Partial Averaging of Fuzzy Differential Equations with Maxima
|
|
- Bernard Harmon
- 5 years ago
- Views:
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.
An Averaging Result for Fuzzy Differential Equations with a Small Parameter
Annual Review o Chaos Theory, Biurcations and Dynamical Systems Vol. 5, 215 1-9, www.arctbds.com. Copyright c 215 ARCTBDS. SSN 2253 371. All Rights Reserved. An Averaging Result or Fuzzy Dierential Equations
More informationFull averaging scheme for differential equation with maximum
Contemporary Analysis and Applied M athematics Vol.3, No.1, 113-122, 215 Full averaging scheme for differential equation with maximum Olga D. Kichmarenko and Kateryna Yu. Sapozhnikova Department of Optimal
More informationFUZZY SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXV 1(26 pp. 119 126 119 FUZZY SOLUTIONS FOR BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS A. ARARA and M. BENCHOHRA Abstract. The Banach fixed point theorem
More informationNumerical Solution of Fuzzy Differential Equations
Applied Mathematical Sciences, Vol. 1, 2007, no. 45, 2231-2246 Numerical Solution of Fuzzy Differential Equations Javad Shokri Department of Mathematics Urmia University P.O. Box 165, Urmia, Iran j.shokri@mail.urmia.ac.ir
More informationEXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVIII, 2(29), pp. 287 32 287 EXISTENCE THEOREMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES A. SGHIR Abstract. This paper concernes with the study of existence
More informationM.S.N. Murty* and G. Suresh Kumar**
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No.1, March 6 THREE POINT BOUNDARY VALUE PROBLEMS FOR THIRD ORDER FUZZY DIFFERENTIAL EQUATIONS M.S.N. Murty* and G. Suresh Kumar** Abstract. In
More informationSolution of the Fuzzy Boundary Value Differential Equations Under Generalized Differentiability By Shooting Method
Available online at www.ispacs.com/jfsva Volume 212, Year 212 Article ID jfsva-136, 19 pages doi:1.5899/212/jfsva-136 Research Article Solution of the Fuzzy Boundary Value Differential Equations Under
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationFunctional Differential Equations with Causal Operators
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.11(211) No.4,pp.499-55 Functional Differential Equations with Causal Operators Vasile Lupulescu Constantin Brancusi
More informationFUZZY STOCHASTIC INTEGRAL EQUATIONS
Dynamic Systems and Applications 19 21 473-494 FUZZY STOCHASTIC INTEGRAL EQUATIONS MAREK T. MALINOWSKI AND MARIUSZ MICHTA Faculty of Mathematics, Computer Science and Econometrics University of Zielona
More informationExistence of Solutions to Boundary Value Problems for a Class of Nonlinear Fuzzy Fractional Differential Equations
232 Advances in Analysis, Vol. 2, No. 4, October 217 https://dx.doi.org/1.2266/aan.217.242 Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Fuzzy Fractional Differential Equations
More informationStolarsky Type Inequality for Sugeno Integrals on Fuzzy Convex Functions
International Journal o Mathematical nalysis Vol., 27, no., 2-28 HIKRI Ltd, www.m-hikari.com https://doi.org/.2988/ijma.27.623 Stolarsky Type Inequality or Sugeno Integrals on Fuzzy Convex Functions Dug
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 15: Nonlinear Systems and Lyapunov Functions Overview Our next goal is to extend LMI s and optimization to nonlinear
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 3. Fundamental properties IST-DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Example Consider the system ẋ = f
More informationInternational Journal of Pure and Applied Mathematics Volume 56 No ,
Inernaional Journal of Pure and Applied Mahemaics Volume 56 No. 2 2009, 165-172 THE GENERALIZED SOLUTIONS OF THE FUZZY DIFFERENTIAL INCLUSIONS Andrej V. Plonikov 1, Naalia V. Skripnik 2 1 Deparmen of Numerical
More informationWEAK AND STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR NONEXPANSIVE MAPPINGS IN HILBERT SPACES
Applicable Analysis and Discrete Mathematics available online at http://pemath.et.b.ac.yu Appl. Anal. Discrete Math. 2 (2008), 197 204. doi:10.2298/aadm0802197m WEAK AND STRONG CONVERGENCE OF AN ITERATIVE
More informationDependence on Initial Conditions of Attainable Sets of Control Systems with p-integrable Controls
Nonlinear Analysis: Modelling and Control, 2007, Vol. 12, No. 3, 293 306 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls E. Akyar Anadolu University, Deartment
More informationExistence And Uniqueness Of Mild Solutions Of Second Order Volterra Integrodifferential Equations With Nonlocal Conditions
Applied Mathematics E-Notes, 9(29), 11-18 c ISSN 167-251 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Existence And Uniqueness Of Mild Solutions Of Second Order Volterra Integrodifferential
More informationNumerical Solution of Hybrid Fuzzy Dierential Equation (IVP) by Improved Predictor-Corrector Method
Available online at http://ijim.srbiau.ac.ir Int. J. Industrial Mathematics Vol. 1, No. 2 (2009)147-161 Numerical Solution of Hybrid Fuzzy Dierential Equation (IVP) by Improved Predictor-Corrector Method
More informationTelescoping Decomposition Method for Solving First Order Nonlinear Differential Equations
Telescoping Decomposition Method or Solving First Order Nonlinear Dierential Equations 1 Mohammed Al-Reai 2 Maysem Abu-Dalu 3 Ahmed Al-Rawashdeh Abstract The Telescoping Decomposition Method TDM is a new
More informationSome Properties of NSFDEs
Chenggui Yuan (Swansea University) Some Properties of NSFDEs 1 / 41 Some Properties of NSFDEs Chenggui Yuan Swansea University Chenggui Yuan (Swansea University) Some Properties of NSFDEs 2 / 41 Outline
More informationFractional Evolution Integro-Differential Systems with Nonlocal Conditions
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 5, Number 1, pp. 49 6 (21) http://campus.mst.edu/adsa Fractional Evolution Integro-Differential Systems with Nonlocal Conditions Amar
More informationEXISTENCE OF SOLUTIONS TO SYSTEMS OF EQUATIONS MODELLING COMPRESSIBLE FLUID FLOW
Electronic Journal o Dierential Equations, Vol. 15 (15, No. 16, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.e or http://ejde.math.unt.e tp ejde.math.txstate.e EXISTENCE OF SOLUTIONS TO SYSTEMS
More informationSOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS
International Journal o Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 2017, 179-187 DOI: 10.28924/2291-8639-15-2017-179 SOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM
More informationNonlocal Cauchy problems for first-order multivalued differential equations
Electronic Journal of Differential Equations, Vol. 22(22), No. 47, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonlocal Cauchy
More informationOn boundary value problems for fractional integro-differential equations in Banach spaces
Malaya J. Mat. 3425 54 553 On boundary value problems for fractional integro-differential equations in Banach spaces Sabri T. M. Thabet a, and Machindra B. Dhakne b a,b Department of Mathematics, Dr. Babasaheb
More informationINITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL DIFFERENTIAL EQUATIONS
Applied Mathematics and Stochastic Analysis, 6:2 23, 9-2. Printed in the USA c 23 by North Atlantic Science Publishing Company INITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL
More informationLYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT. Marat Akhmet. Duygu Aruğaslan
DISCRETE AND CONTINUOUS doi:10.3934/dcds.2009.25.457 DYNAMICAL SYSTEMS Volume 25, Number 2, October 2009 pp. 457 466 LYAPUNOV-RAZUMIKHIN METHOD FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT
More informationO. Karlova LIMITS OF SEQUENCES OF DARBOUX-LIKE MAPPINGS
Математичнi Студiї. Т.40, 2 Matematychni Studii. V.40, No.2 УДК 517.51 O. Karlova LIMITS OF SEQUENCES OF DARBOUX-LIKE MAPPINGS O. Karlova. Limits o sequences o Darboux-like mappings, Mat. Stud. 40 (2013),
More informationNUMERICAL SOLUTIONS OF FUZZY DIFFERENTIAL EQUATIONS BY TAYLOR METHOD
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.2(2002), No.2, pp.113 124 c Institute of Mathematics of the National Academy of Sciences of Belarus NUMERICAL SOLUTIONS OF FUZZY DIFFERENTIAL EQUATIONS
More informationNON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS AND RETARDED EQUATIONS. M. Filali and M. Moussi
Electronic Journal: Southwest Journal o Pure and Applied Mathematics Internet: http://rattler.cameron.edu/swjpam.html ISSN 83-464 Issue 2, December, 23, pp. 26 35. Submitted: December 24, 22. Published:
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationYURI LEVIN AND ADI BEN-ISRAEL
Pp. 1447-1457 in Progress in Analysis, Vol. Heinrich G W Begehr. Robert P Gilbert and Man Wah Wong, Editors, World Scientiic, Singapore, 003, ISBN 981-38-967-9 AN INVERSE-FREE DIRECTIONAL NEWTON METHOD
More informationSeparation Theorems for a Class of Retarded Nonlinear Systems
Separation Theorems or a Class o Retarded Nonlinear Systems A. Germani C. Manes P. Pepe Dipartimento di Ingegneria Elettrica e dell Inormazione 674 Poggio di Roio L Aquila Italy e-mail: alredo.germani@univaq.it
More informationON PERIODIC BOUNDARY VALUE PROBLEMS OF FIRST-ORDER PERTURBED IMPULSIVE DIFFERENTIAL INCLUSIONS
Electronic Journal of Differential Equations, Vol. 24(24), No. 84, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ON PERIODIC
More informationPOSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF SINGULAR FRACTIONAL FUNCTIONAL DIFFERENTIAL EQUATION
International Journal of Pure and Applied Mathematics Volume 92 No. 2 24, 69-79 ISSN: 3-88 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/.2732/ijpam.v92i2.3
More informationCOMPARISON RESULTS OF LINEAR DIFFERENTIAL EQUATIONS WITH FUZZY BOUNDARY VALUES
Journal of Science and Arts Year 8, No. (4), pp. 33-48, 08 ORIGINAL PAPER COMPARISON RESULTS OF LINEAR DIFFERENTIAL EQUATIONS WITH FUZZY BOUNDARY VALUES HULYA GULTEKIN CITIL Manuscript received: 08.06.07;
More informationMidterm Exam Solutions February 27, 2009
Midterm Exam Solutions February 27, 2009 (24. Deine each o the ollowing statements. You may assume that everyone knows the deinition o a topological space and a linear space. (4 a. X is a compact topological
More informationarxiv: v1 [math.oc] 20 Sep 2016
General Viscosity Implicit Midpoint Rule For Nonexpansive Mapping Shuja Haider Rizvi Department of Mathematics, Babu Banarasi Das University, Lucknow 68, India arxiv:169.616v1 [math.oc] Sep 16 Abstract:
More informationProblem Set 2: Solutions Math 201A: Fall 2016
Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that
More informationEXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM. Saeid Shokooh and Ghasem A. Afrouzi. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69 4 (217 271 28 December 217 research paper originalni nauqni rad EXISTENCE OF THREE WEAK SOLUTIONS FOR A QUASILINEAR DIRICHLET PROBLEM Saeid Shokooh and Ghasem A.
More informationProblem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS
Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series
More informationOn Pointwise λ Statistical Convergence of Order α of Sequences of Fuzzy Mappings
Filomat 28:6 (204), 27 279 DOI 0.2298/FIL40627E Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Pointwise Statistical Convergence
More informationOPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS
APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze
More informationNUMERICAL SOLUTION OF A BOUNDARY VALUE PROBLEM FOR A SECOND ORDER FUZZY DIFFERENTIAL EQUATION*
TWMS J. Pure Appl. Math. V.4, N.2, 2013, pp.169-176 NUMERICAL SOLUTION OF A BOUNDARY VALUE PROBLEM FOR A SECOND ORDER FUZZY DIFFERENTIAL EQUATION* AFET GOLAYOĞLU FATULLAYEV1, EMINE CAN 2, CANAN KÖROĞLU3
More informationSome Hermite-Hadamard type integral inequalities for operator AG-preinvex functions
Acta Univ. Sapientiae, Mathematica, 8, (16 31 33 DOI: 1.1515/ausm-16-1 Some Hermite-Hadamard type integral inequalities or operator AG-preinvex unctions Ali Taghavi Department o Mathematics, Faculty o
More informationContents. 1. Introduction
FUNDAMENTAL THEOREM OF THE LOCAL THEORY OF CURVES KAIXIN WANG Abstract. In this expository paper, we present the fundamental theorem of the local theory of curves along with a detailed proof. We first
More informationSOLVING FUZZY DIFFERENTIAL EQUATIONS BY USING PICARD METHOD
Iranian Journal of Fuzzy Systems Vol. 13, No. 3, (2016) pp. 71-81 71 SOLVING FUZZY DIFFERENTIAL EQUATIONS BY USING PICARD METHOD S. S. BEHZADI AND T. ALLAHVIRANLOO Abstract. In this paper, The Picard method
More informationPliska Stud. Math. Bulgar. 12 (1998), STUDIA MATHEMATICA BULGARICA
Pliska Stud. Math. Bulgar. 12 (1998), 191-200 STUDIA MATHEMATICA BULGARICA METHOD OF AVERAGING FOR IMPULSIVE DIFFERENTIAL INCLUSIONS * V. A. Plotnikov, R. P. Ivanov, N. M. Kitanov The paper deals with
More informationLecture 5: The Bellman Equation
Lecture 5: The Bellman Equation Florian Scheuer 1 Plan Prove properties of the Bellman equation (In particular, existence and uniqueness of solution) Use this to prove properties of the solution Think
More informationTheory of Set Differential Equations
Theory of Set Differential Equations V. Lakshmikantham T. Gnana Bhaskar J. Vasundhara Devi Department of Mathematical Sciences Florida Institute of Technology Melbourne FL 32901 USA ii Contents Preface
More informationOn Convexity of Reachable Sets for Nonlinear Control Systems
Proceedings o the European Control Conerence 27 Kos, Greece, July 2-5, 27 WeC5.2 On Convexity o Reachable Sets or Nonlinear Control Systems Vadim Azhmyakov, Dietrich Flockerzi and Jörg Raisch Abstract
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FOURTH-ORDER BOUNDARY-VALUE PROBLEMS IN BANACH SPACES
Electronic Journal of Differential Equations, Vol. 9(9), No. 33, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationSolution of the Synthesis Problem in Hilbert Spaces
Solution o the Synthesis Problem in Hilbert Spaces Valery I. Korobov, Grigory M. Sklyar, Vasily A. Skoryk Kharkov National University 4, sqr. Svoboda 677 Kharkov, Ukraine Szczecin University 5, str. Wielkopolska
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), 79 7 www.emis.de/journals ISSN 176-0091 WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS ADELA MIHAI Abstract. B.Y. Chen
More informationGlobal Weak Solution of Planetary Geostrophic Equations with Inviscid Geostrophic Balance
Global Weak Solution o Planetary Geostrophic Equations with Inviscid Geostrophic Balance Jian-Guo Liu 1, Roger Samelson 2, Cheng Wang 3 Communicated by R. Temam) Abstract. A reormulation o the planetary
More informationBest proximity point results in set-valued analysis
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 3, 293 305 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.3.1 Best proximity point results in set-valued analysis Binayak S. Choudhury a, Pranati
More informationOn Picard value problem of some difference polynomials
Arab J Math 018 7:7 37 https://doiorg/101007/s40065-017-0189-x Arabian Journal o Mathematics Zinelâabidine Latreuch Benharrat Belaïdi On Picard value problem o some dierence polynomials Received: 4 April
More informationConcept of Fuzzy Differential Equations
RESERCH RTICLE Concept of Fuzzy Differential Equations K. yyanar, M. Ramesh kumar M.Phil Research Scholar, sst.professor in Maths Department of Maths, Prist University,Puducherry, India OPEN CCESS bstract:
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationSTATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION
Journal o athematical Analysis ISSN: 227-342, URL: http://www.ilirias.com Volume 4 Issue 2203), Pages 6-22. STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION AAR JYOTI DUTTA, AYHAN
More informationEXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC SCHRÖDINGER EQUATIONS
Electronic Journal of Differential Equations, Vol. 017 (017), No. 15, pp. 1 7. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO ASYMPTOTICALLY PERIODIC
More informationNonlinear Systems Theory
Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we
More informationNonlinear equations. Norms for R n. Convergence orders for iterative methods
Nonlinear equations Norms for R n Assume that X is a vector space. A norm is a mapping X R with x such that for all x, y X, α R x = = x = αx = α x x + y x + y We define the following norms on the vector
More informationOn the Well-Posedness of the Cauchy Problem for a Neutral Differential Equation with Distributed Prehistory
Bulletin of TICMI Vol. 21, No. 1, 2017, 3 8 On the Well-Posedness of the Cauchy Problem for a Neutral Differential Equation with Distributed Prehistory Tamaz Tadumadze I. Javakhishvili Tbilisi State University
More informationOn the effect of α-admissibility and θ-contractivity to the existence of fixed points of multivalued mappings
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 5, 673 686 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.5.7 On the effect of α-admissibility and θ-contractivity to the existence of fixed points
More informationTHE GAMMA FUNCTION THU NGỌC DƯƠNG
THE GAMMA FUNCTION THU NGỌC DƯƠNG The Gamma unction was discovered during the search or a actorial analog deined on real numbers. This paper will explore the properties o the actorial unction and use them
More informationExistence Results for Multivalued Semilinear Functional Differential Equations
E extracta mathematicae Vol. 18, Núm. 1, 1 12 (23) Existence Results for Multivalued Semilinear Functional Differential Equations M. Benchohra, S.K. Ntouyas Department of Mathematics, University of Sidi
More informationReflected Brownian Motion
Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide
More informationEXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO HIGHER-ORDER NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH INTEGRAL BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 212 (212), No. 234, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND UNIQUENESS
More informationAlmost Asymptotically Statistical Equivalent of Double Difference Sequences of Fuzzy Numbers
Mathematica Aeterna, Vol. 2, 202, no. 3, 247-255 Almost Asymptotically Statistical Equivalent of Double Difference Sequences of Fuzzy Numbers Kuldip Raj School of Mathematics Shri Mata Vaishno Devi University
More informationON A TWO-VARIABLES FRACTIONAL PARTIAL DIFFERENTIAL INCLUSION VIA RIEMANN-LIOUVILLE DERIVATIVE
Novi Sad J. Math. Vol. 46, No. 2, 26, 45-53 ON A TWO-VARIABLES FRACTIONAL PARTIAL DIFFERENTIAL INCLUSION VIA RIEMANN-LIOUVILLE DERIVATIVE S. Etemad and Sh. Rezapour 23 Abstract. We investigate the existence
More informationOn the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces
Mathematica Moravica Vol. 14-1 (2010), 113 119 On the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces Amit Singh and R.C. Dimri Abstract. In
More informationLOCAL FIXED POINT THEORY INVOLVING THREE OPERATORS IN BANACH ALGEBRAS. B. C. Dhage. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 24, 24, 377 386 LOCAL FIXED POINT THEORY INVOLVING THREE OPERATORS IN BANACH ALGEBRAS B. C. Dhage Abstract. The present
More informationRigorous pointwise approximations for invariant densities of non-uniformly expanding maps
Ergod. Th. & Dynam. Sys. 5, 35, 8 44 c Cambridge University Press, 4 doi:.7/etds.3.9 Rigorous pointwise approximations or invariant densities o non-uniormly expanding maps WAEL BAHSOUN, CHRISTOPHER BOSE
More informationDifferentiability of solutions with respect to the delay function in functional differential equations
Electronic Journal of Qualitative Theory of Differential Equations 216, No. 73, 1 16; doi: 1.14232/ejqtde.216.1.73 http://www.math.u-szeged.hu/ejqtde/ Differentiability of solutions with respect to the
More informationInternal Stabilizability of Some Diffusive Models
Journal of Mathematical Analysis and Applications 265, 91 12 (22) doi:1.16/jmaa.21.7694, available online at http://www.idealibrary.com on Internal Stabilizability of Some Diffusive Models Bedr Eddine
More informationEXISTENCE OF ISOPERIMETRIC SETS WITH DENSITIES CONVERGING FROM BELOW ON R N. 1. Introduction
EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI Abstract. In this paper, we consider the isoperimetric problem in the space R with
More informationThe (CLR g )-property for coincidence point theorems and Fredholm integral equations in modular metric spaces
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 1, No., 17, 38-54 ISSN 137-5543 www.ejpam.com Published by New York Business Global The (CLR g )-property for coincidence point theorems and Fredholm
More informationSome unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces
An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 211, 331 346 Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces Yonghong Yao, Yeong-Cheng Liou Abstract
More informationSOME RESULTS FOR BOUNDARY VALUE PROBLEM OF AN INTEGRO DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER
Dynamic Systems and Applications 2 (2) 7-24 SOME RESULTS FOR BOUNDARY VALUE PROBLEM OF AN INTEGRO DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER P. KARTHIKEYAN Department of Mathematics, KSR College of Arts
More informationOn (h, k) trichotomy for skew-evolution semiflows in Banach spaces
Stud. Univ. Babeş-Bolyai Math. 56(2011), No. 4, 147 156 On (h, k) trichotomy for skew-evolution semiflows in Banach spaces Codruţa Stoica and Mihail Megan Abstract. In this paper we define the notion of
More informationEXISTENCE AND UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH LINEAR PROGRAMS EMBEDDED
EXISTENCE AND UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH LINEAR PROGRAMS EMBEDDED STUART M. HARWOOD AND PAUL I. BARTON Key words. linear programs, ordinary differential equations, embedded
More information3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?
MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable
More informationCONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES. Jong Soo Jung. 1. Introduction
Korean J. Math. 16 (2008), No. 2, pp. 215 231 CONVERGENCE OF APPROXIMATING FIXED POINTS FOR MULTIVALUED NONSELF-MAPPINGS IN BANACH SPACES Jong Soo Jung Abstract. Let E be a uniformly convex Banach space
More informationTiziana Cardinali Francesco Portigiani Paola Rubbioni. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 2008, 247 259 LOCAL MILD SOLUTIONS AND IMPULSIVE MILD SOLUTIONS FOR SEMILINEAR CAUCHY PROBLEMS INVOLVING LOWER
More informationRecent Trends in Differential Inclusions
Recent Trends in Alberto Bressan Department of Mathematics, Penn State University (Aveiro, June 2016) (Aveiro, June 2016) 1 / Two main topics ẋ F (x) differential inclusions with upper semicontinuous,
More informationNON-ARCHIMEDEAN BANACH SPACE. ( ( x + y
J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. ISSN(Print 6-0657 https://doi.org/0.7468/jksmeb.08.5.3.9 ISSN(Online 87-608 Volume 5, Number 3 (August 08, Pages 9 7 ADDITIVE ρ-functional EQUATIONS
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1. Yong Zhou. Abstract
EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS 1 Yong Zhou Abstract In this paper, the initial value problem is discussed for a system of fractional differential
More informationRemarks on Fuzzy Differential Systems
International Journal of Difference Equations ISSN 097-6069, Volume 11, Number 1, pp. 19 6 2016) http://campus.mst.edu/ijde Remarks on Fuzzy Differential Systems El Hassan Eljaoui Said Melliani and Lalla
More informationA FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS
Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,
More informationChapter 2. Fuzzy function space Introduction. A fuzzy function can be considered as the generalization of a classical function.
Chapter 2 Fuzzy function space 2.1. Introduction A fuzzy function can be considered as the generalization of a classical function. A classical function f is a mapping from the domain D to a space S, where
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI. Received December 14, 1999
Scientiae Mathematicae Vol. 3, No. 1(2000), 107 115 107 ITERATIVE SCHEMES FOR APPROXIMATING SOLUTIONS OF ACCRETIVE OPERATORS IN BANACH SPACES SHOJI KAMIMURA AND WATARU TAKAHASHI Received December 14, 1999
More informationTheorem Let J and f be as in the previous theorem. Then for any w 0 Int(J), f(z) (z w 0 ) n+1
(w) Second, since lim z w z w z w δ. Thus, i r δ, then z w =r (w) z w = (w), there exist δ, M > 0 such that (w) z w M i dz ML({ z w = r}) = M2πr, which tends to 0 as r 0. This shows that g = 2πi(w), which
More informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes
More informationON SOME TOPOLOGICAL PROPERTIES OF NEW TYPE OF DIFFERENCE SEQUENCE SPACES
Acta Universitatis Apulensis ISSN: 1582-5329 No. 32/2012 pp. 61-67 ON SOME TOPOLOGICAL PROPERTIES OF NEW TYPE OF DIFFERENCE SEQUENCE SPACES Çiğdem A. BEKTAŞ and Gülcan ATICİ Abstract. In this paper, we
More informationContinuous Functions on Metric Spaces
Continuous Functions on Metric Spaces Math 201A, Fall 2016 1 Continuous functions Definition 1. Let (X, d X ) and (Y, d Y ) be metric spaces. A function f : X Y is continuous at a X if for every ɛ > 0
More information2 Statement of the problem and assumptions
Mathematical Notes, 25, vol. 78, no. 4, pp. 466 48. Existence Theorem for Optimal Control Problems on an Infinite Time Interval A.V. Dmitruk and N.V. Kuz kina We consider an optimal control problem on
More information