SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 18569 Functional Analysis and Operator Theory A new sequence space and norm of certain matrix operators on this space A new sequence space and norm of certain matrix operators on this space Roopaei Hadi Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran. Foroutannia Davoud Department of Mathematics, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran. 01 02 2016 03 1 1 12 06 10 2015 14 02 2016 Copyright © 2016, دانشگاه مراغه. 2016 https://scma.maragheh.ac.ir/article_18569.html

In the present paper, we introduce the sequence space [{l_p}(E,Delta) = left{ x = (x_n)_{n = 1}^infty : sum_{n = 1}^infty left|  sum_{j in {E_n}} x_j - sum_{j in E_{n + 1}} x_jright| ^p < infty right},] where \$E=(E_n)\$ is a partition of finite subsets of the positive integers and \$pge 1\$. We investigate its topological properties and inclusion relations. Moreover, we consider the problem of finding  the norm of certain matrix operators from  \$l_p\$ into \$ l_p(E,Delta)\$, and apply our results to Copson and Hilbert matrices.

Difference sequence space Matrix domains norm Copson matrix Hilbert matrix
 B. Altay and F. Basar, The ne spectrum and the matrix domain of the di erence operator Δ on the sequence space lp, (0 < p < 1), Commun. Math. Anal., 2(2) (2007) 1-11.  F. Basar, Summability Theory and Its Applications, Bentham Science Publishers, e-books, Monographs, Istanbul, 2012.  F. Basar and B. Altay, On the space of sequences of p-bounded variation and related matrix mappings, Ukr. Math. J., 55(1) (2003) 136-147.  F. Basar, B. Altay, and M. Mursaleen, Some generalizations of the space bvp of p-bounded variation sequences, Nonlinear Anal., 68(2) (2008) 273-287.  D. Foroutannia, On the block sequence space lp(E) and related matrix transfor- mations, Turk. J. Math., 39 (2015) 830-841.  D. Foroutannia, Upper bound and lower bound for matrix opwrators on weighted sequence spaces, Doctoral dissertation, Zahedan, 2007.  G.H. Hardy, J.E. Littlewood, and G. Polya, Inequalities, 2nd edition, Cambridge University press, Cambridge, 2001.  G.J.O. Jameson and R. Lashkaripour, Norms of certain operators on weighted lp spaces and Lorentz sequence spaces, J. Inequal. Pure Appl. Math., 3(1) (2002) Article 6.  H. Kizmaz, On certain sequence spaces I, Canad. Math. Bull., 25(2) (1981) 169-176.  R. Lashkaripour and J. Fathi, Norms of matrix operators on bvp, J. Math. Inequal., 6(4) (2012) 589-592.  M. Mursaleen and A.K. Noman, On some new di erence sequence spaces of non-absolute type, Math. Comput. Modelling, 52 (2010) 603-617.  H. Roopaei and D. Foroutannia, The norm of certain matrix operators on the new di erence sequence spaces, preprint.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 17845 Research Paper The approximate solutions of Fredholm integral equations on Cantor sets within local fractional operators The approximate solutions of Fredholm integral equations on Cantor sets within local fractional operators Kamil Jassim Hassan Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq. 01 02 2016 03 1 13 20 21 08 2015 23 01 2016 Copyright © 2016, دانشگاه مراغه. 2016 https://scma.maragheh.ac.ir/article_17845.html

In this paper, we apply the local fractional Adomian decomposition and variational iteration methods to obtain the analytic approximate solutions of Fredholm integral equations of the second kind within local fractional derivative operators. The iteration procedure is based on local fractional derivative. The obtained results reveal that the proposed methods are very efficient and simple tools for solving local fractional integral equations.

Fredholm integral equation Local fractional Adomian decomposition method Local fractional variational iteration method
 H.K. Jassim, C. Unlu, S.P. Moshokoa, and C.M. Khalique, Local Fractional Laplace Variational Iteration Method for Solving Di usion and Wave Equations on Cantor Sets within Local Fractional Operators, Mathematical Problems in Engineering, 2015 Article ID 309870 (2015) 1-9.  S.S. Ray and P.K. Sahu, Numerical Methods for Solving Fredholm Integral Equations of Second Kind, Abstract and Applied Analysis, 2013 Article ID 42916 (2013)1-17.  C.T. Shieh and V.A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347 (2008) 266-272.  W.H. Su, D. Baleanu, X.J. Yang, and H. Jafari, Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method, Fixed Point Theory and Applications, 2013 Article 89 (2013) 1-7.  A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Springer, New York, NY, USA, 2011.  S.P. Yan, H. Jafari, and H.K. Jassim, Local Fractional Adomian Decomposition and Function Decomposition Methods for Solving Laplace Equation within Local Fractional Operators, Advances in Mathematical Physics, 2014 Article ID 161580 (2014) 1-7.  Y.J. Yang, D. Baleanu, and X.J. Yang, A Local Fractional Variational Iteration Method for Laplace Equation within Local Fractional Operators, Abstract and Applied Analysis, 2013 Article ID 202650 (2013) 1-6.  Y.J. Yang, S.Q. Wang, and H.K. Jassim, Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative, Abstract and Applied Analysis, 2014 Article ID 176395 (2014) 1-7.  X.J. Yang, D. Baleanu, and W.P. Zhong, Approximation solutions for di usion equation on Cantor time-space, Proceeding of the Romanian Academy, 14 (2013) 127-133.  X.J. Yang, Local fractional integral equations and their applications, Advances in Computer Science and its Applications, 1 (2012) 234-239.  X.J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.  W.P. Zhong, F. Gao, X.M. Shen, Applications of Yang-Fourier transform to local Fractional equations with local fractional derivative and local fractional integral operater, Adv. Mat. Res., 461 (2012) 306-310.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 18685 Fuzzy Analysis Some properties of fuzzy real numbers Some properties of fuzzy real numbers Daraby Bayaz Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran. Jafari Javad Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran. 01 02 2016 03 1 21 27 09 08 2015 22 02 2016 Copyright © 2016, دانشگاه مراغه. 2016 https://scma.maragheh.ac.ir/article_18685.html

In the mathematical analysis, there are some theorems and definitions that established for both real and fuzzy numbers. In this study, we try to prove  Bernoulli's inequality in fuzzy real numbers with some of its applications. Also, we prove two other theorems in fuzzy real numbers which are proved before, for real numbers.

Fuzzy real number Bernoulli's inequality Real number
 T. Bag and S.K. Samanta, A comperative study of fuzzy norms on a linear space, Fuzzy Set and Systems, 159(6)(2008), 670-684.  C. Felbin, Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems, 48 (1992), 239-248.  M. Mizumoto and K. Tanaka, Some properties of fuzzy numbers, in: M.M. Gupta et al., Eds., Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979), 153-164.  W. Rudin, Principles of Mathetical Analysis, Mcgraw-Hill, New York, 1976.  I. Sadeqi, F. Moradlou, and M. Salehi, On approximate Cauchy equation in Felbins type fuzzy normed linear spaces, to appear in Iran. J. Fuzzy Syst. 10: 3 (2013), 51-63.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 18094 Complex Analysis Some study on the growth properties of entire functions represented by vector valued Dirichlet series in the light of relative Ritt orders Some study on the growth properties of entire functions represented by vector valued dirichlet series in the light of relative Ritt orders Datta Sanjib Department of Mathematics, University of Kalyani, P.O.-Kalyani, Dist-Nadia, PIN-\ 741235, West Bengal, India. Biswas Tanmay Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.-Krishnagar, Dist-Nadia, PIN-741101, West Bengal, India. Das Pranab Department of Mathematics, University of Kalyani, P.O.-Kalyani, Dist-Nadia, PIN-741235, West Bengal, India. 01 02 2016 03 1 29 35 26 07 2015 31 01 2016 Copyright © 2016, دانشگاه مراغه. 2016 https://scma.maragheh.ac.ir/article_18094.html

For entire functions, the notions of their growth indicators such as Ritt order are classical in complex analysis. But the concepts of relative Ritt order of entire functions and as well as their technical advantages of not comparing with the growths of \$exp exp z\$ are not at all known to the researchers of this area. Therefore the studies of the growths of entire functions in the light of their relative Ritt order are the prime concern of this paper. Actually in this paper we establish some newly developed results related to the growth rates of entire functions on the basis of their relative Ritt order (respectively, relative Ritt lower order).

Vector valued Dirichlet series (VVDS) Relative Ritt order Relative Ritt lower order growth
 Q.I. Rahaman, The Ritt order of the derivative of an entire function, Annales Polonici Mathematici., 17 (1965) 137-140.  C.T. Rajagopal and A.R. Reddy, A note on entire functions represented by Dirichlet series, Annales Polonici Mathematici., 17 (1965) 199-208.  J.F. Ritt, On certain points in the theory of Dirichlet series, Amer. Jour. Math., 50 (1928) 73-86.  G.S. Srivastava, A note on relative type of entire functions represented by vector valued dirichlet series, Journal of Classicial Analysis, 2(1) (2013) 61-72.  G.S. Srivastava and A. Sharma, On generalized order and generalized type of vector valued Dirichlet series of slow growth, Int. J. Math. Archive, 2(12) (2011) 2652-2659.  B.L. Srivastava, A study of spaces of certain classes of vector valued Dirichlet series, Thesis, I. I. T., Kanpur, 1983.  R.P. Srivastav and R.K. Ghosh, On entire functions represented by Dirichlet series, Annales Polonici Mathematici., 13 (1963) 93-100.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 15994 Research Paper Numerical solution of a class of nonlinear two-dimensional integral equations using Bernoulli polynomials Numerical solution of a class of nonlinear two-dimensional integral equations using Bernoulli polynomials Bazm Sohrab Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran. 01 02 2016 03 1 37 51 29 10 2015 02 01 2016 Copyright © 2016, دانشگاه مراغه. 2016 https://scma.maragheh.ac.ir/article_15994.html

In this study, the Bernoulli polynomials are used to obtain an approximate solution of a class of nonlinear two-dimensional integral equations. To this aim, the operational matrices of integration and the product for Bernoulli polynomials are derived and utilized to reduce the considered problem to a system of nonlinear algebraic equations. Some examples are presented to illustrate the efficiency and accuracy of the method.

Nonlinear two-dimensional integral equations Bernoulli polynomials Collocation method Operational matrices
 E. Babolian, S. Bazm, and P. Lima, Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions, Commun. Nonl. Sci. Numer. Simul. 16(3) (2011) 1164{1175.  S. Bazm, Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations, J. Comput. Appl. Math. 275 (2015) 44-60.  A. H. Bhrawy, E. Tohidi, and F. Soleymani, A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-di erential equations with piecewise intervals, Appl. Math. Comput. 219(2) (2012) 482-497.  A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcen-dental Functions, Vol. III, McGraw-Hill, New York, 1955.  H. Guoqiang, K. Hayami, K. Sugihara, and W. Jiong, Extrapolation method of iterated collocation solution for two-dimensional nonlinear Volterra integral equations, App. Math. Comput. 112 (2009) 70-76.  E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1989.  P. Lancaster, The Theory of Matrices: With Applications, second ed., Academic Press, New York, 1984.  Y.L. Luke, The Special Functions and Their Approximations, Vol. I, Academic Press, New York, 1969.  K. Maleknejad, S. Sohrabi, and B. Baranji, Application of 2D-BPFs to nonlinear integral equations, Commun. Nonl. Sci. Numer. Simul. 15 (2010) 527-535.  S. Nemati, P.M. Lima, and Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math. 242 (2013) 53{69.  S. Nemati, and Y. Ordokhani, Solving Nonlinear Two-Dimensional Volterra Integral Equations of the First-kind Using the Bivariate Shifted Legendre Functions, International Journal of Mathematical Modelling & Computations 5(3) (2015) 1-12.  A. Tari, M.Y. Rahimi, S. Shahmorad, and F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the di erential transform method, J. Comput. Appl. Math. 228 (2000) 49-61.  F. Toutounian and E. Tohidi, A new Bernoulli matrix method for solving second order linear partial di erential equations with the convergence analysis, Appl. Math. Comput. 223 (2013) 298-310.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 18096 Functional Analysis and Operator Theory On strongly Jordan zero-product preserving maps On strongly Jordan zero-product preserving maps Khoddami Ali Reza Department of Pure Mathematics, University of Shahrood, P. O. Box 3619995161-316, Shahrood, Iran. 01 02 2016 03 1 53 61 31 07 2015 31 01 2016 Copyright © 2016, دانشگاه مراغه. 2016 https://scma.maragheh.ac.ir/article_18096.html

In this paper, we give a characterization of strongly Jordan zero-product preserving maps on normed algebras as a generalization of  Jordan zero-product preserving maps. In this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly Jordan zero-product preserving maps are completely different. Also, we prove that the direct product and the composition of two strongly Jordan zero-product preserving maps are again  strongly Jordan zero-product preserving maps. But this fact is not the case for tensor product of them in general. Finally, we prove  that every \$*-\$preserving linear map from a normed \$*-\$algebra into a \$C^*-\$algebra that strongly preserves Jordan zero-products is necessarily continuous.

Strongly zero-product preserving map Strongly Jordan zero-product preserving map Zero-product preserving map Jordan zero-product preserving map Tensor product
 M.A. Chebotar, W.-F. Ke, P.-H. Lee and N.-C. Wong, Mappings preserving zero products , Studia Math., 155 1 (2003), 77-94.  H. Ghahramani, Zero product determined triangular algebras , Linear Multilinear Algebra, 61 (2013), 741-757.  A.R. Khoddami and H.R.E. Vishki, The higher duals of a Banach algebra induced by a bounded linear functional, Bull. Math. Anal. Appl. 3 (2011), 118-122.  A.R. Khoddami, Strongly zero-product preserving maps on normed algebras induced by a bounded linear functional, Khayyam J. Math., 1 (2015), no. 1, 107-114.  A.R. Khoddami, On maps preserving strongly zero-products, Chamchuri. J. Math., 7 (2015), 16-23.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 17820 Complex Analysis Parabolic starlike mappings of the unit ball \$B^n\$ Parabolic starlike mappings of the unit ball \$B^n\$ Rahrovi Samira Department of Mathematics, Faculty of Basic Science, University of Bonab, P.O. Box 5551-761167, Bonab, Iran. 01 02 2016 03 1 63 70 13 05 2015 20 01 2016 Copyright © 2016, دانشگاه مراغه. 2016 https://scma.maragheh.ac.ir/article_17820.html

Let \$f\$ be a locally univalent function on the unit disk \$U\$. We consider the normalized extensions of \$f\$ to the Euclidean unit ball \$B^nsubseteqmathbb{C}^n\$ given by \$\$Phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),\$\$  where \$gammain[0,1/2]\$, \$z=(z_1,hat{z})in B^n\$ and \$\$Psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),\$\$ in which \$betain[0,1]\$, \$f(z_1)neq 0\$ and \$z=(z_1,hat{z})in B^n\$. In the case \$gamma=1/2\$, the function \$Phi_{n,gamma}(f)\$ reduces to the well known Roper-Suffridge extension operator. By using different methods, we prove that if \$f\$ is parabolic starlike mapping on \$U\$ then \$Phi_{n,gamma}(f)\$ and \$Psi_{n,beta}(f)\$ are parabolic starlike mappings on \$B^n\$.

Roper-Suffridge extention operator Biholomorphic mapping Parabolic starlike function
 I. Graham and G. Kohr, Univalent mappings associated with the Roper-Su ridge extension operator, J. Anal. Math., 81 (2000) 331-342.  I. Graham and G. Kohr, An extension theorem and subclasses of univalent mappings in several complex variables, Complex Var., 47 (2002) 59-72.  I. Graham and G. Kohr, Geometric function theory in one and higher dimensions, Marcel Dekker, New York, (2003).  I. Graham, G. Kohr, and M. Kohr, Loewner chains and the Roper-Su ridge Extension Operator, J. Math. Anal. Appl., 247 (2000) 448-465.  H. Hamad, T. Honda, and G. Kohr, Parabolic starlike mappings in Several complex variables, Manuscripta math. 123 (2007), 301-324.  W. Hengartner and G. Schober, On schlicht mappings to domains convex in one direction, Comment. Math. Helv., 45 (1970), 303-314.  J. A. Pfaltzgra and T.J. Su ridge, An extension theorem and linear invariant families generated by starlike maps. Ann. Mariae Curie Sklodowska, 53 (1999), 193-207.  K.A. Roper and T.J. Su ridge, Convex mappings on the unit ball Cn, J. Anal. Math., 65 (1995), 333-347.  T.J. Su ridge, Starlike and convex maps in Banach spaces, Pac. J. Math., 46 (1973), 474-489.