ORIGINAL_ARTICLE
Use of the Shearlet Transform and Transfer Learning in Offline Handwritten Signature Verification and Recognition
Despite the growing growth of technology, handwritten signature has been selected as the first option between biometrics by users. In this paper, a new methodology for offline handwritten signature verification and recognition based on the Shearlet transform and transfer learning is proposed. Since, a large percentage of handwritten signatures are composed of curves and the performance of a signature verification/recognition system is directly related to the edge structures, subbands of shearlet transform of signature images are good candidates for input information to the system. Furthermore, by using transfer learning of some pre-trained models, appropriate features would be extracted. In this study, four pre-trained models have been used: SigNet and SigNet-F (trained on offline signature datasets), VGG16 and VGG19 (trained on ImageNet dataset). Experiments have been conducted using three datasets: UTSig, FUM-PHSD and MCYT-75. Obtained experimental results, in comparison with the literature, verify the effectiveness of the presented method in both signature verification and signature recognition.
https://scma.maragheh.ac.ir/article_38395_26252c7c74e40fc0b3676cf37cdf20eb.pdf
2020-07-01
1
31
10.22130/scma.2019.99098.536
Offline handwritten signature
Signature verification
Signature recognition
Shearlet transform
Transfer learning
Atefeh
Foroozandeh
atforoozandeh@yahoo.com
1
Department of Applied Mathematics, Faculty of Sciences and Modern Technology, Graduate University of Advanced Technology, Kerman, Iran.
AUTHOR
Ataollah
Askari Hemmat
askari@uk.ac.ir
2
Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
LEAD_AUTHOR
Hossein
Rabbani
h_rabbanimed@mui.ac.ir
3
Department of Biomedical Engineering, School of Advanced Technologies in Medicine, Isfahan University of Medical Sciences, Isfahan, Iran.
AUTHOR
[1] I. Abroug and N.E. Ben Amara, Off-line signature verification systems: recent advances, Int. Conf. Image Process. Appl. Syst., (2014), pp. 1-6.
1
[2] N.S. Altman, An introduction to kernel and nearest-neighbor nonparametric regression, Amer. Statist., 46 (1992), pp. 175-185.
2
[3] M. Arathi and A. Govardhan, An efficient offline signature verification system, Int. J. Mach. Learn. Comput., 4 (2014), pp. 533-537.
3
[4] A.N. Azmi, D. Nasien and F.S. Omar, Biometric signature verification system based on freeman chain code and k-nearest neighbor, Multimedia Tools Appl., 76 (2017), pp. 15341-15355.
4
[5] Y. Bengio, Learning deep architectures for AI, Found. Trends Mach. Learn., 2 (2009), pp. 1-127.
5
[6] Y. Bengio and A. Courville, Deep learning of representations, in Handbook on Neural Inf. Process., Springer, (2013), pp. 1-28.
6
[7] T.A. Bubba, G. Kutyniok, M. Lassas, M. Marz, W. Samek, S. Siltanen and V. Srinivasan, Learning the invisible: a hybrid deep learning-shearlet framework for limited angle computed tomography, arXiv:1811.04602v1, 2018.
7
[8] E.J. Candes and D.L. Donoho, Continuous curvelet transform: I. Resolution of the wavefront set, Appl. Comput. Harmon. Anal., 19 (2005), pp. 162-197.
8
[9] E.J. Candes and D.L. Donoho, Curvelets: a surprisingly effective nonadaptive representation for objects with edges, Tech. Rep. Department of Statistics, Stanford University, 1999.
9
[10] C. Cortes and V. Vapnik, Support-vector networks, Mach. Learn., 20 (1995), pp. 273-297.
10
[11] B. Cozzens and R. Huang, Signature verification using a convolutional neural network, Las Vegas, Nevada, 2017.
11
[12] I. Daubechies, Ten lectures on wavelets, Soc. Ind. Appl. Math., Philadelphia, 1992.
12
[13] L. Deng and D. Yu, Deep learning: methods and applications, Found. Trends in Signal Process., 7 (2014), pp. 197-387.
13
[14] S. Dey, A. Dutta, J. I. Toledo, S K. Ghosh, J. Llados and U. Pal, SigNet: convolutional siamese network for writer independent offline signature verification, CORR, abs/1707.02131, 2017.
14
[15] M.N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process., 14 (2005), pp. 2091-2106.
15
[16] M.A. Duval-Poo, F. Odone and E. De Vito, Edges and corners with shearlets, IEEE Trans. Image Process., 24 (2015), pp. 3768-3780.
16
[17] G. Easley, D. Labate and Wang-Q Lim, Sparse directional image representations using the discrete shearlet transform, J. Appl. Comput. Harmon. Anal., 25 (2008), pp. 25-46.
17
[18] G. Eskander, R. Sabourin and E. Granger, Hybrid writer-independent-writer-dependent ofﬂine signature veriﬁcation system, IET Biometrics 2(4) (2013), pp. 169-181.
18
[19] M. Fakhlai, H.R. Pourreza, R. Moarefdost and S. Shadroo, Offline signature recognition based on contourlet transform, Int. Conf. Mach. Learn. Comput., 2009.
19
[20] J. Fierrez-Aguilar, N. Alonso-Hermira, G. Moreno-Marquez and J. Ortega-Garcia, An off-Line signature verification system based on fusion of local and global information, Int. Workshop Biom. Authentication, (2004), pp. 295-306.
20
[21] A. Foroozandeh, Y. Akbari, M.J. Jalili and J. Sadri, A novel and practical system for verifying signatures on Persian handwritten bank checks, Int. J. Pattern Recognit. Artif. Intell. (IJPRAI), 26 (2012), pp. 1-27.
21
[22] C. Freitas et al., Bases de dados de cheques bancarios brasileiros, In XXVI Conferencia Latinoamericana de Informatica, 2000.
22
[23] Y. Guerbai, Y. Chibani and B. Hadjadji, The effective use of the one-class SVM classifier for handwritten signature verification based on writer-independent parameters, PR, 48(1) (2015), pp. 103-113.
23
[24] Y. Guo, U. Budak, A. Sengur and F. Smarandache, A retinal vessel detection approach based on shearlet transform and indeterminacy filtering on fundus images, J. Symmetry, 9 (2017).
24
[25] K. Guo and D. Labate, The construction of smooth parseval frames of shearlets, Math. Model. Nat. Phenom., 8 (2013), pp. 82-105.
25
[26] L.G. Hafemann, R. Sabourin and L.S. Oliveira, Learning features for offline handwritten signature verification using deep convolutional neural networks, Pattern Recognit., 70 (2017), pp. 163-176.
26
[27] L.G. Hafemann, R. Sabourin and L.S. Oliveira, Offline handwritten signature verification-literature review, Int. Conf. Image Process. Theory Tool Appl., (2017), pp. 1-8.
27
[28] L.G. Hafemann, R. Sabourin and L.S. Oliveira, Analyzing features learned for Offline Signature Verification using Deep CNNs, 23rd Int. Conf, Pattern Recognit, (ICPR), (2016), pp. 2989-2994.
28
[29] L.G. Hafemann, R. Sabourin and L.S. Oliveira, Writer-independent feature learning for Offline Signature Verification using Deep Convolutional Neural Networks, Int. Joint Conf. Neural Netw., (2016), pp. 2576-2583.
29
[30] S. Hauser and G. Steidl, Fast Finite Shearlet Transform: a tutorial, arXiv:1202.1773v1 [math.NA] 8 Feb 2012.
30
[31] H. Hezil, R. Djemili and H. Bourouba, Signature recognition using binary features and KNN, Int. J. Biometrics, 10 (2018), pp. 1-15.
31
[32] M. Kaboli, A Review of transfer learning algorithms, Research Report, Technische Universitat Munchen, 2017.
32
[33] M.K. Kalera, S. Srihari and A. Xu, Offline signature verification and identification using distance statistics, Int. J. Pattern Recognit. AI., 18 (2004), pp. 1339-1360.
33
[34] G. Kanghui, G. Kutyniok and D. Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, Int. Conf. Interact. Wavelets and Splines, (2005), pp. 189-201.
34
[35] H. Khalajzadeh, M. Mansouri and M. Teshnehlab, Persian signature verification using convolutional neural networks, Int. J. Eng. Res. Technol. 1 (2012), pp. 7-12.
35
[36] M. Khan, A. Jamil, M. Irfan, R. Seungmin and B. Sung, Convolutional neural networks based fire detection in surveillance videos, IEEE Access, 6 (2018), pp. 18174-18183.
36
[37] N. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals, Appl. Comput. Harmon. Anal., 10 (2001), pp. 234-253.
37
[38] N. Kingsbury, Image processing with complex wavelets, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 357 (1999), pp. 2543-2560.
38
[39] J.R. Koza, F.H. Bennett, D. Andre and M.A. Keane, Automated design of both the topology and sizing of analog electrical circuits using genetic programming, J. Artificial Intelligence Des., (1996), pp. 151-170.
39
[40] A. Krizhevsky, I. Sutskever and G.E. Hinton, ImageNet classiﬁcation with deep convolutional neural networks, Adv. Neural Inf. Process. Syst., 25 (2012), pp. 1097-1105.
40
[41] R. Kumar, L. Kundu, B. Chanda and J.D. Sharma, A Writer-independent off-line signature veriﬁcation system based on signature morphology, Proc. Conf. Intell. Interactive Technol. Multimedia, New York, NY, USA, (2010), pp. 261-265.
41
[42] G. Kutyniok and L. Demetrio, Shearlets: multiscale analysis for multivariate data, Springer New York: Birkhauser, 2012.
42
[43] G. Kutyniok, W.Q. Lim and R. Reisenhofer, Shearlab 3D: faithful digital shearlet transforms based on compactly supported shearlets, J. ACM Trans. Math. Softw., 42 (2016), pp. 1-42.
43
[44] G. Kutyniok, M. Shahram and X. Zhuang, ShearLab: a rational design of a digital parabolic scaling algorithm, SIAM J. Imaging Sci., 5 (2012), pp. 1291-1332.
44
[45] S. Mallat and S. Zhong, Characterization of signals from multiscale edges, IEEE Trans. Pattern Anal. Mach. Intell., 14 (1992), pp. 710-732.
45
[46] S. Min, B. Lee and S. Yoon, Deep learning in bioinformatics, Briefings in Bioinformatics, 18 (2016), pp. 851-869.
46
[47] R.A. Mohammed, R.M. Nabi, S.M.R. Mahmood and R.M. Nabi, State-of-the-art in handwritten signature verification system, Int. Conf. Comput. Sci. Comput. Intell., (2015), pp. 519-525.
47
[48] P.N. Narwade, R.R. Sawant and S.V. Bonde, Offline handwritten signature verification using cylindrical shape context, 3D Res. 9 (2018), https://doi.org/10.1007/s13319-018-0200-0.
48
[49] S.Y. Ooi, A.B. J. Teoh, Y.H. Pang and B.Y. Hiew, Image-based handwritten signature veriﬁcation using hybrid methods of discrete radon transform, principal component analysis and probabilistic neural network, Appl. Soft Comput., 40 (2016), pp. 274-282.
49
[50] N. Otsu, A threshold selection method from gray-level histograms, IEEE Trans. Sys. Man. Cyber., 9 (1979), pp. 62-66.
50
[51] S.J. Pan and Q. Yang, A survey on transfer learning, IEEE Trans. Knowledge Data Eng., 22 (2010), pp. 1345-1359.
51
[52] D.N. Perkins and G. Salomon, Transfer of learning, Int. Encyclopedia of Education, 2nd Edition, Oxford, England: Pergamon Press, 1992.
52
[53] M.R. Pourshahabi, M.H. Sigari and H.R. Pourreza, Offline handwritten signature identification and verification using contourlet transform, Int. Conf. Soft Comput. Pattern Recognit., (2009), pp. 670-673.
53
[54] L.Y. Pratt, Discriminability-based transfer between neural networks, Advances in Neural Inf. Process. Syst., 5 (1993), pp. 204-211.
54
[55] S. Rajaraman et al., Pre-trained convolutional neural networks as feature extractors toward improved malaria parasite detection in thin blood smear images, (2018), PeerJ 6:e4568; DOI: 10.7717/peerj.4568.
55
[56] W. Rawat and Z. Wang, Deep convolutional neural networks for image classification: a comprehensive review, Neural Comput., 29 (2017), pp. 2352-2449.
56
[57] S. Razavian, H. Azizpour, J. Sullivan and S. Carlsson, CNN features off-the-shelf: an astounding baseline for recognition, IEEE Conf. Comput. Vis. Pattern Recognit. Workshops (CVPRW), (2014), pp. 512-519.
57
[58] J.D. Regele and O.V. Vasilyev, An adaptive wavelet-collocation method for shock computations, Int. J. Comput. Fluid Dyn., 23 (2009), pp. 503-518.
58
[59] J. Sadri, M.J. Jalili, Y. Akbari and A. Foroozandeh, Designing a new standard structure for improving automatic processing of Persian handwritten bank cheques, Int. J. Pattern Anal. Appl. (PAA), 17 (2014), pp .849-862.
59
[60] M.H. Saffar, M. Fayyaz, M. Sabokrou and M. Fathy, Online signature verification using deep representation: a new descriptor, Int. Comput. Vis. Pattern Recognit. (cs.CV), arXiv:1806.09986.
60
[61] L. Samuel, Some studies in machine learning using the game of checkers, IBM J. Res. Dev., 3 (1959), pp. 210-229.
61
[62] J. Schmidhuber, Deep learning in neural networks: an overview, Neural Networks, 61 (2015), pp. 85-117.
62
[63] M. Sharif, M.A. Khan, M. Faisal, M. Yasmin and S.L. Fernandes, A framework for offline signature verification system: best features selection approach, Pattern Recognit. Lett., 2018.
63
[64] M.H. Sigari, M.R. Pourshahabi and H. R. Pourreza, Offline handwritten signature identification and verification using multi-resolution Gabor wavelet, Int. J. Biometrics and Bioinform. (IJBB), 5 (2011), pp. 234-248.
64
[65] K. Simonyan and A. Zisserman, Very deep convolutional networks for large-scale image recognition, arXiv 1409.1556, 2014.
65
[66] A. Soleimani, B.N. Araabi and K. Fouladi, Deep multitask metric learning for offline signature verification, Pattern Recognit. Lett., 80 (2016), pp. 84-90.
66
[67] A. Soleimani, K. Fouladi and B.N. Araabi, UTSig: a Persian offline signature database, IET Biometrics, 6 (2017), pp. 1-8.
67
[68] A. Soleimani, K. Fouladi and B.N. Araabi, Persian offline signature verification based on curvature and gradient histograms, 6th Int. Conf. Comput. Knowledge Eng., (2016), pp. 147-152.
68
[69] H. Srinivasan, S.N. Srihari and M.J. Beal, Machine learning for signature verification, Computer Vision, Graphics and Image Processing, Springer Berlin Heidelberg, (2006), pp. 761-775.
69
[70] J.L. Starck, F. Murtagh and J. Fadili, Sparse image and signal processing: wavelets, curvelets, morphological diversity, Cambridge University Press, 1st Edition, 2010.
70
[71] S. Tayeb, M. Pirouz, B. Cozzens, R. Huang, M. Jay, K.Khembunjong and S. Paliskara, Toward data quality analytics in signature verification using a convolutional neural network, IEEE Int. Conf. Big Data, (2017), pp. 2644-2651.
71
[72] J. Vargas, M. Ferrer, C. Travieso and J. Alonso, Off-line handwritten signature GPDS-960 corpus, Doc. Anal. Recognit., 9th Int. Conf., 2 (2007), pp. 764-768.
72
[73] K. Weiss, T.M. Khoshgoftaar and D.D. Wang, A survey of transfer learning, J. Big Data, 3 (2016).
73
[74] N. Xiaopeng, W. Zhiliang and P. Zhigeng, Extreme learning machine based deep model for human activity recognition with wearable sensors, Comput. Sci. Eng., 21 (2018), pp. 16-25.
74
[75] M.E. Yahyatabar and J. Ghasemi, Online signature verification using double-stage feature extraction modelled by dynamic feature stability experiment, IET Biometrics, 6 (2017), pp. 393-401.
75
[76] F.Yuan, L-M. Po, M. Liu, X, Xu, W. Jian, K. Wong and K. Cheung, Shearlet based video fingerprint for content-based copy detection, J. Signal Inf. Process., 7 (2016), pp. 84-97.
76
[77] G. Zaccone, M.R. Karim and Menshawy, Deep learning with TensorFlow, explore neural networks and build intelligent systems with Python, Birmingham, England, Mumbai, India, Packt, 2017.
77
[78] G. Zhong, L. Wang and J. Dong, An overview on data representation learning: from traditional feature learning to recent deep learning, J. Financ Data Sci., 2 (2016), pp. 265-278.
78
[79] E.N. Zois, L. Alewijnse and G. Economou, Offline signature verification and quality characterization using poset-oriented grid features, Pattern Recognit., 54 (2016), pp. 162-177.
79
ORIGINAL_ARTICLE
Weighted Composition Operators Between Extended Lipschitz Algebras on Compact Metric Spaces
In this paper, we provide a complete description of weighted composition operators between extended Lipschitz algebras on compact metric spaces. We give necessary and sufficient conditions for the injectivity and the sujectivity of these operators. We also obtain some sufficient conditions and some necessary conditions for a weighted composition operator between these spaces to be compact.
https://scma.maragheh.ac.ir/article_39952_0926650acf7986aa5514b5750bfb1a7a.pdf
2020-07-01
33
70
10.22130/scma.2020.114523.680
Compact operator
Extended Lipschitz algebra
Lipschitz mapping
Supercontactive mapping
Weighted composition operator
Reyhaneh
Bagheri
bagheri.reyhaneh@gmail.com
1
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Arak, Iran.
AUTHOR
Davood
Alimohammadi
alimohammadi.davood@gmail.com
2
Department of Mathematics, Faculty of Science, Arak University, Arak 38156-8-8349, Arak, Iran.
LEAD_AUTHOR
[1] H. Alihoseini and D. Alimohammadi, (-1)-Weak amenability of second dual of real Banach algebras, Sahand Commun. Math. Anal., 12 (2018), pp. 59-88.
1
[2] D. Alimohammadi and S. Daneshmand, Weighted composition operators between Lipschitz algebras of complex-valued bounded functions, Caspian J. Math. Sci., 9 (2020), pp. 100-123.
2
[3] D. Alimohammadi and S. Moradi, Some dense linear subspaces of extended little Lipschitz algebras, ISRN Mathematical Analysis, Article ID 187952, (2011), 10 pages.
3
[4] D. Alimohammadi and S. Moradi, Sufficient conditions for density in extended Lipschitz algebras, Caspian J. Math. Sci., 3 (2014), pp. 141-151.
4
[5] D. Alimohammadi, S. Moradi and E. Analoei, Unital compact homomorphisms between extended Lipschitz algebras, Advances and Applications in Mathematical Sciences, 10 (2011), pp. 307-330.
5
[6] S. Daneshmand and D. Alimohammadi, Weighted composition operators between Lipschitz spaces on pointed metric spaces, Operators and Matrices, 13 (2019), pp. 545-561.
6
[7] A. Golbaharan and H. Mahyar, Weighted composition operators of Lipschitz algebras, Houston J. Math. 42 (2016), pp. 905-917.
7
[8] T.G. Honary and S. Moradi, On the maximal ideal spaces of extended analytic Lipschitz algebras, Quaestiones Mathematicae, 30 (2007), pp. 349-353.
8
[9] A. Jimenez-Vargas and M. Villegas-Vallecillos, Compact composition operators on noncompact Lipschitz spaces, J. Math. Anal. Appl., 398 (2013), pp. 221-229.
9
[10] H. Kamowitz and S. Scheinberg, Some properties of endomorphisms of Lipschitz algebras, Stud. Math., 96 (1990), pp. 255-261.
10
[11] M. Mayghani and D. Alimohammadi, Closed ideals, point derivations and weak amenability of extended little Lipschitz algebras, Caspian J. Math. Sci., 5 (2016), pp. 23-35.
11
[12] M. Mayghani and D. Alimohammadi, The structure of ideals, point derivations, amenability and weak amenability of extended little Lipschitz algebras, Int. J. Nonlinear Anal. Appl., 8 (2017), pp. 389-404.
12
[13] W. Rudin, Real and Complex Analysis, McGraw-Hill, NewYork, Third Edition, 1987.
13
[14] D.R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math., 13 (1963), 1387-1399.
14
[15] 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.
15
[16] N. Weaver, Lipschitz algebras, World Scientific, Singapore, 1999.
16
ORIGINAL_ARTICLE
Strong Convergence of the Iterations of Quasi $\phi$-nonexpansive Mappings and its Applications in Banach Spaces
In this paper, we study the iterations of quasi $\phi$-nonexpansive mappings and its applications in Banach spaces. At the first, we prove strong convergence of the sequence generated by the hybrid proximal point method to a common fixed point of a family of quasi $\phi$-nonexpansive mappings. Then, we give applications of our main results in equilibrium problems.
https://scma.maragheh.ac.ir/article_39052_05e6dbe42b5e8ba9e09f1c6d6e2559a5.pdf
2020-07-01
71
80
10.22130/scma.2019.115400.687
Demiclosed
equilibrium problem
fixed point
hybrid projection
quasi nonexpansive mapping
Resolvent
Rasoul
Jahed
rjahed@iaugermi.ac.ir
1
Department of Mathematics, Sarab Branch, Islamic Azad University, Sarab, Iran.
AUTHOR
Hamid
Vaezi
hvaezi@tabrizu.ac.ir
2
Department of Mathematics, Faculty of Mathematical Science, University of Tabriz, Tabriz, Iran.
LEAD_AUTHOR
Hossein
Piri
h.piri@bonabu.ac.ir
3
Department of Mathematics, University of Bonab, Bonab, Iran.
AUTHOR
[1] Y.I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications. Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996, pp. 15-50.
1
[2] O. Guler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim., 29 (1991), pp. 403-419.
2
[3] A.N. Iusem and M. Nasri, Inexact proximal point methods for equilibrium problems in Banach spaces, Numer. Funct. Anal. Optim., 28 (2007), pp. 1279-1308.
3
[4] S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13 (2002), no. 3, pp. 938-945.
4
[5] H. Khatibzadeh and V. Mohebbi, On the iterations of a sequence of strongly quasi-nonexpansive mappings with applications, Numer. Funct. Anal. Optim., (2019) doi: 10.1080/01630563.2019.1626419.
5
[6] H. Khatibzadeh and V. Mohebbi, On the proximal point method for an infinite family of equilibrium problems in Banach spaces, Bull. Korean Math. Soc., 56 (2019), pp. 757--777.
6
[7] F. Kohsaka and W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. Appl. Anal., (2004), pp. 239-249.
7
[8] Z. Ma, L. Wang and S. Chang, Strong convergence theorem for quasi-$phi$-asymptotically nonexpansive mappings in the intermediate sense in Banach spaces, J. Inequal. Appl., (2013) 2013:306, 13 pp.
8
[9] B. Martinet, Regularisation d'Inequations Variationnelles par Approximations Successives, Revue Francaise d'Informatique et de Recherche Operationnelle, 3 (1970), pp. 154-158.
9
[10] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, (1996), pp. 313-318.
10
[11] R.T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math., 17 (1966), pp. 497-510.
11
[12] R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
12
[13] R.T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), pp. 209-216.
13
ORIGINAL_ARTICLE
Uniform Convergence to a Left Invariance on Weakly Compact Subsets
Let $\left\{a_\alpha\right\}_{\alpha\in I}$ be a bounded net in a Banach algebra $A$ and $\varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $\|aa_\alpha-\varphi(a)a_\alpha\|\to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups such as Segal algebras and $L^1$-algebras are responsive to this concept. It is also shown that $Wap(A)$ has a left invariant $\varphi$-mean if and only if there exists a bounded net $\left\{a_\alpha\right\}_{\alpha\in I}$ in $\left\{a\in A;\ \varphi(a)=1\right\}$ such that $\|aa_\alpha-\varphi(a)a_\alpha\|_{Wap(A)}\to0$ uniformly for all $a$ in weakly compact subsets of $A$. Other results in this direction are also obtained.
https://scma.maragheh.ac.ir/article_40529_52ec68cb33d86279de572c0696818129.pdf
2020-07-01
81
91
10.22130/scma.2019.100548.540
Banach algebra
$varphi$-amenability
$varphi$-means
Weak almost periodic
Weak$^*$ topology
Ali
Ghaffari
aghaffari@semnan.ac.ir
1
Department of Mathematics, Faculty of Science, University of Semnan, P.O.Box 35195-363, Semnan, Iran.
LEAD_AUTHOR
Samaneh
Javadi
2
Faculty of Engineering- East Guilan, University of Guilan, P. O. Box 44891-63157, Rudsar, Iran.
AUTHOR
Ebrahim
Tamimi
3
Department of Mathematics, Faculty of Science, University of Semnan, P.O.Box 35195-363, Semnan, Iran.
AUTHOR
[1] A. Azimifard, $alpha$-amenable hypergroups, Math. Z., 265 (2010), pp. 971-982.
1
[2] A. Azimifard, On the $alpha$-amenability of hypergroups, Monatsh Math., 115 (2008), pp. 1-13.
2
[3] H.G. Dales, Banach algebra and automatic continuity, London Math. Soc. Monogr. Ser. Clarendon Press, 2000.
3
[4] J. Duncan and S.A.R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A, 84 (1979), pp. 309-325.
4
[5] R.E. Edwards, Functional analysis, New-York, Holt, Rinehart and Winston, 1965.
5
[6] F. Filbir, R. Lasser, and R. Szwarc, Reiter's condition $P_1$ and approximate identities for hypergroups, Monatsh Math., 143 (2004), pp. 189-203.
6
[7] G.B. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, FL, 1995.
7
[8] A. Ghaffari, Strongly and weakly almost periodic linear maps on semigroup algebras, Semigroup Forum, 76 (2008), pp. 95-106.
8
[9] Z. Hu, M.S. Monfared, and T. Traynor, On character amenable Banach algebras, Studia Math., 193 (2009), pp. 53-78.
9
[10]B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).
10
[11] E. Kaniuth, A.T. Lau, and J. Pym, On character amenability of Banach algebras, J. Math. Anal. Appl., 344 (2008), pp. 942-955.
11
[12] E. Kaniuth, A.T. Lau and J. Pym, On $varphi$-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc., 144 (2008), pp. 85--96.
12
[13] J.L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955.
13
[14] A.T. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fund. Math., 118 (1983), pp. 161-175.
14
[15] M.S. Monfared, Character amenability of Banach algebras, Math. Proc. Camb. Phil. Soc., 144 (2008), pp. 697-706.
15
[16] J.P. Pier, Amenable locally compact groups, John Wiley And Sons, New York, 1984.
16
[17] H. Reiter, $L^1$-algebras and Segal Algebras, Lecture Notes in Mathematics, Vol. 231, Springer-Verlag, Berlin/ New York, 1971.
17
[18] W. Rudin, Functional analysis, McGraw Hill, New York, 1991.
18
ORIGINAL_ARTICLE
On Some Characterization of Generalized Representation Wave-Packet Frames Based on Some Dilation Group
In this paper we consider (extended) metaplectic representation of the semidirect product $G_{\mathbb{J}}=\mathbb{R}^{2d}\times\mathbb{J}$ where $\mathbb{J}$ is a closed subgroup of $Sp(d,\mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{\mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Finally, we rewrite the dual conditions, by using the Wigner distribution and obtain more reconstruction formulas.
https://scma.maragheh.ac.ir/article_40531_27f20b771699ae668d315a11ac3e008e.pdf
2020-07-01
93
106
10.22130/scma.2019.106144.592
Representation frames
Dilation groups
Dual frames
Continuous frames
Atefe
Razghandi
ateferazghandi@yahoo.com
1
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O.Box 397, Sabzevar, Iran.
AUTHOR
Ali Akbar
Arefijamaal
arefijamaal@gmail.com
2
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O.Box 397, Sabzevar, Iran.
LEAD_AUTHOR
[1] S.T. Ali, J.P. Antoine and J.P. Gazeau, Coherent states, wavelets and their generalizations, Springer-Verlag, New York, 2000.
1
[2] S.T. Ali, J.P. Antoine and J.P. Gazeau, Continuous frames in Hilbert spaces, Ann. Physics, 222 (1993), pp. 1-37.
2
[3] J.P. Antoine, The continuous wavelet transform in image processing, CWI Quarterly, 1 (1998), pp. 323-346.
3
[4] A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk. J. Math. 37 (2013), pp. 71-79.
4
[5] P.G. Casazza, G. Kutyniok and M.C. Lammers, Duality principles in frame theory, J. Fourier Anal. Appl. 10 (2004), pp. 383-408.
5
[6] O. Christensen, Frames and Bases: An Introductory Course, Birkhauser, Boston, 2008.
6
[7] O. Christensen and S.S. Goh, From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa, Appl. Comput. Harmon. Anal., 36 (2014), pp. 198-214.
7
[8] E. Cordero, F.D. Mari, K. Nowak and A. Tabacco, Analytic features of reproducing groups for the metaplectic representation, J. Fourier Anal. appl. 12 (2006), pp. 157-180.
8
[9] E. Cordero, E.D. Mari, K. Nowak and A. Tabacco, Dimensional upper bounds for admissible subgroups for the metaplectic representation, Math. Nachr. 283 (2010), pp. 982-993.
9
[10] E. Cordero and A. Tabacco, Triangular Subgroups of Sp(d;R) and Reproducing Formulae, J. Funct. Anal. 264 (2013), pp. 2034-2058.
10
[11] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), pp. 961-1005.
11
[12] M.N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process, 14 (2005), pp. 2091-2016.
12
[13] M. Duval-Destin, M.A. Muschietti and B. Torresani, Continuous wavelet decompositions, multiresolution and contrast analysis, SIAM J. Math. Anal. 24 (1993), pp. 739-755.
13
[14] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989.
14
[15] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press: Boca Raton, 1995.
15
[16] M. Frazier, G. Garrigos, K. Wang and G. Weiss, A characterization of functions that generate wavelet and related expansions, J. Fourier Anal. Appl. 3 (1997), 883-906.
16
[17] A. Ghaani Farashahi, Square-integrability of multivariate metaplectic wave-packet representations, J. Phys. A, 50 (2017), pp. 115-202.
17
[18] A. Ghaani Farashahi, Square-integrability of metaplectic wave-packet representations on $L^2(mathbb{R})$}, J. Math. Anal. Appl., 449 (2017), pp. 769-792.
18
[19] A. Ghaani Farashahi, Abstract harmonic analysis of wave packet transforms over locally compact abelian groups, Banach J. Math. Anal., 11 (2017), pp. 50-71.
19
[20] A. Ghaani Farashahi, Multivariate wave-packet transforms, J. Anal. Appl., 36 (2017), pp. 481-500.
20
[21] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489 (2016), pp. 75-92.
21
[22] A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra, 30 (2015), pp. 507-529.
22
[23] D. Gosson, Symplectic Geometry and Quantum Mechanics, Birkhauser, Basel, 2006.
23
[24] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2001.
24
[25] D. Han, Frame representations and parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc., 360 (2008), pp. 3307-3326.
25
[26] G. Kaiser, A Firendly Guide to wavelets, Birkhauser, Boston, 1994.
26
[27] V.P. Maslov and M.V. Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981.
27
ORIGINAL_ARTICLE
About Subspace-Frequently Hypercyclic Operators
In this paper, we introduce subspace-frequently hypercyclic operators. We show that these operators are subspace-hypercyclic and there are subspace-hypercyclic operators that are not subspace-frequently hypercyclic. There is a criterion like to subspace-hypercyclicity criterion that implies subspace-frequent hypercyclicity and if an operator $T$ satisfies this criterion, then $T\oplus T$ is subspace-frequently hypercyclic. Additionally, operators on finite spaces can not be subspace-frequently hypercyclic.
https://scma.maragheh.ac.ir/article_43323_c259331d9443082f0a692ce44718f592.pdf
2020-07-01
107
116
10.22130/scma.2020.117046.707
Subspace-frequently hypercyclic operators
Subspace-hypercyclic operators
Frequently hypercyclic operators
Hypercyclic operators
Mansooreh
Moosapoor
mosapor110@gmail.com
1
Assistant Professor, Department of Mathematics, Farhangian University, Tehran, Iran.
LEAD_AUTHOR
Mohammad
Shahriari
shahriari@maragheh.ac.ir
2
Department of Mathematics, Faculty of Science, University of Maragheh, P.O. Box55181-83111, Maragheh, Iran.
AUTHOR
[1] N. Bamerni, V. Kadets and A. Kilicman, Hypercyclic operators are subspace-hypercyclic, J. Math. Anal. Appl., 435(2)(2016), pp. 1812-1815.
1
[2] F. Bayart and S. Grivaux, Frequently hypercyclic operators, Trans. Amer. Math. Soc., 358(11) (2006), pp. 5083-5117.
2
[3] F. Bayart and S. Grivaux, Invariant Gaussian measures for operators on Banach spaces and linear dynamics, Proc. London Math. Soc., 94(3) (2007), pp. 181-210.
3
[4] A. Bonilla and K.G. Grosse-Erdmann, Frequently hypercyclic operators and vectors, Ergod. Theor. Dyn. Syst., 27 (2007), pp. 383-404.
4
[5] S. Grivaux, Frequently hypercyclic operators with irregularly visiting orbits, J. Math. Anal. Appl., 462 (2018), pp. 542-553.
5
[6] K.G. Grosse-Erdmann, Frequently hypercyclic bilateral shifts, Glasgow. Math. J., 61(2) (2019), pp. 271-286.
6
[7] K.G. Grosse-Erdmann and A. Peris, Frequently dense orbits, C. R. Acad. Sci. Paris, Ser. I, 341 (2005), pp. 123-128.
7
[8] K.G. Grosse-Erdmann and A. Peris Manguillot, Linear chaos, Springer, 2011.
8
[9] B.F. Madore and R.A. Martinez-Avendano, Subspace hypercyclicity, J. Math. Anal. Appl., 373(2) (2011), pp. 502-511.
9
[10] R.A. Martinez-Avendano and O. Zatarain-Vera, Subspace-hypercyclicity for Toeplitz operators, J. Math. Anal. Appl., 422(1) (2015), pp. 772-775.
10
[11] Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc., 369(7) (2017), pp. 4977-4994.
11
[12] H. Rezaei, Notes on subspace-hypercyclic operators, J. Math. Anal. Appl., 397(1) (2013), pp. 428-433.
12
[13] S. Shkarin, On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc., 137(1) (2009), pp. 123-134.
13
[14] T.K. Subrahmonian Moothathu, Two remarks on frequent hypercyclicity, J. Math. Anal. Appl., 408 (2013), pp. 843-845.
14
[15] S. Talebi and M. Moosapoor, Subspace-chaotic operators and subspace-weakly mixing operators, Int. J. of Pure and Applied Math., 78 (2012), pp. 879-885.
15
ORIGINAL_ARTICLE
On the Spaces of $\lambda _{r}$-almost Convergent and $\lambda _{r}$-almost Bounded Sequences
The aim of the present work is to introduce the concept of $\lambda _{r}$-almost convergence of sequences. We define the spaces $f\left( \lambda _{r}\right) $ and $f_{0}\left( \lambda _{r}\right) $ of $ \lambda _{r}$-almost convergent and $\lambda _{r}$-almost null sequences. We investigate some inclusion relations concerning those spaces with examples and we determine the $\beta $- and $\gamma $-duals of the space $f\left( \lambda _{r}\right) $. Finally, we give the characterization of some matrix classes.
https://scma.maragheh.ac.ir/article_40579_85759bdb155dcfb274e7ac69b0e053fb.pdf
2020-07-01
117
130
10.22130/scma.2019.111716.644
Almost convergence
Matrix domain
$beta $-
$gamma $-duals
Matrix transformations
Sinan
Ercan
sinanercan45@gmail.com
1
Department of Mathematics, Faculty of Science, Firat University, 23119, Elazig, Turkey.
LEAD_AUTHOR
[1] A. Sonmez, Almost convergence and triple band matrix, Math. Comput. Modelling, 57 (2013), pp. 2393-2402.
1
[2] M. Candan, Almost convergence and double sequential band matrix, Acta. Math. Sci., 34 (2014), pp. 354-366.
2
[3] M. Kirisci, Almost convergence and generalized weighted mean II, J. Inequal. Appl., 1 (2014), pp. 1-13.
3
[4] M. Kirisci, Almost convergence and generalized weighted mean, In: AIP Conference Proceedings, AIP (2012), pp. 191-194.
4
[5] M. Sengonul and K. Kayaduman, On the Riesz almost convergent sequences space, Abstr. Appl. Anal., 2012 (2012), Article ID 691694, 18 pages.
5
[6] A. Karaisa and F. Ozger, Almost difference sequence space derived by using a generalized weighted mean, J. Comput. Anal. Appl., 19 (2015), pp. 27-38.
6
[7] F. Basar and R. Colak, Almost-conservative matrix transformations, Turkish J. Math., 13 (1989), pp. 91-100.
7
[8] M. Kirisci, On the spaces of Euler almost null and Euler almost convergent sequences, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat, 62 (2013), pp. 1-16.
8
[9] Qamaruddin, S.A. Mohuiddine, Almost convergence and some matrix transformations, Filomat, 21 (2007), pp. 261-266.
9
[10] M. Candan and K. Kayaduman, Almost convergent sequence space derived by generalized Fibonacci matrix and Fibonacci core, Brithish J. Math. Comput. Sci., 7 (2015), pp. 150-167.
10
[11] F. Basar and M. Kirisci, Almost convergence and generalized difference matrix, Comput. Math. Appl., 61 (2011), pp. 602-611.
11
[12] M. Mursaleen and A. K. Noman, On the spaces of $lambda$-convergent sequences and bounded sequences, Thai J. Math, 8 (2010), pp. 311-329.
12
[13] P.N. Ng and P.Y. Lee, Cesaro sequence spaces of non-absolute type, Comment. Math. Prace Mat., 20 (1978), pp. 429-433.
13
[14] M. Sengonul and F. Basar, Some new Cesaro sequence spaces of non-absolute type which include the spaces $c_0$ and $c$, Soochow J. Math., 31 (2005), pp. 107-119.
14
[15] M. Yesilkayagil and F. Basar, Space of $A_lambda $-almost null and $A_lambda $-almost convergent sequences, J. Egypt. Math. Soc., 23 (2015), pp. 119-126.
15
[16] A. Wilansky, Summability Through Functional Analysis, in: North-Holland Mathematics Studies, Elsevier Science Publishers, Amsterdam, New York, 1984.
16
[17] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math., 80 (1948), pp. 167-190.
17
[18] A. M. Jarrah and E. Malkowsky, BK spaces, bases and linear operators, Ren. Circ. Mat. Palermo II, 52 (1990), pp. 177-191.
18
[19] H.I. Miller and C. Orhan, On almost convergent and statistically convergent subsequences, Acta Math. Hungar., 93 (2001), pp. 135-151.
19
[20] G.M. Petersen, Regular Matrix Transformations, McGraw-Hill, New York-Toronto-Sydney, 1970.
20
[21] J.A. Siddiqi, Infinite matrices summing every almost periodic sequences, Pac. J. Math., 39 (1971), pp. 235-251.
21
[22] J.P. Duran, Infinite matrices and almost convergence, Math. Z., 128 (1972), pp. 75-83.
22
[23] J.P. King, Almost summable sequences, Proc. Am. Math. Soc., 17 (1966), pp. 1219-1225.
23
[24] F. Basar and I. Solak, Almost-coercive matrix transformations, Rend. Mat. Appl., 11 (1991), pp. 249-256.
24
[25] F. Basar, Summability Theory and Its Applications, Bentham Science Publishers, Istanbul, 2012.
25
[26] S. Nanda, Infinite matrices and almost convergence, J. Indian Math. Soc., 40 (1976), pp. 173-184.
26
[27] P. Korus, On $Lambda ^r$-strong convergence of numerical sequences and Fourier series, J. Class. Anal., 9 (2016), pp. 89-98.
27
[28] S. Ercan, On $lambda_r$-Convergence and $lambda_r$-Boundedness, Journal of Advanced Physics, 7 (2018), pp. 123-129.
28
ORIGINAL_ARTICLE
Almost Multi-Cubic Mappings and a Fixed Point Application
The aim of this paper is to introduce $n$-variables mappings which are cubic in each variable and to apply a fixed point theorem for the Hyers-Ulam stability of such mapping in non-Archimedean normed spaces. Moreover, a few corollaries corresponding to some known stability and hyperstability outcomes are presented.
https://scma.maragheh.ac.ir/article_40581_6b923b481dfe99fa2884509279aee928.pdf
2020-07-01
131
143
10.22130/scma.2019.113393.665
Multi-cubic mapping
Hyers-Ulam stability
Fixed point
non-Archimedean normed space
Nasrin
Ebrahimi Hoseinzadeh
nasrin_ebrahimi_h@yahoo.com
1
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
AUTHOR
Abasalt
Bodaghi
abasalt.bodaghi@gmail.com
2
Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran.
LEAD_AUTHOR
Mohammad Reza
Mardanbeigi
mrmardanbeigi@srbiau.ac.ir
3
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
AUTHOR
[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan., 2 (1950), pp. 64-66.
1
[2] A. Bahyrycz, K. Cieplinski and J. Olko, On Hyers-Ulam stability of two functional equations in non-Archimedean spaces, J. Fixed Point Theory Appl., 18 (2016), pp. 433-444.
2
[3] A. Bodaghi, Ulam stability of a cubic functional equation in various spaces, Mathematica, 55(2) (2013), pp. 125-141.
3
[4] A. Bodaghi, Cubic derivations on Banach algebras, Acta Math. Vietnam., 38(4) (2013) , pp. 517-528.
4
[5] A. Bodaghi, Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, J. Intel. Fuzzy Syst., 30 (2016), pp. 2309-2317.
5
[6] A. Bodaghi, I.A. Alias and M.H. Ghahramani, Approximately cubic functional equations and cubic multipliers, J. Inequal. Appl., 53 (2011):53, doi:10.1186/1029-242X-2011-53.
6
[7] A. Bodaghi, S.M. Moosavi and H. Rahimi, The generalized cubic functional equation and the stability of cubic Jordan $*$-derivations, Ann. Univ. Ferrara, 59 (2013), pp. 235-250.
7
[8] A. Bodaghi, C. Park and O.T. Mewomo, Multiquartic functional equations, Adv. Difference Equ., 2019, 2019:312, https://doi.org/10.1186/s13662-019-2255-5
8
[9] A. Bodaghi and B. Shojaee, On an equation characterizing multi-cubic mappings and its stability and hyperstability, Fixed Point Theory, to appear, arXiv:1907.09378v2
9
[10] J. Brzdek and K. Cieplinski, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal., 74 (2011), pp. 6861-6867.
10
[11] J. Brzdek and K. Cieplinski, Hyperstability and Superstability, Abstr. Appl. Anal., 2013, Art. ID 401756, 13 pp.
11
[12] K. Cieplinski, Generalized stability of multi-additive mappings, Appl. Math. Lett., 23 (2010), pp. 1291-1294.
12
[13] K. Cieplinski, On the generalized Hyers-Ulam stability of multi-quadratic mappings, Comput. Math. Appl., 62 (2011), pp. 3418-3426.
13
[14] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), pp. 431-436.
14
[15] K. Hensel, Uber eine neue Begrndung der Theorie der algebraischen Zahlen, Jahresber, Deutsche Mathematiker-Vereinigung, 6 (1897), pp. 83-88.
15
[16] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., U.S.A., 27 (1941), pp. 222-224.
16
[17] K.W. Jun and H.M. Kim, The generalized Hyers-Ulam-Russias stability of a cubic functional equation, J. Math. Anal. Appl., 274 (2002), no. 2, 267-278.
17
[18] K.W. Jun and H.M. Kim, On the Hyers-Ulam-Rassias stability of a general cubic functional equation, Math. Inequ. Appl., 6(2) (2003), pp. 289-302.
18
[19] A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Mathematics and its Applications, Vol. 427, Kluwer Academic Publishers, Dordrecht, 1997.
19
[20] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy's Equation and Jensen's Inequality, Birkhauser Verlag, Basel, 2009.
20
[21] C. Park and A. Bodaghi, Two multi-cubic functional equations and some results on the stability in modular spaces, J. Inequal. Appl., 2020, 6 (2020).
21
[22] J.M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glasnik Matematicki. Serija III., 36(1) (2001), pp. 63-72.
22
[23] J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), pp. 126-130.
23
[24] Th.M. Rassias, On the stability of the linear mapping in Banach Space, Proc. Amer. Math. Soc., 72(2) (1978), pp. 297-300.
24
[25] S. Salimi and A. Bodaghi, A fixed point application for the stability and hyperstability of multi-Jensen-quadratic mappings, J. Fixed Point Theory Appl., (2020) 22:9, https://doi.org/10.1007/s11784-019-0738-3.
25
[26] S.M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1964.
26
[27] T.Z. Xu, Stability of multi-Jensen mappings in non-Archimedean normed spaces, J. Math. Phys., 53, 023507 (2012); doi: 10.1063/1.368474.
27
[28] T.Z. Xu, Ch. Wang and Th.M. Rassias, On the stability of multi-additive mappings in non-Archimedean normed spaces, J. Comput. Anal. Appl., 18 (2015), pp. 1102-1110.
28
[29] S.Y. Yang, A. Bodaghi and K.A.M. Atan, Approximate cubic $*$-derivations on Banach $*$-algebras, Abstr. Appl. Anal., 2012, Art. ID 684179, 12 pp.
29
[30] X. Zhao, X. Yang and C.-T. Pang, Solution and stability of the multiquadratic functional equation, Abstr. Appl. Anal., 2013, Art. ID 415053, 8 pp.
30
ORIGINAL_ARTICLE
Continuous $ k $-Frames and their Dual in Hilbert Spaces
The notion of $k$-frames was recently introduced by G\u avru\c ta in Hilbert spaces to study atomic systems with respect to a bounded linear operator. A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous super positions. In this manuscript, we construct a continuous $k$-frame, so called c$k$-frame along with an atomic system for this version of frames. Also we introduce a new method for obtaining the dual of a c$k$-frame and prove some new results about it.
https://scma.maragheh.ac.ir/article_40583_8baad0d22b5a702687eb584294dd9657.pdf
2020-07-01
145
160
10.22130/scma.2019.115719.691
c-frame
$K$-frame
c$K$-frame
c$k$-atom
c$k$-dual
Gholamreza
Rahimlou
grahimlou@gmail.com
1
Department of Mathematics, Shabestar Branch,Islamic Azad University, Shabestar, Iran.
AUTHOR
Reza
Ahmadi
rahmadi@tabrizu.ac.ir
2
Institute of Fundamental Science, University of Tabriz, Tabriz, Iran.
LEAD_AUTHOR
Mohammad Ali
Jafarizadeh
jafarizadeh@tabrizu.ac.ir
3
Faculty of Physic, University of Tabriz, Tabriz, Iran.
AUTHOR
Susan
Nami
s.nami@tabrizu.ac.ir
4
Faculty of Physic, University of Tabriz, Tabriz, Iran.
AUTHOR
[1] S.T. Ali, J.P. Antoine, and J.P. Gazeau, Continuous frames in Hilbert spaces, Ann.Phys., 222 (1993), pp. 1-37.
1
[2] S.T. Ali, J.P. Antoine, and J.P. Gazeau, Coherent States, Wavelets and their Generalizations, Springer Graduate Texts in Contemporary Physics, 1999.
2
[3] F. Arabyani and A.A. Arefijamal, Some constructions of $k$-frames and their duals, Rocky Mountain., 47(6)(2017), pp. 1749-1764.
3
[4] J. Benedetto, A. Powell, and O. Yilmaz, Sigma-Delta quantization and finite frames, IEEE Trans. Inform.Th., 52(2006), pp. 1990-2005.
4
[5] H. Bolcskel , F. Hlawatsch, and H.G Feichyinger, Frame-Theoretic analysis of oversampled filter bank, IEEE Trans. Signal Processing.,46(12)(1998), pp. 3256- 3268.
5
[6] E.J. Candes and D.L. Donoho, New tight frames of curvelets and optimal representation of objects with piecwise $C^2$ singularities, Comm. Pure and App. Math.,56 (2004), pp. 216-266.
6
[7] P.G. Casazza and G. Kutyniok, Frame of subspaces, Contemp. Math. 345, Amer. Math. Soc., Providence, RI., (2004), pp. 87-113.
7
[8] P.G. Casazza, G. Kutyniok, and S. Li, Fusion frames and Distributed Processing, Appl. Comput. Harmon. Anal.,25 (2008), pp. 114-132.
8
[9] P.G. Casazza and J. Kovacevic, Equal-norm tight frames with erasures, Adv. Comput. Math., 18 (2003), pp. 387-430.
9
[10] O. Christensen, An Introduction to Frames and Riesz Bases, 2nd ed. Birkhauser,Boston, 2016.
10
[11] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal Expansions, J. Math. Phys., 27(1986), pp. 1271-1283.
11
[12] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert spaces, Proc. Amer. Math. Soc., 17(2) (1966), pp. 413-415.
12
[13] R.J. Duffin and A.C. Schaeffer, A class of nonharmonik Fourier series, Trans. Amer. Math. Soc.,72 (1952), pp. 341-366.
13
[14] M.H. Faroughi and E. Osgooei, C-Frames and C-Bessel Mappings, Bull. Iranian Math. Soc., 38(1) (2012), pp. 203-222.
14
[15] M. Fornasier and H. Rauhut, Continuous frames, function spaces, and the discretization problem, J. Fourier Anal. Appl., 11(3) (2005), pp.245-287.
15
[16] J.P. Gabardo and D. Han, Frames associated with measurable spaces, Adv. Comput. Math., 18(2003), pp.127-147.
16
[17] L.Gavruta, Frames for operators, Appl. Comput. Harmon. Anal.,32 (2012), pp. 139-144.
17
[18] B. Hassibi, B. Hochwald, A. Shokrollahi, and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform.Theory., 47 (2001), pp. 2335-2367.
18
[19] G. Kaiser, A Friendly Guide to Wavelets, Birkhuser Boston, 2011.
19
[20] M. Mirzaee, M. Rezaei, and M.A. Jafarizadeh, Quantum tomography with wavelet transform in Banach space on homogeneous space, Eur. Phys. J. B., 60 (2007), pp. 193-201.
20
[21] A. Rahimi A. Najati, and Y.N. Dehgan, Continuous frame in Hilbert space, Methods Func. Anal. Top., 12 (2006), pp. 170-182.
21
[22] A. Rahimi, A. Najati, and M H. Faroughi, Continuous and discrete frames of subspaces in Hilbert spaces, Southeast Asian Bull. Math., 32 (2008), pp. 305-324.
22
[23] M. Rahmani, On some properties of c-frames, J. Math. Research with Appl., 37(4) (2017), pp. 466-476.
23
[24] W. Rudin, Functional Analysis, New York, Tata Mc Graw-Hill Editions, 1973.
24
[25] W. Rudin, Real and Complex Analysis, New York, Tata Mc Graw-Hill Editions, 1987.
25
[26] S. Sakai, $C^*$-Algebras and $W^*$-Algebras, New York, Springer-Verlag, 1998.
26
[27] X. Xiao, Y. Zhu, and L. Gavruta, Some Properties of $k$-frames in Hilbert Spaces, Results. Math., 63 (2012), pp.1243-1255.
27
ORIGINAL_ARTICLE
$n$-factorization Property of Bilinear Mappings
In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:X\times Y\to Z$, depended on a natural number $n$ and a cardinal number $\kappa$; which is called $n$-factorization property of level $\kappa$. Then we study the relation between $n$-factorization property of level $\kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity and also strong Arens irregularity. These results may help us to prove some previous problems related to strong Arens irregularity more easier than old. These include some results proved by Neufang in ~\cite{neu1} and ~\cite{neu}. Some applications to certain bilinear mappings on convolution algebras, on a locally compact group, are also included. Finally, some solutions related to the Ghahramani-Lau conjecture is raised.
https://scma.maragheh.ac.ir/article_40584_cbae0f5dc8463173efac6e2a2c9b9cae.pdf
2020-07-01
161
173
10.22130/scma.2019.116000.696
Bilinear map
Factorization property
Strongly Arens irregular
Automatically bounded and $w^*$-$w^*$-continuous
Sedigheh
Barootkoob
s.barutkub@ub.ac.ir
1
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran.
LEAD_AUTHOR
[1] G.R. Allan and A.M. Sinclair, Bounded approximate identities, factorization, and a convolution algebra, J. Funct. Anal., 29 (1978), pp. 308-318.
1
[2] G.R. Allan and A.M. Sinclair, Power factorization in Banach algebras with bounded approximate identity, Studia Math., 56 (1976), pp. 31-38.
2
[3] A. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), pp. 839-848.
3
[4] S. Barootkoob, Topological centers and factorization of certain module actions, Sahand Commun. Math. Anal., 15 (1) (2019), pp. 203-215.
4
[5] P. Cohen, Factorization in group algebras, Dllke Math. J., 26 (1959), pp. 199-205.
5
[6] H.G. Dales, Banach algebras and automatic continuity, Vol. 24 of London Mathematical Society Monographs, The Clarendon Press, Oxford, UK, 2000.
6
[7] F. Ghahramani and A.T.-M. Lau, Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal., 132 (1) (1995), pp. 170-191.
7
[8] F. Ghahramani and J.P. McClure, Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull., 35 (2) (1992), pp. 180-185.
8
[9] M. Eshaghi Gordji and M. Filali, Arens regularity of module actions, Studia Math., 181 (3) (2007), pp. 237-254.
9
[10] N. Gronbaek, Power factorization in Banach modules over commutative radical Banach algebras, Math. Scand., 50 (1982), pp. 123-134.
10
[11] K. Haghnejad Azar, Arens Regularity and Factorization Property, J. Sci. Kharazmi University, 13 (2) (2013), pp. 321-336.
11
[12] K. Haghnejad Azar, Factorization properties and generalization of multipliers in module actions, Journal of Hyperstructures, 4 (2) (2015), pp. 142-155.
12
[13] K. Haghnejad Azar and Masoud Ghanji, Factorization properties and topologicalL centers of module actions and $*$-involution algebras, U.P.B. Sci. Bull., Series A, 75 (1) (2013), pp. 35-46.
13
[14] E. Hewitt and K. A. Ross, Abstract harmonic analysts, Volume II: Structure and analysts for compact groups, analysis on locally compart Abeltan gnmps, Springer-Verlag, Berlin, Heidelberg, and New York, 1970.
14
[15] H. Hofmeier and G. Wittstock, A bicommutant theorem for completely bounded module homomorphisms, Math. Ann., 308 (1) (1997), pp. 141-154.
15
[16] Z. Hu and M. Neufang, Decomposability of von Neumann algebras and the Mazur property of higher level, Canad. J. Math., 58 (4) (2006), pp. 768-795.
16
[17] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).
17
[18] A. T.-M Lau and A. Ulger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc., 348 (3) (1996), pp. 1191-1212.
18
[19] V. Losert, M. Neufang, J. Pachl, and J. Steprans, Proof of the Ghahramani–Lau conjecture, Advanc. Math., 290 (2016), pp. 709-738.
19
[20] M. Neufang, On a conjecture by Ghahramani-Lau and related problems concerning topological centers, J. Funct. Anal., 224 (1) (2005), pp. 217-229.
20
[21] M. Neufang, On Mazur's property and property (X), J. Operat. Theory, 60 (2) (2008), pp. 301-316.
21
[22] M. Neufang, Solution to a conjecture by Hofmeier-Wittstock, J. Funct. Anal., 217 (1) (2004), pp. 171-180.
22
[23] D. Poulin, Characterization of amenability by a factorization property of the group Von Neumann algebra, arXiv:1108.3020v1 [math.OA] (2011).
23
ORIGINAL_ARTICLE
Joint and Generalized Spectral Radius of Upper Triangular Matrices with Entries in a Unital Banach Algebra
In this paper, we discuss some properties of joint spectral {radius(jsr)} and generalized spectral radius(gsr) for a finite set of upper triangular matrices with entries in a Banach algebra and represent relation between geometric and joint/generalized spectral radius. Some of these are in scalar matrices, but some are different. For example for a bounded set of scalar matrices,$\Sigma$, $r_*\left(\Sigma\right)= \hat{r}\left(\Sigma\right)$, but for a bounded set of upper triangular matrices with entries in a Banach algebra($\Sigma$), $r_*\left(\Sigma\right)\neq\hat{r}\left(\Sigma\right)$. We investigate when the set is defective or not and equivalent properties for having a norm equal to jsr, too.
https://scma.maragheh.ac.ir/article_37420_7f123da849ba22fd0f27bd234a383849.pdf
2020-07-01
175
188
10.22130/scma.2018.77951.362
Banach algebra
Upper Triangular Matrix
Generalized Spectral Radius
Joint Spectral Radius
Geometric Joint Spectral Radius
Hamideh
Mohammadzadehkan
mohammadzadeh83@gmail.com
1
Department of Mathematics, Faculty of Science, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.
LEAD_AUTHOR
Ali
Ebadian
ebadian.ali@gmail.com
2
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran.
AUTHOR
Kazem
Haghnejad Azar
haghnejad@uma.ac.ir
3
Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran.
AUTHOR
[1] B.A. Barnes, The spectral theory of upper triangular matrices with entries in a Banach algebra, Math. Nachr., 241 (2002), pp. 5-20.
1
[2] M. Berger and Y. Wang, Bounded semigroups of matrices, Lin. Alg. Appl., 166 (1992), pp. 21-27.
2
[3] V. Blondel and Y. Nestero, Computationally efficient approximations of the joint spectral radius, SIAM J. Matrix Anal. Appl., 27.1 (2005), pp. 256-272.
3
[4] L. Elsner, The generalized spectral-radius theorem: an analytic-geometric proof, Lin. Alg. Appl., 220 (1995), pp. 151-159.
4
[5] L. Gurvits, Stability of discrete linear inclusion, Lin. Alg. Appl., 231 (1995), pp. 47-85.
5
[6] R.E. Harte, Spectral mapping theorems, Proc. Roy. lrish Acad. sect. A 72 (1972), pp. 89-107.
6
[7] R. Jungers, The joint spectral radius, Lecture Notes in Control and Information Sciences, vol. 385, Springer-Verlag, Berlin, 2009.
7
[8] H. Mohammadzadehkan, A. Ebadian, and K. Haghnejad Azar, Joint spectrum of n-tuple of upper triangular matrices with entries in a unitall Banach algebra, Math. Rep., 19 (2017), pp. 21-29.
8
[9] P. Rosenthal and A. Soltysiak, Formulas for the joint spectral radius of noncommutating Banach algebra elements, Proc. Amer. Math. Soc., 123 (1995), pp. 2705-2708.
9
[10] G. C. Rota and W. G. Strang, A note on the joint spectral radius, Indag. Math. 22 (1960), pp. 379-381.
10
[11] J. Theys, Joint spectral radius: theory and approximations, PhD Thesis, University of Louvain, (2005).
11
ORIGINAL_ARTICLE
On Fixed Point Results for Hemicontractive-type Multi-valued Mapping, Finite Families of Split Equilibrium and Variational Inequality Problems
In this article, we introduced an iterative scheme for finding a common element of the set of fixed points of a multi-valued hemicontractive-type mapping, the set of common solutions of a finite family of split equilibrium problems and the set of common solutions of a finite family of variational inequality problems in real Hilbert spaces. Moreover, the sequence generated by the proposed algorithm is proved to be strongly convergent to a common solution of these three problems under mild conditions on parameters. Our results improve and generalize many well-known recent results existing in the literature in this field of research.
https://scma.maragheh.ac.ir/article_39955_fccbe118a27b5cd60bf3c5d750ffab92.pdf
2020-07-01
189
217
10.22130/scma.2019.99206.533
Fixed point
multi-valued, hemicontractive-type, variational inequality, split equilibrium problems
strong convergence
Monotone mapping
Tesfalem Hadush
Meche
tesfalemh78@gmail.com
1
Department of Mathematics, College of Natural and Computational Sciences, Aksum University, P.O.Box 1020, Aksum, Ethiopia.
AUTHOR
Habtu
Zegeye
habtuzh@yahoo.com
2
Department of Mathematics and Statistical Sciences, Faculty of Sciences, Botswana International University of Science and Technology, Private Mail Bag 16, Palapye, Botswana.
LEAD_AUTHOR
[1] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), pp. 123-145.
1
[2] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. algorithms, 59 (2012), pp. 301-323.
2
[3] C.E. Chidume, C.O. Chidume, N. Djitte and M.S. Minjibir, Convergence theorems for fixed points of multi-valued strictly pseudocontractive mapping in Hilbert spaces, Abstr. Appl. Anal., 2013, Article ID 629468, 10 pages.
3
[4] S.Y. Cho, Approximation of solutions of a generalized variational inequality problem based on iterative methods, Commun. Korean. Math. Soc., 25 (2010), pp. 207-214.
4
[5] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), pp. 117-136.
5
[6] M. Eslamian, General algorithms for split common fixed point problem of demicontractive mappings, Optimization, 65 (2016), pp. 443-465.
6
[7] Z. He, The split equilibrium problem and its convergence algorithms, J. inequal. Appl., 2012, 2012: 162.
7
[8] J.U. Jeong, Nonlinear algorithms for a common solution of a system of variational inequalities, a split equilibrium problem and fixed point problems, Korean J. Math. 24 (2016), pp. 495-524.
8
[9] S.B. Jeong, A. Raﬁq and S.M. Kang, On implicit mann type iteration process for strictly hemicontractive mappings in real smooth Banach spaces, Int. J. Pure and Applied Math., 89 (2013), pp. 95-103.
9
[10] K.R. Kazmi and S.H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egypt. Math. Soc., 21 (2013), pp. 44-51.
10
[11] J.K. Kim and N. Buong, An iterative method for common solution of a system of equilibrium problems in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), 15 pages.
11
[12] R. Kraikaew and S. Saejung, On a hybrid extragradient-viscosity method for monotone operator and fixed point problems, Numerical Fun. Anal. Optim., 35 (2014), pp. 32-49.
12
[13] P.E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), pp. 899-912.
13
[14] T.H. Meche, M.G. Sangago and H. Zegeye, Iterative methods for a fixed point of hemicontractive-type mapping and a solution of a variational inequality problem, Creat. Math. Inform., 25 (2016), pp. 183-196.
14
[15] T.H. Meche, M.G. Sangago and H. Zegeye, Approximating a common solution of a finite family of generalized equilibrium and fixed point problems, SINET: Ethiop. J. Sci., 38(2015), pp. 17-28.
15
[16] T.H. Meche, M.G. Sangago and H. Zegeye, Iterative methods for common solution of split equilibrium, variational inequality and fixed point problems of multi-valued nonexpansive mapping, (2017) (in press).
16
[17] C. Mongkolkeha, Y.J. Cho and P. Kumam, Convergence theorems for $k-$demicontractive mapping in Hibert spaces, Math. inequal. App., 16 (2013), pp. 1065-1082.
17
[18] S.B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math., 30 (1969), pp. 475-487.
18
[19] C.C. Okeke and O.T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multi-valued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), pp. 223-248.
19
[20] S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), pp. 742-750.
20
[21] K.P.R. Sastry and G.V.R. Babu, Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Math. J., 55 (2005), pp. 817-826.
21
[22] N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal-Theory, 71 (2009), pp. 838-844.
22
[23] Y. Shehu and O. Iyiola, Nonlinear iteration method for proximal split feasibility problems, Wiley, (2016).
23
[24] G. Stampacchia, Formes bilineaires coercivites sur les ensembles convexes, C. R. Acad. Sciences, Paris, 258 (1964), pp. 4413-4416.
24
[25] W. Takahashi, Nonlinear functional analysis, Yokohama Publishere, Yokohama, Japan, 2000.
25
[26] S. Takahashi and W. Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65 (2016), pp. 281-287.
26
[27] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), pp. 417-428.
27
[28] G.C. Ugwunnadi and B. Ali, Approximation methods for solutions of system of split equilibrium problems, Adv. Oper. Theory, 1 (2016), pp. 164-183.
28
[29] J. Vahid, A. Latif and M. Eslamian, New iterative scheme with strict pseudo-contractions and multi-valued nonexpansive mappings for fixed point problems and variational inequality problems, Fixed Point Theory Appl., (2013) 2013:213.
29
[30] S.H. Wang and M.J. Chen, Iterative algorithm for a split equilibrium problem and fixed point problem for finite family of asymptotically nonexpansive mappings in Hilbert space, Filomat, 31:5 (2017), pp. 1423-1434.
30
[31] S. Wang, X. Gong, A.A. Abdou and Y.J. Cho, Iterative algorithm for a family of split equilibrium problems and fixed point problems in Hilbert spaces with applications, Fixed Point Theory Appl., 2016:4 (2016).
31
[32] F. Wang and H.K. Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74 (2011), pp. 4105-4111.
32
[33] S. Wang and C. Zhou, New iterative schemes for finite families of equilibrium, variational inequality and fixed point problems in Banach spaces, Fixed point Theory Appl., Vol. 2011, Article ID 372975, 18 pages.
33
[34] S. Wang, H. Zhou and J. Song, Viscosity approximation methods for equilibrium problems and fixed point problems of nonexpansive mappings and inverse-strongly monotone mappings, Method Appl. Anal., 14 (2007), pp. 405-420.
34
[35] S.T. Woldeamanuel, M.G. Sangago and H. Zegeye, Strong convergence theorems for a common fixed point of a finite family of Lipchitz hemicontractive-type multi-valued mappings, Adv. Fixed Point Theory, 5 (2015), pp. 228-253.
35
[36] H.K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65 (2002), 109-113.
36
[37] H. Zegeye, An iterative approximation method for a common fixed point of two pseudocontractive mappings, ISRN Math. Anal., 2011 (2011), 14 pages.
37
[38] H. Zegeye, T.H. Meche and M.G. Sangago, Algorithms of common solutions for a fixed point of hemicontractive-type mapping and a generalized equilibrium problem, Inter. J. Adv. Math. Sci., 5 (2017), pp. 20-26.
38
[39] H. Zegeye and N. Shahzad, Convergence of Manns type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), pp. 4007-4014.
39
[40] H. Zegeye and N. Shahzad, A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems, Nonlinear Anal., 74 (2011), pp. 263-272.
40
[41] X. Zheng, Y. Yao, Y.C. Liou and L. Leng, Fixed point algorithms for split problem of demicontractive operators, J. Nonlinear Sci. Appl., 10 (2017), pp. 1263-1269.
41