ORIGINAL_ARTICLE
Max-Plus algebra on tensors and its properties
In this paper we generalize the max plus algebra system of real matrices to the class of real tensors and derive its fundamental properties. Also we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverses of tensors is characterized.
http://wala.vru.ac.ir/article_19923_88ef62feab4333580c4d29bc8a94d75f.pdf
2016-06-01T11:23:20
2018-02-18T11:23:20
1
11
Max plus algebra
Tensor
Hamid Reza
Afshin
afshin@mail.vru.ac.ir
true
1
Department of Mathematics, Vali-e-Asr University, Rafsanjan, Islamic
Republic of Iran
Department of Mathematics, Vali-e-Asr University, Rafsanjan, Islamic
Republic of Iran
Department of Mathematics, Vali-e-Asr University, Rafsanjan, Islamic
Republic of Iran
AUTHOR
Ali Reza
Shojaeifard
ashojaeifard@ihu.ac.ir
true
2
Department of Mathematics, Faculty of Sciences, Imam Hossein
Comprehensive University, Tehran, Islamic Republic of Iran
Department of Mathematics, Faculty of Sciences, Imam Hossein
Comprehensive University, Tehran, Islamic Republic of Iran
Department of Mathematics, Faculty of Sciences, Imam Hossein
Comprehensive University, Tehran, Islamic Republic of Iran
AUTHOR
[1] H.R. Afshin, A.R. Shojaeifard, A max version of Perron Frobenuos theorem for nonnegative tensor, Ann. Funct. Anal., 6 (2015).
1
[2] M. Akian, R. Bapat, and S. Gaubert, Max-plus algebras, in Handbook of Linear Algebra, Discrete Mathematics and Its Applications 39, L. Hogben, ed., Chapman and Hall/CRC, Boca Raton, FL, 2006.
2
[3] F. Baccelli, G. Cohen. G. Olsder. J. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event
3
Systems, Wiley, Chichester. 1992.
4
[4] P. Butkovic, Max-linear Systems: Theory and Algorithms, Springer Monogr. Math., SpringerVerlag, London,
5
[5] C. Bu, X. Zhang, J. Zhou, W. Wang, Y. Wei, The inverse, rank and product of tensors, Linear Algebra Appl.,
6
446 (2014) 269-280.
7
[6] R.A. Cuninghame-Green, Minimax Algbera, Lecture notes in Economics and Mathematical Systems, 166
8
Springer, 1979.
9
[7] B.D. Shutter, On the ultimate behavior of the sequence of consecutive powers of a matrix in the max-plus
10
algebra, Linear Algebra Appl., 30 (2000), 103-117.
11
[8] S. Gaubert, Methods and applications of (max,+) linear algebra, Lecure Notes in Computer Science 500,
12
Springer Verlag, Berlin, 1997, 261-282.
13
[9] N. Ghasemizadeh and Gh. Aghamollaei, Some results on matrix polynomials in the max algebra, Banach J. Math. Anal., 40 (2015), 17-26.
14
[10] R.G. Halburd, N.J. Southall, Tropical nevanlinna theory and ultradisctete equations, Loughborough University, 2007.
15
[11] B. Heidergott, G. Olsder and J. Van Der Woude, Max Plus at Work: Modeling and Analysis of Synchronized Systems, Princeton University Press, 2005.
16
[12] L.H. Lim, Singular values and eigenvalues of tensors: a variational approach, Proceedings 1st IEEE International Workshop on Computational Advances of Multitensor Adaptive Processing, (2005), 129-132.
17
[13] L. Qi, Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput., 40 (2005), 1302-1324.
18
[14] J.Y. Shao, A general product of tensors with applications, Linear Algebra Appl., 439 (2013), 2350-2366.
19
ORIGINAL_ARTICLE
A computational wavelet method for numerical solution of stochastic Volterra-Fredholm integral equations
A Legendre wavelet method is presented for numerical solutions of stochastic Volterra-Fredholm integral equations. The main characteristic of the proposed method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Legendre wavelets basis are investigated. The efficiency and accuracy of the proposed method was demonstrated by some non-trivial examples and comparison with the block pulse functions method.
http://wala.vru.ac.ir/article_19924_03eb06bb455bb32b286246d39fdeb99f.pdf
2016-06-01T11:23:20
2018-02-18T11:23:20
13
25
Legendre wavelets, Brownian motion process, Stochastic Volterra-Fredholm integral equations,
Stochastic operational matrix,
Fakhrodin
Mohammadi
f.mohammadi62@hotmail.com
true
1
Hormozgan University
Hormozgan University
Hormozgan University
AUTHOR
[1] P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, in: Applications of Mathematics, Springer-Verlag, Berlin, 1999.
1
[2] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, ﬁfth ed., springer-Verlag, New York, 1998.
2
[3] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Re., 43(3)(2001), 525–546.
3
[4] K. Maleknejad, M. Khodabin, M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Modelling, 55(2012), 791–800.
4
[5] M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri, Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput. Math. Appl., Part A, 64(2012), 1903–1913.
5
[6] M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek, C. Cattani, A computational method for solving stochastic Ito-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys., 270(2014), 402–415.
6
[7] J . C. Cortes, L. Jodar, L. Villafuerte, Numerical solution of random differential equations: a mean square approach, Math. Comput. Modelling, 45 (7)(2007), 757–765.
7
[8] G. Strang, Wavelets and dilation equations, SIAM Rev., 31(1989), 613–627.
8
[9] A. Boggess, F. J. Narcowich, A ﬁrst course in wavelets with Fourier analysis, John Wiley and Sons, 2001.
9
[10] M. Razzaghi, S. Youseﬁ, The Legendre wavelets operational matrix of integration, Int. J. Syst. Sci., 32(4)(2001), 495–502.
10
[11] M. Razzaghi, S. Youseﬁ, Legendre wavelets direct method for variational problems, Math. Comput. Simul., 53(3)(2000), 185–192.
11
[12] F. Mohammadi, M. M. Hosseini, and Syed Tauseef Mohyud-Din. Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution, Int. J. Syst. Sci.,42(4)(2011), 579–585.
12
[13] F. Mohammadi, M. M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst., 348(8)(2011), 1787–1796.
13
[14] Z. H. Jiang, W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, Springer-Verlag, 1992.
14
[15] L. Nanshan, E. B. Lin, Legendre wavelet method for numerical solutions of partial differential equations, Numer.Methods Partial Differ. Equations, 26(1)(2010), 81–94.
15
ORIGINAL_ARTICLE
*-frames for operators on Hilbert modules
$K$-frames which are generalization of frames on Hilbert spaces, were introduced to study atomic systems with respect to a bounded linear operator. In this paper, $*$-$K$-frames on Hilbert $C^*$-modules, as a generalization of $K$-frames, are introduced and some of their properties are obtained. Then some relations between $*$-$K$-frames and $*$-atomic systems with respect to an adjointable operator are considered and some characterizations of $*$-$K$-frames are given. Finally perturbations of $*$-$K$-frames are discussed.
http://wala.vru.ac.ir/article_19952_c7bf18d637cfcab1233eb3974de29b9a.pdf
2016-06-01T11:23:20
2018-02-18T11:23:20
27
43
K-framesep *-frame
Hilbert
C^*-module
Bahram
Dastourian
bdastorian@gmail.com
true
1
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
LEAD_AUTHOR
Mohammad
Janfada
janfada@um.ac.ir
true
2
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
AUTHOR
[1] L. Arambasic, On frames for countably generated Hilbert C*-modules, Proc. Amer. Math. Soc. 135 (2007) 469–478.
1
[2] A. Alijani and M.A. Dehghan, -Frames in Hilbert C*-Modules, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 73(4) (2011), 89–106.
2
[3] M.S. Asgari and H. Rahimi, Generalized frames for operators in Hilbert spaces, Inﬁn. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2) (2014), 1450013 (20 pages).
3
[4] H. Bolcskei, F. Hlawatsch, H. G. Feichtinger, Frame-theoretic analyssis of oversampled ﬁlter banks, IEEE Trans. Signal Process., 46 (1998), 3256–3268.
4
[5] B. Dastourian and M. Janfada, Frames for operators in Banach spaces via semi-inner products, Int. J. Wavelets Multiresolut. Inf. Process., 14 (3) (2016), 1650011.
5
[6] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer.
6
Math. Soc., 17 (2) (1966), 413–415.
7
[7] J. Dun, A.C. Schaeer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341–366.
8
[8] N.E. Dudey Ward, J.R. Partington, A construction of rational wavelets and frames in Hardy-Sobolev space with applications to system modelling, SIAM J. Control Optim., 36(1998), 654–679.
9
[9] Y.C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors. J. Fourier. Anal. Appl. 9 (1) (2003) 77–96.
10
[10] Y.C. Eldar and T. Werther, General framework for consistent sampling in Hilbert spaces, Int. J. Wavelets Multi. Inf. Process., 3 (3) (2005), 347–359.
11
[11] X. Fang, J. Yu and H. Yao, Solutions to operators equation on Hilbert C*-modules, Linear Algebra Appl., 431(11)(2009), 2142–2153.
12
[12] H.G. Feichtinger and T. Werther, Atomic systems for subspaces, in:L. Zayed (Ed.), Proceedings SampTA 2001, Orlando, FL, 2001, 163–165.
13
[13] P.J.S.G. Ferreira, Mathematics for multimedia signal processing II: Discrete ﬁnite frames and signal reconstruction, In:Byrnes, J.S. (ed.) Signal processing for multimedia, IOS Press, Amsterdam, (1999), 35–54.
14
[14] M. Frank and D. R. Larson, A module frame concept for Hilbert C*-modules, Functional and Harmonic Analysis of Wavelets (San Antonio, TX, Jan. 1999), Contemp. Math., 247 (2000), 207–233.
15
[15] M. Frank and D. R. Larson, Frames in Hilbert C*-modules and C*-algebra, J. Operator theory, 48 (2002) 273–314.
16
[16] L. Gavrut¸a, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012) 139–144.
17
[17] W. Jing, Frames in Hilbert C*-modules, Ph.D. Thesis, University of Central Florida, 2006.
18
[18] A. Khosravi and B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert C*-modules,
19
Proc. Indian Acod. Sci., 117 (1) (2007), 1–12.
20
[19] E. C. Lance, Hilbert C*-modules, University of Leeds, Cambridge University Press, London, 1995.
21
[20] B. Magajna, Hilbert C*-modules in which all closed submodules are complemented, Proc. Amer. Math. Soc.,
22
125(3) (1997), 849–852.
23
[21] V. M. Manuilov, Adjointability of operators on Hilbert C*-modules, Acta Math. Univ. Comenianae, LXV (2)
24
(1996), 161–169.
25
[22] J. G. Murphy,Operator Theory and C*-Algebras, Academic Press, San Diego, 1990.
26
[23] M. Skeide, Generalised matrix C*-algebras and representations of Hilbert modules,Math. Proc. R. Ir. Acad.,
27
100(1) (2000), 11–38.
28
[24] M. Pawlak and U. Stadtmuller, Recovering band-limited signals under noise, IEEE Trans. Info. Theory,
29
42(1994), 1425–1438.
30
[25] T. Strohmer and R. Heath Jr., Grassmanian frames with applications to coding and communications, Appl.
31
Comput. Harmon. Anal., 14 (2003) 257–275.
32
[26] N. E. Wegge-Olsen, K-Theory and C*-Algebras-A Friendly Approach, Oxford Uni. Press, Oxford, England,
33
[27] X. Xiao, Y. Zhu and L. Gavrut¸a, Some properties of K-frames in Hilbert spaces, Results. Math., 63(3-4) (2013),
34
1243–1255.
35
[28] X. Xiao, Y. Zhu, Z. Shu, M. Ding, G-frames with bounded linear operators, Rocky Mountain J. Math., 45 (2)(2015), 675–693.
36
[29] L. C. Zhang, The factor decomposition theorem of bounded generalized inverse modules and their topological
37
continuity, J. Acta Math. Sin., 23 (2007), 1413-1418.
38
ORIGINAL_ARTICLE
Inverse Young inequality in quaternion matrices
Inverse Young inequality asserts that if $nu >1$, then $|zw|ge nu|z|^{frac{1}{nu}}+(1-nu)|w|^{frac{1}{1-nu}}$, for all complex numbers $z$ and $w$, and equality holds if and only if $|z|^{frac{1}{nu}}=|w|^{frac{1}{1-nu}}$. In this paper the complex representation of quaternion matrices is applied to establish the inverse Young inequality for matrices of quaternions. Moreover, a necessary and sufficient condition for equality is given.
http://wala.vru.ac.ir/article_19953_35469da2a28c83b1b25944b53b87c748.pdf
2016-06-01T11:23:20
2018-02-18T11:23:20
45
52
Inverse Young inequality
Quaternion matrix
Right eigenvalue
Complex representation
Seyd Mahmoud
Manjegani
manjgani@cc.iut.ac.ir
true
1
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Islamic Republic of Iran
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Islamic Republic of Iran
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Islamic Republic of Iran
LEAD_AUTHOR
Asghar
Norouzi
true
2
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Islamic Republic of Iran
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Islamic Republic of Iran
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Islamic Republic of Iran
AUTHOR
[1] T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl. 75 (1995), 33–38.
1
[2] M. Argerami and D. R. Farenick, Young’s inequality in trace-class operators, Math. Ann., 325 (2003), 727–744.
2
[3] R. Bhatia and F. Kittaneh, On the singular values of a product of operators, SIAM J. Matrix Anal. Appl. 11
3
(1990), 727–277.
4
[4] J. Erlijman, D. R. Farenick and R. Zeng, Young’s inequality in compact operators, Oper. Theory Adv. Appl., 130(2001), 171–184.
5
[5] D. R. Farenick and S. M. Manjegani, Young’s inequality in operator algebras, J. Ramanujan Math. Soc., 20(2)(2005), 107–124.
6
[6] D. R. Farenick and B. A .F. Pidkowich, The spectral theorem in quaternions, Linear Algebra Appl., 371 (2003), 75–102.
7
[7] H. Glockner, Functions operating on positive semideﬁnite quaternionic matrices, Monatsh. Math., 132 (2001), 303–324.
8
[8] H. C. Lee, Eigenvalues and canonical forms of matrices with quaternion coecients, Proc. Roy. Irish. Acad.
9
Sec. A., 52 (1949), 253–260.
10
[9] S. M. Manjegani and A. Norouzi, Martix form for the inverse Young inequality, Linear Algebra Appl., 486
11
(2015), 484 - 493..
12
[10] R. C. Thompson, Convex and concave functions of singular values of matrix sums, Paciﬁc J. Math., 66 (1976), 285–290.
13
[11] R. C. Thompson, The case of equality in the matrix-valued triangle inequality, Paciﬁc J. Math., 82 (1979),
14
279–280.
15
[12] R. C. Thompson, Matrix-valued triangle inequality: quaternion version, Linear and multilinear Algebra,
16
25(1989), 85–91.
17
[13] R. Zeng, The quaternion matrix-valued Young’s inequality, J. Inequal. Pure Appl. Math. (6) , Art. 89, (2005).
18
ORIGINAL_ARTICLE
A note on $lambda$-Aluthge transforms of operators
Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and only if $A=tilde{A}_lambda^ast$, emph{ii}) if $A$ is normaloid and either $sigma(A)$ has only finitely many distinct nonzero value or $U$ is unitary, then from $A=ctilde{A}_lambda$ for some complex number $c$, we can conclude that $A$ is quasinormal, emph{iii}) if $A^2$ is self-adjoint and any one of the $Re(A)$ or $-Re(A)$ is positive definite then $A$ is self-adjoint, emph{iv}) and finally we show that $$opnorm{|A|^{2lambda}+|A^ast|^{2-2lambda}oplus0}leqopnorm{|A|^{2-2lambda}oplus|A|^{2lambda}}+ opnorm{tilde{A}_lambdaoplus(tilde{A}_lambda)^ast}$$ where $opnorm{cdot}$ stand for some unitarily invariant norm. From that we conclude that $$||A|^{2lambda}+|A^ast|^{2-2lambda}|leqmax(|A|^{2lambda},|A|^{2-2lambda})+|tilde{A}_lambda|.$$
http://wala.vru.ac.ir/article_19955_e84b542e36ddd38d3218cc0eb4ef380f.pdf
2016-06-01T11:23:20
2018-02-18T11:23:20
53
60
Aluthge transform, Self-adjoint operators, Unitarily invariant norm
Schatten p-norm
Seyed Mohammad Sadegh
Nabavi Sales
sadegh.nabavi@gmail.com
true
1
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran
LEAD_AUTHOR
[1] A. Aluthge. On p-hyponormal operators for 0 < p < 1, Integral Equations Oper. Theory, 13 (1990), 307–315.
1
[2] T. Ando. Aluthge transforms and the convex hull of the eigenvalues of a matrix, Linear Multilinear Algebra, 52(2004), 281–292.
2
[3] A. Antezana, P. Massey and D. Stojano, $lambda$-Aluthge transforms and Shatten ideals, Linear Algebra Appl., 405(2005), 177–199.
3
[4] T. Furuta. Invitation to linear operators; from matrices to bounded linear operators on a Hilbert space, Taylor and Francis, London, 2001.
4
[5] I. B. Jung, E. Ko and C. Pearcy. Aluthge transforms of operators, Integral Equations Oper. Theory 37 (2000), 437–448.
5
[6] F. Kittaneh.Norm inequalities for some of positive operators, J. Oper. Theory, 48 (2002), 95–103.
6
[7] O. Hirzallah and F. Kittaneh. Matrix Young inequalities for the Hilbert- Schmidt norm , Linear Algebra Appl.,
7
308 (2000), 77–84.
8
[8] M.S. Moslehian and S.M.S. Nabavi Sales. Some conditions implying normality of operators, C. R. Acad. Sci.
9
Paris, Ser. I, 349 (2011), 251–254.
10
[9] M.S. Moslehian and S.M.S. Nabavi Sales. Fuglede–Putnam type theorem via the Aluthge transform, Positivity, 349 (2013), 151–162.
11
[10] A. Oloomi and M. Rajabalipour.Operators with normal Aluthge transforms, C. R. Acad. Sci. Paris, Ser. I, 350(2012), 263–266.
12
[11] B. Simon. Trace ideals and their applications , in: London Mathematical Society Lecture Note Series, Cam-
13
bridge University Press, Cambridge–NewYork, 1979.
14
[12] A. Uchiyama and K. Tanahashi.Fuglede–Putnam theorem for p-hyponorma or log-hyponormal operators, Glasgow Math. J., 44 (2002), 397–410.
15
[13] T. Yamazaki.An expression of spectral radius via Aluthge transformation, Proc. Amer. Math. Soc., 130 (2002), 1131–1137.
16
[14] Jian Yang and Hong-Ke Du. A note on commutativity up to a factor of bounded operators, Proc. Amer. Math. Soc., 132 (2004), 1713–1720.
17
ORIGINAL_ARTICLE
Some results on functionally convex sets in real Banach spaces
We use of two notions functionally convex (briefly, F--convex) and functionally closed (briefly, F--closed) in functional analysis and obtain more results. We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$, then $bigcup_{alphain I}A_{alpha}$ is F--convex. Moreover, we introduce new definition of notion F--convexiy.
http://wala.vru.ac.ir/article_19956_c84c54eb7ec49c507aa1fd6074db65fe.pdf
2016-06-01T11:23:20
2018-02-18T11:23:20
61
67
convex set
F--convex set
F--closed set
Madjid
Eshaghi
madjid,eshaghi@gmail.com
true
1
Department of Mathematics‎, ‎Semnan University‎, ‎P‎. ‎O‎. ‎Box 35195-363‎, ‎Semnan‎, ‎Iran,
Department of Mathematics‎, ‎Semnan University‎, ‎P‎. ‎O‎. ‎Box 35195-363‎, ‎Semnan‎, ‎Iran,
Department of Mathematics‎, ‎Semnan University‎, ‎P‎. ‎O‎. ‎Box 35195-363‎, ‎Semnan‎, ‎Iran,
AUTHOR
Hamidreza
Reisi
hamidreza.reisi@gmail.com
true
2
PhD student of semnan univercity
PhD student of semnan univercity
PhD student of semnan univercity
LEAD_AUTHOR
Alireza
Moazzen
ar,moazzen@yahoo.com
true
3
Department of mathematics‎, ‎Kosar University of Bojnourd‎, ‎Bojnourd‎, ‎Iran
Department of mathematics‎, ‎Kosar University of Bojnourd‎, ‎Bojnourd‎, ‎Iran
Department of mathematics‎, ‎Kosar University of Bojnourd‎, ‎Bojnourd‎, ‎Iran
AUTHOR
[1] D. Aliprantis and C. Border, Infinite Dimensional Analysis, 2th. edition. Springer, 1999.
1
[2] J. B. Conway. A Course in Functionall Analysis, Springer-verlag, 1985.
2
[3] N. Dunford and J. T. Schwartz, Linear operators. Part 1, Interscience, New York 1958.
3
[4] E. Zeidler. Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems, Springer-Verlog New
4
York, 1986.
5
[5] M. Eshaghi, H. Reisi Dezaki and A. Moazzen, Functionally convex sets and functionally closed sets in real
6
Banach spaces, Int. J. Nonlinear Anal. Appl., 7(1)(2016), 289–294.
7