Partial Averaging of Fuzzy Differential Equations with Maxima

Transcription

1 Advances in Dynamical Systems and Applications ISSN , Volume 6, Number 2, pp Partial Averaging o Fuzzy Dierential Equations with Maxima Olga Kichmarenko and Natalia Skripnik Odessa National University named ater I.I. Mechnikov Department o Optimal Control and Economic Cybernetics Odessa, Ukraina olga.kichmarenko@gmail.com and talie@ukr.net Abstract In this paper, a scheme o partial averaging o uzzy dierential equations with ima is considered. AMS Subject Classiications: 3E72, 34C29, 34K5. Keywords: Averaging method, uzzy dierential equation with ima. 1 Introduction The study o uzzy dierential equations FDEs orms a suitable setting or the mathematical modelling o real world problems in which uncertainty or vagueness pervades. Fuzzy dierential equations were irst ormulated by Kaleva [4, 5]. He used the concept o H-dierentiability which was introduced by Puri and Ralescu [13], and obtained the existence and uniqueness theorem or a solution o FDE under the Lipschitz condition. Since then there appeared a lot o papers concerning the theory and applications o uzzy dierential equations, uzzy dynamics and uzzy dierential inclusions [2, 9, 1, 12]. In this paper, a scheme o partial averaging o uzzy dierential equations with ima is considered that continues researches devoted to the uzzy dierential equations with delay [7, 8]. 2 Main Deinitions Let convr n be a amily o all nonempty compact convex subsets o R n with Hausdor metric ha, B = { a b, a b }, min a A b B min b B a A Received December 1, 21; Accepted June 1, 211 Communicated by Martin Bohner

2 2 O. Kichmarenko and N. Skripnik where denotes the usual Euclidean norm in R n. Denote by A = ha,. Let E n be a amily o mappings x : R n [, 1] satisying the ollowing conditions: 1 x is normal, i.e., there exists y R n such that xy = 1; 2 x is uzzy convex, i.e., xλy + 1 λz min{xy, xz} whenever y, z R n and λ [, 1]; 3 x is upper semicontinuous, i.e., or any y R n and > there exists δy, > such that xy < xy + whenever y y < δ, y R n ; 4 the closure o the set {y R n : xy > } is compact. { i y, Let be a uzzy mapping deined by y = 1 i y =. Deinition 2.1 See [11]. The set {y R n : xy α} is called an α level [x] α o a mapping x E n or α, 1]. A closure o the set {y R n : xy > } is called a - level [x] o a mapping x E n. Deine the metric D : E n E n R + by the equation Let I be an interval in R. Dx, y = sup h[x] α, [y] α. α [,1] Deinition 2.2. A mapping : I E n is called weakly continuous at point t I i a multivalued mapping α t = [t] α is continuous or any α [, 1]. Deinition 2.3 See [11]. A mapping : I E n E n is called weakly continuous at point t, x I E n i or any α [, 1] and > there exists δ, α > such that h α t, x, α t, x < or all t, x I E n satisying the condition t t < δ, α, h[x] α, [x ] α < δ, α. Deinition 2.4 See [11]. A mapping : I E n is called measurable on I i a multivalued mapping α s Lebesgue measurable or any α [, 1]. Deinition 2.5 See [11]. An element g E n is called anintegral o : I E n over I i [g] α = A α tdt or any α, 1], where A α tds the Aumann integral [1]. I I Deinition 2.6 See [11]. A mapping : I E n is called dierentiable at point t I i the multivalued mapping α s Hukuhara dierentiable at point t [3] or any α [, 1] and the amily {D H α t : α [, 1]} deines a uzzy number t E n which is called a uzzy derivative o t at point t.

3 Partial Averaging o Fuzzy Dierential Equations with Maxima 21 Consider a uzzy dierential equation with delay x t = t, xt, xαt, xt = x, 2.1 where t I is time; x S E n is a phase variable; the initial conditions t I, x S; a uzzy mapping : I S S E n ; a delay unction αt [t, t]. Deinition 2.7. A uzzy mapping x : I E n, t I I, is called a solution o equation 2.1 i is weakly continuous and or all t I satisies the integral equation xt = x + t s, xs, xαsds. Theorem 2.8 See [8]. Let t, x, y be a weakly continuous unction in the neighborhood o the point t, x, x and satisy the Lipschitz condition in x, y with constant λ. Then there exists a unique solution xt o equation 2.1 or t [t, t + σ], where σ is small enough. 3 Main Results Consider the uzzy dierential equation with ima x t = t, xt, τ [γt,gt] xτ, x = x, 3.1 where t is time; x S E n is a phase variable; > is a small parameter; the initial condition x S; a uzzy mapping : R + S R + E n ; the unctions γt gt t. I there exists a uzzy mapping t, x, z such that or any t, z, x S E n 1 lim T T D t+t t s, x, zds, t+t t s, x, zds =, 3.2 then in the correspondence to equation 3.1 we will set the ollowing partially averaged equation y t = t, yt, yτ, y = x. 3.3 τ [γt,gt] Theorem 3.1. Len the domain Q = {t, z, x S E n } the ollowing hold: 1. the uzzy mappings t, x, z, t, x, z are uniormly bounded by M, weakly continuous in t and satisy the Lipschitz condition in x, z with constant λ, i.e., Dt, x, z, M, Dt, x, z, M, D t, x, z, t, x, z λ [Dx, x + z z ], D t, x, z, t, x, z λ [Dx, x + z z ] ;

4 22 O. Kichmarenko and N. Skripnik 2. the limit 3.2 exists uniormly with respect to t, z and x S; 3. the unctions γt, gt are uniormly continuous and γt gt t; 4. the solution yt o equation 3.3, y = x S S belongs to the domain S together with a ρ-neighborhood or all t. Then or any η, ρ] and L > there exists η, L > such that or all, ] and t [, L 1 ] the ollowing estimate holds: D xt, yt η, 3.4 where xt, yt are solutions o equations 3.1 and 3.3 such that x = y = x. Proo. According to Deinition 2.7, the solutions o equations 3.1 and 3.3 are weakly continuous uzzy mappings that satisy the integral equations Then xt = x + yt = x + Dxt, yt = D x + = D D + D λ x + s, xs, τ [γs,gs] xτ s, xs, xτ ds, τ [γs,gs] s, ys, τ [γs,gs] yτ ds, [ Dxs, ys + s, xs, τ [γs,gs] xτ ds, s, ys, τ [γs,gs] yτ ds. s, xs, τ [γs,gs] xτ ds, s, ys, τ [γs,gs] yτ ds ds, s, ys, yτ ds τ [γs,gs] s, ys, τ [γs,gs] ds s, ys, τ [γs,gs] ds xτ τ [γs,gs] ] yτ τ [γs,gs] ds + σ,

5 Partial Averaging o Fuzzy Dierential Equations with Maxima 23 where β t, = D σ = s, ys, τ [γs,gs] yτ Let δt = Dxs, ys. As s [,t] τ [γs,gs] xτ τ [γs,gs] yτ βt,, t [,L 1 ] ds, s, ys, τ [γs,gs] yτ ds. xτ yτ Dxτ, yτ δs, τ [γs,gs] τ [γs,gs] we have δt 2λ δsds + σ. 3.5 Divide the interval [, L 1 ] on the partial intervals with the points = Li m, i =, m, m N. Let us estimate β t,, using the properties o the metric D or t [, +1 ]: βt, = D i= +1 i= +1 s, ys, τ [γs,gs] yτ s, ys, τ [γs,gs] yτ ds + +1 D i= + D i= +1 s, ys, τ [γs,gs] yτ s, ys, τ [γs,gs] yτ ds, ds + s, ys, τ [γs,gs] yτ ds, s, ys, τ [γs,gs] yτ ds +1 ds, D s, ys, yτ, τ [γs,gs] s, ys, yτ τ [γs,gs] s, ys, yτ ds τ [γs,gs] s, y, yτ ds τ [γ,g ]

6 24 O. Kichmarenko and N. Skripnik + +1 D + D + D s, ys, τ [γs,gs] yτ, s, y, τ [γ,g ] yτ ds +1 2λ i= + D + D +1 s, y, τ [γ,g ] yτ ds, +1 s, ys, τ [γs,gs] yτ ds, +1 [ Dys, y + τ [γs,gs] s, y, τ [γ,g ] yτ ds, s, ys, τ [γs,gs] yτ ds, s, y, τ [γ,g ] yτ ds s, ys, τ [γs,gs] yτ ] yτ yτ τ [γ,g ] ds +1 Let us estimate every summand in 3.6 separately: +1 Dys, y ds = +1 = D y + D +1 s 2 M 2 D s s ds s, y, τ [γ,g ] yτ ds s, ys, yτ τ [γs,gs] ds. τ, yτ, χ [γτ,gτ] yχ dτ, y ds τ, yτ, yχ dτ, ds χ [γτ,gτ] τ, yτ, yχ, dτds χ [γτ,gτ] L = ML2 2m Using the properties o the modulus o continuity ωψ, σ = sup{ ψτ 1 ψτ 2, τ 1 τ 2 < σ}

7 Partial Averaging o Fuzzy Dierential Equations with Maxima 25 o unctions γt, gt, we get +1 τ [γs,gs] +1 = LM m LM m M { yτ yτ τ [γ,g ] ds { ω γ, ω γ, L L, ω, ω g, g, L L } } ds {ω γ, L, ω g, L}. 3.8 From the uniorm convergence to the average in 3.2 it ollows that there exists a monotone decreasing unction Θt as t such that D +1 s, y, τ [γ,g ] yτ ds, +1 s, y τ [γ,g ] yτ ds L L m Θ. 3.9 As the mappings and are uniormly bounded by constant M, we have D s, ys, yτ ds, s, ys, yτ τ [γs,gs] τ [γs,gs] + D s, ys, τ [γs,gs] yτ, D s, ys, yτ, ˆ ds τ [γs,gs] D s, ys, τ [γs,gs] yτ, ˆ Hence using , we have βt, λml2 m + 2λLM 1 m + {ωγ, L, ωg, L} ds s, ys, τ [γs,gs] yτ ds ML ds 2M ds = 2 m.3.1

8 26 O. Kichmarenko and N. Skripnik Let us choose m to satisy the inequality + 2ML m L + LΘ λml 2 m + 2λLM m {ωγ, L, ωg, L} + 2ML m < η 2e 2λL 3.12 and then choose such that L 2λLM {ωγ, L, ωg, L} + LΘ < η e2λL From and 3.5 using the Gronwall Bellman lemma, we get the statement o the theorem. Remark 3.2. From the deinition o xt it ollows that xt = Dxt, ˆ = sup h[xt] α, {} = h[xt], {} = [xt]. α [,1] So the uzzy dierential equation 3.1 is equivalent to the ollowing x t = t, xt, τ [γt,gt] [xτ], x = x Is easy to show thanstead o -level set o the uzzy mapping xt any α-level set, α [, 1], can be taken. The substantiation o the averaging scheme will be almost the same. 4 Conclusion In this paper the substantiation o one scheme o averaging or uzzy dierential equations with ima is considered. These results generalize the results o [6] or dierential equations with Hukuhara derivative with ima. Reerences [1] R.J. Aumann. Integrals o set-valued unctions. J. Math. Anal. Appl., 12:1 12, [2] P. Diamond and P. Kloeden. Metric spaces o uzzy sets. World Scientiic, Singapore, New Jersey, London, Hong Kong, [3] M. Hukuhara. Integration des applications mesurables dont la valeur est un compact convexe. Func. Ekvacioj, 1:25 223, 1967.

9 Partial Averaging o Fuzzy Dierential Equations with Maxima 27 [4] O. Kaleva. Fuzzy dierential equations. Fuzzy Sets Syst., 243:31 317, [5] O. Kaleva. The cauchy problem or uzzy dierential equations. Fuzzy Sets Syst., 35: , 199. [6] O.D. Kichmarenko. Averaging o dierential equations with Hukuhara derivative with ima. Int. J. Pure Appl. Math., 573: , 29. [7] O.D. Kichmarenko and N.V. Skripnik. Averaging o uzzy dierential equations with delay. Nonlinear Oscill., 113: , 28. Russian. [8] O.D. Kichmarenko and N.V. Skripnik. Fuzzy dierential equations with delay. Bull. Kiev. Nat. Univ. Math. Mechanics, 19:18 23, 28. Ukrainian. [9] V. Lakshmikantham, T. Granna Bhaskar, and J. Vasundhara Devi. Theory o set dierential equations in metric spaces. Cambridge Scientiic Publishers, Cambridge, 26. [1] V. Lakshmikantham and R. Mohapatra. Theory o uzzy dierential equations an inclusions. Taylor & Frances, London, 23. [11] J.Y. Park and H.K. Han. Existence and uniqueness theorem or a solution o uzzy dierential equations. Internat. J. Math. Sci., 222: , [12] A.V. Plotnikov and N.V. Skripnik. Dierential equations with clear and uzzy multivalued right-hand side: Asymptotical methods. Astroprint, Odessa, 29. Russian. [13] M.L. Puri and D.A. Ralescu. Dierentielle d une onction loue. C.R. Acad. Sc. Paris, 293-I: , 1981.