# On State-Dependent Delay Intuitionistic Fuzzy Partial Functional Differential Equations with Integral Boundary Conditions

Authors: Bouchra Ben Amma; Said Melliani; Lalla Saadia Chadli
Journal: Journal of Mathematics
 (2021)

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## Abstract

This paper addresses the issue of the existence and uniqueness of intuitionistic fuzzy solutions for some classes of partial functional differential equations with state-dependent delay in a new weighted complete metric space. Theorems on the existence and uniqueness of intuitionistic fuzzy solutions for these problems with integral boundary conditions are established under some sufficient assumptions. Some numerical examples of applications of the main result of this work are presented.


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## Body

## 1. Introduction

In 1965, Zadeh [1] proposed the concept of fuzzy sets with the purpose to model ambiguity, uncertainty, and vagueness in complicated systems. It can be considered as an extension of the usual (crisp) set theory. It has greater pliability to capture different aspects of incompleteness, imperfection, and uncertainty in data about many situations. The membership μ of an element of a fuzzy set is a single value between 0 and 1. Therefore, in reality, it may not always be true that the degree of nonmembership ν of an element in a fuzzy set is equal to 1 minus the membership degree 1−μ because there may be some uncertainty degree. Thus, since the fuzzy set has no means to characterize the neutral state, neither support nor oppose, Atanassov [2] included the nonmembership function and defined the degree of uncertainty as 1−μ−ν. He introduced the topic of intuitionistic fuzzy sets (IFSs) as an extension of the standard fuzzy sets [3]. Several applications of IFS theory in diverse fields have been carried out; they are a very necessary and powerful tool in modeling imprecision; IFSs have been flourished in many different fields, an intuitionistic fuzzy approach to artificial intelligence [4, 5], medical diagnosis [6], drug selection [7], along with pattern recognition [8], microelectronic fault analysis [9], and decision-making problems [10, 11].The concept of intuitionistic fuzzy differential equations has a rich history and is one of the actively developing topics of intuitionistic fuzzy set theory. The first paper on intuitionistic fuzzy differential equations was by Melliani and Chadli [12], and recently, the authors established some intuitionistic fuzzy solutions for such equations. They proved the existence and uniqueness of intuitionistic fuzzy solutions under some assumptions for these intuitionistic fuzzy differential equations using different procedures, see [13, 14]. The topic of intuitionistic fuzzy functional differential equations with delay is very rare: in [15], they first elaborated conditions that can be satisfied by intuitionistic fuzzy functional differential equations with delay and that guarantee the existence and uniqueness of local and global intuitionistic fuzzy solutions for such equations. Numerical algorithms for treatment of IFDEs are still evolving, see [16–20] for applications.The notion of intuitionistic fuzzy partial differential equations was first studied in 2000 [21], and intuitionistic fuzzy partial differential equations with integral boundary conditions were first introduced in 2019 [22], where they proved existence and uniqueness results and defined a procedure to calculate the intuitionistic fuzzy solutions of such problems. However, according to our knowledge, research on the properties of solutions of intuitionistic fuzzy partial functional differential equations is very limited. The study of intuitionistic fuzzy partial differential equations with state-dependent delay has been initiated recently in 2019 [23], in which they considered the existence and uniqueness of intuitionistic fuzzy solutions for these types of equations with local, nonlocal, and integral boundary conditions, based on some complicated assumptions on the domain. But those hypotheses can be reduced to be milder. In the present paper, our models can be considered as an extension and complementation of the previous results into new weighted metric spaces of intuitionistic fuzzy valued functions. We study state-dependent delay intuitionistic fuzzy partial functional differential equations with integral boundary conditions of the following forms:(1)∂2ϕ1,ϕ2λ,θ∂λ∂θ=G1λ,θ,ϕ1,ϕ2λ,θ,λ,θ∈Ic×Id,ϕ1,ϕ2λ,0+∫0θL1λϕ1,ϕ2λ,σdσ=f1λ,θ,λ,θ∈Ic×Id,ϕ1,ϕ20,θ+∫0λL2θϕ1,ϕ2ζ,θdζ=f2λ,θ,λ,θ∈Ic×Id,(2)∂2ϕ1,ϕ2λ,θ∂λ∂θ=∂qλ,θϕ1,ϕ2λ,θ∂θ+G2λ,θ,ϕ1,ϕ2λ,θ,λ,θ∈Ic×Id,ϕ1,ϕ2λ,0+∫0θL1λϕ1,ϕ2λ,σdσ=f1λ,θ,λ,θ∈Ic×Id,ϕ1,ϕ20,θ+∫0λL2θϕ1,ϕ2ζ,θdζ=f2λ,θ,λ,θ∈Ic×Id,where G1,G2:Ic×Id×IFn⟶IFn is continuous and q∈CIc×Id,ℝ, L1∈CIc,ℝ, L2∈CId,ℝ, f1∈CIc×Id,IFn, and f2∈CIc×Id,IFn are given functions.Motivated by this consideration, first, we shall discuss the existence and uniqueness of the intuitionistic fuzzy solutions to those problems by using some appropriate Banach fixed point theorems under suitable conditions. The main contributions of this work are as follows: introducing the integral boundary value problem for intuitionistic fuzzy partial differential equations with the state-dependent delays and defining their solution; further developing theoretical results on the existence and uniqueness of the intuitionistic fuzzy solution; defining the intuitionistic fuzzy solution to these problems with integral boundary conditions through a corresponding parametric problem.Delay differential equations have been used in many fields for a long time. Researchers have considered state-dependent delay differential equations (SDDEs) for the last 50 years. However, more precisely integral initial boundary value problems with state-dependent delays constitute a very interesting and important class of applied problems; they have been used in modeling scientific phenomena in various applications in different fields such as biology, neural networks, control theory, physics, and medicine, see [24–33], for details. Linking the two aspects introduced, intuitionistic fuzzy mathematics and partial functional differential equations with state-dependent delays, we obtain intuitionistic fuzzy partial functional differential equations (IFPFDEs) with state-dependent delays, which will attract great interest both in mathematics and in applications. No result has been reported on the existing models of state-dependent delay IFPFDEs with integral boundary initial conditions, and among the purposes of this work is to fill this gap. This paper initiates the study of such equations: solving state-dependent delay IFPFDEs with integral boundary initial conditions, providing some definitions for intuitionistic fuzzy solutions of such problems, progressing the main results on the existence and uniqueness of intuitionistic fuzzy solutions under suitable hypothesizes in new weighted metric spaces, without any restraints in the data, displaying rigorous well-posedness and regularity analysis of the state-dependent delay intuitionistic fuzzy PFDEs, and establishing numerical examples that can be considered as a contribution to the subject of our present work and will be introduced elsewhere.The rest of the paper is organised as follows: inSection 2, we will recall some basic definitions and preliminary facts which will be used throughout the following sections. In Section 3, we provide the existence and uniqueness of intuitionistic fuzzy solutions for partial functional differential equations with integral boundary conditions under suitable conditions in a new weighted metric space. In Section 4, we illustrate our abstract results by numerical applications. Finally, Section 5 gives the main conclusions and possible future works in this area.


## 2. Preliminaries

We defineIFn as follows:(3)IFn=IFℝn=ϕ1,ϕ2:ℝn⟶0,1×0,1,|∀λ∈ℝn,0≤ϕ1λ+ϕ2λ≤1.We calledϕ1,ϕ2∈IFn as an intuitionistic fuzzy number if it verifies the following assumptions:(a)
ϕ1,ϕ2 is normal, i.e., there exists λ0 and λ1∈ℝn such that ϕ1λ0=1 and ϕ1=2λ1=1(b)
ϕ1 is fuzzy convex, and ϕ2 is fuzzy concave(c)
ϕ1 is upper semicontinuous, and ϕ2 is lower semicontinuous(d)
suppϕ1,ϕ2=clλ∈ℝn:|ϕ2λ<1 is boundedWe define the upper and lowerα-cuts of ϕ1,ϕ2∈IFn with α∈0,1 by the following forms:(4)ϕ1,ϕ2α=λ∈ℝn:ϕ2λ≤1−α,ϕ1,ϕ2α=λ∈ℝn:ϕ1λ≥α.We define01,0∈IFn as(5)01,0s=1,0,s=0,0,1,s≠0.Letϕ1,ϕ2 and ϕ1′,ϕ2′∈IFn and ζ∈ℝ; then,(6)ϕ1,ϕ2+ϕ1′,ϕ2′r=supr=m+nminϕ1m,ϕ1′n,infr=m+nmaxϕ2m,ϕ2′n,ζϕ1,ϕ2=ζv,ζw,ifζ≠0,01,0,ifζ=0.Forϕ1,ϕ2 and ϕ1′,ϕ2′∈IFn and ζ∈ℝ, therefore(7)ϕ1,ϕ2+ϕ1′,ϕ2′α=ϕ1,ϕ2α+ϕ1′,ϕ2′α,ζϕ1,ϕ2α=ζϕ1,ϕ2α,ϕ1,ϕ2+ϕ1′,ϕ2′α=ϕ1,ϕ2α+ϕ1′,ϕ2′α,ζϕ1,ϕ2α=ζϕ1,ϕ2α.Definition 1.
Letα∈0,1 and ϕ1′,ϕ2′∈IFn; the following sets are defined by(8)ϕ1,ϕ2l+α=infλ∈ℝn|ϕ1λ≥α,ϕ1,ϕ2r+α=supλ∈ℝn|ϕ1λ≥α,ϕ1,ϕ2l−α=infλ∈ℝn|ϕ2λ≤1−α,ϕ1,ϕ2r−α=supλ∈ℝn|ϕ2λ≤1−α.Remark 1.
(9)ϕ1,ϕ2α=ϕ1,ϕ2l+α,ϕ1,ϕ2r+α,ϕ1,ϕ2α=ϕ1,ϕ2l−α,ϕ1,ϕ2r−α.Proposition 1 (see [34]).
For allα1 and α2∈0,1 and ϕ1,ϕ2∈IFn,(a)
ϕ1,ϕ2α1⊂ϕ1,ϕ2α1(b)
ϕ1,ϕ2α1 and ϕ1,ϕ2α1 are nonempty compact convex sets in ℝn(c)
ifα1≤α2, therefore ϕ1,ϕ2α2⊂ϕ1,ϕ2α1 and ϕ1,ϕ2α2⊂ϕ1,ϕ2α1(d)
Ifαn↗α1, therefore ϕ1,ϕ2α1=∩nϕ1,ϕ2αn and ϕ1,ϕ2α1=∩nϕ1,ϕ2αnLetα∈0,1; we denote N (any set) as follows:(10)Nα=λ∈ℝn:ϕ1λ≥α,Nα=λ∈ℝn:ϕ2λ≤1−α.Lemma 1 (see [34]).
LetNα,α∈0,1 and Nα,α∈0,1 be two families of subsets of ℝn; verify the assumptions of Proposition 1 if ϕ1 and ϕ2 are defined as follows:(11)ϕ1λ=0,ifλ∉N0,supα∈0,1:λ∈Nα,ifλ∈N0,ϕ2λ=1,ifλ∉N0,1−supα∈0,1:λ∈Nα,ifλ∈N0.Then,ϕ1,ϕ2∈IFn.We define the following metric by(12)d∞nϕ1,ϕ2,ϕ1′,ϕ2′=14sup0<α≤1ϕ1,ϕ2r+α−ϕ1′,ϕ2′r+α+14sup0<α≤1ϕ1,ϕ2l+α−ϕ1′,ϕ2′l+α+14sup0<α≤1ϕ1,ϕ2r−α−ϕ1′,ϕ2′r−α+14sup0<α≤1ϕ1,ϕ2l−α−ϕ1′,ϕ2′l−α.where . is the usual Euclidean norm in ℝn.Theorem 1 (see [34]).
d∞n is a metric on IFn.Theorem 2 (see [34]).
IFn,d∞n is a complete metric space.We denote(i)
Ic=0,c, Id=0,d, I0=−r,0×−r,0, Ir=−r,c×−r,d, and Ir^=Ir∖0,c×0,d(ii)
CIi,IFn: the space of all continuous mappings defined over Ii into IFn (i = c, d, 0, and r)Forβ>0 arbitrary, it can be shown that CIr,IFn,Dβ is a complete metric space, where the supremum weighted metric Dβ on CIr,IFn is defined by(13)Dβϕ1,ϕ2,ϕ1′,ϕ2′=supr1,r2∈Ird∞nϕ1,ϕ2r1,r2,ϕ1′,ϕ2′r1,r2e−βr1+r2.Definition 2.
A mappingG1:Ir⟶IFn is continuous at point λ0,θ0∈Ir provided for any arbitrary ε>0; there exists an σε such that(14)d∞nG1λ,θ,G1λ0,θ0<ε,whenever maxλ−λ0,θ−θ0<σε∀λ,θ∈Ir.Definition 3.
A mappingG1:Ir×IFn⟶IFn is called continuous at point λ0,θ0,ϕ1,ϕ20∈Ir×IFn provided for any arbitrary ε>0; there exists an σε such that(15)d∞nG1λ,θ,ϕ1,ϕ2,G1λ0,θ0,ϕ1,ϕ20<ε,whenever maxλ−λ0,θ−θ0<σε and d∞nϕ1,ϕ2,ϕ1,ϕ20<σε∀λ,θ∈Ir, ϕ1,ϕ2∈IFn.Definition 4.
A mappingG1:Ic×Id⟶IFn is called strongly measurable if for ∀α∈0,1, the set-valued mappings gα:Ic×Id⟶Pkℝn defined by gαλ,θ=G1λ,θα and gα:Ic×Id⟶Pkℝn defined by gαλ,θ=G1λ,θα are (Lebesgue) measurable.Definition 5.
G1:Ic×Id⟶IFn is integrably bounded if there exists an integrable function h:Ic×Id⟶ℝn such that ζ,σ≤hλ,θ holds for any ζ,σ∈suppG1λ,θ,λ,θ∈Ic×Id.Definition 6.
SupposeG1:Ic×Id⟶IFn is integrably bounded and strongly measurable for each α∈0,1; then,(16)∫0c∫0dG1λ,θdθdλα=∫0c∫0dG1λ,θαdθdλ=∫0c∫0dGλ,θdθdλ|G:Ic×Id⟶ℝnis a measurable selection forG1α,∫0c∫0dG1λ,θdθdλα=∫0c∫0dG1λ,θαdθdλ=∫0c∫0dGλ,θdθdλ|G:Ic×Id⟶ℝnis a measurable selection forG1α.LetG1,G2:Ic×Id⟶IFn be integrable and k∈ℝ. The integral has the elementary properties as follows:(i)
∫0c∫0dG1λ,θ+G2λ,θdθdλ=∫0c∫0dG1λ,θdθdλ+∫0c∫0dG2λ,θdθdλ(ii)
∫0c∫0dkG1λ,θdθdλ=k∫0c∫0dG1G1λ,θdθdλ(iii)
d∞n∫0c∫0dG1λ,θdθdλ,∫0c∫0dG2λ,θdθdλ≤∫0c∫0dd∞nG1λ,θ,G2λ,θdθdλDefinition 7.
Letϕ1,ϕ2 and ϕ1′,ϕ2′∈IFn; we call the Hukuhara difference as an intuitionistic fuzzy number φ1,φ2∈IFn; if it exists, such that(17)ϕ1,ϕ2−ϕ1′,ϕ2′=ϕ1,ϕ2⟺ϕ1,ϕ2=ϕ1′,ϕ2′+ϕ1,ϕ2.Definition 8.
LetG1:c,d⟶IFn be differentiable, and suppose that the derivative g′ is integrable over c,d; if the Hukuhara difference G1d−G1c exists in IFn, then we have ∫cdG1′tdt=G1d−G1c.Definition 9.
LetG1:Ic×Id⟶IFn. The intuitionistic fuzzy partial derivative of g with respect to x at the point λ0,θ0∈Ic×Id is the intuitionistic fuzzy quantity ∂G1λ0,θ0/∂λ∈IFn such that ∀△t>0 sufficiently small, the H-differences G1λ0+△t,θ0−G1λ0,θ0 and G1λ0,θ0−G1λ0−△t,θ0 exist in IFn and the limits(18)∂G1λ0,θ0∂λ=lim△t⟶0+G1λ0+△t,θ0−G1λ0,θ0△t,∂G1λ0,θ0∂λ=lim△t⟶0+G1λ0,θ0−G1λ0−△t,θ0△t.
The intuitionistic fuzzy partial derivative ofg with respect to θ at the point λ0,θ0∈Ic×Id and higher order of intuitionistic fuzzy partial derivative of g are defined similarly.


## 3. The Main Results

We prove the existence-uniqueness of the intuitionistic fuzzy solution for the following intuitionistic fuzzy functional partial differential equation with integral boundary conditions:(19)∂2ϕ1,ϕ2λ,θ∂λ∂θ=G1λ,θ,ϕ1,ϕ2λ,θ,λ,θ∈Ic×Id≔0,c×0,d,ϕ1,ϕ2λ,θ=ηλ,θ,λ,θ∈Ir^,ϕ1,ϕ2λ,0+∫0θL1λϕ1,ϕ2λ,σdσ=f1λ,θ,λ,θ∈Ic×Id,ϕ1,ϕ20,θ+∫0λL2θϕ1,ϕ2ζ,θdζ=f2λ,θ,λ,θ∈Ic×Id.Assume thatG1:Ir×CI0,IFn⟶IFn and η∈CI0,IFn is continuous; L1∈CIc,ℝ, L2∈CId,ℝ, f1∈CIc×Id,IFn, and f2∈CIc×Id,IFn are given functions.For eachλ,θ∈Ic×Id, we define the state-dependent delays ϕ1,ϕ2λ,θr1,r2 by(20)ϕ1,ϕ2λ,θr1,r2=ϕ1,ϕ2λ+r1,θ+r2,r1,r2∈I0.In the 2nd part of this section, we will prove the existence-uniqueness of intuitionistic fuzzy solution for the following state-dependent delay intuitionistic fuzzy functional partial differential equation of the general model:(21)∂2ϕ1,ϕ2λ,θ∂λ∂θ=∂qλ,θϕ1,ϕ2λ,θ∂θ+G2λ,θ,ϕ1,ϕ2λ,θ,λ,θ∈Ic×Id≔0,c×0,d,ϕ1,ϕ2λ,θ=ηλ,θ,λ,θ∈Ir^,ϕ1,ϕ2λ,0+∫0θL1λϕ1,ϕ2λ,σdσ=f1λ,θ,λ,θ∈Ic×Id,ϕ1,ϕ20,θ+∫0λL2θϕ1,ϕ2ζ,θdζ=f2λ,θ,λ,θ∈Ic×Id,where G2:Ir×CI0,IFn⟶IFn, L1, L2, f1, and f2 are as in problem (19), and q∈CIc×Id,ℝ.Definition 10.
A functionϕ1,ϕ2∈CIr,IFn can be a solution of model (19) if ϕ1,ϕ2 verifies the following equation:(22)ϕ1,ϕ2λ,θ=Pλ,θ∫0θL1λϕ1,ϕ2λ,σdσ−∫0λL2θϕ1,ϕ2ζ,θdζ−L10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ+∫0λ∫0θG1ζ,σ,ϕ1,ϕ2ζ,σdσdζ,if λ,θ∈Ic×Id where Pλ,θ=f1λ,θ+f2λ,θ−f10,θ+L10∫0θf2λ,σdσ and ϕ1,ϕ2λ,θ=ηλ,θ, if λ,θ∈Ir^.LetL1=supλ∈IcL1λ and L2=supθ∈IdL2θ. By applying Banach fixed point theorem, we will prove the following results.Theorem 3.
Suppose that(1)
G1:Ir×CI0,IFn⟶IFn is continuous(2)
Forλ,θ∈Ic×Id, ϕ1,ϕ2andϕ1′,ϕ2′∈CIr,IFn, and r1,r2∈I0, we have(23)d∞nG1λ,θ,ϕ1,ϕ2λ,θ,G1λ,θ,ϕ1′,ϕ2′λ,θ≤Ld∞nϕ1,ϕ2λ+r1,θ+r2,ϕ1′,ϕ2′λ+r1,θ+r2,where L>0 is a given constant. Moreover, for all β>0 satisfying(24)β2−βL1+L2−L1L2−L>0.Therefore, problem (19) has a unique intuitionistic fuzzy solution in CIr,IFn with the metric Dβ.Proof.
Consider the operatorT1:CIr,IFn⟶CIr,IFn defined as follows:(25)T1ϕ1,ϕ2λ,θ≔ηλ,θ,λ,θ∈Ir^,Pλ,θ−∫0θL1λϕ1,ϕ2λ,σdσ−∫0λL2θϕ1,ϕ2ζ,θdζ−L10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ+∫0λ∫0θG1ζ,σ,ϕ1,ϕ2ζ,σdσdζ,ifλ,θ∈Ic×Id,where Pλ,θ=f1λ,θ+f2λ,θ−f10,θ+L10∫0θf2λ,σdσ.
We will prove thatT1 is a contraction operator.
Therefore, considerϕ1,ϕ2 and ϕ1′,ϕ2′∈CIr,IFn, and for λ,θ∈Ic×Id, we have(26)T1ϕ1,ϕ2λ,θ≔Pλ,θ−∫0θL1λϕ1,ϕ2λ,σdσ−∫0λL2θϕ1,ϕ2ζ,θdζ−L10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ+∫0λ∫0θG1ζ,σ,ϕ1,ϕ2ζ,σdσdζ,T1ϕ1′,ϕ2′λ,θ≔Pλ,θ−∫0θL1λϕ1′,ϕ2′λ,σdσ−∫0λL2θϕ1′,ϕ2′ζ,θdζ−L10∫0θ∫0λL2σϕ1′,ϕ2′ζ,σdζdσ+∫0λ∫0θG1ζ,σ,ϕ1′,ϕ2′ζ,σdσdζ.
We have the following inequality:(27)d∞nT1ϕ1,ϕ2λ,θ,T1ϕ1′,ϕ2′λ,θ≤d∞n∫0θL1λϕ1,ϕ2λ,σdσ,∫0θL1λϕ1′,ϕ2′λ,σdσ+d∞n∫0λL2θϕ1,ϕ2ζ,θdζ,∫0λL2θϕ1′,ϕ2′ζ,θdζ+d∞nL10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ,L10∫0θ∫0λL2σϕ1′,ϕ2′ζ,σdζdσ+d∞n∫0λ∫0θG1ζ,σ,ϕ1,ϕ2ζ,σdσdζ,∫0λ∫0θG1ζ,σ,ϕ1′,ϕ2′ζ,σdσdζ≤L1∫0θd∞nϕ1,ϕ2λ,σ,ϕ1′,ϕ2′λ,σdσ+L2∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ+L10supθ∈IdL2θ∫0θ∫0λd∞nϕ1,ϕ2ζ,σ,ϕ1′,ϕ2′ζ,σdζdσ+∫0λ∫0θd∞nG1ζ,σ,ϕ1,ϕ2ζ,σ,G1ζ,σ,ϕ1′,ϕ2′ζ,σdσdζ.
Multiplying (27) by e−βλ+θ, then we get(28)e−βλ+θd∞n∫0λ∫0θG1ζ,σ,ϕ1,ϕ2ζ+r1,σ+r2,∫0λ∫0θG1ζ,σ,ϕ1′,ϕ2′ζ+r1,σ+r2dσdζe−βλ+θ≤L∫0λ∫0θd∞nϕ1,ϕ2ζ+r1,σ+r2,ϕ1′,ϕ2′ζ+r1,σ+r2dσdζe−βλ+θ≤LDβϕ1,ϕ2,ϕ1′,ϕ2′∫0λ∫0θeβr1+r2+ζ+σdσdζe−βλ+θ≤LDβϕ1,ϕ2,ϕ1′,ϕ2′1βeβr1+r2∫0λeβζ+θ−eβζdζe−βλ+θ≤Lβ2Dβϕ1,ϕ2,ϕ1′,ϕ2′eβr1+r2eβλ−1eβθ−1e−βλ+θ≤Lβ2Dβϕ1,ϕ2,ϕ1′,ϕ2′.
The same, we have(29)e−βλ+θL10supy∈IdL2θ∫0θ∫0λd∞nϕ1,ϕ2ζ,σ,ϕ1′,ϕ2′ζ,σdζdσ≤L1L2β2Dβϕ1,ϕ2,ϕ1′,ϕ2′,(30)e−βλ+θL1∫0θd∞nϕ1,ϕ2λ,σ,ϕ1′,ϕ2′λ,σdσ≤L1Dβϕ1,ϕ2,ϕ1′,ϕ2′∫0θeβλ+σdσe−βλ+θ≤L1Dβϕ1,ϕ2,ϕ1′,ϕ2′e−βλ+θ−eβθe−βλ+θ≤L1βDβϕ1,ϕ2,ϕ1′,ϕ2′.
Similarly, we get(31)e−βλ+θL2∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ≤L2βDβϕ1,ϕ2,ϕ1′,ϕ2′.
From (28)–(31), we obtain(32)e−βλ+θd∞nT1ϕ1,ϕ2λ,θ,T1ϕ1′,ϕ2′λ,θ≤L1+L2β+L1L2+Lβ2Dβϕ1,ϕ2,ϕ1′,ϕ2′.
Ifλ,θ∈I^r,(33)d∞nT1ϕ1,ϕ2λ,θ,T1ϕ1′,ϕ2′λ,θ=0.
Hence, for eachλ,θ∈Ir,(34)DβT1ϕ1,ϕ2,T1ϕ1′,ϕ2′≤L1+L2β+L1L2+Lβ2Dβϕ1,ϕ2,ϕ1′,ϕ2′.
So, for allβ>0 satisfying (31), we have L1+L2/β+L1L2+L/β2<1. Then, T1 is a contraction operator.
Therefore,T1 has a unique fixed point, which is the solution of integral boundary problem (21).Definition 11.
A functionϕ1,ϕ2∈CIr,IFn is called an intuitionistic fuzzy solution of problem (21) if ϕ1,ϕ2 verifies the following integral equation:(35)ϕ1,ϕ2λ,θ=Pλ,θ−∫0θL1λϕ1,ϕ2λ,σdσ−∫0λL2θϕ1,ϕ2ζ,θdζ+∫0λqζ,θϕ1,ϕ2ζ,θdζ−L10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ+∫0λqζ,0ϕ1,ϕ2ζ,0dζ+∫0λ∫0θG2ζ,σ,ϕ1,ϕ2ζ,σdσdζ,if λ,θ∈Ic×Id where Pλ,θ=f1λ,θ+f2λ,θ−f10,θ+L10∫0θf2λ,σdσ and ϕ1,ϕ2λ,θ=ηλ,θ if λ,θ∈Ir^.LetL1=supλ∈IcL1λ, L2=supθ∈IdL2θ, and q=supλ,θ∈Ic×Idqλ,θ. By applying the Banach fixed point theorem, we will prove the following result.Theorem 4.
Suppose that(1)
A mapG2:Ir×CI0,IFn⟶IFn is continuous(2)
Forλ,θ∈Ic×Id, ϕ1,ϕ2andϕ1′,ϕ2′∈CIr,IFn, and r1,r2∈I0, we have(36)d∞nG2λ,θ,ϕ1,ϕ2λ,θ,G2λ,θ,ϕ1′,ϕ2′λ,θ≤Ldn∞ϕ1,ϕ2λ+r1,θ+r2,ϕ1′,ϕ2′λ+r1,θ+r2,where L>0 is a given constant. Then, for all β>0 satisfying(37)β2−βL1+L2+2q−L1L2−L>0.Therefore, problem (21) has a unique intuitionistic fuzzy solution in CIr,IFn with the metric Dβ.Proof.
To prove the existence-uniqueness result, we will show thatT2 has a unique fixed point.
Define operatorT2:CIr,IFn⟶CIr,IFn by(38)T2ϕ1,ϕ2λ,θ≔ηλ,θ,λ,θ∈Ir^,Pλ,θ−∫0θL1λϕ1,ϕ2λ,σdσ−∫0λL2θϕ1,ϕ2ζ,θdζ+∫0λqζ,θϕ1,ϕ2ζ,θdζ−L10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ+∫0λqζ,0ϕ1,ϕ2ζ,0dζ+∫0λ∫0θG2ζ,σ,ϕ1,ϕ2ζ,σdσdζ,ifλ,θ∈Ic×Id,where(39)Pλ,θ=f1λ,θ+f2λ,θ−f10,θ+L10∫0θf2λ,σdσ.
We prove thatT2 is a contraction operator. Then, consider ϕ1,ϕ2 and ϕ1′,ϕ2′∈CIr,IFn.
We can see thatd∞nT2ϕ1,ϕ2λ,θ,T2ϕ1′,ϕ2′λ,θ=0 if λ,θ∈I0. On the contrary, on λ,θ∈Ic×Id, we have(40)T2ϕ1,ϕ2λ,θ≔Pλ,θ−∫0θL1λϕ1,ϕ2λ,σdσ−∫0λL2θϕ1,ϕ2ζ,θdζ+∫0λqζ,θϕ1,ϕ2ζ,θdζ−L10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ+∫0λqζ,0ϕ1,ϕ2ζ,0dζ+∫0λ∫0θG2ζ,σ,ϕ1,ϕ2ζ,σdσdζ,T2ϕ1′,ϕ2′λ,θ≔Pλ,θ−∫0θL1λϕ1′,ϕ2′λ,σdσ−∫0λL2θv′,w′ζ,θdζ+∫0λqζ,θϕ1′,ϕ2′ζ,θdζ−L10∫0θ∫0λL2σϕ1′,ϕ2′ζ,σdζdσ+∫0λqζ,0ϕ1′,ϕ2′ζ,0dζ+∫0λ∫0θG2ζ,σ,ϕ1′,ϕ2′ζ,σdσdζ.
From the properties of supremum metric, we have(41)d∞nT2ϕ1,ϕ2λ,θ,T2ϕ1′,ϕ2′λ,θ≤d∞n∫0θL1λϕ1,ϕ2λ,σdσ,∫0θL1λϕ1′,ϕ2′λ,σdσ+d∞n∫0λL2θϕ1,ϕ2ζ,θdζ,∫0λL2θϕ1′,ϕ2′ζ,θdζ+d∞nL10∫0θ∫0λL2σϕ1,ϕ2ζ,σdζdσ,L10∫0θ∫0λL2σϕ1′,ϕ2′ζ,σdζdσ+d∞n∫0λqζ,θϕ1,ϕ2ζ,θdζ,∫0λqζ,θϕ1′,ϕ2′ζ,θdζ+d∞n∫0λqζ,0ϕ1,ϕ2ζ,0dζ,∫0λqζ,0ϕ1′,ϕ2′ζ,0dζ+d∞n∫0λ∫0θG2ζ,σ,ϕ1,ϕ2ζ,σdσdζ,∫0λ∫0θG2ζ,σ,ϕ1′,ϕ2′ζ,σdσdζ,≤L1∫0θd∞nϕ1,ϕ2λ,σ,ϕ1′,ϕ2′λ,σdσ+L2∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ+L10supθ∈IdL2θ∫0θ∫0λd∞nϕ1,ϕ2ζ,σ,ϕ1′,ϕ2′ζ,σdζdσ+supζ,σ∈Ic×Idqζ,σ∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ+supζ,σ∈Ic×Idqζ,0∫0λd∞nϕ1,ϕ2ζ,0,ϕ1′,ϕ2′ζ,0dζ+∫0λ∫0θd∞nG2ζ,σ,ϕ1,ϕ2ζ,σ,G2ζ,σ,ϕ1′,ϕ2′ζ,0dσdζ.
SetL1=supλ∈IcL1λ, L2=supθ∈IdL2θ, and q=supλ,θ∈Ic×Idqλ,θ. We get the following assessment:(42)supζ,σ∈Ic×Idqζ,σ∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ=q∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ,≤qDβϕ1,ϕ2,ϕ1′,ϕ2′∫0λeβζ+θdζ,≤qDβϕ1,ϕ2,ϕ1′,ϕ2′1βeβζ+θ−eβθ.
Multiplying (42) by exp−βλ+θ, we get(43)e−βλ+θsupζ,σ∈Ic×Idqζ,σ∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ≤qDβϕ1,ϕ2,ϕ1′,ϕ2′1βeβλ+θ−eβθe−βλ+θ,≤qβDβϕ1,ϕ2,ϕ1′,ϕ2′.
Similarly, we obtain(44)e−βλ+θsupζ,0∈Ic×Idqζ,0∫0λd∞nϕ1,ϕ2ζ,0,ϕ1′,ϕ2′ζ,0dσ≤qβDβϕ1,ϕ2,ϕ1′,ϕ2′.
From (30) and (31), we imply(45)e−βλ+θL1∫0θd∞nϕ1,ϕ2λ,σ,ϕ1′,ϕ2′λ,σdσ≤L1βDβϕ1,ϕ2,ϕ1′,ϕ2′,(46)e−βλ+θL2∫0λd∞nϕ1,ϕ2ζ,θ,ϕ1′,ϕ2′ζ,θdζ≤L2βDβϕ1,ϕ2,ϕ1′,ϕ2′.and from (28), we also get(47)e−βλ+θd∞n∫0λ∫0θG2ζ,σ,ϕ1,ϕ2ζ+r1,σ+r2,∫0λ∫0θG2ζ,σ,ϕ1′,ϕ2′ζ+r1,σ+r2dσdζe−βλ+θ≤Lβ2Dβϕ1,ϕ2,ϕ1′,ϕ2′,(48)e−βλ+θL10supy∈IdL2θ∫0θ∫0λd∞nϕ1,ϕ2ζ,σ,ϕ1′,ϕ2′ζ,σdζdσ≤L1L2β2Dβϕ1,ϕ2,ϕ1′,ϕ2′.
From (43)–(48), we have(49)e−βλ+θd∞nT2ϕ1,ϕ2λ,θ,T2ϕ1′,ϕ2′λ,θ≤L1+L2+2qβ+L1L2+Lβ2Dβϕ1,ϕ2,ϕ1′,ϕ2′.
Ifλ,θ∈I^r,(50)d∞nT2ϕ1,ϕ2λ,θ,T2ϕ1′,ϕ2′λ,θ=0.
Hence, for eachλ,θ∈Ir,(51)DβT2ϕ1,ϕ2,T2ϕ1′,ϕ2′≤L1+L2+2qβ+L1L2+Lβ2Dβϕ1,ϕ2,ϕ1′,ϕ2′.
Then, for allβ>0 verifying (36), it implies L1+L2+2q/β+L1L2+L/β2<1; therefore, T2 is a contraction operator.
Thus,T2 has a unique fixed point, which is the solution of (21).


## 4. Examples

We propose the following two examples: to illustrate the usefulness of our main results and to find the intuitionistic fuzzy solution, we will use the method of steps proposed in [22, 23].Example 1.
Now, we present a first example to illustrate Theorem3; consider the following state-dependent delay intuitionistic fuzzy partial functional differential equations for some w>0 in the following form:(52)∂2ϕ1,ϕ2λ,θ∂λ∂θ=w2ew/2ϕ1,ϕ2λ−14,θ−14,λ,θ∈0,1×0,1.
And, the integral boundary initial conditions are(53)ϕ1,ϕ2λ,0+18∫0θϕ1,ϕ2λ,σdσ=C8wewλewθ+8w−1,(54)ϕ1,ϕ20,θ+18∫0λϕ1,ϕ2ζ,θdζ=C8wewθewλ+8w−1,where C∈IF1 is a triangular intuitionistic fuzzy number, and the initial condition is(55)ϕ1,ϕ2λ,θ=Cewλ+ewθ+λθ,for λ,θ∈−1/4,1×−1/4,1∖0,1×0,1.Since(56)G1λ,θ,ϕ1,ϕ2λ,θ=w2ew/2ϕ1,ϕ2λ−14,θ−14,it satisfies hypothesises (1) and (2) of Theorem 3.Indeed, it is easy to see thatG1 is continuous and(57)d∞G1λ,θ,ϕ1,ϕ2λ,θ,G1λ,θ,ϕ1′,ϕ2′λ,θ≤w2ew/2d∞ϕ1,ϕ2λ,θ−14,−14,ϕ1′,ϕ2′λ,θ−14,−14.Therefore,G1 satisfies 2 with an positive number L=w2ew/2, L1=L2=1/8, and c=d=1.Then, from Theorem3, we see that if we choose positive weighted number β satisfying β>1/8+1/32+w2ew/2, Hence, all the assumptions of Theorem 3 are satisfied; therefore, we have a unique intuitionistic fuzzy solution of problems (45)–(47).We try to calculate the intuitionistic fuzzy solution of this problem by using the method of steps proposed in [22, 23].We write the deterministic solution of the crisp equation as follows:v=cewλ+θ.Assume that the parametric form of corresponding intuitionistic fuzzy numberC is(58)Cα=Cl+α,Cr+α,Cα=Cl−α,Cr−α,where it verifies the conditions of Lemma 1.The functionG1:0,π/2×0,1×IF1⟶IF1 is defined as(59)G1λ,θ,ϕ1,ϕ2λ,θ=w2ew/2ϕ1,ϕ2λ−14,θ−14.Theα-cuts of G1 are(60)G1α=w2ew/2ϕ1,ϕ2λ−14,θ−14l+α,w2ew/2ϕ1,ϕ2λ−14,θ−14r+α,G1α=w2ew/2ϕ1,ϕ2λ−14,θ−14l−α,w2ew/2ϕ1,ϕ2λ−14,θ−14r−α.If we denote(61)ϕ1,ϕ2λ,θα=ϕ1,ϕ2λ,θl+α,ϕ1,ϕ2λ,θr+α,ϕ1,ϕ2λ,θα=ϕ1,ϕ2λ,θl−α,ϕ1,ϕ2λ,θr−α.Then,(62)∂2ϕ1,ϕ2λ,θ∂λ∂θα=∂2ϕ1,ϕ2λ,θ∂λ∂θl+α,∂2ϕ1,ϕ2λ,θ∂λ∂θr+α,∂2ϕ1,ϕ2λ,θ∂λ∂θα=∂2ϕ1,ϕ2λ,θ∂λ∂θl−α,∂2ϕ1,ϕ2λ,θ∂λ∂θr−α.Therefore, we solve the following state-dependent delay partial functional differential equations forλ,θ∈0,1×0,1:(63)∂2ϕ1,ϕ2λ,θ∂λ∂θl+α=w2ew/2ϕ1,ϕ2λ−14,θ−14l+α,∂2ϕ1,ϕ2λ,θ∂λ∂θr+α=w2ew/2ϕ1,ϕ2λ−14,θ−14r+α,∂2ϕ1,ϕ2λ,θ∂λ∂θl−α=w2ew/2ϕ1,ϕ2λ−14,θ−14l−α,∂2ϕ1,ϕ2λ,θ∂λ∂θr−α=w2ew/2ϕ1,ϕ2λ−14,θ−14r−α,with initial conditions for λ,θ∈−1/4,1×−1/4,1∖0,1×0,1:(64)ϕ1,ϕ2λ,θl+α=Cl+ewλ+ewθ+λθ,ϕ1,ϕ2λ,θr+α=Cr+ewλ+ewθ+λθ,ϕ1,ϕ2λ,θl−α=Cl−ewλ+ewθ+λθ,ϕ1,ϕ2λ,θr−α=Cr−ewλ+ewθ+λθ,and for λ,θ∈0,1×0,1, we have(65)ϕ1,ϕ2λ,0l+α+18∫0θϕ1,ϕ2λ,σl+αdσ=Cl+α18wewλewθ+8w−1,ϕ1,ϕ2λ,0r+α+18∫0θϕ1,ϕ2λ,σr+αdσ=Cr+α18wewλewθ+8w−1,ϕ1,ϕ2λ,0l−α+18∫0θϕ1,ϕ2λ,σl−αdσ=Cl−α18wewλewθ+8w−1,ϕ1,ϕ2λ,0r−α+18∫0θϕ1,ϕ2λ,σr−αdσ=Cr−α18wewλewθ+8w−1,ϕ1,ϕ2λ,0l+α+18∫0λϕ1,ϕ2ζ,θl+αdζ=Cl+α18wewθewλ+8w−1,ϕ1,ϕ2λ,0r+α+18∫0λϕ1,ϕ2ζ,θr+αdζ=Cr+α18wewθewλ+8w−1,ϕ1,ϕ2λ,0l−α+18∫0λϕ1,ϕ2ζ,θl−αdζ=Cl−α18wewθewλ+8w−1,ϕ1,ϕ2λ,0r−α+18∫0λϕ1,ϕ2ζ,θr−αdζ=Cr−α18wewθewλ+8w−1.We get(66)ϕ1,ϕ2λ,θl+α=Cl+αewλ+θ,λ,θ∈0,1×0,1,ϕ1,ϕ2λ,θr+α=Cr+αewλ+θ,λ,θ∈0,1×0,1,ϕ1,ϕ2λ,θl−α=Cl−αewλ+θ,λ,θ∈0,1×0,1,ϕ1,ϕ2λ,θr−α=Cr−αewλ+θ,λ,θ∈0,1×0,1.Then,(67)ϕ1,ϕ2λ,θα=Cl+αewλ+θ,Cr+αewλ+θϕ1,ϕ2λ,θα=Cl−αewλ+θ,Cr−αewλ+θ.Now we denote the familiesNα and Nα as follows:(68)Cl+αewλ+θ,Cr+αewλ+θ=Nα,Cl−αewλ+θ,Cr−αewλ+θ=Nα,and the families Nα′ and N′α by(69)w2ew/2Cl−αewλ+θ,w2ew/2Cr−αewλ+θ=Nα′,w2ew/2Cl−αewλ+θ,w2ew/2Cr−αewλ+θ=N′α.It is easy to see thatNα,Nα and Nα′,N′α satisfy the conditions of Proposition 1, and by applying Lemma 1, we can build the intuitionistic fuzzy solution ϕ1,ϕ2λ,θ∈IF1 for equations (52)–(54) for every α∈0,1 as follows:(70)ϕ1,ϕ2λ,θα=Cl+αewλ+θ,Cr+αewλ+θ,ϕ1,ϕ2λ,θα=Cl−αewλ+θ,Cr−αewλ+θ.Then,ϕ1,ϕ2x,y is an intuitionistic fuzzy solution which satisfies the initial conditions (53)–(54) and is written as follows:(71)ϕ1,ϕ2x,y=Cewλ+θ.Figures1 and 2 show the membership and nonmembership functions of triangular intuitionistic fuzzy number C=−0.75,0,0.75;−2,0,2 and the simulation of α-cuts of the intuitionistic fuzzy solution ϕ1,ϕ2x,y at some values of λ,θ with w=0.45.Figure 1
C=−0.75,0,0.75;−2,0,2.Figure 2
The curve of intuitionistic fuzzy solutionsϕ1,ϕ2x,y at different values of λ,θ.In Figure3 we present the surface of the intuitionistic fuzzy solution with triangular intuitionistic fuzzy numbers C=−0.75,0,0.75;−2,0,2 and w=0.45.Figure 3
The surface of the intuitionistic fuzzy solution.Example 2.
Now, we present a second example to illustrate Theorem4. Consider the following state-dependent delay intuitionistic fuzzy partial functional differential equations in the general form:(72)∂2ϕ1,ϕ2λ,θ∂λ∂θ=∂1/2e−2/3ϕ1,ϕ2λ,θ∂θ−12ϕ1,ϕ2λ−13,θ−13+c1eλ+θ+12c2cosλ−13.
And, the integral boundary initial conditions are(73)ϕ1,ϕ2λ,0+12∫0θϕ1,ϕ2λ,σdσ=c12eλ1+eθ+c2cosλ1+θ2,(74)ϕ1,ϕ20,θ+12∫0λϕ1,ϕ2ζ,θdζ=c12eθ1+eλ+c21+sinλ2,where c1andc2∈IF1 are triangular intuitionistic fuzzy numbers and λ,θ∈0,π/2×0,1, and for λ,θ∈−1/3,π/2×−1/3,1∖0,π/2×0,1, the initial condition is(75)ϕ1,ϕ2λ,θ=c1eλ+λθ+c2eθ+cosλθ.Since(76)G2λ,θ,ϕ1,ϕ2λ,θ=−12ϕ1,ϕ2λ−13,θ−13+c1eλ+θ+12c2cosλ−13,it verifies conditions 1 and 2 of Theorem 1.Indeed, it is easy to see thatG2 is continuous:(77)d∞G2λ,θ,ϕ1,ϕ2λ,θ,G2λ,θ,ϕ1′,ϕ2′λ,θ≤12d∞ϕ1,ϕ2λ,θ−13,−13,ϕ1′,ϕ2′λ,θ−13,−13,so the assumptions are satisfied with a positive number L=1/2, L1=L2=1/2, c=π/2, d=1, and qλ,θ=1/2e−2/3.If we choose the positive weighted numberβ satisfying β2−β1+e−2/3−3/4>0, then β>1/2+e−2/3/2+1+1/4e−4/3+1/2e−2/3; hence, all the hypothesis of Theorem 4 hold; therefore, we have a unique intuitionistic fuzzy solution of equations (72)–(74).We will use the method of steps proposed in [22, 23] to find an intuitionistic fuzzy solution of equations (72)–(74).The classical solution of the crisp equation isw=C1eλ+θ+C2cosλ.Assume that the parametric forms of corresponding intuitionistic fuzzy numbercii=1,2 are(78)ciα=cil+α,cir+α,ciα=cil−α,cir−α,where they verify the conditions of Lemma 1.The functionG2:0,π/2×0,1×IF1⟶IF1 is defined by(79)G2λ,θ,ϕ1,ϕ2λ,θ=−12ϕ1,ϕ2λ−13,θ−13+c1eλ+θ+12c2cosλ−13.Theα-cuts of G2 are(80)G2α=G2l+α,G2r+α,G2α=G2l−α,G2r−α,where(81)G2l+α=−12ϕ1,ϕ2λ−13,θ−13l+α+c1l+αeλ+θ+12c2l+αcosλ−13,G2r+α=−12ϕ1,ϕ2λ−13,θ−13r+α+c1r+αeλ+θ+12c2r+αcosλ−13,G2l−α=−12ϕ1,ϕ2λ−13,θ−13l−α+c1l−αeλ+θ+12c2l−αcosλ−13,G2r−α=−12ϕ1,ϕ2λ−13,θ−13r−α+c1r−αeλ+θ+12c2r−αcosλ−13.If(82)ϕ1,ϕ2λ,θα=ϕ1,ϕ2λ,θl+α,ϕ1,ϕ2λ,θr+αϕ1,ϕ2λ,θα=ϕ1,ϕ2λ,θl−α,ϕ1,ϕ2λ,θr−α,then(83)∂2ϕ1,ϕ2λ,θ∂λ∂θα=∂2ϕ1,ϕ2λ,θ∂λ∂θl+α∂2ϕ1,ϕ2λ,θ∂λ∂θr+α,∂2ϕ1,ϕ2λ,θ∂λ∂θα=∂2ϕ1,ϕ2λ,θ∂λ∂θl−α∂2ϕ1,ϕ2λ,θ∂λ∂θr−α.Then, we solve the following state-dependent delay partial functional differential equations:(84)∂2ϕ1,ϕ2λ,θ∂λ∂θl+α=∂1/2e−2/3ϕ1,ϕ2λ,θ∂θl+α−12ϕ1,ϕ2λ−13,θ−13l+α+c1l+αeλ+θ+12c2l+αcosλ−13,∂2ϕ1,ϕ2λ,θ∂λ∂θr+α=∂1/2e−2/3ϕ1,ϕ2λ,θ∂θr+α−12ϕ1,ϕ2λ−13,θ−13r+α+c1r+αeλ+θ+12c2r+αcosλ−13,∂2ϕ1,ϕ2λ,θ∂λ∂θl−α=∂1/2e−3/4ϕ1,ϕ2λ,θ∂θl−α−12ϕ1,ϕ2λ−13,θ−13l−α+c1l−αeλ+θ+12c2l−αcosλ−13,∂2ϕ1,ϕ2λ,θ∂λ∂θr−α=∂1/2e−2/3ϕ1,ϕ2λ,θ∂θr−α−12ϕ1,ϕ2λ−13,θ−13r−α+c1r−αeλ+θ+12c2r−αcosλ−13,where λ,θ∈0,π/2×0,1 with initial conditions for λ,θ∈−1/2,π/2×−1/2,1∖0,π/2×0,1:(85)ϕ1,ϕ2λ,θl+α=c1l+αeλ+λθ+c2l+αeθ+cosλθ,ϕ1,ϕ2λ,θr+α=c1r+αeλ+λθ+c2r+αeθ+cosλθ,ϕ1,ϕ2λ,θl−α=c1l−αeλ+λθ+c2l−αeθ+cosλθ,ϕ1,ϕ2λ,θr−α=c1r−αeλ+λθ+c2r−αeθ+cosλθ,and for λ,θ∈0,π/2×0,1,(86)ϕ1,ϕ2λ,0l+α+12∫0θϕ1,ϕ2λ,σl+αdσ=12c1l+αeλ1+eθ+c2l+αcosλ1+θ2,ϕ1,ϕ2λ,0r+α+12∫0θϕ1,ϕ2λ,σr+αdσ=12c1r+αeλ1+eθ+c2r+αcosλ1+θ2,ϕ1,ϕ2λ,0l−α+12∫0θϕ1,ϕ2λ,σl−αdσ=12c1l−αeλ1+eθ+c2l−αcosλ1+θ2,ϕ1,ϕ2λ,0r−α+12∫0θϕ1,ϕ2λ,σr−αdσ=12c1r−αeλ1+eθ+c2r−αcosλ1+θ2,ϕ1,ϕ20,θl+α+12∫0λϕ1,ϕ2ζ,θl+αdζ=12c1l+αeθ1+eλ+c2l+α1+sinλ2,ϕ1,ϕ20,θr+α+12∫0λϕ1,ϕ2ζ,θr+αdζ=12c1r+αeθ1+eλ+c2r+α1+sinλ2,ϕ1,ϕ20,θl−α+12∫0λϕ1,ϕ2ζ,θl−αdζ=12c1l−αeθ1+eλ+c2l−α1+sinλ2,ϕ1,ϕ20,θr−α+12∫0λϕ1,ϕ2ζ,θr−αdζ=12c1r−αeθ1+eλ+c2r−α1+sinλ2.We get(87)ϕ1,ϕ2λ,θl+α=c1l+αeλ+θ+c2l+αcosλ,λ,θ∈0,π2×0,1,ϕ1,ϕ2λ,θr+α=c1r+αeλ+θ+c2r+αcosλ,λ,θ∈0,π2×0,1,ϕ1,ϕ2λ,θl−α=c1l−αeλ+θ+c2l−αcosλ,λ,θ∈0,π2×0,1,ϕ1,ϕ2λ,θr−α=c1r−αeλ+θ+c2r−αcosλ,λ,θ∈0,π2×0,1.Then,(88)ϕ1,ϕ2λ,θα=c1l+αeλ+θ+c2l+αcosλ,c1r+αeλ+θ+c2r+αcosλ,ϕ1,ϕ2λ,θα=c1l−αeλ+θ+c2l−αcosλ,c1r−αeλ+θ+c2r−αcosλ.Now we denote(89)c1l+αeλ+θ+c2l+αcosλ,c1r+αeλ+θ+c2r+αcosλ=Nα,c1l−αeλ+θ+c2l−αcosλ,c1r−αeλ+θ+c2r−αcosλ=Nα,c1l+αeλ+θ,c1r+αeλ+θ=Nα′,c1l−αeλ+θ,c1r−αeλ+θ=N′α.It easy to see thatNα,Nα and Nα′,N′α satisfy the conditions of Proposition 1, and by Lemma 1, we can build the intuitionistic fuzzy solution ϕ1,ϕ2λ,θ∈IF1 for equations (52)–(54) by the form as follows for every α∈0,1:(90)ϕ1,ϕ2λ,θα=c1l+αeλ+θ+c2l+αcosλ,c1r+αeλ+θ+c2r+αcosλ,ϕ1,ϕ2λ,θα=c1l−αeλ+θ+c2l−αcosλ,c1r−αeλ+θ+c2r−αcosλ.Then,ϕ1,ϕ2x,y is an intuitionistic fuzzy solution which verifies the initial conditions (53)–(54) and can be written as follows:(91)ϕ1,ϕ2x,y=c1eλ+θ+c2cosλ.Figures4 and 5 represent the membership and nonmembership functions of triangular intuitionistic fuzzy numbers c1=−2,0,2;−2.75,0,2.75 and c2=−1.25,0,1.25;−3,0,3 and the simulation of α-cuts of the intuitionistic fuzzy solution ϕ1,ϕ2x,y at some values of λ,θ.Figure 4
c1=−2,0,2;−2.75,0,2.75 and c2=−1.25,0,1.25;−3,0,3.Figure 5
c1=−2,0,2;−2.75,0,2.75 and c2=−1.25,0,1.25;−3,0,3.In Figure6, we show the surface of the intuitionistic fuzzy solution with triangular intuitionistic fuzzy numbers c1=−2,0,2;−2.75,0,2.75 and c2=−1.25,0,1.25;−3,0,3.Figure 6
The surface of the intuitionistic fuzzy solution.


## 5. Conclusion

This research work proposed an integral boundary value problem for an intuitionistic fuzzy partial functional differential equation with state-dependent delay. We have proved the existence-uniqueness of intuitionistic fuzzy solutions for these problems with integral boundary conditions by applying the Banach fixed point theorem. These results are illustrated by numerical examples. Studying local and nonlocal intuitionistic fuzzy delay partial differential equations is the next step that will be considered.



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