Hybrid Ćirić Type Graphic (Υ, Λ)-Contraction Mappings with Applications to Electric Circuit and Fractional Differential Equations

Eskandar Ameer 1,2, Hassen Aydi 3,4,* , Muhammad Arshad 1 and Manuel De la Sen 5,* 1 Department of Mathematics, International Islamic University, H-10, Islamabad 44000, Pakistan; eskandarameer@gmail.com (E.A.); marshad_zia@yahoo.com (M.A.) 2 Department of Mathematics, Taiz University, Taiz P.O.Box 6803, Yemen 3 Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam; 4 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 758307, Vietnam 5 Institute of Research and Development of Processes, University of the Basque Country, Leioa (Bizkaia), 48940 Leioa , Spain * Correspondence: hassen.aydi@tdtu.edu.vn (H.A.); manuel.delasen@ehu.eus (M.D.l.S.)

As in [31], a function Υ : (0, ∞) −→ (0, ∞) satisfies: (i) Υ is monotone increasing, that is, t 1 < t 2 =⇒ Υ (t 1 ) ≤ Υ (t 2 ); (ii) lim n→∞ Υ n (t) = 0 for all t> 0, where Υ n stands for the nth iterate of Υ, is called a comparison function. Clearly, if Υ is a comparison function, then Υ(t) < t for each t > 0. (Φ3) Λ is continuous. Liu et al. [2] initiated the concept of (Υ, Λ)-Suzuki contractions. Definition 8. Let (M, d) be a MS. A mapping T : M → M is said to be a (Υ, Λ)-Suzuki contraction, if there exist comparison functions Υ and Λ ∈ Φ such that, for all ζ, η ∈ M with T (ζ) = T (η), Moreover, let (M, P b ) be a partial metric space, and ∆ denotes the diagonal of M × M. Let G be a directed graph, which has no parallel edges such that the set V(G) of its vertices coincides with M, and E(G) ⊆ M × M contains all loops (i.e., ∆ ⊆ E(G)). Hence, G is identify by the pair (V(G), E(G)). Denote by G −1 the graph obtained from G by reversing the direction of its edges. That is, It is more adaptable to treat ∼ G a directed graph for which the set of its edges is symmetric. Under this convention, we have that In V(G), we define the relation R in the following way: for ζ, η ∈ V(G), we have ζRη if and only if there is a path in G from ζ to η. If G is such that E(G) is symmetric, then for ζ ∈ V(G), the equivalence Recall that if φ:M → M is an operator; then, by Fix(φ) we denote the set of all fixed points of φ. Let Property: A graph is said to satisfy property (E * ) if for any sequence {ζ n } in V(G) with ζ n → ζ as n → ∞, (ζ n , ζ n+1 ) ∈ E(G) for n ∈ N implies that there is a subsequence ζ n(k) of {ζ n } with an edge between ζ n(k) and ζ for k ∈ N. Throughout this paper, G is a weighted graph such that the weight of each vertex ζ is P b (ζ, η), and the weight of each edge (ζ, η) is P b (ζ, η). Since (M, P b ) is a partial b-metric space, the weight assigned to each vertex ζ need not to be zero, and whenever a zero weight is assigned to some edge (ζ, η), it reduces to a loop (ζ, ζ).
Our first main result is the following.

) is a complete set if and only if Fix(φ) is a singleton set.
If s = 1 in Theorem 4, we obtain the following result.

Remark 4. 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.

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): Then the following assertions hold: (a) F ix( ) 6 = ; or F ix( ) 6 = ; if and only if F ix( ) \ F ix( ) 6 = ;; (b) if 2 F ix( ) \ F ix( ); then the weight assigned to the vertex is 0; (c) F ix( ) \ F ix( ) 6 = ;, provided that G satis…es property (E ); (d) F ix( ) \ F ix( ) is a complete set if and only if F ix( ) \ F ix( ) is a singleton set.

Applications
4.1. Application to Electric Circuit Equations. In this section, we study the solution of electric circuit equation (see [17]), which is in the second-order di¤erential equation form. The electric circuit ( as in the following …gure): 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 I = dq dt and (1) V = IR; (2) V = qC; (3) V = L dI dt : By law of Kirchho¤s voltage, the sum of these voltage drops is equal to the supplied voltage, i.e, 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 I = dq dt and By law of Kirchhoffs voltage, the sum of these voltage drops is equal to the supplied voltage; i.e, The Green function associated to (23) is given by where the constant τ > 0 is calculated in terms of R and L.
Let M = C([0, 1]) be the set of all continuous functions defined on [0, 1]. The partial b-metric P b on M is defined by Moreover, we define the graph G with the partial ordered relation: where t ∈ [0, 1], and τ ≥ 1, Then the problem (23) has a unique solution.
Then the system (23) has a solution.
Therefore, all hypotheses of Theorem 4 are satisfied. Hence, φ and ψ have a common fixed point; that is, the system (26) has at least one solution.

Conclusions
In this paper, we introduced the concept of aĆirić type rational graphic (Υ, Λ)-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.