# Intuitionistic Fuzzy Fixed Point Theorems in Complex-Valuedb-Metric Spaces with Applications to Fractional Differential Equations

Authors: Rehana Tabassum; Mohammed Shehu Shagari; Akbar Azam; OM Kalthum S. K. Mohamed; Awad A. Bakery
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Among various efforts in advancing fuzzy mathematics, a lot of attentions have been paid to examine novel intuitionistic fuzzy analogues of the classical fixed point results. Along this direction, the idea of intuitionistic fuzzy mapping (IFM) is used in this paper to establish some fixed point (FP) results in complex-valuedb-metric spaces. Moreover, from application perspective, one of our results is rendered to provide an existence condition for a solution of Caputo-type fractional differential equations. A few nontrivial illustrations are also furnished to authenticate and indicate the usability of the presented results.


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## Body

## 1. Introduction

Many FP results of contraction type mappings are specifically beneficial to find the existence and uniqueness of solution to different mathematical problems. In this regard, the Banach contraction principle [1] has become a very powerful source in various fields of applied mathematical analysis. Afterwards, several researchers improved and refined this result using different mappings in various generalized metric spaces (MS). For instance, Nadler refined the Banach FP result for multivalued mappings (MVM). Bakhtin [2] initiated the notion of b-MS as a refinement of classical MS, and Czerwik [3] proved the contraction mapping principle in b-MS. Therefore, a large amount of research has been brought up to obtain FPs of several mappings in b-MS.It is a familiar fact that fixed point results concerning rational contractions cannot be improved or even meaningless in cone metric spaces. To circumvent this problem, Azam et al. [4] have given a brilliant idea of complex-valued MS and improved the Banach contraction principle for a pair of mappings obeying the rational inequality in the bodywork of complex-valued MS. In sequel, Sintunavarat and Kumam [5] presented some common FP theorems using the control functions in contractive condition instead of constants. Subsequently, Ahmad et al. [6] have reported some new FP results for multivalued mappings obeying the greatest lower bound property in the context of complex-valued MS. Later, there has been much progress in the study of complex-valued MS by many authors [6–9]. In continuation to this, Rao et al. [10] brought in a new view of complex-valued b-MS and proved some common FP results in complex-valued b-MS. Meanwhile, Mukheimer [11] refined the results of Azam et al. [4] and Bhatt et al. [9].It is well-known that most of the problems in existing nature have several uncertainties, and to handle these uncertainties, there are various theories including fuzzy set [12], the soft set [13], the fuzzy soft set [14], and intuitionistic fuzzy set (IF-set) [15]. However, many physical problems with vague information can be tackled more precisely by means of IF-set approach. Keeping this in view, some FP results for MVM are refined to IFMs given by Shen et al. [16]. However, research on IFMs to establish the fixed point results is relatively recent, and few work has been done for IFMs on various spaces (e.g., see [17–22]).Stirred by the above-mentioned investigations, we present some FP results for IFMs in the bodywork ofT,N,α~−level set of an IF-set [23] in complete complex-valued b-MS. Further, some nontrivial examples and an application are given for the reliability of our main results.


## 2. Preliminaries

We launch here some basic definitions and results which will be useful in what comes hereafter. Letℂ be the set of complex numbers and I1,I2∈ℂ. We depict a partial order ≺ and ⪯ on ℂ as follows:
(i)
I1≺I2 if and only if ℝeI1<ℝeI2 and ImI1<ImI2(ii)
I1⪯I2 if and only if ℝeI1≤ℝeI2 and ImI1≤ImI2Definition 1 (see [10]).
LetΓ be a nonempty set and Y≥1 be a real number. A function dc:Γ×Γ⟶ℂ is termed a complex-valued b-MS, if for all I,ℓ,℘∈Γ, the following conditions hold:
(i)
0⪯dcI,ℓ and dcI,ℓ=0 if and only if I=ℓ(ii)
dcI,ℓ=dcℓ,I(iii)
dcI,ℓ⪯YdcI,℘+dc℘,ℓ
Then,dc is termed a complex-valued b-metric on Γ and the pair Γ,dc is termed a complex-valued b-MS.Remark 2.
LetΓ,dc be a complex-valued b-MS. If Y=1, then Γ,dc is a complex-valued MS. If Y=1 and ℂ=ℝ, then Γ,dc is a MS.Example 1.
LetΓ=0,1 and a mapping dc:Γ×Γ⟶ℂ is given by
(1)dcI,ℓ=I−ℓ2+iI−ℓ2,for all I,ℓ∈Γ. Then, Γ,dc is a complex-valued b-MS with Y=2.Definition 3 (see [10]).
LetΓ,dc be a complex-valued b-MS. A point ȷ∈Γ is an interior point of a set M⊆Γ, whenever we can find 0≺r∈ℂ:
(2)BI,r=ℓ∈Γ:dcI,ℓ≺r⊆M.A pointȷ∈Γ is said to be a limit point of a set M⊆Γ whenever for every 0≺r∈ℂ,(3)BI,r∩M\I≠ϕ.M⊆Γ is termed an open set if each element of M is an interior point of M.Definition 4 (see [10]).
LetIp be a sequence in Γ,dc and I∈Γ. If for every c∈ℂ with 0≺c there is po∈ℕ: dcIp,ȷ≺c for all p>po, then Ip is said to be a convergent sequence which converges to I, and we denote this by limp⟶∞ȷp=I. If for every c∈ℂ with 0≺c there is po∈ℕ: dcIp,Ip+q≺c for all p,q>po, and p,q∈ℕ, hence Ip is said to be a Cauchy sequence in Γ,dc. A complex-valued b-MS Γ,dc is termed a complete space if every Cauchy sequence is convergent in Γ,dc.Lemma 5 (see [10]).
LetΓ,dc be a complex-valued b-MS and Ip be a sequence in Γ,dc. Then, Ip converges to I if and only if dcIp,I⟶0 as p⟶∞.Lemma 6 (see [10]).
LetΓ,dc be a complex-valued b-MS and Ip be a sequence in Γ,dc. Then, Ip is a Cauchy sequence if and only if dcIp,Ip+q⟶0 as p,q⟶∞.
LetℂBΓ be the collection of all nonempty closed and bounded subsets of Γ,dc. We denote
(4)sz1=z2∈ℂ:z1⪯z2,for z1∈ℂ and
(5)sI,ℕ=⋃n∈ℕsdcI,n=⋃n∈ℕz∈ℂ:dcI,n⪯z,for I∈Γ and ℕ∈ℂBΓ. For M,ℕ∈ℂBΓ, we have
(6)sM,ℕ=⋂m∈Msm,ℕ∩⋂n∈ℕsn,M.Definition 7.
LetΓ,dc be a complex-valued b-MS and S:Γ⟶ℂBΓ be a MVM. Define
(7)WIM=dcI,m:m∈M,for I∈Γ and M∈ℂBΓ. Moreover,
(8)WȷSℓ=dcI,s:s∈Sℓ,for I,ℓ∈Γ and Sℓ∈ℂBΓ.Definition 8.
LetΓ,dc be a complex-valued b-MS. A nonempty subset M of Γ is said to be bounded from below if we can find some z∈ℂ: z⪯m for all m∈M.Definition 9.
LetΓ,dc be a complex-valued b-MS. A MVM T:Γ⟶2C is said to be bounded from below if for each I∈Γ we can find zȷ∈ℂ:
(9)zI⪯℘forall℘∈Tȷ.Definition 10.
LetΓ,dc be a complex-valued b-MS. A MVM S:Γ⟶ℂBΓ is said to have a lower bound property on Γ,dc if for any I∈Γ, the MVM TI:Γ⟶2C given by
(10)TISℓ=WISℓis termed bounded from below. This means for I,ℓ∈Γ, there is an element IISℓ∈C: IȷSℓ⪯a for all a∈WISℓ, where IISℓ is a lower bound of S associated with J,ℓ.Definition 11.
LetΓ,dc be a complex-valued b-MS. A MVM S:Γ⟶ℂBΓ has the greatest lower bound (g.l.b) property on Γ,dc if the g.l.b of WISℓ exists in ℂ for all I,ℓ∈Γ. We denote dcI,Sℓ by the g.l.b of WISℓ, i.e.,
(11)dcI,Sℓ=infdcI,s:s∈Sℓ.Definition 12 (see [15]).
LetW~ be a universal set. An IF-set F~ in W~ is an object of the form F~=℘,μF~℘,υF~℘:℘∈W~, where μF~℘ and υF~℘ denote the membership and nonmembership values of ℘ in F~ obeying 0≤μF~℘+υF~℘≤1 for every ℘∈W~.Definition 13 (see [15]).
LetF~ be an IF-set. Then, the α~−level set of F~ is a crisp set depicted by F~α~ and is given by
(12)F~α~=℘∈W~:μF~℘≥α~andυF~℘≤1−α~ifα~∈0,1.Definition 14 (see [24]).
A mappingT:0,12⟶0,1 is termed a triangular norm (t-norm), if the following conditions are obeyed:
(i)
TI,Tℓ,℘=TTI,ℓ,℘ for all I,ℓ,℘∈W~(ii)
TI,ℓ=Tℓ,J for all J,ℓ∈W~(iii)
IfI,ℓ,℘∈0,1 and I≤ℓ, then TI,℘≤Tℓ,℘(iv)
TJ,1=J for all ȷ∈W~
Minimumt-norm depicted by TM is given by TMI,ℓ=minI,ℓ for all I,ℓ∈0,1.Definition 15 (see [24]).
Fuzzy negation is a decreasing mapN:0,1⟶0,1 such that N0=1,N1=0. If N is continuous and strictly nonincreasing, then it is termed strict. Fuzzy negations with NN℘=℘, for all ℘∈0,1, are termed strong fuzzy negations. The example of fuzzy negation is a standard negation given by NS℘=1−℘, for all ℘∈W~.Definition 16 (see [23]).
LetF~ be an IF-set of W~ and T and N be a triangular norm and a fuzzy negation, respectively. Then, T,N,α~−level set of F~ is a crisp set depicted by F~T,N,α~ and is given by
(13)F~T,N,α~=℘∈W~:TμF~z,NυF~z≥α~ifα~∈0,1.Remark 17.
If we takeT=TM and N=NS, then T,N,α~−level set is reduced into original idea of a cut set by Atanassov [15].Definition 18 (see [16]).
LetW~ be an arbitrary set and Γ be a MS. A mapping F is termed an IFM if F is a mapping from W~ into IFSΓ (class of all intuitionistic fuzzy subsets of Γ).


## 3. Main Results

In this section, first we present few definitions which will be useful in the proof of our main ideas and then establish illustrations to validate their hypotheses.Definition 19.
A pointȷ∗∈Γ is said to be an intuitionistic fuzzy FP of an IFM F:Γ⟶IFSΓ if we can find α~∈0,1: ȷ∗∈Fȷ∗T,N,α~.Definition 20.
LetΓ,dc be a complex-valued b-MS and F:Γ⟶IFSΓ be an intuitionistic fuzzy map. Suppose that for each ℓ∈Γ, we can find α~Fℓ∈0,1: FℓT,N,α~Fℓ∈ℂBΓ. We can depict
(14)UȷFℓT,N,α~Fℓ=dcI,℘:℘∈FIT,N,α~Fℓ,for ȷ,ℓ∈Γ.Definition 21.
LetΓ,dc be a complex-valued b-metric space and F:Γ⟶IFSΓ be an intuitionistic fuzzy map. Suppose that for each ℓ∈Γ, we can find α~Fℓ∈0,1: FℓT,N,α~Fℓ∈ℂBΓ. An IFM F is said to have the greatest lower bound (g.l.b) property on Γ,dc, if the g.l.bofUȷFℓT,N,α~Fℓ exists in ℂ for all I,ℓ∈Γ. We denote the g.l.b of UIFℓT,N,α~Fℓ by dcI,FℓT,N,α~Fℓ and is given by
(15)dcI,FℓT,N,α~Fℓ=infdcI,ℓ1:b∈FℓT,N,α~Fℓ.Theorem 22.
LetΓ,dc be a complete complex-valued b-MS and F,G:Γ⟶IFSΓ be a pair of IFMs obeying the g.l.b property. Assume that for each I∈Γ, we can find α~FI,α~GI∈0,1: FIT,N,α~FI,GIT,N,α~GI∈ℂBΓ. If for all I,ℓ∈Γ,
(16)B∗I,ℓ,F,G,α~∈sFIT,N,α~FI,GℓT,N,α~Gℓ,where
(17)B∗I,ℓ,F,G,α~=βdcI,ℓ+μdcI,FIT,N,α~FIdcℓ,GℓT,N,α~Gℓ1+dcI,ℓ+λdcℓ,FIT,N,α~FIdcI,GℓT,N,α~Gℓ1+dcI,ℓ,and β,μ,λ are nonnegative real numbers with Yβ+μ+λ<1, where Y≥1. Then, F and G have a common FP in Γ.Proof.
Suppose thatI0 is an arbitrary and fixed element of Γ; then by assumption FI0T,N,α~FI0∈ℂBΓ, so we can take ȷ1∈FI0T,N,α~FI0, and therefore from (16),
(18)B∗I0,I1,F,G,α~∈sFI0T,N,α~FI0,GI1T,N,α~GI1.
It follows that(19)B∗I0,I1,F,G,α~∈⋂a′∈FI0T,N,α~FI0sa′,GI1T,N,α~GI1,B∗I0,I1,F,G,α~∈sa′,GI1T,N,α~GI1∀a′∈FI0T,N,α~FI0.
Asȷ1∈FI0T,N,α~FI0, then we obtain
(20)B∗I0,I1,F,G,α~∈sI1,GI1T,N,α~GI1.
Therefore,(21)B∗I0,I1,F,G,α~∈⋃b′∈GI1T,N,α~GI1sdcI1,b′.
Thus, we can find someI2∈GI1T,N,α~GI2:
(22)B∗I0,I1,F,G,α~∈sdcI1,I2.
It yields(23)dcI1,I2⪯B∗I0,I1,F,G,α~.
Using the g.l.b property ofF and G, we get
(24)dcI1,I2⪯βdcI0,I1+μdcI0,I1dcI1,I2+λdcI1,I1dcI0,I21+dcI0,I1⪯βdcI0,I1+μdcI0,I1dcI1,I2+λdcI1,I1dcI0,I21+dcI0,I1.
This implies(25)dcI1,I2≤βdcI0,I1+μdcI0,I1dcI1,I21+dcI0,I1=βdcI0,I1+μdcI1,I2≤β1−μdcI0,I1.
Inductively, we can develop a sequenceȷp in Γ: for p=0,1,2⋯,(26)dcIp,Ip+1≤κpdcI0,I1,where κ=β/1−μ<1.
Now forq∈ℕ and using the triangular inequality of Γ,dc,(27)dcIp,Ip+q⪯YdcIp,Ip+1+dcIp+1,Ip+q⪯YdcIp,Ip+1+Y2dcIp+1,Ip+2+Y2dcIp+2,Ip+q⪯YdcIp,Ip+1+Y2dcIp+1,Ip+2+Y3dcIp+2,Ip+3+⋯+YqdcIp+q−1,Ip+q⪯YκpdcIo,I1+Y2κp+1dcIo,I1+Y3κp+2dcIo,I1+⋯+Yqκp+q−1dcIo,I1⪯YκpdcIo,I11+Yκ+Yκ2+⋯+Yκq−1.
Therefore,(28)dcIp,Ip+q≤YκpdcIo,I11+Yκ+Yκ2+⋯+Yκq−1≤YκpdcIo,I11+Yκ+Yκ2+⋯+Yκq−1≤Yκp1−YκdcIo,I1,since Yβ+μ+λ<1⇒Yβ/1−μ<1 and dcIp,Ip+q⟶0 as p,q⟶∞.
By Lemma6, ξp is a Cauchy sequence in Γ, which is complete; so we can find some ω∈Γ,: limp⟶∞Ip=ω.
However, from (16), we have
(29)B∗I2p,ω,F,G,α~∈sFI2pT,N,α~FI2p,GωT,N,α~Gω.
This implies that(30)B∗I2p,ω,F,G,α~∈⋂a′∈FI2pT,N,α~FI2psa′,GωT,N,α~Gω.
SinceI2p+1∈FI2pT,N,α~FI2p,
(31)B∗I2p,ω,F,G,α~∈sI2p+1,GωT,N,α~Gω.
Therefore,(32)B∗I2p,ω,F,G,α~∈⋃b′∈GωT,N,α~GωsdcI2p+1,b′.
This implies that we can find someωp∈GωT,N,α~Gω:
(33)B∗I2p,ω,F,G,α~∈sdcI2p+1,ωp.
It yields(34)dcI2p+1,ωp⪯B∗I2p,ω,F,G,α~.
Next,(35)dcω,ωp⪯Ydcω,I2p+1+dcI2p+1,ωp⪯Ydcω,I2p+1+YB∗I2p,ω,F,G,α~.
However,(36)dcω,ωp≤Ydcω,I2p+1+YβdcI2p,ω+YμdcI2p,I2p+1dcω,ωp+Yλdcω,I2p+1dcI2p,ωp1+dcI2p,ω,since dcω,ωp⟶0 as p⟶∞.
By Lemma5, we have ωp⟶ω. Since GωT,N,α~Gω is closed, so ω∈GωT,N,α~Gω. Similarly, it follows that ω∈FωT,N,α~Fω. Thus, F and G have a common FP in Γ.
By settingλ=0 in Theorem 22, we have the following result.Corollary 23.
LetΓ,dc be a complete complex-valued b-MS and F,G:Γ⟶IFSΓ be a pair of IFMs obeying the g.l.b property. Assume that for each ȷ∈Γ, we can find α~FI,α~GI∈0,1: FIT,N,α~FI,GIT,N,α~GI∈ℂBΓ. If for all I,ℓ∈Γ,(37)BoI,ℓ,F,α~∈sFIT,N,α~FI,GℓT,N,α~Gℓ,where
(38)BoI,ℓ,F,G,α~=βdcI,ℓ+μdcI,FIT,N,α~FIdcℓ,GℓT,N,α~Gℓ1+dcI,ℓ,and β,μ are nonnegative real numbers with Yβ+μ<1, where Y≥1. Then, F and G have a common FP in Γ.By lettingF=G in Theorem 22, we have the following corollary.Corollary 24.
LetΓ,dc be a complete complex-valued b-MS and F:Γ⟶IFSΓ be an IFM obeying the g.l.b property. Assume that for each I∈Γ, we can find α~FI∈0,1: FIT,N,α~FI∈ℂBΓ. If for all I,ℓ∈Γ,(39)B∗I,ℓ,F,α~∈sFIT,N,α~Fȷ,FℓT,N,α~Fℓ,where
(40)B∗I,ℓ,F,α~=βdcI,ℓ+μdcI,FIT,N,α~FIdcℓ,FℓT,N,α~Fℓ1+dcI,ℓ+λdcℓ,FIT,N,α~FIdcI,FℓT,N,α~Fℓ1+dcI,ℓ,and β,μ,λ are nonnegative real numbers with Yβ+μ+λ<1, where Y≥1. Then, F has a FP in Γ.Corollary 25.
LetΓ,dc be a complete complex-valued b-MS and F,G:Γ⟶ℂBΓ be a pair of MVM obeying the g.l.b property
(41)βdcI,ℓ+μdcI,FIdcℓ,Gℓ1+dcI,ℓ+λdcℓ,FȷdcI,Gℓ1+dcI,ℓ∈sFI,Gℓ,for all I,ℓ∈Γ, and β,μ,λ are nonnegative real numbers with Yβ+μ+λ<1, where Y≥1. Then, F and G have a common FP in Γ.Example 2.
LetΓ=0,1,dcI,ℓ=I−ℓ2eiψ, where ψ=tan−1ℓ/I, for I,ℓ∈Γ and ζ1,ζ2,σ1,σ2∈0,1.
Then,Γ,dc is a complete complex-valued b-MS with Y=2. Assume that ζ1,ζ2,σ1,σ2∈0,1 and a pair of IFMs F=μF,υF,G=μG,υG:Γ⟶IFSΓ are given by
(42)μFIt=ζ1,if0≤t≤I15,ζ15,ifI15<t≤I13,ζ18,ifI11<t<I,0,ifI≤t<∞,υFIt=0,if0≤t≤I15,ζ26,ifI15<t≤I12,ζ23,ifI10<t<I,ζ2,ifI≤t<∞,μGIt=σ1,if0≤t≤I20,σ14,ifI20<t≤I18,σ19,ifI16<t<I,0,ifI≤t<∞,υGIt=0,if0≤t≤I20,σ27,ifI20<t≤I17,σ22,ifI14<t<I,σ2,ifI≤t<∞.Ifα~FI=ζ1 and α~GI=σ1, then we have
(43)FIT,N,ζ1=t∈Γ:TμFIt,NυFIt=ζ1=0,I15,GIT,N,σ1=t∈Γ:TμGIt,NυGIt=σ1=0,I20.Thus, the contractive condition of Theorem22 becomes trivial when ȷ=ℓ=0.Assume that without loss of generality,I,ℓ≠0, and I<ℓ, and then, we have
(44)dcI,ℓ=I−ℓ2eιψ,dcI,FIT,N,α~Fȷ=I−I152eiψ,dcℓ,GℓT,N,α~Gℓ=ℓ−ℓ202eiψ,dcℓ,FIT,N,α~FI=ℓ−I152eiψ,dcI,GℓT,N,α~Gℓ=I−ℓ202eiψ,sdcFIT,N,α~FI,GℓT,N,α~Gℓ=sI15−ℓ202eiψ.Thus, clearly forα~=1/225 and any value of μ and λ, we have
(45)I15−ℓ202≤1225I−ℓ2+μI−I/152ℓ−ℓ/202+λℓ−I/152I−ℓ/2021+dcI,ℓ.Hence, all the conditions of Theorem22 are obeyed to obtain a common FP of F and G.Theorem 26.
LetΓ,dc be a complete complex-valued b-MS and F,G:Γ⟶IFSΓ be a pair of IFMs obeying the g.l.b property. Assume that for each I∈Γ, we can find α~FI,α~GI∈0,1: FIT,N,α~FI,GIT,N,α~GI∈ℂBΓ. If for all I,ℓ∈B¯ȷ0,r,
(46)B∗I,ℓ,F,G,α~∈sFIT,N,α~Fȷ,GℓT,N,α~Gℓ,(47)1−ϵr∈sI0,Fȷ0T,N,α~FI0,where
(48)B∗I,ℓ,F,G,α~=βdcI,ℓ+μdcI,FIT,N,α~FIdcℓ,GℓT,N,α~Gℓ1+dcI,ℓ+λdcℓ,FIT,N,α~FIdcI,GℓT,N,α~Gℓ1+dcI,ℓ,and β,μ,λ are nonnegative real numbers with Yβ+μ+λ<1 and ϵ=τβ/1−μ<1 for any Y≥1. Thus, we can find ω in B¯I0,r:
(49)ω∈FωT,N,α~Fω∩GωT,N,α~Gω.Proof.
Letȷ0 be an arbitrary but fixed element in Γ. Then by hypothesis, Fȷ0T,N,α~Fȷ0∈ℂBΓ. So we can find I1∈FI0T,N,α~Fȷ0: 1−ϵr∈sdcI0,I1. From (47), it is easy to see that
(50)dcI0,I1⪯1−ϵr.
This implies(51)dcI0,I1≤1−ϵr.
Therefore,ȷ1∈B¯ȷ0,r. From (46), we have
(52)B∗I0,I1,F,G,α~∈sFI0T,N,α~FI0,GI1T,N,α~GI1.
From here, by following the remaining steps in the proof of Theorem26, we obtain
(53)dcI1,I2≤ϵdcI0,I1.
From (51), we have
(54)dcI1,I2≤ϵdcI0,I1≤ϵ1−ϵr.Using the triangular inequality of Γ,dc,(55)dcI0,I2≤YdcI0,I1+dcI1,I2≤Y1−ϵr+ϵ1−ϵr≤Y1−ϵ2r.
It follows thatI2∈B¯I0,r. From (46), we have
(56)B∗I1,I2,G,F,α~∈sGI1T,N,α~GJ1,Fȷ2T,N,α~FI2.
By repetition of the above steps and using the fact thatΓ,dc is a complex-valued b-MS, we can generate a sequence Inn∈ℕ in B¯ȷ0,r:
(57)dcI2n,I2n+1≤ϵ2ndcI0,I1,dcI2n+1,I2n+2≤ϵ2n+1dcI0,I1,I2n+1∈FI2nT,N,α~FI2n,I2n+2∈GI2n+1T,N,α~GI2n+1.
Inductively, we can construct a sequenceȷnn∈ℕ in Γ:
(58)dcIn,In+1≤ϵndcI0,I1.
Now, form,n∈ℕ with n<m, using triangle inequality and the iterative scheme (58), we have
(59)dcIn,Im≤YdcIn,In+1+dcIn+1,In+2+⋯+dcIm−1,Im≤Yϵn+ϵn+1+⋯+ϵm−1dcI0,I1≤Yϵn1−ϵdcI0,I1.
Consequently,(60)dcIn,Im≤Yϵn1−ϵdcI0,I1⟶0⋯asn,m⟶∞.
Hence,ȷnn∈ℕ is a Cauchy sequence in B¯I0,r. Since B¯ȷ0,r is a closed subspace of a complete MS Γ, therefore, we can find ω∈B¯I0,r: In⟶ω as n⟶∞. Now, to show that ω∈FωT,N,α~Fω∩GωT,N,α~Gω, from (46), we have
(61)B∗I2n,ω,F,G,α~∈sFI2nT,N,α~FI2n,GωT,N,α~Gω.
This implies(62)B∗I2n,ω,F,G,α~∈sI2n+1,GωT,N,α~Gω.
By definition, we can find someωn∈GωT,N,α~Gω:
(63)B∗I2n,ω,F,G,α~∈sdcI2n+1,ωn.Using the g.l.b property of F and G, we get
(64)dcI2n+1,ωn⪯βdcI2n,ω+μdcI2n,I2n+1dcω,ωn+λdcω,I2n+1dcI2n,ωn1+dcI2n,ω.
From triangle inequality, it follows that(65)dcω,ωn⪯Ydcω,I2n+1+dcI2n+1,ωn⪯Ydcω,I2n+1+YβdcI2n,ωn+YμdcI2n,I2n+1dcω,ωn1+dcI2n,ω+Yλdcω,I2n+1dcI2n,ωn1+dcI2n,ω.
Therefore,(66)dcω,ωn≤Ydcω,I2n+1+YβdcI2n,ω+YμdcI2n,I2n+1dcω,ωn1+dcI2n,ω+Yλdcω,I2n+1dcI2n,ωn1+dcI2n,ω.
Applyingn⟶∞ in (66), we have dcω,ωn⟶0. This implies ωn⟶ω as n⟶∞. Since GωT,N,α~Gω is closed, therefore ω∈GωT,N,α~Gω. Similarly, one can show that ω∈FωT,N,α~Fω.
Hence,(67)ω∈FωT,N,α~Fω∩GωT,N,α~Gω.
By settingF=G in Theorem 26, we obtain the following result as a corollary.Corollary 27.
LetΓ,dc be a complete complex-valued b-MS and F:Γ⟶IFSΓ be an IFM obeying the g.l.b property. Assume that for each j∈Γ, we can find α~Fȷ∈0,1: FȷT,N,α~Fȷ∈ℂBΓ. If for all I,ℓ∈B¯ȷ0,r,(68)B∗I,ℓ,F,α~∈sFIT,N,α~FI,FℓT,N,α~Fℓ,1−ϵr∈sI0,FI0T,N,α~FI0,where
(69)B∗I,ℓ,F,G,α~=βdcI,ℓ+μdcI,FIT,N,α~FIdcℓ,GℓT,N,α~Gℓ1+dcI,ℓ+λdcℓ,FIT,N,α~FIdcI,GℓT,N,α~Gℓ1+dcI,ℓ,and β,μ,λ are nonnegative real numbers with Yβ+μ+λ<1 and ϵ=τβ/1−μ<1 for any Y≥1. Thus, we can find ω in B¯j0,r:
(70)ω∈FωT,N,α~Fω.Corollary 28.
LetΓ,dc be a complete complex-valued b-MS and F,G:Γ⟶CBΓ be a pair of MVM obeying the g.l.b property
(71)βdcI,ℓ+μdcI,FIdcℓ,Gℓ1+dcI,ℓ+λdcℓ,FIdcI,Gℓ1+dcI,ℓ∈sFI,Gℓ,1−ϵr∈sI0,FI0,for all I,ℓ∈B¯I0,r, and β,μ,λ are nonnegative real numbers with Yβ+μ+λ<1, and ϵ=τβ/1−μ<1 for any Y≥1. Thus, we can find ω in B¯I0,r:
(72)ω∈Fω∩Gω.Theorem 29.
LetΓ,dc be a complete complex-valued b-MS and F:Γ⟶IFSΓ be an IFM obeying the g.l.b property. Assume that for each ȷ∈Γ, we can find α~FI∈0,1: FIT,N,α~FI∈ℂBΓ. If for all I,ℓ∈Γ,(73)λ1dcI0,I1+λ2KoI0,I1,T,N,α~+λ3CoI0,I1,T,N,α~+λ4DoI0,I1,T,N,α~+λ5EoI0,I1,T,N,α~∈sFI0T,N,α~FI0,FI1T,N,α~FI1,where
(74)KoI,ℓ,T,N,α~=dcI,FIT,N,α~FIdcℓ,FℓT,N,α~Fℓ1+dcI,ℓ,CoI,ℓ,T,N,α~=dcℓ,FIT,N,α~FIdcI,FℓT,N,α~Fℓ1+dcI,ℓ,DoI,ℓ,T,N,α~=dcI,FIT,N,α~FIdcI,FℓT,N,α~Fℓ1+dcI,ℓ,EoI,ℓ,T,N,α~=dcℓ,FIT,N,α~FIdcℓ,FℓT,N,α~Fℓ1+dcI,ℓ,and λ1,λ2,λ3,λ4,λ5 are nonnegative real numbers with λ1+λ2+λ3+2Yλ4+λ5<1. Then, F has a FP in Γ.Proof.
Suppose thatI0 is an arbitrary and fixed element of Γ, then by assumption, FI0T,N,α~FI0∈ℂBΓ, so we can take ȷ1∈FI0T,N,α~FI0; therefore from Theorem 26,
(75)λ1dcI0,I1+λ2KoI0,I1,T,N,α~+λ3CoI0,I1,T,N,α~+λ4DoI0,I1,T,N,α~+λ5EoI0,I1,T,N,α~∈sFI0T,N,α~FI0,FI1T,N,α~FI1.
This implies(76)λ1dcI0,I1+λ2KoI0,I1,T,N,α~+λ3CoI0,I1,T,N,α~+λ4DoI0,I1,T,N,α~+λ5EoI0,I1,T,N,α~∈⋂a∈FI0T,N,α~FI0sa,FI1T,N,α~FI1.
Thus,(77)λ1dcJ0,J1+λ2KoJ0,J1,T,N,α~+λ3CoJ0,J1,T,N,α~+λ4DoJ0,J1,T,N,α~+λ5EoJ0,J1,T,N,α~∈sa,FJ1T,N,α~FJ1,for all a∈FJ0T,N,α~FJ0.
SinceJ1∈FJ0T,N,α~FJ0, then we obtain
(78)λ1dcJ0,J1+λ2KoJ0,J1,T,N,α~+λ3CoJ0,J1,T,N,α~+λ4DoJ0,J1,T,N,α~+λ5EoJ0,J1,T,N,α~∈sJ1,FJ1T,N,α~FJ1.
It yields(79)λ1dcJ0,J1+λ2KoJ0,J1,T,N,α~+λ3CoJ0,J1,T,N,α~+λ4DoJ0,J1,T,N,α~+λ5EoJ0,J1,T,N,α~∈⋃b∈FJ1T,N,α~FJ1sdcJ1,b.
Therefore, we can find someI2∈Fȷ1T,N,α~FI1:
(80)λ1dcJ0,J1+λ2KoJ0,J1,T,N,α~+λ3CoJ0,J1,T,N,α~+λ4DoJ0,J1,T,N,α~+λ5EoJ0,J1,T,N,α~∈sdcJ1,J2.
This implies(81)dcJ1,J2°λ1dcJ0,J1+λ2KoJ0,J1,T,N,α~+λ3CoJ0,J1,T,N,α~+λ4DoJ0,J1,T,N,α~+λ5EoJ0,J1,T,N,α~.
Using the g.l.b property ofF, we have
(82)dcJ1,J2⪯λ1dcJ0,J1+λ2dcJ0,J1dcJ1,J21+dJ0,J1+λ3dcJ1,J1dJ0,J11+dcJ0,J1+λ4dcJ0,J1dcJ0,J11+dJ0,J1+λ5dcJ1,J1dJ1,J21+dJ0,J1.
However,(83)dcJ1,J2⪯λ1dcJ0,J1+λ2dcJ1,J2+λ4dcJ0,J2,1−λ2−Yλ4dcJ1,J2≤λ1+Yλ4dcJ0,J1.
This implies(84)dcJ1,J2≤ΩdcJ0,J1,where Ω=λ1+Yλ4/1−λ2−Yλ4<1.
Inductively, we develop a sequencejp in Γ:
(85)dcJ1,J2≤ΩpdcJ0,J1.
Now forq∈ℕ and using the triangle inequality of Γ,dc, we have
(86)dcJp,Jp+q⪯YdcJp,Jp+1+dcJp+1,Jp+q⪯YdcJp,Jp+1+Y2dcJp+1,Jp+2+Y2dcJp+2,Jp+q⪯YdcJp,Jp+1+Y2dcJp+1,Jp+2+Y3dcJp+2,Jp+3+⋯+YqdcJp+q−1,Jp+q⪯YΩpdcJ0,J1+Y2Ωp+1dcJ0,J1+Y3Ωp+2dcJ0,J1+⋯+YqΩp+q−1dcJ0,J1⪯YΩpdcJ0,J11+YΩ+YΩ2+⋯+YΩq−1.
However,(87)dcJp,Jp+q≤YΩpdcJ0,J11+YΩ+YΩ2+⋯+YΩq−1≤YΩpdcJ0,J11+YΩ+YΩ2+⋯+YΩq−1≤YΩp1−YΩdcJ0,J1.
Lettingp,q⟶∞, therefore, dcJp,Jp+q⟶0.
By Lemma6, jp is a Cauchy sequence in Γ, which is complete, so we can find some ω∈Γ,: limp⟶∞jp=ω. Next, we show that ω∈FωT,N,α~Fω. From Theorem 26, we have
(88)λ1dcJ2p,ω+λ2KoJ2p,ω,T,N,α~+λ3CoJ2p,ω,T,N,α~+λ4DoJ2p,ω,T,N,α~+λ5EoJ2p,ω,T,N,α~∈sFJ2pT,N,α~FJ2p,FωT,N,α~Fω.
This implies(89)λ1dcJ2p,ω+λ2KoJ2p,ω,T,N,α~+λ3CoJ2p,ω,T,N,α~+λ4DoJ2p,ω,T,N,α~+λ5EoJ2p,ω,T,N,α~∈⋂a∈FJ2pT,N,α~FJ2psa,FωT,N,α~Fω.
SinceJ2p+1∈FJ2pT,N,α~FJ2p, then we obtain
(90)λ1dcJ2p,ω+λ2KoJ2p,ω,T,N,α~+λ3CoJ2p,ω,T,N,α~+λ4DoJ2p,ω,T,N,α~+λ5EoJ2p,ω,T,N,α~∈sJ2p+1,FωT,N,α~Fω.
It yields(91)λ1dcJ2p,ω+λ2KoJ2p,ω,T,N,α~+λ3CoJ2p,ω,T,N,α~+λ4DoJ2p,ω,T,N,α~+λ5EoJ2p,ω,T,N,α~∈⋃b∈FωT,N,α~FωsdcJ2p+1,b.
So, we can findΩp∈FΩT,N,α~FΩ:
(92)λ1dcJ2p,ω+λ2KoJ2p,ω,T,N,α~+λ3CoJ2p,ω,T,N,α~+λ4DoJ2p,ω,T,N,α~+λ5EoJ2p,ω,T,N,α~∈sdcJ2p+1,ωp.
Therefore,(93)dcJ2p+1,ωp⪯λ1dcJ2p,ω+λ2KoJ2p,ω,T,N,α~+λ3CoJ2p,ω,T,N,α~+λ4DoJ2p,ω,T,N,α~+λ5EoJ2p,ω,T,N,α~.
Using the g.l.b property, we have(94)dcJ2p+1,ωp⪯λ1dcJ2p,ω+λ2dcJ2p,J2p+1dcω,ωp1+dJ2p,ω+λ3dcω,J2p+1dJ2p,ωp1+dcJ2p,ω+λ4dcJ2p,J2p+1dcJ2p,ωp1+dcJ2p,ω+λ5dcω,J2p+1dcω,ωp1+dcJ2p,ω.
By triangle inequality,(95)dcω,ωp⪯Ydcω,J2p+1+dcJ2p+1,ωp.
Thus,(96)dcI2p+1,ωp⪯Ydcω,J2p+1+Yλ1dcJ2p,ω+Yλ2dcJ2p,J2p+1dcω,ωp1+dJ2p,ω+Yλ3dcω,J2p+1dcJ2p,ωp1+dcJ2p,ω+Yλ4dcJ2p,J2p+1dcJ2p,ωp1+dcJ2p,ω+Yλ5dcω,J2p+1dcω,ωp1+dcJ2p,ω.
This implies(97)dcI2p+1,ωp≤Ydcω,J2p+1+Yλ1dcI2p,ω+Yλ2dcI2p,I2p+1dcω,ωp1+dI2p,ω+Yλ3dcω,I2p+1dcI2p,ωp1+dcJ2p,ω+Yλ4dcI2p,I2p+1dcI2p,ωp1+dcI2p,ω+Yλ5dcω,I2p+1dcω,ωp1+dcI2p,ω.
Asp⟶∞, we get
(98)dcI2p+1,ωp⟶0.
By Lemma5, we have ωp⟶ω as p⟶∞. Since FωT,N,α~Fω is closed, therefore, F has a FP.Corollary 30.
LetΓ,dc be a complete complex-valued b-MS and F:Γ⟶ℂBΓ be a MVM obeying the g.l.b property such that
(99)λ1dcJ,ℓ+λ2dcI,FJdcℓ,Fℓ1+dcJ,ℓ+λ3dcℓ,FIdcI,Fℓ1+dcJ,ℓ+λ4dcI,FJdcI,Fℓ1+dcI,ℓ+λ5dcℓ,FIdcℓ,Fℓ1+dcI,ℓ∈sFI,Fℓ,for all j,ℓ∈Γ, and λ1,λ2,λ3,λ4,λ5 are nonnegative real numbers with λ1+λ2+λ3+2Yλ4+λ5<1. Then, F has a FP in Γ.


## 4. Applications to Fractional Mixed Volterra-Fredholm Integrodifferential Equations with Integral Boundary Conditions

In this section, first some definitions and results are given from the existing literature and then Corollary24 of Theorem 22 is applied to study the existence of a solution of fractional mixed Volterra-Fredholm integrodifferential equation with integral boundary conditions.Definition 31 [25, 26].
Letg be a function given on a,b; then, the Caputo fractional order derivative of g is given by
(100)aDtα~gt=1Γn−α~∫att−sn−α~−1gnsds,where n=α~+1 and α~ denotes the integer part of α~.Definition 32 [25].
Letg be a function given almost everywhere on a,b, for α~>0; we depict
(101)aDb−α~g=1Γα~∫abb−tα~−1gtdt,provided that the Lebesgue integral exists.Lemma 33 [27].
Letg:0,δ×X⟶X be a continuous function, Dα~ be the Caputo fractional derivative, and 1<α~≤2; then, the solution of fractional mixed Volterra-Fredholm integrodifferential equation with boundary conditions,
(102)Dα~ℓt=gt,ℓt,∫0tkt,s,ℓsds,∫0δr1t,s,ℓsds,(103)ℓ0−ℓ′0=∫0δfℓsds,ℓδ−ℓ′δ=∫0δrℓsds,is given by
(104)ℓt=1+tδ∫0δrℓsds+1−1+tδ∫0δfℓsds−1+tδ∫0δδ−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+1+tδ∫0δδ−sα~−2Γα~−1gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+∫0tt−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds.Theorem 34.
LetΓ be a complete complex-valued b-metric space endowed with the metric dc:Γ×Γ⟶ℂ given by dcI,ℓ=J−ℓ2eiθ and C=0,δ,Γ be the space of all continuous functions from 0,δ into Γ. Suppose that the function g:0,δ×Γ×Γ×Γ⟶Γ,k,r,r1:0,δ×0,δ×Γ⟶Γ and f,r:Γ⟶Γ be chosen: the following conditions are obeyed:
(C1) For eachℓ∈Γ and t∈0,δ, we have Λℓ∈Γ, where
(105)Λℓ=1+tδ∫0δrℓsds+1−1+tδ∫0δfℓsds−1+tδ∫0δδ−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+1+tδ∫0δδ−sα~−2Γα~−1gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+∫0tt−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds
(C2) We can find a constantβ∈0,1: for all J,ℓ∈Γ,(106)I−ℓ2⪯βI−ℓ2+ΛJ−Λℓ
(C3) We can find some constantsa1,a2: for all J,ℓ∈Γ,(107)rJ−rℓ≤a1J−ℓandfJ−fℓ≤a2J−ℓ
(C4) We can find continuous functionsΘ1,Θ2:0,δ⟶0,∞:
(108)∫0tkt,s,I−kt,s,ℓds≤Θ1I−ℓ,∫0δr1t,s,I−r1t,s,ℓds≤Θ2I−ℓ
(C5) We can find a continuous functionK:0,δ⟶0,∞ and a continuous nondecreasing function L:0,∞⟶0,∞:
(109)gt,J1,ℓ1,z1−gt,J2,ℓ2,z2≤KtLJ1−J2+ℓ1−ℓ2+z1−z2,where the function L satisfies LξtJ≤ξtLJ, provided ξ:0,δ⟶0,∞ is continuous
(C6) We can find a multivalued functionM:Γ⟶ℂBΓ and some constants η,β,Y,μ with η∈0,1 and Y≥1:
(110)ηI−ℓ⪯μIJ,ℓt+λEJ,ℓt,where
(111)JJ,ℓ=j−Mu2ℓ−Mv2eiθ1+I−ℓ2eiθ,EI,ℓ=ℓ−Mu2I−Mv2eiθ1+I−ℓ2eiθ,θ=tan−1ℓJ,Yβ+μ+λ<1
Then, the Caputo integrodifferential equations (102) and (103) have a solution.Proof.
By Lemma33, the equivalent integral reformulation of problem (100)–(101) is given by
(112)ℓt=1+tδ∫0δrℓsds+1−1+tδ∫0δfℓsds−1+tδ∫0δδ−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+1+tδ∫0δδ−sα~−2Γα~−1gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+∫0tt−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds.By (C1), for each ℓ∈Γ, we have Λℓ∈Γ, where
(113)Λℓ=1+tδ∫0δrℓsds+1−1+tδ∫0δfℓsds−1+tδ∫0δδ−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+1+tδ∫0δδ−sα~−2Γα~−1gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+∫0tt−sα~−1Γα~gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds.
By (C2), for allJ,ℓ∈Γ and each t∈0,δ, we have
(114)I−ℓ2⪯βI−ℓ2+ΛJ−Λℓ⪯βJ−ℓ2+1+tδ∫0δrI−rℓds+1−1+tδ∫0δfJ−fℓds+1+tδ∫0δδ−sα~−1Γα~gs,Js,∫0sks,τ,Jτdcτ,∫0δr1s,τ,Iτdcτ−gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+1+tδ∫0δδ−sα~−2Γα~−1gs,Js,∫0sks,τ,Jτdcτ,∫0δr1s,τ,Jτdcτ−gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds+∫0tt−sα~−1Γα~gs,Js,∫0sks,τ,Jτdcτ,∫0δr1s,τ,Jτdcτ−gs,ℓs,∫0sks,τ,ℓτdcτ,∫0δr1s,τ,ℓτdcτds.
By (C2)-(C5), we have(115)I−ℓ2⪯βI−ℓ2+a11+tδ∫0δI−ℓds+a21−1+tδ∫0δI−ℓds+1+tδ∫0δδ−sα~−2Γα~−1KSLIs−ℓs+∫0sks,τ,Jτ−ks,τ,ℓτdcτ+∫0δr1s,τ,Jτ−r1s,τ,ℓτdcτ+1+tδ∫0δδ−sα~−1Γα~Ks×LJs−ℓs+∫0sks,τ,Iτ−ks,τ,ℓτdcτ+∫0δr1s,τ,Jτ−r1s,τ,ℓτdcτ+∫0tt−sα~−1Γα~Ks×LJs−ℓs+ks,τ,Iτ−ks,τ,ℓτdcτ+∫0δr1s,τ,Iτ−r1s,τ,ℓτdcτ⪯βI−ℓ2+a11+tδ∫0δI−ℓds+a21−1+tδ∫0δI−ℓds+1+tδ∫0δδ−sα~−2Γα~−1×KsLI−ℓ+Θ1sI−ℓ+Θ2I−ℓds+1+tδ∫0δδ−sα~−1Γα~Ks×LI−ℓ+Θ1sI−ℓ+Θ2sI−ℓ+∫0tt−sα~−1Γα~KsLI−ℓ+Θ1sI−ℓ+Θ2sI−ℓ⪯βI−ℓ2+a11+tδ∫0δI−ℓds+a21−1+tδ∫0δI−ℓds+1+tδ∫0δδ−sα~−2Γα~−1Ks1+Θ1s+Θ2sLI−ℓds+1+tδ∫0δδ−sα~−1Γα~×Ks1+Θ1s+Θ2sLI−ℓds+∫0tt−sα~−1Γα~Ks1+Θ1s+Θ2sLI−ℓds.
LetΔt=Kt1+Θ1t+Θ2t and Δ∗=supΔt:t∈0,δ. Then,
(116)I−ℓ2⪯βI−ℓ2+a11+tδ∫0δI−ℓds+a21−1+tδ∫0δI−ℓds+LΔ∗1+tδ∫0δδ−sα~−2Γα~−1I−ℓds+LΔ∗1+tδ∫0δδ−sα~−1Γα~I−ℓds+LΔ∫0tt−sα~−1Γα~I−ℓds⪯βI−ℓ2+a11+δ+a2δ−1+LΔ∗ýΓα~+1δ2−α~I−ℓ,where ý=2δ2+δ+α~1+δ.
Choosea1,a2: a11+δ+a2δ−1+LΔ∗ý/Γα~+1δ2−α~=η<1. Then by (C6), we have
(117)I−ℓ2⪯βI−ℓ2+ηI−ℓ⪯βI−ℓ2+μI−Mu2ℓ−Mv2eiθ1+I−ℓeiθ+λℓ−Mu2I−Mv2eiθ1+I−ℓ2eiθ⪯βI−ℓ2+μI−Mu2ℓ−Mv2eiθ+λℓ−Mu2I−Mv2eiθ1+I−ℓeiθ.
LetP,Q:Γ⟶0,1 be any two arbitrary mappings and M:Γ⟶ℂBΓ be the given multivalued mapping. Consider an intuitionistic fuzzy mapping G:Γ⟶IFSΓ as follows:
(118)μGIω=PI,ω∈Mu,0,otherwise,νGIω=0,ω∈Mu,QI,otherwise..
By lettingα~FI=Pj and νFI=0, we obtain
(119)FIT,N,α~FI=ω∈Γ:TμFIω,NνFIω=PI=Mu.
Therefore, the inequality (117) can be written as
(120)I−ℓ2⪯βI−ℓ2+μj−FIT,N,α~FI2ℓ−FℓT,N,α~Fℓ2eiθ1+I−ℓ2eiθ+λℓ−FIT,N,α~GI2I−FℓT,N,α~Gℓ2eiθ1+I−ℓ2eiθ.
Multiplying inequality (120) by eiθ, we get
(121)dcI,ℓ⪯βdcI,ℓ+μdcI,FIT,N,α~FIdcℓ,FℓT,N,α~Fℓ1+dcI,ℓ+λdcℓ,FIT,N,α~FIdcI,FℓT,N,α~Fℓ1+dcI,ℓ.
By definition, inequality (121) implies
(122)βdcI,ℓ+μdcI,FIT,N,α~FIdcℓ,FℓT,N,α~Fℓ1+dcI,ℓ+λdcℓ,FIT,N,α~FIdcI,FℓT,N,α~Fℓ1+dcI,ℓ=B∗I,ℓ,F,α~∈sFIT,N,α~FI,FℓT,N,α~Fℓ.
Hence, the conclusion of Theorem34 follows the application of Corollary 24.Example 3.
Consider the following fractional integrodifferential equation:(123)ℓ95t=112+112+ℓt+∫0te−112ℓsds+∫01ℓse−s12+ℓs2ds,t∈0,1,with integral boundary conditions
(124)ℓ0−ℓ′0=∫01112+ℓsds,ℓ1−ℓ′1=∫01112+e−ℓsds.Here,(125)rI−rℓ=112+e−It−112+e−ℓt=e−ℓt−e−It12+e−It12+e−ℓt≤1144I−ℓ,fI−fℓ=112+It−112+ℓt=ℓt−It12+It12+ℓt≤1144I−ℓ,∫0tkt,s,I−kt,s,ℓds=∫0te−112Is−e−112ℓsds≤112I−ℓ,∫01r1t,s,I−r1t,s,ℓds=∫01Ise−s12+ℓs2−ℓses12+ℓs2≤1144I−ℓ.However,(126)gt,I1,ℓ1,z1−gtI2,ℓ2,z2≤112+tI1−I2+ℓ1−ℓ2+z1−z2.Notice that by Lemma33, Condition (C2) of Theorem 34 can be verified by direct calculation.LetΓ=0,1 and dc:Γ×Γ⟶ℂ be given by
(127)dcI,ℓ=I−ℓ2eiθ,θ=tan−1ℓ/I.Then,Γ,dc is a complete complex-valued b-MS. Suppose that M:Γ⟶ℂBΓ is given by
(128)Mu=0,I7.Assume without loss of generality that forI,ℓ∈Γ, I≠ℓ, and I<ℓ. Thus, we have
(129)dcI,Mu=I−I72eiθ,dcℓ,Mv=ℓ−ℓ72eiθ,dcℓ,Mu=ℓ−I72eiθ,dcI,Mv=I−ℓ72eiθ.Clearly, for any value ofμ and λ and η=1/7, we have
(130)I7−ℓ7≤17I−ℓ+μI−I/72ℓ−ℓ/72eiθ1+I−ℓ2eiθ+λℓ−I/72eiθ1+I−ℓ2eiθ.Hence, all the conditions of Theorem34 are obeyed. We conclude that the fractional mixed Volterra-Fredholm integrodifferential equation has a solution in Γ.


## 5. Conclusion

A number of practical and theoretic problems in economics, engineering, management sciences, and medical science and a substantial number of other fields involve vagueness and the complexity of modeling data possessing nonstatistical uncertainties. Prototypal mathematical techniques are not usually successful because the imprecisions in these domains may be of various kinds. Several innovative models such as fuzzy set theory, rough set theory, intuitionistic fuzzy set theory, and other related mathematical tools have been established to manipulate data with incomplete information. On the other hand, fractional differential equations (FDEs) have instigated, in recent years, considerable interests due to their enormous applications in both physical sciences, chemical process in engineering, and social sciences. Different methods for solving FDEs and investigating the existence of their solutions were proposed by many authors; some of these techniques are series methods, the iteration method, the operational calculus, and so on. In this direction, one may see the textbooks [25, 26, 28] for both introductory and detail analysis.Motivated by the above developments, in this article, we have used the ideas of complex-valuedb-MS to establish sufficient conditions for the existence of common FPs of a pair of IFMs obeying some contractive conditions involving rational inequalities. In Theorem 22, we presented a Banach type results involving rational expression, while the Banach type locally contractive condition is discussed in Corollary 24. On the other hand, Theorem 26 is a slight improvement of Definition 19. Moreover, in continuation of the role of the classical Banach contraction theorem in the study of existence of solutions of nonlinear integrodifferential equations, we included the application section. In this regard, Corollary 24 is applied to provide some conditions for the existence of a solution of mixed Volterra-Fredholm integrodifferential equation with integral boundary conditions. In each case of the above-mentioned key results, examples are supplied to support or authenticate their usability. While the current paper is theoretical, at least a note justifying its study is in order. To start with, in several MS, e.g., the cone MS, FP results involving rational expressions cannot be refined, whereas, in complex-valued MS, this is feasible. Thus, by incorporating the notion of complex-valued IFMs into the existing concepts, our remarkable results will be beneficial and significant in the modeling and solution of optimization problems in mathematical analysis.



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