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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 30, 1994 - Issue 4
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Original Articles

Variational-like inequalities

M. Aslam Noor Mathematics Department, College of Science King Saud University, P O Box 2455, Riyadh, 11451, Saudi Arabia
Pages 323-330 | Published online: 10 Jun 2010

Citations (75)

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Read on this site (4)

Harsh Narayan Singh & Vivek Laha. (2023) On minty variational principle for quasidifferentiable vector optimization problems. Optimization Methods and Software 38:2, pages 243-261.
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Muhammad Aslam Noor, Khalida Inayat Noor, Muhammad Uzair Awan & Feng Qi. (2015) Integral inequalities of Hermite–Hadamard type for logarithmically h-preinvex functions. Cogent Mathematics 2:1.
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Muhammad Aslam Noor, Khalida Inayat Noor & Vijay Gupta. (2007) On equilibrium-like problems. Applicable Analysis 86:7, pages 807-818.
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M. Aslam Noor. (1996) General nonlinear mixed variational-link inequalities. Optimization 37:4, pages 357-367.
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Articles from other publishers (71)

HURİYE KADAKAL & MAHİR KADAKAL. (2023) STRONGLY HYPERBOLIC TYPE CONVEXITY AND SOME NEW INEQUALITIES. Journal of Science and Arts 23:3, pages 587-602.
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Muhammad Tariq, Sotiris K. Ntouyas & Asif Ali Shaikh. (2023) New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator. Fractal and Fractional 7:5, pages 405.
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Muhammad Tariq, Asif Ali Shaikh & Sotiris K. Ntouyas. (2023) Some New Fractional Hadamard and Pachpatte-Type Inequalities with Applications via Generalized Preinvexity. Symmetry 15:5, pages 1033.
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Tareq Saeed, Muhammad Bilal Khan, Savin Treanță, Hamed H. Alsulami & Mohammed Sh. Alhodaly. (2023) Study of Log Convex Mappings in Fuzzy Aunnam Calculus via Fuzzy Inclusion Relation over Fuzzy-Number Space. Mathematics 11:9, pages 2043.
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. (2023) General biconvex functions and bivariational inequalities. Numerical Algebra, Control and Optimization 13:1, pages 11-27.
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Muhammad Bilal Khan, Gustavo Santos-García, Savin Treanțǎ, Muhammad Aslam Noor & Mohamed S. Soliman. (2022) Perturbed Mixed Variational-like Inequalities and Auxiliary Principle Pertaining to a Fuzzy Environment. Symmetry 14:12, pages 2503.
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Muhammad Bilal Khan, Gustavo Santos-García, Muhammad Aslam Noor & Mohamed S. Soliman. (2022) New Class of Preinvex Fuzzy Mappings and Related Inequalities. Mathematics 10:20, pages 3753.
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Muhammad Tariq, Soubhagya Kumar Sahoo, Sotiris K. Ntouyas, Omar Mutab Alsalami, Asif Ali Shaikh & Kamsing Nonlaopon. (2022) Some Hermite–Hadamard and Hermite–Hadamard–Fejér Type Fractional Inclusions Pertaining to Different Kinds of Generalized Preinvexities. Symmetry 14:10, pages 1957.
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Badreddine MeftahAbdourazek Souahi. (2022) Hermite-Hadamard’s inequalities for ( h, r )-preinvex functions . General Mathematics 30:1, pages 33-54.
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Muhammad Aslam Noor, Hayat Ali & Khalida Inayat Noor. (2022) Some Iterative Schemes for Solving Mixed Equilibrium Variational-like Inequalities. Earthline Journal of Mathematical Sciences, pages 67-84.
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Zeng-bao Wu, Xing Wang, Nan-jing Huang, Yi-bin Xiao & Guang-hui Zhang. (2022) On a new system of fractional delay differential equations coupled with fuzzy variational inequalities. Fuzzy Sets and Systems 436, pages 55-81.
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Muhammad NoorAslamAslam & Khalida NoorInayatInayat. (2022) Some new classes of preinvex functions and directional variational-like inequalities. Filomat 36:12, pages 3995-4008.
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Muhammad Aslam Noor, Khalida Inayat Noor, Bandar Mohsen, Michael Th. Rassias & Andrei Raigorodskii. 2022. Approximation and Computation in Science and Engineering. Approximation and Computation in Science and Engineering 643 666 .
Jiangming Ma, Muhammad Aslam Noor & Khalida Inayat Noor. (2021) Generalized Higher Order Preinvex Functions and Equilibrium-like Problems. Symmetry 13:10, pages 1875.
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Saad Ihsan Butt, Hüseyin Budak, Muhammad Tariq & Muhammad Nadeem. (2021) Integral inequalities for n ‐polynomial s ‐type preinvex functions with applications . Mathematical Methods in the Applied Sciences 44:14, pages 11006-11021.
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Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed & Juan L. G. Guirao. (2021) Fuzzy Mixed Variational-like and Integral Inequalities for Strongly Preinvex Fuzzy Mappings. Symmetry 13:10, pages 1816.
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S. Ghomrani, B. Meftah, W. Kaidouchi & M. Benssaad. (2021) Fractional Hermite–Hadamard type integral inequalities for functions whose modulus of the mixed derivatives are co-ordinated $$(log,(\alpha ,m))$$-preinvex. Afrika Matematika 32:5-6, pages 925-940.
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Peide Liu, Muhammad Bilal Khan, Muhammad Aslam Noor & Khalida Inayat Noor. (2021) On Strongly Generalized Preinvex Fuzzy Mappings. Journal of Mathematics 2021, pages 1-16.
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Muhammad Bilal Khan, Muhammad Aslam Noor, Khalida Inayat Noor, Ahmad Termimi Ab Ghani & Lazim Abdullah. (2021) Extended Perturbed Mixed Variational-Like Inequalities for Fuzzy Mappings. Journal of Mathematics 2021, pages 1-16.
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Muhammad Aslam Noor & Khalida Inayat Noor. (2021) Properties of higher order preinvex functions. Numerical Algebra, Control & Optimization 11:3, pages 431.
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Muhammad Aslam Noor & Khalida Inayat Noor. 2021. Mathematical Analysis in Interdisciplinary Research. Mathematical Analysis in Interdisciplinary Research 573 588 .
Muhammad Aslam Noor, Khalida Inayat Noor & Michael Th. Rassias. (2020) New Trends in General Variational Inequalities. Acta Applicandae Mathematicae 170:1, pages 981-1064.
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Muhammad Bilal Khan, Muhammad Aslam Noor & Khalida Inayat Noor. (2020) On Some Characterization of Preinvex Fuzzy Mappings. Earthline Journal of Mathematical Sciences, pages 17-42.
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Huriye KADAKAL. (2020) Differentiable Preinvex and Prequasiinvex Functions. Universal Journal of Mathematics and Applications 3:2, pages 69-77.
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Misbah Iram Bloach & Muhammad Aslam Noor. (2020) Perturbed mixed variational-like inequalities. AIMS Mathematics 5:3, pages 2153-2162.
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Serap Özcan. (2020) Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions. AIMS Mathematics 5:2, pages 1505-1518.
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Badreddine Meftah & Abdourazek Souahi. (2019) Fractional Hermite-Hadamard type integral inequalities for functions whose modulus of the mixed derivatives are co-ordinated s -preinvex in the second sense . Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 18:1, pages 67-83.
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Badreddine MEFTAH & Abdourazek SOUAHİ. (2019) Fractional Hermite-Hadamard Type Inequalities for Functions Whose Derivatives are s-Preinvex. Mathematical Sciences and Applications E-Notes 7:2, pages 128-138.
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Saima Rashid, Ahmet Ocak Akdemir, Muhammad Aslam Noor & Khalida Inayat Noor. (2019) Generalization of Inequalities Analogous to Preinvex Functions via Extended Generalized Mittag-Leffler Functions. Generalization of Inequalities Analogous to Preinvex Functions via Extended Generalized Mittag-Leffler Functions.
Ting Xie & Zengtai Gong. (2019) Variational-like inequalities for n-dimensional fuzzy-vector-valued functions and fuzzy optimization. Open Mathematics 17:1, pages 627-645.
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Artion Kashuri, Muhammad Adil Khan & Rozana Liko. (2019) Some new $k$-fractional trapezium-like integral inequalities via generalized relative semi-$(r;m,h_{1},h_{2})$-preinvex mappings and applications. Tbilisi Mathematical Journal 12:3.
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Izhar Ahmad, Meraj Ali Khan & Amira A. Ishan. (2019) Generalized Geodesic Convexity on Riemannian Manifolds. Mathematics 7:6, pages 547.
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B. Meftah & A. Souahi. (2019) Fractional Hermite-Hadamard type inequalities for co-ordinated prequasiinvex functions. Journal of Applied Mathematics, Statistics and Informatics 15:1, pages 19-32.
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Muhammad Aslam Noor & Khalida Inayat Noor. (2019) New classes of strongly exponentially preinvex functions. AIMS Mathematics 4:6, pages 1554-1568.
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Saima Rashid, Ahmet Ocak Akdemir, Fahd Jarad, Muhammad Aslam Noor & Khalida Inayat Noor. (2019) Simpson’s type integral inequalities for <em>ĸ</em>-fractional integrals and their applications. AIMS Mathematics 4:4, pages 1087-1100.
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Muhammad Noor, Khalida Noor & Saima Rashid. (2018) Some New Classes of Preinvex Functions and Inequalities. Mathematics 7:1, pages 29.
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Babli Kumari & Anurag Jayswal. (2018) Some properties of geodesic E-preinvex function and geodesic semi E-preinvex function on Riemannian manifolds. OPSEARCH 55:3-4, pages 807-822.
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İmdat İŞCAN, Mahir KADAKAL & Huriye KADAKAL. (2018) On two times differentiable preinvex and prequasiinvex functions. Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 68:1, pages 950-963.
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Meftah Badreddine. (2018) Ostrowski's inequalities for functions whose first derivatives are s-logarithmically preinvex in the second sense. Mathematica Moravica 22:2, pages 11-28.
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DAH-YAN HWANG & SILVESTRU SEVER DRAGOMIR. (2017) EXTENSIONS OF THE HERMITE–HADAMARD INEQUALITY FOR -PREINVEX FUNCTIONS ON AN INVEX SET. Bulletin of the Australian Mathematical Society 95:3, pages 412-423.
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Muhammad Aslam Noor, Themistocles M. Rassias, Khalida Inayat Noor & Muhammad Uzair Awan. 2017. Progress in Approximation Theory and Applicable Complex Analysis. Progress in Approximation Theory and Applicable Complex Analysis 237 268 .
Badreddine Meftah. (2016) Ostrowski Inequalities for Functions Whose First Derivatives Are Logarithmically Preinvex. Chinese Journal of Mathematics 2016, pages 1-10.
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Akhlad Iqbal & V. Samhita. 2016. Applied Analysis in Biological and Physical Sciences. Applied Analysis in Biological and Physical Sciences 373 384 .
Muhammad Aslam Noor, Khalida Inayat Noor & Muhammad Uzair Awan. (2015) Some quantum integral inequalities via preinvex functions. Applied Mathematics and Computation 269, pages 242-251.
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Muhammad Amer Latif & Muhammad Shoaib. (2015) Hermite–Hadamard type integral inequalities for differentiable m-preinvex and (α,m)-preinvex functions. Journal of the Egyptian Mathematical Society 23:2, pages 236-241.
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Saleh Al-Mezel, Falleh Al-Solamy & Qamrul AnsariQamrul Ansari. 2014. Fixed Point Theory, Variational Analysis, and Optimization. Fixed Point Theory, Variational Analysis, and Optimization 207 245 .
Shuhong Wang & Ximin Liu. (2014) New Hermite-Hadamard Type Inequalities for -Times Differentiable and -Logarithmically Preinvex Functions . Abstract and Applied Analysis 2014, pages 1-11.
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Muhammad Amer Latif & Sever Silvestru Dragomir. (2013) Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications. Journal of Inequalities and Applications 2013:1.
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S. K. Mishra & Vivek Laha. (2012) On Approximately Star-Shaped Functions and Approximate Vector Variational Inequalities. Journal of Optimization Theory and Applications 156:2, pages 278-293.
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Wasim Ul-Haq & Javed Iqbal. (2013) Hermite-Hadamard-Type Inequalities for r -Preinvex Functions . Journal of Applied Mathematics 2013, pages 1-5.
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Muhammad Aslam Noor, Khalida Inayat Noor, Saira Zainab & Eisa Al-Said. (2012) Regularized Mixed Variational-Like Inequalities. Journal of Applied Mathematics 2012, pages 1-14.
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Yeol Je Cho, Ioannis K. Argyros & Narin Petrot. (2010) Approximation methods for common solutions of generalized equilibrium, systems of nonlinear variational inequalities and fixed point problems. Computers & Mathematics with Applications 60:8, pages 2292-2301.
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M. Soleimani-damaneh. (2010) Nonsmooth Optimization Using Mordukhovich's Subdifferential. SIAM Journal on Control and Optimization 48:5, pages 3403-3432.
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Ali Farajzadeh, Muhammad Aslam Noor & Khalida Inayat Noor. (2009) Vector nonsmooth variational-like inequalities and optimization problems. Nonlinear Analysis: Theory, Methods & Applications 71:7-8, pages 3471-3476.
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Muhammad Aslam Noor, Khalida Inayat Noor & Saira Zainab. (2009) On a predictor–corrector method for solving invex equilibrium problems. Nonlinear Analysis: Theory, Methods & Applications 71:7-8, pages 3333-3338.
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Xiao Gang & Sanyang Liu. (2009) Existence of solutions for generalized vector quasi-variational-like inequalities without monotonicity. Computers & Mathematics with Applications 58:8, pages 1550-1557.
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Jie Shen & Li-Ping Pang. (2008) A bundle-type auxiliary problem method for solving generalized variational-like inequalities. Computers & Mathematics with Applications 55:12, pages 2993-2998.
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Muhammad Aslam Noor & Themistocles M. Rassias. 2009. Encyclopedia of Optimization. Encyclopedia of Optimization 4007 4011 .
S. K. Mishra, S. Y. Wang & K. K. Lai. (2007) On non-smooth α-invex functions and vector variational-like inequality. Optimization Letters 2:1, pages 91-98.
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Muhammad Aslam Noor & Khalida Inayat Noor. (2006) Hemiequilibrium-like problems. Nonlinear Analysis: Theory, Methods & Applications 64:12, pages 2631-2642.
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S.K. Mishra & S.Y. Wang. (2006) Vector variational-like inequalities and non-smooth vector optimization problems. Nonlinear Analysis: Theory, Methods & Applications 64:9, pages 1939-1945.
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Rais Ahmad & Syed Shakaib Irfan. (2006) On generalized nonlinear variational-like inequality problems. Applied Mathematics Letters 19:3, pages 294-297.
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S.K. Mishra & M.A. Noor. (2005) On vector variational-like inequality problems. Journal of Mathematical Analysis and Applications 311:1, pages 69-75.
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Muhammad Aslam Noor. (2005) Hemivariational-like inequalities. Journal of Computational and Applied Mathematics 182:2, pages 316-326.
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Muhammad Aslam Noor. (2005) Invex equilibrium problems. Journal of Mathematical Analysis and Applications 302:2, pages 463-475.
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Muhammad Aslam Noor. (2004) Generalized mixed quasi-variational-like inequalities. Applied Mathematics and Computation 156:1, pages 145-158.
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Q. H. Ansari & J. C. Yao. (2001) Iterative Schemes for Solving Mixed Variational-Like Inequalities. Journal of Optimization Theory and Applications 108:3, pages 527-541.
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Muhammad Aslam Noor & Themistocles M. Rassias. 2001. Encyclopedia of Optimization. Encyclopedia of Optimization 2684 2689 .
C.-H. Lee, Q. H. Ansari & J.-C. Yao. (2009) A perturbed algorithm for strongly nonlinear variational-like inclusions. Bulletin of the Australian Mathematical Society 62:3, pages 417-426.
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Nan-Jing Huang, Ya-Ping Liu, Ya-Yong Tang & Min-Ru Bai. (1999) The generalized set-valued strongly nonlinear implicit variational inequalities. Computers & Mathematics with Applications 37:10, pages 29-36.
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M. A. Noor. (1995) Nonconvex functions and variational inequalities. Journal of Optimization Theory and Applications 87:3, pages 615-630.
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