ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Existence and Uniqueness of The Solution of Nonlinear Volterra Fuzzy Integral Equations Eman A. Hussain Ayad W. Ali Department of Mathematics/College of Science/ University of AL-Mustansiriyah Abstract In this paper, we proved the existence and uniqueness of the solution of nonlinear Volterra fuzzy integral equations of the second kind. Keywords: Volterra fuzzy integral equations. 348 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Introduction The concept of integration of fuzzy functions has been introduced by Dubois and Prade [1], Goetschel and Voxman [2], Kaleva [3] and others. However, if the fuzzy function is continuous, all the various procedures yield the same result. The fuzzy integral is applied in fuzzy integral equations, such that there is a growing interest in fuzzy integral equations particularly in the past decade. The fuzzy integral equations have been studied by authors of [4,5,6,7] and others. In this paper, the existence and uniqueness theorem is proved for nonlinear Volterra fuzzy integral equation under the Lipschitz condition and arbitrary kernels by means of the successive iterations involving fuzzy set-valued function of a real variable where values are normal, convex, upper semi continuous and compactly supported fuzzy sets in Rn. The authors of [8] proved the existence and uniqueness of the solution of linear Volterra fuzzy integral equations of the second kind. P. Prakash and V. Kalaiselvi [9] proved the existence and uniqueness of the solution of nonlinear Volterra fuzzy integral equations with infinite delay of the form ú(x) = f�x, u(x)� + ∫ g(x, t, u(t)) x −∞ dt, x ∈ J = (−∞, ∞) (1.1) where f: J × En → En and g: J × J × En → En are levelwise continuous and satisfy the generalized Lipschitz condition. K. Balachandran and K. Kanagarajan in [10] proved the existence and uniqueness of the solution of general nonlinear Volterra-Fredholm fuzzy integral equations of the form u(x) = F(x, u(x), ∫ f1(x, t, u(t) x 0 dt, … , ∫ fm(x, t, u(t) x 0 dt, (1.2) ∫ g1(x, t, u(t) b 0 dt, … , ∫ gm(x, t, u(t) b 0 dt) , 0 ≤ x ≤ b, The purpose of this paper is to prove the existence and uniqueness of the solution of nonlinear Volterra fuzzy integral equations of the second kind u(x) = f(x) + ∫ K�x, t, u(t)�dt x a (1.3) where f is fuzzy continuous function on I = [a, b], K is continuous fuzzy function over the region ∆= I × I × En = {(x, t, u(t))|a ≤ t ≤ x ≤ b, u(t) ∈ En} and u(x) is the solution of equation (1.3) to prove its existence and uniqueness. Preliminaries By PK(Rn), we denote the family of all nonempty compact convex subsets of Rn. Let I = [a, b] be a compact interval and denote [3] En = {p ∶ Rn → [0,1]} such that p satisfies (i) through (iv) below i) p is normal i.e. there exists an x0 ∈ Rn such that p(x0) = 1, ii) p is fuzzy convex, iii) p is upper semi continuous, i.e the α − level sets [p]α are closed for each α ∈ [0,1], iv) [p]0 = cl{x ∈ Rn|p(x) > 0} is compact. where the α − level sets [p]α is defined by [p]α = {x ∈ Rn|p(x) ≥ α} for 0 < α ≤ 1 and [p]0 for α = 0. Then from (i)-(iv), it follows that [p]α ∈ PK(Rn) for all 0 ≤ α ≤ 1. 349 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 If g: Rn × Rn → Rn is a function, then using Zadeh's extension principle, we can extend g to g:� En × En → En by the relation g�(p, q)(z) = sup z=g(x,y) min {p(x), q(y)} for each p, q ∈ En, 0 ≤ α ≤ 1 and continuous function g. It is well known that [g(p, q)]α = g([p]α, [q]α) Moreover, we have [p + q]α = [p]α + [q]α, [kp]α = k[p]α, where k ∈ R. Define D: En × En → R+ by D(p, q) = sup 0≤α≤1 d([p]α, [q]α), where d is the Hausdorf metric defined in PK(Rn) by d(A, B) = max {sup x∈A inf y∈B |x − y|, sup y∈B inf x∈A |x − y|} for each A, B ∈ PK(Rn), then D is a metric in En. Definition 2.1 [3] A function F: I → En is called strongly measurable, if for all α ∈ [0,1] the set-valued function Fα: I → PK(Rn) is defined by Fα(x) = [F(x)]α is Lebesgue measurable, where PK(Rn) has the topology induced by the Hausdorf metric d. Definition 2.2 [3] A function F: I → En is called integrably bounded if there exists an integrable function h such that ‖y‖ < h(x)for all y ∈ F0(x). Definition 2.3 [3] Let F: I → En. The integral of F over I, denoted by ∫ F(x) dxI or ∫ F(x) dx b a , is defined levelwise by [� F(x)dx I ]α = � Fα(x)dx I = {� f(x)dx I |f: I → Rn is a measurable function for Fα} for all 0 ≤ α ≤ 1. Definition 2.4 [3] A function F: I → En is called levelwise continuous at t0 ∈ I if the set- valued function Fα(x) = [F(x)]α is continuous at t = t0 with respect to the Hausdorf metric d for all α ∈ [0,1]. Proposition 2.1 [3] Let F, G: I → En be integrable and θ ∈ R. Then 1. ∫(F + G) = ∫ F + ∫ G, 2. ∫θF = θ∫ F, 3. D(F, G) is integrable, 350 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 4. D(∫ F , ∫ G) ≤ ∫ D(F, G). Theorem 2.1 [3,11] For any p, q, r, s ∈ En and θ ∈ R, then the following hold • (En, D) is a complete metric space, • D(θp, θq) = |θ|D(p, q), • D(p + r, q + r) = D(p, q), • D(p + q, r + s) ≤ D(p, r) + D(q, s). Definition 2.5 [12] A function F: I → En is called bounded if there exists a constant M > 0 such that D(F(x), 0�) ≤ M for all x ∈ I. Definition 2.6 [7] A function F: I → En is said to be continuous if for arbitrary fixed x0 ∈ I and ε > 0 there exists δ > 0 such that if |x − x0| < δ, then D(F(x), F(x0)) < ε for each x ∈ I. Main Results Theorem 3.1 (Existence and uniqueness) Assume the following conditions are satisfied i) nEbaf →],[: is continuous and bounded, ii) nEK →∆: is a continuous function, iii) if nEbavu →],[:, are continuous, then the Lipschitz condition ( ) ))(),(())(,,()),(,,( xvxuLDtvtxKtutxKD ≤ (3.1) is satisfied, with ab L − << 1 0 . where ∆= I × I × En = {(x, t, u(t))|a ≤ t ≤ x ≤ b, u(t) ∈ En}. Then there exists a unique fuzzy solution )(xu of (1.3) and the successive iterations 0,))(,,()()( )()( 1 1 11 ≥+= = ∑∫ + = −+ ndttutxKxfx xfx n i x a in o ψ ψ (3.2) are uniformly convergent to )(xu on ],[ ba ; where 1,))(,,()( )()( 1 ≥= = ∫ − ndttutxKxu xfxu x a nn o (3.3) First we prove the following Lemma. Lemma 3.1 If the conditions of Theorem (3.1) are hold and nu is given by (3.3) then for 0≥n I) )(xun is bounded, II) )(xun is continuous. 351 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 Proof: I) Clearly )()( xfxuo = is bounded by part (i) of theorem (3.1). Assume )(1 xun− is bounded. From (3.1), (3.3) and proposition (2.1) we have ( ) ),0 ~ ),((sup)( )0 ~ ),((sup )0 ~ ,)(( 0 ~ )),(,,(0 ~ ,))(,,()0 ~ ),(( 1 ],[ 1 ],[ 1 11 xuDLab dtxuDL dtxuDL dttutxKDdttutxKDxuD n bax x anbax x a n x a n x a nn − ∈ − ∈ − −− −≤ ≤ ≤ ≤    = ∫ ∫ ∫∫ where 0 ~ is the zero function. Hence by induction )(xun is bounded. II) To prove the continuity of )(xun , we suppose bxxa ≤≤≤ ˆ , hence by proposition (2.1) and theorem (2.1) we have ( ) ( )dttutxKDdttutxKtutxKD dttutxKDdttutxKdttutxKD dttutxKdttutxKdttutxKD dttutxKdttutxKDxuxuD x x n x a nn x x n x a n x a n x x n x a n x a n x a n x a nnn ∫∫ ∫∫∫ ∫∫∫ ∫∫ −−− −−− −−− −− +≤     +    ≤      +=     = ˆ 111 ˆ 111 ˆ 111 ˆ 11 0 ~ )),(,,ˆ())(,,ˆ()),(,,( 0 ~ ,))(,,ˆ())(,,ˆ(,))(,,( ))(,,ˆ())(,,ˆ(,))(,,( ))(,,ˆ(,))(,,())ˆ(),(( ( ) ( ) ( ) ( )0~),(sup)ˆ())(,,ˆ()),(,,(sup)( 0 ~ ),())(,,ˆ()),(,,(sup)( 1 ],[ 11 ],[ ˆ 111 ],[ xuDxxLtutxKtutxKDab dtxuLDtutxKtutxKDab n bax nn bat x xnnnbat − ∈ −− ∈ −−− ∈ −+−≤ +−≤ ∫ Since K is continuous, we obtain 0))ˆ(),(( →xuxuD nn as xx ˆ→ . Thus )(xun is continuous on ],[ ba . Proof of Theorem (3.1) We shall prove that all )(xnψ , 0≥n are bounded on ],[ ba . It is clear that )()( xfxo =ψ is bounded by the assumption. Suppose that )(1 xn−ψ is bounded. From (3.2) and theorem (2.1) we have       ++=       += ∫∑∫ ∑∫ − − = − = − 0 ~ ,))(,,())(,,()( 0 ~ ,))(,,()()0 ~ ),(( 1 1 1 1 1 1 x a n n i x a i n i x a in dttutxKdttutxKxfD dttutxKxfDxD ψ 352 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ( ) )0 ~ ),(()0 ~ ),(( 0 ~ ,))(,,(0 ~ ),( 0 ~ ,))(,,()( 1 11 11 xuDxD dttutxKDxD dttutxKxD nn x a nn x a nn +=     +≤      += − −− −− ∫ ∫ ψ ψ ψ From induction and Lemma (3.1) part (I) we have that )(xnψ is bounded. Consequently, ∞ =0)}({ nn xψ is a sequence of bounded functions on ],[ ba . In the following, we prove that )(xnψ are continuous on ],[ ba . By Lemma (3.1) part (II) and theorem (2.1) and proposition (2.1) for bxxa ≤≤≤ ˆ , we have ( ) ( ) )0 ~ ,))(,,ˆ((sup)ˆ( ))(,,ˆ(,))(,,(sup)()ˆ(),( 0 ~ ,))(,,ˆ( ))(,,ˆ(,))(,,()ˆ(),( 1 1 ],[ 1 1 1 1 ],[ ˆ 1 1 1 1 1 1 ∑ ∑∑ ∫ ∑ ∫ ∑∑ = − ∈ = − = − ∈ = − = − = − −+       −+≤       +       +≤ n i i bat n i i n i i bat x x n i i x a n i i n i i tutxKDxx tutxKtutxKDabxfxfD dttutxKD dttutxKtutxKDxfxfD Finally we obtain 0))ˆ(),(( →xxD nn ψψ as xx ˆ→ . Therefore the sequence ∞=0)}({ nn xψ is continuous on ],[ ba . To prove uniform convergence of the sequence ∞=0)}({ nn xψ , for 0≥n we have ( )      +=       ++=       += ∫ ∑ ∫∫ ∑∫ = − + = −+ )(,))(,,()( )(,))(,,())(,,()( )(,))(,,()()(),( 1 1 1 1 11 xdttutxKxD xdttutxKdttutxKxfD xdttutxKxfDxxD n x a nn n n i x a n x a i n n i x a inn ψψ ψ ψψψ ( ) ( ) ( ) ( )       +       +≤       ++=       +≤       ++= ∑∫ ∑∫∑∫ ∑∫∑∫∑∫ ∑∫∑∫ ∑∫∑∫ = − = − = − = − = − = − = − = − = − = − 0 ~ ,))(,,ˆ( ))(,,ˆ(,))(,,()ˆ(),( ))(,,ˆ())(,,ˆ(,))(,,()ˆ(),( ))(,,ˆ(,))(,,()ˆ(),( ))(,,ˆ()ˆ(,))(,,()()ˆ(),( 1 ˆ 1 1 1 1 1 1 ˆ 1 1 1 1 1 1 ˆ 1 1 1 1 ˆ 1 1 1 n i x x i n i x a i n i x a i n i x x i n i x a i n i x a i n i x a i n i x a i n i x a i n i x a inn dttutxKD dttutxKdttutxKDxfxfD dttutxKdttutxKdttutxKDxfxfD dttutxKdttutxKDxfxfD dttutxKxfdttutxKxfDxxD ψψ 353 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 ( ) ( ) ( ).0~),(sup)( 0 ~ ),( 0 ~ )),(,,( 0 ~ ,))(,,( ],[ xuDLab dtxuLD dttutxKD dttutxKD n bax x a n x a n x a n ∈ −≤ ≤ ≤     = ∫ ∫ ∫ Hence we obtain )0 ~ ),((sup)())(),((sup ],[ 1 ],[ xuDLabxxD n bax nn bax ∈ + ∈ −≤ψψ (3.4) On the other hand, by (3.1) we can obtain for 1≥n , ( ) )0 ~ ),(()( 0 ~ )),(,,( 0 ~ ,))(,,()0 ~ ),(( 1 1 1 xuLDab dttutxKD dttutxKDxuD n x a n x a nn − − − −≤ ≤     = ∫ ∫ by the same way we have )0 ~ ),(()()0 ~ ),(( 21 xuLDabxuD nn −− −≤ Thus we obtain )0 ~ ),((sup}){( )0 ~ ),((}){()0 ~ ),((}){( . . . )0 ~ ),((}){( )0 ~ ),(()()0 ~ ),(( ],[ 0 2 2 1 xfDLab xfDLabxuDLab xuDLab xuLDabxuD bax n nn n nn ∈ − − −≤ −=−≤ −≤ −≤ this implies that n n bax LabQxuD }){()0 ~ ),((sup ],[ −≤ ∈ (3.5) where )0 ~ ),((sup ],[ xfDQ bax∈ = . For 0≥n , from (3.4) and (3.5) we obtain 1 1 ],[ 1 }){())(),((sup))(),(( + + ∈ + −≤≤ n nn bax nn LabQxxDxxD ψψψψ The series ∑ ∞ = −− 0 }){()( n nLabLabQ is convergent, hence the series ∑ ∞ = + 0 1 ))(),(( n nn xxD ψψ is convergent uniformly on ],[ ba by the comparison test, this implies the uniform convergence 354 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 of the sequence ∞=0)}({ nn xψ . If we denote )(lim)( xxu nn ψ∞→= , then )(xu satisfies (1.3). It is obviously continuous and bounded on ],[ ba . At last, we prove the uniqueness of solution. Let )(xu and )(xv be two continuous solutions of (1.3) on ],[ ba , then ( ) ( ) ( ) ( ))(),()(),( )()(),()()(),(0 xxvDxxuD xxvxxuDxvxuD nn nn ψψ ψψ +≤ ++=≤ and since )(xnψ is convergent to solution of (1.3), then ( ) ( ) ,0)(),( ,0)(),( → → xxuD xxuD n n ψ ψ when ∞→n , then ( ) 0)(),( =xvxuD that is )()( xvxu = . This completes the proof. References 1. Towards Fuzzy Differential Calculus, I,II,III, Dubois, D. and Prade, H., (1982), Fuzzy Sets and Systems, 8, 1-7, 105-116, 225-233. 2. Elementary Calculus, Goetschel, R. and Voxman, W., (1986), Fuzzy Sets and Systems, 18, 31-43. 3. Fuzzy differential equations, Kaleva, O., (1987), Fuzzy sets and Systs., 24, 301-317. 4. Numerical Solution of Linear Fredholm Fuzzy Integral Equations of The Second Kind by Adomian method, Babolian, E., Goghary, H. S. and Abbasbandy S., (2005), App. Math. and Comput., 161, 733-744. 5. A Note on Fuzzy Integral Equations, Park, J. Y. and Ug Jeong J., (1999), Fuzzy Sets and Systems, 108, 193-200. 6. Existence and Uniqueness of Fuzzy Solution for Semi Linear Fuzzy Integro- Differential Equations with Nonlocal Conditions, Balasubramaniam, P. and Muralisankar, S., (2004), Comp. and Math. with Appls., 47, 1115-1122. 7. Numerical Solution of Fuzzy Linear Volterra Integral Equations of the Second Kind by Homotopy Analysis Method, Ghanbari, M., (2010), Int. J. Industrial Math. 2, 2, 73-87. 8. Existence and Uniqueness of Fuzzy Solution for Linear Volterra Fuzzy Integral Equations, Proved by Adomian Decomposition Method, Rouhparvar, H., Allahviranloo, T. and Abbasbandy, S., (2009), Romai J., 5, 2, 153-161 9. Fuzzy Volterra Integral Equations with Infinite Delay, Prakash, P. and Kalaiselvi, V., (2009), Tamkang Jornal of Math., Spring, 40, 1, 19-29. 10. Existence of Solutions of General Nonlinear Fuzzy Volterra-Fredholm Integral Equations, Balachandran, K. and Kanagarajan, K., (2005), Journal of Appl. Math. and Stochastic Analysis, 3, 333–343. 355 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 11. Fuzzy Random Variables, Puri, M. L. and Ralescu, D. A., (1986), J. Math. Anal. Appl., 114, , 409-422. 12. Metric Spaces of Fuzzy Sets Theory and Applications, Diamond, P. and Kloeden, P. E., (1994), World Scientific, Singapore. 356 | Mathematics ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Ü‹1a26@@ÖÜ»€a@I2@‚b«@H2013 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 26 (2) 2013 وجود ووحدانیة الحل لمعادالت فولتیرا التكاملیة الضبابیة الالخطیة إیمان علي حسین أیاد ولي علي الجامعة المستنصریة / كلیة العلوم / قسم الریاضیات الخالصة .وجود ووحدانیة الحل لمعادالت فولتیرا التكاملیة الضبابیة الالخطیة من النوع الثانيفي ھذا البحث برھنّا .معادالت فولتیرا التكاملیة الضبابیة : الكلمات المفتاحیة 357 | Mathematics