Abstract
The purpose of this paper is to present some existence results for coupled fixed points of contraction type operators in metric spaces endowed with a directed graph. Our results generalize the results obtained by Gnana Bhaskar and Lakshmikantham in (Nonlinear Anal. 65:1379-1393, 2006). As an application, the existence of a continuous solution for a system of Fredholm and Volterra integral equations is obtained.
MSC:47H10, 54H25.
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1 Preliminaries
In fixed point theory, the importance of study of coupled fixed points is due to their applications to a wide variety of problems. Bhaskar and Lakshmikantham [1] gave some existence results for coupled fixed point for a mixed monotone type mapping in a metric space endowed with partial order, using a weak contractivity type of assumption.
The purpose of this paper is to generalize these results using the context of metric spaces endowed with a graph. This new research direction in the theory of fixed points was initiated by Jachymski [2] and Gwóźdź-Lukawska, Jachymski [3]. Other results for single-valued and multivalued operators in such metric spaces were given by Beg, Butt, Radojevic in [4], Chifu, Petrusel in [5].
Our results also generalize and extend some coupled fixed points theorems in partially ordered complete metric spaces given by Harjani, Sadarangani [6], Nieto, Rodríguez-López [7] and [8], Nieto, Pouso, Rodríguez-López [9], O’Regan, Petruşel [10], Ran, Reurings [11], Gnana Bhaskar, Lakshmikantham [1], and Chifu, Petrusel [12].
Let be a metric space and Δ be the diagonal of . Let G be a directed graph, such that the set of its vertices coincides with X and , where is the set of the edges of the graph. Assume also that G has no parallel edges and, thus, one can identify G with the pair .
Throughout the paper we shall say that G with the above-mentioned properties satisfies standard conditions.
Let us denote by the graph obtained from G by reversing the direction of edges. Thus,
Let us consider the function .
Definition 1.1 An element is called coupled fixed point of the mapping F, if and .
We shall denote by the set of all coupled fixed points of mapping F, i.e.
Definition 1.2 We say that is edge preserving if
Definition 1.3 The operator is called G-continuous if for all , and for any sequence of positive integers, with , , as , and , , we have that
Definition 1.4 Let be a complete metric space and G be a directed graph. We say that the triple has the property (), if for any sequence with , as , and , for , we have .
Definition 1.5 Let be a complete metric space and G be a directed graph. We say that the triple has the property (), if for any sequence with , as , and , for , we have .
2 Coupled fixed point theorems
Let be a metric space endowed with a directed graph G satisfying the standard conditions.
We consider the set denoted by and defined as
Proposition 2.1 If is edge preserving, then:
-
(i)
and implies and ;
-
(ii)
implies and , for all ;
-
(iii)
implies , for all .
Proof (i) Let and . Because F is edge preserving we have and , which, using the same property, will imply that and .
By induction we shall obtain and .
(ii) Let and .
Using (i) we have
(iii) From (ii) we have
This is equivalent to , for all . □
Definition 2.1 The mapping is called a G-contraction if:
-
(i)
F is edge preserving;
-
(ii)
there exists such that
Lemma 2.1 Let be a metric space endowed with a directed graph G and let be a G-contraction with constant k. Then
Proof Let . Because F is edge preserving we have and .
From Proposition 2.1(i), it follows that and .
Since F is a G-contraction, we shall obtain
and
Hence, by induction, we have
□
Lemma 2.2 Let be a metric space endowed with a directed graph G and let be a G-contraction with constant k. Then, given , there exists such that
Proof Let and .
If in Lemma 2.1 we consider and we shall obtain
which is
If we consider , then
In a similar way we obtain
□
Lemma 2.3 Let be a complete metric space endowed with a directed graph G and let be a G-contraction with the constant k. Then for each , there exist and such that converges to and converges to , as .
Proof Let . From Lemma 2.2, we have
Let . We have
Similarly we shall obtain
Hence and are Cauchy sequences. Since is a complete metric space we shall obtain the result that there exist and such that converges to and converges to , as . □
Now we shall prove the main results of this section.
Theorem 2.1 Let be a complete metric space endowed with a directed graph G and let be a G-contraction with the constant k. Suppose that:
-
(i)
F is G-continuous;
or
-
(ii)
The triple has the properties () and ().
Under these conditions if and only if .
Proof Suppose that .
Let .
We have and .
Hence and which means that and thus .
Suppose now that . Let : and .
Let be a sequence of positive integers. From Proposition 2.1(ii), we know that
Moreover, from Lemma 2.3, there exist and such that
We shall prove that and .
Suppose that we have (i). Since F is G-continuous we shall obtain that
Now
Using the G-continuity of F and the convergence of we obtain , which means that .
In a similar way it can be proved that .
Thus is a coupled fixed point of the mapping F; therefore, .
Suppose now that we have (ii). From (2.1) and (2.2), using properties () and () of the triple , we shall obtain
Hence and , which means that and .
Thus . □
Let us suppose now that for every there exists such that
Theorem 2.2 Adding condition (2.3) to the hypothesis of Theorem 2.1 we obtain the uniqueness of the coupled fixed point of F.
Proof Let us suppose that there exist two coupled fixed points of F. From (2.3) we find that there exists such that
Using Lemma 2.1, we shall have
Hence .
In a similar way it can be proved that . □
Remark 2.1 It is obvious that if and , then and .
Theorem 2.3 Under the conditions of Theorem 2.1, if with , then .
Proof Since , , from the G-contraction condition we have
Hence and thus . □
Theorem 2.4 Under the conditions of Theorem 2.1, if we suppose that for each , there exists such that and , then if we have .
Proof Let . According to Theorem 2.1, we have that . Hence, there exists such that and .
Now let and . Using Proposition 2.1(i) we shall have and .
If then and using the same property we shall have and .
Finally if and then and ,
Using Lemma 2.2, we shall obtain
Hence, we have
Thus . □
Theorem 2.5 Under the conditions of Theorem 2.1, if we suppose that there exists , such that and the sequence converges to and converges to , as , then .
Proof Let , such that .
Hence and
Using the property of G-contraction we shall have
Hence
Hence if , then , and thus . □
Remark 2.2 Let be a partially ordered set and d be a metric on X such that is a complete metric space. Let . In this case we obtain all the results from [1].
Remark 2.3 It is obvious that if we consider a function , all these results concerning the coupled fixed point of the mapping F result in the existence and uniqueness results for the fixed point of f.
3 Applications
Let us consider the following integral systems:
and
The purpose of this section is to give existence and uniqueness results for the solution of the systems (3.1) and (3.2) using Theorem 2.1.
We shall discuss first the system (3.1).
Let us consider with the usual supremum norm, i.e. , for .
Consider also the graph G defined by the partial order relation, i.e.
Hence is a complete metric space endowed with a directed graph G.
If we consider , then the diagonal Δ of is included in . On the other hand .
Moreover, has the properties () and ().
In this case .
Theorem 3.1 Consider the system (3.1). Suppose
-
(i)
and are continuous;
-
(ii)
for all with we have , for all ;
-
(iii)
there exists such that
-
(iv)
there exists such that
Then there exists at least one solution of the integral system (3.1).
Proof Let , , where
In this way, the system (3.1) can be written as
It can be seen, from (3.4), that a solution of this system is a coupled fixed point of the mapping F.
We shall verify if the conditions of Theorem 2.1 are fulfilled.
Let such that and . We have
Hence, if and , then and , which, according to the definition of , shows that F is edge preserving.
On the other hand
Hence, there exists such that
Thus we see that F is a G-contraction.
Condition (iv) from Theorem 3.1 shows that there exists such that and , which implies that .
On the other hand, because of (i) and of the fact that has the properties () and () we find that either (i) or (ii) from Theorem 2.1 is fulfilled.
In this way, we see that , defined by (3.3), verifies the conditions of Theorem 2.1.
Thus, there exists , a coupled fixed point of the mapping F, which also is a solution of (3.1). □
Let us consider now the system (3.2) and let us endow with a Bielecki type norm, i.e. , for , where is arbitrarily chosen.
Consider, as previously, the graph G defined by the partial order relation, i.e.
Hence is a complete metric space endowed with a directed graph G.
If we consider , then the diagonal Δ of is included in . On the other hand .
Moreover, has the properties () and ().
In this case .
Theorem 3.2 Consider the system (3.2). Suppose
-
(i)
and are continuous;
-
(ii)
for all with we have , for all ;
-
(iii)
there exists such that
-
(iv)
there exists such that
Then there exists at least one solution of the integral system (3.2).
Proof Let , , where
In this way, the system (3.2) can be written as
As previously, from (3.6), the solution of this system is a coupled fixed point of the mapping F.
Let such that and ,
Hence, if and , then and , which, according to the definition of , shows that F is edge preserving.
On the other hand
Multiplying the above relation with and choosing , we find that there exists such that
Hence we see that F is a G-contraction.
Condition (iv) from Theorem 3.1, shows that there exists such that and , which implies that .
On the other hand, because of (i) and of the fact that has the properties () and () we see that either (i) or (ii) from Theorem 2.1 is fulfilled.
In this way, we see that , defined by (3.5), verifies the conditions of Theorem 2.1.
Thus, there exists , a coupled fixed point of the mapping F, which also is a solution of (3.2). □
In what follows we shall suppose that the conditions of Theorems 3.1 and 3.2 are satisfied.
Remark 3.1 In both cases we are working with . Equation (2.2) becomes the following.
For every there exists such that
It is obvious that we can find such a pair simply considering
In this respect, if the conditions of Theorems 3.1 and 3.2 are fulfilled, (3.7) implies the uniqueness for the solution of both systems.
Remark 3.2 If with , then .
Remark 3.3 If we suppose that for each , there exists such that and , then if we have .
Remark 3.4 If we suppose that there exists , such that and the sequence converges to and converges to , as , then .
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Chifu, IC, Petrusel, GR: Coupled fixed point theorems in metric spaces endowed with a graph (submitted)
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Chifu, C., Petrusel, G. New results on coupled fixed point theory in metric spaces endowed with a directed graph. Fixed Point Theory Appl 2014, 151 (2014). https://doi.org/10.1186/1687-1812-2014-151
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DOI: https://doi.org/10.1186/1687-1812-2014-151