# Quadrature Rules and Iterative Method for Numerical Solution of Two-Dimensional Fuzzy Integral Equations

Authors: S. M. Sadatrasoul; R. Ezzati
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We introduce some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy-number-valued functions. We also give error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, we propose an iterative procedure based on quadrature formula to solve two-dimensional linear fuzzy Fredholm integral equations of the second kind (2DFFLIE2), and we present the error estimation of the proposed method. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.


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## Body

## 1. Introduction

The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh and others. The topic of fuzzy integrations is discussed in [1]. The Henstock and Riemann integral for fuzzy-number-valued functions was introduced and studied in [2, 3]. Their numerical computation was also proposed; see, for example, [3–6]. In [6], the authors obtained the upper estimates of error of some fuzzy quadrature rules for mappings of bounded variation and of Lipschitz type and gave some applications. In [7], the authors studied the Gaussian quadrature rules for fuzzy integrals. Also, in [8], Wu presented some optimal fuzzy quadrature formula for classes of fuzzy-number-valued functions of Lipschitz type. To study other works, see [9–12].Since many real-valued problems in engineering and mechanics can be brought in the form of two-dimensional fuzzy integral equations, it is important that we develop quadrature rules and numerical methods for such integral equations. In this paper, we introduce two-dimensional fuzzy integrals and propose some generalized quadrature rules and their dependent theorems for mappings of bounded variation. Also, we present the conditions for existence of unique solution for2DFFLIE2. Finally, we introduce an iterative method for solving2DFFLIE2. The rest of the paper is organized as follows. In Section 2, we give basic information about the fuzzy set theory and develop them to two-dimensional space. Also, we define two-dimensional fuzzy integral equation and some other properties of it in this section. In Section 3, we derive the proposed method to obtain numerical solutions of2DFFLIE2 based on an iterative procedure. The error estimation of the introduced method is presented in Section 4 in terms of uniform modulus of continuity to prove the convergence of the method. Some numerical experiments are presented in Section 5.


## 2.  Preliminaries

In this section, we review some necessary basic definitions on fuzzy numbers, fuzzy-number-valued functions, and fuzzy integrals.Definition 1 (see [13, 14]).
A fuzzy number is a functionu:R→[0,1] having the following properties: (i)
uis normal; that is, ∃x0∈R, such that u(x0)=1;(ii)
u is fuzzy convex set (i.e., u(λx+(1-λ)y)≥min⁡{u(x),u(y)}, for all x,y∈R,λ∈[0,1]);(iii)
u is upper semicontinuous on R;(iv)
the support{x∈R:u(x)>0}¯ is a compact set, where A¯ denotes the closure of A.
The set of all fuzzy numbers is denoted byRϜ. According to [2], any real number α∈R can be interpreted as a fuzzy number α=χ{α}, and therefore R⊂RϜ. Also, the neutral element with respect to ⊕ in RϜ is denoted by 0~=χ{0}.Definition 2 (see [2, 15]).
For any0<r≤1, an arbitrary fuzzy number is represented in parametric form, by an ordered pair of functions (u_(r),u¯(r)), which satisfies the following properties: (i)
u_(r) is bounded left continuous nondecreasing function over [0,1];(ii)
u¯(r) is bounded left continuous nonincreasing function over [0,1];(iii)
u_(r)≤u¯(r).
Moreover, the addition and scalar multiplication of fuzzy numbers inRϜ are defined as follows: (i)
(1)(u⊕v)(r)=(u_(r)+v_(r),u¯(r)+v¯(r)),(ii)
(2)(λ⊙v)(r)={(λu_(r),λu¯(r))λ≥0,(λu¯(r),λu_(r))λ<0.
Also, according to [2, 16], the following algebraic properties for any u,v,w∈RϜ hold:(i)
u⊕(v⊕w)=(u⊕v)⊕w;(ii)
u⊕0~=0~⊕u=u;(iii)
with respect to0~, none of u∈(RϜ-R), u≠0~   has opposite in (RϜ,+);(iv)
(a⊕b)⊙u=a⊙u⊕b⊙u, for all a,b∈R with ab≥0 or ab≤0;(v)
a⊙(u⊕v)=a⊙u⊕a⊙v, for all a∈R;(vi)
a⊙(b⊙u)=(ab)⊙u, for all a∈R and 1⊙u=u.Definition 3 (see [2, 17]).
For arbitrary fuzzy numbersu=(u_(r),u¯(r)), v=(v_(r),v¯(r)), the quantity D(u,v)=sup⁡r∈[0,1]max⁡{|u_(r)-v_(r)|,|u¯(r)-v¯(r)|} is the distance between u and v. Also, the following properties hold [6]:(i)
(RϜ,D) is a complete metric space;(ii)
D(u⊕w,v⊕w)=D(u,v) for allu,v,w∈RϜ;(iii)
D(k⊙u,k⊙v)=|k|D(u,v) for all u,v∈RϜforallk∈R;(iv)
D(u⊕v,w⊕e)≤D(u,w)+D(v,e) for all u,v,w,e∈RϜ;(v)
D(k1⊙u,k2⊙u)=|k1-k2|D(u,0~) for all k1,k2∈R with k1k2≥0 and for all u∈RϜ.
Throughout this paper, we denote that∥·∥Ϝ=D(·,0).Theorem 4 (see [14]).
(
i
)  (RϜ,D) is a complete metric space.
(
i
i
) The pair (RϜ,D) is a commutative semigroup with 0~=χ0 zero elements but cannot be a group for pure fuzzy numbers.
(
i
i
i
)  ∥·∥Ϝ has the properties of a usual norm on RϜ; that is, ∥·∥Ϝ=0 if and only if u=0, ∥λ⊙u∥Ϝ=|λ|∥u∥Ϝ, and ∥u⊕v∥Ϝ≤∥u∥Ϝ+∥v∥Ϝ.
(
i
v
)  |∥u∥Ϝ-∥v∥Ϝ|≤D(u,v) and D(u,v)≤∥u∥Ϝ+∥v∥Ϝ for any u,v∈RϜ.In [2], the authors introduced the concept of the Henstock integral for a fuzzy-number-valued function. We present a generalized definition of this concept for two-dimensional Henstock integrability for bivariate fuzzy-number-valued functions.Definition 5.
Suppose thatf:[a,b]×[c,d]→RϜ is a bounded mapping, and then the function ω[a,b]×[c,d](f,·):R+∪0→R+ defined by
(3)ω[a,b]×[c,d](f,δ)=sup⁡{(x-s)2+(y-t)2D(f(x,y),f(s,t));hhhhhhhx,s∈[a,b];y,t∈[c,d];hhhhhh(x-s)2+(y-t)2≤δ}
is called the modulus of oscillation of f on [a,b]×[c,d].
Also, iff∈CϜ([a,b]×[c,d])  (i.e.,  f:[a,b]×[c,d]→RϜ is continuous on [a,b]×[c,d]), then ω[a,b]×[c,d](f,δ) is called uniform modulus of continuity of f. The following properties will be very useful in what follows. The proofs of these properties in one-dimensional case are presented in [14] and those in two-dimensional case will be obtained in a similar way.Theorem 6.
The following properties hold:(i)
D(f(x,y),f(s,t))≤ω[a,b]×[c,d](f,(x-s)2+(y-t)2) for any x,s∈[a,b] and y,t∈[c,d];(ii)
ω[a,b]×[c,d](f,δ) is a nondecreasing mapping in δ;(iii)
ω[a,b]×[c,d](f,0)=0;(iv)
ω[a,b]×[c,d](f,δ1+δ2)≤ω[a,b]×[c,d](f,δ1)+ω[a,b]×[c,d](f,δ2) for any δ1,δ2≥0;(v)
ω[a,b]×[c,d](f,nδ)≤nω[a,b]×[c,d](f,δ) for any δ≥0 and n∈N;(vi)
ω[a,b]×[c,d](f,λδ)≤(λ+1)ω[a,b]×[c,d](f,δ) for any δ,λ≥0.Definition 7.
Letf:[a,b]×[c,d]→RϜ, for Δxn:a=x0<x1<⋯<xn=b and Δyn:c=y0<y1<⋯<yn=d, be two partitions of the intervals [a,b] and [c,d], respectively. Let one consider the intermediate points ξi∈[xi-1,xi] and ηi∈[yj-1,yj], i=1,…,n; j=1,…,n, and δ:[a,b]→R+ and σ:[c,d]→R+. The divisions Px=([xi-1,xi];ξi), i=1,…,n, and Py=([yj-1,yj];ηj), j=1,…,n, denoted shortly by Px=(Δn,ξ) and Py=(Δn,η) are said to be δ-fine and σ-fine, respectively, if [xi-1,xi]⊆(ξi-δ(ξi),ξi+δ(ξi)) and [yj-1,yj]⊆(ηj-σ(ηj),ηj+σ(ηj)).The functionf is said to be two-dimensional Henstock integrable to I∈RϜ if for every ε>0 there are functions δ:[a,b]→R+ and σ:[c,d]→R+ such that for any δ-fine and σ-fine divisions we have D(∑i=0n‍∑j=0n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj),I)<ϵ, where ∑‍ denotes the fuzzy summation. Then, I is called the two-dimensional Henstock integral of f and is denoted by I(f)=(FH)∫cd‍∫ab‍f(s,t)dsdt.If the aboveδ and σ are constant functions, then one recaptures the concept of Riemann integral. In this case, I∈RϜ will be called two-dimensional integral of f on [a,b]×[c,d] and will be denoted by (FR)∫cd‍∫ab‍f(s,t)dsdt.Corollary 8.
In [13], the authors proved that if f∈CϜ[a,b], its definite integral exists, also (FR)∫ab‍f(t;r)dt_=∫ab‍f_(t,r)dt, and (FR)∫ab‍f(t;r)dt¯=∫ab‍f¯(t,r)dt. In a similar way, we can prove that if f∈CϜ([a,b]×[c,d]), its definite integral exists, and one has
(4)(FR)∫cd‍∫ab‍f(s,t;r)dsdt_=∫cd‍∫ab‍f_(s,t,r)dsdt,(FR)∫cd‍∫ab‍f(s,t;r)dsdt¯=∫cd‍∫ab‍f¯(s,t,r)dsdt.Theorem 9.
Iff and g are Henstock integrable mappings on [a,b]×[c,d] and if D(f(s,t),g(s,t)) is Lebesgue integrable, then
(5)D((FH)∫cd‍∫ab‍f(s,t)dsdt,(FH)∫cd‍∫ab‍g(s,t)dsdt)≤(L)∫cd‍∫ab‍D(f(s,t),g(s,t))dsdt.Proof.
In [2, 17], the authors demonstrated that for any integrable functions h,r:[α,β]→RϜ we have D((FH)∫αβ‍h(x)dx,(FH)∫αβ‍r(x)dx)≤(L)∫αβ‍D(h(x),r(x))dx, and, clearly, we obtain
(6)D((FH)∫cd‍∫ab‍f(s,t)dsdt,(FH)∫cd‍∫ab‍g(s,t)dsdt)≤(L)∫cd‍D((FH)∫ab‍f(s,t)ds,(FH)∫ab‍g(s,t)ds)dt≤(L)∫cd‍∫ab‍D(f(s,t),g(s,t))dsdt,
which completes the proof.Theorem 10.
Iff:[a,b]×[c,d]→RϜ is an integrable bounded mapping, then for any fixed u∈[a,b] and v∈[c,d] the function φuv:[a,b]×[c,d]→R+, defined by φuv(s,t)=D(f(u,v),f(s,t)), is Lebesgue integrable on [a,b]×[c,d].Proof.
Regarding [6], Lemma 1, part (ii), it is easy to see that if f is two-dimensional Henstock integrable and bounded on [a,b]×[c,d], then f-r(s,t) and f+r(s,t) as real functions of (s,t)∈[a,b]×[c,d] are two-dimensional integrable and uniformly bounded with respect to r∈[0,1]; that is, f-r(s,t) and f+r(s,t) are Lebesgue measurable (as functions of (s,t)) and uniformly bounded with respect to r∈[0,1] by
(7)φuv(s,t)=D(f(u,v),f(s,t))=sup⁡r∈[0,1]max⁡{|f-r(u,v)-f-r(s,t)|,hhhhhhhhhhhhh|f+r(u,v)-f+r(s,t)|}=sup⁡rn∈[0,1]max⁡{|f-rn(u,v)-f-rn(s,t)|,hhhhhhhhhhhhh|f+rn(u,v)-f+rn(s,t)|},
where rn,n∈N, represent all the rational numbers in [0,1]. By Lebesgue’s theorem of dominated convergence, it follows that φuv(s,t) is Lebesgue integrable on [a,b]×[c,d], and this ends the proof.Definition 11.
A functionf:[a,b]×[c,d]→RϜ is said to be bounded if there exists M such that ∥f(x,y)∥Ϝ≤M for any (x,y)∈[a,b]×[c,d].Definition 12.
A functionf:[a,b]×[c,d]→RϜ is said to be of bounded variation if
(8)sup⁡(x,y)∈[a,b]×[c,d]VΔxyn<∞,
where
(9)VΔxyn=∑i=0n-1‍∑j=0n-1‍D(f(xi+1,yj+1),f(xi,yj))
is the variation of f related to partitions Δxn, Δyn. The total variation of f is defined to be, in this case, the number
(10)⋁(f)=sup⁡(x,y)∈[a,b]×[c,d]VΔxyn∈R.
It is known also that a function of bounded variation is Riemann integrable (see [18]), so it is Henstock integrable too.Theorem 13.
(
i
) If [a,b]×[c,d]⊆[e,f]×[g,h], then ω[a,b]×[c,d](f,δ)≤ω[e,f]×[g,h](f,δ) for all δ≥0.
(
i
i
) If f is of bounded variation, then ω[a,b]×[c,d](f,δ)≤⋁(f) for all δ≥0.Proof.
(i) It is easy to see that(11)sup⁡{(x-s)2+(y-t)2D(f(x,y),f(s,t))∣x,s∈[a,b],y,t∈[c,d],hhhh(x-s)2+(y-t)2≤δ}≤sup⁡{(x-s)2+(y-t)2D(f(x,y),f(s,t))∣x,s∈[e,f],y,t∈[g,h],ggggggg(x-s)2+(y-t)2≤δ},
and, therefore, we obtain the required inequality.
(ii) Letx,s∈[a,b] and y,t∈[c,d]; assume that a<x<s<b, c<y<t<d, VΔx=a=x0<x1=x<x2=s<b, and VΔy=c=y0<y1=y<y2=t<d. Taking supremum for any x,s∈[a,b] and y,t∈[c,d] with (x-s)2+(y-t)2≤δ, we obtain the required inequality. It is obvious now that under this condition f is bounded; therefore, we obtain
(12)∥f(x,y)∥Ϝ=D(f(x,y),0~)≤D(f(x,y),f(a,c))+d(f(a,c),0~)≤⋁(f)+∥f(a,c)∥Ϝ,
which completes the proof.Definition 14.
A functionf:[a,b]×[c,d]→RϜ is said to beL-Lipschitz, if
(13)D(f(x,y),f(s,t))≤L(x-s)2+(y-t)2,
for any x,s∈[a,b] and y,t∈[c,d].Definition 15.
A functionf:[a,b]×[c,d]→RϜ is said to beM-Condition, if
(14)D(f(x,y),f(s,t))≤M(b-a)(d-c),
for any x,s∈[a,b] and y,t∈[c,d].Remark 16.
We see that iff isM-Condition function, then f is of bounded variation and
(15)⋁(f)≤M(b-a)(d-c).
Indeed, we have(16)VΔxyn=∑i=0n-1‍∑j=0n-1‍D(f(xi+1,yj+1),f(xi,yj))≤∑i=0n-1‍∑j=0n-1‍(M(xi+1-xi)(yj+1-yj))=M(b-a)(d-c),
and since
(17)⋁(f)=sup⁡(x,y)∈[a,b]×[c,d]VΔxyn,
we obtain the required result.


## 3. Quadrature Rules for Two-Dimensional (2D) Henstock Integrals

In this section, we present some quadrature rules for 2D Henstock integral. The following theorem gives a unified approach to quadrature rules in 2D Henstock integrals.Theorem 17.
Letf:[c,d]×[c,d]→RϜ be Henstock integrable, bounded mappings. Then, for any divisions a=x0<x1<⋯<xn=b and c=y0<y1<⋯<yn=d and any points ξi∈[xi-1,xi] and ηj∈[yj-1,yj], one has
(18)D(∑j=1n‍∑i=1n‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)ω[xi-1,xi]×[yj-1,yj]·(f,(xi-xi-1)2+(yj-yj-1)2).Proof.
It is known that the Henstock integrals are additive related to interval. This leads us to(19)D(∑j=1n‍∑i=1n‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))=D(∑j=1n‍∑i=1n‍(FH)∫cd‍(∑i=1n‍(FH)∫xi-1xi‍f(s,t)ds)dt,hhhhhhhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))=D((FH)∑j=1n‍∫yj-1yj‍(∑i=1n‍(FH)∫xi-1xi‍f(s,t)ds)dt,gggggghh∑j=1n‍∑i=1n‍(FH)∫yj-1yj‍(FH)∫xi-1xi‍f(ξi,ηj)dsdt),
and, by Definition 3 part (iv) and Theorem 9, we have
(20)D(∑j=1n‍(FH)∫yj-1yj‍(∑i=1n‍(FH)∫xi-1xi‍f(s,t)ds)dt,hhh∑j=1n‍∑i=1n‍(FH)∫yj-1yj‍(FH)∫xi-1xi‍f(ξi,ηj)dsdt)≤∑j=1n‍D((FH)∫yj-1yj‍∑i=1n‍(FH)∫xi-1xi‍f(s,t)dsdt,hhhhhhhhhh∑i=1n‍(FH)∫yj-1yj‍(FH)∫xi-1xi‍f(ξi,ηj)dsdt)≤∑j=1n‍∑i=1n‍D((FH)∫yj-1yj‍(FH)∫xi-1xi‍f(s,t)dsdt,hhhhhhhhhhhhhhh(FH)∫yj-1yj‍(FH)∫xi-1xi‍f(ξi,ηj)dsdt)≤∑j=1n‍∑i=1n‍(L)∫yj-1yj‍(L)∫xi-1xi‍D(f(s,t),f(ξi,ηj))dsdt.
From part (i) of Theorem6, we conclude that
(21)∑j=1n‍∑i=1n‍(L)∫yj-1yj‍(L)∫xi-1xi‍D(f(s,t),f(ξi,ηj))dsdt≤∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)ω[xi-1,xi]×[yj-1,yj]·(f,(xi-xi-1)2+(yj-yj-1)2),
which completes the proof.From the above inequality, we infer some generalization of well-known trapezoidal-type, midpoint-type, and three-point-type inequalities with error estimations.Corollary 18.
Assume thatf:[a,b]×[c,d]→RϜ is a Henstock integrable, bounded mapping. Then, with the notation
(22)ωxy¯×zt¯=ω[x,y]×[z,t](f,(y-x)2+(t-z)2),
one has(i)
(23)D((FH)∫cd(FH)∫abf(s,t)dsdt,hhhh(b-a)(d-c)⊙f(x,y)∫cd(FH))≤(x-a)(y-c)wax-×cy-+(b-x)(y-c)wab-×cy-+(x-a)(d-y)wax-×yd-+(b-x)(d-y)wxb-×yd-,for any (x,y)∈[a,b]×[c,d];(ii)
(24)D((FH)∫cd(FH)∫abf(s,t)dsdt,[(x-a)(y-c)hh⊙f(u,α)⊕(x-a)(d-y)hh⊙f(u,β)⊕(b-x)(d-y)hh⊙f(c,β)⊕(b-x)(d-y)hhhhhh⊙f(v,β)]∫cd(FH))≤(x-a)(y-c)wax-×cy-+(b-x)(y-c)wxb-×cy-+(x-a)(d-y)wax-×yd-+(b-x)(d-y)wxb-×yd-,for any x∈[a,b], y∈[c,d], u∈[a,x], v∈[x,b], α∈[c,y], and β∈[y,d];(iii)
(25)D((FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh[(α-a)(θ-c)⊙f(u,r)hh⊕(α-a)(γ-θ)⊙f(u,p)hh⊕(α-a)(d-γ)⊙f(u,z)hh⊕(β-α)(θ-c)⊙f(v,r)hh⊕(β-α)(γ-θ)⊙f(v,p)hh⊕(β-α)(d-γ)⊙f(u,z)hh⊕(b-β)(θ-c)⊙f(w,r)hh⊕(b-β)(γ-θ)⊙f(w,p)hh⊕(b-β)(d-γ)⊙f(w,z)]∫ab‍)≤(α-a)(θ-c)ωaα¯×cθ¯+(α-a)(γ-θ)ωaα¯×θγ¯+(α-a)(d-γ)ωaα¯×γd¯+(β-α)(θ-c)ωαβ¯×cθ¯+(β-α)(γ-θ)ωαβ¯×θγ¯+(β-α)(d-γ)ωαβ¯×γd¯+(b-β)(θ-c)ωβb¯×cθ¯+(b-β)(γ-θ)ωβb¯×θγ¯+(b-β)(d-γ)ωβb¯×γd¯,for any u, v, w, α, β, r, p, θ, γ, and z with a<u<α<v<β<w<b and c<r<θ<p<γ<z<d.Proof.
(i) Taking in the previous theorem thatn=2,  x1=ξ1=ξ2=x, and y1=η1=η2=y, we obtain the required inequality. Indeed,
(26)D(∑j=12‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh∑j=12‍∑i=12‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))=D(∑j=12‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhhhhhh∑j=12‍[(x-a)(yj-yj-1)⊙f(x,y)hhhhhhhhhhh⊕(b-x)(yj-yj-1)⊙f(x,ηj)]∑j=12‍)=D(∑j=12‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhhhhhh∑j=12[(x-a)(y-c)⊙f(x,y)hhhhhhhh⊕(b-x)(y-c)⊙f(x,y)hhhhhhhh⊕(x-a)(d-y)⊙f(x,y)hhhhhhhh⊕(b-x)(d-y)⊙f(x,y)]∑j=12‍)=D(∑j=12‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhhhhh∑j=12‍[(b-a)(d-c)⊙f(x,y)]∑j=12‍)≤∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)ω[xi-1,xi]×[yj-1,yj]·(f,(xi-xi-1)2+(yj-yj-1)2)=(x-a)(y-c)ωax¯×cy¯+(b-x)(y-c)ωxb¯×cy¯+(x-a)(d-y)ωax¯×yd¯+(b-x)(d-y)ωxb¯×yd¯.
(ii) Taking thatn=2, x1=x, ξ1=u, ξ2=v, y1=y, η1=α, and η2=β in Theorem 17, we obtain the required inequality. Indeed,
(27)D(∑j=12‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh∑j=12‍∑i=12‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤D((FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhhhhh∑j=12‍[(x-a)(yj-yj-1)⊙f(u,ηj)hhhhhh⊕(b-x)(yj-yj-1)⊙f(v,ηj)]∫cd‍)=D((FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhhhhhh[(x-a)(y-c)⊙f(u,α)hhhh⊕(x-a)(d-y)⊙f(u,β)hhhh⊕(b-x)(y-c)⊙f(v,α)hhhh⊕(b-x)(d-y)⊙f(v,β)]∫cd‍)≤∑j=12‍∑i=12‍(xi-xi-1)(yj-yj-1)ωxi-1,xi¯×yj-1,yj¯=(x-a)(y-c)ωax¯×cy¯+(b-x)(y-c)ωxb¯×cy¯+(x-a)(d-y)ωax¯×yd¯+(b-x)(d-y)ωxb¯×yd¯.
(iii) Consideringn=4 and performing the similar way in part (ii), it is obvious that the inequality in previous theorem becomes the inequality stated above.Corollary 19.
Letf:[a,b]×[c,d]→RϜ be a two-dimensional Henstock integrable, bounded mapping. Then, the following inequalities hold:
(28)(i)D((FH)∫cd(FH)∫abf(s,t)dsdt,hhhh(b-a)(d-c)⊙f(a+b2,c+d2))≤(b-a)(d-c)w[a,b]×[c,d](f,(b-a)(d-c)4),(ii)D((FH)∫cd(FH)∫abf(s,t)dsdt,hhhh(b-a)(d-c)4⊙[f(a,c)⊕f(a,d)ggggggggggggggggh⊕f(b,c)⊕f(b,d)]∫d(FH))≤(b-a)(d-c)w[a,b]×[c,d](f,(b-a)(d-c)4),(iii)D((FH)∫abf(s,t)dsdt,(b-a)(d-c)36⊙[f(a,c)⊕f(a,d)hhhhhhhhhhhhhhhhhhh⊕4⊙f(a,c+d2)hhhhhhhhhhhhhhhhhhh⊕4⊙f(a+b2,c)hhhhhhhhhhhhhhhhh⊕16⊙f(a+b2,a+c2)hhhhhhhhhhhhhhh⊕4⊙f(a+b2,d)hhhhhhhhhhhhhhh⊕4⊙f(b,c+d2)hhhhhhhhhhhhhihhhhh⊕f(b,c)⊕f(b,d)]∫abf(s,t)dsdt)≤(b-a)(d-c)w[a,b]×[c,d](f,(b-a)(d-c)36).Proof.
(i) If we takex=(a+b)/2 and y=(c+d)/2 in the assertion (i) of Corollary 18, we obtain the required inequality. In other words, we have
(29)D((FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhhh(b-a)(d-c)⊙f(a+b2,c+d2))≤(a+b2-a)(c+d2-c)ωa((a+b)/2¯)×c((c+d)/2)¯+(b-a+b2)(c+d2-c)ω((a+b)/2)b¯×c((c+d)/2)¯+(a+b2-a)(d-c+d2)ωa((a+b)/2)¯×((c+d)/2)d¯+(b-a+b2)(d-c+d2)ω((a+b)/2)b¯×((c+d)/2)d¯=(b-a)(d-c)4×[ωa((a+b)/2)¯×c((c+d)/2)¯+ω((a+b)/2)b¯×c((c+d)/2)¯hhh+ωa((a+b)/2)¯×((c+d)/2)d¯+ω((a+b)/2)b¯×((c+d)/2)d¯]≤(b-a)(d-c)ω[a,b]×[c,d](f,(b-a)(d-c)4).
(ii) Takingx=(a+b)/2, y=(c+d)/2, u=a, v=b, α=c, and β=d in the assertion (ii) of the previous corollary, we obtain
(30)D((FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh(a+b2-a)(c+d2-c)⊙f(a,c)h⊕(a+b2-a)(d-c+d2)⊙f(a,d)h⊕(b-a+b2)(c+d2-c)⊙f(b,c)hhh⊕(b-a+b2)(d-c+d2)⊙f(a,d))=(b-a)(d-c)4×[ωa((a+b)/2)¯×c((c+d)/2)¯+ω((a+b)/2)b¯×c((c+d)/2)¯hh+ωa((a+b)/2)¯×((c+d)/2)d¯+ω((a+b)/2)b¯×((c+d)/2)d¯]≤(b-a)(d-c)ω[a,b]×[c,d](f,(b-a)(d-c)4).
(iii) It is easy to see that the inequality follows from the corresponding assertion (iii) of the previous corollary by takingn=4, α=(5a+b)/6, β=(a+5b)/6, u=a, v=(a+b)/2, w=b, θ=(5c+d)/6, γ=(c+5d)/6, r=c, p=(c+d)/2, and z=d. Indeed, we have
(31)D((FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh(b-a)(d-c)36⊙[(a+b2,d)f(a,c)⊕f(a,d)⊕4⊙f(a,c+d2)hhhhhhh⊕4⊙f(a+b2,c)hhhhhhh⊕16⊙f(a+b2,a+c2)⊕4⊙f(a+b2,d)hhhhhhh⊕4⊙f(b,c+d2)⊕f(b,c)⊕f(b,d)]∫cd‍)≤(b-a)(d-c)ω[a,b]×[c,d](f,(b-a)(d-c)36).The next corollaries present simpler error estimation for the inequality stated in Theorem17.Corollary 20.
Letf:[a,b]×[c,d]→RϜ be a two-dimensional Henstock integrable, bounded mapping. Then, for any divisions Δxn:a=x0<x1<⋯<xn=b and Δyn:c=y0<y1<⋯<yn=d, ξi∈[xi-1,xi] and ηi∈[yj-1,yj], i=1,…,n; j=1,…,n, one has
(32)D(∑j=1n‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤(b-a)(d-c)ω[a,b]×[c,d](f,v(Δxy)),
where v(Δxy)=max⁡i,j=1,…,n{(xi-xi-1)(yj-yj-1)} is the norm of the divisions Δxn and Δyn.Proof.
Considering Theorem17 and parts (i), (ii) of Theorem 13 and by regarding the definition of ⋁(f), we infer that
(33)D(∑j=1n‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)ω[xi-1,xi]×[yj-1,yj]·(f,(xi-xi-1)2+(yj-yj-1)2)≤∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)ω[a,b]×[c,d](f,v(Δxy))=(b-a)(c-d)ω[a,b]×[c,d](f,v(Δxy)).Corollary 21.
Letf:[a,b]×[c,d]→RϜ be a two-dimensional Henstock integrable, bounded mapping. Then, for any divisions Δxn:a=x0<x1<⋯<xn=b and Δyn:c=y0<y1<⋯<yn=d, ξi∈[xi-1,xi] and ηi∈[yj-1,yj], i=1,…,n; j=1,…,n, one has
(34)D(∑j=1n‍(FH)∫cd‍(FH)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤v(Δxy)∑j=1n‍∑i=1n‍ω[a,b]×[c,d]·(f,(xi-xi-1)2+(yj-yj-1)2).Proof.
Sincev(Δxy) is the least upper bound of partitions Δxn and Δyn, we conclude that (xi-xi-1)(yj-yj-1)≤v(Δxy) for any i,j=1,n¯. Hence, the required inequality holds.Remark 22.
Iff:[a,b]×[c,d]→RϜ is a two-dimensional Riemann integrable function, it is also Henstock integrable function. Therefore, the above quadrature rules hold for Riemann integrable function too.Theorem 23.
Letf:[a,b]×[c,d]→RϜ be a mapping of bounded variation. Then, for any divisions Δxn:a=x0<x1<⋯<xn=b and Δyn:c=y0<y1<⋯<yn=d, ξi∈[xi-1,xi] and ηi∈[yj-1,yj], i=1,…,n; j=1,…,n, one has
(35)D(∑j=1n‍(FR)∫cd‍(FR)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤v(Δxy)·⋁(f).Proof.
If we defineφxy:[a,b]×[c,d]→R+ such that φxy(s,t)=D(f(s,t),f(x,y)) for any (x,y)∈[a,b]×[c,d], we see that φ is of bounded variation and we have
(36)⋁φxy(s,t)≤⋁(f),(x,y)∈[a,b]⊗[c,d];
in other words,
(37)VΔxyn(D(f(s,t),f(x,y)))=∑k=0n-1‍∑m=0n-1‍|D(f(tk+1,sm+1),f(x,y))hhhhhhfhh-D(f(tk,sm),f(x,y))|≤∑k=0n-1‍∑m=0n-1‍(D(f(tk+1,sm+1),f(x,y)))=VΔxyn(f).
Considering Theorem 13, Theorem 17, Corollary 21, and [18] and since any real valued function of bounded variation is Lebesgue integrable, we observe that
(38)∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)ω[xi-1,xi]×[yj-1,yj]·(f,(xi-xi-1)2+(yj-yj-1)2)≤v(Δxy)∑j=1n‍∑i=1n‍ω[xi-1,xi]×[yj-1,yj]·(f,(xi-xi-1)2+(yj-yj-1)2)=v(Δxy)∑j=1n‍∑i=1n‍⋁[xi-1,xi][yj-1,yj](f)=v(Δxy)·⋁(f).Theorem 24.
Iff:[a,b]×[c,d]→RϜ isL-Lipschitz mapping, then, for any divisions Δxn:a=x0<x1<⋯<xn=b and Δyn:c=y0<y1<⋯<yn=d, ξi∈[xi-1,xi] and ηi∈[yj-1,yj], i=1,…,n;j=1,…,n, one has
(39)D(∑j=1n‍(FR)∫cd‍(FR)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤L∑j=1n‍∑i=1n‍((xi-xi-1)2(yj-yj-1)2).Proof.
Analogous to the proof of Theorem17 and by definition ofL-Lipschitz mapping, we infer that
(40)D(∑j=1n‍(FR)∫cd‍(FR)∫ab‍f(s,t)dsdt,hhh∑j=1n‍∑i=1n‍(xi-xi-1)(yj-yj-1)⊙f(ξi,ηj))≤∑j=1n‍∑i=1n‍(L)∫yj-1yj‍(L)∫xi-1xi‍D(f(s,t),f(ξi,ηj))dsdt≤L∑j=1n‍∑i=1n‍(L)∫yj-1yj‍(L)∫xi-1xi‍(s-ξi)2+(t-ηj)2dsdt≤L∑j=1n‍∑i=1n‍(L)∫yj-1yj‍(L)hhhhhhhhhhhhh·∫xi-1xi‍(+(yj-yj-1)2(xi-xi-1)2hhhhhhhhhhhhhhhhhhhh+(yj-yj-1)2)1/2dsdt=L∑j=1n‍∑i=1n‍((xi-xi-1)2+(yj-yj-1)2).


## 4. 2D Fuzzy Fredholm Integral Equations

Here, we consider the two-dimensional fuzzy Fredholm integral equations as follows:(41)F(s,t)=f(s,t)⊕λ⊙∫cd‍∫ab‍K(s,t,x,y)⊙F(x,y)dxdy,
where λ>0, K(s,t,x,y) is an arbitrary positive kernel on [a,b]×[c,d]×[a,b]×[c,d] and f:[a,b]×[c,d]→RϜ. We assume that K is continuous, and therefore it is uniformly continuous with respect to (s,t). This property implies that there exists M>0 such that
(42)M=max⁡a≤s,x≤bc≤t,y≤d|K(s,t,x,y)|.Now, we will prove the existence and uniqueness of the solution of (41) by the method of successive approximations. Let X={f:[a,b]×[c,d]→RϜ;fiscontinuous} be the space of two-dimensional fuzzy continuous functions with the metric
(43)D*(f,g)=sup⁡a≤s≤bc≤t≤dD(f(s,t),g(s,t))
that is called the uniform distance between two-dimensional fuzzy-number-valued functions. We define the operator A:X→X by
(44)A(F)(s,t)=f(s,t)⊕λ⊙∫cd‍∫ab‍K(s,t,x,y)⊙F(x,y)dxdy,hhhhhhhhhhhhhhhhhh∀(s,t)∈[a,b]×[c,d],∀f∈X.Sufficient conditions for the existence of a unique solution of (41) are given in the following result.Theorem 25.
LetK(s,t,x,y) be continuous and positive for a≤s, x≤b, and c≤t,y≤d, and let f:[a,b]×[c,d]→RϜ be continuous on [a,b]×[c,d]. If B=λM(b-a)(d-c)<1, then the iterative procedure
(45)F0(s,t)=f(s,t),Fm(s,t)=f(s,t)⊕λ⊙(FR)Fm(s,t)·∫cd‍(FR)∫ab‍K(s,t,x,y)⊙Fm-1(x,y)dxdy,hhhhhFm(s,t)hhhhhhhhhhhhhhhhhhhhhhhhhhhm≥1,
converges to the unique solution F* of (41).
Moreover, the following error bound holds:(46)D*(F*,Fm)≤Bm+11-BM1,
where
(47)M1=sup⁡a≤s≤bc≤t≤d∥F(s,t)∥Ϝ.Proof.
To prove this theorem, we investigate the conditions of the Banach fixed point principle. We first show thatA maps X into X (i.e., A(X)⊂X). To the end, we show that the operator A is uniformly continuous. Since f is continuous on compact set of [a,b]×[c,d], we deduce that it is uniformly continuous, and hence for ε1>0 exists δ1>0 such that
(48)D(f(s1,t1),f(s2,t2))<ε1hhhhwhenever(t2-t1)2+(s2-s1)2<δ1,hhhhhh∀s1,s2∈[a,b],∀t1,t2∈[c,d].
As mentioned above, K also is uniformly continuous; thus, for ε2>0 exists δ2>0 such that
(49)|K(s1,t1,x,y)-K(s2,t2,x,y)|<ε2whenever(t2-t1)2+(s2-s1)2<δ2,hhhhhh∀s1,s2∈[a,b],∀t1,t2∈[c,d].
Let δ=min⁡{δ1,δ2}, s1,s2∈[a,b], and t1,t2∈[c,d], with (t2-t1)2+(s2-s1)2<δ. According to Definition 3 and Theorem 9, we obtain
(50)D(A(F)(s1,t1),A(F)(s2,t2))≤D(f(s1,t1),f(s2,t2))+D(λ⊙(FR)∫cd‍(FR)∫ab‍K(s1,t1,x,y)hhhhhhhhhhhhhhhhhhhhhh⊙F(x,y)dxdy,hhhhhhhhgλ⊙(FR)∫cd‍(FR)∫abK(s2,t2,x,y)‍hhhhhhhhjjjjjjjjjjjjjjjjjjjjjjjjj⊙F(x,y)dxdy∫cd‍)≤ε1+λ⊙(FR)∫cd‍D((FR)∫ab‍K(s1,t1,x,y)hhhhhhhhhhhhhhhhhhhhhhh⊙F(x,y)dx,hhhhhhhhhhhhhhhhhhhh(FR)∫abK(s2,t2,x,y)‍hhhhhhhhhhhhhhhhhhhhh⊙F(x,y)dx∫cd‍)dy≤ε1+λ⊙(FR)·∫cd‍(FR)∫ab‍D(K(s1,t1,x,y)⊙F(x,y),hhhhhhhhhhhhhhhhK(s2,t2,x,y)⊙F(x,y))dxdy≤ε1+λ|K(s1,t1,x,y)-K(s2,t2,x,y)|⊙(FR)∫cd‍(FR)∫ab‍(F(x,y),0~)dxdy≤ε1+λε2⊙(FR)∫cd‍(FR)∫ab‍∥F(x,y)∥Ϝdxdy≤ε1+λε2(b-a)(d-c)∥F(x,y)∥Ϝ≤ε1+λ(b-a)(d-c)M1ε2hhhhhhhwhereM1=sup⁡a≤s≤bc≤t≤d∥F(s,t)∥Ϝ,
and by choosing ε1=ε/2 and ε2=(1/2M1λ(b-a)(d-c))ε, we derive
(51)D(A(F)(s1,t1),A(F)(s2,t2))≤ε.
This shows thatA(F) is uniformly continuous for any F∈X and so continuous on [a,b]×[c,d], and hence A(X)⊂X.
Now, we prove that the operatorA is a contraction map. So, for F1,F2∈X, s∈[a,b], and t∈[c,d], we have
(52)D(A(F1)(s,t),A(F2)(s,t))≤D(f(s,t),f(s,t))+D(λ⊙(FR)∫cd‍(FR)∫ab‍K(s,t,x,y)hhhhhhhhhhhhhhhhhh⊙F1(x,y)dxdy,hhhλ⊙(FR)∫cd‍(FR)∫ab‍K(s,t,x,y)hhhhhhhhhhhhhhhhh⊙F2(x,y)dxdy∫cd‍)≤λ⊙(FR)∫cd‍(FR)∫ab‍D(K(s,t,x,y)⊙F1(x,y),hhhhhhhhhhhhhhhhhhhhhhhK(s,t,x,y)hhhhhhhhhhhhhhhhhhhhhh⊙F2(x,y))dxdy=λ|K(s,t,x,y)|⊙(FR)·∫cd‍(FR)∫ab‍D(F1(x,y),F2(x,y))dxdy≤λM⊙(FR)∫cd‍(FR)∫ab‍D(F1(x,y),hhhhhhhhhhhhhhhhhhhhhhhhF2(x,y))dxdy≤λM⊙(FR)∫cd‍(FR)∫ab‍D*(F1,F2)dxdy=λM(b-a)(d-c)D*(F1,F2)=BD*(F1,F2).
Therefore, we obtained(53)D*(A(F1)(s,t),A(F2)(s,t))≤BD*(F1,F2).
Since B<1, the operator A is a contraction on Banach space (X,D*). Consequently, Banach’s fixed point principle implies that (41) has a unique solution F* in X and we also have
(54)D(F*(s,t),Fm(s,t))≤D*(F*,Fm)≤λM(b-a)(d-c)D*(F*,Fm-1)=BD*(F*,Fm-1)≤BD*(F*,Fm)+BD*(Fm-1,Fm)≤BD*(F*,Fm)+BmD*(F0,F1);
therefore,
(55)D*(F*,Fm)≤Bm1-BD*(F0,F1);
on the other hand,
(56)D*(F0,F1)=sup⁡a≤s≤bc≤t≤dD(∫ab‍f(s,t)⊕0~,f(s,t)⊕λ⊗(FR)hhhhhhh·∫cd‍(FR)∫ab‍K(s,t,x,y)hhhhhhhhhhhhhhhhh⊙F0(x,y)dxdy∫ab‍)≤sup⁡a≤s≤bc≤t≤dλ⊙(FR)·∫cd‍(FR)∫ab‍D(0~,K(s,t,x,y)⊙F0(x,y)dxdy)≤Mλ⊙(FR)∫cd‍(FR)∫ab‍sup⁡a≤s≤bc≤t≤dD(0~,F0(x,y))dxdy=λM(b-a)(d-c)M1=M1B,
so by (55) and (56), we obtained inequality (46), which completes the proof.Now, we introduce a numerical method to solve (41). We consider (41) with continuous kernel K(s,t,x,y) having positive sign on [a,b]×[c,d]×[a,b]×[c,d] and uniform partitions
(57)Dx:a=s0<s1<s2<⋯<sn-1<sn=b,Dy:b=t0<t1<t2<⋯<tn-1<tn=d,
with si=a+ih, tj=c+jh′, where h=(b-a)/n, h′=(d-c)/n. Then, the following iterative procedure gives the approximate solution of (41) in point (s,t):
(58)u0(s,t)=f(s,t),um(s,t)=f(s,t)⊕λhh′4um(s,t)h⊙[(∑i=0n-1‍K(s,t,s0,t0)⊙um-1(s0,t0)hum(s,t)hhhh⊕K(s,t,s0,tn)⊙um-1(s0,tn)hhum(s,t)hhh⊕K(s,t,sn,t0)⊙um-1(sn,t0)hhum(s,t)hhh⊕K(s,t,sn,tn)hhum(s,t)hhh⊙um-1(sn,tn)∑i=0n-1‍)hhhhhhh⊕2(∑i=0n-1‍K(s,t,si,t0)hhhhhhum(s,t)hhh⊙um-1(si,t0)hhhum(s,t)hhhhhh⊕∑j=0n-1‍K(s,t,s0,tj)hhhhhhhhhhhhhhh⊙um-1(s0,tj)∑i=0n-1‍)hhhhhhh⊕4(∑i=0n-1‍∑j=0n-1‍K(s,t,si,tj)hhhhhum(s,t)hh⊙um-1(si,tj)∑i=0n-1‍)].
The above recursive relation can be written as follows:
(59)u0(s,t)=f(s,t),um(s,t)=f(s,t)⊕λhh′4hhhhhhh⊙∑i=0n-1‍∑j=0n-1‍(K(s,t,si,tj)⊙um-1(si,tj)hhhhhhhhhhhhhh⊕K(s,t,si,tj+1)hhhhhhhhhhhh⊙um-1(si,tj+1)hhhhhhhhhhhh⊕K(s,t,si+1,tj)hhhhhhhhhhhh⊙um-1(si+1,tj)hhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhh⊙um-1(si+1,tj+1)).


### 4.1. Error Estimation

Here, we obtain an error estimate between the exact solution and the approximate solution for the given fuzzy Fredholm integral equation (41).Theorem 26.
Consider the2DFFLIE2 (41) with continuous kernel K(s,t,x,y) having positive sign on [a,b]×[c,d]×[a,b]×[c,d] and suppose that f is continuous on [a,b]×[c,d]. If B=λM(b-a)(d-c)<1, where M=max⁡a≤s,x≤bc≤t,y≤d|K(s,t,x,y)|, then the iterative procedure (59) converges to the unique solution of (41), F, and the following error estimate holds true:
(60)D*(F,um)≤(Bm+11-m)Γ0+(B4(1-B))ω[a,b]×[c,d](f,hh′)+(μB2+4τB4M(1-B))ωst(K,h+h′),
where
(61)ωst(K,δ)=sup⁡{(s2-s1)2+(t2-t1)2|K(s1,t1,x,y)-K(s2,t2,x,y)|;hhhhh(s2-s1)2+(t2-t1)2≤δ}∀δ≥0,a≤s1,s2≤b,c≤t1,t2≤d,(62)Mk=sup⁡(s,t)∈[a,b]×[c,d]∥uk(s,t)∥Ϝ,Γk=sup⁡(s,t)∈[a,b]×[c,d]∥Fk(s,t)∥Ϝ,(63)τ=max⁡i=0,1,…,m-1{Mi},μ=max⁡i=0,1,…,m-2{Γi}.Proof.
Considering iterative procedure (59), for all  (s,t)∈[a,b]×[c,d], we have
(64)D(F1(s,t),u1(s,t))=D(f(s,t),f(s,t))+D(λhh′4λ⊙(FR)hhhhhhh·∫cd‍(FR)∫ab‍K(s,t,x,y)hhhhhhhhhhhhhh⊙F0(x,y)dxdy,hhhhλhh′4⊙∑i=0n-1‍∑j=0n-1‍[K(s,t,si,tj)⊙F0(si,tj)hhhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙F0(si,tj+1)hhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙F0(si+1,tj)hhhhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)∫cd‍hhhhhhhhhhhhhhhhh⊙F0(si+1,tj+1)]∫cd‍)=D(λ⊙∑i=0n-1‍∑j=0n-1‍(FR)hhhhhhh·∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)gghhhhhhhhhhhhhhh⊙f(x,y)dxdy,hhλhh′4⊙∑i=0n-1‍∑j=0n-1‍[K(s,t,si,tj)⊙f(si,tj)hhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙f(si,tj+1)hhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙f(si+1,tj)hhhhhjjjjjffhjhdjfjjfjgjbg⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhh⊙f(si+1,tj+1)]∑i=0n-1‍)≤λ⊙∑i=0n-1‍∑j=0n-1‍D(hh′4(FR)∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhhhhhh⊙f(x,y)dxdy,hhhhhhhhhhhhhhh′4⊙[K(s,t,si,tj)⊙f(si,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙f(si,tj+1)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙f(si+1,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhh⊙f(si+1,tj+1)]∫sisi+1‍)≤λ⊙∑i=0n-1‍∑j=0n-1‍D(hh′4(FR)∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhhhhhhhhhhh⊙f(x,y)dxdy,hhhhhhhhhhhhhhh′4⊙[K(s,t,x,y)⊙f(si,tj)hhhhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si,tj+1)hhhhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si+1,tj)hhhhhhhhhhhhhhhhhhhhhhh⊕K(s,t,x,y)hhhhhhhhhhhhhhhhhhh⊙f(si+1,tj+1)]∫sisi+1‍)+λ⊙∑i=0n-1‍∑j=0n-1‍D(hh′4[K(s,t,x,y)⊙f(si,tj)hhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si,tj+1)hhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si+1,tj)hhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si+1,tj+1)],hhhhhhhhhhhhhh′4⊙[K(s,t,si,tj)⊙f(si,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙f(si,tj+1)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙f(si+1,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhhhh⊙f(si+1,tj+1)]hh′4)≤λ⊙∑i=0n-1‍∑j=0n-1‍|K(s,t,x,y)|h×D((FR)∫sisi+1‍(FR)∫tjtj+1‍f(x,y)dxdy,hhhhhhhhhh′4⊙[f(si,tj)⊕f(si,tj+1)hhhhhhhhhhh⊕f(si+1,tj)⊕f(si+1,tj+1)]∫sisi+1‍)+λhh′4⊙∑i=0n-1‍∑j=0n-1‍[f(si+1,tj+1)D(K(s,t,x,y)⊙f(si,tj),hhhhhhhhhhhhhhhhhhggK(s,t,si,tj)⊙f(si,tj))+D(K(s,t,x,y)⊙f(si,tj+1),hhhhhhhhK(s,t,si,tj+1)⊙f(si,tj+1))+D(K(s,t,x,y)⊙f(si+1,tj),hhhhhhhhK(s,t,si+1,tj)⊙f(si+1,tj))+D(K(s,t,x,y)⊙f(si+1,tj+1),hhhhhhhhK(s,t,si+1,tj+1)⊙f(si+1,tj+1))].
Using part (ii) of Corollary19, part (v) of Definition 3, and part (i) of Theorem 13, we obtain
(65)D(F1(s,t),u1(s,t))≤λMhh′4∑i=0n-1‍∑j=0n-1‍(4ω[si,si+1]×[tj,tj+1](f,hh′4))+λhh′4∑i=0n-1‍∑j=0n-1‍[|K(s,t,x,y)-K(s,t,si,tj)|hhhhhhhhhhhhhhhhh×D(f(si,tj),0~)hhhhhhhhhhhhh+|K(s,t,x,y)-K(s,t,si,tj+1)|hhhhhhhhhhhhhhhhh×D(f(si,tj+1),0~)hhhhhhhhhhhhh+|K(s,t,x,y)-K(s,t,si+1,tj)|hhhhhhhhhhhhhhhhh×D(f(si+1,tj),0~)hhhhhhhhhhhhh+|K(s,t,x,y)-K(s,t,si,tj)|hhhhhhhhhhhhhhhhh×D(f(si+1,tj+1),0~)].
By part (ii) of Theorem6 and direct computation, it follows that
(66)D(F1(s,t),u1(s,t))≤λM(b-a)(d-c)4ω(f,hh′)+λhh′∑i=0n-1‍∑j=0n-1‍(|K(s,t,x,y)-K(s,t,si,tj)|)hh·sup⁡(s,t)∈[a,b]×[c,d]D(f(s,t),0~)≤λM(b-a)(d-c)4ω(f,hh′)+λ(b-a)(d-c)M0ωst(K,h+h′)=B4ω(f,hh′)+BMM0ωst(K,h+h′);
therefore, we obtain
(67)D(F1(s,t),u1(s,t))≤B4ω(f,hh′)+BMM0ωst(K,h+h′).
Now, sinceF2(s,t)=f(s,t)+λ⊙(FR)∫cd‍(FR)∫ab‍K(s,t,x,y)⊙F1(x,y)dxdy, we infer that
(68)D(F2(s,t),u2(s,t))=D(f(s,t),f(s,t))+λD((FR)∫cd‍(FR)∫ab‍K(s,t,x,y)⊙F1(x,y)dxdy,hhhhhhhhhhhh′4⊙∑i=0n-1‍∑j=0n-1‍[K(s,t,si,tj)⊙F1(si,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙F1(si,tj+1)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙F1(si+1,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhhhh⊙F1(si+1,tj+1)]∫cd‍)≤λ∑i=0n-1‍∑j=0n-1‍[D((FR)∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhggggggffffffffff⊙F1(x,y)dxdy,hhhhhhhhhhhhh′4⊙K(s,t,x,y)hhhhhhhhhhh⊙[F1(si,tj)⊕F1(si,tj+1)⊕F1(si+1,tj)hhhhhhhhhhhhhhhhhh⊕F1(si+1,tj+1)]∫sisi+1‍)hhhhhhhg+hh′4D(∫sisi+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhhh[F1(si,tj)⊕F1(si,tj+1)hhhhhhhhhhhhhhhh⊕F1(si+1,tj)⊕F1(si+1,tj+1)],hhhhhhhhhhhhhhhhhhhhK(s,t,x,y)hhhhhhhhhhhhhhhhhhhh×[u1(si,tj)hhhhhhhhhhhhhhhhh⊕u1(si,tj+1)⊕u1(si+1,tj)hhhhhhhhhhhhhhhhh⊕u1(si+1,tj+1)]∫sisi+1‍)hhhhhhh+hh′4D(∫sisi+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhh×[u1(si,tj)hjhhhhhhhhhhhhhhhhh⊕u1(si,tj+1)⊕u1(si+1,tj)hhhhhhhhhhhhhhhhhh⊕u1(si+1,tj+1)],hhhhhhhhhhhhhhhhhhh[u1(si+1,tj+1)K(s,t,si,tj)⊙u1(si,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙u1(si,tj+1)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙u1(si+1,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhh⊙u1(si+1,tj+1)]∫sisi+1‍)∫sisi+1‍]≤λM(b-a)(d-c)4ω[a,b]×[c,d](F1,hh′)+λM(b-a)(d-c)4[D(F1(si,tj),u1(si,tj))hhhhhhhhhhhhhhhhh+D(F1(si,tj+1),u1(si,tj+1))hhhhhhhhhhhhhhhhh+D(F1(si+1,tj),u1(si+1,tj))hhhhhhhhhhhhhhhhh+D(F1(si+1,tj+1),hhhhhhhhhhhhhhhhhhhhhhhhhhu1(si+1,tj+1))]+λ(b-a)(d-c)M1ωst(K,h+h′);
therefore, we have
(69)D(F2(s,t),u2(s,t))≤B4ω[a,b]×[c,d](F1,hh′)+B4[D(F1(si,tj),u1(si,tj))+D(F1(si,tj+1),u1(si,tj+1))+D(F1(si+1,tj),u1(si+1,tj))+D(F1(si+1,tj+1),u1(si+1,tj+1))]+λ(b-a)(d-c)M1ωst(K,h+h′).
By induction for m≥3, using (45), (46), (59), and (62), we see that
(70)D(Fm(s,t),um(s,t))≤B4ω[a,b]×[c,d](F1,hh′)+B4[D(Fm-1(si,tj),um-1(si,tj))hhhh+D(F1(si,tj+1),u1(si,tj+1))hhhh+D(Fm-1(si+1,tj),um-1(si+1,tj))hhhh+D(Fm-1(si+1,tj+1),um-1(si+1,tj+1))]+BMMm-1ωst(K,h+h′);
taking supremum for (t,s)∈[a,b]×[c,d] from (70), we conclude that
(71)D*(Fm,um)≤B4ω[a,b]×[c,d](Fm-1,hh′)+BD*(Fm-1,um-1)+BMMm-1ωst(K,h+h′),D*(Fm-1,um-1)≤B4ω[a,b]×[c,d](Fm-2,hh′)+BD*(Fm-2,um-2)+BMMm-2ωst(K,h+h′),D*(Fm-2,um-2)≤B4ω[a,b]×[c,d](Fm-3,hh′)+BD*(Fm-3,um-3)+BMMm-3ωst(K,h+h′),⋮D*(F1,u1)≤B4ω[a,b]×[c,d](F0,hh′)+BD*(F0,u0)+BMM0ωst(K,h+h′),
and multiplying the above inequalities by 1,B,B2,…,Bm-1, respectively, and summing them, we obtain
(72)D*(Fm,um)≤B4(ω[a,b]×[c,d](Fm-1,hh′)+Bω[a,b]×[c,d]hhhhhhhhhh×(Fm-2,hh′)hhhhhhhhhh+⋯+Bm-1ω[a,b]×[c,d](f,hh′))+BMωst(K,h+h′)(Mm-1+BMm-2+B2Mm-3hhhhhhhhhhhhhhh+⋯+Bm-1M0).
Since, for (s1,t1), (s2,t2)∈[a,b]×[c,d] with|s1-s2|≤h, |t1-t2|≤h′, we have
(73)D(Fm(s1,t1),Fm(s2,t2))=D(∫cd‍f(s1,t1)⊕λhhhh⊗∫cd‍∫ab‍K(s1,t1,x,y)hhhh⊗Fm-1(x,y)dxdy,hhhhf(s2,t2)⊕λ⊗∫cd‍∫ab‍K(s2,t2,x,y)hhhhhhhhhhhhhhhhhhhhh⊗Fm-1(x,y)dxdy∫cd‍)≤D(f(s1,t1),f(s2,t2))⊕λ⊗∫cd‍∫ab‍|K(s1,t1,x,y)-K(s2,t2,x,y)|hhhhhhh×D(Fm-1(x,y),0~)dxdy≤D(f(s1,t1),f(s2,t2))+BMwst(K,h+h’)Γm-1,
we infer that
(74)ω[a,b]×[c,d](Fm,hh′)≤ω[a,b]×[c,d](f,hh′)+BMωst(K,h+h′)Γm-1.
By this inequality and (72), we see that
(75)D*(Fm,um)≤B4(1+B+B2+⋯+Bm-1)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)(BΓm-2+B2Γm-3+⋯+Bm-1Γ0)+BMωst(K,h+h′)(Mm-1+BMm-2+B2Mm-3hhhhhhhhhhhhhhhh+⋯+Bm-1M0)=B4(1-Bm1-B)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)×[(BΓm-2+B2Γm-3+⋯+Bm-1Γ0)hhhh+4(Mm-1+BMm-2+B2Mm-3+⋯+Bm-1M0)].
By (62) and (63) and since B<1, we obtain
(76)D*(Fm,um)≤B4(1-Bm1-B)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)(B(1-Bm)1-Bμ+4(1-Bm)1-Bτ)≤B4(1-B)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)(μB+4τ1-B);
therefore, we obtain
(77)D*(Fm,um)≤(B4(1-B))ω[a,b]×[c,d](f,hh′)+(μB2+4τB4M(1-B))ωst(K,h+h′).
By inequalities (77) and (46), we deduce that
(78)D*(F,um)≤D*(F,Fm)+D*(Fm,um)≤(Bm+11-B)Γ0+(B4(1-B))ω[a,b]×[c,d](f,hh′)+(μB2+4τB4M(1-B))ωst(K,h+h′).Remark 27.
SinceB<1, it is easy to see that
(79)lim⁡m→∞h,h′→0D*(F,um)=0,
which shows the convergence of the method.


## 4.1. Error Estimation

Here, we obtain an error estimate between the exact solution and the approximate solution for the given fuzzy Fredholm integral equation (41).Theorem 26.
Consider the2DFFLIE2 (41) with continuous kernel K(s,t,x,y) having positive sign on [a,b]×[c,d]×[a,b]×[c,d] and suppose that f is continuous on [a,b]×[c,d]. If B=λM(b-a)(d-c)<1, where M=max⁡a≤s,x≤bc≤t,y≤d|K(s,t,x,y)|, then the iterative procedure (59) converges to the unique solution of (41), F, and the following error estimate holds true:
(60)D*(F,um)≤(Bm+11-m)Γ0+(B4(1-B))ω[a,b]×[c,d](f,hh′)+(μB2+4τB4M(1-B))ωst(K,h+h′),
where
(61)ωst(K,δ)=sup⁡{(s2-s1)2+(t2-t1)2|K(s1,t1,x,y)-K(s2,t2,x,y)|;hhhhh(s2-s1)2+(t2-t1)2≤δ}∀δ≥0,a≤s1,s2≤b,c≤t1,t2≤d,(62)Mk=sup⁡(s,t)∈[a,b]×[c,d]∥uk(s,t)∥Ϝ,Γk=sup⁡(s,t)∈[a,b]×[c,d]∥Fk(s,t)∥Ϝ,(63)τ=max⁡i=0,1,…,m-1{Mi},μ=max⁡i=0,1,…,m-2{Γi}.Proof.
Considering iterative procedure (59), for all  (s,t)∈[a,b]×[c,d], we have
(64)D(F1(s,t),u1(s,t))=D(f(s,t),f(s,t))+D(λhh′4λ⊙(FR)hhhhhhh·∫cd‍(FR)∫ab‍K(s,t,x,y)hhhhhhhhhhhhhh⊙F0(x,y)dxdy,hhhhλhh′4⊙∑i=0n-1‍∑j=0n-1‍[K(s,t,si,tj)⊙F0(si,tj)hhhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙F0(si,tj+1)hhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙F0(si+1,tj)hhhhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)∫cd‍hhhhhhhhhhhhhhhhh⊙F0(si+1,tj+1)]∫cd‍)=D(λ⊙∑i=0n-1‍∑j=0n-1‍(FR)hhhhhhh·∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)gghhhhhhhhhhhhhhh⊙f(x,y)dxdy,hhλhh′4⊙∑i=0n-1‍∑j=0n-1‍[K(s,t,si,tj)⊙f(si,tj)hhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙f(si,tj+1)hhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙f(si+1,tj)hhhhhjjjjjffhjhdjfjjfjgjbg⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhh⊙f(si+1,tj+1)]∑i=0n-1‍)≤λ⊙∑i=0n-1‍∑j=0n-1‍D(hh′4(FR)∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhhhhhh⊙f(x,y)dxdy,hhhhhhhhhhhhhhh′4⊙[K(s,t,si,tj)⊙f(si,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙f(si,tj+1)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙f(si+1,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhh⊙f(si+1,tj+1)]∫sisi+1‍)≤λ⊙∑i=0n-1‍∑j=0n-1‍D(hh′4(FR)∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhhhhhhhhhhh⊙f(x,y)dxdy,hhhhhhhhhhhhhhh′4⊙[K(s,t,x,y)⊙f(si,tj)hhhhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si,tj+1)hhhhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si+1,tj)hhhhhhhhhhhhhhhhhhhhhhh⊕K(s,t,x,y)hhhhhhhhhhhhhhhhhhh⊙f(si+1,tj+1)]∫sisi+1‍)+λ⊙∑i=0n-1‍∑j=0n-1‍D(hh′4[K(s,t,x,y)⊙f(si,tj)hhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si,tj+1)hhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si+1,tj)hhhhhhhhhhhhhhhhh⊕K(s,t,x,y)⊙f(si+1,tj+1)],hhhhhhhhhhhhhh′4⊙[K(s,t,si,tj)⊙f(si,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙f(si,tj+1)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙f(si+1,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhhhh⊙f(si+1,tj+1)]hh′4)≤λ⊙∑i=0n-1‍∑j=0n-1‍|K(s,t,x,y)|h×D((FR)∫sisi+1‍(FR)∫tjtj+1‍f(x,y)dxdy,hhhhhhhhhh′4⊙[f(si,tj)⊕f(si,tj+1)hhhhhhhhhhh⊕f(si+1,tj)⊕f(si+1,tj+1)]∫sisi+1‍)+λhh′4⊙∑i=0n-1‍∑j=0n-1‍[f(si+1,tj+1)D(K(s,t,x,y)⊙f(si,tj),hhhhhhhhhhhhhhhhhhggK(s,t,si,tj)⊙f(si,tj))+D(K(s,t,x,y)⊙f(si,tj+1),hhhhhhhhK(s,t,si,tj+1)⊙f(si,tj+1))+D(K(s,t,x,y)⊙f(si+1,tj),hhhhhhhhK(s,t,si+1,tj)⊙f(si+1,tj))+D(K(s,t,x,y)⊙f(si+1,tj+1),hhhhhhhhK(s,t,si+1,tj+1)⊙f(si+1,tj+1))].
Using part (ii) of Corollary19, part (v) of Definition 3, and part (i) of Theorem 13, we obtain
(65)D(F1(s,t),u1(s,t))≤λMhh′4∑i=0n-1‍∑j=0n-1‍(4ω[si,si+1]×[tj,tj+1](f,hh′4))+λhh′4∑i=0n-1‍∑j=0n-1‍[|K(s,t,x,y)-K(s,t,si,tj)|hhhhhhhhhhhhhhhhh×D(f(si,tj),0~)hhhhhhhhhhhhh+|K(s,t,x,y)-K(s,t,si,tj+1)|hhhhhhhhhhhhhhhhh×D(f(si,tj+1),0~)hhhhhhhhhhhhh+|K(s,t,x,y)-K(s,t,si+1,tj)|hhhhhhhhhhhhhhhhh×D(f(si+1,tj),0~)hhhhhhhhhhhhh+|K(s,t,x,y)-K(s,t,si,tj)|hhhhhhhhhhhhhhhhh×D(f(si+1,tj+1),0~)].
By part (ii) of Theorem6 and direct computation, it follows that
(66)D(F1(s,t),u1(s,t))≤λM(b-a)(d-c)4ω(f,hh′)+λhh′∑i=0n-1‍∑j=0n-1‍(|K(s,t,x,y)-K(s,t,si,tj)|)hh·sup⁡(s,t)∈[a,b]×[c,d]D(f(s,t),0~)≤λM(b-a)(d-c)4ω(f,hh′)+λ(b-a)(d-c)M0ωst(K,h+h′)=B4ω(f,hh′)+BMM0ωst(K,h+h′);
therefore, we obtain
(67)D(F1(s,t),u1(s,t))≤B4ω(f,hh′)+BMM0ωst(K,h+h′).
Now, sinceF2(s,t)=f(s,t)+λ⊙(FR)∫cd‍(FR)∫ab‍K(s,t,x,y)⊙F1(x,y)dxdy, we infer that
(68)D(F2(s,t),u2(s,t))=D(f(s,t),f(s,t))+λD((FR)∫cd‍(FR)∫ab‍K(s,t,x,y)⊙F1(x,y)dxdy,hhhhhhhhhhhh′4⊙∑i=0n-1‍∑j=0n-1‍[K(s,t,si,tj)⊙F1(si,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙F1(si,tj+1)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙F1(si+1,tj)hhhhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhhhh⊙F1(si+1,tj+1)]∫cd‍)≤λ∑i=0n-1‍∑j=0n-1‍[D((FR)∫sisi+1‍(FR)∫tjtj+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhggggggffffffffff⊙F1(x,y)dxdy,hhhhhhhhhhhhh′4⊙K(s,t,x,y)hhhhhhhhhhh⊙[F1(si,tj)⊕F1(si,tj+1)⊕F1(si+1,tj)hhhhhhhhhhhhhhhhhh⊕F1(si+1,tj+1)]∫sisi+1‍)hhhhhhhg+hh′4D(∫sisi+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhhh[F1(si,tj)⊕F1(si,tj+1)hhhhhhhhhhhhhhhh⊕F1(si+1,tj)⊕F1(si+1,tj+1)],hhhhhhhhhhhhhhhhhhhhK(s,t,x,y)hhhhhhhhhhhhhhhhhhhh×[u1(si,tj)hhhhhhhhhhhhhhhhh⊕u1(si,tj+1)⊕u1(si+1,tj)hhhhhhhhhhhhhhhhh⊕u1(si+1,tj+1)]∫sisi+1‍)hhhhhhh+hh′4D(∫sisi+1‍K(s,t,x,y)hhhhhhhhhhhhhhhhhhh×[u1(si,tj)hjhhhhhhhhhhhhhhhhh⊕u1(si,tj+1)⊕u1(si+1,tj)hhhhhhhhhhhhhhhhhh⊕u1(si+1,tj+1)],hhhhhhhhhhhhhhhhhhh[u1(si+1,tj+1)K(s,t,si,tj)⊙u1(si,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si,tj+1)⊙u1(si,tj+1)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj)⊙u1(si+1,tj)hhhhhhhhhhhhhhhh⊕K(s,t,si+1,tj+1)hhhhhhhhhhhhhhhh⊙u1(si+1,tj+1)]∫sisi+1‍)∫sisi+1‍]≤λM(b-a)(d-c)4ω[a,b]×[c,d](F1,hh′)+λM(b-a)(d-c)4[D(F1(si,tj),u1(si,tj))hhhhhhhhhhhhhhhhh+D(F1(si,tj+1),u1(si,tj+1))hhhhhhhhhhhhhhhhh+D(F1(si+1,tj),u1(si+1,tj))hhhhhhhhhhhhhhhhh+D(F1(si+1,tj+1),hhhhhhhhhhhhhhhhhhhhhhhhhhu1(si+1,tj+1))]+λ(b-a)(d-c)M1ωst(K,h+h′);
therefore, we have
(69)D(F2(s,t),u2(s,t))≤B4ω[a,b]×[c,d](F1,hh′)+B4[D(F1(si,tj),u1(si,tj))+D(F1(si,tj+1),u1(si,tj+1))+D(F1(si+1,tj),u1(si+1,tj))+D(F1(si+1,tj+1),u1(si+1,tj+1))]+λ(b-a)(d-c)M1ωst(K,h+h′).
By induction for m≥3, using (45), (46), (59), and (62), we see that
(70)D(Fm(s,t),um(s,t))≤B4ω[a,b]×[c,d](F1,hh′)+B4[D(Fm-1(si,tj),um-1(si,tj))hhhh+D(F1(si,tj+1),u1(si,tj+1))hhhh+D(Fm-1(si+1,tj),um-1(si+1,tj))hhhh+D(Fm-1(si+1,tj+1),um-1(si+1,tj+1))]+BMMm-1ωst(K,h+h′);
taking supremum for (t,s)∈[a,b]×[c,d] from (70), we conclude that
(71)D*(Fm,um)≤B4ω[a,b]×[c,d](Fm-1,hh′)+BD*(Fm-1,um-1)+BMMm-1ωst(K,h+h′),D*(Fm-1,um-1)≤B4ω[a,b]×[c,d](Fm-2,hh′)+BD*(Fm-2,um-2)+BMMm-2ωst(K,h+h′),D*(Fm-2,um-2)≤B4ω[a,b]×[c,d](Fm-3,hh′)+BD*(Fm-3,um-3)+BMMm-3ωst(K,h+h′),⋮D*(F1,u1)≤B4ω[a,b]×[c,d](F0,hh′)+BD*(F0,u0)+BMM0ωst(K,h+h′),
and multiplying the above inequalities by 1,B,B2,…,Bm-1, respectively, and summing them, we obtain
(72)D*(Fm,um)≤B4(ω[a,b]×[c,d](Fm-1,hh′)+Bω[a,b]×[c,d]hhhhhhhhhh×(Fm-2,hh′)hhhhhhhhhh+⋯+Bm-1ω[a,b]×[c,d](f,hh′))+BMωst(K,h+h′)(Mm-1+BMm-2+B2Mm-3hhhhhhhhhhhhhhh+⋯+Bm-1M0).
Since, for (s1,t1), (s2,t2)∈[a,b]×[c,d] with|s1-s2|≤h, |t1-t2|≤h′, we have
(73)D(Fm(s1,t1),Fm(s2,t2))=D(∫cd‍f(s1,t1)⊕λhhhh⊗∫cd‍∫ab‍K(s1,t1,x,y)hhhh⊗Fm-1(x,y)dxdy,hhhhf(s2,t2)⊕λ⊗∫cd‍∫ab‍K(s2,t2,x,y)hhhhhhhhhhhhhhhhhhhhh⊗Fm-1(x,y)dxdy∫cd‍)≤D(f(s1,t1),f(s2,t2))⊕λ⊗∫cd‍∫ab‍|K(s1,t1,x,y)-K(s2,t2,x,y)|hhhhhhh×D(Fm-1(x,y),0~)dxdy≤D(f(s1,t1),f(s2,t2))+BMwst(K,h+h’)Γm-1,
we infer that
(74)ω[a,b]×[c,d](Fm,hh′)≤ω[a,b]×[c,d](f,hh′)+BMωst(K,h+h′)Γm-1.
By this inequality and (72), we see that
(75)D*(Fm,um)≤B4(1+B+B2+⋯+Bm-1)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)(BΓm-2+B2Γm-3+⋯+Bm-1Γ0)+BMωst(K,h+h′)(Mm-1+BMm-2+B2Mm-3hhhhhhhhhhhhhhhh+⋯+Bm-1M0)=B4(1-Bm1-B)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)×[(BΓm-2+B2Γm-3+⋯+Bm-1Γ0)hhhh+4(Mm-1+BMm-2+B2Mm-3+⋯+Bm-1M0)].
By (62) and (63) and since B<1, we obtain
(76)D*(Fm,um)≤B4(1-Bm1-B)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)(B(1-Bm)1-Bμ+4(1-Bm)1-Bτ)≤B4(1-B)ω[a,b]×[c,d](f,hh′)+B4Mωst(K,h+h′)(μB+4τ1-B);
therefore, we obtain
(77)D*(Fm,um)≤(B4(1-B))ω[a,b]×[c,d](f,hh′)+(μB2+4τB4M(1-B))ωst(K,h+h′).
By inequalities (77) and (46), we deduce that
(78)D*(F,um)≤D*(F,Fm)+D*(Fm,um)≤(Bm+11-B)Γ0+(B4(1-B))ω[a,b]×[c,d](f,hh′)+(μB2+4τB4M(1-B))ωst(K,h+h′).Remark 27.
SinceB<1, it is easy to see that
(79)lim⁡m→∞h,h′→0D*(F,um)=0,
which shows the convergence of the method.


## 5. Numerical Experiments

The proposed iterative method of successive approximations was tested on three numerical examples to provide the accuracy and the convergence of the method and illustrate the correctness of the theoretical results. In these examples, we assumed that[a,b]×[c,d]=[0,1]×[0,1], λ=1, and we performed the algorithm in point [s0,t0]=[0.5,0.5].Example 1.
Assume that(80)F(s,t)=f(s,t)⊕∬01K(s,t,x,y)⊙F(x,y)dxdy,
where
(81)f(s,t,r)=(f_(s,t,r),f¯(s,t,r)),f_(s,t,r)=(r2+r)ssint2,f¯(s,t,r)=(4-r3-r)ssint2,K(s,t,x,y)=s2tx;
the exact solution is given by
(82)F(s,t,r)=(F_(s,t,r),F¯(s,t,r)),F_(s,t,r)=(r2+r)(ssint2-1621(cos⁡12-1)s2t),F¯(s,t,r)=(4-r3-r)(ssint2-1621(cos⁡12-1)s2t).
To obtain numerical solution, we apply the proposed method. To compare numerical and exact solutions, see Table1.Table 1
Numerical results on the level sets for Example1 in (s0,t0)=(0.5,0.5).
r-level
m
=
5, n=10
m
=
5, n=20
m
=
7, n=10
m
=
7, n=20
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
0.00
0.000000
0.000657
0.000000
0.000661
0.000000
0.000008
0.000000
0.000003
0.25
0.000067
0.000427
0.000051
0.000617
0.000011
0.000053
0.000001
0.000005
0.50
0.000086
0.000586
0.000024
0.000258
0.000066
0.000815
0.000043
0.000001
0.75
0.000150
0.000423
0.000022
0.000367
0.000069
0.000499
0.000008
0.000006
1.00
0.000229
0.000329
0.000131
0.000221
0.000154
0.000154
0.000005
0.000005Example 2.
Consider (80) with
(83)f(s,t,r)=(f_(s,t,r),f¯(s,t,r)),f_(s,t,r)=r(13r+83)(1+s+t-712st),f¯(s,t,r)=(2r2-4r+5)(1+s+t-712st),ggggggK(s,t,x,y)=stxy
and exact solution
(84)F(s,t,r)=(F_(s,t,r),F¯(s,t,r)),=(r(13r+83)(s+t+1),hhh(2r2-4r+5)(s+t+1)13).
We perform the proposed method and obtain numerical solution. Comparison of these two results is presented in Table 2.Table 2
Numerical results on the level sets for Example2 in (s0,t0)=(0.5,0.5).
r-level
m
=
4, n=10
m
=
8, n=20
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
0.0
0.000000
0.000000
0.000000
0.000000
0.2
0.000003
0.000008
0.000000
0.000002
0.4
0.000007
0.000011
0.000000
0.000008
0.6
0.000004
0.000006
0.000001
0.000000
0.8
0.000000
0.000005
0.000000
0.000000
1.0
0.000000
0.000000
0.000000
0.000000Example 3.
The integral equation (80) with
(85)f(s,t,r)=(f_(s,t,r),f¯(s,t,r)),f_(s,t,r)=(2rcos⁡(1-r)-1)(1+s2+t-1324(s+t)),f¯(s,t,r)=(2-sinrπ2)(1+s2+t-1324(s+t)),K(s,t,x,y)=(s+t)xy
has the exact solution
(86)F(s,t,r)=(F_(s,t,r),F¯(s,t,r)),=(rπ2(2rcos⁡(1-r)-1)(s2+t+1),hhh(2-sinrπ2)(s2+t+1)).
For this linear example, we apply our proposed iterative method and obtain numerical results that can be viewed in Table 3.Table 3
Numerical results on the level sets for Example3 in (s0,t0)=(0.5,0.5).
r-level
m
=
5, n=20
m
=
5, n=40
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
|
F
_
-
u
_
m
|
|
F
¯
-
u
¯
m
|
0.0
0.000047
0.000028
0.000008
0.000003
0.2
0.000039
0.000086
0.000006
0.000009
0.4
0.000009
0.000008
0.000003
0.000000
0.6
0.000004
0.000005
0.000001
0.000000
0.8
0.000012
0.000033
0.000007
0.000002
1.0
0.000006
0.000004
0.000000
0.000000


## 6.  Conclusions

In this paper, we introduced 2D fuzzy mappings and defined 2D fuzzy integrals. Quadrature rules to approximate the solution of 2D fuzzy integrals are given. We established the theorem of existence of unique solution of2DFFLIE2, and we have proved it by using Banach’s fixed point principle. Moreover, to approximate the solution of2DFFLIE2, we have proposed an iterative algorithm based on method of successive approximations. The convergence to the unique solution in our iterative method is investigated. The presented numerical experiments show that the method applies well for2DFFLIE2.



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*Source: 413570-2014-05-19.xml*