Advertisement

From Representation Theorems to Variational Inequalities

  • Muhammad Aslam Noor
  • Khalida Inayat Noor
Chapter
  • 92 Downloads
Part of the Springer Optimization and Its Applications book series (SOIA, volume 159)

Abstract

We start with an historical introduction and then give an overview of the most important representation theorems for the linear(nonlinear) continuous functionals by the bifunction. Then we extensively study representations theorems for nonlinear operators. These representation results contain the difference of two(more) monotone operators, complementarity problems, systems of the absolute values equations and difference of two convex functions as special cases. These problems are very important and significant, which provide a unified and general framework of studying a wide class of unrelated problems.

Notes

Acknowledgements

The authors would like to thank the Rector, COMSATS University Islamabad, Pakistan, for providing excellent research and academic environments. Prof. Dr. Muhammad Aslam Noor would like to express his sincere gratitude to Prof. Dr. C. F. Schubert, Mathematics Department, Queens University, Kingston, Ontario, Canada, for introducing the most interesting and fascinating field of Variational Inequalities. Authors are grateful to Prof. Dr. Themistocles M. Rassias for his kind invitation and support.

References

  1. 1.
    C. Baiocchi, A. Capelo, Variational and Quasi-Variational Inequalities (Wiley, New York, 1984)zbMATHGoogle Scholar
  2. 2.
    E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems. Math. Student 63(1994), 325–333MathSciNetzbMATHGoogle Scholar
  3. 3.
    R.W. Cottle, Nonlinear programs with positively bounded Jacobians. SIAM J. Appl. Math. 14, 147–158 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    R.W. Cottle, J.S. Pang, R.E. Stone, The Linear Complementarity Problem (Academic Press, New York, 1992)zbMATHGoogle Scholar
  5. 5.
    J. Crank, Free and Moving Boundary Problems (Clarendon Press, Oxford, 1984)zbMATHGoogle Scholar
  6. 6.
    G. Cristescu, L. Lupsa, Non-Connected Convexities and Applications (Kluwer Academic Publishers, Dordrecht, 2002)zbMATHCrossRefGoogle Scholar
  7. 7.
    G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics (Springer, Berlin, 1976)zbMATHCrossRefGoogle Scholar
  8. 8.
    I. Ekland, R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976)Google Scholar
  9. 9.
    R. Eskandari, M. Frank, V. M. Manuilov, M.S. Moslehian, Extensions of the Lax-Milgram theorem to Hilbert C -modules (preprint, 2019)Google Scholar
  10. 10.
    W. Fechner, Functional inequalities motivated by the Lax-Milgram lemma. J. Math. Anal. Appl. 402, 411–414 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    G. Fichera, Problemi elastostatici con vincoli unilaterali: il problema di Signorini con ambique condizione al contorno. Atti. Acad. Naz. Lincei. Mem. Cl. Sci. Nat. Sez. Ia 7(8), 91–140 (1963/1964)zbMATHGoogle Scholar
  12. 12.
    V.M. Filippov, Variational Principles for Nonpotential Operators, vol. 77 (American Mathematical Society, Providence, 1989)zbMATHCrossRefGoogle Scholar
  13. 13.
    M. Frechet, Sur les ensembles des fonctions et les operations lineaires. C. R. Acad. Sci. Paris 144, 1414–1416 (1907)zbMATHGoogle Scholar
  14. 14.
    T.L. Friesz, D.H. Bernstein, N.J. Mehta, S. Ganjliazadeh, Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper. Res. 42, 1120–1136 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    T.L. Friesz, D.H. Bernstein, R. Stough, Dynamic systems, variational inequalities and control theoretic models for predicting time-varying urban network flows. Trans. Sci. 30, 14–31 (1996)zbMATHCrossRefGoogle Scholar
  16. 16.
    M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    F. Giannessi, A. Maugeri, Variational Inequalities and Network Equilibrium Problems (Plenum Press, New York, 1995)zbMATHCrossRefGoogle Scholar
  18. 18.
    F. Giannessi, A. Maugeri, P.M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models (Kluwer Academic Publishers, Dordrecht, 2001)zbMATHGoogle Scholar
  19. 19.
    R. Glowinski, J.J. Lions, R. Tremolieres, Numerical Analysis of Variational Inequalities (North-Holland, Amsterdam, 1981)zbMATHGoogle Scholar
  20. 20.
    P.T. Harker, J.S. Pang, Finite dimensional variational inequalities and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990)zbMATHCrossRefGoogle Scholar
  21. 21.
    N. Kikuchi, J.T. Oden, Contact problems in Elasticity (SIAM, Philadelphia, 1988)zbMATHCrossRefGoogle Scholar
  22. 22.
    D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications (SIAM, Philadelphia, 2000)zbMATHCrossRefGoogle Scholar
  23. 23.
    G.M. Korpelevich, The extragradient method for finding saddle points and other problems. Matecon 12, 747–756 (1976)MathSciNetzbMATHGoogle Scholar
  24. 24.
    H. Kozono, Lax-Milgram theorem in Banach spaces and its generalization to the elliptic system of boundary value problems. Manuscripta Math. 141, 637–662 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    P.D. Lax, A.N. Milgram, Parabolic equations. Ann. Math. Study 33, 167–190 (1954)zbMATHGoogle Scholar
  26. 26.
    J.L. Lions, G. Stampacchia, Variational inequalities. Commun. Pure Appl. Math. 20, 493–512 (1967)zbMATHCrossRefGoogle Scholar
  27. 27.
    D.T. Luc, M.A. Noor, Local uniqueness of solutions of general variational inequalities. J. Optim. Theory Appl. 117, 103–119 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    O.L. Mangasarian, A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    O. Mangasarian, Absolute value equation solution via dual complementarity. Optim. Lett. 7, 625–630 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    O.L. Mangasarian, R.R. Meyer, Absolute value equations. Linear Algebra Appl. 419(2), 359–367 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    B. Martinet, Regularization d’inequations variationnelles par approximations successive. Revue Fran. d’Informat. Rech. Oper. 4, 154–159 (1970)zbMATHGoogle Scholar
  32. 32.
    U. Mosco, Implicit Variational Problems and Quasi Variational Inequalities. Lecture Notes in Mathematics, vol. 543 (Springer, Berlin, 1976), pp. 83–126Google Scholar
  33. 33.
    A. Moudafi, M.A. Noor, Sensitivity analysis for variational inclusions by Wiener-Hopf equations technique. J. Appl. Math. Stoch. Anal. 12, 223–232 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    A. Nagurney, Network Economics, A Variational Inequality Approach (Kluwer Academics Publishers, Boston, 1999)zbMATHGoogle Scholar
  35. 35.
    C.P. Niculescu, L.E. Persson, Convex Functions and Their Applications (Springer, New York, 2018)zbMATHCrossRefGoogle Scholar
  36. 36.
    M.A. Noor, The Riesz-Frechet theorem and monotonicity. M.Sc. Thesis, Queen’s University, Kingston (1971)Google Scholar
  37. 37.
    M.A. Noor, Bilinear forms and convex set in Hilbert space. Boll. Union. Math. Ital. 5, 241–244 (1972)MathSciNetzbMATHGoogle Scholar
  38. 38.
    M.A. Noor, On variational inequalities, Ph.D. Thesis, Brunel University, London (1975)Google Scholar
  39. 39.
    M.A. Noor, Variational inequalities and approximations. Punjab Univer. J. Math. 8, 25–40 (1975)MathSciNetGoogle Scholar
  40. 40.
    M.A. Noor, Mildly nonlinear variational inequalities. Math. Anal. Numer. Theory Approx. 24, 99–110 (1982)MathSciNetzbMATHGoogle Scholar
  41. 41.
    M.A. Noor, Strongly nonlinear variational inequalities. C. R. Math. Rep. 4, 213–218 (1982)MathSciNetzbMATHGoogle Scholar
  42. 42.
    M.A. Noor, On the nonlinear complementarity problems. J. Math. Anal. Appl. 123, 455–460 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    M.A. Noor, The quasi complementarity problem. J. Math. Anal. Appl. 130, 344–353 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    M.A. Noor, Fixed-point approach for complementarity problems. J. Math. Anal. Appl. 133, 437–448 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    M.A. Noor, General variational inequalities. Appl. Math. Lett. 1, 119–121 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    M.A. Noor, Quasi variational inequalities. Appl. Math. Lett. 1, 367–370 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    M.A. Noor, Wiener-Hopf equations and variational inequalities. J. Optim. Theory Appl. 79, 197–206 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    M.A. Noor, Variational inequalities in physical oceanography, in Ocean Wave Engineering ed. by M. Rahman. (Computer Mechanics Publications, Southampton, 1994), pp. 201–226Google Scholar
  49. 49.
    M.A. Noor, Sensitivity analysis for quasi variational inequalities. J. Optim. Theory Appl. 95, 399–407 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    M.A. Noor, Some recent advances in variational inequalities, Part I, basic concepts. New Zealand J. Math. 26, 53–80 (1997)MathSciNetzbMATHGoogle Scholar
  51. 51.
    M.A. Noor, Some recent advances in variational inequalities, Part II, other concepts. New Zealand J. Math. 26, 229–255 (1997)MathSciNetzbMATHGoogle Scholar
  52. 52.
    M.A. Noor, Some iterative techniques for variational inequalities. Optimization 46, 391–401 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    M.A. Noor, Generalized quasi variational inequalities and implicit Wiener-Hopf equations. Optimization 45, 197–222 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    M.A. Noor, A modified extragradient method for general monotone variational inequalities. Comput. Math. Appl. 38, 19–24 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    M.A. Noor, Some algorithms for general monotone mixed variational inequalities. Math. Comput. Model. 29, 1–9 (1999)MathSciNetzbMATHGoogle Scholar
  56. 56.
    M.A. Noor, Set-valued mixed quasi variational inequalities and implicit resolvent equations. Math. Comput. Model. 29, 1–11 (1999)MathSciNetzbMATHGoogle Scholar
  57. 57.
    M.A. Noor, New approximation schemes for general variational inequalities. J. Math. Anal. Appl. 251, 217–229 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    M.A. Noor, Three-step iterative algorithms for multivalued quasi variational inclusions. J. Math. Anal. Appl. 255, 589–604 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    M.A. Noor, Modified resolvent algorithms for general mixed variational inequalities. J. Comput. Appl. Math. 135, 111–124 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    M.A. Noor, A class of new iterative methods for general mixed variational inequalities. Math. Comput. Model. 31(13), 11–19 (2001)MathSciNetCrossRefGoogle Scholar
  61. 61.
    M.A. Noor, A predictor-corrector method for general variational inequalities. Appl. Math. Lett. 14, 53–87 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    M.A. Noor, Projection-splitting algorithms for general monotone variational inequalities. J. Comput. Anal. Appl. 4, 47–61 (2002)MathSciNetzbMATHGoogle Scholar
  63. 63.
    M.A. Noor, Proximal methods for mixed variational inequalities. J. Optim. Theory Appl. 115, 447–451 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    M.A. Noor, Implicit dynamical systems and quasi variational inequalities. Appl. Math. Comput. 134, 69–81 (2002)MathSciNetzbMATHGoogle Scholar
  65. 65.
    M.A. Noor, Implicit resolvent dynamical systems for quasi variational inclusions. J. Math. Anal. Appl. 269, 216–226 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    M.A. Noor, Sensitivity analysis framework for general quasi variational inequalities. Comput. Math. Appl. 44, 1175–1181 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    M.A. Noor, A Wiener-Hopf dynamical system for variational inequalities. New Zealand J. Math. 31, 173–182 (2002)MathSciNetzbMATHGoogle Scholar
  68. 68.
    M.A. Noor, New extragradient-type methods for general variational inequalities. J. Math. Anal. Appl. 277, 379–395 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    M.A. Noor, Extragradient method for pseudomonotone variational inequalities. J. Optim. Theory Appl. 117, 475–488 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    M.A. Noor, Well-posed variational inequalities. J. Appl. Math. Comput. 11, 165–172 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    M.A. Noor, Mixed quasi variational inequalities. Appl. Math. Comput. 146, 553–578 (2003)MathSciNetzbMATHGoogle Scholar
  72. 72.
    M.A. Noor, Multivalued general equilibrium problems. J. Math. Anal. Appl. 283(1), 140–149 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    M.A. Noor, Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122(2), 371–386 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    M.A. Noor, Some dvelopments in general variational inequalites. Appl. Math. Comput. 152, 199–277 (2004)MathSciNetGoogle Scholar
  75. 75.
    M.A. Noor, Extended general variational inequalities. Appl. Math. Lett. 22(2), 182–185 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    K.I. Noor, M.A. Noor, A generalization of the Lax-Milgram lemma. Can. Math. Bull. 23(2), 179–184 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    M.A. Noor, K.I. Noor, Multivalued variational inequalities and resolvent equations. Math. Comput. Model. 26(4), 109–121 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    M.A. Noor, K.I. Noor, Sensitivity analysis for quasi variational inclusions. J. Math. Anal. Appl. 236, 290–299 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    M.A. Noor, K.I. Noor, Self-adaptive projection algorithms for general variational inequalities. Appl. Math. Comput. 151, 659–670 (2004)MathSciNetzbMATHGoogle Scholar
  80. 80.
    M.A. Noor, W. Oettli, On general nonlinear complementarity problems and quasi equilibria. Le Matematiche 49, 313–331 (1994)MathSciNetzbMATHGoogle Scholar
  81. 81.
    M.A. Noor, E. Al-Said, Change of variable method for generalized complementarity problems. J. Optim. Theory Appl. 100, 389–395 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    M.A. Noor, E.A. Al-Said, Finite difference method for a system of third-order boundary value problems. J. Optim. Theory Appl. 112, 627–637 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    M.A. Noor, T.M. Rassias, A class of projection methods for general variational inequalities. J. Math. Anal. Appl. 268, 334–343 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    M.A. Noor, K.I. Noor, T.M. Rassias, Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    M.A. Noor, K.I. Noor, T.M. Rassias, Set-valued resolvent equations and mixed variational inequalities. J. Math. Anal. Appl. 220, 741–759 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    M.A. Noor, Y.J. Wang, N.H. Xiu, Some projection iterative schemes for general variational inequalities. J. Inequal. Pure Appl. Math. 3(3), 1–8 (2002)zbMATHGoogle Scholar
  87. 87.
    M.A. Noor, K.I. Noor, S. Batool, On generalized absolute value equations. U. P. B. Sci. Bull. Ser. A 80(4), 63–70 (2018)MathSciNetzbMATHGoogle Scholar
  88. 88.
    N. Nyamoradi, M.R. Hamidi, An extension of the Lax-Milgram theorem and its application to fractional differential equations. Electron. J. Differ. Equ. 2015, 95 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    M. Pappalardo, M. Passacantando, Stability for equilibrium problems: from variational inequalities to dynamical systems. J. Optim. Theory Appl. 113, 567–582 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    M. Patriksson, Nonlinear Programming and Variational Inequality Problems: A Unified Approach (Kluwer Academic Publishers, Dordrecht, 1998)zbMATHGoogle Scholar
  91. 91.
    J. Pecaric, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications (Academic Press, New York, 1992)zbMATHGoogle Scholar
  92. 92.
    B.T. Polyak, Introduction to Optimization (Optimization Software, New York, 1987)zbMATHGoogle Scholar
  93. 93.
    T.M. Rassias, M.A. Noor, K.I. Noor, Auxiliary principle technique for the general Lax-Milgram lemma. J. Nonlinear Funct. Anal. 2018, 34 (2018)Google Scholar
  94. 94.
    F. Riesz, Sur une espace de geometric alytique des aystemes fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411 (1907)zbMATHGoogle Scholar
  95. 95.
    F. Riesz, Zur theorie des Hilbertschen rauemes. Acta Sci. 7, 34–38 (1934/1935)zbMATHGoogle Scholar
  96. 96.
    P. Shi, Equivalence of variational inequalities with Wiener-Hopf equations. Proc. Am. Math. Soc. 111, 339–346 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    M. Sibony, Methodes iteratives pour les equations et inequations aux derivees partielles nonlineaires de type monotone. Calcolo 7, 65–183 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    M.V. Solodov, P. Tseng, Modified projection type methods for monotone variational inequalities. SIAM J. Control Optim. 34, 1814–1830 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 99.
    G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes. C. R. Acad. Paris 258, 4413–4416 (1964)MathSciNetzbMATHGoogle Scholar
  100. 100.
    E. Tonti, Variational formulation for every nonlinear problem. Int. J. Eng. Sci. 22, 1343–1371 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    R.L. Tobin, Sensitivity analysis for variational inequalities. J. Optim. Theory Appl. 48, 191–204 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    P. Tseng, A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    P. Tseng, On linear convergence of iterative methods for variational inequality problem. J. Comput. Appl. Math. 60, 237–252 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Y.J. Wang, N.H. Xiu, C.Y. Wang, Unified framework of projection methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 111, 643–658 (2001)Google Scholar
  105. 105.
    Y.J. Wang, N.H. Xiu, C.Y. Wang, A new version of extragradient projection method for variational inequalities. Comput. Math. Appl. 42, 969–979 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Y.S. Xia, J. Wang, On the stability of globally projected dynamical systems. J. Optim. Theory Appl. 106, 129–150 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  107. 107.
    N. Xiu, J. Zhang, M.A. Noor, Tangent projection equations and general variational equalities. J. Math. Anal. Appl. 258, 755–762 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  108. 108.
    N.H. Xiu, J. Zhang, Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 109.
    X.Q. Yang, G.Y. Chen, A class of nonconvex functions and variational inequalities. J. Math. Anal. Appl. 169, 359–373 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    N.D. Yen, Holder continuity of solutions to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    N.D. Yen, G.M. Lee, Solution sensitivity of a class of variational inequalities. J. Math. Anal. Appl. 215, 46–55 (1997)MathSciNetCrossRefGoogle Scholar
  112. 112.
    E.A. Youness,E-convex sets, E-convex functions and E-convex programming. J. Optim. Theory Appl. 102, 439–450 (1999)Google Scholar
  113. 113.
    D. Zhang, A. Nagurney, On the stability of the projected dynamical systems. J. Optim. Theory Appl. 85, 97–124 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Y.B. Zhao, Extended projection methods for monotone variational inequalities. J. Optim. Theory Appl. 100, 219–231 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    D.L. Zhu, P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6, 714–726 (1996)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Muhammad Aslam Noor
    • 1
  • Khalida Inayat Noor
    • 1
  1. 1.COMSATS University IslamabadIslamabadPakistan

Personalised recommendations