scholarly journals On Strongly Generalized Preinvex Fuzzy Mappings

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Peide Liu ◽  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Optimality Conditions ◽  
Generalized Monotonicity ◽  
Fuzzy Mapping ◽  
Invex Set ◽  
Special Cases

In this article, we introduce a new notion of generalized convex fuzzy mapping known as strongly generalized preinvex fuzzy mapping on the invex set. Firstly, we have investigated some properties of strongly generalized preinvex fuzzy mapping. In particular, we establish the equivalence among the strongly generalized preinvex fuzzy mapping, strongly generalized invex fuzzy mapping, and strongly generalized monotonicity. We also prove that the optimality conditions for the sum of G-differentiable preinvex fuzzy mappings and non-G-differentiable strongly generalized preinvex fuzzy mappings can be characterized by strongly generalized fuzzy mixed variational-like inequalities, which can be viewed as a novel and innovative application. Several special cases are discussed. Results obtained in this paper can be viewed as improvement and refinement of previously known results.

scholarly journals On Some Characterization of Preinvex Fuzzy Mappings

2020 ◽  
pp. 17-42
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Fuzzy Mapping ◽  
Invex Set ◽  
Special Cases

In this paper, a new notion of generalized convex fuzzy mapping is introduced, which is called α-preinvex fuzzy mapping on the α-invex set. We have investigated the characterization of preinvex fuzzy mappings using α-preinvex fuzzy mappings, which can be viewed as a novel and innovative application. Some important and significant special cases are discussed. We have also investigated that the minimum of α-preinvex fuzzy mappings can be characterized by fuzzy α-variational like inequalities.

Some new fractional integral inequalities for generalized relative semi-m-(r; h1, h2)-preinvex mappings via generalized Mittag-Leffler function

2019 ◽  
Vol 26 (1/2) ◽  
pp. 41-55 ◽  
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integral ◽  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Invex Set ◽  
Special Means ◽  
Special Cases ◽  
Differentiable Mappings

The authors discover a new identity concerning differentiable mappings defined on m-invex set via fractional integrals. By using the obtained identity as an auxiliary result, some fractional integral inequalities for generalized relative semi- m-(r;h1,h2)-preinvex mappings by involving generalized Mittag-Leffler function are presented. It is pointed out that some new special cases can be deduced from main results of the paper. Also these inequalities have some connections with known integral inequalities. At the end, some applications to special means for different positive real numbers are provided as well.

scholarly journals Some New Hermite-Hadamard Type Inequalities Via k-Fractional Integrals Concerning Differentiable Generalized Relative Semi-(r;m,p,q,h1,h2)-Preinvex Mappings

2018 ◽  
Vol 60 (1) ◽  
pp. 59-78
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integrals ◽  
Auxiliary Result ◽  
Invex Set ◽  
Special Cases ◽  
Differentiable Mappings

Abstract In this article, we first presented a new identity concerning differentiable mappings defined on m-invex set via k-fractional integrals. By using the notion of generalized relative semi-(r;m,p,q,h1,h2)-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard type inequalities via k-fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article.

scholarly journals Some New Hermite-Hadamard-Fejer Type Inequlaties via k-Fractional Integrals Concerning Differentiable Generalized Relative Semi-(r; m, h1, h2)-Preinvex Mappings

2018 ◽  
Vol 11 (1) ◽  
pp. 51
Author(s):  
Miftar Ramosacaj ◽  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integrals ◽  
Auxiliary Result ◽  
Invex Set ◽  
Special Cases ◽  
Differentiable Mappings

In this article, we first presented a new identity concerning differentiable mappings defined on m-invex set via k-fractional integrals. By using the notion of generalized relative semi-(r;m,h1,h2)-preinvexity and the obtained identity as an auxiliary result, some new estimates with respect to Hermite-Hadamard-Fejer type inequalities via k-fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article.

scholarly journals Fractional trapezium-like inequalities involving generalized relative semi--preinvex mappings on an -invex set

2020 ◽  
Vol 72 (12) ◽  
pp. 1633-1350
Author(s):  
T. S. Du ◽  
C. Y. Luo ◽  
Z. Z. Huang ◽  
A. Kashuri
Keyword(s):  
Fractional Integral ◽  
Second Order ◽  
Fractional Integrals ◽  
Invex Set ◽  
Special Cases ◽  
Integral Equality ◽  
Differentiable Mappings

UDC 517.5 The authors derive a fractional integral equality concerning twice differentiable mappings defined on -invex set. By using this identity, the authors obtain new estimates on generalization of trapezium-like inequalities for mappings whose second order derivatives are generalized relative semi--preinvex via fractional integrals. We also discuss some new special cases which can be deduced from our main results.

scholarly journals Optimality criteria in set-valued optimization

2003 ◽  
Vol 75 (2) ◽  
pp. 221-232 ◽  
Author(s):  
C. S. Lalitha ◽  
J. Dutta ◽  
Misha G. Govil
Keyword(s):  
Optimality Conditions ◽  
Optimization Problem ◽  
Convex Set ◽  
Necessary Optimality Conditions ◽  
Optimality Criteria ◽  
New Class ◽  
Invex Set

AbstractThe main aim of this paper is to obtain optimality conditions for a constrained set-valued optimization problem. The concept of Clarke epiderivative is introduced and is used to derive necessary optimality conditions. In order to establish sufficient optimality criteria we introduce a new class of set-valued maps which extends the class of convex set-valued maps and is different from the class of invex set-valued maps.

scholarly journals Fuzzy Mixed Variational-Like and Integral Inequalities for Strongly Preinvex Fuzzy Mappings

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1816
Author(s):  
Muhammad Bilal Khan ◽  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Juan L. G. Guirao
Keyword(s):  
Optimality Conditions ◽  
Critical Role ◽  
Fuzzy Optimization ◽  
Strong Relationship ◽  
The Other ◽  
Hadamard Inequality ◽  
New Class ◽  
Invex Set ◽  
Tight Connection ◽  
The Subject

It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite–Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings.

scholarly journals General biconvex functions and bivariational inequalities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Convergence Analysis ◽  
Optimality Conditions ◽  
Banach Spaces ◽  
Higher Order ◽  
Implicit Method ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
New Concepts ◽  
Special Cases

<p style='text-indent:20px;'>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.</p>

scholarly journals Hermite–Hadamard type inequalities via k-fractional integrals concerning differentiable generalized η-convex mappings

2020 ◽  
Vol 24 (1) ◽  
pp. 19-35
Author(s):  
Artion Kashuri ◽  
Rozana Liko
Keyword(s):  
Fractional Integrals ◽  
Auxiliary Result ◽  
Real Numbers ◽  
Positive Real ◽  
Convex Mappings ◽  
Invex Set ◽  
Special Means ◽  
Special Cases ◽  
Differentiable Mappings

The authors discover a new identity concerning differentiable mappings dened on (m; g; θ)-invex set via k-fractional integrals. By using the obtained identity as an auxiliary result, some new estimates with respect to Hermite–Hadamard type inequalities via k-fractional integrals for generalized-m-(((h1 ∘g)p; (h2 ∘g)q); (η1; η2))-convex mappings are presented. It is pointed out that some new special cases can be deduced from the main results. Also, some applications to special means for different positive real numbers are provided.

scholarly journals A Class of SemilocalE-Preinvex Functions and Its Applications in Nonlinear Programming

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Hehua Jiao ◽  
Sanyang Liu ◽  
Xinying Pai
Keyword(s):  
Nonlinear Programming ◽  
Optimality Conditions ◽  
Convex Functions ◽  
Convex Set ◽  
Basic Properties ◽  
New Class ◽  
Duality Results ◽  
Invex Set ◽  
Preinvex Functions ◽  
Class Of Functions

A kind of generalized convex set, called as local star-shapedE-invex set with respect toη,is presented, and some of its important characterizations are derived. Based on this concept, a new class of functions, named as semilocalE-preinvex functions, which is a generalization of semi-E-preinvex functions and semilocalE-convex functions, is introduced. Simultaneously, some of its basic properties are discussed. Furthermore, as its applications, some optimality conditions and duality results are established for a nonlinear programming.

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