doi: 10.3934/naco.2021041
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General biconvex functions and bivariational inequalities

Department of Mathematics, COMSATS University Islamabad, Pakistan

* Corresponding author: Muhammad Aslam Noor

Received  June 2021 Revised  August 2021 Early access September 2021

In this paper, we define and introduce some new concepts of the higher order strongly general biconvex functions involving the arbitrary bifunction and a function. Some new relationships among various concepts of higher order strongly general biconvex functions have been established. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine general biconvex functions, which is itself an novel application. It is proved that the optimality conditions of the higher order strongly general biconvex functions are characterized by a class of variational inequalities, which is called the higher order strongly general bivariational inequality. Auxiliary principle technique is used to suggest an implicit method for solving strongly general bivariational inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. Some special cases also discussed. Results obtained in this paper can be viewed as refinement and improvement of previously known results.

Citation: Muhammad Aslam Noor, Khalida Inayat Noor. General biconvex functions and bivariational inequalities. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021041
References:
[1]

O. AlabdaliA. Guessab and G. Schmeisser, Characterization of uniform convexity for differentiable functions, Appl. Anal. Discrete Math., 13 (2019), 721-732.  doi: 10.2298/aadm190322029a.  Google Scholar

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W. L. Bynum, Weak parallelogram laws for Banach spaces, Canad. Math. Bull., 19 (1976), 269-275.  doi: 10.4153/CMB-1976-042-4.  Google Scholar

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R. Cheng and C. B. Harris, Duality of the weak parallelogram laws on Banach spaces, J. Math. Anal. Appl., 404 (2013), 64-70.  doi: 10.1016/j.jmaa.2013.02.064.  Google Scholar

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R. Cheng and W. T. Ross, Weak parallelogram laws on Banach spaces and applications to prediction, Period. Math. Hung., 71 (2015), 45-58.  doi: 10.1007/s10998-014-0078-4.  Google Scholar

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G. Cristescu M.and Găianu, Shape properties of Noors' convex sets, Proceed. of the Twelfth Symposium of Mathematics and its Applications, Timisoara, (2009), 1–13.  Google Scholar

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G. Cristescu and L. Lupsa, Non-Connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, 2002. doi: 10.1007/978-1-4615-0003-2.  Google Scholar

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G. H. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 118 (2003), 67-80.  doi: 10.1023/A:1024787424532.  Google Scholar

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J. Mako and Z. Pales, On $\varphi$-convexity, Publ. Math. Debrecen, 80 (2012), 107-126.  doi: 10.1057/9780230226203.3173.  Google Scholar

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B. B. MohsenM. A. NoorK. I. Noor and M. Postolache, Strongly convex functions of higher order involving bifunction, Mathematics, 7 (2019), 1028.  doi: 10.3390/math7111028.  Google Scholar

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M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7.  Google Scholar

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M. A. Noor, Differeniable nonconvex functions and general variational inequalities, Appl. Math. Comput., 199 (2008), 623-630.  doi: 10.1016/j.amc.2007.10.023.  Google Scholar

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M. A. Noor and K. I. Noor, On generalized strongly convex functions involving bifunction, Appl. Math. Inform. Sci., 13 (2019), 411-416.  doi: 10.18576/amis/130313.  Google Scholar

[23]

M. A. Noor and K. I. Noor, Some characterization of strongly preinvex functions, J. Math. Anal. Appl., 316 (2006), 697-706.  doi: 10.1016/j.jmaa.2005.05.014.  Google Scholar

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M. A. Noor and K. I. Noor, Higher-order strongly-generalized convex functions, Appl. Math. Inf. Sci., 14 (2020), 133-139.  doi: 10.18576/amis/140117.  Google Scholar

[25]

M. A. Noor and K. I. Noor, Higher order strongly general convex functions and variational inequalities, AIMS Math., 5 (2020), 3646-3663.  doi: 10.3934/math.2020236.  Google Scholar

[26]

M. A. Noor and K. I. Noor, Higher order variational inequalities, Inform. Sci. Lett., 11 (2022), 1-5.   Google Scholar

[27]

M. A. Noor and K. I. Noor, Properties of higher order preinvex functions, Numer. Algebr. Control. Optim., 11 (2021), 431-441.  doi: 10.3934/naco.2020035.  Google Scholar

[28]

M. A. NoorK. I. Noor and I. M. Baloch, Auxiliary principle technqiue for strongly mixed variational-like inequalities, U. P. B. Sci. Bull. Series A, 80 (2018), 93-100.   Google Scholar

[29]

M. A. NoorK. I. Noor and M. T. Rassias, New trends in general variational inequalities, Acta Appl. Mathematica, 170 (2020), 981-1064.  doi: 10.1007/s10440-020-00366-2.  Google Scholar

[30]

M. A. Noor, K. I. Noor and M. T. Rassias, Strongly biconvex functions and bivariational inequaliies, in Mathematical Analysis, Optimization, Aproximation and Applications(eds: Panos M. Pardalos and Th. M. Rassias), World Scientific Pulshing Compnay, Singapore. Google Scholar

[31]

A. Olbrys, A support theorem or genralized convexity and its applications, J. Math. Anal. Appl., 458 (2018), 1044-1058.  doi: 10.1016/j.jmaa.2017.09.038.  Google Scholar

[32]

B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 2-75.   Google Scholar

[33]

G. Qu and N. Li, On the exponentially stability of primal-dual gradeint dynamics, IEEE Control Syst. Letters, 3 (2019), 43-48.   Google Scholar

[34]

G. Stampacchia, Formes bilieaires coercives sur les ensembles convexes, Comput. Rend.l'Acad. Sciences, Paris, 258 (1964), 4413-4416.   Google Scholar

[35]

H. K. Xu, Inequalities in Banach spaces with applications, Nonl. Anal.Theory, Meth. Appl., 16 (1991), 1127-1138.  doi: 10.1016/0362-546X(91)90200-K.  Google Scholar

[36]

E. A. Youness, $E$-convex sets, $E$-convex functions and $E$-convex programming, J. Optim. Theory Appl., 102 (1999), 439-450.  doi: 10.1023/A:1021792726715.  Google Scholar

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D. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim., 6 (1996), 714-726.  doi: 10.1137/S1052623494250415.  Google Scholar

show all references

References:
[1]

O. AlabdaliA. Guessab and G. Schmeisser, Characterization of uniform convexity for differentiable functions, Appl. Anal. Discrete Math., 13 (2019), 721-732.  doi: 10.2298/aadm190322029a.  Google Scholar

[2]

M. Adamek, On a problem connected with strongly convex functions, Math. Inequ. Appl., 19 (2016), 1287-1293.  doi: 10.7153/mia-19-94.  Google Scholar

[3]

H. AnguloJ GimenezA. M. Moeos and K. Nikodem, On strongly $h$-convex functions, Anal. Funct. Anal., 2 (2011), 85-91.  doi: 10.15352/afa/1399900197.  Google Scholar

[4]

W. L. Bynum, Weak parallelogram laws for Banach spaces, Canad. Math. Bull., 19 (1976), 269-275.  doi: 10.4153/CMB-1976-042-4.  Google Scholar

[5]

R. Cheng and C. B. Harris, Duality of the weak parallelogram laws on Banach spaces, J. Math. Anal. Appl., 404 (2013), 64-70.  doi: 10.1016/j.jmaa.2013.02.064.  Google Scholar

[6]

R. Cheng and W. T. Ross, Weak parallelogram laws on Banach spaces and applications to prediction, Period. Math. Hung., 71 (2015), 45-58.  doi: 10.1007/s10998-014-0078-4.  Google Scholar

[7]

G. Cristescu M.and Găianu, Shape properties of Noors' convex sets, Proceed. of the Twelfth Symposium of Mathematics and its Applications, Timisoara, (2009), 1–13.  Google Scholar

[8]

G. Cristescu and L. Lupsa, Non-Connected Convexities and Applications, Kluwer Academic Publisher, Dordrechet, 2002. doi: 10.1007/978-1-4615-0003-2.  Google Scholar

[9]

G. Glowinski, J. L. Lions and R. Tremileres, Numerical Analysis of Variational Inequalities, , NortHolland, Amsterdam, Holland, 1981.  Google Scholar

[10]

S. Karamardian, The nonlinear complementarity problems with applications, Part 2, J. Optim. Theory Appl., 4 (1969), 167-181.  doi: 10.1007/BF00927414.  Google Scholar

[11]

G. H. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 118 (2003), 67-80.  doi: 10.1023/A:1024787424532.  Google Scholar

[12]

J. Mako and Z. Pales, On $\varphi$-convexity, Publ. Math. Debrecen, 80 (2012), 107-126.  doi: 10.1057/9780230226203.3173.  Google Scholar

[13]

B. B. MohsenM. A. NoorK. I. Noor and M. Postolache, Strongly convex functions of higher order involving bifunction, Mathematics, 7 (2019), 1028.  doi: 10.3390/math7111028.  Google Scholar

[14]

C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, Springer-Verlag, New York, 2018. doi: 10.1007/978-3-319-78337-6.  Google Scholar

[15]

K. Nikodem and Z. S. Pales, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal., 1 (2011), 83-87.  doi: 10.15352/bjma/1313362982.  Google Scholar

[16]

M. A. Noor, General variational inequalities, Appl. Math. Letters, 1 (1988), 119-121.  doi: 10.1016/0893-9659(88)90054-7.  Google Scholar

[17]

M. A. Noor, Variational-like inequalities, Optimization, 30 (1994), 323-330.  doi: 10.1080/02331939408843995.  Google Scholar

[18]

M. A. Noor, Invex Equilibrium problems, J. Math. Anal. Appl., 302 (2005), 463-475.  doi: 10.1016/j.jmaa.2004.08.014.  Google Scholar

[19]

M. A. Noor, New approximation schemses for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229.  doi: 10.1006/jmaa.2000.7042.  Google Scholar

[20]

M. A. Noor, Some developments in general variational inequalities, Appl. Math. Comput., 152 (2004), 199-277.  doi: 10.1016/S0096-3003(03)00558-7.  Google Scholar

[21]

M. A. Noor, Differeniable nonconvex functions and general variational inequalities, Appl. Math. Comput., 199 (2008), 623-630.  doi: 10.1016/j.amc.2007.10.023.  Google Scholar

[22]

M. A. Noor and K. I. Noor, On generalized strongly convex functions involving bifunction, Appl. Math. Inform. Sci., 13 (2019), 411-416.  doi: 10.18576/amis/130313.  Google Scholar

[23]

M. A. Noor and K. I. Noor, Some characterization of strongly preinvex functions, J. Math. Anal. Appl., 316 (2006), 697-706.  doi: 10.1016/j.jmaa.2005.05.014.  Google Scholar

[24]

M. A. Noor and K. I. Noor, Higher-order strongly-generalized convex functions, Appl. Math. Inf. Sci., 14 (2020), 133-139.  doi: 10.18576/amis/140117.  Google Scholar

[25]

M. A. Noor and K. I. Noor, Higher order strongly general convex functions and variational inequalities, AIMS Math., 5 (2020), 3646-3663.  doi: 10.3934/math.2020236.  Google Scholar

[26]

M. A. Noor and K. I. Noor, Higher order variational inequalities, Inform. Sci. Lett., 11 (2022), 1-5.   Google Scholar

[27]

M. A. Noor and K. I. Noor, Properties of higher order preinvex functions, Numer. Algebr. Control. Optim., 11 (2021), 431-441.  doi: 10.3934/naco.2020035.  Google Scholar

[28]

M. A. NoorK. I. Noor and I. M. Baloch, Auxiliary principle technqiue for strongly mixed variational-like inequalities, U. P. B. Sci. Bull. Series A, 80 (2018), 93-100.   Google Scholar

[29]

M. A. NoorK. I. Noor and M. T. Rassias, New trends in general variational inequalities, Acta Appl. Mathematica, 170 (2020), 981-1064.  doi: 10.1007/s10440-020-00366-2.  Google Scholar

[30]

M. A. Noor, K. I. Noor and M. T. Rassias, Strongly biconvex functions and bivariational inequaliies, in Mathematical Analysis, Optimization, Aproximation and Applications(eds: Panos M. Pardalos and Th. M. Rassias), World Scientific Pulshing Compnay, Singapore. Google Scholar

[31]

A. Olbrys, A support theorem or genralized convexity and its applications, J. Math. Anal. Appl., 458 (2018), 1044-1058.  doi: 10.1016/j.jmaa.2017.09.038.  Google Scholar

[32]

B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 2-75.   Google Scholar

[33]

G. Qu and N. Li, On the exponentially stability of primal-dual gradeint dynamics, IEEE Control Syst. Letters, 3 (2019), 43-48.   Google Scholar

[34]

G. Stampacchia, Formes bilieaires coercives sur les ensembles convexes, Comput. Rend.l'Acad. Sciences, Paris, 258 (1964), 4413-4416.   Google Scholar

[35]

H. K. Xu, Inequalities in Banach spaces with applications, Nonl. Anal.Theory, Meth. Appl., 16 (1991), 1127-1138.  doi: 10.1016/0362-546X(91)90200-K.  Google Scholar

[36]

E. A. Youness, $E$-convex sets, $E$-convex functions and $E$-convex programming, J. Optim. Theory Appl., 102 (1999), 439-450.  doi: 10.1023/A:1021792726715.  Google Scholar

[37]

D. Zhu and P. Marcotte, Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities, SIAM J. Optim., 6 (1996), 714-726.  doi: 10.1137/S1052623494250415.  Google Scholar

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