SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.30023 Vactor and Tensor Analysis On Some Properties of the Max Algebra System Over Tensors On Some Properties of the Max Algebra System Over Tensors Shojaeifard Ali Reza Department of Mathematics and Statistics, Faculty of Basic Sciences, Imam Hossein Comprehensive University, Tehran, Iran. Afshin Hamid Reza Department of Mathematics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran. 01 11 2018 12 1 1 14 06 04 2017 29 01 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_30023.html

Recently we generalized the max algebra system to the class of nonnegative  tensors. In this paper we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverse of tensors is characterized. Also we generalize the direct product of matrices to the direct product of tensors (of the same order, but may be different dimensions) and investigate its properties relevant to the spectral theory.

‎Tensor Max algebra Left (right) inverse Direct Product Eigenvalue
 H.R. Afshin and A.R. Shojaeifard,  A max version of Perron Frobenuos theorem for nonnegative tensor, Ann. Funct. Anal., 6 (2015), pp. 145-154.  R. Bapat,  A max version of the Perron Frobenius theorem, Linear Algebra Appl., (1998),  pp. 3-18.  F. Baccelli, G. Cohen, G. Olsder, and J. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley, 1992.  P. Butkovic and M. Fiedler, Tropical tensor product and beyond, School of Mathematics University of Birmingham, 2011.  C. Bu, X. Zhang, J. Zhou, W. Wang, and Y. Wei,  The inverse, rank and product of tensors,  Linear Algebra Appl., 446 (2014), pp. 269-280.  K.C. Chang, K. Pearson, and T. Zhang,  Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), pp. 507-520.  R.A. Cuninghame-Green,  Minimax Algbera, Springer-Verlag, 1979.  V. Loan,  Future directions in tensor based computation and modeling, NSF Workshop Report in Arlington, Virginia, USA, 2009.  K. Pearson,  Essentially positive tensors,  Int. J. Algebra.,  4 (2010), pp. 421-427.  L. Qi,  Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput.,  40 (2005), pp. 1302-1324.  J.Y. Shao,  A general product of tensors with applications, Linear Algebra Appl., 439 (2013), pp. 2350-2366.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.46703.100 Functional Analysis and Operator Theory Inequality Problems of Equilibrium Problems with Application Inequality Problems of Equilibrium Problems with Application Eleiwis Hashoosh Ayed Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq. Alimohammady Mohsen Department of mathematics, University of Mazandaran, Babolsar, Iran. Mohsen Buite Haiffa Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq. 01 11 2018 12 1 15 26 05 08 2016 22 04 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_30860.html

This paper aims at establishing the existence of results for a nonstandard equilibrium problems \$(EP_{N})\$. The solutions of this inequality are discussed in a subset \$K\$ (either bounded or unbounded) of a Banach spaces \$X\$. Moreover, we enhance the main results by application of some differential inclusion.

Monotone bifunction Equilibrium problem KKM technique Differential inclusion
 M. Ait Mansour, Z. Chbani, and H. Riahi, Recession bifunction and solvability of noncoercive equilibrium problems, Commun. Appl. Anal., 7, (2003), pp. 369 -377.  M. Alimohammady and A.E. Hashoosh, Existence Theorems For \$alpha(u,v)\$-monotone of nonstandard Hemivariational Inequality, Advances in Math., 10, (2015), pp. 3205-3212.  Q.H. Ansari and J.C. Yao, An existence result for the generalized vector equlibrium problem, Appl. Math. Lett., 19, (1999), pp. 53-56.  M.Bianchi and S. Schaible, Equilibrium problems under generalized convexity and generalized monotonicity, J. Global Optim., 30, (2004), pp. 121-134.  E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63, (1994), pp. 123-145.  F.E. Browder, The solvability of non-linear functional equations, Duke Math. J., 30, (1963), pp. 557-566.  S. Carl, V. Khoi Le, and D. Motreanu, Nonsmooth variational problems and their inequalities, Springer Monographs in Mathematics, Springer, New York, (2007).  O. Chadli, Y. Chiang, and S. Huang, Topological pseudomonotonicity and vector equilibm problems, J. Math. Anal. Appl., 270, (2002), pp. 435-450.  K. Fan, A generalization of Tychonoffs fixed point theorem, Math. Ann., 142, (1961), pp. 305-310.  Y.P. Fang and N.J. Huang, Variational-like inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl., 118, (2003), pp. 327-338.  N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, optimization, 59, (2010), pp. 147-160.  A.E. Hashoosh and M. Alimohammady, On Well-Posedness Of Generalized Equilibrium Problems Involving α-Monotone Bifunction, J. Hyperstruct., 5, (2016), pp. 151-168.  A.E. Hashoosh and M. Alimohammady, Bα,β-Operator and Fitzpatrick Functions, Jordan J. Math. Stat., 1, (2017), pp. 259-278.  A.E. Hashoosh, M. Alimohammady, and M.K. Kalleji, Existence Results for Some Equilibrium Problems involving α-Monotone Bifunction, Int. J. Math. Math. Sci., 2016, (2016), pp. 1-5.  U. Kamraksa and R. Wangkeeree, Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces,  J. Global Optim., 51, (2011), pp. 689 -714.  B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe, Fund. Math., 14, (1929), pp. 132-137.  N.K. Mahato and C. Nahak, Mixed equilibrium problems with relaxed α-monotone mapping in Banach spaces, Rend. Circ. Mat. Palermo, (2013).  J.W. Peng and J.Yao, A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings., Nonlinear Anal., 71, (2009), pp. 6001-6010.  A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and equilibrium problem,  J. Optim. Theory Appl., 133, (2007), pp. 359-370.  R.U. Verma, On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, J. Math. Anal. Appl. , 213, (1997), pp. 387-392.  R.U. Verma, On monotone nonlinear variational inequality problems, Comment. Math. Univ. Carolin., 39, (1998), pp. 91-98.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.67135.257 Topology and Methods On Regular Generalized \$delta\$-closed Sets in Topological Spaces On Regular Generalized \$delta\$-closed Sets in Topological Spaces Rajakumar Somasundaram Department of Science and Humanities, Krishnasamy College of Engineering and Technology, Cuddalore, Tamil Nadu-608 002, India. 01 11 2018 12 1 27 37 29 06 2017 21 05 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_31670.html

In this paper a new class of sets called  regular generalized  \$delta\$-closed set (briefly rg\$delta\$-closed set)is introduced and its properties are studied. Several examples are provided to illustrate the behaviour of these new class of sets.

\$rgdelta\$-closed set \$delta\$-closed set \$gdelta\$-closed set
 A.A. Omari and M.S.M. Noorani, On Generalized \$b\$-closed sets, Bull. Malays. Math. Sci. Soc., 32 (2009), pp. 19-30.  S.P. Arya and T. Nour, Characterizations of \$s\$-normal spaces, Indian J. Pure and Applied Maths., 21 (1990), pp. 717-719.  P. Bhattacharya and B.K. Lahiri, Semi-generalized closed sets on topology, Indian J. Maths., 29 (1987), pp. 375-382.  Y. Gnanambal, On Generalized Pre-regular closed sets in topological spaces, Indian J. Pure Appl. Math., 28 (1997), pp. 351-360.  D. Iyappan and N. Nagaveni, On Semi generalized \$b\$-closed set, Nat. Sem. On Mat. Comp. Sci., Jan., (2010), Proc.6.  N. Levine, Generalized closed sets in topology, Rend Circ., Mat. Palermo., 19 (1970), pp. 89-96.  N. Levine, Semi-open sets and Semi-continuity in topological spaces, Amer. Math. Monthly., 70 (1963), pp. 36-41.  H. Maki, R. Devi, and K. Balachandran, Associated topologies of generalized \$alpha\$-closed sets and \$alpha\$-generalized closed sets, Mem. Fac. Sci. Kochi. Univ. Ser. A. Math., 15 (1994), pp. 51-63.  H. Maki, R.J. Umehara, and T. Noiri, Every topological space is Pre-\$T\$, Mem. Fac. Sci. Kochi. Univ. Ser. A. Math., 17 (1996), pp. 33-42.  A.S. Mashhour Abd El-Monsef. M. E. and Ei-Deeb S.N, On Pre-continuous and weak Pre-continuous mappings, Proc. Math. Phys. Soc. Egypt., 53 (1982), pp. 47-53.  O. Njastad, On Some classes of nearly open sets, Pacific J. Math., 15 (1965), pp. 961-970.  N. Nagaveni, Studies on generalized homeomorphisms in topological spaces, Ph. D., Thesis, Bharathiar University, Coimbatore 1999.  L. Vinayagamoorthi and N. Nagaveni, On Generalized-\$alpha b\$ closed sets, Proceedings, ICMD-Pushpa Publications, Vol. 1., 2010-11.  M.K.R.S. Veerakumar, Between closed sets and \$g\$-closed sets, Mem. Fac. Sci.Kochi Univ., Math., 21 (2000), pp. 1-19.  N.V. Velico, H-closed topological spaces, Amer. Math. Soc. Transl., 78 (1968), pp. 103-118.  N. Palaniappan and K. C. Rao, Regular generalized closed sets, Kyungpook Math. J., 33 (1993), pp. 211-219.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.46462.98 Nonlinear Analysis The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving \$p\$-Laplacian and Critical Sobolev Exponent The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving \$p\$-Laplacian and Critical Sobolev Exponent Khademloo Somayeh Department of Basic Sciences, Babol Noushirvani University of Technology, 47148-71167, Babol, Iran. Khanjany Ghazi Saeed Department of Basic Sciences, Babol Noushirvani University of Technology, 47148-71167, Babol, Iran. 01 11 2018 12 1 39 57 31 07 2016 09 04 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_30802.html

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

Variational methods Nehari manifold Dirichlet boundary condition Critical Sobolev exponent
 G.A. Afrouzi and S.H. Rasouli, A remark on the existence of multiple solutions to a multiparameter nonlinear elliptic system, Nonlin. Anal., 71 (2009), pp. 445-455.  A. Ambrosetti, H. Brezis, and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), pp.519-543.  A. Ambrosetti, J. Garcia-Azorero, and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), pp. 219-242.  C.O. Alves, D.C. de Morais Filho, and M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlin. Anal., 42 (2000), pp. 771-787.  G. Azorero and I. Peral, Some results about the existence of a second positive solution in a quasilinear critical problem, Indiana Univ. Math. J., 43 (1994), pp. 941-957.  T. Barstch and M. Willem, On a elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), pp. 3555-3561.  P.A. Binding, P. Drabek, and Y.X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron J. Differ. Eqs, 5 (1997), pp. 1-11.  H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), pp. 437-477.  H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), pp. 486-490.  K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Eq.s, 193 (2003), pp. 481-499.  K.J. Brown and T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function, J. Math. Anal. Appl., 337 (2008), pp. 1326-1336.  P. Han, The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), pp. 1241-1257.  T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities, Nonlin. Anal., 71 (2009), pp. 2688-2698.  T.S. Hsu, Multiplicity results for P-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions, Abs. and Appl Anal. Article ID 652109, 24 pages, 2009.  G. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), pp. 281-304.  T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlin. Anal., 68 (2008), pp. 1733-1745.  T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), pp. 253-270.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.88929.466 Banach Algebra \$(-1)\$-Weak Amenability of Second Dual of Real Banach Algebras \$(-1)\$-Weak Amenability of Second Dual of Real Banach Algebras Alihoseini Hamidreza Department of Mathematics, Faculty of Science, Arak University, 38156-8-8349, Arak, Iran. Alimohammadi Davood Department of Mathematics, Faculty of Science, Arak University, 38156-8-8349, Arak, Iran. 01 11 2018 12 1 59 88 30 06 2018 27 11 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_34113.html

Let \$ (A,| cdot |) \$ be a real Banach algebra, a complex algebra \$ A_mathbb{C} \$ be a complexification of \$ A \$ and \$ | | cdot | | \$ be an algebra norm on  \$ A_mathbb{C}  \$  satisfying a simple condition together with the norm \$ | cdot | \$ on \$ A\$.  In this paper we first show that \$ A^* \$ is a real Banach \$ A^{**}\$-module if and only if \$ (A_mathbb{C})^* \$ is a complex Banach \$ (A_mathbb{C})^{**}\$-module. Next  we prove that \$ A^{**} \$ is \$ (-1)\$-weakly  amenable if and only if \$ (A_mathbb{C})^{**} \$ is \$ (-1)\$-weakly  amenable. Finally, we give some examples of real Banach algebras which their second duals of some them are and of others are not \$ (-1)\$-weakly  amenable.

Banach algebra‎ ‎Banach module‎ ‎Complexification‎ ‎Derivation‎ ‎\$(-1)\$-Weak amenability
 D. Alimohammadi and A. Ebadian, Hedberg's theorem in real Lipschitz algebras, Indian J. Pure Appl. Math., 32 (2010), pp. 1470-1493.  D. Alimohammadi and T.G. Honary, Contractibility, amenability and weak amenability of real Banach algebras, J. Aanalysis, (9)(2001), pp. 69-88.  R. Arens, The adjoint of a bilinear operation, Proc. Math. Amer. Soc., 2 (1951), pp. 839-848.  W.G. Bade, P.C. Curtis, and H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc., 55 (1987), pp. 359-377.  F.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer Verlag, New York, 1973.  H.G. Dales, Banach Algebras and Automatic Continuity, Oxford University Press, 2000.  J. Duncan and S.A.R. Hosseinioun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburg Sect. A., 84 (1979), pp. 309-325.  M. Eshaghi Gordji, S.A.R. Hosseinioun, and A. Valadkhani, On (-1)-weak amenability of Banach algebras, Math. Reports, 15 (65), (2013), pp. 271-279.  T.G. Honary and S. Moradi, On the maximal ideal space of extended analytic Lipschitz algebras, Quaestiones Mathematicae, 30 (2007), pp. 349-353.  S.A.R. Hosseinioun and A. Valadkhani, (-1)-Weak amenability of unitized Banach algebras, Europ. J. Pure Appl. Math., 9 (2016), pp. 231-239.  S.A.R. Hosseinioun and A. Valadkhani, Weak and (-1)-weak amenability of second dual of Banach algebras, Int. J. Nonlinear Anal. Appl., 7 (2016), pp. 39-48.  B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).  B.E. Johnson, Derivations from \$L^1(G)\$ into \$L^1(G)\$ and \$L^infty (G)\$, Proc. International conference on Harmonic Analysis, Luxembourg, (Lecture note in Math. Springer-Verlag), 1359 (1987), pp. 191-198.  S.H. Kulkarni and B.V. Limaye, Gleason parts of real function algebras, Canad. J. Math., 33 (1981), pp. 181-200.  S.H. Kulkarni and B.V. Limaye, Real Function Algebras, Marcel Dekker, Inc. New York, 1992.  M. Mayghani and D. Alimohammadi, The Structure of ideals, point derivations, amenability and weak amenability of extended Lipschitz algebras, Int. J. Nonlinear Anal. Appl., 2017, pp. 389-404.  A. Medghalchi and T. Yazdanpanah, Problems concerning n-weak amenability of a Banach algebra, Czechoslovak Math. J., 55 (2005), pp. 863-876.  T.W. Palmer, Banach Algebras, the General Theory of *-Algebras, Vol. 1: Algebras and Banach Algebras, Cambridge University Press, Cambridge, 1994.  D.R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., (13) (1963), pp. 1387-1399.  D.R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc, 111 (1964), pp. 240-272.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.68371.262 Complex Analysis The Norm Estimates of Pre-Schwarzian Derivatives of Spirallike Functions and Uniformly Convex \$alpha\$-spirallike Functions The Norm Estimates of Pre-Schwarzian Derivatives of Spirallike Functions and Uniformly Convex \$alpha\$-spirallike Functions Orouji Zahra Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran. Aghalary Rasul Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran. 01 11 2018 12 1 89 96 18 07 2017 16 05 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_31361.html

For a constant \$alphain left(-frac{pi}{2},frac{pi}{2}right)\$,  we definea  subclass of the spirallike functions, \$SP_{p}(alpha)\$, the setof all functions \$fin mathcal{A}\$[releft{e^{-ialpha}frac{zf'(z)}{f(z)}right}geqleft|frac{zf'(z)}{f(z)}-1right|.]In  the present paper, we shall give the estimate of the norm of the pre-Schwarzian derivative  \$mathrm{T}_f=f''/f'\$ where \$|mathrm{T}_f|=sup_{zin Delta} (1-|z|^2)|mathrm{T}_f(z)|\$ for the functions in  \$SP_{p}(alpha)\$.

Pre-Schwarzian derivative Spiral-like function Uniformly convex function
 J. Becker and CH. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math., 354 (1984), pp. 74-94.  P.L. Duren, Univalent Functions, Grundlehren Math. Wiss., 259 (1983), Springer-Verlay. New York.  A.W. Goodman, On uniformly convex functions, Ann. Pol. Math., 57 (1991), pp. 87-92.  A.W. Goodman, On uniformly convex functions, J. Math. Anal. Appl., 155 (1991), pp. 364-370.  Y.C. Kim and T. Sugawa, Growth and coefficient estimates for uniformly locally univalent functions on the unit disk, Rocky Mountain J. Math., 32 (2002), pp. 179-200.  Y.C. Kim and T. Sugawa, Norm estimates of the pre-Schwarzian derivative for certain classes of univalent functions, Proc. Edinburgh Math. Soc., 49 (2006), pp. 131-143.  W. Ma and D. Minda, Uniformly convex functions, Ann. Pol. Math., 57 (1992), pp. 165-175.  Y. Okuyama, The norm estimates of pre-Schwarzian derivatives of spiral-like functions, Complex Variables Theory Appl., 42 (2000), pp. 225-239.  V. Ravichandran, C. Selvaraj, and R. Rajagopal, On uniformly convex spiral functions and uniformly spirallike functions, Soochow J. Math., 29 (2003), pp. 393-405.  F. RΦnning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), pp. 189-196.  S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math., 28 (1999), pp. 217-230.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.62911.238 Fixed Point Theory Numerical Reckoning Fixed Points in \$CAT(0)\$ Spaces Numerical Reckoning Fixed Points in \$CAT(0)\$ Spaces Ullah Kifayat Department of Mathematics, University of Science and Technology Bannu, KPK Pakistan. Khan Hikmat Department of Mathematics, University of Science and Technology Bannu, KPK Pakistan. Arshad Muhammad Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan. 01 11 2018 12 1 97 111 27 04 2017 20 08 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_34179.html

In this paper, first we use an example to show the efficiency of \$M\$ iteration process introduced by Ullah and Arshad  for approximating fixed points of Suzuki generalized nonexpansive mappings. Then by using \$M\$ iteration process, we prove some strong and \$Delta -\$convergence theorems for Suzuki generalized nonexpansive mappings in the setting of \$CAT(0)\$ Spaces. Our results are the extension, improvement and generalization of many known results in \$CAT(0)\$ spaces.

Suzuki generalized nonexpansive mapping, \$CAT(0)\$ space iteration process, \$Delta\$-convergence, Strong convergence
 M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Math. Vesn., 66 (2014), pp. 223-234.  R.P. Agarwal, D. O'Regan, and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal., 8 (2007), pp. 61-79.  M. Bridson and A. Heaflinger, Metric Space of Non-positive Curvature, Springer-Verlag, Berlin, 1999.  F. Bruhat and J. Tits, Groupes reductifs sur un corps local. I, Donnees radicielles valuees Inst Hauts Etudea Sci. Publ. Math., 41 (1972), pp. 5-251.  D. Burago, Y. Burago and S. Inavo, A course in Metric Geometry, Vol. 33, Americal Mathematical Socity, Providence, RI, 2001.  R. Chugh, V. Kumar, and S. Kumar, Strong Convergence of a new three step iterative scheme in Banach spaces, Amer. J. Comp. Math., 2 (2012), pp. 345-357.  S. Dhompongsa, W.A. Kirk, and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), pp. 762-772.  S. Dhompongsa, W.A. Kirk, and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear and convex Anal., 8 (2007), pp. 35-45.  S. Dhompongsa and B. Panyanak, On \$Delta \$-convergence theorem in \$CAT(0)\$ Spaces, Comput. Math. Appl., 56 (2008), pp. 2572-2579.  A. Gharajelo and H. Dehghan, Convergence Theorems for Strict Pseudo-Contractions in \$CAT(0)\$ Metric Spaces, Filomat, 31 (2017), pp. 1967-1971.  F. Gursoy and V. Karakaya, A Picard-S hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2 (2014).  S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc., 44 (1974), 147-150.  I. Karahan and M. Ozdemir, A general iterative method for approximation of fixed points and their applications, Advances in Fixed Point Theory, 3 (2013), pp. 510-526.  S.H. Khan, A Picard-Mann hybrid iterative process, Fixed Point Th. Appl., 2013, Article ID 69 (2013).  W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4 (1953), pp. 506-510.  A. Moudafi, Krasnoselski-Mann iteration for hierarchical fixed point problems, Inverse Probl., 23 (2007), pp. 1635-1640.  M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), pp. 217-229.  W. Phuengrattana, Approximating fixed points of Suzuki-generalized nonexpansive mappings, Nonlinear Anal. Hybrid Syst. 5 (2011), pp. 583-590.  W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comp. Appl. Math., 235 (2011), pp. 3006-3014.  H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Am. Math. Soc., 44 (1974), pp. 375-380.  D.R. Sahu and A. Petrusel, Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces, NonlinearAnalysis: Theory, Methods and Applications, 74 (2011), pp. 6012-6023.  T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340 (2008), pp. 1088-1095.  B.S Thakur, D. Thakur and M. Postolache, A new iterative scheme for numerical reckoning fixed points of Suzuki's generalized nonexpansive mappings, App. Math. Comp., 275 (2016), pp. 147-155.  K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki generalized nonexpansive mapping via new iteration process, Filomat, 32 (2018), pp. 187-196.  R. Wangkeeree, H. Dehghan, Strong and \$Delta\$-convergence of Moudafi's iterative scheme in \$CAT(0)\$ spaces, J. Nonlinear Convex Anal., 16 (2015), 299-309.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.78500.364 Real and Harmonic Analysis A Certain Class of Character Module Homomorphisms on Normed Algebras A Certain Class of Character Module Homomorphisms on Normed Algebras Khoddami Ali Reza Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran. 01 11 2018 12 1 113 120 01 01 2018 26 05 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_31199.html

For two normed algebras \$A\$ and \$B\$ with the character space   \$bigtriangleup(B)neq emptyset\$  and a left \$B-\$module \$X,\$  a certain class of bounded linear maps from \$A\$ into \$X\$ is introduced. We set \$CMH_B(A, X)\$  as the set of all non-zero \$B-\$character module homomorphisms from \$A\$ into \$X\$. In the case where \$bigtriangleup(B)=lbrace varphirbrace\$ then \$CMH_B(A, X)bigcup lbrace 0rbrace\$ is a closed subspace of \$L(A, X)\$  of all bounded linear operators from \$A\$ into \$X\$.   We  define an  equivalence  relation on  \$CMH_B(A, X)\$ and use it  to show that  \$CMH_B(A, X)bigcuplbrace 0rbrace \$ is  a union of closed subspaces of \$L(A, X)\$.  Also some basic results and some hereditary properties are presented. Finally some relations between \$varphi-\$amenable Banach algebras and character module homomorphisms are examined.

Character space‎ ‎Character module homomorphism‎ ‎Arens products‎ ‎\$varphi-\$amenability‎ ‎\$varphi-\$contractibility‎
 Z. Hu, M.S. Monfared and T. Traynor, On character amenable Banach algebras, Studia Math., 193 (2009), pp. 53-78.  E. Kaniuth, A.T.-M. Lau and J. Pym, On character amenability of Banach algebras, J. Math. Anal. Appl., 344 (2008), pp. 942-955.  A.R. Khoddami, On maps preserving strongly zero-products, Chamchuri J. Math., 7 (2015), pp. 16-23.  A.R. Khoddami, On strongly Jordan zero-product preserving maps, Sahand Commun. Math. Anal., 3 (2016), pp. 53-61.  A.R. Khoddami, Strongly zero-product preserving maps on normed algebras induced by a bounded linear functional, Khayyam J. Math., 1 (2015), pp. 107-114.  A.R. Khoddami, The second dual of strongly zero-product preserving maps, Bull. Iran. Math. Soc., 43 (2017), pp. 1781-1790.  A.R. Khoddami and H.R. Ebrahimi Vishki, The higher duals of a Banach algebra induced by a bounded linear functional, Bull. Math. Anal. Appl., 3 (2011), pp. 118-122.  M.S. Monfared, Character amenability of Banach algebras, Math. Proc. Camb. Phil. Soc., 144 (2008), pp. 697-706.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.66851.256 Real and Harmonic Analysis \$L^p\$-Conjecture on Hypergroups \$L^p\$-Conjecture on Hypergroups Tabatabaie Seyyed Mohammad Department of Mathematics, University of Qom, Qom 3716146611, Iran. Haghighifar Faranak Department of Mathematics, University of Qom, Qom 3716146611, Iran. 01 11 2018 12 1 121 130 24 06 2017 10 06 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_31386.html

In this paper, we study \$L^p\$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions  for a weighted Lebesgue space  \$L^p(K,w)\$ to be a convolution Banach algebra, where \$1<p<infty\$, \$K\$ is a locally compact hypergroup and \$w\$ is a weight function on \$K\$.  Among the other things, we also show that if \$K\$ is a locally compact hypergroup and \$p\$ is greater than 2, \$K\$ is compact if and only if \$m(K)\$ is finite and \$fast g\$ exists for all \$f,gin L^p(K)\$, where \$m\$ is a left Haar measure for \$K\$, and in particular, if \$K\$ is discrete, \$K\$ is finite if and only if  the convolution of any two elements of \$L^p(K)\$ exists.

Locally compact hypergroup Weight function Banach algebra \$L^p\$-space
 F. Abtahi, R. Nasr-Isfahani, and A. Rejali, On the \$L^p\$-conjecture for locally compact groups, Arch. Math., 89 (2007), pp. 237-242.  F. Abtahi, R. Nasr-Isfahani, and A. Rejali, Weighted \$L^P\$-conjecture for locally compact groups, Periodica Math. Hun., 60 (2010), pp. 1-11.  W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, 1995.  W.R. Bloom and P. Ressel, Exponentially bounded positive-definite functions on a commutative hypergroup, J. Austral. Math. Soc., (Series A) 61 (1996), pp. 238-248.  C.F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc., 179 (1973), pp. 331-348.  F. Ghahramani and A.R. Medghalchi, Compact multipliers on weighted hypergroup algebras, Math. Proc. Camb. Phil. Soc., 98 (1985), pp. 493-500.  F. Ghahramani and A.R. Medghalchi, Compact multipliers on weighted hypergroup algebras. II, Math. Proc. Camb. Phil. Soc., 100 (1986), pp. 145-149.  R.I. Jewett, Spaces with an abstract convolution of measures, Adv. Math., 18 (1975), pp. 1-101.  M. Lashkarizade Bami, The semisimplicity of \$L^1(K,w)\$ of a weighted commutative hypergroup \$K\$, Acta Math. Sinica, English Series Apr., 24 (2008), pp. 607-610.  Kh. Pourbarat, Amenable weighted hypergroups, J. Sci. I.R. Iran, 7 (1996), pp. 273-276.  M. Rajagopalan, \$L^p\$-conjecture for locally compact groups I, Trans. Amer. Math. Soc., 125 (1966), pp. 216-222.  S. Saeki, The \$L^p\$-conjecture and Young's inequality, Illinois. J. Math., 34 (1990), pp. 615-627.  R. Spector, Apercu de la theorie des hypergroups, Analyse Harmonique sur les Groups de Lie, 643-673, Lec. Notes Math. Ser., 497, Springer, 1975.  R. Spector, Measures invariantes sur les hypergroups, Trans. Amer. Math. Soc., 239 (1978), pp. 147-165.  S.M. Tabatabaie and F. Haghighifar, The weighted KPC-hypergroups, Gen. Math. Notes, 34 (2016), pp. 29-38.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2017.28223 Fuzzy Topology On Fuzzy \$e\$-open Sets, Fuzzy \$e\$-continuity and Fuzzy \$e\$-compactness in Intuitionistic Fuzzy Topological Spaces On Fuzzy \$e\$-open Sets, Fuzzy \$e\$-continuity and Fuzzy \$e\$-compactness in Intuitionistic Fuzzy Topological Spaces Chandrasekar Veerappan Department of Mathematics, Kandaswami Kandar&#039;s College, P-velur-638 182, Tamil Nadu, India. Sobana Durairaj Department of Mathematics, Kandaswami Kandar&#039;s College, P-velur-638 182, Tamil Nadu, India. Vadivel Appachi Department of Mathematics, Annamalai University, Annamalainagar, Tamil Nadu-608 002. 01 11 2018 12 1 131 153 08 01 2017 30 10 2017 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_28223.html

The purpose of this paper is to introduce and study the concepts of fuzzy \$e\$-open set, fuzzy \$e\$-continuity and fuzzy \$e\$-compactness in intuitionistic fuzzy topological spaces. After giving the fundamental concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces, we present intuitionistic fuzzy \$e\$-open sets and intuitionistic fuzzy \$e\$-continuity and other results related topological concepts. Several preservation properties and some characterizations concerning intuitionistic fuzzy \$e\$-compactness have been obtained.

Intuitionistic fuzzy set Intuitionistic fuzzy point Intuitionistic fuzzy topological space Intuitionistic fuzzy \$e\$-open set Intuitionistic fuzzy \$e\$-continuity Intuitionistic fuzzy \$e\$-closure Intuitionistic fuzzy \$e\$-interior Intuitionistic fuzzy \$e\$-compact spaces
 K. Atanassov, Intuitionistic fuzzy sets, VII ITKR's Session, Sofia, 1983, (in Bulgarian).  K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), pp. 87-96.  C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), pp. 182-190.  D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88 (1997), pp. 81-89.  D. Coker and M. Demirci, On intuitionistic fuzzy points NIFS, 1 (1995), pp. 79-84.  E. Ekici, New forms of contra-continuity, Carpathian J. Math., 24 (2008), pp. 37-45.  E. Ekici, On \$e\$-open sets, \$DP^*\$-sets and \$DPepsilon^*\$-sets and decompositions of continuity, Arab. J. Sci. Eng., 33 (2008), pp. 269-282.  E. Ekici, On \$a\$-open sets \$A^*\$-sets and decompositions of continuity and super-continuity, Annales Univ. Sci. Dudapest. Eotvos Sect. Math., 51 (2008), pp. 39-51.  E. Ekici, Some generalizations of almost contra-super-continuity, Filomat, 21 (2007), pp. 31-44.  E. Ekici, On \$e^*\$-open sets and \$(D,S)^*\$-sets, Mathematica Moravica, 13 (2009), pp. 29-36.  H. Gurcay, D. Coker, and A.H. Es, On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math., 5 (1997), pp. 365-378.  V. Seenivasan and K. Kamala, Fuzzy \$e\$-continuity and fuzzy \$e\$-open sets, Ann. Fuzzy Math. Inform., 8 (2014), pp. 141-148.  P. Smets, The degree of belief in a fuzzy event, Information Sciences, 25 (1981), pp. 1-19.  M. Sugeno, An introductory survey of fuzzy control, Information Science, 36 (1985), pp. 59-83.  S.S. Thakur and S. Singh, On fuzzy semi-pre open sets and fuzzy semi-pre continuity, Fuzzy Sets and Systems, (1998), pp. 383-391.  L.A. Zadeh, Fuzzy Sets, Inform. and Control, 8 (1965), pp. 338-353.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2017.28401 Research Paper On Generators in Archimedean Copulas On Generators in Archimedean Copulas Najjari Vadoud Young Researchers and Elite Club, Maragheh branch, Islamic Azad University, Maragheh, Iran. 01 11 2018 12 1 155 166 06 02 2017 31 10 2017 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_28401.html

This study after reviewing  construction methods of generators in Archimedean copulas (AC),  proposes several useful lemmas related with generators of AC. Then a new trigonometric Archimedean family will be shown which is based on cotangent function. The generated new family is able to model the low dependence structures.

Copulas Generator Dependence concepts Measures of association Tails
 T. Bacigal, M. Juranova, and R. Mesiar, On some new constructions of Archimedean copulas and applications to fitting problems, Neural Network World, 20 (2010) pp. 81-90.  T. Bacigal, R. Mesiar, and V. Najjari, Generators of copulas and aggregation, Information science, 306 (2015), pp. 81-87.  T. Bacigal, V. Najjari, R. Mesiar, and Hasan Bal, Additive generators of copulas, Fuzzy Sets and Systems, 264 (2015), pp. 42-50.  F. Durante, R. Foschi, and P. Sarkoci, Distorted copulas: constructions and tail dependence, Comm. Statist. Theory and Methods, 39 (2010), pp. 2288-2301.  F. Durante and C. Sempi, Copula and semicopula transforms, Int. J. Math. Sci., (2005), pp. 645-655.  V. Jagr, M.  Komornikova, and R. Mesiar, Conditioning stable copulas, Neural Network World, 20 (2010), pp. 69-79.  M. Junker and A. May, Measurement of aggregate risk with copulas, Econom. J., 8 (2005), pp. 428-454.  E.P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000.  R. Mesiar, V. Jagr, M. Juranova, and M.  Komornikova, Univariate Conditioning Of Copulas, Kybernetika, 44 (2008), pp. 807-816.  F. Michiels and A. De Schepper, How to improve the fit of Archimedean copulasby means of transforms, Stat Papers, 53 (2012), pp. 345-355.  F. Michiels and A. De Schepper, Understanding copula transforms: a review of dependence properties, Working Paper, 2009.  F. Michiels, I. Koch, and A. De Schepper, A New Method for the Construction of Bivariate Archimedean Copulas Based on the \$lambda\$ Function, Comm. Statist. Theory Methods, 40 (2011), pp. 2670-2679.  V. Najjari and A. Rahimi, A note on ''generalized bivariate copulas and their properties'', Sahand Commun. Math. Anal., 2 (2015), pp. 61-64.  R.B. Nelsen, An introduction to copulas, Second edition, Springer, New York, 2006.  M. Pekarova, Construction of copulas with predetermined properties, PhD. Dissertation, 2012.  B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, 1983.  A. Sklar, Fonctions de repartitiona n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris. 8 (1959), pp. 229-231.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2017.27822 Functional Analysis and Operator Theory Some Properties of Reproducing Kernel Banach and Hilbert Spaces Some Properties of Reproducing Kernel Banach and Hilbert Spaces Hashemi Sababe Saeed Department of Mathematics, Payame Noor University (PNU), P.O. Box, 19395-3697, Tehran, Iran. Ebadian Ali Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran. 01 11 2018 12 1 167 177 22 10 2016 11 10 2017 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_27822.html

This paper is devoted to the study of reproducing kernel Hilbert spaces. We focus on multipliers of reproducing kernel Banach and Hilbert spaces. In particular, we try to extend this concept and prove some related theorems. Moreover, we focus on reproducing kernels in vector-valued reproducing kernel Hilbert spaces. In particular, we extend reproducing kernels to relative reproducing kernels and prove some theorems in this subject.

Reproducing kernel Multipliers Vector-valued spaces
 D. Alpay, P. Jorgensen, and D. Volok, Rlative reproducing kernel Hilbert spaces, Proc. Amer. Math. Soc., 142 (2014), pp. 3889-3895.  N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), pp. 337-404.  A. Berlinet and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Kluwer Academic Publishers, Boston, 2004.  S.S. Dragomir, Semi- inner products and application, Nova Science Publishers, 2004.  G.E. Fasshauer and Q. Ye, Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators, Numer. Math., 119 (2011), pp. 585-611.  K. Fukumizu, G.R. Lanckriet, and B.K. Sriperumbudur, Learning in Hilbert vs. Banach Spaces: A measure embedding viewpoint, Advances in Neural Information Processing Systems, 24 (2011).  J.R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc., 129 (1967), pp. 436-446.  G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc., 100 (1961), pp. 29-43.  B.D. Malviya, A note on semi-inner product algebras, Math.Nachr., 47 (1970), pp. 127-129.  P.V. Pethe and N.K. Thakare, Applications of Riesz's representation theorem in semi-inner product spaces, Indian J. Pure Appl. Math., 7 (1976), pp. 1024-1031.  B. Scholkopf and A.J. Smola, Learning with kernels, MIT Press, Cambridge, Massachusetts, 2002.  A. Smola and S.V.N. Vishwanathan, Introduction to machine learning, Cambridge University Press, 2008.  S. Tsui, Hilbert \$C^*\$-modules: a useful tool, Taiwanese Journal of Mathematics, 1 (1997), pp. 111-126.  Y. Xu and Q. Ye, Constructions of reproducing kernel Banach spaces via generalized Mercer kernels, arXiv:1412.8663v1, 30 (2014).  H. Zhang, Y. Xu, and J. Zhang, Reproducing kernel Banach spaces for machine learning, Journal of Machine Learning Research, 10 (2009), pp. 2741-2775.  D.X. Zhou, Capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory, 49 (2003), pp. 1743-1752.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2017.23647 Complex Analysis On Some Results in the Light of Generalized Relative Ritt Order of Entire Functions Represented by Vector Valued Dirichlet Series On Some Results in the Light of Generalized Relative Ritt Order of Entire Functions Represented by Vector Valued Dirichlet Series Kumar 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. 01 11 2018 12 1 179 186 25 06 2016 12 01 2017 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_23647.html

In this paper, we study some growth properties of entire functions represented by a vector valued Dirichlet series on the basis of generalized relative Ritt order and generalized relative Ritt lower order.

Vector valued Dirichlet series (VVDS) Generalized relative Ritt order Generalized relative Ritt lower order growth
 Q.I. Rahman, The Ritt order of the derivative of an entire function, Ann. Polon. Math. , 17 (1965), pp. 137-140.  C.T. Rajagopal and A.R. Reddy, A note on entire functions represented by Dirichlet series, Ann. Polon. Math., 17 (1965), pp. 199-208.  J.F. Ritt, On certain points in the theory of Dirichlet series, Amer. Jour. Math., 50 (1928), pp. 73-86.  G.S. Srivastava, A note on relative type of entire functions represented by vector valued dirichlet series, Journal of Classicial Analysis, 2 (2013), pp. 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, 12 (2011), pp. 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, Ann. Polon. Math., 13 (1963), pp. 93-100.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2017.27982 Banach Algebra On Character Space of the Algebra of BSE-functions On Character Space of the Algebra of BSE-functions Fozouni Mohammad Department of Mathematics and Statistics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, P.O.Box 163, Gonbad Kavous, Iran. 01 11 2018 12 1 187 194 31 10 2016 17 10 2017 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_27982.html

Suppose that \$A\$ is a semi-simple and commutative Banach algebra. In this paper we try to characterize the character space of the Banach algebra \$C_{rm{BSE}}(Delta(A))\$ consisting of all  BSE-functions on \$Delta(A)\$ where \$Delta(A)\$ denotes the character space of \$A\$. Indeed, in the case that \$A=C_0(X)\$ where \$X\$ is a non-empty locally compact Hausdroff space, we give a complete characterization of \$Delta(C_{rm{BSE}}(Delta(A)))\$ and in the general case we give a partial answer.  Also, using the Fourier algebra, we show that \$C_{rm{BSE}}(Delta(A))\$ is not a \$C^*\$-algebra in general. Finally for some subsets \$E\$ of \$A^*\$, we define the subspace of BSE-like functions on \$Delta(A)cup E\$ and give a nice application of this space related to Goldstine's theorem.

Banach algebra BSE-function Character space Locally compact group
 C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, Springer-Verlag Berlin Heidelberg, edition 3, 2006.  H.G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.  Z. Kamali and M.L. Bami, The Bochner-Schoenberg-Eberlein Property for \${L}^{1}(mathbb{R}^{+})\$, J. Fourier Anal. Appl., 20 (2014), pp. 225-233.  E. Kaniuth, A Course in Commutative Banach Algebras, Springer Verlag, Graduate texts in mathematics, 2009.  E. Kaniuth and A. Ulger, The Bochner-Schoenberg-Eberlein property for commutative Banach algebras, especially Fourier and Fourier-Stieltjes algebras, Trans. Amer. Math. Soc., 362 (2010), pp. 4331-4356.  J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup., 33 (2000), pp. 837–934.  R. Larsen, Functional Analysis: an introduction, Marcel Dekker, New York, 1973.  G.J. Murphy, \$C^*\$-Algebras and Operator Theory, Academic Press Inc, 1990.  J.P. Pier, Amenable Locally Compact Groups, Wiley Interscience, New York, 1984.  W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, New York, 1962.  S.E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein-type theorem, Proc. Amer. Math. Soc., 110 (1990), pp. 149-158.  S.E. Takahasi and O. Hatori, Commutative Banach algebras and BSE-inequalities, Math. Japonica, 37 (1992), pp. 607-614.
SCMA دانشگاه مراغه Sahand Communications in Mathematical Analysis 2322-5807 دانشگاه مراغه 102 10.22130/scma.2018.63978.244 Non-standard Analysis A New Approach to Nonstandard Analysis A New Approach to Nonstandard Analysis Abdeljalil Saghe Cpge de Settat, Lycee Qualifiant Technique PB: 576-Maroc. 01 11 2018 12 1 195 254 13 05 2017 20 08 2018 Copyright © 2018, دانشگاه مراغه. 2018 https://scma.maragheh.ac.ir/article_34187.html

In this paper, we propose a new approach to nonstandard analysis without using the ultrafilters. This method is very simple in practice. Moreover, we construct explicitly the total order relation in the new field of the infinitesimal numbers. To illustrate the importance of this work, we suggest comparing  a few applications of this approach with the former methods.

Nonstandard analysis Hyppereals Internal set theory
 F. Bagarello, Nonstandard variational calculus with applications to classical mechanics. I. An existence criterion, Internat. J. Theoret. Phys., 38 (1999), pp. 1569-1592.  F. Bagarello, Nonstandard variational calculus with applications to classical mechanics. II. The inverse problem and more, Internat. J. Theoret. Phys., 38 (1999), pp. 1593-1615.  F. Bagarello and S. Valenti, Nonstandard analysis in classical physics and quantum formal scattering, Int. J. Theor. Phys., 27 (1988), pp. 557-566.  V. Benci and M. Di Nasso, Alpha-theory: an elementary axiomatics for nonstandard analysis, Expo. Math., 21 (2003), pp. 355-386.  V. Benci, M. Di Nasso, and M. Forti, The eightfold path to nonstandard analysis, Nonstandard Methods and Applications in Mathematics, 25 (2006), pp. 3-44.  J.M. Borwein, D.M. Bradley, and R.E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math., 121 (2000), pp. 247-296.  N. Cutland, Nonstandard Analysis and its Applications, Cambridge University Press, Cambridge, 1988.  F. Diener and G. Reeb, Analyse non standard, Hermann, 1989.  R. Fittler, Asymptotic nonstandard quantum electrodynamics, J. Math. Phys., 34 (1993), pp. 1692-1724.  P. Fletcher, K. Hrbacek, V. Kanovei, M. Katz, C. Lobry, and S. Sanders, Approaches to analysis with infinitesimals following Robinson, Nelson, and others, Real Analysis Exchange, 42 (2017), pp. 193-252.  R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Springer, Graduate Texts in Mathematics 188, 1998.  A. Granville, Harald Cramer and the distribution of prime numbers, Scand. Actuar. J., 1 (1995), pp. 12-28.  E. Nelson, Internal Set Theory: a new approach to Nonstandard Analysis, Bull. A.M.S., 83 (1977), pp. 1165-1198.  A. Robinson, Non-standard Analysis, North-Holland, Amsterdam, 1966.  W. Rudin, Analyse reelle et complexe, Masson, 1978.  P. Tauvel, Cours dalgebre: agregation de mathematiques, Dunod, 1999.