Type 2 solutions of radom fuzy wave equantion under generalized hukuhara diferntiability

) ∈ I if 
 (ii) u is (ii)gH differentiable with respect to x at ( x0,y 0) ∈ I if 
 The fuzzy (i)gH and (ii)gH derivative of u with respect to y and higher order of fuzzy 
partial derivative of u at the point ( x0,y 0) ∈ I are defined similarly. 
 Definition 2.4. [1] For any fixed x0, we say that ( x0,y ) ∈ I is a switching point for the 
differentiability of u with respect to x, if in any neighborhood V of ( x0,y ) ∈ I, there exist 
points A(x1,y ),B (x2,y ) such that x1 < x 0 < x 2 and: 
 (type I) u is (i)gH differentiable at A while u is (ii)gH differentiable at B for all y, or 
 (type II) u is (i)gH differentiable at B while u is (ii)gH differentiable at A for all y. 
 Definition 2.5. Let u: I → E be gHdifferentiable with respect to x and ∂u/∂x is 
gHdifferentiable at ( x0,y 0) ∈ I with respect to y. We say that u is gHdifferentiable of order 
 2
2 with respect to x,y in type 2 at ( x0,y 0) ∈ I, denoted by D xy u(x0,y 0), if the type of 
gHdifferentiability of both u and ∂u/∂x are different. Then: 
for all 0 ≤ α ≤ 1 . 
TẠP CHÍ KHOA HỌC −−− SỐ 18/2017 161 
3. PROBLEM FORMULATION 
 Let ( ,F,P) be a complete probability space. 
 Definition 3.1. [21] A function u:  → E is called a random fuzzy variable, if for all α 
∈ [0 ,1], the setvalued mapping uα:  → K C is a measurable multifunction, i.e { ω ∈ 
|[ u(ω)] α ∩ C 6= ∅} ∈ F for every closed set C ⊂ R. 
 Let U ⊂ Rm. A mapping u: U × → E is said to be a fuzzy stochastic process if u(.,ω ) 
is a fuzzyvalued function with any fixed ω ∈  and u(ν,. ) is a random fuzzy variable for 
any fixed ν ∈ U. 
 A fuzzy stochastic process u: U × → E is called continuous if for almost every ω ∈ 
, the trajectory u(.,ω ) is a continuous function on U with respect to metric d∞. 
 In this paper, we consider following boundary valued problem of nonlinear wave 
equations: 
 (1) 
with local condition: 
 (2) 
where ν1 and ν2 are fuzzy continuous stochastic processes satisfying: 
exists with P.1 for all y ∈ [0 ,b ] and fω(x,y, (x,y,ω )) satisfies following hypothesis: 
 (H1 ) fω(x,y, ):  → E is a random fuzzy variable for all ( x,y ) ∈ J, ∈ E, and the 
mapping fω(.,.,. ): J × E → E is a fuzzy jointly continuous mapping with P.1. 
 (H2 ) There exist a real continuous stochastic process L: J ×  → (0 ,∞) and a 
nonnegative random variable M:  → R+ such that: 
 And: 
 Here, for convenience, the formula η(ω) P.1 = (ω) means that P(ω ∈ | η(ω) = (ω)) = 1 
(or η(ω) = (ω) almost everywhere) and similarly for inequalities. Also if we have 
P(ω ∈ | u(ν,ω ) = v(ν,ω ), ∀ν ∈ U) = 1, where u,v are fuzzy stochastic processes, then we 
will write u(ν,ω ) U=P.1 v(ν,ω ) for short, similarly for the inequalities and other relations. 
162 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
 Thanks for Lemma 4.4 in [15], we have following definition. 
 Definition 3.2. A fuzzy continuous stochastic process u: J ×  → E is called a 
random fuzzy solution (in type 2) of the problem (1)(2) if it satisfies following random 
integral equation 
 (3) 
 Where 
4. MAIN RESULTS 
 Following result shows the solvability of the problem (1)(2) by using the method of 
successive approximations. 
 Theorem 4.1. Assume hypotheses (H1) and (H2) are satisfied. Moreover, assume that 
there exists a sequence u n: J ×  → E, n ∈ 0,1,2,..., defined by 
 (4) 
 in E. Then, the Problem (1)(2) has a unique random fuzzy solution (in type 2) on J × . 
 Proof . From the hypothesis, the Hukuhara ifferences 
exist with P.1 for all (x, y) ∈ J, n ∈ N, then from Theorem 5.1 in [8] we have 
 Since: 
is measurable and [ q(x,y,ω )] α is also measurable, then 
are fuzzy stochastic processes for all n ∈ N. 
TẠP CHÍ KHOA HỌC −−− SỐ 18/2017 163 
 Since f satisfies ( H1), applying to Lemma 5.3, it is easy to see that the functions 
un(.,.,ω ): J → E are continuous with P.1. Then un(x,y,ω ) are also continuous fuzzy 
stochastic processes for all n ∈ N∗. 
 We now prove that the sequence { un(x,y,ω )} is uniformly convergent with P.1 on J. 
 Denote 
 Observe that 
when ( xm,y m) → ( x,y ) with P.1 (see Lemma 5.2). Hence, Tn is a continuous function on J 
with P.1. 
 For all n > m > 0, from estimations of Lemma 5.2, we obtain 
 The almost sure convergence of the series implies that the (E,d ∞) is a complete metric 
space, there exists c ⊂  such that P(c) = 1 and for every ω ∈ c the sequence 
{un(.,.,ω )} is uniformly convergent. For ω ∈ c denote its limit by 
 Define u: J ×  → E by 
 It is easy to see that u(.,., ω) is continuous with P.1. From 
we infer that [u(x, y,.)]α is a measurable multivalued function. Therefore u is a continuous 
fuzzy stochastic process. 
164 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
 In another way, for any n ∈ N, fω(x, y, un(x, y, ω)) are continuous fuzzy stochastic 
processes and for all n > m > 0 
 Then the sequence { fω (x, y, un (x, y, ω ))} is a Cauchy sequence on J with P.1 and it 
converges to fω (x, y, u (x, y, ω )) when n → ∞ for all ( x, y ) ∈ J with P.1. Then 
 Therefore u(x,y,ω ) satisfies random fuzzy integral equation (3) or u is a random fuzzy 
solution in type 2 of the Problem (1)(2). 
 Assume that u,v: J× → E are two continuous stochastic processes which are 
solutions of the problem. Note that 
 Thanks for the Gronwall’s inequality in Lemma 5.1, we obtain: 
 (5)
 The theorem is proved completely. 
 Now we consider the Darboux problems for (1) with following local condition: 
where εk(.,ω ), k = 1 ,2, are small noisy fuzzy random variables. Following theorem gives 
continuous dependence of random fuzzy solutions to data of the problems and the stability 
of behavior of solutions. 
TẠP CHÍ KHOA HỌC −−− SỐ 18/2017 165 
 Theorem 4.2. Assume that all the hypotheses of Theorem 4.1 are satisfied. And 
assume that u (.,.,. ) is a random fuzzy solution of (1) with local boundary condition (2) and 
v(.,.,. ) is a fuzzy stochastic processes which satisfies 
 (6) 
where q (x,y,ω) = q(x,y,ω) + ε(x,y,ω), ε(x,y,ω):= ε1(x,ω) + ε2(y,ω) for all (x,y ) ∈ J. 
 Then 
 (7) 
where C is a positive constant which does not depend on u (.,.,. ) or v (.,.,. ). 
 Proof. Denote 
 P(x,y,ω) = d∞(u(x,y,ω),v (x,y,ω)) 
for ω ∈ , ( x,y ) ∈ J. It is easy to see from hypothesis (H1) that P(x,y,ω ) is a real stochastic 
process. Thanks for hypothesis (H2) we have: 
 Applying Gronwall’s inequality in Lemma 5.1 we receive 
 From (6) we have 
 Since (x, y) ∈ J, then 
 Thus (7) holds. The theorem is proved completely. 
5. APPENDIX 
 Lemma 5.1. (Gronwall’s Lemma) Let (,F,P) be a probability space, A:  → [0 ,+∞) 
be a real random variable and u,c: U ×  → R be real stochastic processes such that 
166 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
 a) u( ,,ω ) is nonnegative and continuous with P.1 on U; 
 b) c( ,,ω ) is nonnegative, locally Lebesgue integrable on U with P.1; 
 c) furthermore following inequality hold 
 (8) 
 Then we have: 
 (9) 
 Proof. Let for ( x,y ) ∈ U. 
 From (8) we have: 
is nonnegative with P.1 then v(.,.,ω ) is nondecreasing in each variable x,y and v(0 ,y,ω ) = 
A(ω). We have: 
 Therefore: 
 It follows: 
TẠP CHÍ KHOA HỌC −−− SỐ 18/2017 167 
 Or: 
 Thus: 
 It completes the proof of this lemma. 
 Lemma 5.2. Suppose that hypotheses (H1) and (H2) are satisfied. Following 
estimations hold for all n ≥ 1 
 (10) 
where u n(.,.,ω ): J → E, n ≥ 0 are defined by (4) and 
 Proof. Denote 
 By mathematical induction, we will prove (10) for every n ≥ 1. In fact, we observe that 
 Moreover, 
168 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
 Thus (10) is true for n = 1. Now, we assume that the inequality (10) is true for any 
n ≥ 1. We will prove that it is also true for n + 1. Indeed 
 Therefore (10) holds for all n + 1, the proof is completed. 
 Lemma 5.3. Under hypotheses (H1) and (H2), un(.,.,ω ): J → E, n ≥ 0 defined by (4) 
are continuous on J with P.1. 
 Proof. Indeed, u0(x,y,ω ) is natural continuous on J. Fixed ( x,y ) ∈ J, consider an 
arbitrary sequence {( xm,y m)} that converges to ( x,y ) as m → ∞. For fixed
 , there are four cases happening. 
 Case 1. When x < x m, y < y m, one has following presentation 
 (11) 
TẠP CHÍ KHOA HỌC −−− SỐ 18/2017 169 
 Case 2. If x ≥ xm, y ≥ ym then 
 Case 3. If x < x m, y ≥ ym then 
 (12) 
 Case 4. If x ≥ xm, y < y m then 
 Now for n ≥ 1, from presentation (11) in Case 1, we have 
 (13) 
 From the hypothesis (H2) and the inequality (10) in Lemma 5.2 we have 
 (14) 
 Therefore 
170 TRƯỜNG ĐẠI HỌC THỦ ĐÔ H NỘI 
 Do the same arguments to the second and the third items of (13), we receive following 
estimates for all n ∈ N∗ 
 (15) 
 Now we consider Case 3: x < x m,y ≥ ym. Using presentation (12) we have: 
 (16) 
for all n ∈ N. 
 Repeating all the arguments in (15) and (16) for Case 2 and Case 4, we receive the 
same estimations. Now let ( xm,y m) tends to ( x,y ) then | x − xm|,|y − ym| tend to zero, too. It 
implies from (15) and (16) that for every n ∈ N, functions un(.,ω ): J → E are continuous 
with P.1. 
6. CONCLUSION 
 Random fuzzy local boundary valued problems for partial hyperbolic equations are 
studied under gHdifferentiability. We prove the existence and uniqueness of random fuzzy 
solutions in type 2. The uniqueness here is understood that each considering solution does 
not have switching points. The method of successive approximations is used instead of 
applying the frequently used fixed point method, which were applied in [13][20]. This 
research provides the foundations for the further studying on the asymptotic behavior of 
random fuzzy 135 solutions of partial differential equations. 
TẠP CHÍ KHOA HỌC −−− SỐ 18/2017 171 
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 NGHIM LOI 2 CA PHƯƠNG TRÌNH TRUYN SÓNG M 
 NGU NHIÊN DƯI ĐO HÀM HUKUHARA TNG QUÁT 
 Tóm tttttt: Bài báo nghiên cu v phương trình truyn sóng m ngu nhiên dưi ño hàm 
 Hukuhara tng quát. Thông qua phương pháp xp x liên tip, s tn ti, tính duy nht và 
 s ph thuc liên tc vào các d kin ban ñu ca nghim m ngu nhiên loi 2 ñưc 
 chng minh. Khó khăn chính trong hưng nghiên cu này không ch ph thuc vào khái 
 nim ca quá trình ngu nhiên m  trong ñó yêu cu tính ño ñưc ca các hàm nhiu 
 bin ña tr, mà còn ph thuc vào các phép toán gii tích m liên quan ñn ño hàm 
 Hukuhara tng quát ca hàm m nhiu bin. Khi các khó khăn ñó ñưc gii quyt, chúng 
 ta nhn ñưc nghim m ngu nhiên có bán kính tp mc gim theo thi gian, phù hp 
 vi nhiu bài toán ñt ra trong thc t. 
 TTTT kkhhóóaakhóa:khóa Phương trình truyn sóng ngu nhiên, ño hàm gH, b ñ Gronwall, s tn ti, 
 tính duy nht, tính gii ñưc, tính b chn, nghim m.