2021
18
1
0
136
1

Some Properties of Lebesgue Fuzzy Metric Spaces
https://scma.maragheh.ac.ir/article_46667.html
10.22130/scma.2020.120854.743
1
In this paper, we establish a sequential characterisation of Lebesgue fuzzy metric and explore the relationship between Lebesgue, weak $G$complete and compact fuzzy metric spaces. We also discuss the Lebesgue property of several wellknown fuzzy metric spaces.
0

1
14


Sugata
Adhya
Department of Mathematics, The Bhawanipur Education Society College. 5, Lala Lajpat Rai Sarani, Kolkata 700020, West Bengal, India.
India
sugataadhya@yahoo.com


Atasi
Deb Ray
Department of Pure Mathematics, University of Calcutta. 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India.
India
atasi@hotmail.com
Fuzzy metric space
Lebesgue property
Weak $G$complete
[[1] S. Adhya and A. Deb Ray, On Lebesgue property for fuzzy metric spaces, TWMS J. Appl. Eng. Math., (appear).##[2] G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Am. Math. Soc., 95 (4) (1985), pp. 653658.##[3] A. George and P.V. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), pp. 395399.##[4] A. George and P.V. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst., 90 (1997), pp. 365368.##[5] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 27 (1988), pp. 385389.##[6] V. Gregori, A. LópezCrevillén, S. Morillas and A. Sapena, On convergence in fuzzy metric spaces, Topology Appl., 156 (2009), pp. 30023006.##[7] V. Gregori, J.J. Minana and S. Morillas, Uniform continuity in fuzzy metric spaces, Rend. Ist. Mat. Univ. Trieste, 32 Suppl. 2 (2001), pp. 8188.##[8] V. Gregori, J.J. Minana and A. Sapena, Completable fuzzy metric spaces, Topology Appl., 225 (2017), pp. 103111.##[9] V. Gregori, J.J. Minana and A. Sapena, On Banach contraction principles in fuzzy metric spaces, Fixed Point Theory, 19 (2018), pp. 235248.##[10] V. Gregori and S. Romaguera, Characterizing completable fuzzy metric spaces, Fuzzy Sets Syst., 144 (2004), pp. pp. 411420.##[11] V. Gregori and S. Romaguera, Some properties of fuzzy metric spaces, Fuzzy Sets Syst., 115 (2000), pp. 485489.##[12] V. Gregori, S. Romaguera and A. Sapena, Some questions in fuzzy metric spaces, Fuzzy Sets Syst., 204 (2012), pp. 7185.##[13] O. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetica, 11 (1975), pp. 326334.##[14] A. Sapena, A contribution to the study of fuzzy metric spaces, Appl. Gen. Topol., 2 (1) (2001), pp. 6375.##[15] B. Schweizer and A. Sklar, Statistical metric spaces, Pac. J. Math., 10 (1960), pp. 314334.##[16] P. Tirado, On compactness and $G$completeness in fuzzy metric spaces, Iran. J. Fuzzy Syst., 9 (4) (2012), pp. 151–158.##[17] G. Toader, On a problem of Nagata, Mathematica, Cluj, 20 (43) (1978), pp. 7879.##[18] S. Willard, General Topology, Reading, MA: AddisonWesley Publishing (1970).##]
1

A Note on Some Results for $C$controlled $K$Fusion Frames in Hilbert Spaces
https://scma.maragheh.ac.ir/article_46575.html
10.22130/scma.2020.123056.766
1
In this manuscript, we study the relation between Kfusion frame and its local components which leads to the definition of a $C$controlled $K$fusion frames, also we extend a theory based on Kfusion frames on Hilbert spaces, which prepares exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we define the analysis, synthesis and frame operator for $C$controlled $K$fusion frames, which even yield a reconstruction formula. Also, we define dual of $C$controlled $K$fusion frames and study some basic properties and perturbation of them.
0

15
34


Habib
Shakoory
Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran.
Iran
habibshakoory@yahoo.com


Reza
Ahmadi
Research Institute for Fundamental Sciences, University of Tabriz, Tabriz, Iran.
Iran
rahmadi@tabrizu.ac.ir


Naghi
Behzadi
Research Institute for Fundamental Sciences, University of Tabriz, Tabriz,
Iran.
Iran
n.behzadi@tabrizu.ac.ir


Susan
Nami
Faculty of Physic, University of Tabriz, Tabriz, Iran.
Iran
s.nami@tabrizu.ac.ir
Frame
$k$fusion frame
Controlled fusion frame
Controlled $K$fusion frame
[[1] Y. Alizadeh and M. Abdollahpour, Controlled Continuous G Frames and Their Multipliers in Hilbert Spaces, Sahand Commun. Math. Anal., 15(1) (2019), pp. 3748.##[2] F. Arabyani and A. Arefijamal, Some constructions of Kframes and their duals, Rocky Mountain., 47 (2017), pp. 17491764.##[3] P. Balazs, J.P. Antoine and A. Grybo's, Weighted and Controlled frames: mutual relationship and first numerical properties, Int. J. Wavelets, Multi. Info. Proc., 8(1) (2010), pp. 109132.##[4] B.G. Bodmann and V.I. Paulsen, Frame paths and error bounds for sigmadelta quantization, Appl Comput Harmon Anal., 22 (2007), pp. 176197.##[5] I. Bogdanova, P. Vandergheynst, J.P. Antoine, L. Jacques and M. Morvidone, Stereographic wavelet frames on the sphere, Appl. Comput. Harmon. Anal., 16 (2005), pp. 223252.##[6] O. Christensen, An Introduction to Frames and Riesz Basesو Birkhauser, 2016.##[7] R.G. Douglas, On majorization, Factorization and Range Inclusion of Operators on Hilbert Spaces, Proc Amer. Math. Soc., 17 (1996), pp. 413415.##[8] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc., 72 (1952), pp. 341366.##[9] A. Fang and P. Tong Li, Kfusion Frames and the Corresponding Generators for Unitary Systems, Acta Math. Sci., (2018), pp.843854.##[10] H. Jamali and M. Kolahdouz, Richardson and Chebyshev Iterative Methods by Using Gframes, Sahand Commun. Math. Anal., 13(1) (2019), pp. 129139.##[11] D. Hua and Y. Huang, Controlled KGFrames in Hilbert Spaces, Results. Math., 523 (2016), pp. 152168.##[12] A. Khosravi and K. Musazadeh, Controlled fusion frames, Meth. Func. Anal. Topol.,18(3), (2012), pp. 256265.##[13] K. Musazadeh and K. Khandani, Some results on controlled frames in Hilbert space, Acta Math. Sci., 36(3)(2016), pp. 655665.##[14] A. Rahimi and A. Fereydooni, Controlled GFrames and Their GMultipliers in Hilbert spaces, Analele Stiintifice ale Universitatii Ovidius Constanta., 2(12), (2012), pp. 223236.##[15] A. Rahimi , SH. Najafzadeh and M.Nouri, Controlled Kframes in Hilbert spaces, International journal of Analysis and Applications., 4(2), (2015), pp. 3950.##[16] G. Rahimlou, R. Ahmadi, M. Jafarizadeh and S. Nami, Continuous K Frames and their Dual in Hilbert Spaces, Sahand Commun. Math. Anal., 17(3), (2020), pp. 145160.##[17] M. RashidiKouchi, A. Rahimi and Firdous A. Shah, Duals and multipliers of controlled frames in Hilbert spaces, Int. J. Wavelets Multiresolut. Inf. Process., 16(5), (2018), pp. 113.##[18] M. RashidiKouchi, Frames in super Hilbert modules, Sahand Commun. Math. Anal., 9(1), (2018), pp. 129142.##[19] V. Sadri, R. Ahmadi, M. Jafarizadeh and S. Nami, Continuous K Fusion Frames in Hilbert Spaces, Sahand Commun. Math. Anal., 17(1), (2020), pp. 3955.##[20] W. Sun, Gframes and gRiesz bases, J. Math. Anal., 322 (2006), pp. 437452.##]
1

On Approximation of Some Mixed Functional Equations
https://scma.maragheh.ac.ir/article_46665.html
10.22130/scma.2020.127585.801
1
In this paper, we have improved some of the results in [C. Choi and B. Lee, Stability of Mixed AdditiveQuadratic and AdditiveDrygas Functional Equations. Results Math. 75 no. 1 (2020), Paper No. 38]. Indeed, we investigate the HyersUlam stability problem of the following functional equationsbegin{align*} 2varphi(x + y) + varphi(x  y) &= 3varphi(x)+ 3varphi(y) \ 2psi(x + y) + psi(x  y) &= 3psi(x) + 2psi(y) + psi(y).end{align*}We also consider the Pexider type functional equation [2psi(x + y) + psi(x  y) = f(x) + g(y),] and the additive functional equation[2psi(x + y) + psi(x  y) = 3psi(x) + psi(y).]
0

35
46


Abbas
Najati
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.
Iran
a.nejati@yahoo.com


Batool
Noori
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.
Iran
noori.batool@yahoo.com


Mohammad Bagher
Moghimi
Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.
Iran
mbfmoghimi@yahoo.com
HyersUlam stability
Additive
Quadratic
Drygas
Functional equation
Lebesgue measure zero
Pexider equation
[[1] B. Batko, Stability of an alternative functional equation, J. Math. Anal. Appl., 339~ (2008), pp. 303311.##[2] C. Choi and B. Lee, Stability of Mixed AdditiveQuadratic and AdditiveDrygas Functional Equations, Results Math., 75 (2020), Paper No. 38.##[3] J. Chung, Stability of conditional Cauchy functional equations, Aequat. Math., 83 (2012), pp. 313320.##[4] J. Chung and J.M. Rassias, Quadratic functional equations in a set of Lebesgue measure zero, J. Math. Anal. Appl., 419 (2014), pp. 10651075.##[5] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publ. Co., 2002.##[6] D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), pp. 222224.##[7] D.H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, 1998.##[8] S.M. Jung, HyersUlamRassias Stability of Functional Equations in Nonlinear Analysis, Springer, 2011.##[9] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009.##[10] P. Kaskasem, A. Janchada and C. Klineam, On approximate solutions of the generalized radical cubic functional equation in quasi $beta$Banach spaces, Sahand Commun. Math. Anal., 17 (2020), pp. 6990.##[11] B. Khosravi, M.B. Moghimi and A. Najati, Asymptotic aspect of Drygas, quadratic and Jensen functional equations in metric abelian groups, Acta Math. Hungar., 155 (2018), pp. 248265.##[12] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publ. and Silesian Univ. Press, Warsaw, 1985.##[13] Y.H. Lee, S.M. Jung and M.Th. Rassias, On an $n$dimensional mixed type additive and quadratic functional equation, Appl. Math. Comput., 228 (2014), pp. 1316.##[14] Y.H. Lee, S.M. Jung and M.Th. Rassias, Uniqueness theorems on functional inequalities concerning cubicquadraticadditive equation, J. Math. Inequal., 12 (2018), pp. 4361.##[15] M. Maysami Sadr, Stability of additive functional equation on discrete quantum semigroups, Sahand Commun. Math. Anal., 8 (2017), pp. 7381.##[16] D. Molaei and A. Najati, Hyperstability of the general linear equation on restricted domains, Acta Math. Hungar., 149 (2016), pp. 238253.##[17] A. Najati and SoonMo Jung, Approximately quadratic mappings on restricted domains, J. Inequal. Appl., (2010), Art. ID 503458, 10 pages.##[18] A. Najati and Th.M. Rassias, Stability of the Pexiderized Cauchy and Jensen's equations on restricted domains, Sahand Commun. Math. Anal., 8 (2010), pp. 125135.##[19] J.C. Oxtoby, Measure and Category, Springer, New York, 1980.##[20] J. Senasukh and S. Saejung, On the hyperstability of the Drygas functional equation on a restricted domain, Bull. Aust. Math. Soc., 102 (2020), pp. 126137.##]
1

Gabor Dual Frames with Characteristic Function Window
https://scma.maragheh.ac.ir/article_46666.html
10.22130/scma.2020.121704.751
1
The duals of Gabor frames have an essential role in reconstruction of signals. In this paper we find a necessary and sufficient condition for two Gabor systems $left(chi_{left[c_1,d_1right)},a,bright)$ and $left(chi_{left[c_2,d_2right)},a,bright)$ to form dual frames for $L_2left(mathbb{R}right)$, where $a$ and $b$ are positive numbers and $c_1,c_2,d_1$ and $d_2$ are real numbers such that $c_1<d_1$ and $c_2<d_2$.
0

47
57


Mohammad Ali
Hasankhani Fard
Department of Mathematics, ValieAsr University of Rafsanjan, P.O.Box 546, Rafsanjan, Iran.
Iran
m.hasankhani@vru.ac.ir
Frame
Dual frame
Gabor system
Gabor frame
[[1] A.A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal., 35 (2013), pp. 535540.##[2] A.A. Arefijamaal and E. Zekaee, Image processing by alternate dual Gabor frames, Bull. Iranian Math. Soc., 42(6), (2016), pp. 1305 1314.##[3] A. Askari Hemmat, A. Safapour and Z. Yazdani Fard, Coherent Frames, Sahand Commun. Math. Anal., 11(1) (2018), pp. 111.##[4] J.J. Benedetto, Frame decomposition, sampling and uncertainty principle inequalities in wavelets, Mathematics and applications (Eds. J.J. Benedetto and M. W. Frazier.), CRC Press., Boca Raton, FL, 1994 , Chapt. 7.##[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser., Boston, Basel, Berlin, 2002.##[6] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341366.##[7] H.G. Feichtinger and T. Strohmer, Eds, Gabor Analysis and AlgorithmsTheory and Applications, Birkhauser., Boston, 1998.##[8] A. Ghaani Farashahi, Continuous partial Gabor transform for semidirect product of locally compact groups, Bull. Malays. Math. Sci. Soc., 38(2) (2015), pp. 779803.##[9] A. Ghaani Farashahi and R.A. KamyabiGol, Continuous Gabor transform for a class of nonAbelian groups, B. Belg. Math. SocSim., 19(4) (2012), pp. 683701.##[10] F. Ghobadzadeh and A. Najati, Gdual Frames in Hilbert $C^*$module Spaces, Sahand Commun. Math. Anal., 11(1) (2018), pp. 6579.##[11] K. Grochenig, Aspects of Gabor analysis on locally compact Abelian groups, Gabor analysis and Algorithms, ANHA, Birkhauser., Boston MA, 1998, 211231.##[12] C. Heil and D. Walnut, Continuous and discrete wavelet transform, SIAM Rev., 31 (1969), pp. 628666.##[13] M. Mirzaee Azandaryani, Approximate Duals of gframes and Fusion Frames in Hilbert $C^*$modules, Sahand Commun. Math. Anal., 15(1) (2019), pp. 135146.##[14] A.J.E.M. Janssen, The duality condition for WeylHeisenberg frames, In Gabor analysis: theory and application (Eds. H.G. Feichtinger and T. Strhmer). Birkhauser., Boston, 1998.##[15] A.J.E.M. Janssen, Zak transforms with few zeros and the tie, In: Advances in Gabor Analysis (Eds.: H.G. Feichtinger and T. Strohmer), Birkhauser., Boston, 2003.##[16] M. RashidiKouchi, Frames in super Hilbert modules, Sahand Commun. Math. Anal., 09(1) (2018), pp. 129142.##[17] A. Ron and Z. Shen, WeylHeisenberg systems and Riesz bases in $L_2(mathbb{R^d)$, Duke. Math. J., 89 (1997), pp. 237282.##[18] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press., New York, 1980.##]
1

$K$orthonormal and $K$Riesz Bases
https://scma.maragheh.ac.ir/article_47114.html
10.22130/scma.2020.130958.827
1
Let $K$ be a bounded operator. $K$frames are ordinary frames for the range $K$. These frames are a generalization of ordinary frames and are certainly different from these frames. This research introduces a new concept of bases for the range $K$. Here we define the $K$orthonormal basis and the $K$Riesz basis, and then we describe their properties. As might be expected, the $K$bases differ from the ordinary ones mentioned in this article.
0

59
72


Ahmad
Ahmdi
Department of Mathematics, Faculty of Science, University of Hormozgan, P.O.Box 7916193145, Bandar Abbas, Iran.
Iran
ahmadi_a@hormozgan.ac.ir


Asghar
Rahimi
Department of Mathematics, Faculty of Science, University of Maragheh, P.O.Box 55136553, Maragheh, Iran.
Iran
rahimi@maragheh.ac.ir
$K$frame
Riesz basis
Orthonormal basis
Atomic system
[[1] F. Arabyani Neyshaburi and A. Arefijamaal, Some constructions of $K$frames and their duals, Rocky Mt. J. Math., 46 (2017), pp. 17491764.##[2] A. Askari Hemmat, A. Safapour and Z. Yazdani Fard, Coherent Frames, Sahand Commun. Math. Anal., 11 (2018), pp. 111.##[3] T. Bemrose, P.G. Casazza, K. Grochenig, M.C. Lammers and R.G. Lynch, Weaving frames, OAM, 10 (2016), pp. 10931116.##[4] P.G. Casazza and G. Kutyniok, Frames of subspaces, in: Wavelets, Frames and Operator Theory (College Park, MD, 2003), Contemp. Math. 345, Amer. Math. Soc., Providence, RI, (2004), pp. 87113.##[5] O. Christensen, Frames and bases, Birkhauser, 2008.##[6] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. phys., 27 (1986), pp. 12711283.##[7] S.G. Deepshikha and L.K. Vashisht, Weaving $K$frames in Hilbert spaces, Results. Math, 73 (2018), pp. 81100.##[8] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966), pp. 413415.##[9] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341366.##[10] P.A. Fillmore and J.P. Williams, On Operator Ranges, Adv. in Math., 7 (1971), pp. 254281.##[11] D. Gabor, Theory of communication. Part 1: The analysis of information, J. IEE, London, 93 (1946), pp. 429457.##[12] L. Gavruta, Frames for operators, App. and Comp. Harm. Anal., 32 (2012), pp. 139144.##[13] M. Jia and Y.C. Zhu, Some results about the operator perturbation of a $K$frame, Results. Math, 73 (2018), pp. 138148.##[14] M. Nouri, A. Rahimi and SH. Najafzadeh, Some results on controlled $K$frames in Hilbert spaces, Int. J. of Anal. and App., 16 (2018), pp. 6274.##[15] G. Ramu and P.S. Johnson, Frame operators of $K$frames, SeMA Journal, 73 (2016), pp. 171181.##[16] M. RashidiKouchi, Frames in super Hilbert modules, Sahand Commun. Math. Anal., 9 (2018), pp. 129142.##[17] W. Sun, gframes and gRiesz bases, J. Math. Anal. Appl., 322 (2006), pp. 437452.##[18] X. Xiao, Y. Zhu and L. Gavruta, Some properties of $K$frames in Hilbert spaces, Results. Math, 63 (2013), pp. 12431255.##]
1

On New Extensions of HermiteHadamard Inequalities for Generalized Fractional Integrals
https://scma.maragheh.ac.ir/article_239415.html
10.22130/scma.2020.121963.759
1
In this paper, we establish some Trapezoid and Midpoint type inequalities for generalized fractional integrals by utilizing the functions whose second derivatives are bounded . We also give some new inequalities for $k$RiemannLiouville fractional integrals as special cases of our main results. We also obtain some HermiteHadamard type inequalities by using the condition $f^{prime }(a+bx)geq f^{prime }(x)$ for all $xin left[ a,frac{a+b}{2}right] $ instead of convexity.
0

73
88


Huseyin
Budak
Department of Mathematics, Faculty of Science and Arts, Duzce
University, Duzce, Turkey
Turkey
hsyn.budak@gmail.com


Ebru
Pehlivan
Department of Mathematics, Faculty of Science and Arts, Duzce
University, Duzce, Turkey
Turkey
ebrpehlivan.1453@gmail.com


Pınar
Kosem
Department of Mathematics, Faculty of Science and Arts, Duzce
University, Duzce, Turkey
Turkey
pinarksm18@gmail.com
HermiteHadamard inequality
convex function
Bounded function
[[1] M.U. Awan, M.A. Noor, T.S. Du and K.I. Noor, New refinements of fractional HermiteHadamard inequality, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 113(1), (2019), pp. 2129.##[2] H. Budak, M.Z. Sarikaya and M.K. Yildiz, HermiteHadamard type inequalities for Fconvex function involving fractional integrals, Filomat, 32(16),(2018), pp. 55095518.##[3] H. Budak, On refinements of HermiteHadamard type inequalities for RiemannLiouville fractional integral operators, Int. J. Optim. Control. Theor. Appl. IJOCTA, 9(1), (2019), pp. 4148.##[4] H. Budak, On Fejer type inequalities for convex mappings utilizing fractional integrals of a function with respect to another function, Results Math., 74(1), (2019), 29.##[5] H. Budak, H. Kara, M.Z. Sarikaya and M.E. Kiris, New extensions of the HermiteHadamard inequalities involving RiemannLiouville fractional integrals, Miskolc Math. Notes, 21(2), 2020.##[6] H. Budak, F. Ertugral and M.Z. Sarikaya, New generalization of HermiteHadamard type inequalities via generalized fractional integrals, An. Univ. Craiova Ser. Mat. Inform., 2020.##[7] F.X. Chen, Extensions of the HermiteHadamard inequality for convex functions via fractional integrals, J. Math. Inequal, (2016), 10(1), pp. 7581.##[8] F.X. Chen, On the generalization of some HermiteHadamard Inequalities for functions with convex absolute values of the second derivatives via fractional integrals, Ukrainian Math. J., 12(70), (2019), pp. 19531965.##[9] S.S. Dragomir and C.E.M. Pearce, Selected topics on HermiteHadamard inequalities and applications, RGMIA Monographs, Victoria University, 2000. Online: https://rgmia.org/papers/monographs/Master.pdf.##[10] S.S. Dragomir, Some inequalities of HermiteHadamard type for symmetrized convex functions and RiemannLiouville fractional integrals, RGMIA Res. Rep. Coll., 20 (2017).##[11] S.S. Dragomir, P. Cerone and A. Sofo, Some remarks on the midpoint rule in numerical integration, Stud. Univ. Babe¸sBolyai Math., XLV(1), (2000), pp. 6374.##[12] S.S. Dragomir, P. Cerone and A. Sofo, Some remarks on the trapezoid rule in numerical integration, Indian J. Pure Appl. Math., 31(5), (2000), pp. 475494.##[13] A. Gozpinar, E. Set and S.S. Dragomir, Some generalized HermiteHadamard type inequalities involving fractional integral operator for functions whose second derivatives in absolute value are sconvex, Acta Math. Univ. Comenian., 88(1), (2019), pp. 87100.##[14] S.R. Hwang and K.L. Tseng, New HermiteHadamardtype inequalities for fractional integrals and their applications, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 112(4), (2018), pp. 12111223.##[15] M. Jleli and B. Samet, On HermiteHadamard type inequalities via fractional integrals of a function with respect to another function, J. Nonlinear Sci. Appl., 9(3), (2016), pp. 12521260.##[16] M.A. Khan, A. Iqbal, M. Suleman and Y.M. Chu, HermiteHadamard type inequalities for fractional integrals via Green's function, J. Inequal. Appl., 2018 (2018), Article ID 161.##[17] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.##[18] K. Liu, J. Wang and D. O'Regan, On the HermiteHadamard type inequality for $psi$RiemannLiouville fractional integrals via convex functions, J. Inequal. Appl., 2019 (2019), Article ID 27.##[19] S. Miller and B. Ross, An introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, USA, 1993.##[20] P.O. Mohammed and M.Z. Sarikaya, HermiteHadamard type inequalities for Fconvex function involving fractional integrals, J. Inequal. Appl., 2018 (2018), Article ID 359.##[21] S. Mubeen and G.M. Habibullah, $k$ Fractional integrals and application, Int. J. Contemp. Math. Sciences, 7(2), (2012), pp. 8994.##[22] N. Minculete and FC. Mitroi, Fejertype inequalities, Aust. J. Math. Anal. Appl., 9(1), (2012), Art. 12.##[23] J.E. Pecaric, F. Proschan and Y.L. Tong, Convex functions, partial orderings and statistical applications, Academic Press, Boston, 1992.##[24] S. Qaisar, M. Iqbal, S. Hussain, S. Butt and M.A. Meraj, New inequalities on HermiteHadamard utilizing fractional integrals, Kragujevac J. Math., 42(1), (2018), pp. 1527.##[25] K. Qiu and J.R. Wang, A fractional integral identity and its application to fractional HermiteHadamard type inequalities, Journal of Interdisciplinary Mathematics, 21(1), (2018), pp. 116.##[26] M.Z. Sarikaya and N. Aktan, On the generalization some integral inequalities and their applications, Math. Comput. Model., 54 (2011), pp. 21752182.##[27] M.Z. Sarikaya and H. Yildirim, On HermiteHadamard type inequalities for RiemannLiouville fractional integrals, Miskolc Math. Notes, 17(2), (2016), pp. 10491059.##[28] M.Z. Sarikaya and F. Ertugral, On the generalized HermiteHadamard inequalities, Annals of the University of CraiovaMathematics and Computer Science Series, 47(1), (2020), pp. 193–213.##[29] M.Z. Sarikaya, On Fejer type inequalities via fractional integrals, J. Interdisciplinary Math., 21(1), (2018), pp. 143155.##[30] M.Z. Sarikaya, E. Set, H. Yaldiz and N., Basak, HermiteHadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), pp. 24032407.##[31] E. Set, A. Akdemir and B. Celik, On generalization of Fejer type inequalities via fractional integral operator, Filomat, 32(16), (2018), pp. 55375547.##[32] T. Tunc, S. Sonmezoglu and M.Z. Sarikaya, On integral inequalities of HermiteHadamard type via Green function and applications, Appl. Appl. Math., 14(1), (2019), pp. 452462.##]
1

Some biHamiltonian Systems and their Separation of Variables on 4dimensional Real Lie Groups
https://scma.maragheh.ac.ir/article_239419.html
10.22130/scma.2020.122380.764
1
In this work, we discuss biHamiltonian structures on a family of integrable systems on 4dimensional real Lie groups. By constructing the corresponding control matrix for this family of biHamiltonian structures, we obtain an explicit process for finding the variables of separation and the separated relations in detail.
0

89
105


Ghorbanali
Haghighatdoost
Department of Mathematics,Azarbaijan Shahid Madani University, 53714161, Tabriz, Iran.
Iran
gorbanali@yahoo.com


Salahaddin
AbdolhadiZangakani
Department of Mathematics, University of Bonab, Tabriz, Iran.
Iran
s.abdolhadi@ubonab.ac.ir


Rasoul
MahjoubiBahman
Department of Mathematics, University of Bonab, Tabriz, Iran.
Iran
r.mahjoubi@ubonab.ac.ir
Integrable system
BiHamiltonian
Control matrix
Variables of separation
[[1] J. AbediFardad, A. RezaeiAghdam and GH. Haghighatdoost, Integrable and superintegrable Hamiltonian systems with 4dimensional real Lie algebras as symmetry of the systems, J.Math. Phys., 55 (2014), pp. 112.##[2] G. Falqui and M. Pedroni, Separation of variables for biHamiltonian systems, Math. Phys. Anal. Geom., 6 (2003), pp. 139179.##[3] I.M. Gel'fand and I. Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct Anal Its Appl, 13 (1979), pp. 248262.##[4] Gh. Haghighatdoost, H. Abbasi Makrani and R. Mahjoubi, On the cyclic Homology of multiplier Hopf algebras, Sahand Commun. Math. Anal., 9 (2018), pp. 113128.##[5] Y. KosmannSchwarzbach and F. Magri, PoissonNijenhuis structures, Ann. Inst. H. Poincare, 53 (1990), pp. 35  81.##[6] F. Magri and C. Morosi, A Geometrical Characterization of Integrable Hamiltonian Systems through the theory of PoissonNijenhuis manifolds, Dipartimento di Matematica F. Enriques, 1984, Pages 176.##[7] F. Magri, A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), pp. 11561162.##[8] G. Ovando, Four dimensional symplectic Lie algebras, Beitr. Algebra Geom., 47 (2006), pp. 419434.##[9] A.V. Tsiganov, New Variables of Separation for the SteklovLyapunov System, SIGMA, Symmetry Integrability Geom. Methods Appl., 8 (2012), pp. 1214.##[10] A.V. Tsiganov, On bihamiltonian structure of some integrable systems on $so*(4)$, J. Nonlinear Math. Phys., 15 (2008), pp. 171185.##[11] A.V. Tsiganov, On bihamiltonian geometry of the Lagrange top, J. Phys. A: Math. Theor., 41 (2008), pp. 112.##[12] A.V. Tsiganov, On the two different biHamiltonian structures for the Toda lattice, J. Phys. A, Math. Theor., 40 (2007), pp. 63956406.##[13] A.V. Vershilov, A.V. Tsiganov, On biHamiltonian geometry of some integrable systems on the sphere with cubic integral of motion, J. Phys. A, Math. Theor., 42 (2009), pp. 112.##]
1

A Fixed Point Theorem for Weakly Contractive Mappings
https://scma.maragheh.ac.ir/article_240244.html
10.22130/scma.2020.124853.778
1
In this paper, we generalize the concepts of weakly Kannan, weakly Chatterjea and weakly Zamfirescu for fuzzy metric spaces. Also, we investigate Banach's fixed point theorem for the mentioned classes of functions in these spaces. Moreover, we show that the class of weakly Kannan and weakly Chatterjea maps are subclasses of the class of weakly Zamfirescu maps.
0

107
122


Morteza
Saheli
Department of Mathematics, ValieAsr University of Rafsanjan, Rafsanjan, Iran.
Iran
saheli@vru.ac.ir


Seyed Ali Mohammad
Mohsenialhosseini
Department of Mathematics, ValieAsr University of Rafsanjan, Rafsanjan, Iran.
Iran
amah@vru.ac.ir
Fixed point
Weakly Zamfirescu mappings
Weakly Kannan mappings
Weakly Chatterjea mappings
Weakly contractive mappings
[[1] A.H. Ansari and A. Razani, Some fixed point theorems for Cclass functions in bmetric spaces, Sahand Commun. Math. Anal., 10 (2018), pp. 8596.##[2] D. ArizaRuiz and A. JiménezMelado, A continuation method for weakly Kannan maps, Fixed Point Theory Appl., 2010 (2010), pp. 112,##[3] D. ArizaRuiza, A. JiménezMeladob and G. LopezAcedo, A fixed point theorem for weakly Zamfirescu mappings, Nonlinear Analysis, 74 (2011), pp. 16281640.##[4] S.K. Chatterjea, Fixedpoint theorems, C. R. Acad. Bulgare Sci., 25 (1972), pp. 727730.##[5] L.B. Ciric, Generalized contractions and fixed point theorem, Publ. Inst. Math., 12 (1971), pp. 1926.##[6] J. Dugundji and A. Granas, Weakly contractive maps and elementary domain invariance theorem, Bull. Soc. Math. Grece (N.S.), 19 (1978), pp. 141151.##[7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), pp. 395399.##[8] M.B. Ghaemi and A. Razani, Fixed and periodic points in the probabilistic normed and metric spaces, Chaos Solitons Fractals, 28 (2006), pp. 11811187.##[9] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 27 (1988), pp. 385389.##[10] V. Gregori, J. Minana and S. Morillas, On completable fuzzy metric spaces, Fuzzy Sets and Systems, 267 (2015), pp. 133139##[11] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), pp. 7176.##[12] E. Rakotch, A note on contractive mappings, Proceedings of the American Mathematical Society, 13 (1962), pp. 459465.##[13] A. Razani, A contraction theorem in fuzzy metric spaces, Fixed Point Theory Appl., 3 (2005), pp. 257265.##[14] A. Razani, A fixed point theorem in the Menger probabilistic metric space, New Zealand J. Math., 35 (2006), pp. 109114.##[15] A. Razani, Existence of fixed point for the nonexpansive mapping of intuitionistic fuzzy metric spaces, Chaos Solitons Fractals, 30 (2006), pp. 367373.##[16] A. Razani, Results in fixed point theory, Andisheh Zarin publisher, Qazvin, August 2010.##[17] A. Razani and R. Moradi, Fixed point theory in modular space, Saieh Ghostar publisher, Qazvin, April 2006.##[18] A. Razani and M. Shirdaryazdi, A common fixed point theorem of compatible maps in Menger space, Chaos Solitons Fractals, 37 (2007), pp. 2634.##[19] A. Razani and M. Shirdaryazdi, Some results on fixed points in the fuzzy metric space, J. Appl. Math. Comput., 20 (2006), pp. 401408.##[20] R. Saadati, A. Razani and H. Adibi, A Common Fixed Point Theorem in Lfuzzy metric spaces, Chaos Solitons Fractals, 33 (2007), pp. 358363.##[21] T. Zamfirescu, Fixed point theorems in metric spaces, Arch. Math., 23 (1972), pp. 292298.##]
1

On Some Coupled Fixed Point Theorems with Rational Expressions in Partially Ordered Metric Spaces
https://scma.maragheh.ac.ir/article_240245.html
10.22130/scma.2020.120323.739
1
The aim of this paper is to prove some coupled fixed point theorems of a self mapping satisfying a certain rational type contraction along with strict mixed monotone property in an ordered metric space. Further, a result is presented for the uniqueness of a coupled fixed point under an order relation in a space. These results generalize and extend known existing results in the literature.
0

123
136


N.
Seshagiri Rao
Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Post Box No.1888, Adama, Ethiopia.
Ethiopia
seshu.namana@gmail.com


K.
Kalyani
Department of Mathematics, Vignan's Foundation for Science, Technology & Research, Vadlamudi522213, Andhra Pradesh, India.
India
kalyani.namana@gmail.com
Ordered metric space
Monotone property
Rational type contraction
Coupled fixed point
[[1] R.P. Agarwal, M.A. ElGebeily and D. O'Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., 87 (2008), pp. 18.##[2] J. Ahmad, M. Arshad and C. Vetro, On a theorem of Khan in a generalized metric space, Int. J. Anal., 2013, Article ID 852727, (2013).##[3] I. Altun, B. Damjanovic and D. Djoric, Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett., 23 (2010), pp. 310316.##[4] A. AminiHarandi and H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., Theory Methods Appl., 72 (2010), pp. 22382242.##[5] M. Arshad, A. Azam and P. Vetro, Some common fixed results in cone metric spaces, Fixed Point Theory Appl., 2009, Article ID 493965 (2009).##[6] M. Arshad, J. Ahmad and E. Karapinar, Some common fixed point results in rectangular metric spaces, Int. J. Anal., 2013, Article ID 307234 (2013).##[7] M. Arshad, E. Karapinar and J. Ahmad, Some unique fixed point theorems for rational contractions in partially ordered metric spaces, Journal of Inequalities and Applications, 2013:248, 2013.##[8] H. Aydi, E. Karapinar and W. Shatanawi, Coupled fixed point results for ($psi, varphi$)weakly contractive condition in ordered partial metric spaces, Comput. Math. Appl., 62(12) (2011), pp. 44494460.##[9] A. Azam, B. Fisher and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim., 32(3) (2011), pp. 243253.##[10] I. Beg and A.R. Butt, Fixed point for setvalued mappings satisfying an implicit relation in partially ordered metric spaces, Nonlinear Anal., 71 (2009), pp. 36993704.##[11] T.G. Bhaskar and V. Lakshmikantham, Fixed point theory in partially ordered metric spaces and applications, Nonlinear Anal., Theory Methods Appl., 65 (2006), pp. 13791393.##[12] S. Chandok, T.D. Narang and M.A. Taoudi, Some coupled fixed point theorems for mappings satisfying a generalized contractive condition of rational type, Palestine Journal of Mathematics, 4(2) (2015), pp. 360366.##[13] B.S. Choudhury and A. Kundu, A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal., Theory Methods Appl., 73 (2010), pp. 25242531.##[14] L. Ciric, M.O. Olatinwo, D. Gopal and G. Akinbo, Coupled fixed point theorems for mappings satisfying a contractive condition of rational type on a partially ordered metric space, Advances in Fixed Point Theory, 2(1) (2012), pp. 18.##[15] Z. Dricia, F.A. McRaeb and J.V. Devi, Fixedpoint theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear Anal., Theory Methods Appl., 67 (2007), pp. 641647.##[16] M. Edelstein, On fixed points and periodic points under contraction mappings, J. Lond. Math. Soc., 37 (1962), pp. 7479.##[17] D. Gopal, M. Abbas, D. K. Patel and C. Vetro, Fixed points of $alpha$type $F$contractive mappings with an application to nonlinear fractional differential equation, Acta Math. Sci., 36(3) (2016), pp. 957970.##[18] G.C. Hardy and T. Rogers, A generalization of fixed point theorem of S. Reich, Can. Math. Bull., 16 (1973), pp. 201206.##[19] S. Hong, Fixed points of multivalued operators in ordered metric spaces with applications, Nonlinear Anal., Theory Methods Appl., 72 (2010), pp. 39293942.##[20] R. Kannan, Some results on fixed pointsII, Am. Math. Mon., 76 (1969), pp. 7176.##[21] E. Karapinar and N.V. Luong, Quadruple fixed point theorems for nonlinear contractions, Comput. Math. Appl., 64(6) (2012), pp. 18391848.##[22] E. Karapinar, Coupled fixed point on cone metric spaces, Gazi Univ. J. Sci., 24(1) (2011), pp. 5158.##[23] P. Kumam, F. Rouzkard, M. Imdad and D. Gopal, Fixed Point Theorems on Ordered Metric Spaces through a Rational Contraction, Abstract and Applied Analysis, 2013, Article ID 206515, 9 pages.##[24] V. Lakshmikantham and L.B. Ciric, Couple fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., Theory Methods Appl., 70 (2009), pp. 43414349.##[25] H. Lakzian, D. Gopal and W. Sintunavarat, New fixed point results for mappings of contractive type with an application to nonlinear fractional differential equations, J. Fixed Point Theory Appl., 18(2) (2016), pp. 251266.##[26] N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal., Theory Methods Appl., 74 (2011), pp. 983992.##[27] B. Monjardet, Metrics on partially ordered setsa survey, Discrete Math., 35 (1981), pp. 173184.##[28] J.J. Nieto and R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), pp. 223239.##[29] J.J. Nieto and R.R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation, Acta Math. Sin. Engl. Ser., 23(12) (2007), pp. 22052212.##[30] J.J. Nieto, L. Pouso and R. RodríguezLópez, Fixed point theorems in ordered abstract spaces, Proc. Am. Math. Soc., 135 (2007), pp. 25052517.##[31] M. Ozturk and M. Basarir, On some common fixed point theorems with rational expressions on cone metric spaces over a Banach algebra, Hacet. J. Math. Stat., 41(2) (2012), pp. 211222.##[32] A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Am. Math. Soc., 132 (2004), pp. 14351443.##[33] S. Reich, Some remarks concerning contraction mappings, Can. Math. Bull., 14 (1971), pp. 121124.##[34] F. Rouzkard and M. Imdad, Some common fixed point theorems on complex valued metric spaces, Comput. Math. Appl., (2012). doi:10.1016/j.camwa.2012.02.063.##[35] B. Samet, Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces, Nonlinear Anal., 74(12) (2010), pp. 45084517.##[36] P.L. Sharma and A.K. Yuel, A unique fixed point theorem in metric space, Bull. Cal. Math. Soc., 76 (1984), pp. 153156.##[37] S. Shukla, D. Gopal and J. MartinezMoreno, Fixed Points of set valued $alphaF$contractions and its application to nonlinear integral equations, Filomat, 31 (11) (2017), pp. 33773390.##[38] D.R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1974.##[39] E.S. Wolk, Continuous convergence in partially ordered sets, Gen. Topol. Appl., 5 (1975), pp. 221234.##[40] C.S. Wong, Common fixed points of two mappings, Pac. J. Math., 48 (1973), pp. 299312.##[41] X. Zhang, Fixed point theorems of multivalued monotone mappings in ordered metric spaces, Appl. Math. Lett., 23 (2010), pp. 235240.##]