Properties of higher order preinvex functions

Noor, Muhammad Aslam; Noor, Khalida İnayat

doi:10.3934/naco.2020035

Cited by 7 publications

(9 citation statements)

References 22 publications

(53 reference statements)

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“…Noor and Noor [21] introduce and study the higher-order preinvex functions. Obviously, when k(λ) = λ, a higher-order k(λ) preinvex function appears, which is a higherorder preinvex function.…”

Section: Definitionmentioning

confidence: 99%

“…Many interesting properties, applications, and generalizations of the variational inequalities, which are proposed by Stampacchia [1], have been discussed in signal processing, linear inverse problems, and machine learning; see References [2][3][4][5][6][7][8][9][10][11][12][13][14][15] and the references therein. The concepts and applications of convex functions and convex sets are investigated; see Hanson [16], Ben-Israel et al [17], and Noor [18][19][20][21], as well as References [22][23][24][25][26] for more details.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

Noor

2021

Symmetry

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Equilibrium problems and variational inequalities are connected to the symmetry concepts, which play important roles in many fields of sciences. Some new preinvex functions, which are called generalized preinvex functions, with the bifunction ζ(.,.) and an arbitrary function k, are introduced and studied. Under the normed spaces, new parallelograms laws are taken as an application of the generalized preinvex functions. The equilibrium-like problems are represented as the minimum values of generalized preinvex functions under the kζ-invex sets. Some new inertial methods are proposed and researched to solve the higher order directional equilibrium-like problem, Convergence criteria of the our methods is discussed, along with some unresolved issues.

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Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

Noor

2021

Symmetry

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“…Noor at el. [22,23] investigated the properties of the higher order preinvex functions and their variant forms. We would like to emphasize that the higher order preinvex functions include the higher order strongly convex functions as special cases.…”

Section: Introductionmentioning

confidence: 99%

New classes of higher order variational-like inequalities

Noor,

Inayat Noor

2023

Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znano

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In this paper, we prove that the optimality conditions of the higher order convex functions are characterized by a class of variational inequalities, which is called the higher order variational inequality. Auxiliary principle technique is used to suggest an implicit method for solving higher order variational 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.

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“…HOS-preinvex functions and their generalization have many applications in different areas such as multilevel games, engineering design, and economical equilibrium. Some of the authors presented different types of HOS-preinvex functions and discussed their characterizations like Alabdali et al [18], Noor and Noor [9,19,20], and Mohsen et al [21] considered different classes and prove that the optimality condition of HOS-preinvex functions can be distinguished by different kinds of variational inequalities like strongly variational inequality, strongly variational-like inequality, HOS-variational inequality, and HOS-variational-like inequality.…”

Section: Introductionmentioning

confidence: 99%

Higher-Order Strongly Preinvex Fuzzy Mappings and Fuzzy Mixed Variational-Like Inequalities

Khan¹,

Noor²,

Noor³

et al. 2021

IJCIS

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A family of fuzzy mappings is called higher-order strongly preinvex fuzzy mappings (HOS-preinvex fuzzy mappings), which take the place of generalization of the notion of nonconvexity is introduced through the "fuzzy-max" order among fuzzy numbers. This family properly includes the family of preinvex fuzzy mappings and is included in the family of quasi preinvex fuzzy mappings. With the support of examples, we have discussed some special cases. Some properties are derived and relations among the HOS-preinvex fuzzy mappings, HOS-invex fuzzy mappings, and fuzzy HOS-monotonicity are obtained under some mild conditions. Then, we have shown that optimality conditions of generalized differentiable HOS-preinvex fuzzy mappings and for the sum of generalized differentiable (briefly, G-differentiable) preinvex fuzzy mappings and nongeneralized differentiable HOSpreinvex fuzzy mappings can be distinguished by HOS-fuzzy variational-like inequalities and HOS-fuzzy mixed variational-like inequalities, respectively which can be viewed as novel applications. These inequalities are very interesting outcome of our main results and appear to be new ones. Several exceptional cases are debated. Presented results in this paper can be considered and development of previously established results.

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Properties of higher order preinvex functions

Cited by 7 publications

References 22 publications

Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

New classes of higher order variational-like inequalities

Higher-Order Strongly Preinvex Fuzzy Mappings and Fuzzy Mixed Variational-Like Inequalities

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