2018
12
1
0
254
On Some Properties of the Max Algebra System Over Tensors
2
2
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.
1

1
14


Ali Reza
Shojaeifard
Department of Mathematics and Statistics, Faculty of Basic Sciences, Imam Hossein Comprehensive University, Tehran, Iran.
Department of Mathematics and Statistics,
Iran
ashojaeifard@ihu.ac.ir


Hamid Reza
Afshin
Department of Mathematics, Faculty of Mathematical Sciences, ValieAsr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran.
Department of Mathematics, Faculty of Mathematical
Iran
afshin@vru.ac.ir
Tensor
Max algebra
Left (right) inverse
Direct Product
Eigenvalue
[[1] H.R. Afshin and A.R. Shojaeifard, A max version of Perron Frobenuos theorem for nonnegative tensor, Ann. Funct. Anal., 6 (2015), pp. 145154.##[2] R. Bapat, A max version of the Perron Frobenius theorem, Linear Algebra Appl., (1998), pp. 318.##[3] F. Baccelli, G. Cohen, G. Olsder, and J. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley, 1992.##[4] P. Butkovic and M. Fiedler, Tropical tensor product and beyond, School of Mathematics University of Birmingham, 2011. ##[5] C. Bu, X. Zhang, J. Zhou, W. Wang, and Y. Wei, The inverse, rank and product of tensors, Linear Algebra Appl., 446 (2014), pp. 269280.##[6] K.C. Chang, K. Pearson, and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), pp. 507520.##[7] R.A. CuninghameGreen, Minimax Algbera, SpringerVerlag, 1979.##[8] V. Loan, Future directions in tensor based computation and modeling, NSF Workshop Report in Arlington, Virginia, USA, 2009. ##[9] K. Pearson, Essentially positive tensors, Int. J. Algebra., 4 (2010), pp. 421427.##[10] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 13021324.##[11] J.Y. Shao, A general product of tensors with applications, Linear Algebra Appl., 439 (2013), pp. 23502366.##]
Inequality Problems of Equilibrium Problems with Application
2
2
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.
1

15
26


Ayed
Eleiwis Hashoosh
Department of Mathematics, Faculty of Education for Pure Sciences, University of ThiQar, Nasiriyah, Iraq.
Department of Mathematics, Faculty of Education
Iran
ayed197991@yahoo.com


Mohsen
Alimohammady
Department of mathematics, University of Mazandaran, Babolsar, Iran.
Department of mathematics, University of
Iran
amohsen@umz.ac.ir


Haiffa
Mohsen Buite
Department of Mathematics, Faculty of Education for Pure Sciences, University of ThiQar, Nasiriyah, Iraq.
Department of Mathematics, Faculty of Education
Iran
alrfayalrfay@gmail.com
Monotone bifunction
Equilibrium problem
KKM technique
Differential inclusion
[[1] M. Ait Mansour, Z. Chbani, and H. Riahi, Recession bifunction and solvability of noncoercive equilibrium problems, Commun. Appl. Anal., 7, (2003), pp. 369 377.##[2] M. Alimohammady and A.E. Hashoosh, Existence Theorems For $alpha(u,v)$monotone of nonstandard Hemivariational Inequality, Advances in Math., 10, (2015), pp. 32053212.##[3] Q.H. Ansari and J.C. Yao, An existence result for the generalized vector equlibrium problem, Appl. Math. Lett., 19, (1999), pp. 5356.##[4] M.Bianchi and S. Schaible, Equilibrium problems under generalized convexity and generalized monotonicity, J. Global Optim., 30, (2004), pp. 121134.##[5] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63, (1994), pp. 123145.##[6] F.E. Browder, The solvability of nonlinear functional equations, Duke Math. J., 30, (1963), pp. 557566.##[7] S. Carl, V. Khoi Le, and D. Motreanu, Nonsmooth variational problems and their inequalities, Springer Monographs in Mathematics, Springer, New York, (2007).##[8] O. Chadli, Y. Chiang, and S. Huang, Topological pseudomonotonicity and vector equilibm problems, J. Math. Anal. Appl., 270, (2002), pp. 435450.##[9] K. Fan, A generalization of Tychonoffs fixed point theorem, Math. Ann., 142, (1961), pp. 305310.##[10] Y.P. Fang and N.J. Huang, Variationallike inequalities with generalized monotone mappings in Banach spaces, J. Optim. Theory Appl., 118, (2003), pp. 327338.##[11] N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, optimization, 59, (2010), pp. 147160.##[12] A.E. Hashoosh and M. Alimohammady, On WellPosedness Of Generalized Equilibrium Problems Involving αMonotone Bifunction, J. Hyperstruct., 5, (2016), pp. 151168.##[13] A.E. Hashoosh and M. Alimohammady, Bα,βOperator and Fitzpatrick Functions, Jordan J. Math. Stat., 1, (2017), pp. 259278.##[14] A.E. Hashoosh, M. Alimohammady, and M.K. Kalleji, Existence Results for Some Equilibrium Problems involving αMonotone Bifunction, Int. J. Math. Math. Sci., 2016, (2016), pp. 15.##[15] 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.##[16] B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur ndimensionale Simplexe, Fund. Math., 14, (1929), pp. 132137.##[17] N.K. Mahato and C. Nahak, Mixed equilibrium problems with relaxed αmonotone mapping in Banach spaces, Rend. Circ. Mat. Palermo, (2013).##[18] 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. 60016010.##[19] A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and equilibrium problem, J. Optim. Theory Appl., 133, (2007), pp. 359370.##[20] R.U. Verma, On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators, J. Math. Anal. Appl. , 213, (1997), pp. 387392.##[21] R.U. Verma, On monotone nonlinear variational inequality problems, Comment. Math. Univ. Carolin., 39, (1998), pp. 9198.##]
On Regular Generalized $delta$closed Sets in Topological Spaces
2
2
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.
1

27
37


Somasundaram
Rajakumar
Department of Science and Humanities, Krishnasamy College of Engineering and Technology, Cuddalore, Tamil Nadu608 002, India.
Department of Science and Humanities, Krishnasamy
Iran
srkumar277@gmail.com
$rgdelta$closed set
$delta$closed set
$gdelta$closed set
[[1] A.A. Omari and M.S.M. Noorani, On Generalized $b$closed sets, Bull. Malays. Math. Sci. Soc., 32 (2009), pp. 1930.##[2] S.P. Arya and T. Nour, Characterizations of $s$normal spaces, Indian J. Pure and Applied Maths., 21 (1990), pp. 717719.##[3] P. Bhattacharya and B.K. Lahiri, Semigeneralized closed sets on topology, Indian J. Maths., 29 (1987), pp. 375382.##[4] Y. Gnanambal, On Generalized Preregular closed sets in topological spaces, Indian J. Pure Appl. Math., 28 (1997), pp. 351360.##[5] D. Iyappan and N. Nagaveni, On Semi generalized $b$closed set, Nat. Sem. On Mat. Comp. Sci., Jan., (2010), Proc.6.##[6] N. Levine, Generalized closed sets in topology, Rend Circ., Mat. Palermo., 19 (1970), pp. 8996.##[7] N. Levine, Semiopen sets and Semicontinuity in topological spaces, Amer. Math. Monthly., 70 (1963), pp. 3641.##[8] 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. 5163.##[9] 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. 3342.##[10] A.S. Mashhour Abd ElMonsef. M. E. and EiDeeb S.N, On Precontinuous and weak Precontinuous mappings, Proc. Math. Phys. Soc. Egypt., 53 (1982), pp. 4753.##[11] O. Njastad, On Some classes of nearly open sets, Pacific J. Math., 15 (1965), pp. 961970.##[12] N. Nagaveni, Studies on generalized homeomorphisms in topological spaces, Ph. D., Thesis, Bharathiar University, Coimbatore 1999.##[13] L. Vinayagamoorthi and N. Nagaveni, On Generalized$alpha b$ closed sets, Proceedings, ICMDPushpa Publications, Vol. 1., 201011.##[14] M.K.R.S. Veerakumar, Between closed sets and $g$closed sets, Mem. Fac. Sci.Kochi Univ., Math., 21 (2000), pp. 119.##[15] N.V. Velico, Hclosed topological spaces, Amer. Math. Soc. Transl., 78 (1968), pp. 103118.##[16] N. Palaniappan and K. C. Rao, Regular generalized closed sets, Kyungpook Math. J., 33 (1993), pp. 211219.##]
The Solvability of ConcaveConvex Quasilinear Elliptic Systems Involving $p$Laplacian and Critical Sobolev Exponent
2
2
In this work, we study the existence of nontrivial multiple solutions for a class of quasilinear elliptic systems equipped with concaveconvex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and PalaisSmale condition, we prove the existence and multiplicity results of positive solutions.
1

39
57


Somayeh
Khademloo
Department of Basic Sciences, Babol Noushirvani University of Technology, 4714871167, Babol, Iran.
Department of Basic Sciences, Babol Noushirvani
Iran
s.khademloo@nit.ac.ir


Saeed
Khanjany Ghazi
Department of Basic Sciences, Babol Noushirvani University of Technology, 4714871167, Babol, Iran.
Department of Basic Sciences, Babol Noushirvani
Iran
s.khanjany@gmail.com
Variational methods
Nehari manifold
Dirichlet boundary condition
Critical Sobolev exponent
[[1] 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. 445455.##[2] A. Ambrosetti, H. Brezis, and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), pp.519543.##[3] A. Ambrosetti, J. GarciaAzorero, and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), pp. 219242.##[4] 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. 771787.##[5] 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. 941957.##[6] T. Barstch and M. Willem, On a elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), pp. 35553561.##[7] 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. 111.##[8] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), pp. 437477.##[9] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), pp. 486490.##[10] K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a signchanging weight function, J. Differ. Eq.s, 193 (2003), pp. 481499.##[11] K.J. Brown and T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and signchanging weight function, J. Math. Anal. Appl., 337 (2008), pp. 13261336.##[12] 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. 12411257.##[13] T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concaveconvex nonlinearities, Nonlin. Anal., 71 (2009), pp. 26882698.##[14] T.S. Hsu, Multiplicity results for PLaplacian with critical nonlinearity of concaveconvex type and signchanging weight functions, Abs. and Appl Anal. Article ID 652109, 24 pages, 2009.##[15] G. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), pp. 281304.##[16] T.F. Wu, The Nehari manifold for a semilinear elliptic system involving signchanging weight functions, Nonlin. Anal., 68 (2008), pp. 17331745.##[17] T.F. Wu, On semilinear elliptic equations involving concaveconvex nonlinearities and signchanging weight function, J. Math. Anal. Appl., 318 (2006), pp. 253270.##]
$(1)$Weak Amenability of Second Dual of Real Banach Algebras
2
2
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.
1

59
88


Hamidreza
Alihoseini
Department of Mathematics, Faculty of Science, University of Arak, 3815688349, Arak, Iran.
Department of Mathematics, Faculty of Science,
Iran
hr_alihoseini@yahoo.com


Davood
Alimohammadi
Department of Mathematics, Faculty of Science, University of Arak, 3815688349, Arak, Iran.
Department of Mathematics, Faculty of Science,
Iran
alimohammadi.davood@gmail.com
Banach algebra
Banach module
Complexification
Derivation
$(1)$Weak amenability
[[1] D. Alimohammadi and A. Ebadian, Hedberg's theorem in real Lipschitz algebras, Indian J. Pure Appl. Math., 32 (2010), pp. 14701493.##[2] D. Alimohammadi and T.G. Honary, Contractibility, amenability and weak amenability of real Banach algebras, J. Aanalysis, (9)(2001), pp. 6988. ##[3] R. Arens, The adjoint of a bilinear operation, Proc. Math. Amer. Soc., 2 (1951), pp. 839848.##[4] 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. 359377. ##[5] F.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer Verlag, New York, 1973.##[6] H.G. Dales, Banach Algebras and Automatic Continuity, Oxford University Press, 2000.##[7] J. Duncan and S.A.R. Hosseinioun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburg Sect. A., 84 (1979), pp. 309325.##[8] M. Eshaghi Gordji, S.A.R. Hosseinioun, and A. Valadkhani, On (1)weak amenability of Banach algebras, Math. Reports, 15 (65), (2013), pp. 271279.##[9] T.G. Honary and S. Moradi, On the maximal ideal space of extended analytic Lipschitz algebras, Quaestiones Mathematicae, 30 (2007), pp. 349353.##[10] S.A.R. Hosseinioun and A. Valadkhani, (1)Weak amenability of unitized Banach algebras, Europ. J. Pure Appl. Math., 9 (2016), pp. 231239.##[11] 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. 3948.##[12] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).##[13] 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. SpringerVerlag), 1359 (1987), pp. 191198.##[14] S.H. Kulkarni and B.V. Limaye, Gleason parts of real function algebras, Canad. J. Math., 33 (1981), pp. 181200.##[15] S.H. Kulkarni and B.V. Limaye, Real Function Algebras, Marcel Dekker, Inc. New York, 1992.##[16] 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. 389404.##[17] A. Medghalchi and T. Yazdanpanah, Problems concerning nweak amenability of a Banach algebra, Czechoslovak Math. J., 55 (2005), pp. 863876.##[18] T.W. Palmer, Banach Algebras, the General Theory of *Algebras, Vol. 1: Algebras and Banach Algebras, Cambridge University Press, Cambridge, 1994.##[19] D.R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., (13) (1963), pp. 13871399.##[20] D.R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc, 111 (1964), pp. 240272.##]
The Norm Estimates of PreSchwarzian Derivatives of Spirallike Functions and Uniformly Convex $alpha$spirallike Functions
2
2
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}geqleftfrac{zf'(z)}{f(z)}1right.]In the present paper, we shall give the estimate of the norm of the preSchwarzian derivative $mathrm{T}_f=f''/f'$ where $mathrm{T}_f=sup_{zin Delta} (1z^2)mathrm{T}_f(z)$ for the functions in $SP_{p}(alpha)$.
1

89
96


Zahra
Orouji
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.
Department of Mathematics, Faculty of Science,
Iran
z.oroujy@yahoo.com


Rasul
Aghalary
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.
Department of Mathematics, Faculty of Science,
Iran
raghalary@yahoo.com
PreSchwarzian derivative
Spirallike function
Uniformly convex function
[[1] J. Becker and CH. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. Reine Angew. Math., 354 (1984), pp. 7494.##[2] P.L. Duren, Univalent Functions, Grundlehren Math. Wiss., 259 (1983), SpringerVerlay. New York.##[3] A.W. Goodman, On uniformly convex functions, Ann. Pol. Math., 57 (1991), pp. 8792.##[4] A.W. Goodman, On uniformly convex functions, J. Math. Anal. Appl., 155 (1991), pp. 364370.##[5] 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. 179200.##[6] Y.C. Kim and T. Sugawa, Norm estimates of the preSchwarzian derivative for certain classes of univalent functions, Proc. Edinburgh Math. Soc., 49 (2006), pp. 131143.##[7] W. Ma and D. Minda, Uniformly convex functions, Ann. Pol. Math., 57 (1992), pp. 165175.##[8] Y. Okuyama, The norm estimates of preSchwarzian derivatives of spirallike functions, Complex Variables Theory Appl., 42 (2000), pp. 225239.##[9] V. Ravichandran, C. Selvaraj, and R. Rajagopal, On uniformly convex spiral functions and uniformly spirallike functions, Soochow J. Math., 29 (2003), pp. 393405.##[10] F. RΦnning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), pp. 189196.##[11] S. Yamashita, Norm estimates for function starlike or convex of order alpha, Hokkaido Math., 28 (1999), pp. 217230.##]
Numerical Reckoning Fixed Points in $CAT(0)$ Spaces
2
2
In this paper, first we use an example to show the efficiency of $M$ iteration process introduced by Ullah and Arshad [4] 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.
1

97
111


Kifayat
Ullah
Department of Mathematics, University of Science and Technology
Bannu, KPK Pakistan.
Department of Mathematics, University of
Iran
kifayatmath@yahoo.com


Hikmat
Khan
Department of Mathematics, University of Science and Technology Bannu, KPK Pakistan.
Department of Mathematics, University of
Iran
hikmatnawazkhan@gmail.com


Muhammad
Arshad
Department of Mathematics, International Islamic University, H10, Islamabad  44000, Pakistan.
Department of Mathematics, International
Iran
marshadzia@iiu.edu.pk
Suzuki generalized nonexpansive mapping, $CAT(0)$ space
iteration process, $Delta$convergence, Strong convergence
[[1] M. Abbas and T. Nazir, A new faster iteration process applied to constrained minimization and feasibility problems, Math. Vesn., 66 (2014), pp. 223234.##[2] 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. 6179.##[3] M. Bridson and A. Heaflinger, Metric Space of Nonpositive Curvature, SpringerVerlag, Berlin, 1999.##[4] F. Bruhat and J. Tits, Groupes reductifs sur un corps local. I, Donnees radicielles valuees Inst Hauts Etudea Sci. Publ. Math., 41 (1972), pp. 5251.##[5] D. Burago, Y. Burago and S. Inavo, A course in Metric Geometry, Vol. 33, Americal Mathematical Socity, Providence, RI, 2001.##[6] 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. 345357.##[7] S. Dhompongsa, W.A. Kirk, and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal., 65 (2006), pp. 762772.##[8] S. Dhompongsa, W.A. Kirk, and B. Panyanak, Nonexpansive setvalued mappings in metric and Banach spaces, J. Nonlinear and convex Anal., 8 (2007), pp. 3545.##[9] S. Dhompongsa and B. Panyanak, On $Delta $convergence theorem in $CAT(0)$ Spaces, Comput. Math. Appl., 56 (2008), pp. 25722579.##[10] A. Gharajelo and H. Dehghan, Convergence Theorems for Strict PseudoContractions in $CAT(0)$ Metric Spaces, Filomat, 31 (2017), pp. 19671971.##[11] F. Gursoy and V. Karakaya, A PicardS hybrid type iteration method for solving a differential equation with retarded argument, arXiv:1403.2546v2 (2014).##[12] S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc., 44 (1974), 147150.##[13] 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. 510526.##[14] S.H. Khan, A PicardMann hybrid iterative process, Fixed Point Th. Appl., 2013, Article ID 69 (2013).##[15] W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc., 4 (1953), pp. 506510.##[16] A. Moudafi, KrasnoselskiMann iteration for hierarchical fixed point problems, Inverse Probl., 23 (2007), pp. 16351640.##[17] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), pp. 217229.##[18] W. Phuengrattana, Approximating fixed points of Suzukigeneralized nonexpansive mappings, Nonlinear Anal. Hybrid Syst. 5 (2011), pp. 583590.##[19] W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SPiterations for continuous functions on an arbitrary interval, J. Comp. Appl. Math., 235 (2011), pp. 30063014.##[20] H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Am. Math. Soc., 44 (1974), pp. 375380.##[21] 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. 60126023.##[22] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl., 340 (2008), pp. 10881095.##[23] 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. 147155.##[24] K. Ullah and M. Arshad, Numerical reckoning fixed points for Suzuki generalized nonexpansive mapping via new iteration process, Filomat, 32 (2018), pp. 187196.##[25] R. Wangkeeree, H. Dehghan, Strong and $Delta$convergence of Moudafi's iterative scheme in $CAT(0)$ spaces, J. Nonlinear Convex Anal., 16 (2015), 299309.##]
A Certain Class of Character Module Homomorphisms on Normed Algebras
2
2
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 nonzero $B$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.
1

113
120


Ali Reza
Khoddami
Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box 3619995161316, Shahrood, Iran.
Faculty of Mathematical Sciences, Shahrood
Iran
khoddami.alireza@shahroodut.ac.ir
Character space
Character module homomorphism
Arens products
$varphi$amenability
$varphi$contractibility
[[1] Z. Hu, M.S. Monfared and T. Traynor, On character amenable Banach algebras, Studia Math., 193 (2009), pp. 5378.##[2] E. Kaniuth, A.T.M. Lau and J. Pym, On character amenability of Banach algebras, J. Math. Anal. Appl., 344 (2008), pp. 942955.##[3] A.R. Khoddami, On maps preserving strongly zeroproducts, Chamchuri J. Math., 7 (2015), pp. 1623.##[4] A.R. Khoddami, On strongly Jordan zeroproduct preserving maps, Sahand Commun. Math. Anal., 3 (2016), pp. 5361.##[5] A.R. Khoddami, Strongly zeroproduct preserving maps on normed algebras induced by a bounded linear functional, Khayyam J. Math., 1 (2015), pp. 107114.##[6] A.R. Khoddami, The second dual of strongly zeroproduct preserving maps, Bull. Iran. Math. Soc., 43 (2017), pp. 17811790. ##[7] 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. 118122.##[8] M.S. Monfared, Character amenability of Banach algebras, Math. Proc. Camb. Phil. Soc., 144 (2008), pp. 697706.##]
$L^p$Conjecture on Hypergroups
2
2
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.
1

121
130


Seyyed Mohammad
Tabatabaie
Department of Mathematics, University of Qom, Qom 3716146611, Iran.
Department of Mathematics, University of
Iran
sm.tabatabaie@qom.ac.ir


Faranak
Haghighifar
Department of Mathematics, University of Qom, Qom 3716146611, Iran.
Department of Mathematics, University of
Iran
f.haghighifar@yahoo.com
Locally compact hypergroup
Weight function
Banach algebra
$L^p$space
[[1] F. Abtahi, R. NasrIsfahani, and A. Rejali, On the $L^p$conjecture for locally compact groups, Arch. Math., 89 (2007), pp. 237242.##[2] F. Abtahi, R. NasrIsfahani, and A. Rejali, Weighted $L^P$conjecture for locally compact groups, Periodica Math. Hun., 60 (2010), pp. 111.##[3] W.R. Bloom and H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, De Gruyter, Berlin, 1995.##[4] W.R. Bloom and P. Ressel, Exponentially bounded positivedefinite functions on a commutative hypergroup, J. Austral. Math. Soc., (Series A) 61 (1996), pp. 238248.##[5] C.F. Dunkl, The measure algebra of a locally compact hypergroup, Trans. Amer. Math. Soc., 179 (1973), pp. 331348.##[6] F. Ghahramani and A.R. Medghalchi, Compact multipliers on weighted hypergroup algebras, Math. Proc. Camb. Phil. Soc., 98 (1985), pp. 493500. ##[7] F. Ghahramani and A.R. Medghalchi, Compact multipliers on weighted hypergroup algebras. II, Math. Proc. Camb. Phil. Soc., 100 (1986), pp. 145149.##[8] R.I. Jewett, Spaces with an abstract convolution of measures, Adv. Math., 18 (1975), pp. 1101.##[9] 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. 607610.##[10] Kh. Pourbarat, Amenable weighted hypergroups, J. Sci. I.R. Iran, 7 (1996), pp. 273276.##[11] M. Rajagopalan, $L^p$conjecture for locally compact groups I, Trans. Amer. Math. Soc., 125 (1966), pp. 216222.##[12] S. Saeki, The $L^p$conjecture and Young's inequality, Illinois. J. Math., 34 (1990), pp. 615627.##[13] R. Spector, Apercu de la theorie des hypergroups, Analyse Harmonique sur les Groups de Lie, 643673, Lec. Notes Math. Ser., 497, Springer, 1975.##[14] R. Spector, Measures invariantes sur les hypergroups, Trans. Amer. Math. Soc., 239 (1978), pp. 147165.##[15] S.M. Tabatabaie and F. Haghighifar, The weighted KPChypergroups, Gen. Math. Notes, 34 (2016), pp. 2938.##]
On Fuzzy $e$open Sets, Fuzzy $e$continuity and Fuzzy $e$compactness in Intuitionistic Fuzzy Topological Spaces
2
2
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.
1

131
153


Veerappan
Chandrasekar
Department of Mathematics, Kandaswami Kandar's College, Pvelur638 182, Tamil Nadu, India.
Department of Mathematics, Kandaswami Kandar'
Iran
vckkc3895@gmail.com


Durairaj
Sobana
Department of Mathematics, Kandaswami Kandar's College, Pvelur638 182, Tamil Nadu, India.
Department of Mathematics, Kandaswami Kandar'
Iran
slmsobana@gmail.com


Appachi
Vadivel
Department of Mathematics, Annamalai University, Annamalainagar, Tamil Nadu608 002.
Department of Mathematics, Annamalai University,
Iran
avmaths@gmail.com
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
[[1] K. Atanassov, Intuitionistic fuzzy sets, VII ITKR's Session, Sofia, 1983, (in Bulgarian).##[2] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), pp. 8796.##[3] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), pp. 182190.##[4] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88 (1997), pp. 8189.##[5] D. Coker and M. Demirci, On intuitionistic fuzzy points NIFS, 1 (1995), pp. 7984.##[6] E. Ekici, New forms of contracontinuity, Carpathian J. Math., 24 (2008), pp. 3745.##[7] E. Ekici, On $e$open sets, $DP^*$sets and $DPepsilon^*$sets and decompositions of continuity, Arab. J. Sci. Eng., 33 (2008), pp. 269282.##[8] E. Ekici, On $a$open sets $A^*$sets and decompositions of continuity and supercontinuity, Annales Univ. Sci. Dudapest. Eotvos Sect. Math., 51 (2008), pp. 3951.##[9] E. Ekici, Some generalizations of almost contrasupercontinuity, Filomat, 21 (2007), pp. 3144.##[10] E. Ekici, On $e^*$open sets and $(D,S)^*$sets, Mathematica Moravica, 13 (2009), pp. 2936.##[11] H. Gurcay, D. Coker, and A.H. Es, On fuzzy continuity in intuitionistic fuzzy topological spaces, J. Fuzzy Math., 5 (1997), pp. 365378.##[12] V. Seenivasan and K. Kamala, Fuzzy $e$continuity and fuzzy $e$open sets, Ann. Fuzzy Math. Inform., 8 (2014), pp. 141148.##[13] P. Smets, The degree of belief in a fuzzy event, Information Sciences, 25 (1981), pp. 119.##[14] M. Sugeno, An introductory survey of fuzzy control, Information Science, 36 (1985), pp. 5983.##[15] S.S. Thakur and S. Singh, On fuzzy semipre open sets and fuzzy semipre continuity, Fuzzy Sets and Systems, (1998), pp. 383391.##[16] L.A. Zadeh, Fuzzy Sets, Inform. and Control, 8 (1965), pp. 338353.##]
On Generators in Archimedean Copulas
2
2
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.
1

155
166


Vadoud
Najjari
Young Researchers and Elite Club, Maragheh branch, Islamic Azad University, Maragheh, Iran.
Young Researchers and Elite Club, Maragheh
Iran
fnajjary@yahoo.com
Copulas
Generator
Dependence concepts
Measures of association
Tails
[[1] 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. 8190.##[2] T. Bacigal, R. Mesiar, and V. Najjari, Generators of copulas and aggregation, Information science, 306 (2015), pp. 8187.##[3] T. Bacigal, V. Najjari, R. Mesiar, and Hasan Bal, Additive generators of copulas, Fuzzy Sets and Systems, 264 (2015), pp. 4250.##[4] F. Durante, R. Foschi, and P. Sarkoci, Distorted copulas: constructions and tail dependence, Comm. Statist. Theory and Methods, 39 (2010), pp. 22882301.##[5] F. Durante and C. Sempi, Copula and semicopula transforms, Int. J. Math. Sci., (2005), pp. 645655.##[6] V. Jagr, M. Komornikova, and R. Mesiar, Conditioning stable copulas, Neural Network World, 20 (2010), pp. 6979.##[7] M. Junker and A. May, Measurement of aggregate risk with copulas, Econom. J., 8 (2005), pp. 428454.##[8] E.P. Klement, R. Mesiar, and E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000.##[9] R. Mesiar, V. Jagr, M. Juranova, and M. Komornikova, Univariate Conditioning Of Copulas, Kybernetika, 44 (2008), pp. 807816.##[10] F. Michiels and A. De Schepper, How to improve the fit of Archimedean copulasby means of transforms, Stat Papers, 53 (2012), pp. 345355.##[11] F. Michiels and A. De Schepper, Understanding copula transforms: a review of dependence properties, Working Paper, 2009.##[12] 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. 26702679.##[13] V. Najjari and A. Rahimi, A note on ''generalized bivariate copulas and their properties'', Sahand Commun. Math. Anal., 2 (2015), pp. 6164.##[14] R.B. Nelsen, An introduction to copulas, Second edition, Springer, New York, 2006.##[15] M. Pekarova, Construction of copulas with predetermined properties, PhD. Dissertation, 2012.##[16] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, 1983.##[17] A. Sklar, Fonctions de repartitiona n dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris. 8 (1959), pp. 229231.##]
Some Properties of Reproducing Kernel Banach and Hilbert Spaces
2
2
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 vectorvalued reproducing kernel Hilbert spaces. In particular, we extend reproducing kernels to relative reproducing kernels and prove some theorems in this subject.
1

167
177


Saeed
Hashemi Sababe
Department of Mathematics, Payame Noor University (PNU), P.O. Box, 193953697, Tehran, Iran.
Department of Mathematics, Payame Noor University
Iran
hashemi_1365@yahoo.com


Ali
Ebadian
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.
Department of Mathematics, Faculty of Science,
Iran
ebadian.ali@gmail.com
Reproducing kernel
Multipliers
Vectorvalued spaces
[[1] D. Alpay, P. Jorgensen, and D. Volok, Rlative reproducing kernel Hilbert spaces, Proc. Amer. Math. Soc., 142 (2014), pp. 38893895.##[2] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), pp. 337404.##[3] A. Berlinet and C. ThomasAgnan, Reproducing kernel Hilbert spaces in probability and statistics, Kluwer Academic Publishers, Boston, 2004.##[4] S.S. Dragomir, Semi inner products and application, Nova Science Publishers, 2004.##[5] 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. 585611.##[6] 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).##[7] J.R. Giles, Classes of semiinnerproduct spaces, Trans. Amer. Math. Soc., 129 (1967), pp. 436446.##[8] G. Lumer, Semiinnerproduct spaces, Trans. Amer. Math. Soc., 100 (1961), pp. 2943.##[9] B.D. Malviya, A note on semiinner product algebras, Math.Nachr., 47 (1970), pp. 127129.##[10] P.V. Pethe and N.K. Thakare, Applications of Riesz's representation theorem in semiinner product spaces, Indian J. Pure Appl. Math., 7 (1976), pp. 10241031.##[11] B. Scholkopf and A.J. Smola, Learning with kernels, MIT Press, Cambridge, Massachusetts, 2002.##[12] A. Smola and S.V.N. Vishwanathan, Introduction to machine learning, Cambridge University Press, 2008.##[13] S. Tsui, Hilbert $C^*$modules: a useful tool, Taiwanese Journal of Mathematics, 1 (1997), pp. 111126.##[14] Y. Xu and Q. Ye, Constructions of reproducing kernel Banach spaces via generalized Mercer kernels, arXiv:1412.8663v1, 30 (2014).##[15] H. Zhang, Y. Xu, and J. Zhang, Reproducing kernel Banach spaces for machine learning, Journal of Machine Learning Research, 10 (2009), pp. 27412775.##[16] D.X. Zhou, Capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory, 49 (2003), pp. 17431752.##]
On Some Results in the Light of Generalized Relative Ritt Order of Entire Functions Represented by Vector Valued Dirichlet Series
2
2
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.
1

179
186


Sanjib
Kumar Datta
Department of Mathematics, University of Kalyani, P.O.Kalyani, DistNadia, PIN741235, West Bengal, India.
Department of Mathematics, University of
Iran
sanjib_kr_datta@yahoo.co.in


Tanmay
Biswas
Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.Krishnagar, DistNadia, PIN741101, West Bengal, India.
Rajbari, Rabindrapalli, R. N. Tagore Road,
Iran
tanmaybiswas_math@rediffmail.com
Vector valued Dirichlet series (VVDS)
Generalized relative Ritt order
Generalized relative Ritt lower order
growth
[[1] Q.I. Rahman, The Ritt order of the derivative of an entire function, Ann. Polon. Math. , 17 (1965), pp. 137140.##[2] C.T. Rajagopal and A.R. Reddy, A note on entire functions represented by Dirichlet series, Ann. Polon. Math., 17 (1965), pp. 199208.##[3] J.F. Ritt, On certain points in the theory of Dirichlet series, Amer. Jour. Math., 50 (1928), pp. 7386.##[4] G.S. Srivastava, A note on relative type of entire functions represented by vector valued dirichlet series, Journal of Classicial Analysis, 2 (2013), pp. 6172.##[5] 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. 26522659.##[6] B.L. Srivastava, A study of spaces of certain classes of vector valued Dirichlet series, Thesis, I. I. T., Kanpur, 1983.##[7] R.P. Srivastav and R.K. Ghosh, On entire functions represented by Dirichlet series, Ann. Polon. Math., 13 (1963), pp. 93100.##]
On Character Space of the Algebra of BSEfunctions
2
2
Suppose that $A$ is a semisimple 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 BSEfunctions 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 nonempty 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 BSElike functions on $Delta(A)cup E$ and give a nice application of this space related to Goldstine's theorem.
1

187
194


Mohammad
Fozouni
Department of Mathematics and Statistics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, P.O.Box 163, Gonbad Kavous, Iran.
Department of Mathematics and Statistics,
Iran
fozouni@hotmail.com
Banach algebra
BSEfunction
Character space
Locally compact group
[[1] C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, SpringerVerlag Berlin Heidelberg, edition 3, 2006.##[2] H.G. Dales, Banach Algebras and Automatic Continuity, Clarendon Press, Oxford, 2000.##[3] Z. Kamali and M.L. Bami, The BochnerSchoenbergEberlein Property for ${L}^{1}(mathbb{R}^{+})$, J. Fourier Anal. Appl., 20 (2014), pp. 225233.##[4] E. Kaniuth, A Course in Commutative Banach Algebras, Springer Verlag, Graduate texts in mathematics, 2009.##[5] E. Kaniuth and A. Ulger, The BochnerSchoenbergEberlein property for commutative Banach algebras, especially Fourier and FourierStieltjes algebras, Trans. Amer. Math. Soc., 362 (2010), pp. 43314356.##[6] J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. Ecole Norm. Sup., 33 (2000), pp. 837–934.##[7] R. Larsen, Functional Analysis: an introduction, Marcel Dekker, New York, 1973.##[8] G.J. Murphy, $C^*$Algebras and Operator Theory, Academic Press Inc, 1990.##[9] J.P. Pier, Amenable Locally Compact Groups, Wiley Interscience, New York, 1984.##[10] W. Rudin, Fourier Analysis on Groups, WileyInterscience, New York, 1962.##[11] S.E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a BochnerSchoenbergEberleintype theorem, Proc. Amer. Math. Soc., 110 (1990), pp. 149158.##[12] S.E. Takahasi and O. Hatori, Commutative Banach algebras and BSEinequalities, Math. Japonica, 37 (1992), pp. 607614.##]
A New Approach to Nonstandard Analysis
2
2
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.
1

195
254


Saghe
Abdeljalil
Cpge de Settat, Lycee Qualifiant Technique PB: 576Maroc.
Cpge de Settat, Lycee Qualifiant Technique
Iran
saghe007@gmail.com
Nonstandard analysis
Hyppereals
Internal set theory
[[1] F. Bagarello, Nonstandard variational calculus with applications to classical mechanics. I. An existence criterion, Internat. J. Theoret. Phys., 38 (1999), pp. 15691592.##[2] F. Bagarello, Nonstandard variational calculus with applications to classical mechanics. II. The inverse problem and more, Internat. J. Theoret. Phys., 38 (1999), pp. 15931615.##[3] F. Bagarello and S. Valenti, Nonstandard analysis in classical physics and quantum formal scattering, Int. J. Theor. Phys., 27 (1988), pp. 557566.##[4] V. Benci and M. Di Nasso, Alphatheory: an elementary axiomatics for nonstandard analysis, Expo. Math., 21 (2003), pp. 355386.##[5] V. Benci, M. Di Nasso, and M. Forti, The eightfold path to nonstandard analysis, Nonstandard Methods and Applications in Mathematics, 25 (2006), pp. 344.##[6] J.M. Borwein, D.M. Bradley, and R.E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math., 121 (2000), pp. 247296.##[7] N. Cutland, Nonstandard Analysis and its Applications, Cambridge University Press, Cambridge, 1988.##[8] F. Diener and G. Reeb, Analyse non standard, Hermann, 1989.##[9] R. Fittler, Asymptotic nonstandard quantum electrodynamics, J. Math. Phys., 34 (1993), pp. 16921724.##[10] 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. 193252.##[11] R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Springer, Graduate Texts in Mathematics 188, 1998.##[12] A. Granville, Harald Cramer and the distribution of prime numbers, Scand. Actuar. J., 1 (1995), pp. 1228.##[13] E. Nelson, Internal Set Theory: a new approach to Nonstandard Analysis, Bull. A.M.S., 83 (1977), pp. 11651198.##[14] A. Robinson, Nonstandard Analysis, NorthHolland, Amsterdam, 1966.##[15] W. Rudin, Analyse reelle et complexe, Masson, 1978.##[16] P. Tauvel, Cours dalgebre: agregation de mathematiques, Dunod, 1999.##]