2016
5
3
3
84
On weakly eRopen functions
2
2
The main goal of this paper is to introduce and study a new class of function via the notions of $e$$theta$open sets and $e$$theta$closure operator which are defined by Özkoç and Aslım [10] called weakly $eR$open functions and $e$$theta$open functions. Moreover, we investigate not only some of their basic properties but also their relationships with other types of already existing topological functions.
1

145
153


M.
Ozkoc
Department of Mathematics, Faculty of Science Mu˘gla Sıtkı Ko¸cman University,
Mente¸seMu˘gla 48000, Turkey
Department of Mathematics, Faculty of Science
Turkey
murad.ozkoc@mu.edu.tr


B. S.
Ayhan
Department of Mathematics, Faculty of Science Mu˘gla Sıtkı Ko¸cman University,
Mente¸seMu˘gla 48000, Turkey
Department of Mathematics, Faculty of Science
Turkey
brcyhn@gmail.com
$e$closed set
$e$$theta$open set
weakly $eR$open function
$e$$theta$open function
[[1] D.Andrijevic, On bopen sets. Mat. Vesnik., 48 (1996), 5964. ##[2] M. Caldas, E. Ekici, S. Jafari, R.M. Latif, On weakly BRopen functions and their characterizations in topological spaces. Demonstratio Math., 44 (1) (2011), 159168. ##[3] E. Ekici, New forms of contra continuity. Carpathian J. Math., 24 (1) (2008), 3745. ##[4] , On eopen sets, DP*sets and DPE*sets and decompositions of continuity. Arab. J. Sci. Eng. Sect. A Sci., 33 (2) (2008), 269282. ##[5] N. Levine, Semiopen sets and semicontinuity in topological spaces. Amer. Math. Monthly., 70 (1963), 3641. ##[6] , Strong continuity in topological spaces. Amer. Math. Monthly., 67 (1960), 269. ##[7] A.S. Mashhour, M.E. Abd ElMonsef and S.N. ElDeeb, On precontinuous and weak precontinuous mappings. Proc. Math. Phys. Soc. Egypt., 53 (1982), 4753. ##[8] O. Njastad, On some classes of nearly open sets. Pacic J. Math., 15 (1965), 961970. ##[9] T. Noiri, A generalization of closed mappings. Atti. Accad. Naz. Lince Rend. Cl. Sci. Fis. Mat. Natur., 8 (1973), 210214. ##[10] M. Ozkoc and G. Aslm, On strongly econtinuous functions. Bull. Korean Math. Soc., 47 (5) (2010), 10251036. ##[11] J.H. Park, Strongly bcontinuous functions. Acta Math. Hungar., 110 (4) (2006), 347359. ##[12] M. Stone, Application of the theory of Boolean ring to general topology. Trans. Amer. Math. Soc., 41 (1937), 374481. ##[13] N.V. Velicko, Hclosed topological spaces. Amer. Math. Soc. Transl., 78 (1968), 103118.##]
The directional hybrid measure of efficiency in data envelopment analysis
2
2
The efficiency measurement is a subject of great interest. The majority of studies on DEA models have been carried out using radial or nonradial approaches regarding the application of DEA for the efficiency measurement. This paper, based on the directional distance function, proposes a new generalized hybrid measure of efficiency under generalized returns to scale with the existence of both radial and nonradial inputs and outputs. It extends the hybrid measure of efficiency from Tone (2004) to a more general case. The proposed model is not only flexible enough for the decisionmaker to adjust the radial and nonradial inputs and outputs to attain the efficiency score but also avoids the computational and interpretive difficulties, thereby giving rise to an important clarification and understanding of the generalized DEA model. Furthermore, several frequentlyused DEA models (such as the CCR, BCC, ERM and SBM models) which depend on the radial or nonradial approaches are derived while their results were compared to the ones obtained from this hybrid model. The empirical examples emphasize the consequence of the proposed measure.
1

155
174


A.
Mirsalehy
University Putra Malaysia, Malaysia
University Putra Malaysia, Malaysia
Malaysia
mirsalehi_ali@yahoo.com


M.
Rizam Abu Baker
Laboratory of Computational Statistics and Operations Research, Institute of Mathematical
Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia,
43400
Laboratory of Computational Statistics and
Malaysia
drrizam@gmail.com


L. S.
Lee
Laboratory of Computational Statistics and Operations Research, Institute of Mathematical
Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

Department of Mathematics, Faculty of Science, Universiti Putra Malaysia,
43400
Laboratory of Computational Statistics and
Malaysia


Gh. R.
Jahanshahloo
Department of Mathematics, Science and Research Branch,
Islamic Azad University, Tehran, Iran
Department of Mathematics, Science and Research
Iran
jahanshahloomath@gmail.com
Data Envelopment Analysis
Directional distance function
Hybrid model
Efficiency score
[[1] R. D. Banker, A. Charnes, and W. W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30 (9) (1984), 10781092. ##[2] R. G. Chambers, Y. Chung and R. Fare, Benet and distance functions. Journal of Economic Theory, 70 (2) (1996), 407419. ##[3] R. G. Chambers, Y. Chung and R. Fare, Prot, directional distance functions, and Nerlovian efficiency. Journal of Optimization Theory and Applications, 98 (2) (1998), 351364. ##[4] A. Charnes, W. W. Cooper, Programming with linear fractional functionals. Naval Research Logistics Quarterly, 9 (34) (1962), 181186. ##[5] A. Charnes, W. W. Cooper, B. Golany, L. Seiford and J. Stutz, Foundations of data envelopment analysis for ParetoKoopmans efficient empirical production functions. Journal of Econometrics, 30 (1) (1985), 91107. ##[6] A. Charnes, W. W. Cooper, and E. Rhodes, Measuring the eciency of decision making units. European Journal of Operational Research, 2 (6) (1978), 429444. ##[7] W. D. Cook, L. M. Seiford, Data envelopment analysis (DEA)Thirty years on. European Journal of Operational Research, 192 (1) (2009), 117. ##[8] W. W. Cooper, K. S. Park, and J. T. P. Ciurana, Marginal rates and elasticities of substitution with additive models in DEA. Journal of Productivity Analysis, 13 (2) (2000), 105123. ##[9] W. W. Cooper, K. S. Park and G. Yu, IDEA and ARIDEA: Models for dealing with imprecise data in DEA. Management Science, 45 (4) (1999), 597607. ##[10] W. W. Cooper, L. M. Seiford and J. Zhu, Data envelopment analysis: History, models, and interpretations. Handbook on data envelopment analysis, Springer, (2011),139. ##[11] A. Emrouznejad, B. R. Parker and G. Tavares, Evaluation of research in efficiency and productivity: A survey and analysis of the rst 30 years of scholarly literature in DEA. SocioEconomic Planning Sciences, 42 (3) (2008), 151157. ##[12] R. Fare, S. Grosskopf, A nonparametric cost approach to scale efficiency. The Scandinavian Journal of Economics, (1985), 594604. ##[13] M. J. Farrell, The measurement of productive efficiency. Journal of the Royal Statistical Society. Series A (General), 120 (3) (1957), 253290. ##[14] T. C. Koopmans, Analysis of production as an ecient combination of activities. Activity Analysis of Production and Allocation, 13 (1951), 3337. ##[15] S. Lertworasirikul, P. Charnsethikul and S. C. Fang, Inverse data envelopment analysis model to preserve relative efficiency values: The case of variable returns to scale. Computers & Industrial Engineering, 61 (4) (2011), 10171023. ##[16] J. S. Liu, L. Y. Y Lu, W. M. Lu and B. J. Y. Lin, Data envelopment analysis 19782010: A citationbased literature survey. Omega, 41 (1) (2013), 315. ##[17] D. G. Luenberger, Benet functions and duality. Journal of Mathematical Economics, 21 (5) (1992), 461481. ##[18] D. G. Luenberger, Microeconomic theory. 486 McGrawHill New York, 1995. ##[19] J. T. Pastor, J. L. Ruiz and I. Sirvent, An enhanced DEA Russell graph eciency measure. 115 (1999), 596607. ##[20] J. T. Pastor, J. L. Ruiz and I. Sirvent, An enhanced DEA Russell graph eciency measure. European Journal of Operational Research,115 (3) (1999), 596607. ##[21] J. T. Pastor, J. L. Ruiz and I. Sirvent, Statistical test for detecting in uential observations in DEA. European Journal of Operational Research,115 (3) (1999), 542554. ##[22] V. V. Podinovski, Bridging the gap between the constant and variable returnstoscale models: selective proportionality in data envelopment analysis. Journal of the Operational Research Society, 55 (3) (2004), 265276. ##[23] R. R. Russell, Measures of technical eciency. Journal of Economic Theory, 35 (1) (1985), 109126. ##[24] L. M. Seiford, R. M. Thrall, Recent developments in DEA: the mathematical programming approach to frontier analysis. Journal of Econometrics, 46 (1) (1990), 738. ##[25] R. W. Shepherd, Theory of cost and production functions. Princeton University Press, 2015. ##[26] K. Tone, A slacksbased measure of eciency in data envelopment analysis. European Journal of Operational Research, 130 (3) (2001), 498509. ##[27] K. Tone, A hybrid measure of eciency in DEA. GRIPS Research Report Series, 2004.##]
Fuzzy soft ideals of nearsubraction semigroups
2
2
Our aim in this paper is to introduce the notion of fuzzy soft nearsubtraction semigroups and fuzzy soft ideals of nearsubtraction semigroups. We discuss some important properties of these new fuzzy algebraic structure and investigate some examples and counter examples.
1

175
186


V.
Chinnadurai
Department of Mathematics, Annamalai University, Annamalainagar,
PO. Code 608002, Tamilnadu, India
Department of Mathematics, Annamalai University,
India
kv.chinnadurai@yahoo.com


S.
Kadalarasi
Department of Mathematics, Annamalai University, Annamalainagar,
PO. Code 608002, Tamilnadu, India
Department of Mathematics, Annamalai University,
India
kadalarasi89@gmail.com
Nearsubtraction semigroup, fuzzy soft set, fuzzy soft nearsubtraction
semigroup, fuzzy soft ideal
[[1] J. C. Abbott, Sets, lattices and Boolean algebras. Allyn and Bacon, Boston, 1969. ##[2] H. Aktas and N. Cagman, Soft sets and soft groups. Information Sciences, 177 (2007), 27262735. ##[3] M. I. Ali, F. Feng, X. Liu, W. K. Min and M. Shabir, On some new operations in soft set theory. Computers and Mathematica with Applications, 57 (9) (2009), 15471553. ##[4] Aygunoglu A and aygun H, Introduction to fuzzy soft groups. Computers and Mathematica with Applications, 58 (2009), 12791286. ##[5] ChengFu Yang, Fuzzy soft semigroup and fuzzy soft ideals. Computers and Mathematica with Applications, 61 (2011), 255261. ##[6] F. Feng, Y. B. Jun and X. Zhao, Soft semirings. Computers and Mathematica with Applications, 56 (2008), 26212628. ##[7] P. Dheena and G. Satheeshkumar, On strongly regular nearsubtraction semigroups. Commun. Korean Math. Soc., 22(3) (2007), 323330. ##[8] P. Dheena and G. Satheeshkumar, Weakly prime left ideals in nearsubtraction semigroups. Commun. Korean Math. Soc., 23(3) (2008), 325331. ##[9] P. Dheena and G. Mohanraj, Fuzzy weakly prime ideals in nearsubtraction semigroups. Annals of Fuzzy Mathematics and Informatics, 4(2) (2012), 235242. ##[10] Y. B. Jun, Kyung Ja Lee and Asghar Khan, Ideal theory of subtraction algebras. Sci. Math. Jpn, 61 (2005), 459464. ##[11] P. K Maji, A. R. Roy and R. Biswas, An application of soft sets in decision making problem. Computers and Mathematics with Applications, 44 (2002), 10771083. ##[12] P. K Maji, R. Biswas and A. R. Roy, Soft set theory. Computers and Mathematics with Applications, 45 (2003), 555563. ##[13] P. K Maji, R. Biswas and A. R. Roy, Fuzzy soft set. The Journal Fuzzy Mathematics, 9 (2001), 586602. ##[14] D. Molodsov, Soft set theoryrst result. Computers and Mathematics with Applications, 37 (1999), 1931. ##[15] D. R. Prince Williams and Arshan Borumand Saeid, Fuzzy soft ideals in subtraction algebras. Neural Comput. and Applic., 21 (2012), S159S169. ##[16] D. R. Prince Williams, Fuzzy ideals in nearsubtraction semigroups. International Scholarly and Scientic Research & Innovation, 2(7) (2008), 625632. ##[17] B. M. Schein, Dierence semigroups. Commun Algebra, 20 (1992), 25132169. ##[18] B. Zelinka, Subtraction semigroups. Math. Bohemia, 120 (1995), 445447.##]
Solving systems of nonlinear equations using decomposition technique
2
2
A systematic way is presented for the construction of multistep iterative method with frozen Jacobian. The inclusion of an auxiliary function is discussed. The presented analysis shows that how to incorporate auxiliary function in a way that we can keep the order of convergence and computational cost of Newton multistep method. The auxiliary function provides us the way to overcome the singularity and illconditioning of the Jacobian. The order of convergence of proposed pstep iterative method is p + 1. Only one Jacobian inversion in the form of LUfactorization is required for a single iteration of the iterative method and in this way, it oers an efficient scheme. For the construction of our proposed iterative method, we used a decomposition technique that naturally provides different iterative schemes. We also computed the computational convergence order that confirms the claimed theoretical order of convergence. The developed iterative scheme is applied to large scale problems, and numerical results show that our iterative scheme is promising.
1

187
198


M.
Nili Ahmadabadi
Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Iran
Department of Mathematics, Najafabad Branch,
Iran
mneely59@hotmail.com


F.
Ahmad
Departament de Fsica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Comte
d'Urgell 187, 08036 Barcelona, Spain
Departament de Fsica i Enginyeria Nuclear,
Spain
fayyaz.ahmad@upc.edu


G.
Yuan
College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi,
530004, P.R. China
College of Mathematics and Information Science,
China
glyuan@gxu.edu.cn


X.
Li
School of Mathematics and Computing Science, Guangxi Colleges and Universities Key
Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology,
Guilin, Guangxi, China
School of Mathematics and Computing Science,
China
lixiangli@guet.edu.cn
systems of nonlinear equations
Decomposition
Order of convergence
Higher order methods
Computational efficiency
[[1] J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, NewYork, 1970. ##[2] C. T. Kelley, Solving Nonlinear Equations with Newtons Method, SIAM, Philadelphia, 2003. ##[3] F. Ahmad, E. Tohidi, J. A. Carrasco, A parameterized multistep Newton method for solving systems of nonlinear equations. Numerical Algorithms, (2015), 10171398. ##[4] F. Ahmad, E. Tohidi, M. Z. Ullah, J. A. Carrasco, Higher order multistep Jarrattlike method for solving systems of nonlinear equations: Application to PDEs and ODEs, Computers & Mathematics with Applications, 70 (4) (2015), 624636. http://dx.doi.org/10.1016/j.camwa.2015.05.012. ##[5] M. Z. Ullah, S. SerraCapizzano, F. Ahmad, An ecient multistep iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs, Applied Mathematics and Computation, 250 (2015), 249259. http://dx.doi.org/10.1016/j.amc.2014.10.103. ##[6] E. S. Alaidarous, M. Z. Ullah, F. Ahmad, A.S. AlFhaid, An Ecient HigherOrder Quasilinearization Method for Solving Nonlinear BVPs ,Journal of Applied Mathematics, vol. 2013 (2013), Article ID 259371, 11 pages. ##[7] F. Ahmed Shah, M, Aslam Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput. 251 (2015) 378386. ##[8] J. M. Gutirrez, M. A. Hernndez, A family of ChebyshevHalley type methods in banach spaces. Bull. Aust. Math. Soc. 55 (1997) 113130. ##[9] M. Palacios, Kepler equation and accelerated Newton method, J. Comput. Appl. Math. 138 (2002) 335346. ##[10] S. Amat, S. Busquier, J. M. Gutierrez, Geometrical constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157 (2003) 197205. ##[11] M. Frontini, E. Sormani, Some variant of Newtons method with thirdorder convergence, Appl. Math. Comput. 140 (2003) 419426. ##[12] M. Frontini, E. Sormani, Thirdorder methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput. 149 (2004) 771782. ##[13] H. H. H. Homeier, A modied Newton method with cubic convergence: the multivariable case. J. Comput. Appl. Math. 169 (2004) 161169. ##[14] M.T. Darvishi, A. Barati, A fourthorder method from quadrature formulae to solve systems of nonlinear equations. Appl. Math. Comput. 188 (2007) 257261. ##[15] A. Cordero, J.R. Torregrosa, Variants of Newtons method using fthorder quadrature formulas. Appl. Math. Comput. 190 (2007) 686698. ##[16] M. A. Noor, M. Waseem, Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57 (2009) 101106. ##[17] M. GrauSanchez, A. Grau, M. Noguera, Ostrowski type methods for solving systems of nonlinear equations. Appl. Math. Comput. 218 (2011) 23772385. ##[18] J.A. Ezquerro, M.GrauSanchez, A. Grau, M.A. Hernandez, M. Noguera, N. Romero, On iterative methods with accelerated convergence for solving systems of nonlinear equations, J. Optim. Theory Appl. 151 (2011) 163174. ##[19] A. Cordero, J.L. Hueso, E. Martinez, J.R. Torregrosa, Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012) 23692374. ##[20] J.R. Sharma, R.K. Guha, R. Sharma, An ecient fourth order weightedNewton method for systems of nonlinear equations, Numer. Algorithms 62 (2013) 307323. ##[21] X Xiao,H Yin, A new class of methods with higher order of convergence for solving systems of nonlinear equations, Appl. Math. Comput. 264 (2015) 300309. ##[22] E. D. Dolan and J. J. More, Benchmarking optimization software with performance proles, Mathematical Programming, 91(2002), 201213. ##[23] V. DaftardarGejji, H. Jafari, An iterative method for solving nonlinear functional equations. J. Math. Anal. Appl. 316 (2006) 753763. ##[24] J. H. He, Variational iteration methodsome recent results and new interpretations. J. Comput. Appl. Math. 207 (2007) 317. ##[25] Sergio Amat, Sonia Busquier, ngela Grau, Miquel GrauSanchez, Maximum eciency for a family of Newtonlike methods with frozen derivatives and some applications. Applied Mathematics and Computation 219 (2013) 79547963. ##[26] Xinyuan Wu, Note on the improvement of Newtons method for system of nonlinear equations. Applied Mathematics and Computation 189 (2007) 14761479.##]
Some topological operators via grills
2
2
In this paper, we define and study two operators $Phi^s$ and $Psi^s$ with grill. Characterization and basic properties of these operators are obtained. Also, we generalize a grill topological spaces via topology $tau^s$ induced from operators $Phi^s$ and $Psi^s$.
1

199
204


A. A.
Nasef
Department of Physics and Engineering Mathematics , Faculty of Engineering,
Kafr ElSheikh University, Kafr ElSheikh, Egypt
Department of Physics and Engineering Mathematics
Egypt
nasefa50@yahoo.com


A.
Azzam
Department of Mathematics, Faculty of Science, Assuit University,
New Valley, Egypt
Department of Mathematics, Faculty of Science,
Egypt
azzam0911@yahoo.com
Grill topological spaces
$Phi^s_G$
$Ψ^s_G$operators and $tau^s_G$
[[1] A. AlOmari, T. Noiri, Decomposition of continuity via grilles. Jordan J. Math and Stat., 4 (1) (2011), 3346. ##[2] C. W. Baker, Slightly precontinuous functions. Acta Math. Hungar., 94 (12)(2002). ##[3] G. Choquet, Sur les notions de lter et. grill. Completes Rendus Acad. Sci. Paris, 224 (1947), 171173. ##[4] K. C. Chattopadhyay, O. Njastad and W. J. Thron, Merotopic spaces and extensions of closure spaces. Can. J. Math., 35 (4) (1983), 613629. ##[5] K. C. Chattopadhyay and W. J. Thron, Extensions of closur spaces. Can. j. Math., 29 (6) (1977), 12771286. ##[6] S. G. Crossely and S. K. Hildebrand, Semiclosure. Texas J. Sci. (2+3) (1971), 99119. ##[7] E. Hater and S. Jafari, On Some new Classes of Sets and a new decomposition of continuity via grills. J. Adv. Math. Studies, 3(1) (2010), 3340. ##[8] R. C. Jain, The Role of Regularly Open Sets in General Topology, Ph.D. Thesis. Meerut Univ., (Meerut, India, 1980). ##[9] D. Mondal and M.N. Mukherjee, On a class of sets via grill: A decomposition of continuity. An. St. Univ. Ovidius Constanta, 20 (1) (2012), 307316. ##[10] N. Levine, Semiopen sets and semicontinuity in topological spaces. Amer, Math. Mounthly, 86 (1961), 4446. ##[11] B. Roy and M. N. Mukherjee, On a Typical Topology Induced by a Grill. Soochow Journal of Math., 33, No. 4 (2007), 771786. ##[12] R. Staum. The algebra of bounded continuous functions into a nonarchimedian eld. Pacic J. Math., 50 (1974), 169185. ##[13] W. J. Thron, Proximity structure and grills. Math. Ann. 206 (1973), 3562.##]
Error estimation for nonlinear pseudoparabolic equations with nonlocal boundary conditions in reproducing kernel space
2
2
In this paper we discuss about nonlinear pseudoparabolic equations with nonlocal boundary conditions and their results. An effective error estimation for this method altough has not yet been discussed. The aim of this paper is to fill this gap.
1

205
214


B.
Zamanifar
Department of Mathematics, Hamedan Branch,
Islamic Azad University, Hamedan, Iran
Department of Mathematics, Hamedan Branch,
Islamic
Iran
behnaz_zamanifar@yahoo.com


T.
Lotfi
Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran
Department of Mathematics, Hamedan Branch,
Iran
Reproducing kernel method
Error estimation
nonlinear pseudoparabolic equation
[[1] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A, 272 (1220) (1972), 4778. ##[2] A. Bouziani, Solvability of nonlinear pseudoparabolic equation with a nonlocal boundary condition. Analysis 55 (2003), 883904. ##[3] A. Bouziani, Initialboundary value problems for a class of pseudoparabolic equations with integral boundary conditions. J. Math. Anal. Appl. 291 (2004), 371386. ##[4] P. J. Chen, M. E. Gurtin, On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys. 19 (1968), 614627. ##[5] M. G. Cui, Y. Z. Lin, Nonlinear numerical analysis in the reproducing kernel space. Nova 2009. ##[6] D. Q. Dai, H. Yu, Nonlocal boundary problems for a thirdorder onedimensional nonlinear pseudoparabolic equation. SIAM J. Nonlinear Anal. 66 (2007), 179191. ##[7] G. V. Demidenko, S. V. Uspenskii, Equations and Systems that are not Solved with Respect to the Highest Derivative. Nauchnaya Kniga, Novosibirsk, 1998, 437443. ##[8] I. E. Egorov, S. G. Pyatkov and S. V. Popov, Nonclassical Operator Dierential Equations. Nauka, Novosibirsk, 2000, 336342. ##[9] A. Favini, A. Yagi, Degenerate Dierential Equations in Banach Spaces. vol. 215, Marcel Dekker, New York, 1999, 313335. ##[10] H. Gajewski, K. Grger, K. Zacharias, Nichtlineare Operatorgleichungen und Operator differential gleichungen. Akademie, Berlin, 1974, 281293. ##[11] E. I. Kaikina, Nonlinear pseudoparabolic type equations on a halfline with large initial data. Nonlinear Anal. 67 (2007), 28392858. ##[12] A. I. Kozhanov, A.I, An initialboundary value problem for equations of the generalized Boussi nesq equation type with a nonlinear source. Mat. Zametki 65 (1) (1999), 7075. ##[13] Y. Z. Lin, Y. F. Zhou, Solving nonlinear pseudoparabolic equations with nonlocal boundary cnditions in reproducing kernel space. Numer Algor, 52 (2009), 173189. ##[14] X. Y. Li, B. Y. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems. App. Math, 243 (2013), 1015. ##[15] S. Mesloub, A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation. J. Math. Anal. Appl. 316 (2006), 189209 . ##[16] V. Padron, Eect of aggregation on population recovery modeled by a forwardbackward pseudoparabolic equation. Trans. Am. Math. Soc. 356 (7) (2004), 27392756. ##[17] R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Dierential Equations. Amer. Math. Soc., Providence 49 (1997), 278282. ##[18] G. A. Sviridyuk, V. E. Fdorov, Analytic semigroups with kernels, and linear equations of Sobolev type. Sib. Mat. Z. 36 (5) (1995),11301145.##]
mProjections involving Minkowski inverse and range symmetric property in Minkowski space
2
2
In this paper we study the impact of Minkowski metric matrix on a projection in the Minkowski Space M along with their basic algebraic and geometric properties.The relation between the mprojections and the Minkowski inverse of a matrix A in the minkowski space M is derived. In the remaining portion commutativity of Minkowski inverse in Minkowski Space M is analyzed in terms of mprojections as an analogous development and extension of the results on EP matrices.
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215
228


M.
Saleem Lone
Department of Mathematics, Annamalai University, Chidambaram,
PO. Code 608002, Tamilnadu, India
Department of Mathematics, Annamalai University,
India
saleemlone9@gmail.com


D.
Krishnaswamy
Department of Mathematics, Annamalai University, Chidambaram,
PO. Code 608002, Tamilnadu, India
Department of Mathematics, Annamalai University,
India
krishna_swamy2004@yahoo.co.in
Minkowski inverse
mprojections
Range Symmetric
EP matrix
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