϕn∥E+∥Tϕn-(Sϕ*)(c)∥E=∥Tϕ*-Tϕn∥E+∥(Sϕn+1)(c)-(Sϕ*)(c)∥E≤β(∥Sϕ*-Sϕn∥E0)∥Sϕ*-Sϕn∥E0+∥(Sϕn+1)(c)-(Sϕ*)(c)∥E,
for all n∈ℕ. Taking limit as n→+∞ in the above inequality, we get
(66)∥Tϕ*-(Sϕ*)(c)∥E=0.
That is,
(67)Tϕ*=(Sϕ*)(c),
which implies that Sϕ* is a PPF dependent fixed point of T in S(ℛc) and hence ϕ* is a PPF dependent coincidence point of S and T.If in Theorem17 we take η(ϕ(c),ξ(c))=1 for all ϕ,ξ∈E0, then we deduce the following corollary.Corollary 18.
LetS:E0→E0, T:E0→E, and α:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair;(iii)
there existsβ∈ℱ such that
(68)α((Sϕ)(c),(Sξ)(c))≥1⟹∥Tϕ-Tξ∥E≤β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0,
for all ϕ,ξ∈ℛc;(iv)
if{Sϕn} is a sequence in ℛc such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥1 for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.If in Theorem13 we take α(ϕ(c),ξ(c))=1 for all ϕ,ξ∈E0, then we deduce the following corollary.Corollary 19.
LetS:E0→E0, T:E0→E, and η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-subadmissible pair;(iii)
there existsβ∈ℱ such that
(69)η((Sϕ)(c),(Sξ)(c))≤1⟹∥Tϕ-Tξ∥E≤β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0,
for all ϕ,ξ∈ℛc;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and η((Sϕn)(c),(Sϕn+1)(c))≤1 for all n∈ℕ∪0, then η((Sϕn)(c),(Sϕ)(c))≤1;(v)
there existsϕ0∈ℛc such that η((Sϕ0)(c),Tϕ0)≤1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.


### 2.1. Consequences of Corollary14

Theorem 20.
LetT:E0→E and α:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping;(iii)
there existsβ∈ℱ such that
(70)α(ϕ(c),ξ(c))∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥1 for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Proof.
Letα(ϕ(c),ξ(c))≥1; then by (iii) we have
(71)∥Tϕ-Tξ∥E≤α(ϕ(c),ξ(c))∥Tϕ-Tξ∥E∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0.
That is, all conditions of Corollary 14 hold and T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 21.
LetT:E0→E and α:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping;(iii)
there existsβ∈ℱ such that
(72)(∥Tϕ-Tξ∥E+ϵ)α(ϕ(c),ξ(c))≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0+ϵ,
for all ϕ,ξ∈E0, where ϵ≥1;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥1 for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Proof.
Letα(ϕ(c),ξ(c))≥1; then by (iii) we have
(73)∥Tϕ-Tξ∥E+ϵ≤(∥Tϕ-Tξ∥E+ϵ)α(ϕ(c),ξ(c))∥Tϕ-Tξ∥E+ϵ≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0+ϵ,
which implies ∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0. That is, all conditions of Corollary 14 hold and T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 22.
LetT:E0→E and α:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping;(iii)
there existsβ∈ℱ such that
(74)(α(ϕ(c),ξ(c))-1+σ)∥Tϕ-Tξ∥E≤ϵβ(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0, where 1<ϵ≤σ;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥1 for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Proof.
Letα(ϕ(c),ξ(c))≥1; then by (iii) we have
(75)σ∥Tϕ-Tξ∥E≤(α(ϕ(c),ξ(c))-1+σ)∥Tϕ-Tξ∥Eσ∥Tϕ-Tξ∥E≤ϵβ(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0≤σβ(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
which implies ∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0. That is, all conditions of Corollary 14 hold and T has a PPF dependent fixed point ϕ*∈ℛc.


### 2.2. Consequences of Corollary15

Theorem 23.
LetT:E0→E and η:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular ηc-subadmissible mapping;(iii)
there existsβ∈ℱ such that
(76)∥Tϕ-Tξ∥E≤η(ϕ(c),ξ(c))β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→∞ and η(ϕn(c),ϕn+1(c))≤1 for all n∈ℕ∪0, then η(ϕn(c),ϕ(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η(ϕ0(c),Tϕ0)≤1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 24.
LetT:E0→E and η:E×E→[0,∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular ηc-subadmissible mapping;(iii)
there existsβ∈ℱ such that
(77)∥Tϕ-Tξ∥E+ϵ≤(β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0+ϵ)η(ϕ(c),ξ(c)),
for all ϕ,ξ∈E0, where ϵ≥1 and ψ∈Ψ;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and η(ϕn(c),ϕn+1(c))≤1 for all n∈ℕ∪0, then η(ϕn(c),ϕ(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η(ϕ0(c),Tϕ0)≤1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 25.
LetT:E0→E and η:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular ηc-subadmissible mapping;(iii)
there existsβ∈ℱ such that
(78)σ∥Tϕ-Tξ∥E≤(η(ϕ(c),ξ(c))+ϵ-1)β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0, where 1<ϵ≤σ and ψ∈Ψ;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and η(ϕn(c),ϕn+1(c))≤1 for all n∈ℕ∪0, then η(ϕn(c),ϕ(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η(ϕ0(c),Tϕ0)≤1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.


### 2.3. Consequences of Corollary18

Theorem 26.
LetS:E0→E0, T:E0→E, and α:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair;(iii)
there existsβ∈ℱ such that
(79)α((Sϕ)(c),(Sξ)(c))∥Tϕ-Tξ∥E≤β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0,
for all ϕ,ξ∈ℛc;(iv)
if{Sϕn} is a sequence in ℛc such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥1 for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Theorem 27.
LetS:E0→E0, T:E0→E, and α:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair;(iii)
there existsβ∈ℱ such that
(80)(∥Tϕ-Tξ∥E+ϵ)α((Sϕ)(c),(Sξ)(c))≤β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0+ϵ,
for all ϕ,ξ∈ℛc where ϵ≥1;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥1 for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Theorem 28.
LetS:E0→E0, T:E0→E, and α:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is topologically closed and algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair;(iii)
there existsβ∈ℱ such that
(81)(α((Sϕ)(c),(Sξ)(c))-1+σ)∥Tϕ-Tξ∥E≤ϵβ(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0
for all ϕ,ξ∈ℛc, where 1<ϵ≤σ;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥1 for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.


### 2.4. Consequences of Corollary19

Corollary 29.
LetS:E0→E0, T:E0→E, and η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-subadmissible pair;(iii)
there existsβ∈ℱ such that
(82)∥Tϕ-Tξ∥E≤η((Sϕ)(c),(Sξ)(c))β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0,
for all ϕ,ξ∈ℛc;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and η((Sϕn)(c),(Sϕn+1)(c))≤1 for all n∈ℕ∪0, then η((Sϕn)(c),(Sϕ)(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η((Sϕ0)(c),Tϕ0)≤1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Corollary 30.
LetS:E0→E0, T:E0→E, and η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-subadmissible pair;(iii)
there existsβ∈ℱ such that
(83)∥Tϕ-Tξ∥E+ϵ≤(β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0+ϵ)η((Sϕ)(c),(Sξ)(c))
for all ϕ,ξ∈ℛc, where ϵ≥1;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and η((Sϕn)(c),(Sϕn+1)(c))≤1 for all n∈ℕ∪0, then η((Sϕn)(c),(Sϕ)(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η((Sϕ0)(c),Tϕ0)≤1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Corollary 31.
LetS:E0→E0, T:E0→E, and η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is topologically closed and algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-subadmissible pair;(iii)
there existsβ∈ℱ such that
(84)σ∥Tϕ-Tξ∥E≤(η((Sϕ)(c),(Sξ)(c))+ϵ-1)β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0
for all ϕ,ξ∈ℛc, where 1<ϵ≤σ;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and η((Sϕn)(c),(Sϕn+1)(c))≤1 for all n∈ℕ∪0, then η((Sϕn)(c),(Sϕ)(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η((Sϕ0)(c),Tϕ0)≤1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.


## 2.1. Consequences of Corollary14

Theorem 20.
LetT:E0→E and α:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping;(iii)
there existsβ∈ℱ such that
(70)α(ϕ(c),ξ(c))∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥1 for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Proof.
Letα(ϕ(c),ξ(c))≥1; then by (iii) we have
(71)∥Tϕ-Tξ∥E≤α(ϕ(c),ξ(c))∥Tϕ-Tξ∥E∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0.
That is, all conditions of Corollary 14 hold and T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 21.
LetT:E0→E and α:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping;(iii)
there existsβ∈ℱ such that
(72)(∥Tϕ-Tξ∥E+ϵ)α(ϕ(c),ξ(c))≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0+ϵ,
for all ϕ,ξ∈E0, where ϵ≥1;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥1 for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Proof.
Letα(ϕ(c),ξ(c))≥1; then by (iii) we have
(73)∥Tϕ-Tξ∥E+ϵ≤(∥Tϕ-Tξ∥E+ϵ)α(ϕ(c),ξ(c))∥Tϕ-Tξ∥E+ϵ≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0+ϵ,
which implies ∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0. That is, all conditions of Corollary 14 hold and T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 22.
LetT:E0→E and α:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular αc-admissible mapping;(iii)
there existsβ∈ℱ such that
(74)(α(ϕ(c),ξ(c))-1+σ)∥Tϕ-Tξ∥E≤ϵβ(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0, where 1<ϵ≤σ;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and α(ϕn(c),ϕn+1(c))≥1 for all n∈ℕ∪0, then α(ϕn(c),ϕ(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α(ϕ0(c),Tϕ0)≥1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Proof.
Letα(ϕ(c),ξ(c))≥1; then by (iii) we have
(75)σ∥Tϕ-Tξ∥E≤(α(ϕ(c),ξ(c))-1+σ)∥Tϕ-Tξ∥Eσ∥Tϕ-Tξ∥E≤ϵβ(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0≤σβ(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
which implies ∥Tϕ-Tξ∥E≤β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0. That is, all conditions of Corollary 14 hold and T has a PPF dependent fixed point ϕ*∈ℛc.


## 2.2. Consequences of Corollary15

Theorem 23.
LetT:E0→E and η:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular ηc-subadmissible mapping;(iii)
there existsβ∈ℱ such that
(76)∥Tϕ-Tξ∥E≤η(ϕ(c),ξ(c))β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→∞ and η(ϕn(c),ϕn+1(c))≤1 for all n∈ℕ∪0, then η(ϕn(c),ϕ(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η(ϕ0(c),Tϕ0)≤1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 24.
LetT:E0→E and η:E×E→[0,∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular ηc-subadmissible mapping;(iii)
there existsβ∈ℱ such that
(77)∥Tϕ-Tξ∥E+ϵ≤(β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0+ϵ)η(ϕ(c),ξ(c)),
for all ϕ,ξ∈E0, where ϵ≥1 and ψ∈Ψ;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and η(ϕn(c),ϕn+1(c))≤1 for all n∈ℕ∪0, then η(ϕn(c),ϕ(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η(ϕ0(c),Tϕ0)≤1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.Theorem 25.
LetT:E0→E and η:E×E→[0,+∞) be two mappings satisfying the following assertions:(i)
there existsc∈I such that ℛc is topologically closed and algebraically closed with respect to difference;(ii)
T is a triangular ηc-subadmissible mapping;(iii)
there existsβ∈ℱ such that
(78)σ∥Tϕ-Tξ∥E≤(η(ϕ(c),ξ(c))+ϵ-1)β(∥ϕ-ξ∥E0)∥ϕ-ξ∥E0,
for all ϕ,ξ∈E0, where 1<ϵ≤σ and ψ∈Ψ;(iv)
if{ϕn} is a sequence in E0 such that ϕn→ϕ as n→+∞ and η(ϕn(c),ϕn+1(c))≤1 for all n∈ℕ∪0, then η(ϕn(c),ϕ(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η(ϕ0(c),Tϕ0)≤1.Then, T has a PPF dependent fixed point ϕ*∈ℛc.


## 2.3. Consequences of Corollary18

Theorem 26.
LetS:E0→E0, T:E0→E, and α:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair;(iii)
there existsβ∈ℱ such that
(79)α((Sϕ)(c),(Sξ)(c))∥Tϕ-Tξ∥E≤β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0,
for all ϕ,ξ∈ℛc;(iv)
if{Sϕn} is a sequence in ℛc such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥1 for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Theorem 27.
LetS:E0→E0, T:E0→E, and α:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair;(iii)
there existsβ∈ℱ such that
(80)(∥Tϕ-Tξ∥E+ϵ)α((Sϕ)(c),(Sξ)(c))≤β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0+ϵ,
for all ϕ,ξ∈ℛc where ϵ≥1;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥1 for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Theorem 28.
LetS:E0→E0, T:E0→E, and α:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is topologically closed and algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-admissible pair;(iii)
there existsβ∈ℱ such that
(81)(α((Sϕ)(c),(Sξ)(c))-1+σ)∥Tϕ-Tξ∥E≤ϵβ(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0
for all ϕ,ξ∈ℛc, where 1<ϵ≤σ;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and α((Sϕn)(c),(Sϕn+1)(c))≥1 for all n∈ℕ∪0, then α((Sϕn)(c),(Sϕ)(c))≥1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that α((Sϕ0)(c),Tϕ0)≥1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.


## 2.4. Consequences of Corollary19

Corollary 29.
LetS:E0→E0, T:E0→E, and η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-subadmissible pair;(iii)
there existsβ∈ℱ such that
(82)∥Tϕ-Tξ∥E≤η((Sϕ)(c),(Sξ)(c))β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0,
for all ϕ,ξ∈ℛc;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and η((Sϕn)(c),(Sϕn+1)(c))≤1 for all n∈ℕ∪0, then η((Sϕn)(c),(Sϕ)(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η((Sϕ0)(c),Tϕ0)≤1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Corollary 30.
LetS:E0→E0, T:E0→E, and η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-subadmissible pair;(iii)
there existsβ∈ℱ such that
(83)∥Tϕ-Tξ∥E+ϵ≤(β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0+ϵ)η((Sϕ)(c),(Sξ)(c))
for all ϕ,ξ∈ℛc, where ϵ≥1;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and η((Sϕn)(c),(Sϕn+1)(c))≤1 for all n∈ℕ∪0, then η((Sϕn)(c),(Sϕ)(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η((Sϕ0)(c),Tϕ0)≤1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.Corollary 31.
LetS:E0→E0, T:E0→E, and η:E×E→[0,+∞) be three mappings satisfying the following assertions:(i)
there existsc∈I such that S(ℛc)⊂ℛc is topologically closed and algebraically closed with respect to difference;(ii)
(S,T) is a triangular αc-subadmissible pair;(iii)
there existsβ∈ℱ such that
(84)σ∥Tϕ-Tξ∥E≤(η((Sϕ)(c),(Sξ)(c))+ϵ-1)β(∥Sϕ-Sξ∥E0)∥Sϕ-Sξ∥E0
for all ϕ,ξ∈ℛc, where 1<ϵ≤σ;(iv)
if{Sϕn} is a sequence in E0 such that Sϕn→Sϕ as n→+∞ and η((Sϕn)(c),(Sϕn+1)(c))≤1 for all n∈ℕ∪0, then η((Sϕn)(c),(Sϕ)(c))≤1 for all n∈ℕ∪0;(v)
there existsϕ0∈ℛc such that η((Sϕ0)(c),Tϕ0)≤1;(vi)
S(ℛc) is complete in ℛc;(vii)
T(ℛc)⊂{(Sϕ)(c):ϕ∈ℛc}.Then, S and T have a PPF dependent coincidence point ϕ*∈ℛc.



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*Source: 673647-2014-02-02.xml*