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In this paper, we initiate the notion of Ćirić type rational graphic -contraction pair mappings and provide some new related common fixed point results on partial b-metric spaces endowed with a directed graph G. We also give examples to illustrate our main results. Moreover, we present some applications on electric circuit equations and fractional differential equations.
The Banach principle [1] has been improved and generalized by several researchers for different kinds of contractions in various spaces. One of these generalizations corresponding to an -contraction, has been established by [2]. Recently, Ameer et al. [3] introduced common fixed point results for generalized multivalued -contractions in -complete partial b-metric spaces. Ameer et al. [4,5] introduced common fixed point results for generalized multivalued - contractions in complete metric, b-metric spaces. Ameer et al. [6] initiated the notion of rational -contractive pair of mappings (where ℜ is a binary relation) and established new common fixed point results for these mappings in complete metric spaces. On the other hand, Bakhtin [7] investigated the concept of b-metric spaces. Subsequently, Czerwik [8] initiated the study of fixed point results in b-metric spaces and proved an analogue of Banach’s fixed point theorem. Matthews [9] gave the concept of a partial metric space and proved and Banach fixed point result. Shukla [10] extended the notion of a partial metric to a partial b-metric. Afterwards, numerous research articles have been dealt with fixed point theorems for various classes of single-valued and multi-valued operators in b-metric and partial b-metric spaces (see, for example, [3,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]). In this article, we shall investigate fixed points of Ćirić type rational graphic -contraction pair mappings on partial b-metric spaces endowed with a directed graph G.
Bakhtin [7] and Czerwik [8] generalized the notion of a metric as follows:
Definition1
([7,8]).Let M be a nonempty set and be a real number. A mapping is said to be a b-metric if for all ,
if and only if ;
;
.
The pair is called a b-metric space (with coefficient s).
Matthews [9] generalized the notion of a metric as follows:
Definition2
([9]).Let M be a nonempty set. A mapping is said to be a partial metric if for all , P satisfies following axioms;
if and only if ;
;
;
.
The pair is called a partial metric space.
Shukla [10] generalized the notion of a partial metric as follows:
Definition3
([10]).Let M be a nonempty set and a real number. A mapping is said to be a partial b-metric if for all , satisfies the following axioms:
if and only if ;
;
;
.
The pair is called a partial b-metric space (with coefficient s).
Remark1.
The self distance , referring to the size or weight of , is a feature used to describe the amount of information contained in M.
Remark2.
Obviously, every partial metric space is a partial b-metric space with coefficient , and every b-metric space is a partial b-metric space with zero self-distance. However, the converse of this fact need not hold.
Definition4
([10]).Let be a partial b-metric space with coefficient . Let be a sequence in M and . Then
(i)
is said to be convergent to if .
(ii)
is said to be Cauchy sequence if exists and is finite.
(iii)
is said to be complete if every Cauchy sequence is convergent in M.
() for all , where stands for the nth iterate of is called a comparison function. Clearly, if is a comparison function, then for each .
Lemma2
([2]).Let be a continuous non-decreasing function such that . Let be a positive sequence. So
Example1
([31]).The following functions are comparison functions:
(i) with , for each ;
() , for each
Denote by the set of functions verifying:
() is non-decreasing;
() for each positive sequence ,
() is continuous. Liu et al. [2] initiated the concept of ()-Suzuki contractions.
Definition8.
Let be a MS. A mapping is said to be a -Suzuki contraction, if there exist comparison functions Υ and such that, for all with ,
where
Moreover, let be a partial metric space, and denotes the diagonal of . Let G be a directed graph, which has no parallel edges such that the set of its vertices coincides with M, and contains all loops (i.e., ). Hence, G is identify by the pair (). Denote by the graph obtained from G by reversing the direction of its edges. That is,
It is more adaptable to treat a directed graph for which the set of its edges is symmetric. Under this convention, we have that
In , we define the relation R in the following way: for , we have if and only if there is a path in G from to . If G is such that is symmetric, then for , the equivalence class in defined by the relation R is . Recall that if : is an operator; then, by we denote the set of all fixed points of . Let
Property: A graph is said to satisfy property if for any sequence in with as , for implies that there is a subsequence of with an edge between and for . Throughout this paper, G is a weighted graph such that the weight of each vertex is , and the weight of each edge is . Since is a partial b-metric space, the weight assigned to each vertex need not to be zero, and whenever a zero weight is assigned to some edge , it reduces to a loop .
2. Main Results
We start with the following definition.
Definition9.
Let be a partial b-metric space endowed with a directed graph and be self-mappings of M. We say that the pair is a Ćirić type rational graphic -contraction pair, if:
(1) For every vertex , we have
(2) There exists a comparison function Υ and such that for all with and we have
where,
Remark3.
If , then we say that ϕ is a Ćirić type rational graphic -contraction.
Our first main result is the following.
Theorem4.
Let be a complete partial b-metric space endowed with a directed graph G. Let be maps such that is a Ćirić type rational graphic -contraction pair. If Υ is continuous, then the following assertions hold:
(a) or if and only if ;
(b) If then the weight assigned to the vertex is 0;
(c) , provided that G satisfies property
(d) is a complete set if and only if is a singleton set.
Proof.
(a) Let , so there exists . Then there is an edge between and , so . Now, we shall prove that ; that is, the weight assigned to the edge is zero. Assume, on the contrary, that a non zero weight is assigned to the edge . As and is a Ćirić type rational graphic -contraction pair, from (1), we have
where
Thus,
It is a contradiction. Hence, the weight assigned to the edge is zero; that is, Thus,
Therefore,
Conversely, let . So there exists such that and then and Thus, the proof of (a) is ended.
(b) Let Suppose on the contrary that the weight assigned to the vertex is nonzero. As and is a Ćirić type rational graphic -contraction pair, we get
where
It implies that
which is a contradiction. Therefore, the weight assigned to the edge is zero. The proof of (b) is completed.
(c) Let . If or , then from (a) the proof is finished. Assume that ; then . Since there is an edge between and , that is, , this implies that there is such that . Similarly, implies . Continuing this process, we can construct the sequence such that is defined by
If the weight assigned to the edge is zero for some then , which implies , and from (a), . Then there is nothing to prove. Assume that the weight assigned to the edge is non zero for all ; that is, for all . By (1), we get
From (8) and (11), we can choose a positive integer such that for all , from (1), we get
where
Taking the upper limit as and using (8), (11), (14) and (15), we get
Thus,
It is a contradiction. Therefore, is Cauchy. Since ( is a complete partial b-metric space, by Lemma 1, is a complete b-metric space. Therefore, the sequence converges to some . Again, by Lemma 1, there exists such that
if and only if
Now, we show that , so the weight assigned to the edge is zero. Suppose that . If , , then we get . By property (), there is a subsequence of with an edge between and for . Using (1), one gets
Taking the upper limit as in (22) and using the continuity of , we have
a contradiction. Therefore, the assigned weight of the edge is zero; that is, Similarly, . Hence, The proof of (c) is completed.
(d) First, we assume that is complete. We shall prove that is a singleton. On the contrary, suppose that there exists such that As so from (1), we have
It is a contradiction. Thus,
Conversely, assume that is a singleton; then, is complete. □
Example2.
Let and defined by , for all Then is a complete partial b-metric space with . Set
Define by
and by
It is easy to show that, for every vertex , we have . Now, for all with
Hence, by Figure 1, (ϕ, ψ ) is a Ćirić type rational graphic -contraction pair. Thus, all the conditions of Theorem 4 are satisfied, and ϕ and ψ have a unique common fixed point (that is, 1). Figure 2 represents the graph with all the possible cases.
If in Theorem 4, we obtain the following result.
Corollary1.
Let be a complete partial b-metric space endowed with a directed graph G and the map such that ϕ is a Ćirić type rational graphic -contraction. If Υ is continuous, then the following assertions hold:
(a) If then the weight assigned to the vertex is 0;
(b) , provided that G satisfies property
(c) is a complete set if and only if is a singleton set.
If in Theorem 4, we obtain the following result.
Theorem5.
Let be a complete partial metric space endowed with a directed graph G. Let be maps such that:
(1) For every vertex , we have
(2) There exist a comparison function Υ and such that for all with and we have
where
If Υ is continuous, then the following assertions hold:
(a) or if and only if ;
(b) If then the weight assigned to the vertex is 0;
(c) , provided that G satisfies the property
(d) is complete set if and only if is a singleton set.
Example3.
Let and be defined by , for all Then, is a complete partial metric space. Set
Define by
and by
It is easy to show that, for every vertex , we have . Now, for all with
Therefore, is a Ćirić type rational graphic -contraction pair. Hence, the conditions of Theorem 5 hold. Moreover, 0 is a common fixed point of and .
3. Some Consequences
Corollary2.
Let be a complete partial b-metric space () endowed with a directed graph G. Let be maps such that:
(1) For every vertex , we have
(2) There exist and such that for all with and we have
where
Then the following assertions hold:
(a) or if and only if ;
(b) If then the weight assigned to the vertex is 0;
(c) , provided that G satisfies property
(d) is a complete set if and only if is a singleton set.
Proof.
It suffices to take in Theorem 4, and □
Corollary3.
Let be a complete partial b-metric space ) endowed with a directed graph G. Let be maps such that:
(1) For every vertex , we have
(2) There exist and such that for all with and we have
where
Then the following assertions hold:
(a) or if and only if ;
(b) If then the weight assigned to the vertex is 0;
(c) , provided that G satisfies property
(d) is a complete set if and only if is a singleton set.
Proof.
The result follows from Theorem 4 by taking and □
Corollary4.
Let be a complete partial b-metric space () endowed with a directed graph G. Let be maps such that:
(1) For every vertex , we have
(2) If for all with ,
where
and is such that for each .
Then the following assertions hold:
(a) or if and only if ;
(b) If then the weight assigned to the vertex is 0;
(c) , provided that G satisfies property
(d) is a complete set if and only if is a singleton set.
Proof.
It follows from Theorem 4 by taking and □
Remark4.
Theorems 4 and 5 generalize and extend results of Liu et al. [2], Jleli and Samet [29] and Wardowski [27] for partial b-metric spaces and partial metric spaces along with a power graphic contraction pair, respectively.
4. Applications
4.1. Application to Electric Circuit Equations
In this section, we study the solution of the electric circuit equation (see [32]), which is in the second-order differential equation form. The electric circuit (as in Figure 3):
Contains an electromotive force E, a resistor R, an inductor L, a capacitor C, and a voltage V in series. If the current I is the rate of change of charge q with respect to time t, we have and
By law of Kirchhoffs voltage, the sum of these voltage drops is equal to the supplied voltage; i.e,
The above problem is equivalent to the integral equation:
where . Consider a mapping defined by
where . Then, is a solution of (24) if and only if is a fixed point of . From Condition (2), it is easy to show that for every , we have ; i.e., . It follows from Condition (2) that . Let ; then, from Condition (1), we have
Thus,
This implies that
Since , we get that
Hence,
Taking and with , one gets
or
Therefore, from Corollary 1, has a fixed point. Consequently, the differential equation arising in the electric circuit Equation (23) has a solution. □
4.2. Application to Fractional Differential Equations
We apply the result given by Theorem 4 to study the existence of a solution for a system of nonlinear fractional differential equations (see [33]). Let be the space of all continuous functions on . The partial b-metric on M is defined by
Moreover, we define the graph G with the partial ordered relation:
for all Let Note that is a complete partial b-metric space with coefficient , including a directed graph G. Clearly, and has property .
Consider the following system of fractional differential equations:
with boundary conditions
Note that denotes the Caputo fractional derivative of order defined by
where
and and denote the Riemann–Liouville fractional integral of order of continuous functions and , given by
The system (23) can be written in the following integral form:
where is the beta function. From the inequality (27), we obtain that
Hence,
This implies that
where and , Since the above inequality holds for all with , it is true for any Hence, we have
Therefore, all hypotheses of Theorem 4 are satisfied. Hence, and have a common fixed point; that is, the system (26) has at least one solution. □
5. Conclusions
In this paper, we introduced the concept of a Ćirić type rational graphic -contraction pair of mappings and established some new results for such contractions in the context of complete partial b-metric spaces endowed with a directed graph. Moreover, we give some examples in support of main theorems. At the end, we applied our main results to provide solutions of electric circuit equations and also of fractional differential equations. The obtained results generalize several corresponding results in metric spaces.
Author Contributions
All authors contributed equally and read and agreed to the published version of the manuscript.
Funding
Basque Government through grant IT1207/19.
Acknowledgments
The authors are grateful to the Basque Government for its support of this work through grant IT1207/19.
Conflicts of Interest
The authors declare that they have no competing interests.
Availability of Data and Material
Not applicable.
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Ameer, E.; Aydi, H.; Arshad, M.; De la Sen, M.
Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations. Symmetry2020, 12, 467.
https://doi.org/10.3390/sym12030467
AMA Style
Ameer E, Aydi H, Arshad M, De la Sen M.
Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations. Symmetry. 2020; 12(3):467.
https://doi.org/10.3390/sym12030467
Chicago/Turabian Style
Ameer, Eskandar, Hassen Aydi, Muhammad Arshad, and Manuel De la Sen.
2020. "Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations" Symmetry 12, no. 3: 467.
https://doi.org/10.3390/sym12030467
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.
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Ameer, E.; Aydi, H.; Arshad, M.; De la Sen, M.
Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations. Symmetry2020, 12, 467.
https://doi.org/10.3390/sym12030467
AMA Style
Ameer E, Aydi H, Arshad M, De la Sen M.
Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations. Symmetry. 2020; 12(3):467.
https://doi.org/10.3390/sym12030467
Chicago/Turabian Style
Ameer, Eskandar, Hassen Aydi, Muhammad Arshad, and Manuel De la Sen.
2020. "Hybrid Ćirić Type Graphic Υ,Λ-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations" Symmetry 12, no. 3: 467.
https://doi.org/10.3390/sym12030467
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.