# NewP

Authors: M. A. Kutbi; N. Hussain; S. Khaleghizadeh
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

We introduce new concepts of anα
c-GF-contractive nonself-mapping, a weak α
c-GF-contractive nonself-mapping, a generalized α
c-GF-contractive nonself-mapping, and Suzuki type GF-contractions and establish the existence of
PPF dependent fixed point theorems for such kind of contractive nonself-mappings in the Razumikhin class. As applications of our results, we derive some PPF dependent fixed point theorems for GF-contractive nonself-mappings whenever the range space is endowed with a graph or a partial order. The obtained results generalize, extend, and modify some PPF dependent fixed point results in the literature.


---

## Body

## 1. Introduction

It is well known that the contraction mapping principle, formulated and proved in the PhD dissertation of Banach, has laid the foundation of metric fixed point theory for contraction mappings on complete metric spaces. Since then, Banach’s fixed point theorem has been generalized, improved, and extended in several directions; see the papers (see [1–11] and references therein). Bernfeld et al. [12] introduced the concept of fixed point for mappings that have different domains and ranges, which is called P
P
F dependent fixed point or the fixed point with P
P
F dependence. Also, they introduced the notion of Banach type contraction for nonself-mapping and established the existence of P
P
F dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings (see [13]). The P
P
F dependent fixed point theorems are useful for proving the solutions of nonlinear functional differential and integral equations which may depend on the past history, present data, and future consideration (see [14–16]). However, as proved in a recent paper by Cho et al. [17], the starting conditions [imposed by the problem setting] relative to the ambient Razumikhin class R
c may be converted into starting conditions relative to the constant class K; so, ultimately, we may arrange for these P
P
F dependent fixed point results holding over K.On the other hand, Samet et al. [18] introduced and studied α-ψ-contractive mappings in complete metric spaces and provided applications of the results to ordinary differential equations. More recently, Salimi et al. [19] modified the notions of α-ψ-contractive and α-admissible mappings and established fixed point theorems to modify the results in [18].Consistent with Wardowski [20], we denote by Δ
F the set of all functions F
:
R
+
→
R satisfying the following conditions:(
F
1)
F is strictly increasing;
(F
2)
for all sequence{
α
n
}
⊆
R
+, lim
n
→
∞
α
n
=
0 if and only if lim
n
→
∞
F
(
α
n
)
=
-
∞;
(F
3)
there exists0
<
k
<
1 such that lim
n
→
0
+
α
k
F
(
α
)
=
0.A mappingT
:
X
→
X is called an F
-contraction if there exists τ
>
0 such that, for all x
,
y
∈
X, we haveΥ
d
T
x
,
T
y
>
0
⟹
τ
+
F
d
T
x
,
T
y
≤
F
d
x
,
y
.For example, from theF-contraction of T, we have the following.Define a mappingF
:
R
+
→
R by F
(
α
)
=
ln
⁡
α for all α
∈
R
+. Then it is clear that F satisfies (F
1)–(F
3). Each mapping T
:
X
→
X satisfying Υ is an F-contraction such that (1)
d
T
x
,
T
y
≤
e
-
τ
d
x
,
y
,for all x
,
y
∈
X with T
x
≠
T
y. Clearly, for all x
,
y
∈
X with T
x
=
T
y, the inequality (2)
d
(
T
x
,
T
y
)
≤
e
-
τ
d
(
x
,
y
)also holds and this implies that T is a Banach contraction.Further, consistent with Hussain and Salimi [21], we denote by Δ
G the set of all functions G
:
R
+
4
→
R
+ satisfying the following.(G)
For allt
1
,
t
2
,
t
3
,
t
4
∈
R
+ with t
1
t
2
t
3
t
4
=
0 there exists τ
>
0 such that G
(
t
1
,
t
2
,
t
3
,
t
4
)
=
τ.Example 1 (see [21]).
IfG
(
t
1
,
t
2
,
t
3
,
t
4
)
=
L
min
⁡
{
t
1
,
t
2
,
t
3
,
t
4
}
+
τ where L
∈
R
+ and τ
>
0, then G
∈
Δ
G.Example 2 (see [21]).
IfG
(
t
1
,
t
2
,
t
3
,
t
4
)
=
τ
e
L
min
⁡
{
t
1
,
t
2
,
t
3
,
t
4
} where L
∈
R
+ and τ
>
0, then G
∈
Δ
G.Example 3 (see [21]).
IfG
(
t
1
,
t
2
,
t
3
,
t
4
)
=
L
ln
⁡
(
min
⁡
{
t
1
,
t
2
,
t
3
,
t
4
}
+
1
)
+
τ where L
∈
R
+ and τ
>
0, then G
∈
Δ
G.In this paper, motivated by the works of Hussain and Salimi, Samet et al., and Wardowski we introduce the concept of a 0-α
c-admissible nonself-mapping, a α
c-G
F-contractive nonself-mapping, a weak α
c-G
F-contractive nonself-mapping, a generalized α
c-G
F-contractive nonself-mapping, and Suzuki type G
F-contractive nonself-mapping and establish the existence of P
P
F dependent fixed point theorems for such kinds of contractive nonself-mappings in the Razumikhin class. As applications of our theorems, we deduce some P
P
F dependent fixed point theorems for G
F-contractive nonself-mapping and integral type contractions whenever the range space is endowed with a graph or a partial order.


## 2. Preliminaries

Throughout this paper, we assume that(
E
,
·
E
) is a Banach space, I denotes a closed interval [
a
,
b
] in R, and E
0
=
C
(
I
,
E
) denotes the set of all continuous E-valued functions on I equipped with the supremum norm ·
E
0 defined by(3)
ϕ
E
0
=
sup
⁡
t
∈
I
ϕ
(
t
)
E
.Here,N
=
{
0,1
,
…
} denotes the set of all natural numbers; in addition, for each h
∈
N, we put N
h
=
{
n
∈
N
;
h
≤
n
}.Definition 4 (see [12]).
A mappingϕ
∈
E
0 is said to be a P
P
F dependent fixed point or a fixed point with P
P
F dependence of mapping T
:
E
0
→
E if T
ϕ
=
ϕ
(
c
) for some c
∈
I.Motivated by results of Agarwal et al. [22], Ćirić et al. [23], and Hussain et al. [15], we give the following notion which is suitable for our main results.Definition 5.
Letc
∈
I and T
:
E
0
→
E and α
:
E
×
E
→
R. T is called a 0-α
c-admissible nonself-mapping if, for any ϕ
,
ξ
∈
E
0,(4)
α
(
ϕ
(
c
)
,
ξ
(
c
)
)
>
0
⟹
α
(
T
ϕ
,
T
ξ
)
>
0
.Example 6.
LetE
=
R be a real Banach space with usual norm and I
=
[
0,1
]. Define T
:
E
0
→
E by T
ϕ
=
1
/
2
ϕ
(
1
) for all ϕ
∈
E
0 and α
:
E
×
E
→
R by(5)
α
(
x
,
y
)
=
2
,
if
x
≥
y
-
1
,
otherwise
;then T is an 0-α
1-admissible nonself-mapping.Definition 7.
LetT
:
E
0
→
E and α
:
E
×
E
→
R be two nonself-mappings and c
∈
I.(i)
T is called a α
c-G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0, (6)
α
(
ϕ
(
c
)
,
ξ
(
c
)
)
G
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
-
ξ
E
0
k
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
(
c
)
-
T
ξ
E
,
ξ
(
c
)
-
T
ϕ
E
kkkkkkkkkkk
+
F
T
ϕ
-
T
ξ
E
-
F
ϕ
-
ξ
E
0
≤
0
;
(ii)
T is called aweak
α
c
-
G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0, (7)
α
(
ϕ
(
c
)
,
ξ
(
c
)
)
·
G
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
-
ξ
E
0
k
k
k
k
k
k
ϕ
c
-
T
ξ
E
,
ξ
c
-
T
ϕ
E
+
F
(
T
ϕ
-
T
ξ
E
)
k
k
k
k
-
F
max
⁡
⁡
ϕ
-
ξ
E
0
,
ϕ
c
-
T
ϕ
E
,
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
-
ξ
E
0
ξ
c
-
T
ξ
E
≤
0
;
(iii)
T is called a generalized α
c
-
G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0, (8)
α
(
ϕ
(
c
)
,
ξ
(
c
)
)
·
G
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
k
k
k
k
k
k
k
ϕ
(
c
)
-
T
ξ
E
,
ξ
(
c
)
-
T
ϕ
E
+
F
(
T
ϕ
-
T
ξ
E
)
k
k
k
k
-
F
max
⁡
ϕ
-
ξ
E
0
,
ϕ
c
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
≤
0
.Definition 8 (see [17]).
The Razumikhin or minimal class (attached toc) is defined as(9)
R
c
=
{
ϕ
∈
E
0
:
ϕ
E
0
=
ϕ
(
c
)
E
}
.
Also, denote, for simplicity,(10)
R
c
0
=
{
ϕ
∈
R
c
:
ϕ
is
constant
function
}
.It will be referred as the constant Razumikhin class. To get a useful representation for this subclass, we need a lot of preliminary facts. For each u
∈
E, let H
[
u
] denote the constant function of E
0, defined as (11)
H
[
u
]
(
t
)
=
u
,
t
∈
I
.Note that, by this definition,(12)
H
[
u
]
E
0
=
u
E
,
H
u
c
=
u
,whence H
[
u
]
∈
R
c. We now claim that(13)
R
c
0
=
{
H
[
u
]
;
u
∈
E
}
;or, in other words, the constant Razumikhin class R
c
0 is just the subclass of all constant functions in E
0. In fact, the right to left inclusion is clear. For the left to right inclusion, it will suffice noting that any constant function ψ in R
c may be written as (14)
ψ
=
H
u
,
for
some
u
∈
E
;and this ends our argument.The following properties of this subclass are almost immediate, so, we do not give details.Proposition 9 (see [17]).
Under the above conventions,(i)
H
[
u
+
v
]
=
H
[
u
]
+
H
[
v
], ∀
u
,
v
∈
E,
(ii)
H
[
λ
u
]
=
λ
H
[
u
], ∀
λ
∈
R, ∀
u
∈
E,
(iii)
u
E
=
H
[
u
]
E
0, ∀
u
∈
E,
(iv)
the mappingu
↦
H
[
u
] is an algebraic and topological isomorphism between (
E
,
·
E
) and (
R
c
0
,
·
E
0
).Definition 10.
Letα
:
E
×
E
→
R be a nonself-mapping.(i)
T is called (
R
c
,
α
)
-starting, if there exists ϕ
0
∈
R
c such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.
(ii)
T is called (
R
c
0
,
α
)
-starting, if there exists ϕ
0
∈
R
c
0 such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.
Clearly, ifT is (
R
c
0
,
α
)-starting, then it is also (
R
c
,
α
)-starting. The reciprocal assertion is also true, under certain regularity conditions upon T. Precisely, we have the following.Proposition 11.
Let the nonself-mappingα
:
E
×
E
→
R be such that (i)
T is 0-α
c-admissible;
(ii)
T is (
R
c
,
α
)-starting.Then, T is (
R
c
0
,
α
)-starting.Proof.
By (ii) there existϕ
0
∈
R
c, such that (15)
α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0
.Since T
ϕ
0
∈
E, we may consider the element ξ
0
=
H
[
T
ϕ
0
] from the constant Razumikhin class R
c
0; this, by definition, means that(16)
ξ
0
t
=
T
ϕ
0
,
∀
t
∈
I
;
hence
,
ξ
0
(
c
)
=
T
ϕ
0
.The condition upon ϕ
0 becomes (17)
α
(
ϕ
0
(
c
)
,
ξ
0
(
c
)
)
>
0
.Since T is 0-α-admissible, this yields (18)
α
(
T
ϕ
0
,
T
ξ
0
)
>
0
,or, equivalently, (19)
α
(
ξ
0
(
c
)
,
T
ξ
0
)
>
0
.This ends the argument.The usefulness of this result is to be judged from the observations below. To state them, two more concepts involving the Razumikhin classR
c are needed.Definition 12.
(i) The classR
c is said to be algebraically closed with respect to difference if ϕ
-
ξ
∈
R
c whenever ϕ
,
ξ
∈
R
c.
(ii) The classR
c is said to betopologically closed if it is closed with respect to the topology on E
0 generated by the norm ·
E
0.


## 3. Main Results

We start with the following proposition which will be crucial to our main results.Proposition 13.
Letc
∈
I, T
:
E
0
→
E, α
:
E
×
E
→
R, and F
∈
Δ
F be such that(i)
T is 0-α
c-admissible,
(ii)
T is a generalized α
c-G
F-contraction,
(iii)
there existsϕ
0
∈
R
c such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.In addition, assume that(Ze)
T has no P
P
F dependent fixed points in R
c
0 (T
ϕ
≠
ϕ
(
c
), for all ϕ
∈
R
c
0).Then, there exist a sequence (
ϕ
n
) in R
c
0, a ϕ
∗
∈
R
c
0, and a h
∈
N, such that (c1)
T
ϕ
n
=
ϕ
n
+
1
(
c
) and α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N;
(c2)
ϕ
n
→
ϕ
∗ as n
→
∞;
(c3)
T
ϕ
n
≠
T
ϕ
∗ (hence, ϕ
n
≠
ϕ
∗), for all n
∈
N
h.Proof.
By a previous observation, condition (iii) may be written as(iv)
there existsϕ
0
∈
R
c
0 such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.Let ϕ
0
∈
R
c
0 be such an element. Since T
ϕ
0
∈
E, we may consider the element ϕ
1
=
H
[
T
ϕ
0
] from the constant Razumikhin class R
c
0; this, by definition, means that(20)
ϕ
1
(
t
)
=
T
ϕ
0
,
∀
t
∈
I
;
hence
,
ϕ
1
(
c
)
=
T
ϕ
0
.Further, since T
ϕ
1
∈
E, we may consider the element ϕ
2
=
H
[
T
ϕ
1
] from the constant Razumikhin class R
c
0; this, by definition, means that(21)
ϕ
2
(
t
)
=
T
ϕ
1
,
∀
t
∈
I
;
hence
,
ϕ
2
(
c
)
=
T
ϕ
1
.The process may continue indefinitely; it gives us a sequence (
ϕ
n
) in the constant Razumikhin class R
c
0, with(22)
∀
n
∈
N
1
:
ϕ
n
t
=
T
ϕ
n
-
1
,
∀
t
∈
I
;
hence
,
ϕ
n
(
c
)
=
T
ϕ
n
-
1
.By the algebraic-topological properties of the constant Razumikhin class R
c
0, it follows that(23)
ϕ
n
-
1
-
ϕ
n
E
0
=
ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
,for all n
∈
N
1. Since T is 0-α
c-admissible and(24)
α
(
ϕ
0
(
c
)
,
ϕ
1
(
c
)
)
=
α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0
,then(25)
α
(
ϕ
1
(
c
)
,
ϕ
2
(
c
)
)
=
α
(
T
ϕ
0
,
T
ϕ
1
)
>
0
.Again since T is 0-α
c-admissible, then(26)
α
(
ϕ
2
(
c
)
,
ϕ
3
(
c
)
)
>
0
.By continuing this process, we have(27)
α
(
ϕ
n
-
1
(
c
)
,
ϕ
n
(
c
)
)
>
0
,for all n
∈
N
1; and this proves the conclusion (c1).
By the imposed condition on nonself-mappingT, a relation like (28)
∃
h
∈
N
:
T
ϕ
h
+
1
=
ϕ
h
+
1
c
=
T
ϕ
his impossible, so that, (29)
T
ϕ
n
≠
T
ϕ
n
+
1
(
hence
,
ϕ
n
≠
ϕ
n
+
1
)
,
∀
n
∈
N
.
Now sinceT is generalized α
c-G
F-contraction, so we have(30)
α
ϕ
n
-
1
c
,
ϕ
n
c
·
G
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
,
ϕ
n
(
c
)
-
T
ϕ
n
E
,
ϕ
n
-
1
c
-
T
ϕ
n
E
2
k
k
k
k
k
k
k
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
,
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
k
k
k
k
+
F
T
ϕ
n
-
1
-
T
ϕ
n
E
k
k
k
k
-
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
,
ϕ
n
-
1
c
-
T
ϕ
n
E
2
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
(
c
)
-
T
ϕ
n
E
,
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
1
c
-
T
ϕ
n
E
+
ϕ
n
c
-
T
ϕ
n
-
1
E
2
≤
0
.On the other hand,(31)
G
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
,
ϕ
n
(
c
)
-
T
ϕ
n
E
,
k
k
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
,
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
=
G
ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
,
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
k
k
k
k
k
k
ϕ
n
-
1
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
ϕ
n
(
c
)
-
ϕ
n
(
c
)
E
=
G
ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
,
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
k
k
k
k
k
k
ϕ
n
-
1
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
0
.Now since ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
·
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
·
ϕ
n
-
1
(
c
)
-
ϕ
n
+
1
(
c
)
E
·
0
=
0, so from (G) there exists τ
>
0 such that,(32)
G
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
,
ϕ
n
(
c
)
-
T
ϕ
n
E
,
k
k
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
,
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
=
τ
.Since α
(
ϕ
1
(
c
)
,
T
ϕ
1
)
>
0, so by (30) and (32) we obtain(33)
τ
+
F
(
T
ϕ
n
-
1
-
T
ϕ
n
E
)
≤
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
,
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
+
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
2
k
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
(
c
)
-
T
ϕ
n
E
,
k
k
l
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
+
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
2
.Therefore,(34)
τ
+
F
ϕ
n
c
-
ϕ
n
+
1
c
E
=
τ
+
F
T
ϕ
n
-
1
-
T
ϕ
n
E
≤
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
,
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
+
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
2
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
(
c
)
-
T
ϕ
n
E
,
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
+
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
2
=
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
,
ϕ
n
-
1
(
c
)
-
ϕ
n
+
1
(
c
)
E
+
ϕ
n
(
c
)
-
ϕ
n
(
c
)
E
2
k
k
k
k
k
k
k
k
l
k
k
k
k
k
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
1
(
c
)
-
ϕ
n
+
1
(
c
)
E
+
ϕ
n
(
c
)
-
ϕ
n
(
c
)
E
2
=
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
1
-
ϕ
n
+
1
E
0
+
ϕ
n
-
ϕ
n
E
0
2
k
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
ϕ
n
+
1
E
0
,
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
1
-
ϕ
n
+
1
E
0
+
ϕ
n
-
ϕ
n
E
0
2
=
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
ϕ
n
+
1
E
0
,
ϕ
n
-
1
-
ϕ
n
+
1
E
0
+
ϕ
n
-
ϕ
n
E
0
2
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
1
-
ϕ
n
+
1
E
0
+
ϕ
n
-
ϕ
n
E
0
2
≤
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
ϕ
n
+
1
E
0
,
ϕ
n
-
1
-
ϕ
n
E
0
+
ϕ
n
-
ϕ
n
+
1
E
0
2
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
-
1
-
ϕ
n
E
0
+
ϕ
n
-
ϕ
n
+
1
E
0
2
=
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
ϕ
n
+
1
E
0
,which implies that(35)
F
ϕ
n
-
ϕ
n
+
1
E
0
≤
F
max
⁡
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
ϕ
n
+
1
E
0
-
τ
.
Now, ifmax
⁡
{
ϕ
n
-
1
-
ϕ
n
E
0
,
ϕ
n
-
ϕ
n
+
1
E
0
}
=
ϕ
n
-
ϕ
n
+
1
E
0, then we have(36)
F
ϕ
n
-
ϕ
n
+
1
E
0
≤
F
ϕ
n
-
ϕ
n
+
1
E
0
-
τ
<
F
ϕ
n
-
ϕ
n
+
1
E
0
,which is a contradiction. Therefore, we have (37)
F
ϕ
n
-
ϕ
n
+
1
E
0
≤
F
ϕ
n
-
1
-
ϕ
n
E
0
-
τ
,and so(38)
F
ϕ
n
-
ϕ
n
+
1
E
0
≤
F
(
ϕ
n
-
1
-
ϕ
n
E
0
)
-
τ
≤
F
(
ϕ
n
-
2
-
ϕ
n
-
1
E
0
)
-
2
τ
≤
⋯
≤
F
(
ϕ
0
-
ϕ
1
E
0
)
-
n
τ
.Hence lim
n
→
∞
F
(
ϕ
n
-
ϕ
n
+
1
E
0
)
=
-
∞ which implies lim
n
→
∞
ϕ
n
-
ϕ
n
+
1
E
0
=
0, since F
∈
Δ
F. Further, there exists k
∈
(
0,1
) such that(39)
lim
n
→
∞
ϕ
n
-
ϕ
n
+
1
E
0
k
F
(
ϕ
n
-
ϕ
n
+
1
E
0
)
=
0
.From (38), it follows that(40)
ϕ
n
-
ϕ
n
+
1
E
0
k
·
F
ϕ
n
-
ϕ
n
+
1
E
0
-
F
ϕ
0
-
ϕ
1
E
0
≤
-
n
τ
ϕ
n
-
ϕ
n
+
1
E
0
k
≤
0
.By taking limit as n
→
∞ we have(41)
lim
n
→
∞
n
ϕ
n
-
ϕ
n
+
1
E
0
k
=
0
.Therefore, there exists n
1
∈
N such that(42)
n
ϕ
n
-
ϕ
n
+
1
E
0
k
≤
1
,for all n
≥
n
1. This implies that(43)
ϕ
n
-
ϕ
n
+
1
E
0
≤
1
n
1
/
k
,for all n
≥
n
1. Now, for all m
>
n
≥
n
1, we obtain (44)
ϕ
n
-
ϕ
m
E
0
≤
∑
i
=
n
m
-
1
‍
ϕ
i
-
ϕ
i
+
1
E
0
≤
∑
i
=
n
m
-
1
‍
1
i
1
/
k
.Since 0
<
k
<
1, the series ∑
i
≥
1
‍
(
1
/
i
1
/
k
) converges. Therefore, ϕ
n
-
ϕ
m
E
0
→
0 as m
,
n
→
∞, which tells us that (
ϕ
n
) is a Cauchy sequence. Completeness of R
c
0 ensures that there exists ϕ
∗
∈
R
c
0 such that ϕ
n
→
ϕ
∗ as n
→
∞. As a consequence, conclusion (c2) holds too. Finally, assume that conclusion (c3) is not true. Consider(45)
∀
n
∈
N
,
∃
m
>
n
:
T
ϕ
∗
=
T
ϕ
m
=
ϕ
m
+
1
c
.This tells us that there exists an infinite sequence (
k
(
n
)
) in N, with (46)
T
ϕ
∗
=
ϕ
k
n
c
,
∀
n
∈
N
.Passing to limit as n
→
∞ we get T
ϕ
∗
=
ϕ
∗
(
c
), which is contradiction to the imposed hypothesis. Hence, the conclusion (c3) holds too and the proof is complete.Theorem 14.
Letc
∈
I, T
:
E
0
→
E, α
:
E
×
E
→
R, and F
∈
Δ
F be such that(i)
T is 0-α
c-admissible,
(ii)
T is an α
c-G
F-contraction,
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N, then α
(
ϕ
n
(
c
)
,
ϕ
(
c
)
)
>
0 for all n
∈
N,
(iv)
there existsϕ
0
∈
R
c such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.Then T has a P
P
F dependent fixed point in R
c
0.Proof.
SinceF is strictly increasing function, every α
c-G
F-contraction is a generalized α
c-G
F-contraction. Thus all conditions of Proposition 13 hold and hence there exist a sequence (
ϕ
n
) in R
c
0, a ϕ
∗
∈
R
c
0 and h
∈
N such that(c1)
T
ϕ
n
=
ϕ
n
+
1
(
c
), α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N,
(c2)
ϕ
n
→
ϕ
∗ as n
→
∞,
(c3)
T
ϕ
n
≠
T
ϕ
∗ (hence, ϕ
n
≠
ϕ
∗), for all n
∈
N
h.Since T is an α
c-G
F-contraction, we have, ∀
n
∈
N
h, (47)
α
ϕ
n
c
,
ϕ
∗
c
·
G
ϕ
n
-
T
ϕ
n
E
,
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
ϕ
n
-
ϕ
∗
E
0
k
k
k
k
k
k
ϕ
n
(
c
)
-
T
ϕ
∗
E
,
ϕ
∗
(
c
)
-
T
ϕ
n
E
k
k
k
+
F
T
ϕ
n
-
T
ϕ
∗
E
-
F
ϕ
n
-
ϕ
∗
E
0
≤
0
,which implies(48)
G
ϕ
n
-
T
ϕ
n
E
,
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
ϕ
n
(
c
)
-
T
ϕ
∗
E
,
k
k
k
ϕ
∗
c
-
T
ϕ
n
E
+
F
(
T
ϕ
n
-
T
ϕ
∗
E
)
≤
F
(
ϕ
n
-
ϕ
∗
E
0
)
,since α
(
ϕ
n
(
c
)
,
ϕ
∗
(
c
)
)
>
0. Now we obtain(49)
F
(
T
ϕ
n
-
T
ϕ
∗
E
)
≤
G
ϕ
n
-
T
ϕ
n
E
,
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
ϕ
n
(
c
)
-
T
ϕ
∗
E
,
k
k
k
k
k
k
l
ϕ
∗
(
c
)
-
T
ϕ
n
E
+
F
(
T
ϕ
n
-
T
ϕ
∗
E
)
≤
F
(
ϕ
n
-
ϕ
∗
E
0
)
.Since F
∈
Δ
F, we get(50)
T
ϕ
n
-
T
ϕ
∗
E
≤
ϕ
n
-
ϕ
∗
E
0
,and hence(51)
T
ϕ
∗
-
ϕ
∗
(
c
)
E
≤
T
ϕ
∗
-
T
ϕ
n
E
+
T
ϕ
n
-
ϕ
∗
(
c
)
E
=
T
ϕ
∗
-
T
ϕ
n
E
+
ϕ
n
+
1
(
c
)
-
ϕ
∗
(
c
)
E
0
≤
ϕ
∗
-
ϕ
n
E
0
+
ϕ
n
+
1
-
ϕ
∗
E
0
.Taking limit as n
→
∞ in the above inequality, we have T
ϕ
∗
-
ϕ
∗
(
c
)
E
=
0; that is, T
ϕ
∗
=
ϕ
∗
(
c
). This completes the proof.Example 15.
Let(
E
,
·
E
) be a Banach space where E
=
R and x
E
=
x and E
0
=
C
(
[
0,1
]
,
E
) denote the set of all continuous E-valued functions on [
0,1
] equipped with the supremum norm ·
E
0 defined by(52)
ϕ
E
0
=
sup
⁡
t
∈
I
ϕ
(
t
)
E
.
DefineT
:
E
0
→
E, α
:
E
×
E
→
[
0
,
∞
), G
:
R
+
4
→
R
+, and F
:
R
+
→
R by(53)
T
ϕ
=
ϕ
1
3
+
1
,
if
ϕ
1
<
0
1
2
e
-
τ
ϕ
1
2
,
if
0
≤
ϕ
1
≤
1
5
ϕ
1
if
ϕ
1
>
1
,
α
x
,
y
=
3
,
if
x
,
y
∈
0,1
0
,
otherwise
,
G
t
1
,
t
2
,
t
3
,
t
4
=
τ
h
h
h
h
h
h
h
h
h
h
h
h
h
h
where
τ
>
0
,
F
r
=
ln
⁡
r
.
Letα
(
ϕ
(
1
)
,
ψ
(
1
)
)
>
0. Then, 0
≤
ϕ
(
1
)
≤
1 and 0
≤
ψ
(
1
)
≤
1 and so 0
≤
T
ϕ
=
1
/
2
e
-
τ
[
ϕ
(
1
)
]
2
≤
1 and 0
≤
T
ψ
=
1
/
2
e
-
τ
[
ψ
(
1
)
]
2
≤
1; that is, α
(
T
ϕ
,
T
ψ
)
>
0. Therefore T is 0-α
1-admissible mapping. Let (
ϕ
n
) be a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and α
(
ϕ
n
(
1
)
,
ϕ
n
+
1
(
1
)
)
>
0 for all n
∈
N. Now since 0
≤
ϕ
n
(
1
)
≤
1 for all n
∈
N and ϕ
n
→
ϕ as n
→
∞, so 0
≤
ϕ
(
1
)
≤
1; that is, α
(
ϕ
n
(
1
)
,
ϕ
(
1
)
)
>
0 for all n
∈
N. Clearly, α
(
0
,
T
0
)
>
0.
Let0
≤
ϕ
(
1
)
≤
1 and 0
≤
ψ
(
1
)
≤
1. Then α
(
ϕ
(
1
)
,
ψ
(
1
)
)
=
3, 0
≤
T
ϕ
=
1
/
2
e
-
τ
[
ϕ
(
1
)
]
2
≤
1,  0
≤
T
ψ
=
1
/
2
e
-
τ
[
ψ
(
1
)
]
2
≤
1, and(54)
T
ϕ
-
T
ψ
E
=
1
2
e
-
τ
ϕ
1
2
-
ψ
1
2
=
1
2
e
-
τ
ϕ
1
-
ψ
1
ϕ
1
+
ψ
1
≤
e
-
τ
ϕ
1
-
ψ
1
≤
e
-
τ
sup
⁡
t
∈
0,1
ϕ
t
-
ψ
t
=
e
-
τ
ϕ
-
ψ
E
0
.Therefore,(55)
α
(
ϕ
(
1
)
,
ψ
(
1
)
)
·
G
ϕ
c
-
T
ϕ
E
,
ψ
c
-
T
ψ
E
,
ϕ
c
-
T
ψ
E
,
ϕ
-
ψ
E
0
k
k
k
k
k
k
k
k
ψ
(
c
)
-
T
ϕ
E
k
k
k
k
+
F
T
ϕ
-
T
ψ
E
-
F
ϕ
-
ψ
E
0
=
3
τ
+
ln
⁡
T
ϕ
-
T
ψ
E
-
ln
⁡
ϕ
-
ψ
E
0
≤
3
τ
+
ln
⁡
e
-
τ
ϕ
-
ψ
E
0
-
ln
⁡
ϕ
-
ψ
E
0
=
0
.Otherwise, α
(
ϕ
(
1
)
,
ψ
(
1
)
)
=
0 implies that(56)
α
(
ϕ
(
1
)
,
ψ
(
1
)
)
·
G
ϕ
(
c
)
-
T
ϕ
E
,
ψ
(
c
)
-
T
ψ
E
,
ϕ
(
c
)
-
T
ψ
E
,
ϕ
-
ψ
E
0
k
k
k
k
k
k
k
ψ
c
-
T
ϕ
E
k
k
k
+
F
(
T
ϕ
-
T
ψ
E
)
-
F
ϕ
-
ψ
E
0
=
0
.Hence, T is an α
c-G
F-contraction and all conditions of Theorem 14 hold. Thus T has a P
P
F dependent fixed point. Here, ϕ
≡
0 is P
P
F dependent fixed point of T.
Further, defineϕ
(
t
)
=
1
-
(
t
-
1
)
2 and ψ
(
t
)
=
1 for all t
∈
[
0,1
]. Clearly, ϕ
,
ψ
∈
R
1. Note that(57)
ψ
-
ϕ
E
0
=
sup
⁡
t
∈
[
0,1
]
t
-
1
2
=
1
≠
0
=
(
ψ
-
ϕ
)
(
1
)
E
.Thus ψ
-
ϕ
∉
R
1 and hence, in this case, R
1 is not algebraically closed with respect to difference. Therefore it is interesting to notice that almost all results on P
P
F dependent fixed points established in [15, 22, 23] cannot be applied to this example.If in Theorem14 we take α
(
ϕ
(
c
)
,
ξ
(
c
)
)
=
1 for all ϕ
,
ξ
∈
E, then we derive following result.Corollary 16.
Let the nonself-mappingT
:
E
0
→
E and F
∈
Δ
F be such that there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0,(58)
G
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
k
k
k
ϕ
(
c
)
-
T
ξ
E
,
ξ
(
c
)
-
T
ϕ
E
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
(
ϕ
-
ξ
E
0
)
.Then T has a P
P
F dependent fixed point in R
c
0.Theorem 17.
Let the nonself-mappingT
:
E
0
→
E, α
:
E
×
E
→
R, and the F
∈
Δ
F be such that(i)
T is 0-α
c-admissible,
(ii)
T is generalized α
c-G
F-contraction such that F and G are continuous,
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N, then α
(
ϕ
n
(
c
)
,
ϕ
(
c
)
)
>
0 for all n
∈
N,
(iv)
there existsϕ
0
∈
R
c such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.Then T has a P
P
F dependent fixed point in R
c
0.Proof.
SinceT is generalized α
c-G
F-contraction, it follows from Proposition 13 that there exist a sequence (
ϕ
n
) in R
c
0, a ϕ
∗
∈
R
c
0, and a h
∈
N such that(c1)
T
ϕ
n
=
ϕ
n
+
1
(
c
), α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N,
(c2)
ϕ
n
→
ϕ
∗ as n
→
∞,
(c3)
T
ϕ
n
≠
T
ϕ
∗ (hence, ϕ
n
≠
ϕ
∗), for all n
∈
N
h.Since T is generalized α
c-G
F-contraction and α
(
ϕ
n
(
c
)
,
ϕ
∗
(
c
)
)
>
0 for all n
∈
N, we obtain(59)
G
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
ϕ
n
(
c
)
-
T
ϕ
∗
E
,
k
k
k
ϕ
∗
(
c
)
-
ϕ
n
+
1
(
c
)
E
+
F
(
ϕ
n
+
1
(
c
)
-
T
ϕ
∗
E
)
=
G
ϕ
n
(
c
)
-
T
ϕ
n
E
,
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
ϕ
n
(
c
)
-
T
ϕ
∗
E
,
k
k
k
k
k
k
k
ϕ
∗
c
-
T
ϕ
n
E
+
F
(
T
ϕ
n
-
T
ϕ
∗
E
)
≤
F
max
⁡
ϕ
n
-
ϕ
∗
E
0
,
ϕ
n
(
c
)
-
T
ϕ
n
E
,
ϕ
n
(
c
)
-
T
ϕ
∗
E
+
ϕ
∗
(
c
)
-
T
ϕ
n
E
2
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
(
c
)
-
T
ϕ
∗
E
+
ϕ
∗
(
c
)
-
T
ϕ
n
E
2
=
F
max
⁡
ϕ
n
-
ϕ
∗
E
0
,
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
ϕ
n
(
c
)
-
T
ϕ
∗
E
+
ϕ
∗
(
c
)
-
ϕ
n
+
1
(
c
)
E
2
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
k
k
k
k
k
k
k
k
k
k
k
ϕ
n
(
c
)
-
T
ϕ
∗
E
+
ϕ
∗
(
c
)
-
ϕ
n
+
1
(
c
)
E
2
.Now, since F and G are continuous, by taking limit as n
→
∞, in the above inequality, we obtain(60)
τ
+
F
(
ϕ
∗
(
c
)
-
T
ϕ
∗
E
)
≤
F
(
ϕ
∗
(
c
)
-
T
ϕ
∗
E
)
,which is a contradiction. Therefore, we have ϕ
∗
(
c
)
-
T
ϕ
∗
E
=
0; that is, ϕ
∗
(
c
)
=
T
ϕ
∗. This completes the proof.If in Theorem17 we take α
(
ϕ
(
c
)
,
ξ
(
c
)
)
=
1 for all ϕ
,
ξ
∈
E, then we derive the following corollary.Corollary 18.
Let the nonself-mappingT
:
E
0
→
E, F
∈
Δ
F, and G
∈
Δ
G be such that, for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0, one has(61)
G
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
,
k
k
k
ξ
(
c
)
-
T
ϕ
E
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
max
⁡
ϕ
-
ξ
E
0
,
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
k
k
k
k
k
k
k
k
k
k
k
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
.
ThenT has a P
P
F dependent fixed point in R
c
0.Similarly, we can prove the following theorem.Theorem 19.
Let the nonself-mappingT
:
E
0
→
E, α
:
E
×
E
→
R, and F
∈
Δ
F be such that(i)
T is 0-α
c-admissible;
(ii)
T is a weak α
c-G
F-contraction and F is continuous;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N, then α
(
ϕ
n
(
c
)
,
ϕ
(
c
)
)
>
0 for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.Then T has a P
P
F dependent fixed point in R
c
0.If in Theorem19 we take α
(
ϕ
(
c
)
,
ξ
(
c
)
)
=
1 for all ϕ
,
ξ
∈
E, then we derive the following corollary.Corollary 20.
Let the nonself-mappingT
:
E
0
→
E, F
∈
Δ
F, and G
∈
Δ
G be such that, for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0, one has(62)
G
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
,
k
k
k
ξ
(
c
)
-
T
ϕ
E
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
max
⁡
ϕ
-
ξ
E
0
,
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
EThen T has a P
P
F dependent fixed point in R
c
0.Now, motivated by Suzuki [24], we prove the following Suzuki type results for α
c-G
F-contractions in Razumikhin class.Theorem 21.
Let the nonself-mappingT
:
E
0
→
E and α
:
E
×
E
→
R be such that(i)
T is 0-α
c-admissible;
(ii)
there existF
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0 and 1
/
2
ϕ
(
c
)
-
T
ϕ
E
≤
ϕ
-
ξ
E
0, one has(63)
α
(
ϕ
(
c
)
,
ξ
(
c
)
)
·
G
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
,
k
k
k
k
k
k
ξ
(
c
)
-
T
ϕ
E
k
k
k
k
+
F
(
T
ϕ
-
T
ξ
E
)
-
F
(
ϕ
-
ξ
E
0
)
≤
0
;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N, then α
(
ϕ
n
(
c
)
,
ϕ
(
c
)
)
>
0 for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.Then T has a P
P
F dependent fixed point in R
c
0.Proof.
By previous observation, condition (iv) may be written as(v)
there existsϕ
0
∈
R
c
0 such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0.Since T
ϕ
0
∈
E, we may consider the element ϕ
1
=
H
[
T
ϕ
0
] from the constant Razumikhin class R
c
0; this, by definition, means that(64)
ϕ
1
(
t
)
=
T
ϕ
0
,
∀
t
∈
I
;
hence
,
ϕ
1
(
c
)
=
T
ϕ
0
.Further, since T
ϕ
1
∈
E, we may consider the element ϕ
2
=
H
[
T
ϕ
1
] from the constant Razumikhin class R
c
0; this, by definition, means that(65)
ϕ
2
(
t
)
=
T
ϕ
1
,
∀
t
∈
I
;
hence
,
ϕ
2
(
c
)
=
T
ϕ
1
.The process may continue indefinitely; it gives us a sequence (
ϕ
n
) in R
c
0, with(66)
∀
n
∈
N
1
:
ϕ
n
t
=
T
ϕ
n
-
1
,
∀
t
∈
I
;
hence
,
ϕ
n
c
=
T
ϕ
n
-
1
.By the algebraic-topological properties of the constant Razumikhin class R
c
0, it follows that(67)
ϕ
n
-
1
-
ϕ
n
E
0
=
ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
,for all n
∈
N
1. Since T is 0-α
c-admissible and(68)
α
(
ϕ
0
(
c
)
,
ϕ
1
(
c
)
)
=
α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0
,so(69)
α
(
ϕ
1
(
c
)
,
ϕ
2
(
c
)
)
=
α
(
T
ϕ
0
,
T
ϕ
1
)
>
0
.Again since T is a 0-α
c-admissible nonself-mapping, so(70)
α
(
ϕ
2
(
c
)
,
ϕ
3
(
c
)
)
>
0
.By continuing this process, we have(71)
α
(
ϕ
n
-
1
(
c
)
,
ϕ
n
(
c
)
)
>
0
,for all n
∈
N
1. By the imposed condition about our nonself-mapping T, a relation like (72)
T
ϕ
h
=
ϕ
h
c
=
T
ϕ
h
-
1
,
for
some
h
∈
N
1
,is impossible, so that, we must have (73)
T
ϕ
n
≠
T
ϕ
n
+
1
hence
,
ϕ
n
≠
ϕ
n
+
1
,
∀
n
∈
N
.
Now, we have(74)
1
2
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
=
1
2
ϕ
n
-
1
c
-
ϕ
n
c
E
=
1
2
ϕ
n
-
1
-
ϕ
n
E
0
≤
ϕ
n
-
1
-
ϕ
n
E
0
.So, by (63), we have(75)
α
ϕ
n
-
1
c
,
ϕ
n
c
·
G
ϕ
n
-
1
(
c
)
-
T
ϕ
n
-
1
E
,
ϕ
n
(
c
)
-
T
ϕ
n
E
,
ϕ
n
-
1
-
ϕ
n
E
0
k
k
k
k
k
k
ϕ
n
-
1
(
c
)
-
T
ϕ
n
E
,
ϕ
n
(
c
)
-
T
ϕ
n
-
1
E
k
k
k
+
F
(
T
ϕ
n
-
1
-
T
ϕ
n
E
)
-
F
ϕ
n
-
1
-
ϕ
n
E
0
≤
0
;and then(76)
α
(
ϕ
n
-
1
(
c
)
,
ϕ
n
(
c
)
)
·
G
ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
,
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
ϕ
n
-
1
-
ϕ
n
E
0
k
k
k
k
k
k
ϕ
n
-
1
c
-
ϕ
n
+
1
c
E
,
0
k
k
k
k
+
F
(
ϕ
n
c
-
ϕ
n
+
1
c
E
)
k
k
k
k
-
F
ϕ
n
-
1
-
ϕ
n
E
0
≤
0
.On the other hand there exists τ
>
0 such that(77)
G
ϕ
n
-
1
(
c
)
-
ϕ
n
(
c
)
E
,
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
,
k
k
k
ϕ
n
-
1
c
-
ϕ
n
+
1
c
E
,
0
=
τ
,and so(78)
α
(
ϕ
n
-
1
(
c
)
,
ϕ
n
(
c
)
)
·
[
τ
+
F
(
ϕ
n
(
c
)
-
ϕ
n
+
1
(
c
)
E
)
-
F
(
ϕ
n
-
1
-
ϕ
n
E
0
)
]
≤
0
;which implies(79)
F
ϕ
n
-
ϕ
n
+
1
E
0
≤
F
ϕ
n
-
1
-
ϕ
n
E
0
-
τ
,for all n
∈
N. Again as in proof of Proposition 13, we can deduce that (
ϕ
n
) is a Cauchy sequence in R
c
0 and there exists ϕ
∗
∈
R
c
0 such that ϕ
n
→
ϕ
∗ as n
→
∞.
Also, from (iii) we haveα
(
ϕ
n
(
c
)
,
ϕ
∗
(
c
)
)
>
0 for all n
∈
N
∪
{
0
}.
From (79), we get(80)
F
ϕ
n
-
ϕ
n
+
1
E
0
≤
F
(
ϕ
n
-
1
-
ϕ
n
E
0
)
-
τ
<
F
(
ϕ
n
-
1
-
ϕ
n
E
0
)
.Now, since F
∈
Δ
F, we have(81)
ϕ
n
-
ϕ
n
+
1
E
0
<
ϕ
n
-
1
-
ϕ
n
E
0
,for all n
∈
N. Suppose that there exists n
0
∈
N such that(82)
1
2
ϕ
n
0
(
c
)
-
T
ϕ
n
0
E
>
ϕ
n
0
-
1
-
ϕ
∗
E
0
,
1
2
ϕ
n
0
+
1
(
c
)
-
T
ϕ
n
0
+
1
E
>
ϕ
n
0
-
ϕ
∗
E
0
.Then, from (81), it follows that(83)
ϕ
n
0
-
1
-
ϕ
n
0
E
0
≤
ϕ
n
0
-
1
-
ϕ
∗
E
0
+
ϕ
n
0
-
ϕ
∗
E
0
<
1
2
ϕ
n
0
(
c
)
-
T
ϕ
n
0
E
+
1
2
ϕ
n
0
+
1
(
c
)
-
T
ϕ
n
0
+
1
E
=
1
2
ϕ
n
0
c
-
ϕ
n
0
+
1
c
E
+
1
2
ϕ
n
0
+
1
c
-
ϕ
n
0
+
2
c
E
=
1
2
ϕ
n
0
-
ϕ
n
0
+
1
E
0
+
1
2
ϕ
n
0
+
1
-
ϕ
n
0
+
2
E
0
≤
1
2
ϕ
n
0
-
ϕ
n
0
-
1
E
0
+
1
2
ϕ
n
0
-
ϕ
n
0
-
1
E
0
=
ϕ
n
0
-
ϕ
n
0
-
1
E
0
,which is a contradiction. Hence either(84)
1
2
ϕ
n
(
c
)
-
T
ϕ
n
E
≤
ϕ
n
-
1
-
ϕ
∗
E
0or(85)
1
2
ϕ
n
+
1
(
c
)
-
T
ϕ
n
+
1
E
≤
ϕ
n
-
ϕ
∗
E
0holds for all n
∈
N. First, suppose that(86)
1
2
ϕ
n
(
c
)
-
T
ϕ
n
E
≤
ϕ
n
-
1
-
ϕ
∗
E
0holds for all n
∈
N. Therefore, from (63), we have(87)
α
(
ϕ
n
(
c
)
,
ϕ
∗
(
c
)
)
·
G
ϕ
n
-
T
ϕ
n
E
,
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
ϕ
n
(
c
)
-
T
ϕ
∗
E
,
ϕ
n
-
ϕ
∗
E
0
k
k
k
k
k
k
ϕ
∗
(
c
)
-
T
ϕ
n
E
+
F
(
T
ϕ
n
-
T
ϕ
∗
E
)
kkkkk
-
F
ϕ
n
-
ϕ
∗
E
0
≤
0
,which implies(88)
F
(
T
ϕ
n
-
T
ϕ
∗
E
)
≤
F
(
ϕ
n
-
ϕ
∗
E
0
)
.Now, since F
∈
Δ
F, we get(89)
T
ϕ
n
-
T
ϕ
∗
E
≤
ϕ
n
-
ϕ
∗
E
0
,and so(90)
T
ϕ
∗
-
ϕ
∗
(
c
)
E
≤
T
ϕ
∗
-
T
ϕ
n
E
+
T
ϕ
n
-
ϕ
∗
(
c
)
E
=
T
ϕ
∗
-
T
ϕ
n
E
+
ϕ
n
+
1
(
c
)
-
ϕ
∗
(
c
)
E
0
≤
ϕ
∗
-
ϕ
n
E
0
+
ϕ
n
+
1
-
ϕ
∗
E
0
.Taking limit as n
→
∞ in the above inequality, we get T
ϕ
∗
-
ϕ
∗
(
c
)
E
=
0; that is, T
ϕ
∗
=
ϕ
∗
(
c
). By the similar method, we can deduce T
ϕ
∗
=
ϕ
∗
(
c
) when(91)
1
2
ϕ
n
+
1
(
c
)
-
T
ϕ
n
+
1
E
≤
ϕ
n
-
ϕ
∗
E
0
.Hence it follows that ϕ
∗ is a P
P
F dependent fixed point of T in R
c
0. This completes the proof.Corollary 22.
Let the nonself-mappingT
:
E
0
→
E, α
:
E
×
E
→
R, and F
∈
Δ
F be such that for all ϕ
,
ξ
∈
E
0 with T
ϕ
-
T
ξ
E
>
0 and 1
/
2
ϕ
(
c
)
-
T
ϕ
E
≤
ϕ
-
ξ
E
0
(92)
G
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
E
,
ϕ
c
-
T
ξ
E
,
k
k
k
ξ
c
-
T
ϕ
E
+
F
T
ϕ
-
T
ξ
E
≤
F
ϕ
-
ξ
E
0holds. Then(I)
T has a P
P
F dependent fixed point in R
c
0;
(I
I)
T has a unique P
P
F dependent fixed point in R
c.Proof.
By taking in Theorem21, (93)
α
u
,
v
=
1
,
∀
u
,
v
∈
E
,we deduce that T has a P
P
F dependent fixed point in R
c
0. For the uniqueness, suppose that ϕ
∗ and ξ
∗ are two P
P
F dependent fixed points of T in R
c such that ϕ
∗
≠
ξ
∗. So, we have(94)
1
2
ϕ
∗
(
c
)
-
T
ϕ
∗
E
=
0
≤
ϕ
∗
-
ξ
∗
E
0
,which implies from (92) that(95)
G
ϕ
∗
(
c
)
-
T
ϕ
∗
E
,
φ
∗
(
c
)
-
T
φ
∗
E
,
ϕ
∗
(
c
)
-
T
φ
∗
E
,
k
k
k
φ
∗
(
c
)
-
T
ϕ
∗
E
+
F
(
T
ϕ
∗
-
T
φ
∗
E
)
≤
F
(
ϕ
∗
-
φ
∗
E
0
)
.Thus,(96)
G
0,0
,
ϕ
∗
c
-
φ
∗
c
E
,
φ
∗
c
-
ϕ
∗
c
E
+
F
(
ϕ
∗
c
-
φ
∗
c
E
)
≤
F
ϕ
∗
-
φ
∗
E
0
.On the other hand, there exists, τ
>
0 such that(97)
G
0,0
,
ϕ
∗
c
-
φ
∗
c
E
,
φ
∗
c
-
ϕ
∗
c
E
=
τ
,and so(98)
τ
+
F
ϕ
∗
-
φ
∗
E
0
=
τ
+
F
(
ϕ
∗
(
c
)
-
φ
∗
(
c
)
E
)
≤
F
(
ϕ
∗
-
φ
∗
E
0
)
,which is a contradiction. Hence ϕ
∗
=
φ
∗. This completes the proof.


## 4. Some Results in Banach Spaces Endowed with a Graph

Consistent with Jachymski [7], let (
E
,
d
) be a metric space where d
(
x
,
y
)
=
x
-
y
E for all x
,
y
∈
E and Δ denotes the diagonal of the Cartesian product of X
×
X. Consider a directed graph G such that the set V
(
G
) of its vertices coincides with X and the set E
(
G
) of its edges contains all loops, that is, E
(
G
)
⊇
Δ. We assume that G has no parallel edges, so we can identify G with the pair (
V
(
G
)
,
E
(
G
)
). Moreover, we may treat G as a weighted graph (see [25], p. 309) by assigning to each edge the distance between its vertices. If x and y are vertices in a graph G, then a path in G from x to y of length p  (
p
∈
N
) is a sequence {
x
i
}
i
=
0
p of p
+
1 vertices such that x
0
=
x, x
p
=
y, and (
x
i
-
1
,
x
i
)
∈
E
(
G
) for i
=
1
,
…
,
p. A graph G is connected if there is a path between any two vertices. G is weakly connected if G
~ is connected (see for more details [4, 6, 7]).Definition 23 (see [7]).
Let(
X
,
d
) be a metric space endowed with a graph G. One says that a self-mapping T
:
X
→
X is a Banach G-contraction or simply a G-contraction if T preserves the edges of G; that is,(99)
∀
x
,
y
∈
X
,
x
,
y
∈
E
G
⟹
T
x
,
T
y
∈
E
G
,and T decreases weights of the edges of G in the following way:(100)
∃
α
∈
0,1
such
that
∀
x
,
y
∈
X
,
(
x
,
y
)
∈
E
(
G
)
⟹
d
(
T
x
,
T
y
)
≤
α
d
(
x
,
y
)
.Definition 24.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a graph G:(i)
T is called a graphic G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with (
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
) and T
ϕ
-
T
ξ
E
>
0, one has(101)
G
(
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
,
ξ
(
c
)
-
T
ϕ
E
)
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
(
ϕ
-
ξ
E
0
)
;
(ii)
T is called a weak graphic G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with (
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
) and T
ϕ
-
T
ξ
E
>
0, one has(102)
G
(
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
,
ξ
(
c
)
-
T
ϕ
E
)
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
(
max
⁡
{
ϕ
-
ξ
E
0
,
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
}
)
;
(iii)
T is called a generalized graphic G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with (
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
) and T
ϕ
-
T
ξ
E
>
0, one has(103)
G
(
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
,
ξ
(
c
)
-
T
ϕ
E
)
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
max
⁡
ϕ
-
ξ
E
0
,
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
k
k
k
k
k
k
k
k
k
k
k
k
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
.Theorem 25.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a graph G. Suppose that the following assertions hold:(i)
if(
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
), then (
T
ϕ
,
T
ξ
)
∈
E
(
G
);
(ii)
T is a graphic G
F-contraction, for some F
∈
Δ
F;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and (
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
∈
E
(
G
) for all n
∈
N, then (
ϕ
n
(
c
)
,
ϕ
(
c
)
)
∈
E
(
G
) for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that (
ϕ
0
(
c
)
,
T
ϕ
0
)
∈
E
(
G
).Then T has a P
P
F dependent fixed point in R
c
0.Proof.
Defineα
:
E
×
E
→
R by(104)
α
(
x
,
y
)
=
1
,
if
(
x
,
y
)
∈
E
(
G
)
0
,
otherwise
.
First, we prove thatT is 0-α
c-admissible nonself-mapping. Assume that α
(
ϕ
(
c
)
,
ξ
(
c
)
)
>
0. Then we have (
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
). From (i), we have (
T
ϕ
,
T
ξ
)
∈
E
(
G
); that is, α
(
T
ϕ
,
T
ξ
)
>
0. Thus T is an 0-α
c-admissible nonself-mapping. From (iv), there exists ϕ
0
∈
R
c such that α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0. Let {
ϕ
n
} be a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and (
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
∈
E
(
G
) for all n
∈
N
∪
{
0
}. Then α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N
∪
{
0
}. Thus from (iii), we get (
ϕ
n
(
c
)
,
ϕ
)
∈
E
(
G
) for all n
∈
N
∪
{
0
}. That is, α
(
ϕ
n
(
c
)
,
ϕ
)
>
0 for all n
∈
N
∪
{
0
}.
Now, if(
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
), then α
(
ϕ
(
c
)
,
ξ
(
c
)
)
=
1. Hence, from definition of graphic G
F-contraction, we have(105)
α
(
ϕ
(
c
)
,
ξ
(
c
)
)
·
G
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
E
,
ϕ
c
-
T
ξ
E
,
k
k
k
k
k
k
ξ
c
-
T
ϕ
E
+
F
T
ϕ
-
T
ξ
E
-
F
ϕ
-
ξ
E
0
≤
0
.
Otherwise,α
(
ϕ
(
c
)
,
ξ
(
c
)
)
=
0. Then(106)
α
(
ϕ
(
c
)
,
ξ
(
c
)
)
·
G
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
E
,
ϕ
c
-
T
ξ
E
,
ϕ
-
ξ
E
0
k
k
k
k
k
k
ξ
c
-
T
ϕ
E
k
k
k
k
+
F
T
ϕ
-
T
ξ
E
-
F
ϕ
-
ξ
E
0
=
0
≤
0
.Therefore, for all ϕ
,
ξ
∈
E
0, we have (107)
α
ϕ
c
,
ξ
c
·
G
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
E
,
ϕ
c
-
T
ξ
E
,
ϕ
-
ξ
E
0
k
k
k
k
k
k
ξ
c
-
T
ϕ
E
+
F
T
ϕ
-
T
ξ
E
-
F
ϕ
-
ξ
E
0
≤
0
,and so all the conditions of Theorem 14 hold and T has a P
P
F dependent fixed point. This completes the proof.Similarly, we can prove the following theorems.Theorem 26.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a graph G. Suppose that the following assertions hold:(i)
if(
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
), then (
T
ϕ
,
T
ξ
)
∈
E
(
G
);
(ii)
T is a graphic weak G
F-contraction, such that F is continuous;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and (
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
∈
E
(
G
) for all n
∈
N, then (
ϕ
n
(
c
)
,
ϕ
(
c
)
)
∈
E
(
G
) for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that (
ϕ
0
(
c
)
,
T
ϕ
0
)
∈
E
(
G
).Then T has a P
P
F dependent fixed point in R
c
0.Theorem 27.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a graph G. Suppose that the following assertions hold:(i)
if(
ϕ
(
c
)
,
ξ
(
c
)
)
∈
E
(
G
), then (
T
ϕ
,
T
ξ
)
∈
E
(
G
);
(ii)
T is a graphic generalized G
F-contraction, such that F and G are continuous;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and (
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
∈
E
(
G
) for all n
∈
N, then (
ϕ
n
(
c
)
,
ϕ
(
c
)
)
∈
E
(
G
) for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that (
ϕ
0
(
c
)
,
T
ϕ
0
)
∈
E
(
G
).Then T has a P
P
F dependent fixed point in R
c
0.


## 5. Some Results in Banach Spaces Endowed with a Partial Order

The study of existence of fixed points in partially ordered sets has been established by Ran and Reurings [11] and Nieto and Rodrıguez-Lopez [26] with applications to matrix and differential equations. Agarwal et al. [1], Ćirić et al. [5], and Hussain et al. [6, 27] and many other authors obtained some new fixed point results for nonlinear contractions in partially ordered Banach and metric spaces with some applications. In this section, as an application of our results we derive some new P
P
F dependent fixed and coincidence point results whenever the range space is endowed with partial order.Definition 28 (see [15]).
Letc
∈
I, T
:
E
0
→
E, and E be endowed with a partial order ⪯. One says that T is a c-increasing nonself-mapping if for ϕ
,
ξ
∈
E
0 with ϕ
(
c
)
⪯
ξ
(
c
) one has T
ϕ
⪯
T
ξ.Definition 29.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a partial order ⪯:(i)
T is called ordered G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with ϕ
(
c
)
⪯
ξ
(
c
) and T
ϕ
-
T
ξ
E
>
0, one has(108)
G
(
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
,
ξ
(
c
)
-
T
ϕ
E
)
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
(
ϕ
-
ξ
E
0
)
;
(ii)
T is called weak ordered G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with ϕ
(
c
)
⪯
ξ
(
c
) and T
ϕ
-
T
ξ
E
>
0, one has(109)
G
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
E
,
ϕ
c
-
T
ξ
E
,
kkk
ξ
c
-
T
ϕ
E
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
(
max
⁡
{
ϕ
-
ξ
E
0
,
ϕ
(
c
)
-
T
ϕ
E
,
ξ
(
c
)
-
T
ξ
E
}
)
;
(iii)
T is called generalized ordered G
F-contraction if there exist F
∈
Δ
F and G
∈
Δ
G such that, for all ϕ
,
ξ
∈
E
0 with ϕ
(
c
)
⪯
ξ
(
c
) and T
ϕ
-
T
ξ
E
>
0, one has(110)
G
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
E
,
ϕ
c
-
T
ξ
E
,
kkk
ξ
c
-
T
ϕ
E
+
F
(
T
ϕ
-
T
ξ
E
)
≤
F
max
⁡
⁡
ϕ
-
ξ
E
0
,
ϕ
c
-
T
ϕ
E
,
ξ
c
-
T
ξ
E
,
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
k
k
k
k
k
k
k
k
k
k
ϕ
(
c
)
-
T
ξ
E
+
ξ
(
c
)
-
T
ϕ
E
2
.Theorem 30.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a partially order ⪯. Suppose that the following conditions hold:(i)
T is c-increasing;
(ii)
T is an ordered G
F-contraction such that F is continuous;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and ϕ
n
(
c
)
⪯
ϕ
n
+
1
(
c
) for all n
∈
N, then ϕ
n
(
c
)
⪯
ϕ
(
c
) for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that ϕ
0
(
c
)
⪯
T
ϕ
0.Then T has a P
P
F dependent fixed point in R
c
0.Proof.
Defineα
:
E
×
E
→
R by(111)
α
(
x
,
y
)
=
1
,
if
x
⪯
y
0
,
otherwise
.
First, we prove thatT is an 0-α
c-admissible nonself-mapping. Assume that α
(
ϕ
(
c
)
,
ξ
(
c
)
)
>
0. Then we have ϕ
(
c
)
⪯
ξ
(
c
). Since T is c-increasing, we get T
ϕ
⪯
T
ξ, that is, α
(
T
ϕ
,
T
ξ
)
≥
0. Hence, T is a 0-α
c-admissible nonself-mapping. From (iv), there exists ϕ
0
∈
R
c such that ϕ
0
(
c
)
⪯
T
ϕ
0. That is, α
(
ϕ
0
(
c
)
,
T
ϕ
0
)
>
0. Let {
ϕ
n
} be a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and α
(
ϕ
n
(
c
)
,
ϕ
n
+
1
(
c
)
)
>
0 for all n
∈
N
∪
{
0
}. Then ϕ
n
(
c
)
⪯
ϕ
n
+
1
(
c
) for all n
∈
N
∪
{
0
}. Thus, from (iii), we get ϕ
n
(
c
)
⪯
ϕ
(
c
) for all n
∈
N
∪
{
0
}. That is, α
(
ϕ
n
(
c
)
,
ϕ
(
c
)
)
>
0 for all n
∈
N
∪
{
0
}. Therefore, all the conditions of Theorem 14 hold and T has a P
P
F dependent fixed point. This completes the proof.Similarly, we can prove the following results.Theorem 31.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a partially order ⪯. Suppose that the following conditions hold:(i)
T is c-increasing;
(ii)
T is an ordered weak G
F-contraction, such that F is continuous;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and ϕ
n
(
c
)
⪯
ϕ
n
+
1
(
c
) for all n
∈
N, then ϕ
n
(
c
)
⪯
ϕ
(
c
) for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that ϕ
0
(
c
)
⪯
T
ϕ
0.Then T has a P
P
F dependent fixed point in R
c
0.Theorem 32.
LetT
:
E
0
→
E be a nonself-mapping and c
∈
I, where E is endowed with a partially order ⪯. Suppose that the following assertions hold:(i)
T is c-increasing;
(ii)
T is an ordered generalized G
F-contraction, such that F and G are continuous;
(iii)
if(
ϕ
n
) is a sequence in E
0 such that ϕ
n
→
ϕ as n
→
∞ and ϕ
n
(
c
)
⪯
ϕ
n
+
1
(
c
) for all n
∈
N, then ϕ
n
(
c
)
⪯
ϕ
(
c
) for all n
∈
N;
(iv)
there existsϕ
0
∈
R
c such that ϕ
0
(
c
)
⪯
T
ϕ
0.Then T has a P
P
F dependent fixed point in R
c
0.



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*Source: 136306-2015-05-18.xml*