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In this paper, we establish the bounded rationality model M for generalized vector equilibrium problems by using a nonlinear scalarization technique. By using the model M, we introduce a new well-posedness concept for generalized vector equilibrium problems, which unifies its Hadamard and Levitin-Polyak well-posedness. Furthermore, sufficient conditions for the well-posedness for generalized vector equilibrium problems are given. As an application, sufficient conditions on the well-posedness for generalized equilibrium problems are obtained. MSC: 49K40, 90C31.

In this paper, we establish an inequality for the q-integral of the bilateral basic hypergeometric function ${}_{r+1}\psi_{r+1}$ . As applications of the inequality, we give some sufficient conditions for the convergence of q-series.

In this study, some new Lyapunov-type inequalities are presented for Caputo-Hadamard fractional Langevin-type equations of the forms $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\beta } \bigl({}_{H}^{C}D_{a + }^{\alpha }+ p(t)\bigr)x(t) + q(t)x(t) = 0,\quad 0 < a < t < b, \end{aligned} $$ and $$ \begin{aligned} &{}_{H}^{C}D_{a + }^{\eta }{ \phi _{p}}\bigl[\bigl({}_{H}^{C}D_{a + }^{\gamma }+ u(t)\bigr)x(t)\bigr] + v(t){\phi _{p}}\bigl(x(t)\bigr) = 0,\quad 0 < a < t < b, \end{aligned} $$ subject to mixed boundary conditions, respectively, where $p(t)$ , $q(t)$ , $u(t)$ , $v(t)$ are real-valued functions and $0 < \beta < 1 < \alpha < 2$ , $1 < \gamma $ , $\eta < 2$ , ${\phi _{p}}(s) = |s{|^{p - 2}}s$ , $p > 1$ . The boundary value problems of fractional Langevin-type equations were firstly converted into the equivalent integral equations with corresponding kernel functions, and then the Lyapunov-type inequalities were derived by the analytical method. Noteworthy, the Langevin-type equations are multi-term differential equations, creating significant challenges and difficulties in investigating the problems. Consequently, this study provides new results that can enrich the existing literature on the topic.

In the present, we first obtain Chen–Ricci inequality for C-totally real warped product submanifolds in cosymplectic space forms. Then, we focus on characterizing spheres and Euclidean spaces, by using the Bochner formula and a second-order ordinary differential equation with geometric inequalities. We derive the characterization for the base of the warped product via the first eigenvalue of the warping function. Also, it proves that there is an isometry between the base $\mathbb{N}_{1}$ and the Euclidean sphere $\mathbb{S}^{m_{1}}$ under some different extrinsic conditions.

In this paper, we propose a low-rank matrix approximation algorithm for solving the Toeplitz matrix completion (TMC) problem. The approximation matrix was obtained by the mean projection operator on the set of feasible Toeplitz matrices for every iteration step. Thus, the sequence of the feasible Toeplitz matrices generated by iteration is of Toeplitz structure throughout the process, which reduces the computational time of the singular value decomposition (SVD) and approximates well the solution. On the theoretical side, we provide a convergence analysis to show that the matrix sequences of iterates converge. On the practical side, we report the numerical results to show that the new algorithm is more effective than the other algorithms for the TMC problem.

This paper deals with higher-order sensitivity analysis in terms of the higher-order adjacent derivative for nonsmooth vector optimization. The relations between the higher-order adjacent derivative of the minima/the proper minima/the weak minima of a multifunction and its profile map are given. Then the relationships between the higher-order adjacent derivative of the perturbation map/the proper perturbation map/the weak perturbation map, and the higher-order adjacent derivative of a feasible map in objective space are considered. Finally, the formulas for estimating the higher-order adjacent derivative of the perturbation map, the proper perturbation map, the weak perturbation map via the adjacent derivative of the constraint map, and the higher-order Fréchet derivative of the objective map are also obtained.

Let be a -algebra of real rank zero and be a -algebra with unit I. It is shown that if is an additive mapping which satisfies for every and for some with , then the restriction of mapping ϕ to is a Jordan homomorphism, where denotes the set of all self-adjoint elements. We will also show that if ϕ is surjective preserving the product and an absolute value, then ϕ is a -linear or -antilinear ∗-homomorphism on . MSC:47B49, 46L05, 47L30.

The aim of this paper is to introduce the normed binomial sequence spaces $b^{r,s}_{p}(\nabla)$ by combining the binomial transformation and difference operator, where $1\leq p\leq\infty$ . We prove that these spaces are linearly isomorphic to the spaces $\ell_{p}$ and $\ell _{\infty}$ , respectively. Furthermore, we compute Schauder bases and the α-, β- and γ-duals of these sequence spaces.