Existence and uniqueness of solutions of some general fuzzy partial hyperbolic functional differential equations

Abstract. In this paper, we investigate the existence and uniqueness of fuzzy solution for a class of general hyperbolic equations with state-dependent delays. We will prove the well-posedness of problem doesn’t depend on the domain and boundary data as well as initial data. Our method is based on Banach fixed point theorem in completely new weighted metric space. Fuzzy solutions in these cases are comprehended in the sense of Buckley and Feuring. Moreover, some examples are presented to illustrated the results. In these we use the continuity of the Zadeh’s extension principle combined with Matlab simulation to show the surface of numerical fuzzy solutions.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0028 Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 14-29 This paper is available online at EXISTENCE AND UNIQUENESS OF SOLUTIONS OF SOME GENERAL FUZZY PARTIAL HYPERBOLIC FUNCTIONAL DIFFERENTIAL EQUATIONS Nguyen Phuong Dong1, Hoang Thi Phuong Thao2, Vu Tuan3 and Pham Thi Thao3 1Hanoi Pedagogical University Number 2 2Foreign Language Specialized School, University of Languages and International Studies, VNU, Hanoi 3Faculty of Information Technology, People’s Police University of Technology and Logistics, Bac Ninh Abstract. In this paper, we investigate the existence and uniqueness of fuzzy solution for a class of general hyperbolic equations with state-dependent delays. We will prove the well-posedness of problem doesn’t depend on the domain and boundary data as well as initial data. Our method is based on Banach fixed point theorem in completely new weighted metric space. Fuzzy solutions in these cases are comprehended in the sense of Buckley and Feuring. Moreover, some examples are presented to illustrated the results. In these we use the continuity of the Zadeh’s extension principle combined with Matlab simulation to show the surface of numerical fuzzy solutions. Keywords: Partial hyperbolic Functional differential equations, fuzzy solution, local conditions, boundary conditions, contraction operator. 1. Introduction Fuzzy partial differential equations (FPDEs) were first introduced by Buckley and Feuring in [1]. They used the concepts Hukuhara-differentiability to obtain a strategy to find BF-solution or S-solution of some elementary FPDEs by crisp solution and the extension principle. Since then there appeared some papers concerning different approaches to not only the theoretical field of FPDEs, but also the applications of FPDEs, fuzzy partial differential inclusions and numerical methods for these equations. For instance, Allahviranloo et al. [2] used the same strategy of Buckley and Feuring to find the exact solutions of fuzzy wave-like equations in one and two dimensions with variable coefficients by a variational iteration method. Bertone et al. [3] applied a fuzzification process using Zadeh’s extension principle to obtain a fuzzy solution for some type of heat, wave and Poisson equations. In this paper, we will investigate the well-posedness of a general class of fuzzy hyperbolic equations with state-dependent delays, known as fuzzy partial hyperbolic functional differential Received July 6, 2016. Accepted August 5, 2016. Contact Nguyen Phuong Dong, e-mail address: phuongdongspt@gmail.com 14 Existence and uniqueness of solutions of some general fuzzy... equations (FPHFDEs), as they can provide good models for dynamics of real phenomena which are subjected to one kind of uncertainties [4, 5, 6]. This kind of equations was initialed by Arara et al. in [7], in which they considered the local and nonlocal initial problem for some classes of hyperbolic equations. However, this results based on some complicated conditions on data and domain. Generally, existence theorems need a condition which may restrict the domain to a small scale. To overcome this restriction, in this paper we introduce new weighted metric space. For which with a suitable choice of weighted numbers we prove some existence and uniqueness theorems for whole domain without any constraint on the boundary of domain. Moreover, in this paper, we introduce one class of FPHFDEs with integral boundary conditions after that we investigate some results on the existence and uniqueness of fuzzy solutions for this class of equations. As we know, integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoplasticity, underground water flow, population dynamics, and so forth. 2. Content 2.1. Preliminaries We denote the set consisting of all nonempty compact, convex subsets of Rn by KnC . Let A and B be two nonempty bounded subsets of KnC . Denote by ||.|| a norm in Rn. The distance between A and B is defined by the Hausdorff metric dH(A,B) = max { sup a∈A inf b∈B ||a− b|| , sup b∈B inf a∈A ||a− b|| } , then (KnC , dH) is a complete metric space. Let En be the space of fuzzy sets on Rn, that are nonempty subsets {(x, u(x)) : x ∈ Rn} in Rn × [0, 1] of certain functions u: Rn → [0, 1] being normal, fuzzy convex, semi-continuous and compact support, where [u]0 = {x ∈ Rn : u(x) > 0} is denoted for support of u. The α-cuts or level sets of u are defined by [u]α = {x ∈ Rn : u(x) ≥ α} for each 0 < α ≤ 1. Supremum metric on En is defined by the Hausdorff metric distance between the level sets of the fuzzy numbers d∞(u, v) = sup 0<α≤1 Hd ([u] α , [v]α) for all u, v ∈ En. It is obviously that (En, d∞) is a complete metric space. Definition 2.1. A mapping f : J = [x1, x2]× [y1, y2]→ En is called continuous at (s0, t0) ∈ J if the multi-valued mapping fα (s, t) = [f (s, t)] α is continuous at (s0, t0) with respect to the Hausdorff metric dH for all α ∈ [0, 1]. For any positive real numbers r, a, b, we denote Ja = [0, a], Jb = [0, b], J0 = [−r, 0] × [−r, 0], Jr = [−r, a]× [−r, b], J¯r = Jr\(0, a] × (0, b] and Jab = [0, a]× [0, b]. The supremum metric d0∞ on C(J0, En) the space of all continuous functions f : J0 → En is defined by d0∞(f, g) := max (ω,θ)∈J0 d∞(f(ω, θ), g(ω, θ)) 15 Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao for all f, g ∈ C(J0, En) and (ω, θ) ∈ J0. In this paper, we also denote by C(Jr, En) the space of all continuous functions f : Jr → En with the supremum weighted metric Hλ defined by Hλ(f, g) = sup (s,t)∈Jr { d∞(f (s, t) , g (s, t))e−λ(s+t) } . Since (En, d∞) is a complete metric space, it can be shown that Proposition 2.1. (C(Jr, E n),Hλ) is a complete metric space for λ > 0 arbitrary. Definition 2.2. A mapping f : J ×En → En is called continuous at point (s0, t0, u0) ∈ J ×En provided that for any fixed α ∈ [0, 1] and arbitrary ǫ > 0, there exists δ(ǫ, α) > 0 such that dH ([f (s, t, u)] α , [f (s0, t0, u0)] α) < ǫ whenever (s, t, u) ∈ J × En satisfying max { |s− s0| , |t− t0| , dH ([u]α , [u0]α) } < δ(ǫ, α). The notion of integrability considered is the Aumann-integrability defined as Definition 2.3. We say that f : J → En is strongly measurable if, for all α ∈ (0, 1], the set-valued mapping fα : J → KnC given by fα(t) = [f(t)]α is Lebesgue-measurable. The integral of f on J is defined levelsetwise by[∫ x2 x1 ∫ y2 y1 f (s, t) dsdt ]α = ∫ x2 x1 ∫ y2 y1 [f (s, t)]αdsdt = { ∫ x2 x1 ∫ y2 y1 v (s, t) dsdt|v : J → Rnis a measurable selection for [f (s, t)]α} for all α ∈ (0, 1]. We denote it by ∫ x2x1 ∫ y2y1 f (s, t) dsdt. A function f : J → En is integrable on J if ∫ x2 x1 ∫ y2 y1 f (s, t) dsdt ∈ En. Definition 2.4. Given u, v ∈ En, if there exists w ∈ En such that u = v + w, we call w = u− v the Hukuhara difference of u and v. Definition 2.5. Given mapping f : J → En, we say that f is Hukuhara partial differentiable with respect to x at (x0, y0) ∈ J if for each h > 0 the Hukuhara-difference f(x0 +∆t, y) − f(x0, y) and f(x0, y)−f(x0−∆t, y0) exists inEn for every 0 < ∆t < h and if it exists ∂f (x0, y0) ∂x ∈ En such that lim h→0+ d∞ ( f (x0 +∆t, y0)− f (x0, y0) h , ∂f (x0, y0) ∂x ) = 0 and lim h→0+ d∞ ( f (x0, y0)− f (x0 −∆t, y0) h , ∂f (x0, y0) ∂x ) = 0. In this case, ∂f (x0, y0) ∂x ∈ En is called the Hukuhara partial derivative of f at (x0, y0). The fuzzy partial derivative of f with respect to y and higher order of fuzzy partial derivative of f at the point (x0, y0) ∈ J are defined similarly. 16 Existence and uniqueness of solutions of some general fuzzy... 2.2. Fuzzy partial hyperbolic functional differential equations with local conditions For (x, y) ∈ Jab, we denote the state-dependent delays u(x,y) : J0 → En, by u(x,y)(s, t) = u(x+ s, y + t), (s, t) ∈ J0. In this section, we consider the FPHFDEs which is the generalization models of Arara et al. [7] ∂2u(x, y) ∂x∂y = ∂(p(x, y)u(x, y)) ∂y + ∂(q(x, y)u(x, y)) ∂x + c(x, y)u(x, y) + f(x, y, u(x,y)) (2.1) for (x, y) ∈ Jab with the initial condition u(x, y) = ϕ(x, y), (x, y) ∈ J¯r (2.2) and the local conditions are u(x, 0) = η1(x), x ∈ Ja, u(0, y) = η2(y), y ∈ Jb, ϕ(0, 0) = η1(0) = η2(0), (2.3) where p, q, c ∈ C(Jab,R), η1 ∈ C(Ja, En), η2 ∈ C(Jb, En) are given functions, ϕ ∈ C(J¯r, En) and f : Jab × C(J0, En)→ En is a L1− Carathéodory function. Definition 2.6. Mapping f : Jab × C(J0, En) → En is called satisfying Lipschitz condition if there exists a positive real number K such that d∞(f(x, y, u), f(x, y, v)) ≤ Kd0∞(u, v) (2.4) holds for all u, v ∈ C(J0, En), (x, y) ∈ Jab. Definition 2.7. A function u ∈ C(Jr, En) is called a solution of the problem (2.1)- (2.3) if it satisfies u(x, y) =ψ(x, y) + ∫ x 0 p(s, y)u(s, y)ds + ∫ y 0 q(x, t)(x, t)dt + ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt + ∫ x 0 ∫ y 0 f ( s, t, u(s,t) ) dsdt, where ψ(x, y) is a fuzzy function that satisfies ψ(x, y) + ϕ(0, 0) + ∫ x 0 p(s, 0)η1(s)ds + ∫ y 0 q(0, t)η2(t)dt = η1(x) + η2(y) for all (x, y) ∈ Jab and u(x, y) = ϕ(x, y), (x, y) ∈ J¯r . Theorem 2.1. Suppose that function f : Jab × C(J0, En) → En satisfies Lipschitz condition (2.4). Then the problem (2.1)- (2.3) has a unique solution in C(Jr, En). 17 Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao Proof. We transform problem (2.1)-(2.3) into a fixed point problem as flowing steps Step 1. Base on Definition 2.7, we construct an operator N1 : C(Jr, En) → C(Jr, En) defined by N1(u(x, y)) =   ϕ(x, y), if (x, y) ∈ J¯r ψ(x, y) + ∫ x 0 p(s, y)u(s, y)ds + ∫ y 0 q(x, t)u(x, t)dt + ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt + ∫ x 0 ∫ y 0 f ( s, t, u(s,t) ) dsdt, if (x, y) ∈ Jab where ψ(x, y) + ϕ(0, 0) + ∫ x 0 p(s, 0)η1(s)ds+ ∫ y 0 q(0, t)η2(t)dt = η1(x) + η2(y). It can be seen that the local conditions (2.3) imply the continuity of operator N1. Step 2.We will prove that N1 is a contraction operator. Indeed, with u, v ∈ C(Jr, En) and (x, y) ∈ Jab we have d∞(N1(u(x, y)), N1(v(x, y))) ≤ d∞( ∫ x 0 p(s, y)u(s, y)ds, ∫ x 0 p(s, y)v(s, y)ds) + d∞( ∫ y 0 q(x, t)u(x, t)dt, ∫ y 0 q(x, t)v(x, t)dt) + d∞( ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt, ∫ x 0 ∫ y 0 c(s, t)v(s, t)dsdt) + d∞( ∫ x 0 ∫ y 0 f(s, t, u(s,t))dsdt, ∫ x 0 ∫ y 0 f(s, t, v(s,t))dsdt). (2.5) For simplicity, we denote p = sup (s,t)∈Jab |p(s, t)|, q = sup (s,t)∈Jab |q(s, t)|, c = sup (s,t)∈Jab |c(s, t)|, and obtain following assessment d∞( ∫ x 0 p(s, y)u(s, y)ds, ∫ x 0 p(s, y)v(s, y)ds) ≤ sup (s,t)∈Ja×Jb |p(s, t)|d∞( ∫ x 0 u(s, y)ds, ∫ x 0 v(s, y)ds) ≤ p ∫ x 0 d∞(u(s, y), v(s, y))ds. Then we receive e−λ(x+y)d∞( ∫ x 0 p(s, y)u(s, y)ds, ∫ x 0 p(s, y)v(s, y)ds) ≤ pe−λ(x+y) ∫ x 0 d∞(u(s, y), v(s, y))e−λ(s+y)eλ(s+y)ds ≤ p λ Hλ(u, v). (2.6) By doing the same arguments, we have e−λ(x+y)d∞( ∫ y 0 q(x, t)u(x, t)dt, ∫ y 0 q(x, t)v(x, t)dt) ≤ q λ Hλ(u, v), (2.7) 18 Existence and uniqueness of solutions of some general fuzzy... and e−λ(x+y)d∞( ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt, ∫ x 0 ∫ y 0 c(s, t)v(s, t)dsdt) ≤ ce−λ(x+y) ∫ x 0 ∫ y 0 d∞(u(s, t), v(s, t))dsdt ≤ c λ2 Hλ(u, v). (2.8) Since f satisfies Lipschitz condition that leads to e−λ(x+y)d∞( ∫ x 0 ∫ y 0 f(s, t, u(s,t))dsdt, ∫ x 0 ∫ y 0 f(s, t, v(s,t))dsdt) ≤ Ke−λ(x+y) ∫ x 0 ∫ y 0 d0∞(u(s,t), v(s,t))dsdt ≤ K λ2 Hλ(u, v). (2.9) Otherwise, when (x, y) ∈ J¯r we have Hλ(N1(u)(x, y), N1(v)(x, y)) = 0. (2.10) Hence, from (2.5)- (2.10) we can see that e−λ(x+y)d∞(N1(u(x, y)), N1(v(x, y))) ≤ [p+ q λ + c+K λ2 ]Hλ(u, v) holds for all (x, y) ∈ Jr. That leads to Hλ(N1(u(x, y)), N1(v(x, y))) ≤ [p+ q λ + c+K λ2 ]Hλ(u, v), (2.11) for all u, v ∈ C(Jr, En). If we choose weighted number λ > 0 satisfying p+ q λ + c+K λ2 p+ q + √ (p+ q)2 + 4c+ 4K 2 (2.12) then N1 is a contraction operator. Step 3. By Banach’s fixed point theorem, N1 has a unique fixed point u with N1(u) = u, i.e., u is a solution of Problem (2.1)-(2.3). From Definition 2.7, we can see that the fixed point u is a unique solution of the problem (2.1)- (2.3). Example 2.1.We consider the following FPHFDEs Uxy − Uy + (x exp(−x− y)U)x + U(x, y) = exp(−x− y + 2 3 )U(x,y)(− 1 3 ,−1 3 ) + C(x+ 2) exp(x+ y) + Cx+ 1 3 C, (2.13) 19 Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao for all (x, y) ∈ J12 = [0, 1] × [0, 2], with the local conditions U(x, 0) = C(x+ 1) exp(x), U(0, y) = C exp(y), U(0, 0) = C (2.14) for x ∈ [0, 1], y ∈ [0, 2], and C is a fuzzy number in universal interval I = [0,M ],M > 0. And U(x, y) = C(x exp(y) + xy + 1) exp(x+ y) (2.15) for (x, y) ∈ J¯ 1 3 := [−1 3 , 1] × [−1 3 , 2]\(0, 1] × (0, 2]. From (2.13), we have F (x, y, U(x,y)) = exp(−x− y + 2 3 )U(x− 1 3 , y − 1 3 ) + C(x+ 2) exp(x+ y) + Cx+ 1 3 C, satisfies Lipschitz condition with K = exp( 2 3 ). Due to p = 1, q = 1 exp(1) , c = 1,K = exp( 2 3 ) we can choose λ > 1 + exp(1) + √ 1 + 2 exp(1)− 4 exp(83)− 3 exp(2) 2 exp(1) , It implies that the conditions of Theorem 3.1 are hold. Therefore, there exists a unique fuzzy solution U of this problem. The deterministic solution of the crisp hyperbolic functional PDEs corresponding to (2.13)-(2.15) is u(x, y) = g(x, y, c) = c(x+ 1) exp(x+ y). We now fuzzify this crisp solution to find fuzzy solution of FPDEs (2.13)-(2.15). To this end, let us denote h(x, y, c) = c(x+2) exp(x+ y)+ cx+ 1 3 c. We apply the fuzzification in c, and supposed that the parametric form of corresponding fuzzy number C is [C]α = [c1(α), c2(α)], α ∈ [0, 1]. By using the extension principle, we compute H from h and Y from g. We will show that Y is the fuzzy solution of this problem. Indeed, since all the partials of h and g with respect to k are positive. We see that [H(x, y, C)]α =[H1(x, y, α),H2(x, y, α)] =[c1(α)(x+ 2) exp(x+ y) + c1(α)x+ 1 3 c1(α), c2(α)(x + 2) exp(x+ y) + c2(α)x + 1 3 c2(α)] satisfied conditions h1-h3. Therefore [H(x, y, C)]α is the α−cuts of fuzzy number C(x + 2) exp(x+ y) + Cx+ 1 3 C. Similarly [Y (x, y, C)]α = [Y1(x, y, α), Y2(x, y, α)] = [c1(α)(x + 1) exp(x+ y), c2(α)(x+ 1) exp(x+ y)] 20 Existence and uniqueness of solutions of some general fuzzy... is the α−cuts of fuzzy number C(x+ 1) exp(x+ y). Define differential operator ϕ(Dx,Dy)U(x, y) =Uxy − Uy + (x exp(−x− y)U)x + U(x, y) − exp(−x− y + 2 3 )U(x− 1 3 , y − 1 3 ) and S(x, y, α) = [ϕ(Dx,Dy)Y1(x, y, α), ϕ(Dx,Dy)Y2(x, y, α)] We first check to see if Y (x, y) is differentiable. We compute [ϕ(Dx,Dy)Y1(x, y, α), ϕ(Dx,Dy)Y2(x, y, α)] which equals [c1(α)(x + 2) exp(x+ y) + c1(α)x + 1 3 c1(α), c2(α)(x + 2) exp(x+ y) + c2(α)x+ 1 3 c2(α)] which are the α-cuts of fuzzy number C(x + 2) exp(x + y) + Cx + 1 3 C. Hence,Y (x, y) is differentiable. Because all partials of h and g with respect to c are positive. Therefore, Y (x, y) is a the fuzzy solution of (2.13)-(2.15) in the sense of Buckley and Feuring (see in [2, 1]), which satisfies the boundary conditions. This solution may be written as U(x, y) = C(x+ 1) exp(x+ y). In order to visualize the surface of fuzzy solution, we represent some α−cut of the solutions by consider fuzzy number C with membership function is Laplace function [8] C(t) = 2 ( 1 + exp (π|t− c| σ √ 6 ))−1 , t > 0. We can transform this fuzzy number into parametric form [C]α = [c− σ √ 6 π ln ( 2 α − 1), c+ σ √ 6 π ln ( 2 α − 1)], α ∈ [0, 1]. The continuity of extension principle states that the α-cuts of U(x, y) = C(x+ 1) exp(x+ y) (for any fixed pair (x, y)) are [U(x, y)]α = (x+ 1) exp(x+ y) [ c− σ √ 6 π ln ( 2 α − 1), c+ σ √ 6 π ln ( 2 α − 1)]. Obviously, the deterministic solution is the preferred solution [U(x, y)]1, which means that it has membership degree 1. So the membership function of U(x, y) is U(x, y)(t) = 2 ( 1 + exp (π| t(x+1) exp(x+y) − c| σ √ 6 ))−1 . Figure 1 shows the surface of fuzzy solutions U(x, y)(t) in the whole domain, the dark area corresponds to surface of solution in phase space J0, the light area is the surface of sulution in J12. 21 Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao Figure 1. Surface of fuzzy solutions U(x, y)(t) by fuzzifying Laplacian fuzzy number C 2.3. Fuzzy solutions of fuzzy partial hyperbolic functional differential equations with boundary conditions In this section, we consider the general FPHFDEs in the form ∂2u(x, y) ∂x∂y = ∂(p(x, y)u(x, y)) ∂y + ∂(q(x, y)u(x, y)) ∂x + c(x, y)u(x, y) + f(x, y, u(x,y)) (2.16) for (x, y) ∈ Jab, with the following integral boundary conditions u(x, 0) + ∫ y 0 k1(x)u(x, t)dt = g1(x, y), (x, y) ∈ Jab (2.17) u(0, y) + ∫ x 0 k2(y)u(s, y)ds = g2(x, y), (x, y) ∈ Jab (2.18) and u(x, y) = ϕ(x, y), (x, y) ∈ J¯r (2.19) where p, q, c ∈ C(Jab,R), g1, g2 ∈ C(Jab, En), k1 ∈ C(Ja, En), k2 ∈ C(Jb, En) are given functions, , ϕ ∈ C(J¯r, En) and f : Jab × C(J0, En)→ En is a L1− Carathéodory function and satisfies the condition L. Definition 2.8. A function u ∈ C(Jr, En) is called a solution of the problem (2.16) - (2.19) if it satisfies integral equation u(x, y) = φ(x, y)− k1(0) ∫ y 0 ∫ x 0 k2(t)u(s, t)dsdt− ∫ y 0 k1(x)u(x, t)dt − ∫ x 0 k2(y)u(s, y)ds + ∫ x 0 p(s, y)u(s, y)ds − ∫ x 0 p(s, 0)u(s, 0)ds + ∫ y 0 q(x, t)u(x, t)dt − ∫ y 0 q(0, t)u(0, t)dt + ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt + ∫ x 0 ∫ y 0 f(s, t, u(s,t))dsdt 22 Existence and uniqueness of solutions of some general fuzzy... where φ(x, y) is a fuzzy function that satisfies φ(x, y) + g1(0, y) = g1(x, y) + g2(x, y) + k1(0) ∫ y 0 g2(x, t)dt, for all (x, y) ∈ Jab and u(x, y) = ϕ(x, y), (x, y) ∈ J¯r . Theorem 2.2. The problem (2.16) - (2.19) has a unique solution in C(Jr, En) if the function f : Jab × C(J0, En)→ En satisfies the condition L. Proof. By doing the same arguments in Theorem 2.1 we transform problem (2.16) - (2.19) into fixed point problem. Step 1. Consider operator N2 : C(Jr, En)→ C(Jr, En) defined by N2(u(x, y)) =   ϕ(x, y) if (x, y) ∈ J¯r φ(x, y) − k1(0) y∫ 0 x∫ 0 k2(t)u(s, t)dsdt − y∫ 0 k1(x)u(x, t)dt − x∫ 0 k2(y)u(s, y)ds + x∫ 0 p(s, y)u(s, y)ds − x∫ 0 p(s, 0)u(s, 0)ds + y∫ 0 q(x, t)u(x, t)dt − y∫ 0 q(0, t)u(0, t)dt + x∫ 0 y∫ 0 c(s, t)u(s, t)dsdt+ x∫ 0 y∫ 0 f ( s, t, u(s,t) ) dsdt if (x, y) ∈ Jab where φ(x, y) + g1(0, y) = g1(x, y) + g2(x, y) + k1(0) ∫ y 0 g2(x, t)dt, Step 2. We will prove that N2 is a contraction operator. In fact, for u, v ∈ C(Jr, En) and 23 Nguyen Phuong Dong, Vu Tuan and Hoang Thi Phuong Thao when (x, y) ∈ Jab we have d∞(N2(u(x, y)), N2(v(x, y)) ≤ d∞( ∫ y 0 k1(x)u(x, t)dt, ∫ y 0 k1(x)v(x, t)dt) + d∞( ∫ x 0 k2(y)u(s, y)ds, ∫ x 0 k2(y)v(s, y)ds) + d∞(k1(0) ∫ y 0 ∫ x 0 k2(t)u(s, t)dsdt, k1(0) ∫ y 0 ∫ x 0 k2(t)v(s, t)dsdt) + d∞( ∫ x 0 p(s, y)u(s, y)ds, ∫ x 0 p(s, y)v(s, y)ds) + d∞( ∫ x 0 p(s, 0)u(s, 0)ds, ∫ x 0 p(s, 0)v(s, 0)ds) + d∞( ∫ y 0 q(x, t)u(x, t)dt, ∫ y 0 q(x, t)v(x, t)dt) + d∞( ∫ x 0 q(0, t)u(0, t)dt, ∫ y 0 q(0, t)v(0, t)dt) + d∞( ∫ x 0 ∫ y 0 c(s, t)u(s, t)dsdt, ∫ x 0 ∫ y 0 c(s, t)v(s, t)dsdt) + d∞( ∫ x 0 ∫ y 0 f(s, t, u(s,t))dsdt, ∫ x 0 ∫ y 0 f(s, t, v(s,t))dsdt) (2.20) Let p = sup (s,t)∈Jab |p(s, t)|, q = sup (s,t)∈Jab |q(s, t)|, c = sup (s,t)∈Jab |c(s, t)|, k1 = sup x∈Ja |k1(x)| and k2 = sup y∈Jb |k2(y)|. First of all, we estimate the first term in the right side of (2.20) after multiplying with e−λ(x+y) e−λ(x+y)d∞( ∫ y 0 k1(x)u(x, t)dt, ∫ y 0 k1(x)v(x, t)dt) ≤ k1e−λ(x+y) ∫ y 0 d∞(u(x, t), v(x, t))dt ≤ k1 λ Hλ(u, v)e −λ(x+y)(eλ(x+y) − eλx) ≤ k1 λ Hλ(u, v) (2.21) Similarly, we obtai