]>
2014
7
6
ISSN 2008-1898
66
On the General Solution of a Quadratic Functional Equation and its Ulam Stability in Various Abstract Spaces
On the General Solution of a Quadratic Functional Equation and its Ulam Stability in Various Abstract Spaces
en
en
In this paper, we establish the general solution of a new quadratic functional equation
\(f ( x -\frac{ y+z}{ 2} ) + f ( x + \frac{y-z}{ 2}) +f(x+z) = 3f(x)+ \frac{1}{ 2}f(y)+ \frac{3}{ 2}f(z)\). Next, the Ulam stability of this equation in a real normed
space and a non-Archimedean space is studied, respectively.
368
378
Yonghong
Shen
School of Mathematics and Statistics
Tianshui Normal University
P. R. China
shenyonghong2008@hotmail.com
Yaoyao
Lan
Department of Mathematics and Finance
Chongqing University of Arts and Sciences
P. R. China
yylan81@hotmail.com
General solution
Ulam stability
Quadratic functional equation
Normed space
Non-Archimedean space.
Article.1.pdf
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P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436
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S. M. Jung, On the Hyers-Ulam-Rassias stability of a quadratic functional equation , J. Math. Anal. Appl., 232 (1999), 384-393
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S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, (2011)
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H. A. Kenary, C. Park, H. Rezaei, S. Y. Jang, Stability of a generalized quadratic functional equation in various spaces: A fixed point alternative approach, Adv. Differ. Equ., 2011 (2011), 1-62
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H. A. Kenary, Y. J. Cho, Stability of mixed additive-quadratic Jensen type functional equation in various spaces, Comput. Math. Appl., 61 (2011), 2704-2724
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H. M. Kim, K. W. Jun, E. Son, Generalized Hyers-Ulam stability of quadratic functional inequality, Abstr. Appl. Anal., Article ID 564923, 2013 (2013), 1-8
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Y. S. Lee, S. Y. Chung, Stability of a quadratic Jensen type functional equation in the spaces of generalized functions, J. Math. Anal. Appl., 324 (2006), 1395-1406
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Y. W. Lee, On the stability of a quadratic Jensen type functional equation, J. Math. Anal. Appl., 270 (2002), 590-601
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M. S. Moslehian, Th. M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discr. Math., 1 (2007), 325-334
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C. Park, H. A. Kenary, Th. M. Rassias, Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non- Archimedean Banach spaces, J. Inequal. Appl., 2012 (2012), 1-174
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Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
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P. K. Sahoo, P. Kannappan, Introduction to Functional Equations, CRC Press, Boca Raton (2011)
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S. M. Ulam, Problems in Modern Mathematics, Wiley, New York (1960)
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D. Yang, The stability of the equadratic functional equation on amenable groups, J. Math. Anal. Appl., 291 (2004), 666-672
]
Asymptotic behavior of solutions of a rational system of difference equations
Asymptotic behavior of solutions of a rational system of difference equations
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en
We consider a two-dimensional autonomous system of rational difference equations with three positive
parameters. It was conjectured by Ladas that every positive solution of this system converges to a finite
limit. Here we confirm this conjecture.
379
382
Miron B.
Bekker
Department of Mathematics
University of Pittsburgh at Johnstown
USA
bekkerm@umr.edu
Martin J.
Bohner
Department of Mathematics and Statistics
Missouri S&T
USA
bohner@mst.edu
Hristo D.
Voulov
Department of Mathematics and Statistics
University of Missouri-Kansas City
USA
voulovh@umkc.edu
Systems of rational difference equations
global attractors.
Article.2.pdf
[
[1]
E. Camouzis, G. Ladas, When does local asymptotic stability imply global attractivity in rational equations?, J. Difference Equ. Appl., 12 (2006), 863-885
##[2]
E. Camouzis, G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, Advances in Discrete Mathematics and Applications, 5. Chapman & Hall CRC, Boca Raton, FL (2008)
##[3]
E. Camouzis, M. R. S. Kulenović, G. Ladas, O. Merino, Rational systems in the plane, J. Difference Equ. Appl., 15 (2009), 303-323
##[4]
E. Camouzis, G. Ladas , Global results on rational systems in the plane, part 1, J. Difference Equ. Appl., 16 (2010), 975-1013
##[5]
E. Camouzis, C. M. Kent, G. Ladas, C. D. Lynd, On the global character of solutions of the system: \(x_{n+1} = \frac{\alpha_1+y_n}{ x_n}\) and \(y_{n+1} = \frac{\alpha_2+\beta_2x_n+\gamma_2y_n}{ A_2+B_2x_n+C_2y_n}\), J. Difference Equ. Appl., 18 (2012), 1205-1252
]
Multivariate Fuzzy Perturbed Neural Network Operators Approximation
Multivariate Fuzzy Perturbed Neural Network Operators Approximation
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en
This article studies the determination of the rate of convergence to the unit of each of three newly introduced
here multivariate fuzzy perturbed normalized neural network operators of one hidden layer. These are given
through the multivariate fuzzy modulus of continuity of the involved multivariate fuzzy number valued
function or its high order fuzzy partial derivatives and that appears in the right-hand side of the associated
fuzzy multivariate Jackson type inequalities. The multivariate activation function is very general, especially
it can derive from any sigmoid or bell-shaped function. The right hand sides of our multivariate fuzzy
convergence inequalities do not depend on the activation function. The sample multivariate fuzzy functionals
are of Stancu, Kantorovich and Quadrature types. We give applications for the first fuzzy partial derivatives
of the involved function.
383
406
George A.
Anastassiou
Department of Mathematical Sciences
University of Memphis
U.S.A
ganastss@memphis.edu
Multivariate neural network fuzzy approximation
fuzzy partial derivative
multivariate fuzzy modulus of continuity
multivariate fuzzy operator.
Article.3.pdf
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G. A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, J. Math. Anal. Appli., 212 (1997), 237-262
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G. A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit, J. Comp. Math. Appl., 40 (2000), 1-19
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G. A. Anastassiou, Fuzzy Approximation by Fuzzy Convolution type Operators, Computers and Mathematics, 48 (2004), 1369-1386
##[5]
G. A. Anastassiou, Higher order Fuzzy Korovkin Theory via inequalities, Commun. Appl. Anal., 10 (2006), 359-392
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G. A. Anastassiou, Fuzzy Korovkin Theorems and Inequalities, J. Fuzzy Math., 15 (2007), 169-205
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G. A. Anastassiou , Fuzzy Mathematics: Approximation Theory, Springer, Heidelberg, New York (2010)
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G. A. Anastassiou , Higher order multivariate fuzzy approximation by multivariate fuzzy wavelet type and neural network operators, J. Fuzzy Math., 19 (2011), 601-618
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G. A. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, revisited, Vesnik, 65 (2013), 511-518
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G. A. Anastassiou, Rate of convergence of some multivariate neural network operators to the unit , revisited, J. Comput. Anal. and Appl., 15 (2013), 1300-1309
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G. A. Anastassiou, Fuzzy fractional approximations by fuzzy normalized bell and squashing type neural network operators, J. Fuzzy Math., 22 (2014), 139-156
##[12]
G. A. Anastassiou, Higher Order Multivariate Fuzzy Approximation by basic Neural Network Operators, Cubo, accepted (2013)
##[13]
G. A. Anastassiou , Approximation by Perturbed Neural Network Operators, , submitted (2014)
##[14]
G. A. Anastassiou, Approximations by Multivariate Perturbed Neural Network Operators, , submitted (2014)
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P. Cardaliaguet, G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Networks, 5 (1992), 207-220
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S. Gal, Approximation Theory in Fuzzy Setting, Chapter 13 in Handbook of Analytic-Computational Methods in Applied Mathematics, 617-666, edited by G. Anastassiou, Chapman & Hall/CRC, Boca Raton, New York (2000)
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]
Asymptotic Behavior of Neutral Stochastic Partial Functional Integro--Differential Equations Driven by a Fractional Brownian Motion
Asymptotic Behavior of Neutral Stochastic Partial Functional Integro--Differential Equations Driven by a Fractional Brownian Motion
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en
This paper deals with the existence, uniqueness and asymptotic behavior of mild solutions to neutral stochastic
delay functional integro-differential equations perturbed by a fractional Brownian motion BH, with Hurst
parameter \(H \in ( \frac{1}{2} , 1)\). The main tools for the existence of solution is a fixed point theorem and the theory of
resolvent operators developed in Grimmer [R. Grimmer, Trans. Amer. Math. Soc., 273 (1982), 333-349.],
while a Gronwall-type lemma plays the key role for the asymptotic behavior. An example is provided to
illustrate the results of this work.
407
421
Tomás
Caraballo
Dpto. Ecuaciones Diferenciales y Análisis Numérico
Universidad de Sevilla. Apdo. de Correos
Spain
caraball@us.es
Mamadou Abdoul
Diop
Département de Mathématiques
Université Gaston Berger de Saint-Louis,
Sénégal
mamadou-abdoul.diop@ugb.edu.sn
Abdoul Aziz
Ndiaye
Département de Mathématiques
Université Gaston Berger de Saint-Louis
Sénégal
abdoulazizndiaye@yahoo.fr
Resolvent operators
\(C_0\)-semigroup
Wiener process
Mild solutions
Fractional Brownian motion
Exponential decay of solutions.
Article.4.pdf
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[1]
E. Alos, O. Mazet, D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab., 29 (1999), 766-801
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J. Bao, Z. Hou, Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients, Computers and Mathematics with Applications, 59 (2010), 207-214
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B. Boufoussi, S. Hajji, Neutral stochastic functional differential equations driven by a fractional Brownian motion in a Hilbert space, Statistics and Probability Letters, 82 (2012), 1549-1558
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G. Cao, K. He, X. Zhang, Successive approximations of infinite dimensional SDES with jump, Stoch. Syst., 5 (2005), 609-619
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T. Caraballo, M. A. Diop, Neutral stochastic delay partial functional integro-differential equations driven by a fractional Brownian motion , Front. Math. China, 8 (2013), 745-760
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T. Caraballo, A. M. Márquez-Durán, Existence, uniqueness and asymptotic behavior of solutions for a nonclassical diffusion equation with delay, Dyn. Partial Differ. Equ., 10 (2013), 267-281
##[8]
T. Caraballo, J. Real, T. Taniguchi , The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Syst., 18 (2007), 295-313
##[9]
H. Chen, Integral inequality and exponential stability for neutral stochastic partial differential equations, Journal of Inequalities and Applications, 2009 (2009), 1-297478
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T. E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics , 77 (2005), 139-154
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R. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349
##[13]
F. Jiang, Y. Shen , A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients, Computers and Mathematics with Applications , 61 (2011), 1590-1594
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Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Topics, in: Lecture Notes in Mathematics no. 1929, (2008)
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D. Nualart, The Malliavin Calculus and Related Topics, second edition, Springer-Verlag, Berlin (2006)
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S. Tindel, C. Tudor, F. Viens, Stochastic evolution equations with fractional brownian motion, Probab. Theory Related Fields, 127 (2) (2003), 186-204
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]
Stability in Nonlinear Delay Volterra Integro-differential Systems
Stability in Nonlinear Delay Volterra Integro-differential Systems
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en
We employ Lyapunov functionals to the system of Volterra integro-differential equations of the form
\[x'(t) = Px(t) - \int^ t _{t-r} C(t; s)g(x(s))ds,\]
and obtain conditions for the stability of the of the zero solution. In addition, we will furnish an example
as an application.
422
428
Youssef
Raffoul
Department of Mathematics
University of Dayton
USA
yraffoul1@udayton.edu
Mehmet
Ünal
Faculty of Education
Sinop University
Tutkey
munal@sinop.edu.tr
Volterra integro-differential equations
zero solution
stability
Lyapunov functional
Article.5.pdf
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H. Brown, K. Ergen, A theorem on rearrangements and its application to certain delay differential equations, J. Rat. Mech. Anal., 3 (1954), 565-579
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T. A. Burton, Fixed points and stability of a nonconvolution equation, Proc. Amer. Math. Soc., 132 (2004), 3679-3687
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T. A. Burton, Stability by fixed point theory or Liapunov theory: A comparison, Fixed Point Theory, 4 (2003), 15-32
##[4]
T. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publication, New York (2006)
##[5]
M. Cable, Y. Raffoul , Lyapunov functionals and exponential stability and instability in multi delay differential equations, Int. J. Math. Sci. Appl., 1 (2011), 961-970
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Y. Raffoul , Exponential Stability and Instability in Finite Delay nonlinear Volterra Integro-differential Equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 20 (2013), 95-106
##[10]
T. Wang , Inequalities and stability for a linear scalar functional differential equation, J. Math. Anal. Appl., 298 (2004), 33-44
]
Global Bifurcation Analysis of the Lorenz System
Global Bifurcation Analysis of the Lorenz System
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en
We carry out the global bifurcation analysis of the classical Lorenz system. For many years, this system has
been the subject of study by numerous authors. However, until now the structure of the Lorenz attractor
is not clear completely yet, and the most important question at present is to understand the bifurcation
scenario of chaos transition in this system. Using some numerical results and our bifurcational geometric
approach, we present a new scenario of chaos transition in the Lorenz system.
429
434
Valery A.
Gaiko
United Institute of Informatics Problems
National Academy of Sciences of Belarus
Belarus
valery.gaiko@gmail.com
Lorenz system
bifurcation
singular point
limit cycle
chaos.
Article.6.pdf
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H. W. Broer, F. Takens, Dynamical Systems and Chaos, Springer, New York (2011)
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E. J. Doedel, B. Krauskopf, H. M. Osinga, Global invariant manifolds in the transition to preturbulence in the Lorenz system, Indagationes Math., 22 (2011), 222-240
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V. A. Gaiko, Global Bifurcation Theory and Hilbert's Sixteenth Problem, Kluwer, Boston (2003)
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V. A. Gaiko, Limit cycles of quadratic systems, Nonlinear Anal., 69 (2008), 2150-2157
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V. A. Gaiko, A quadratic system with two parallel straight-line-isoclines, Nonlinear Anal., 71 (2009), 5860-5865
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V. A. Gaiko, Limit cycle bifurcations of a special Liénard polynomial system, Adv. Dyn. Syst. Appl., 9 (2014), 109-123
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Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York (2004)
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N. A. Magnitskii, S. V. Sidorov , New Methods for Chaotic Dynamics, World Scientific, New Jarsey (2006)
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L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part I, World Scientific, New Jarsey (1998)
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L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, New Jarsey (2001)
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]