Search Results (16)

This article introduces certain algebraic properties of generalized $({\tilde{h}}_1,{\tilde{h}}_2)$-pre-invex functions on ${\mathbb{R}}^{\mathsf{\aleph }}$$(0<\mathsf{\aleph }\le 1)$. A new fractal weighted integral identity is established and further employed to obtain several Ostrowski-type results in the fractal setting for functions whose first derivatives in the modulus belong to the generalized $({\tilde{h}}_1,{\tilde{h}}_2)$-pre-invex functions’s class. An illustrative example is presented to validate the theoretical findings. Moreover, applications of the main results are derived in connection with generalized random variables and various special means, highlighting the effectiveness and potential scope of the proposed approach. Full article

In this manuscript, we define the $(E,F)$-invex set, $(E,F)$-invex functions, and $(E,F)$-preinvex functions on Euclidean space, i.e., simply vector space. We extend these concepts on the Riemannian manifold. We also detail the fundamental properties of $(E,F)$-preinvex functions and provide some examples that illustrate the concepts well. We have established a relation between $(E,F)$-invex and $(E,F)$-preinvex functions on Riemannian manifolds. We introduce the conditions $\mathcal{A}$ and define the $(E,F)$-proximal sub-gradient. $(E,F)$-preinvex functions are also used to demonstrate their applicability in optimization problems. In the last, we establish the points of extrema of a non-smooth $(E,F)$-preinvex functions on $(E,F)$-invex subset of the Riemannian manifolds by using the $(E,F)$-proximal sub-gradient. Full article

In this paper, a class of controlled variational control models is studied by considering the notion of $(q,w)-\pi$-invexity. Our aim is to investigate a solution set in the considered interval-valued controlled models. To achieve this, we establish some characterization results of solutions in the controlled interval-valued variational models. More precisely, necessary and sufficient conditions of optimality are highlighted as part of a feasible solution. To prove that the optimality conditions are sufficient, we impose generalized invariant convexity hypotheses for the involved multiple integral functionals. Finally, a duality result is provided in order to better describe the problem under study. The methodology used in this paper is a combination of techniques from the Lagrange–Hamilton theory, calculus of variations, and control theory. This study could be immediately improved by including an analysis of this class of optimization problems by using curvilinear integrals instead of multiple integrals. The independence of path imposed to these functionals and their physical significance would increase the applicability and importance of the paper. Full article

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher’s interest even more. Full article

The symmetric function class interacts heavily with other types of functions. One of these is the pre-invex function class, which is strongly related to symmetry theory. This paper proposes a novel fuzzy fractional extension of the Hermite-Hadamard, Hermite-Hadamard-Fejér, and Pachpatte type inequalities for up and down pre-invex fuzzy-number-valued mappings. Using fuzzy fractional operators, several generalizations have been developed, where well-known results fit as particular cases. Additionally, some non-trivial examples are included to support the discussion and the applicability of the key findings. The approach appears trustworthy and effective for dealing with various nonlinear problems in science and engineering. The findings are general and may constitute contributions to complex waveform theory. Full article

In this paper, by using the invexity (or pseudoinvexity) and Fréchet differentiability of some integral functionals of curvilinear type, we state some relations between the solutions of a new non-linear optimization problem and the associated variational inequality. In order to prove the results derived in this paper, we use the new notion of invex set by considering some given functions. To justify the effectiveness and outstanding applicability of this work, some illustrative examples are provided. Full article

Fuzzy variational problems have received significant attention over the past decade due to a number of successful applications in fields such as optimal control theory and image segmentation. Current literature on fuzzy variational problems focuses on the necessary optimality conditions for finding the extrema, which have been studied under several differentiability conditions. In this study, we establish the sufficient conditions for the existence of minimizers for fuzzy variational problems under a weaker notion of convexity, namely preinvexity and Buckley–Feuring differentiability. We further discuss their application in a cost minimization problem. Full article

In this paper, by considering the notions of the invex set, Fréchet differentiability, invexity and pseudoinvexity for the involved functionals of curvilinear integral type, we establish some relations between the solutions of a class of weak vector variational inequalities and (weak) efficient solutions of the associated control problem. Full article

The connection between generalized convexity and symmetry has been studied by many authors in recent years. Due to this strong connection, generalized convexity and symmetry have arisen as a new topic in the subject of inequalities. In this paper, we introduce the concept of interval-valued preinvex functions on the coordinates in a rectangle from the plane and prove Hermite–Hadamard type inclusions for interval-valued preinvex functions on coordinates. Further, we establish Hermite–Hadamard type inclusions for the product of two interval-valued coordinated preinvex functions. These results are motivated by the symmetric results obtained in the recent article by Kara et al. in 2021 on weighted Hermite–Hadamard type inclusions for products of coordinated convex interval-valued functions. Our established results generalize and extend some recent results obtained in the existing literature. Moreover, we provide suitable examples in the support of our theoretical results. Full article

The purpose of this study is to prove the existence of fractional integral inclusions that are connected to the Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for χ-pre-invex fuzzy-interval-valued functions. Some of the related fractional integral inequalities are also proved via Riemann–Liouville fractional integral operator, where integrands are fuzzy-interval-valued functions. To prove the validity of our main results, some of the nontrivial examples are also provided. As specific situations, our findings can provide a variety of new and well-known outcomes which can be viewed as applications of our main results. The results in this paper can be seen as refinements and improvements to previously published findings. Full article

The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two years, convexity and symmetry have emerged as a new field due to considerable association. In this paper, we study a new version of interval-valued functions (I-V·Fs), known as left and right $\chi$-pre-invex interval-valued functions (LR-$\chi$-pre-invex I-V·Fs). For this class of non-convex I-V·Fs, we derive numerous new dynamic inequalities interval Riemann–Liouville fractional integral operators. The applications of these repercussions are taken into account in a unique way. In addition, instructive instances are provided to aid our conclusions. Meanwhile, we’ll discuss a few specific examples that may be extrapolated from our primary findings. Full article

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 $\zeta (.,.)$ 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 $k\zeta$-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. Full article

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. Full article

In the paper, we analyze the necessary efficiency conditions for scalar, vectorial and vector fractional variational problems using curvilinear integrals as objectives and we establish sufficient conditions of efficiency to the above variational problems. The efficiency sufficient conditions use of notions of the geodesic invex set and of (strictly, monotonic) ( $\rho$ , b)-geodesic quasiinvex functions. Full article

By using the contemporary theory of inequalities, this study is devoted to proposing a number of refinements inequalities for the Hermite-Hadamard’s type inequality and conclude explicit bounds for two new definitions of $(p_1{p}_2,q_1{q}_2)$ -differentiable function and $(p_1{p}_2,q_1{q}_2)$ -integral for two variables mappings over finite rectangles by using pre-invex set. We have derived a new auxiliary result for $(p_1{p}_2,q_1{q}_2)$ -integral. Meanwhile, by using the symmetry of an auxiliary result, it is shown that novel variants of the the Hermite-Hadamard type for $(p_1{p}_2,q_1{q}_2)$ -differentiable utilizing new definitions of generalized higher-order strongly pre-invex and quasi-pre-invex mappings. It is to be acknowledged that this research study would develop new possibilities in pre-invex theory, quantum mechanics and special relativity frameworks of varying nature for thorough investigation. Full article