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This research paper investigated weighted Milne-type inequalities utilizing Riemann-Liouville fractional integrals across diverse function classes. A key contribution lies in the establishment of a fundamental integral equality, facilitated by the use of a nonnegative weighted function, which is pivotal for deriving the main results. The paper systematically proved weighted Milne-type inequalities for various function classes, including differentiable convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation. The obtained results not only contribute to the understanding of Milne-type inequalities but also offer insights that pave the way for potential future research in the considered topics. Furthermore, it is evident that the results obtained encompass numerous findings that were previously presented in various studies as special cases. [ABSTRACT FROM AUTHOR]

In this paper, only under the q - k -Keller–Osserman conditions, we consider the existence and global estimates of innumerable radial q - k -convex positive solutions to the q - k -Hessian equation and systems. Our conditions are strictly weaker than those in previous papers. [ABSTRACT FROM AUTHOR]

This paper focuses on introducing and investigating the class of generalized n-fractional polynomial s-type convex functions within the framework of fractional calculus. Relationships between the novel class of functions and other kinds of convex functions are given. New integral inequalities of Hermite-Hadamard and Ostrowski-type are established for our novel generalized class of convex functions. Using some identities and fractional operators, new refinements of Ostrowskitype inequalities are presented for generalized n-fractional polynomial s-type convex functions. Some special cases of the newly obtained results are discussed. It has been presented that, under some certain conditions, the class of generalized n-fractional polynomial s-type convex functions reduces to a novel class of convex functions. It is interesting that, our results for particular cases recaptures the RiemannLiouville fractional integral inequalities and quadrature rules. By extending these particular types of inequalities, the objective is to unveil fresh mathematical perspectives, attributes, and connections that can enhance the evolution of more resilient mathematical methodologies. This study aids in the progression of mathematical instruments across diverse scientific fields. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the optimality conditions for a class of nonsmooth multiobjective fractional optimization problems subject to vanishing constraints. In particular, necessary and sufficient conditions for (weak) Pareto solution are presented in terms of the Clark subdifferential. Furthermore, we construct Wolfe and Mond–Weir-type dual models and derive some duality theorems by using generalized quasiconvexity assumptions. Some examples to show the validity of our conclusions are provided. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the distributed approximate subgradient-type method for minimizing a sum of differentiable and non-differentiable convex functions subject to nondifferentiable convex functional constraints in a Euclidean space. We establish the convergence of the sequence generated by our method to an optimal solution of the problem under consideration. Moreover, we derive a convergence rate of order O(N1−a) for the objective function values, where a ∈ (0.5, 1). Finally, we provide a numerical example illustrating the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We prove a weighted Lp boundedness of Marcinkiewicz integral operators along surfaces on product domains. For various classes of surfaces, we prove the boundedness of the corresponding operators on the weighted Lebsgue space Lp(Rn ×Rm, ω1(x)dx, ω2(y)dy), provided that the weights ω1 and ω2 are certain radial weights and that the kernels are rough in the optimal space L(log L)(Sn-1 × Sm-1). In particular, we prove the boundedness of Marcinkiewicz integral operators along surfaces determined by mappings that are more general than polynomials and convex functions. Also, in this paper we prove the weighted Lp boundedness of the related square and maximal functions. Our weighted Lp inequalities extend as well as generalize previously known Lp boundedness results. [ABSTRACT FROM AUTHOR]

This paper discussed the decay of a thermoelastic laminated beam subjected to nonlinear delay and nonlinear structural damping. We provided explicit and general energy decay rates of the solution by imposing suitable conditions on both weight delay and wave speeds. To achieve this, we leveraged the properties of convex functions and employed the multiplier technique as a specific approach to demonstrate our stability results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a von Karman plate system with general type of relaxation functions on the boundary. We derive the general decay rate result without requiring the assumption that the initial value w0 = 0 on the boundary, using the multiplier method and some properties of the convex functions. Here we consider the resolvent kernels ki(i = 1, 2), namely k"i (t) = -ξi(t)Gi(-k'i(t)), where Gi are convex and increasing functions near the origin and ξi are positive nonincreasing functions. Moreover, the energy decay rates depend on the functions ξi and Gi. These general decay estimates allow for certain relaxation functions which are not necessarily of exponential or polynomial decay and therefore improve earlier results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a variance reduced forward-reflected-backward algorithm for solving nonconvex finite-sum mixed variational inequalities, where $ F $ is Lipschitz continuous, $ h $ is proper lower semicontinuous prox-convex, and they satisfy a generalized monotonicity condition on $ C $. We prove that the algorithm is almost surely convergent and exhibits $ R $-linear rate when $ F $ is strongly monotone. [ABSTRACT FROM AUTHOR]

In this paper, we study the decay estimate of the trajectories of the second order differential equation:$ \ddot{x}(t)+\gamma (t) \dot{x}(t)+\nabla F(x(t)) = g(t), $where $ \gamma (t) = \frac{\alpha }{t} $ is a viscous damping coefficient with $ \alpha >0 $, $ F $ is a convex differentiable function, and $ g(t) $ is a small perturbation term. We are concerned in the optimal convergence rates for damped inertial gradient dynamics with flat geometries, which is proposed in Sebbouh and Dossal (SIAM J Optim 30: 1850-1877, 2020). Making use of some Lyapunov functions and iteration method, we get some results. More specifically, we obtained the convergence rate $ t^{-{\frac{2\gamma \alpha }{\gamma +2}} } $ of $ F(x(t))-F^* $ when $ 1+\frac{2}{\gamma } < \alpha <\frac{\gamma +2}{\gamma -2} $. In addition, for the case $ \alpha<1+\frac{2}{\gamma} $, we proved that the convergence rate cannot be better than $ t^{-\frac{2\bar{\gamma } \alpha }{\gamma +2} } $ for $ \bar{ \gamma }\le\gamma+2 $. [ABSTRACT FROM AUTHOR]

In the paper, a class of cooperative differential games on networks is considered. In such games, the new characteristic function is introduced based on the possibility of stopping interaction by players outside the coalition in each time instant or imposing sanction on players from the coalition. This gives the real possibility for the computation of characteristic function. Thus, the characteristic function is evaluated along the cooperative trajectory. It measures the worth of coalitions under the process of cooperation instead of under minimax confrontation or the Nash non-cooperative stance. The approach essentially simplifies the construction of the characteristic function and cooperative solutions such as the Shapley value, Core, $ \tau $-value and others. Also, it is proved that the proposed characteristic function is convex, time consistent, and as a result, the Shapley value belongs to the Core and is time consistent. Also, a modification of the dynamic game on networks, namely, dynamic network game with partner sets is considered. In this case, payoffs of a given player depend on his actions and the actions of the players from his partner set. Using previous ideas, the special type of characteristic function is introduced, and cooperative solutions are proposed. [ABSTRACT FROM AUTHOR]

Let $ F: \mathbb{R}^n\rightarrow [0, \, \infty) $ be a convex function of class $ C^2(\mathbb{R}^n\backslash\{0\}) $, which is even and positively homogeneous of degree $ 1 $. In this paper, we prove that$ \sup\limits_{u\in W^{1, n}(\mathbb{R}^n), \, \displaystyle{\int}_{\mathbb{R}^n}(F^n(\nabla u)+|u|^n)dx\leq1}\displaystyle{\int}_{\mathbb{R}^n}\frac{\Phi(\lambda_{n}(1-\frac{\beta}{n})(1+\alpha\|u\|^{n}_n)^{\frac{1}{n-1}}|u|^{\frac{n}{n-1}})}{F^o(x)^\beta}dx $is finite for $ 0\leq\alpha<1 $, and the supremum is infinity for $ \alpha\geq1 $, where $ F^o(x) $ is the polar function of $ F $, $ \Phi(t) = e^t-\sum_{j = 0}^{n-2}\frac{t^j}{j!} $, $ \beta\in[0, n) $, $ \lambda_n = n^{\frac{n}{n-1}}\kappa_n^{\frac{1}{n-1}} $ and $ \kappa_n $ is the volume of the unit Wulff ball. Moreover, by using the method of blow-up analysis, we also obtain the existence of extremal functions for the supremum when $ 0\leq\alpha<1 $. [ABSTRACT FROM AUTHOR]

Optimization problems under uncertainty has achieved much attention because of inaccurate or incomplete data that often appear in real-world applications. In order to address this challenging issue, several types of methodologies have been developed in optimization theory and one of them is fuzzy optimization. In this paper, an attempt is taken to use the minimax exact penalty function method for solving convex nondifferentiable optimization problems with fuzzy objective functions and with both inequality and equality constraints. The most important property of exact penalty function methods, that is, exactness of the penalization, is defined and analyzed if the minimax penalty function method is applicable for solving a convex nondifferentiable optimization problem with a fuzzy objective function. It is proved that a (weak) KKT point of the considered fuzzy optimization problem is a (weakly) nondominated solution of its associated fuzzy penalized optimization problem constructed in the used penalized approach. Further, the conditions are also derived under which there is the equivalence between (weakly) nondominated solutions of the aforesaid fuzzy optimization problems. These results are established under assumption that the functions involved in the considered nondifferentiable fuzzy optimization problem are convex. Further, the algorithm of solving the considered nondifferentiable fuzzy optimization problem by using the minimax exact penalty fuzzy function method is proposed and its convergence is also established. [ABSTRACT FROM AUTHOR]

We considered a convex bilevel optimization problem when the outer level problem was to find a minimizer of a strongly convex function over the set of solutions of the inner level problem which was in the form of minimization of the sum of a convex differentiable function and a nonsmooth convex function. In this work, we proposed a novel accelerated algorithm by employing both linesearch and inertial techniques for solving a convex bilevel optimization problem. Then, we proved the strong convergence of the sequence generated by our proposed algorithm to an optimal solution of the convex bilevel optimization problems without the continuity assumption on the gradient of the objective function. Moreover, we presented the convergence behavior of the proposed method by some numerical experiments addressing image restoration problems and data classification problems with least squares constraints. Finally, the performances of the restorative image and the data classification of the proposed method were compared with other existing algorithms in the literature. According to the experiment, our proposed algorithm had a better convergence behavior than the others in the literature. [ABSTRACT FROM AUTHOR]

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of p-valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy p-valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a q-calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized q-calculus operator, defined here and having the (p, q)-Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator. [ABSTRACT FROM AUTHOR]

We consider the class of functions u ∈ C²((0, ∞)) satisfying second-order differential inequalities in the form u'(x) + k/x²u(x) ≥ 0 for all x > 0. For this class of functions, we establish Hermite-Hadamard-type inequalities in both cases (k = 1/4 and 0 < k < 1/4 ). We next extend our obtained results to the two-dimensional case. In the limit case k → 0+ we deriver some existing results from the literature that are related to convex functions and convex functions on the coordinates. In our approach, we make use of some tools from ordinary differential equations. [ABSTRACT FROM AUTHOR]

In this study, we develop a nonmonotone variable metric Barzilai-Borwein method for minimizing the sum of a smooth function and a convex, possibly nondifferentiable, function. At each step, the descent direction is obtained by taking the difference between the minimizer of the scaling proximal function and the current iteration point. An adaptive nonmonotone line search is proposed for determining the step length along this direction. We also show that the limit point of the iterates sequence is a stationary point. Numerical results with parallel magnetic resonance imaging, Poisson, and Cauchy noise deblurring demonstrate the effectiveness of the new algorithm. [ABSTRACT FROM AUTHOR]

This study focused on deriving Milne-type inequalities using expanded fractional integral operators. We began by establishing a key equality associated with these operators. Using this equality, we explored Milne-type inequalities for functions with convex derivatives, supported by an illustrative example for clarity. Additionally, we investigated Milne-type inequalities for bounded and Lipschitzian functions utilizing fractional expanded integrals. Finally, we extended our exploration to Milne-type inequalities involving functions of bounded variation. [ABSTRACT FROM AUTHOR]

In this study, some new Hermite-Hadamard type inequalities for co-ordinated convex functions were obtained with the help of conformable fractional integrals. We have presented some remarks to give the relation between our results and earlier obtained results. Moreover, an identity for partial differentiable functions has been established. By using this equality and concept of co-ordinated convexity, we have proven a trapezoid type inequality for conformable fractional integrals. [ABSTRACT FROM AUTHOR]

The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval [ b 0 , b 1 ] ⊂ ℜ , we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at b 0 ∈ [ b 0 , b 1 ] ⊂ ℜ. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point b 1 , and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus. The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval , we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at . We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point , and provide some examples as well. Some justification with graphical analysis is provided as well. Finally, with the assistance of these outcomes, we give a midpoint type inequality and some of its approximations for convex functions in symmetric quantum calculus. [ABSTRACT FROM AUTHOR]

The main objective of this study is to establish error estimates of the new parameterized quadrature rule similar to and covering the second Simpson formula. To do this, we start by introducing a new parameterized identity involving the right and left Riemann-Liouville integral operators. On the basis of this identity, we establish some fractional Simpson-type inequalities for functions whose absolute value of the first derivatives are s-convex in the second sense. Also, we examine the special cases m = 1 / 2 and m = 3 / 8 , as well as the two cases s = 1 and α = 1 , which respectively represent the classical convexity and the classical integration. By applying the definition of convexity, we derive larger estimates that only used the extreme points. Finally, we provide applications to quadrature formulas, special means, and random variables. [ABSTRACT FROM AUTHOR]

In this work, we focused on minimizing a strongly convex smooth function over the common fixed-point constraints. We proposed an extrapolated fixed-point optimization method, which is a modified version of the extrapolated sequential constraint method with conjugate gradient direction. We proved the convergence of the generated sequence to the unique solution to the considered problem without boundedness assumption. We also investigated some numerical experiments to underline the effectiveness and performance of the proposed method. [ABSTRACT FROM AUTHOR]

Local minimizers of nonhomogeneous quasiconvex variational integrals with standard pgrowth of the type ... feature almost everywhere BMO-regular gradient provided that f belongs to the borderline Marcinkiewicz space L(n, ∞). [ABSTRACT FROM AUTHOR]

The features of analytical functions were mostly studied using a fuzzy subset and a q -difference operator in this study, as we investigate many fuzzy differential subordinations related to the q -analogue multiplier-Noor integral operator. By applying fuzzy subordination to univalent functions whose range is symmetric with respect to the real axis, we create a few new subclasses of analytical functions. We define numerous classes related to the family of linear q -operators and introduce them. Here, we focus on the inclusion results and other integral features. The features of analytical functions were mostly studied using a fuzzy subset and a -difference operator in this study, as we investigate many fuzzy differential subordinations related to the -analogue multiplier-Noor integral operator. By applying fuzzy subordination to univalent functions whose range is symmetric with respect to the real axis, we create a few new subclasses of analytical functions. We define numerous classes related to the family of linear -operators and introduce them. Here, we focus on the inclusion results and other integral features. [ABSTRACT FROM AUTHOR]