Hybrid Fixed Point Theorem with Applications to Forced Damped Oscillations and Infinite Systems of Fractional Order Differential Equations

In this manuscript, hybrid common fixed point results in the setting of a 
 
 b
 
 -metric space are established. Our results generalized the results of Fisher, Khan, and Piri et al. for set-valued mapping in 
 
 b
 
 -metric spaces. Applications to forced damped oscillations, infinite systems of fractional order differential equations, and system of functional equations are also studied. We construct an example to support our main result.


Introduction and Preliminaries
The idea of a metric space was generalized by Czerwik [1] and Bakhtin [2]. They presented metric spaces called b-metric spaces. Several researchers took the idea of Czerwik and illustrated interesting results. For details, see [3][4][5][6][7]. For recent generalizations to b-metric spaces by employing control functions in the triangle inequality to replace the constant of the b-metric triangle inequality, we refer to [8][9][10][11][12] and the references therein.
In 1973, the contraction was introduced by Geraghty [13] in which the contraction constant was quite changed by mapping owing to its interesting properties. After that, several papers for rational Geraghty contractive mappings have appeared (for details, see [14][15][16][17][18]). Khan [19] introduced one of the best works in this line, and Fisher [20] modified it. By rational expressions, Khan [19] and Fisher [20] results were lately extended by Piri et al. [21] by introducing a new general contractive condition. Fixed point results via F-Khan contractions were studied by Piri et al. [22] on complete metric spaces, and they discuss their application to integral equations. In [23], Ullah et al. established fixed point results and discuss the application to an infinite system of fractional order differential equations.
Nadler [24] elaborated and extended the Banach contraction principle [25] to set-valued mapping by using the Hausdorff metric. After different generalizations of the Nadler contraction principle, Wardowski [26] introduced a contraction called F-contraction. In this way, Wardowski generalized the Banach contraction principle (BCP) in a different manner from the known results of literature. Following this direction, Sgroi and Vetro [27] studied set-valued F-contractions and discussed their application on certain functional and integral equations.
Cosentino and Vetro [25] extended F-contraction in the setting of b-metric spaces and proved some fixed point results. Ali et al. [28] studied the fixed point, generalize the result of Cosentino et al. [25] for a new class of F-contractions in the set-ting of b-metric spaces, and apply the result to obtain existence results for Volterra-type integral inclusion in b-metric spaces. Several authors generalized F-contraction by combining it with some existing contractive conditions (see [27,[29][30][31][32]).
In the current work, we derive a hybrid (single and multivalued) common fixed point result for the F-Khan-type contraction in the b-metric space. Also, we shall provide an example and applications for the validity of the established result. Throughout this paper, CBðΛÞ indicates the family of nonempty subsets of Λ, which is bounded and closed. ℝ + , ℕ 0 , and ℕ signify the set of any nonnegative real numbers, the set of nonnegative integers, and the set of positive integers. Now, we recall a few basic results and definitions.
Definition 2 [1]. Assume ðΛ, d, sÞ is a b-metric space, where s ≥ 1: Let σ n be a sequence in Λ. Then, σ ∈ Λ is said to be the limit of the sequence σ n if and the sequence σ n is said to be convergent in Λ.
Definition 3 [1]. If for each ϵ > 0, there is a positive integer ℕ such that dðσ n , σ m Þ < ϵ for all n, m > ℕ, then a sequence σ n is said to be a b-Cauchy sequence.
Definition 4 [1]. A b-metric space ðΛ, d, sÞ is said to be complete (or a b-complete metric space) if every Cauchy sequence in ðΛ, d, sÞ is convergent in Λ.
Definition 5 [33]. Assume s ≥ 1 is a real number and F represents the family of functions F : ℝ + ⟶ ℝ, with the below conditions: (F 1 ) For each fγ n g ⊂ ℝ + which is a positive term sequence, F is strictly increasing.
then, λ has a fixed point in Λ.

Main Results
Definition 11. Let ðΛ, d, sÞ be a b-metric space. (i) θ and Θ have a common fixed point if θθα = θα, and θ is occasionally Θ -weakly commuting at α. Then, θ and Θ have a common fixed point Proof. Let ζ 0 ∈ Λ and ζ 1 ∈ Θζ 0 . Let ζ 2 ≔ θζ 1 . By Lemma 8, there exists ζ 3 ∈ Θζ 2 such that dðζ 3 , ζ 2 Þ ≤ ℍðΘζ 2 , fθζ 1 gÞ. Inductively, we let ζ 2n ≔ θζ 2n−1 , and by Lemma 8, we choose ζ 2n+1 ∈ Θζ 2n such that Using equation (8), we have which implies We deduce that Let Q n = dðζ 2n+1 , ζ 2n+2 Þ > 0, ∀n ∈ ℕ. It follows from (12) and axiom F 4 that Thus, by equation (13), which implies that Applying limit n ⟶ 1, we have lim n→∞ n s n p n ð Þ k = 0: From (21), there exists n 1 ∈ ℕ such that nðs n Q n Þ k < 1 such that To show that fζ n g is a b-Cauchy sequence, consider m, n ∈ ℕ such that m > n > n 1 , using triangular inequality, and using (18) 3 Journal of Function Spaces By taking limit, we get dðζ n , ζ m Þ ⟶ 0. Hence, fζ n g is a b -Cauchy sequence, but a b-metric space ðΛ, d, sÞ is a complete space so there exists ζ ∈ Λ such that ζ n ⟶ ζ as n ⟶ ∞: The next step is to show that ζ is a common fixed point of the mapping Θ and θ. We have which implies that Since F is strictly increasing, therefore Adding τ to both sides and using equation (7), we have Since τ ∈ ℝ + , we have Since F is strictly increasing, therefore Applying limit n ⟶ ∞, we get By taking limit and using continuity of θ, we have which implies θζ ∈ Θζ. Since θθζ = θζ and θζ ∈ Θθζ, therefore γ = θγ ∈ Θγ. By putting θ = Θ in Theorem 12, we get the following. (i) θ and Θ have a common fixed point if θθα = θα, and θ is occasionally Θ -weakly commuting at α. Then, θ and Θ have a common fixed point Remarks. Our result extended the results of (i) Fisher [20] for set-valued mapping in the setting of b-metric spaces (ii) Khan [19] for set-valued mapping in b-metric spaces (iii) Piri et al. [21,22] for set-valued mapping in b-metric spaces

Journal of Function Spaces
Example 17. Consider the sequence ffS q g: q ∈ f1, 2 ⋯ 100gg as follows: Let Λ = fS q : q ∈ f1, 2,⋯,100gg and d : Λ × Λ ⟶ ½0, ∞Þ be defined by Then, ðΛ, d, sÞ is a complete b-metric space. Define the mapping Θ : Λ ⟶ CBðΛÞ by and θ : Λ ⟶ Λ by Let us consider the following calculation. First, observe that max fdðx q , θx p Þ, dðθx q , For each p ∈ ℕ, p > 2, we have For each p, q ∈ N, p > q > 1, we have Multiplying 1.01 on both sides and taking log to the base e on both sides, we get inequality (7) and also find that τ = 0:004365 ⋯ .Therefore, θ and Θ have a fixed point.
Assume that an object of mass m moves to and fro on the x-axis around an equilibrium position x = 0 (see Figure 1). The object has position xðtÞ at time t. It undergoes a force due to a spring: Furthermore, a damping force that resists the movement of the object is shown: Now, by the second law of motion, where m, k, and b are all positive constants. Up to that feature, the system is simply the damped harmonic oscillator. Now, suppose the additional time-dependent force f ðtÞ is applied to the object. Then, by Newton's second law, The problem (43) can be written in the form of the Fredholm integral equation: Here, Green's function for critically damping oscillation is defined as where τ can be found in terms of m, b, and k. Let Λ = C½0, 1 be the set of all continuous functions defined on ½0, 1. For u ∈ Cð½0, 1Þ, define the supremum norm as v k k = sup Let Cð½0, 1, ℝÞ be endowed with the b-metric: With these contexts, Cð½0, 1, R, k:kÞ becomes the Banach space. We give the following theorem.
Theorem 18. Assume the assumptions given below hold.
Similarly, we can apply our theorem for the existence of underdamping oscillation and overdamping oscillation.

Application to Infinite Systems of Fractional Order Differential Equations
Now, we have to derive sufficient conditions for the solutions in space c to the following nonlinear infinite systems of fractional order differential equations: with the initial condition ϑ 0 = ϑ 0 i , where t ∈ J, i, j = 1, 2 ⋯ , and τ is the positive real number. J is any fixed interval on the real line. Let Λ = c be the space of all real sequences whose limit is finite.