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

Authors: Humaira; Muhammad Sarwar; Thabet Abdeljawad
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The main 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.


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## Body

## 1. 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.The notion of fuzzy metric was established by Kramosil and Michalek [2]. They 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. This 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 [5–7] 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. Thus, 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. The 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.The 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]. The 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 [16–24].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.(1)D0Ctjμgt=Kt,gt,D0Ctjμgt,t∈J,t≠tj,j=1,2,…,q,Δgtj=ℑjgt,Δg′tj=ℑj˜gt,j=1,2,…,q,g0=−g′0,g1=−g′ρ,ρ∈0,1,ρ≠tj,j=1,2,…,q,where the notation D0Ctjμ stands for Caputo fractional derivative of order μ∈1,2,J=0,1, and K:J×ℝ2⟶ℝ is a continuous function. Further, the nonlinear functions ℑj,ℑj˜:J⟶ℝ, are also continuous for j=1,2,…,q and Δgtj=gtj+−gtj− and Δg′tj=g′tj+−g′tj−, where gtj+,gtj− represent the right and left-hand limit of the function gt, respectively. Also, 0=t0<t1<t2<⋯<tq<tq+1=1,q∈Z+, where Z+ is the set of positive integers. Finally, we gave an illustrative example to make our results strong.Throughout the paper, we have denoted the set of complex numbers byℂ. Let P=x,y:0≤x<∞,0≤y<∞⊂ℂ. The elements 0,0,1,1∈P are denoted by θ and ℓ, respectively.Define a partial ordering≺¯ on ℂ by ξ1≺¯ξ2 if ξ2−ξ1∈P. The relations ξ1≺¯ξ2 and ξ1≺ξ2 indicate that Reξ1≤Reξ2, Imξ1≤Imξ2 and Reξ1≤Reξ2, Imξ1≤Imξ2, respectively. A sequence is monotonic with respect to ≺¯ if either ξb≺¯ξb+1 or ξb+1≺¯ξb,∀q∈N. Let the unit closed complex interval be denoted by I=z,y:0≤z≤1,0≤y≤1, the open unit complex interval be denoted by I0=z,y:0≤z<1,0≤y<1, and the set z,y:0<z<∞,0<y<∞ be denoted by Pθ. Clearly, for ξ,z∈ℂ,z≺¯ξ if z−ξ∈Pθ.LetD⊂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 μQx 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(2)μQx=rQxℓ,where rQx∈0,1. The complex fuzzy set may be written as(3)M=x,μQxx∈X.Definition 2.
(see [13]). A binary equation ∗:I×I⟶I is said to be complex valued t-norm if the following conditions hold:(1)
w1∗w2=w2∗w1(2)
w1∗w2≺¯w3∗w4 whenever w1≺¯w3,w2≺¯w4(3)
w1∗w2∗w3=w1∗w2∗w3(4)
w∗θ=θ,w∗ℓ=wfor all w,w1,w2,w3,w4∈I.Example 1.
Let∗a,∗b,∗c:I×I⟶I be three binary operations defined, respectively, by(1)
o1∗ao2=e1e2,c1,c2, for all o1=e1,c1,o2=e2,c2∈I(2)
o1∗bo2=mine1,e2,minc1,c2, for all o1=e1,c1,o2=e2,c2∈I(3)
o1∗co2=maxe1+e2−1,0,maxc1+c2−1,0, for all o1=e1,c1,o2=e2,c2∈I
Then,∗a,∗b,∗c are complex valued t-norms.
Indeed ifIR=0,1 is the real unit closed interval and ∗a,∗b:IR×IR⟶IR are two t-norms, then ∗:I×I⟶I defined by(4)o1∗o2=e1∗ae2,c1∗bc2,for allo1=e1,c1,o2=e2,c2∈I,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×Pθ⟶I satisfying the following conditions:(1)
0≺¯Mw,z,t(2)
Mw,z,t=ℓ for every t∈Pθ if and only if w=z(3)
Mw,z,t=Mz,w,t(4)
Mw,z,t∗Mz,y,t′≺¯Mw,y,t+t′(5)
Mw,z,∗:Pθ⟶I is continuous for all w,z,y∈Q and t,t′∈Pθ
Then, the tripletQ,M,∗ is said to be a complex valued fuzzy metric space and M is called a complex valued fuzzy metric on Q. The functions Mw,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.Example 2.
Consider any metric spaceQ,σ. Define ∗ by o1∗o2=e1e2,c1,c2, for all o1=e1,c1,o2=e2,c2∈I. Let the complex fuzzy set M defined be given by(5)Mz,ξ,t=e1+c1e1+c1+σz,ξℓ,for all z,ξ∈Pθ,t=e,c. Then, Q,M,∗ is a complex valued fuzzy metric space.
Indeed in above example, ifg:Pθ⟶0,∞ is continuous and nondecreasing function, that is, o1≺¯o2 implies that go1≤go2, then Q,M,∗ is a complex valued fuzzy metric space, where(6)Mz,ξ,t=gtgt+σz,ℓ.
Similarly, it is obvious that for the fuzzy set defined by(7)Mz,ξ,t=expσz,ξgt−1ℓ,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 zn∈Q converges to z∈Q, if for each ν∈I0 and t∈Pθ, there exists k0∈N with(8)ℓ−ν≺Mzb,z,t,∀b>k0.Definition 5.
(see [13]). Let Q,M,∗ be a complex valued fuzzy metric space. A sequence xb in Q is known as a Cauchy sequence if(9)limb⟶∞infd>bMxb,xd,t=ℓ,∀t∈Pθ.
The complex valued fuzzy metric spaceQ,M,∗ is called complete if every Cauchy sequence is convergent in Q.Lemma 1 (see [13]).
LetQ,M,∗ be a complex valued fuzzy metric space. If t,t′∈Pθ and t≺¯t′, then Mz,ξ,t≺¯Mz,ξ,t′,∀z,ξ∈Q.Lemma 2 (see [13]).
LetQ,M,∗ be complex valued fuzzy metric space. A sequence zb in Q converges to v∈Q if limb⟶∞Mzb,v,t=ℓ holds ∀t∈Pθ.Remark 1.
(see [13]). Let zb∈P∀n∈N.(a)
If the sequencezb is monotonic with respect to ≺¯ and there exist γ,η∈P with γ≺¯zb≺¯η,∀b∈N, then there exists z∈P such that limb⟶∞zb=z.(b)
Although the partial ordering≺¯ is not a linear order on ℂ, the pair ℂ,≺¯ is a lattice.(c)
IfQ⊂ℂ and there exists γ,η∈C with γ≺¯s≺¯η,∀s∈Q, then infQ and supQ both exist.Remark 2.
(see [13]). Let xb,xb′,w∈P, ∀b∈N.(a)
Ifxb≺¯xb′≺¯ℓ,∀b∈N and limb⟶∞xb=ℓ, then limb⟶∞xb′=ℓ.(b)
Ifxb≺¯w,∀b∈N and limb⟶∞xb=x, then x≺¯w.(c)
Ifw≺¯xb,∀b∈N and limb⟶∞xb=x, then w≺¯x.


## 2. Common Fixed-Point Results

Theorem 1.
LetQ,M,∗ be a complete complex valued fuzzy metric space and let Ϝ,Ξ:Q⟶Q be self-mappings, such that(10)ℓ−MϜz,Ξξ,t≺¯αℓ−Ωz,ξ,t,for all z∈Q, where α∈0,1.(11)Ωz,ξ,t=maxMz,ξ,t,Mz,Ϝz,tMξ,Ξξ,tℓ+Mz,ξ,t,Mξ,Ϝz,tMz,Ξξ,tℓ+Mz,ξ,t.Then, the pair of mappingsϜ,Ξ has a unique common fixed point.Proof.
Letz0∈Q. Define a sequence zb in Q by(12)z2b+1=Ϝz2b,z2b+2=Ξz2b+1,for allb∈0,1,2,….
By using (10), we have(13)ℓ−Mz2b+1,z2b+2,t=ℓ−MϜz2b,Ξz2b+1,t≺¯αℓ−maxMz2b,z2b+1,t,Mz2b,Ϝz2b,tMz2b+1,Ξz2b+1,tℓ+Mz2b,z2b+1,t,Mz2b+1,Ϝz2b,tMz2b,Ξz2b+1,tℓ+Mz2b,z2b+1,t≺¯αℓ−maxMz2b,z2b+1,t,Mz2b,z2b+1,tMz2b+1,z2b+2,tℓ+Mz2b,z2b+1,t,Mz2b+1,z2b+1,tMz2b,z2b+2,tℓ+Mz2b,z2b+1,t≺¯αℓ−maxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t≺¯ℓ−maxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t,which yields(14)Mz2b+1,z2b+2,t≽maxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t.
IfmaxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t=Mz2b,z2b+2,t, then(15)Mz2b+1,z2b+2,t≽Mz2b,z2b+2,t≽Mz2b,z2b+1,t2∗Mz2b+1,z2b+2,t2.
By Lemma1, this leads to a contradiction; therefore, let(16)maxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t=Mz2b,z2b+2,t,which implies that(17)Mz2b+1,z2b+2,t≽Mz2b,z2b+1,t,for allb=0,1,2,…,
LetMz2b,z2b+1,t=ρb; we shall show that ρb=ℓ. Therefore, by definition, we have(18)ℓ≽ρb+1≽ρb≽θ.
Hence,ρb is monotonic sequence in P, and using Remark 1 and (18), there exists ℓ′∈P such that(19)limb⟶∞ρb=ℓ′.
Inequality (13) suggests that(20)1−αℓ+αmaxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t≺¯Mz2b+1,z2b+2,t1−αℓ+αMz2b,z2b+1,t≺¯Mz2b+1,z2b+2,t1−αℓ+αρb≺¯ρb+1.
By using (19), we obtain(21)1−αℓ≺¯1−αℓ′.
Sinceα∈0,1 and utilizing Remark 2, we must obtain ℓ′=ℓ. Thus,(22)limb⟶∞ρb=ℓ.
To show thatzb is a Cauchy sequence, define(23)Ob=Mzb,zd,t:d>b⊆I,for b∈0,1,2,… and fixed t∈Pθ. Since θ≺zb,zd,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,(24)Mzb,zb+d,t≽Mzb,zb+1,td∗Mzb+1,zb+2,td∗⋯∗Mzb+d−1,zb+d,td.
It follows that(25)limb⟶∞infb+d>bMzb,zb+d,t≽ℓ∗ℓ∗⋯∗ℓ=ℓ.
Therefore,(26)limb⟶∞infb+d>bMzb,zb+d,t=ℓ,for allt∈P0.
Therefore, from (26), we have showed that zb is a Cauchy sequence in Q. Since Q is complete, by Lemma 2, there exists an element τ∈Q such that(27)limb⟶∞infb+d>bMzb,τ,t=ℓ,for allt∈Pθ.
Forb∈R and for any t∈Pθ, we obtain from (10) that(28)ℓ−MϜτ,Ξz2b+1,t2≺¯αℓ−maxMτ,z2b+1,t2,Mτ,Ϝτ,t/2Mz2b+1,Ξz2b+1,t/2ℓ+Mτ,z2b+1,t/2,Mτ,Ξz2b+1,t/2Mz2b+1,Ϝτ,t/2ℓ+Mτ,z2b+1,t/2≺¯αℓ−maxMτ,z2b+1,t2,Mτ,Ϝτ,t/2Mz2b+1,z2b+2,t/2ℓ+Mτ,z2b+1,t/2,Mτ,z2b+2,t/2Mz2b+1,Ϝτ,t/2ℓ+Mτ,z2b+1,t/2≺¯ℓ−maxMτ,z2b+1,t2,Mτ,Ϝτ,t/2,Mz2b+1,z2b+2,t/2ℓ+Mτ,z2b+1,t/2,Mτ,z2b+2,t/2Mz2b+1,Ϝτ,t/2ℓ+Mτ,z2b+1,t/2,which implies that(29)MϜτ,Ξz2b+1,t2≽maxMτ,z2b+1,t2,Mτ,Ϝτ,t/2Mz2b+1,z2b+2,t/2ℓ+Mτ,z2b+1,t/2,Mτ,z2b+2,t/2Mz2b+1,Ϝτ,t/2ℓ+Mτ,z2b+1,t/2.
Now, for anyt∈Pθ,(30)Mτ,Ϝτ,t≽Mτ,z2b+2,t2∗Mz2b+2,Ϝτ,t2≽Mτ,z2b+2,t2∗MϜτ,Ξz2b+1,t2≽Mτ,z2b+2,t2∗maxMτ,z2b+1,t2,Mτ,Ϝτ,t/2Mz2b+1,z2b+2,t/2ℓ+Mτ,z2b+1,t/2,Mτ,z2b+1,t/2,Mz2b+1,Ϝτ,t/2ℓ+Mτ,z2b+1,t/2.
By taking limit asb⟶∞ and using Remark 2 and (27), we have(31)Mτ,Ϝτ,t≽ℓ∗maxℓ,Mτ,Ϝτ,t/2ℓℓ+ℓ,ℓMz2b+1,Ϝτ,t/2ℓ+ℓ=ℓ∗ℓ=ℓ.
Thus, we obtain thatMτ,Ϝτ,t=ℓ for all t∈Pθ, that is, Ϝτ=τ. Similarly, it follows that Mτ,Ξτ,t=ℓ and so Ξτ=τ. Hence, the pair Ϝ,Ξ has a common fixed point. Assume that υ is any other common fixed point of Ϝ,Ξ and there exists t∈Pθ with Mτ,υ,t≠ℓ; then,(32)ℓ−Mτ,υ,t=ℓ−MϜτ,Ξυ,t≺¯αℓ−maxMτ,υ,t,Mτ,Ϝτ,tMυ,Ξυ,tℓ+Mτ,υ,t,Mτ,Ξυ,tMυ,Ϝτ,tℓ+Mτ,υ,t≺¯αℓ−Mτ,υ,t.
This leads to a contradiction. Therefore,Mτ,υ,t=ℓ for all t∈P0, that is, τ=υ.Theorem 2.
LetQ,M,∗ be a complete complex valued fuzzy metric space with limb⟶∞infξ∈QMz,ξ,tb=ℓ for all z∈Q, where tb is a sequence in Pθ. Let Ϝ,Ξ:Q⟶Q be self-mappings, such that(33)MϜz,Ξξ,αt≽Ωz,ξ,t,for all z∈Q, where α∈0,1.(34)Ωz,ξ,t=maxMz,ξ,t,Mz,Ϝz,tMξ,Ξξ,tℓ+Mz,ξ,t,Mξ,Ϝz,tMz,Ξξ,tℓ+Mz,ξ,t.Then, the pair of mappingsϜ,Ξ has a unique common fixed point.Proof.
Letz0 be any arbitrary point in Q. Define a sequence zb in Q by(35)z2b+1=Ϝz2bandz2b+2=Ξz2b+1,for allb∈0,1,2….Case 1.
Whenz2b+1≠Ϝz2b, for n=0,1,2,… by using (33) with z=z2b and ξ=z2b+1, we have(36)Mz2b+1,z2b+2,αt=MϜz2b,Ξz2b+1,αt≽maxMz2b,z2b+1,t,Mz2b,Ϝz2b,tMz2b+1,Ξz2b+1,tℓ+Mz2b,z2b+1,t,Mz2b+1,Ϝz2b,tMz2b,Ξz2b+1,tℓ+Mz2b,z2b+1,t≽maxMz2b,z2b+1,t,Mz2b,z2b+1,tMz2b+1,z2b+2,tℓ+Mz2b,z2b+1,t,Mz2b+1,z2b+1,tMz2b,z2b+2,tℓ+Mz2b,z2b+1,t≽maxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t.
Now suppose that ifmaxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t=Mz2b,z2b+2,t, then using (36), we have(37)Mz2b+1,z2b+2,αt≽Mz2b,z2b+2,t≽Mz2b,z2b+1,1−αt∗Mz2b+1,z2b+2,αt.
By Lemma1, this leads to a contradiction; therefore, let(38)maxMz2b,z2b+1,t,Mz2b+1,z2b+2,t,Mz2b,z2b+2,t=Mz2b,z2b+1,t,and this implies that(39)Mz2b+1,z2b+2,αt≽Mz2b,z2b+1,t,for allb=0,1,2,….
To show thatzn is a Cauchy sequence, define(40)Ob=Mzb,zd,t:d>b⊆I,for b∈0,1,2,… and fixed t∈Pθ. Since θ≺zb,zd,t≺¯ℓ for all b∈0,1,2,…, using Remark 1, we obtain that for all b∈0,1,2,… the infimum, infOb=ρb (say), exists. For t∈Pθ and b,d∈N with d>b, we obtain the following from (39) and Lemma 1:(41)Mz2b+1,z2d+2,αt≽Mz2b,z2d+1,t≽Mz2b,z2d+1,αt,which yields(42)Mz2b+1,z2d+2,αt≽Mz2b,z2d+1,αt.
Therefore, by definition, we have(43)ℓ≽ρb+1≽ρb≽θ.
Hence,ρb is monotonic sequence in P, and using Remark 1 and (43), there exists ℓ′∈P such that(44)limb⟶∞ρb=ℓ1.
Again from (39), we have the following for t∈P and b∈N:(45)ρ2b+1=infd>bMz2b+1,z2d+1,t≽infd>bMz2b,z2d,tα.
Similarly, we get the following ford>b:(46)Mz2b+1,z2d+1,t≽Mz2b,z2d,tα≽MϜz2b−1,Ξz2d−1,tα≽Mz2b−1,z2d−1,tα2≽MϜz2b−2,Ξz2d−2,tα2≽Mz2b−2,z2d−2,tα3≽⋯≽Mz0,z2d−2b,tα2b+1.
Hence, for allt∈Pθ and b∈N,(47)ρ2b+1=infd>bMz2b+1,z2d+1,t≽infd>bMz0,z2d−2b,tα2b+1≽infξ∈QMz0,ξ,tα2b+1.
Sincelimb⟶∞t/α2b+1=∞, by (44) and by the hypothesis, we have(48)ℓ1≽limb⟶∞infξ∈QMz0,ξ,tα2b+1=ℓ.
From (41) and (48), we obtain(49)limb⟶∞ρb=ℓ.
Hence,zb is a Cauchy sequence in Q. Since Q is complete and using Lemma 2, there exists μ∈Q such that(50)limb⟶∞Mzb,μ,t=ℓ,for allt∈Pθ.
For anyt∈Q, it follows from (33) that(51)Mμ,Ϝμ,t≽Mμ,z2b+2,t2∗Mz2b+2,Ϝμ,t2≽Mμ,z2b+2,t2∗MϜμ,Ξz2b+1,t2≽Mμ,z2b+2,t2∗maxMμ,z2b+1,t2,Mμ,Ϝμ,t/2Mz2b+1,z2b+2,t/2ℓ+Mμ,z2b+1,t/2,Mμ,z2b+2,t/2,Mz2b+1,Ϝμ,t/2ℓ+Mμ,z2b+1,t/2.
By taking limit asb⟶∞ and using Remark 2 and (50), we obtain that Mμ,Ϝμ,t=ℓ for all t∈Pθ, that is, Ϝμ=μ. Similarly, it follows that Mμ,Ξμ,t=ℓ, and so Ξμ=μ. Hence, the pair Ϝ,Ξ has a common fixed point. Assume that υ is any other common fixed point of Ϝ,Ξ and there exists t∈Pθ with Mμ,υ,t≠ℓ; then,(52)Mμ,υ,t=MϜμ,Ξυ,t≽Mμ,υ,t≽Mμ,υ,tα≽MϜμ,Ξυ,tα≽Mμ,υ,tα2≽⋯≽Mμ,υ,tαb,for all b∈N. Applying limb⟶∞t/αb=∞ and Mμ,υ,t/αb≽infξ∈QMμ,ξ,t/αb, it follows that from the above inequality, we have Mμ,υ,t≽ℓ. This leads to a contradiction. Therefore, Mμ,υ,t=ℓ for all t∈P0, that is, μ=υ.Case 2.
Ifz2b=z2b+1,b=0,1,2,…. It implies that the sequence zb is constant and so convergent. The rest of the proof can be completed on the steps of Case 1. This completes the proof.Example 3.
LetX=0∪1/b:b∈N with the metric δ defined by(53)δw,z=w−z,for allw,z∈X.
Define at-norm “∗” by a∗e=mina,e where a,e∈I. Let M be the complex valued fuzzy set given by(54)Mw,z,c=cc+δw,zℓ,for c∈P0. Clearly, X,M,∗ is a complex valued fuzzy metric space.
Certainly, for any sequencecb∈P0,cb=mb,nb with limb⟶∞cb=∞ and for each fixed w∈Z, we have, w−z≤1, for all z∈0,1, then(55)ℓ≽infz∈Xw,z,cb=infz∈Xmbnbmbnb+w−zℓ=infz∈X11+w−z/mbnbℓ=11+supz∈Xw−z/mbnbℓ≽11+w−z/mbnbℓ.
Therefore,(56)ℓ≽limb⟶∞infz∈Xw,z,cb≽limb⟶∞11+w−z/mbnbℓ=ℓ,so that(57)limb⟶∞infz∈Xw,z,cb=ℓ.
DefineϜ,Ξ=w/5. Note that with α=2/7 and by routine calculation, one can easily verify that Ϝ,Ξ satisfy condition (33). Hence, all the assumptions of Theorem 2 are satisfied. Moreover, w=0 remains fixed under Ϝ,Ξ; therefore, they have a common fixed point.Remark 3.
In Theorem1, the contraction condition (10) for the pair of self-mappings Ϝ,Ξ can be replaced by the following one, with analogous proof:(58)ℓ−MϜz,Ξξ,t≺¯αtℓ−Ωz,ξ,t,for all z∈Q, where α is a real valued function α:Pθ⟶0,1.(59)Ωz,ξ,t=maxMz,ξ,t,Mz,Ϝz,tMξ,Ξξ,t1+Mz,ξ,t,Mξ,Ϝz,tMz,Ξξ,t1+Mz,ξ,t.Corollary 1.
LetQ,M,∗ be a complete complex valued fuzzy metric space and let Ϝ:Q⟶Q be self-mapping, such that(60)ℓ−MϜz,Ϝξ,t≺¯αℓ−Ωz,ξ,t,for all z∈Q, where α∈0,1.(61)Ωz,ξ,t=maxMz,ξ,t,Mz,Ϝz,tMξ,Ϝξ,tℓ+Mz,ξ,t,Mξ,Ϝz,tMz,Ϝξ,tℓ+Mz,ξ,t.
Then, the mappingϜ has a unique fixed point.Proof.
The proof is immediate from Theorem1 by putting Ϝ=Ξ.Corollary 2.
LetQ,M,∗ be a complete complex valued fuzzy metric space and let Ϝ:Q⟶Q be self-mapping, such that(62)ℓ−MϜbz,Ϝbξ,t≺¯αℓ−Ωz,ξ,t,for all z∈Q, where α∈0,1.(63)Ωz,ξ,t=maxMz,ξ,t,Mz,Ϝbz,tMξ,Ϝbξ,tℓ+Mz,ξ,t,Mξ,Ϝbz,tMz,Ϝbξ,tℓ+Mz,ξ,t.Then, the mappingϜ has a unique fixed point.Proof.
From Corollary1, we obtain that that there exists τ∈Q such that(64)Ϝbτ=τ.
So,(65)ℓ−MϜτ,τ,t=ℓ−MϜϜbτ,Ϝbτ,t=ℓ−MϜbϜτ,Ϝbτ,t≺¯αℓ−maxMϜτ,τ,t,MϜτ,ϜbϜτ,tMτ,Ϝbτ,tℓ+MϜτ,τ,t,Mτ,ϜbϜτ,tMϜτ,Ϝbτ,tℓ+MϜτ,τ,t≺¯αℓ−maxMϜτ,τ,t,MϜτ,ϜϜbτ,tMτ,Ϝbτ,tℓ+MϜτ,τ,t,Mτ,ϜϜbτ,tMϜτ,Ϝbτ,tℓ+MϜτ,τ,t≺¯αℓ−MϜτ,τ,t≺¯ℓ−αMϜτ,τ,t.
This yields(66)MϜτ,τ,t≽αMϜτ,τ,t.
SinceϜb satisfies all the conditions of Corollary 1, Ϝb has a unique common fixed τ∈Q. But ϜbϜτ=Ϝτ implies that Ϝτ is another fixed point of Ϝb. As common fixed point is unique, we have Ϝτ=τ. Since the fixed point of Ϝ is also a fixed point of Ϝb, the fixed point of Ϝ is unique.


## 3. Existence Theorem for Nonlinear Impulsive Fractional Differential Equations

In this section, we present a situation where our obtained results can be applied. Precisely, we study the existence of solution for a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations (1). This problem is equivalent to the integral equation(67)g=1Γμ∫0tt−ϱμ−1Kϱ,gϱ,D0Cϱjμgϱdϱ+G1−t1Γμ∫tjtt−ϱμ−1Kϱ,gϱ,D0Cϱjμgϱdϱ+1Γμ∑j=1r∫tj−1tjt−ϱμ−1Kϱ,gϱ,D0Cϱjμgϱdϱ+1Γμ−1∑j=1rt−tj∫tj−1tjtj−ϱμ−2Kϱ,gϱ,D0Cϱjμgϱdϱ+∑j=1rt−tjℑj˜gtj+∑j=1rℑjgtj+G1−t,t∈Jr,r=1,2,3,…,q,where(68)G=1Γμ∑j=1q+1∫tj−1tjtj−ϱμ−1Kϱ,gϱ,D0Cϱjμgϱdϱ+1Γμ−1∑j=1q1−tj∫tj−1tjtj−ϱμ−2Kϱ,gϱ,D0Cϱjμgϱdϱ+1Γμ−1∫tj−1ρρ−ϱμ−2Kϱ,gϱ,D0Cϱjμgϱdϱ+1Γμ−1∑j=1q∫tj−1tjtj−ϱμ−2Kϱ,gϱ,D0Cϱjμgϱdϱ+∑j=1q1−tjℑj˜gtj+∑j=1qℑj˜gtj+∑j=1qℑjgtj.For simplicity, we use the notationθgt=Kt,gt,D0Cϱjμgt.If necessary, the reader can refer to [16] for a more detailed explanation of the background of the problem. Here, we shall prove our result by establishing the existence of a common fixed point for a pair of integral operators ℑ1,ℑ2, defined on CJ,ℝ as(69)ℑigt=1Γμ∫tjtt−ϱμ−1θiϱdϱ+1Γμ∑0≤tj≤t∫tj−1tjtj−ϱμ−1θiϱdϱ+1Γμ−1∑0≤tj≤tt−tj∫tj−1tjtj−ϱμ−2θi,gϱdϱ+∑0≤tj≤tt−tjℑj˜gtj+∑0≤tj≤tt−tjℑjgtj+1−t1Γμ∑j=1q+1∫tj−1tjtj−ϱμ−1θi,gϱdϱ+1Γμ−1∑j=1q1−tj∫tj−1tjtj−ϱμ−2θi,gϱdϱ+1Γμ−1∫tj−1ρρ−ϱμ−2θi,gϱdϱ+1Γμ−1∑j=1q∫tj−1tjtj−ϱμ−2θi,gϱdϱ+∑j=1q1−tjℑj˜gtj+∑j=1qℑj˜gtj+∑j=1qℑjgtj,t∈Jr,i=1,2.Theorem 3.
Assume the following hypotheses are satisfied forj=1,2,3,…,q.(i)
H1 The nonlinear function Ki:0,1×ℝ2⟶ℝ is continuous.(ii)
H2 Let us have constants γ,δ∈0,1, which satisfy(70)K1t,g,g¯−K2t,f,f¯≤γgt−ft+δg¯t−f¯t.(iii)
(H3) ℑjgtj−ℑjg¯tj=hgtj−g¯tj, with h>0.(iv)
H4ℑj˜gtj−ℑj˜g¯tj=h∗gtj−g¯tj, with h∗>0.(v)
(H5) Λ=γ/1−δ2q+3/Γϱ−1+3q+1/Γϱ+q2h+h∗≤1 and(71)Mg,g¯,c=maxMg,g¯,c,Mg,ℑ1g,cMg¯,ℑ2g¯,cℓ+Mg,g¯,c,Mg¯,ℑ1g,cMg,ℑ2g¯,cℓ+Mg,g¯,c.Then, the pair of nonlinear integral equations(72)gt=1Γμ∫tjtt−ϱμ−1θi,gϱdϱ+1Γμ∑0≤tj≤t∫tj−1tjtj−ϱμ−1θi,gϱdϱ+1Γμ−1∑0≤tj≤tt−tj∫tj−1tjtj−ϱμ−2θi,gϱdϱ+∑0≤tj≤tt−tjℑj˜gtj+∑0≤tj≤tt−tjℑjgtj+1−t1Γμ∑j=1q+1∫tj−1tjtj−ϱμ−1θi,gϱdϱ+1Γμ−1∑j=1q1−tj∫tj−1tjtj−ϱμ−2θi,gϱdϱ+1Γμ−1∫tj−1ρρ−ϱμ−2θi,gϱdϱ+1Γμ−1∑j=1q∫tj−1tjtj−ϱμ−2θi,gϱdϱ+∑j=1q1−tjℑj˜gtj+∑j=1qℑj˜gtj+∑j=1qℑjgtj,t∈Jr,i=1,2.has a common solution in CJ,ℝ.Proof.
ConsiderΘ=CJ,ℝ with the metric(73)σg,f=maxt∈Jgt−ft.
Definet-norm as(74)c1∗c2=a1,b1,ifa2,b2=ℓ,a2,b2,ifa1,b1=ℓ,0,otherwise,for c1=a1,b1,c2=a2,b2∈I and M:Θ×Θ×Pθ⟶I defined by(75)Mg,f,c=ℓ−σg,f1+abℓ,for c=a,b∈Pθ. It is obvious that Θ,M,∗ is a complex valued fuzzy metric space. Let g,g¯∈Θ and t∈J; then, one has(76)ℑ1gt−ℑ2g¯t=1Γμ∫tjtt−ϱμ−1θ1,gϱ−θ2,g¯ϱdϱ+1Γμ∑0≤tj≤t∫tj−1tjtj−ϱμ−1θ1,gϱ−θ2,g¯ϱdϱ+1Γμ−1∑0≤tj≤tt−tj∫tj−1tjtj−ϱμ−2θ1,gϱ−θ2,g¯ϱdϱ+∑0≤tj≤tt−tjℑj˜gtj−ℑj˜g¯tj+∑0≤tj≤tℑjgtj−ℑjg¯tj+1Γμ∑j=1q+1∫tj−1tjtj−ϱμ−1θ1,gϱ−θ2,g¯ϱdϱ+1Γμ−1∑j=1q1−tj∫tj−1tjtj−ϱμ−2θ1,gϱ−θ2,g¯ϱdϱ+1Γμ−1∫tj−1ρρ−ϱμ−2θ1,gϱ−θ2,g¯ϱdϱ+1Γμ−1∑j=1q∫tj−1tjtj−ϱμ−2θ1,gϱ−θ2,g¯ϱdϱ+∑j=1q1−tjℑj˜gtj−ℑj˜g¯tj+∑j=1qℑj˜gtj−ℑj˜g¯tj+∑j=1qℑjgtj−ℑjg¯tj.
With the use ofH2, one has(77)θ1,gϱ−θ2,g¯ϱ≤γ1−δg−g¯.
Therefore, using (77) in (76), we obtain(78)ℑ1g−ℑ2g¯=γg−g¯Γμ1−δ∫tjtt−ϱμ−1dϱ+2γg−g¯Γμ1−δ∑j=1q+1∫tj−1tjtj−ϱμ−1dϱ+3γg−g¯Γμ−11−δ∑j=1q∫tj−1tjtj−ϱμ−2dϱ+γg−g¯Γμ−11−δ∫tj−1ρρ−ϱμ−2dϱ+2qhg−g¯+3qh∗g−g¯≤Λg−g¯.
Now,(79)ℓ−Mℑ1g,ℑ2g¯,c=ℓ−ℓ−σℑ1g,ℑ2g¯1+abℓ=σℑ1g,ℑ2g¯1+abℓ∗ℓ≺¯Λσg,g¯1+abℓ∗ℓ=Λσg,g¯1+abℓ≺¯Λℓ−ℓ−σg,g¯1+abℓ≺¯Λℓ−Mg,g¯,c=Λℓ−maxMg,g¯,c,Mg,ℑ1g,cMg¯,ℑ2g¯,c1+Mg,g¯,c,Mg¯,ℑ1g,cMg,ℑ2g¯,c1+Mg,g¯,c.
Thus, Theorem1 applies to ℑ1,ℑ2, which have a common fixed point g∗∈Θ, that is, g∗ is a common solution of (72).
As an immediate consequence of Theorem3, in the case ℑ1=ℑ2, we find that integral equation (67) has a solution in Θ, and hence the nonlinear implicit boundary value problems of impulsive fractional differential equations (1) have a solution.


## 4. Conclusion

By successful applications of our derived results, we have studied the existence and uniqueness of common solution to the proposed class of implicit impulsive differential equations. Further, some useful results were also obtained that ensure the generalization of some essential results from metric spaces to complex valued fuzzy metric spaces. We finally hope to study classes of fuzzy fractional differential problems in future works.



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