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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 and 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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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. 

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[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.

[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.

[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.

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.

[2]

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

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[17]

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

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[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.

[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. 

[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.

[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.

[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.

[32]

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

[33]

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

[34]

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

[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.

[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.

[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.

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