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PPF dependent fixed point theorems for -admissible rational type contractive mappings in Banach spaces
Fixed Point Theory and Applications volume 2014, Article number: 197 (2014)
Abstract
In this paper, we prove some PPF dependent fixed point theorems in the Razumikhin class for some rational type contractive mappings involving -admissible mappings where the domain and range of the mappings are not the same. As applications of these results, we derive some PPF dependent fixed point theorems for these nonself-contractions whenever the range space is endowed with a graph. Our results extend and generalize some results in the literature.
MSC:46N40, 47H10, 54H25, 46T99.
1 Introduction
The fixed point theory in Banach spaces plays an important role and is useful in 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 (see [1–37]). In 1977, Bernfeld et al. [2] introduced the concept of PPF dependent fixed point or the fixed point with PPF dependence which is a fixed point for mappings that have different domains and ranges. They also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings. Very recently, some authors established the existence and uniqueness of PPF dependent fixed point for different types of contractive mappings and generalized some results of Bernfeld et al. [2] (see [1, 4, 12, 15, 20], and [33]).
In order to generalize the Banach contraction principle, Geraghty [9] proved the following theorem.
Theorem 1 (Geraghty [9])
Let be a complete metric space and be an operator. Suppose that there exists satisfying the condition
If T satisfies the following inequality:
then T has a unique fixed point.
Throughout this paper, let be a Banach space, I denotes a closed interval in ℝ and denotes the set of all continuous E-valued functions on I equipped with the supremum norm defined by
For a fixed element , the Razumikhin or minimal class of functions in is defined by
Clearly, every constant function from I to E is a member of . It is easy to see that the class is algebraically closed with respect to difference, i.e., when . Also the class is topologically closed if it is closed with respect to the topology on generated by the norm .
Definition 1 ([2])
A mapping is said to be a PPF dependent fixed point or a fixed point with PPF dependence of mapping if for some .
Definition 2 ([2])
The mapping is called a Banach type contraction if there exists such that
for all .
Samet in 2012 introduced the concepts of α-ψ-contractive and α-admissible mappings. Karapınar and Samet generalized these notions to obtain other fixed point results. Many authors generalized these notions to obtain fixed point results (see [18, 19, 21–23], and [32]).
Samet et al. [31], defined the notion of α-admissible mappings as follows:
Definition 3 ([31])
Let T be a self-mapping on X and be a function. We say that T is an α-admissible mapping if
Definition 4 ([17])
Let and . We say that f is a triangular α-admissible mapping if
(T1) implies , ,
(T2) implies , .
The concept of -admissible mapping was introduced by Agarwal et al. in 2013 (see [1]).
Definition 5 ([1])
Let , , and . We say T is an -admissible mapping if for
Definition 6 ([4])
Let , , and . We say T is a triangular -admissible mapping if
(T1) implies ,
(T2) and implies
for .
Lemma 1 ([4])
Let be a triangular -admissible mapping. Define the sequence in the following way:
for all , where is such that . Then
2 Main results
Let ℱ denotes the class of all functions satisfying the following condition:
Definition 7 Let be a nonself-mapping and be a function. We say T is a rational Geraghty contraction of type I if there exist and such that
for all , where
Theorem 2 Let and be two mappings satisfying the following assertions:
-
(a)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(b)
T is an -admissible,
-
(c)
T is a rational Geraghty contractive mapping of type I,
-
(d)
if is a sequence in such that as and , then for all ,
-
(e)
there exists such that .
Then T has a unique PPF dependent fixed point . Moreover, for a fixed , if the sequence of iterates of T is defined by for all , then converges to .
Proof Let is a point in such that . Since , there exists such that . Choose such that . Since and, by hypothesis, we get . This implies that there exists such that . Thus, we can choose such that . Continuing this process, by induction, we can build the sequence in such that for all . It follows from the fact that is algebraically closed with respect to difference
Since T is -admissible and , we deduce that
By continuing this process, we get for all . Since T is a rational Geraghty contraction of type I, we have
On the other hand,
If
from (2.2) we have
which is a contradiction. So,
By (2.2) we conclude
for all . This implies that the sequence is decreasing in . So, it is convergent. Suppose that there exists such that . Assume that . Taking the limit as from (2.4) we conclude
which implies . So,
and since , , which is a contradiction. Hence, . This means
We prove that the sequence is a Cauchy sequence in . Assume that is not a Cauchy sequence, then
Since T is a rational Geraghty contraction of type I, we have
Taking the limit when in the above inequality and applying (2.5) we deduce
where
Letting in the above inequality and applying (2.5), we get
So, by (2.7) and (2.9), we have
and hence from (2.6) we get . This means
and since , we conclude
which is a contradiction. Consequently,
and hence is a Cauchy sequence in . By Completeness of , we find that converges to a point , this means , as . Since is topologically closed, we deduce, . By condition b, we have . Now, since T is a rational Geraghty contraction of type I, we have
Taking the limit as in the above inequality, we get
But
Therefore, from (2.10) and (2.11), we deduce
that is,
which implies that is a PPF dependent fixed point of T in . Now, we show that T has a unique PPF dependent fixed point in . Suppose on the contrary that and are two PPF dependent fixed points of T in such that . Then
where
Therefore,
which is a contradiction. Hence, . Then T has a unique PPF dependent fixed point in . □
Definition 8 Let and . We say that T is a rational Geraghty contraction of type II if there exist and such that
for all , where
Theorem 3 Let and be two mappings satisfying the following assertions:
-
(a)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(b)
T is an -admissible,
-
(c)
T is a rational Geraghty contractive mapping of type II,
-
(d)
if is a sequence in such that as and , then for all ,
-
(e)
there exists such that .
Then T has a unique PPF dependent fixed point . Moreover, for a fixed , if the sequence of iterates of T is defined by for all , then converges to .
Proof Suppose that is a point in such that . Since , there exists such that . Choose such that . Since and, by hypothesis, we get . This implies that there exists such that . Thus, we can choose such that . Continuing this process, by induction, we can build the sequence in such that for all . It follows from the fact that is algebraically closed with respect to difference
Since T is -admissible and , we deduce that
Continuing this process, we get for all . Since T is a rational Geraghty contraction of type II, we have
On the other hand,
From (2.12) we conclude
for all . So, the sequence is decreasing in and there exists such that . Reviewing the proof of Theorem 2, we can show that , i.e.,
Now, we prove that the sequence is Cauchy in . If not, then
From the fact that T is a rational Geraghty contraction of type II, we have
Letting in the above inequality and applying (2.14) we deduce
where
Letting in the above inequality and applying (2.14), we get
So, from (2.16) and (2.18), we obtain
and so by (2.15) we get, . That is,
and since , we deduce
which is a contradiction. Consequently,
and hence is a Cauchy sequence in . By completeness of , we find that converges to a point , this means that , as . Since is topologically closed, we deduce that . Now, since T is a rational Geraghty contraction of type II, we have
Taking the limit as in the above inequality, we get
But
So,
and by (2.19) and (2.20), we conclude
that is,
which implies that is a PPF dependent fixed point of T in . Finally, we prove the uniqueness of the PPF dependent fixed point of T in . Let and be two PPF dependent fixed points of T in such that . So, we obtain
where
Therefore,
which is a contradiction. Hence, . Therefore, T has a unique PPF dependent fixed point in . This completes the proof. □
Definition 9 Let and . We say that T is a rational Geraghty contraction of type III if there exist and such that
for all , where
Theorem 4 Let and be two mappings satisfying the following assertions:
-
(a)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(b)
T is an -admissible,
-
(c)
T is a rational Geraghty contractive mapping of type III,
-
(d)
if is a sequence in such that as and , then for all ,
-
(e)
there exists such that .
Then T has a unique PPF dependent fixed point . Moreover, for a fixed , if the sequence of iterates of T defined by for all , then converges to the PPF dependent fixed point of T in .
Proof Suppose that be a point in such that . Since , there exists such that . Choose such that . Since and, by hypothesis, we get . This implies that there exists such that . Thus, we can choose such that . Repeating this process, by induction, we can construct the sequence in such that for all . From the fact that is algebraically closed with respect to difference it follows that
Since T is -admissible and , we deduce
Continuing this process, we get for all . By the fact that T is a rational Geraghty contraction of type III, we have
On the other hand,
From (2.21) we conclude
for all . This implies that the sequence is decreasing in . Then there exists such that . Repeating the proof of Theorem 2, we conclude that . That is,
Now, we prove that the sequence is Cauchy in . If not, then
Since T is a rational Geraghty contraction of type III, we have
Making in the above inequality and applying (2.23) we have
Also,
Letting in the above inequality and applying (2.23), we get
Hence, from (2.25) and (2.26), we obtain
and so by (2.24) we get . That is,
and since , we deduce
which is a contradiction. Consequently,
and hence is a Cauchy sequence in . Completeness of shows that converges to a point , this means that , as . Since is topologically closed, we deduce that . Now, since T is a rational Geraghty contraction of type III, we have
Taking the limit as in the above inequality, we get
But
Therefore, from (2.27) and (2.28), we deduce that
that is,
which implies that is a PPF dependent fixed point of T in . Suppose that and are two PPF dependent fixed points of T in such that . So,
where
Therefore,
which is a contradiction. Hence, . Then T has a unique PPF dependent fixed point in . □
Corollary 1 Let and be two mappings satisfying the following assertions:
-
(a)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(b)
T is an -admissible,
-
(c)
assume that
for all , where and
or
or
-
(d)
if is a sequence in such that as and , then for all ,
-
(e)
there exists such that .
Then T has a unique PPF dependent fixed point .
Corollary 2 Let , be two mappings satisfying the following assertions:
-
(a)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(b)
T is an -admissible,
-
(c)
assume that
or
or
for all , where , and ,
-
(d)
if is a sequence in such that as and , then for all ,
-
(e)
there exists such that .
Then T has a unique PPF dependent fixed point . Moreover, for a fixed , if the sequence of iterates of T is defined by for all , then converges to a PPF dependent fixed point of T in .
Let Ψ be the family of all nondecreasing functions such that
for all .
Lemma 2 (Berinde [3], Rus [28])
If , then the following are satisfied:
-
(a)
for all ;
-
(b)
.
As an example for all , where and for all , are in Ψ.
Theorem 5 Let and be two mappings satisfying the following assertions:
-
(a)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(b)
T is a triangular -admissible,
-
(c)
suppose that there exists such that
(2.29)
where
for all ,
-
(d)
if is a sequence in such that as and , then for all ,
-
(e)
there exists such that .
Then T has a unique PPF dependent fixed point . Moreover, for a fixed , if the sequence of iterates of T is defined by for all , then converges to the PPF dependent fixed point of T in .
Proof Suppose that is a point in such that . Since , there exists such that . Choose such that . Since and, by hypothesis, we get . This implies that there exists such that . Thus, we can choose such that . Inductively, we can build the sequence in such that for all . From Lemma 1, we have for all with . It follows from the fact that is algebraically closed with respect to difference that
Now, by (2.29) we have
where
If
from (2.30) we have
which is a contradiction. So,
By (2.30), we conclude
By induction, we get
for all . As , we conclude
We prove that the sequence is a Cauchy sequence in . Assume that is not a Cauchy sequence, then
By (2.29), we have
Letting in the above inequality and applying (2.33) we have
where
Letting in the above inequality and applying (2.33), we get
So, by (2.35) and (2.37), we have
which is a contradiction. Consequently,
Hence, is a Cauchy sequence in . Completeness of shows that converges to a point , that is, as . Since is topologically closed, we deduce, . Now, by (2.29), we get
Taking the limit as in the above inequality, we get
But
Therefore, from (2.38) and (2.39), we deduce
that is,
which implies that is a PPF dependent fixed point of T in . Suppose that and are two PPF dependent fixed points of T in such that . So,
where
Therefore,
which is a contradiction. Hence, . Then T has a unique PPF dependent fixed point in . □
Now, in Theorem 5 we take , where and we have the following corollary.
Corollary 3 Let , be two mappings satisfying the following assertions:
-
(a)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(b)
T is a triangular -admissible,
-
(c)
(2.40)
where
for all ,
-
(d)
if is a sequence in such that as and , then for all ,
-
(e)
there exists such that .
Then T has a unique PPF dependent fixed point . Moreover, for a fixed , if the sequence of iterates of T is defined by for all , then converges to a PPF dependent fixed point of T in .
3 Some results in Banach spaces endowed with a graph
Consistent with Jachymski [13], let be a metric space and Δ denotes the diagonal of the Cartesian product . Consider a directed graph G such that the set of its vertices coincides with X, and the set of its edges contains all loops, that is, . We suppose that G has no parallel edges, so we can identify G with the pair . Moreover, we may treat G as a weighted graph (see [[14], 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 N () is a sequence of vertices such that , , and for . Recently, some results have appeared providing sufficient conditions for a mapping to be a Picard operator if is endowed with a graph. The first result in this direction was given by Jachymski [13].
Definition 10 ([13])
Let be a metric space endowed with a graph G. We say that a self-mapping is a Banach G-contraction or simply a G-contraction if T preserves the edges of G, that is,
and T decreases the weights of the edges of G in the following way:
Theorem 6 Let and E endowed with a graph G. Suppose that the following assertions hold:
-
(i)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(ii)
if , then ,
-
(iii)
assume that
where
or
or
-
(iv)
if is a sequence in such that as and , then for all ,
-
(v)
there exists such that .
Then T has a unique PPF dependent fixed point in .
Proof Define by
Now, we show that T is an -admissible mapping. Suppose that . Therefore, we have . From (ii), we get . So, and T is an -admissible mapping. By the definition of α and from (iii), we have
From (v), there exists such that . Let is a sequence in such that as and for all , then . Thus, from (iv) we get, . That is, . Therefore all conditions of Theorems 2, 3, 4 hold true and T has a PPF dependent fixed point. □
Theorem 7 Let and E be endowed with a graph G and for all and , we have . Suppose that the following assertions hold:
-
(i)
there exists such that is topologically closed and algebraically closed with respect to difference,
-
(ii)
if , then ,
-
(iii)
assume that for , we have
(3.1)
where
for all ,
-
(iv)
if is a sequence in such that as and , then , for all ,
-
(v)
there exists such that .
Then T has a unique PPF dependent fixed point in .
4 Application
In this section, we present an application of our Theorem 5 to establish PPF dependent solution of a nonlinear integral equation. Let where with . is a Banach space with the following norm:
For consider the following nonlinear integral problem:
where , with and and .
Let
and
This means that
In [20], it is shown that is complete. Next, we define the function by
We will consider (4.1) under the following assumptions:
-
(i)
,
-
(ii)
there exist and such that for all , with we have
-
(iii)
if , then ,
-
(iv)
if and , then ,
-
(v)
there exists such that ,
-
(vi)
if is a sequence in such that as and for all n, then .
Theorem 8 Under assumptions (i)-(vi), the integral equation (4.1) has a solution on .
Proof For with from (ii), we have
Now, we define by
Hence, for all , we have
where
From the conditions (iii) and (iv), we deduce that S is a triangular -admissible mapping. By the condition (vi), we conclude that if a sequence is such that as and for all n, then and by (v), there is such that . The Razumikhin is , which is topologically closed and algebraically closed with respect to difference. Hence, the hypotheses of Theorem 5 are satisfied with . So, there exists a fixed point such that . This means that
□
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Zabihi, F., Razani, A. PPF dependent fixed point theorems for -admissible rational type contractive mappings in Banach spaces. Fixed Point Theory Appl 2014, 197 (2014). https://doi.org/10.1186/1687-1812-2014-197
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DOI: https://doi.org/10.1186/1687-1812-2014-197
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