Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations via Common Fixed-Point Techniques in Complex Valued Fuzzy Metric Spaces

emain purpose of this paper is to study the existence theorem for a common solution to a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations. In this respect, we study the fuzzy version of some essential common fixed-point results from metric spaces in the newly introduced notion of complex valued fuzzy metric spaces. Also, we provide an illustrative example to demonstrate the validity of our derived results.


Introduction and Preliminaries
Zadeh initiated the concept of fuzzy sets in 1965 [1], which introduced a deep research activity leading to the improvement of attractive theory of fuzzy system. Afterwards, several researchers have contributed towards some basic significant results in fuzzy sets. e notion of fuzzy metric was established by Kramosil and Michalek [2]. ey generalized the concepts of probabilistic metric spaces to the fuzzy situation. George and Veeramani [3] amended the notion of fuzzy metric to derive a Hausdorff topology initiated by fuzzy metric. is obtained a milestone in the existence theory of fixed point in fuzzy metric spaces. Afterwards, a number of different generalizations appeared for the existence theory of fixed point in fuzzy metric. Garbiec [4] established the fuzzy version of Banach contraction principle in fuzzy metric spaces. For some necessary definitions, examples, and basic results, we refer to [5][6][7] and the references herein.
Fixed-point theory has a broad set of applications in modern mathematics. Banach contraction principle is the most basic and widely used technique in mathematical analysis. Due to the constructive nature of Banach contraction principle, it is the most useful tool to solve several existence problems in mathematics. Several generalizations of the Banach contraction technique are made in many directions satisfying different types of contractive conditions and having many applications in mathematical disciplines [8,9].
In the meanwhile, researchers realized that due to vector division, rational type contraction is not meaningful in cone metric spaces.
us, many results cannot be extended to cone metric spaces. To overcome this problem, very recently, in 2011, Azam et al. [10] initiated a new setting of metric fixed-point theory which is known as complex valued metric spaces. Here, they considered the set of complex number instead of set of positive real numbers as a ground set endowing with a partial structure. e authors obtained fixed-point results satisfying rational contraction and discussed its applications in the said setting. Moreover, Rouzkard and Imdad [11] generalized the applications of complex valued metric spaces and wrote some beautiful remarks. Furthermore, Sintunavarat and Kumam obtained existence results of fixed point for single valued mappings involving control functions instead of constants in contractive condition [12].
Very recently, Shukla et al. [13] introduced an innovative concept of complex valued fuzzy metric spaces where they defined several associated topological features for complex valued fuzzy metric spaces. Moreover, they established the fuzzy version of the well-celebrated Banach contraction principle in different directions and discussed its applications. e theory of impulsive functional differential equations is emerging as an important area of investigation since such equations appear to represent a natural framework for mathematical modeling of many real processes and phenomena studied in optimal control, electronics, economics, and so on. To further study on impulsive functional differential equations, we refer the readers to [14,15]. e researchers used a set of different techniques to discuss the existence of solution of such models such as the homotopy perturbation method, Laplace transform method, Adomian decomposition method, and different types of approximation methods. Among such techniques, fixed-point theory is one of the main tools to investigate the analytical and numerical solution of mathematical models. Many researchers have utilized the existence results of fixed point to investigate the analytical solution of different types of differential and integral equations in different spaces. For instance, we refer to [16][17][18][19][20][21][22][23][24].
In our work, we extend fixed-point results under the general contractive condition in [25] to the setting of complex valued fuzzy metric spaces. Moreover, we studied a result of existence and uniqueness of the solution of nonlinear impulsive fractional differential equations.
Define a partial ordering ≺ on C by ξ 1 ≺ ξ 2 if ξ 2 − ξ 1 ∈ P. e relations ξ 1 ≺ ξ 2 and ξ 1 ≺ξ 2 indicate that Re(ξ 1 ) ≤ Re(ξ 2 ), Let D ⊂ C. If there exists infD such that it is the lower bound of D, that is infD ≺ c, ∀c ∈ D and v ≺ infD for every lower bound v of D, then infD is called the greatest lower bound of D. In the same way, we define supD, (lub) the least upper bound of D.
Definition 1 (see [13]). Let Q⊆X be a nonempty set. A complex fuzzy set M is characterized by a mapping μ Q (x) with domain Q and the range in the closed unit complex interval I, which assigns each element x ∈ Q, a grade of membership in Q, and is thus of the form where r Q (x) ∈ [0, 1]. e complex fuzzy set may be written as Definition 2 (see [13]). A binary equation * : I × I ⟶ I is said to be complex valued t-norm if the following conditions hold: Example 1. Let * a , * b , * c : I × I ⟶ I be three binary operations defined, respectively, by is the real unit closed interval and * a , * b : I R × I R ⟶ I R are two t-norms, then * : Mathematical Problems in Engineering is a complex valued t-norm.
Definition 3 (see [13]). Let Q be a nonempty set, * be a continuous complex valued t-norm, and M be a complex fuzzy set on Q × Q × P θ ⟶ I satisfying the following conditions: and t, t ′ ∈ P θ en, the triplet (Q, M, * ) is said to be a complex valued fuzzy metric space and M is called a complex valued fuzzy metric on Q. e functions M(w, z, t) denote the degree of nearness and the degree of non-nearness between w and z with respect to the complex parameter t, respectively.
Let the complex fuzzy set M defined be given by for all z, ξ ∈ P θ , t � (e, c). en, (Q, M, * ) is a complex valued fuzzy metric space.
Indeed in above example, if g: P θ ⟶ (0, ∞) is continuous and nondecreasing function, that is, Similarly, it is obvious that for the fuzzy set defined by (Q, M, * ) is a complex valued fuzzy metric space.
Definition 4 (see [13]). Let (Q, M, * ) be a complex valued fuzzy metric space. A sequence z n ∈ Q converges to z ∈ Q, if for each ] ∈ I 0 and t ∈ P θ , there exists k 0 ∈ N with Definition 5 (see [13]). Let (Q, M, * ) be a complex valued fuzzy metric space. A sequence x b in Q is known as a Cauchy sequence if e complex valued fuzzy metric space (Q, M, * ) is called complete if every Cauchy sequence is convergent in Q.
(a) If the sequence z b is monotonic with respect to ≺ and there exist c, η ∈ P with c ≺ z b ≺ η, ∀b ∈ N, then there exists z ∈ P such that lim b⟶∞ z b � z. (b) Although the partial ordering ≺ is not a linear order on C, the pair (C, ≺ ) is a lattice. (c) If Q ⊂ C and there exists c, η ∈ C with c ≺ s ≺ η, ∀s ∈ Q, then infQ and supQ both exist.
en, the pair of mappings (5, Ξ) has a unique common fixed point.
Proof. Let z 0 ∈ Q. Define a sequence z b in Q by Mathematical Problems in Engineering By using (10), we have which yields By Lemma 1, this leads to a contradiction; therefore, let which implies that Hence, ρ b is monotonic sequence in P, and using Remark 1 and (18), there exists ℓ ′ ∈ P such that Inequality (13) suggests that By using (19), we obtain Since α ∈ [0, 1) and utilizing Remark 2, we must obtain ℓ ′ � ℓ. us, To show that z b is a Cauchy sequence, define for b ∈ 0, 1, 2, . . . { } and fixed t ∈ P θ . Since θ≺(z b , z d , t) ≺ ℓ for all b ∈ 0, 1, 2, . . . { }, using Remark 1, we obtain that for all b ∈ 0, 1, 2, . . . the infimum exists. For d > b, by (10), we have now for each positive integer d, It follows that erefore, erefore, from (26), we have showed that z { } b is a Cauchy sequence in Q. Since Q is complete, by Lemma 2, there exists an element τ ∈ Q such that For b ∈ R and for any t ∈ P θ , we obtain from (10) that which implies that Now, for any t ∈ P θ , By taking limit as b ⟶ ∞ and using Remark 2 and (27), we have Mathematical Problems in Engineering us, we obtain that M(τ, 5τ, t) � ℓ for all t ∈ P θ , that is, 5τ � τ. Similarly, it follows that M(τ, Ξτ, t) � ℓ and so Ξτ � τ. Hence, the pair (5, Ξ) has a common fixed point.

(39)
To show that z n is a Cauchy sequence, define for b ∈ 0, 1, 2, . . . { } and fixed t ∈ P θ . Since θ≺(z b , z d , t) ≺ ℓ for all b ∈ 0, 1, 2, . . . { }, using Remark 1, we obtain that for all b ∈ 0, 1, 2, . . . the infimum, infO b � ρ b (say), exists. For t ∈ P θ and b, d ∈ N with d > b, we obtain the following from (39) and Lemma 1: which yields erefore, by definition, we have Hence, ρ b is monotonic sequence in P, and using Remark 1 and (43), there exists ℓ ′ ∈ P such that Again from (39), we have the following for t ∈ P and b ∈ N: Similarly, we get the following for d > b: Hence, for all t ∈ P θ and b ∈ N, Since lim b⟶∞ (t/α 2b+1 ) � ∞, by (44) and by the hypothesis, we have From (41) and (48), we obtain Hence, z b is a Cauchy sequence in Q. Since Q is complete and using Lemma 2, there exists μ ∈ Q such that lim b⟶∞ M z b , μ, t � ℓ, for all t ∈ P θ . (50) For any t ∈ Q, it follows from (33) that By taking limit as b ⟶ ∞ and using Remark 2 and (50), we obtain that M(μ, 5μ, t) � ℓ for all t ∈ P θ , that is, 5μ � μ. Similarly, it follows that M(μ, Ξμ, t) � ℓ, and so Ξμ � μ.
Case 2. If z 2b � z 2b+1 , b � 0, 1, 2, . . .. It implies that the sequence z b is constant and so convergent. e rest of the proof can be completed on the steps of Case 1. is completes the proof. (53) Define a t-norm " * " by a * e � min a, e { } where a, e ∈ I. Let M be the complex valued fuzzy set given by for c ∈ P 0 . Clearly, (X, M, * ) is a complex valued fuzzy metric space. Certainly, for any sequence c b ∈ P 0 , c b � (m b , n b ) with lim b⟶∞ c b � ∞ and for each fixed w ∈ Z, we have, |w − z| ≤ 1, for all z ∈ 0, 1 { }, then Define 5, Ξ � (w/5). Note that with α � (2/7) and by routine calculation, one can easily verify that 5, Ξ satisfy condition (33). Hence, all the assumptions of eorem 2 are satisfied. Moreover, w � 0 remains fixed under 5, Ξ; therefore, they have a common fixed point.
(61) en, the mapping 5 has a unique fixed point.