In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results.

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for $2$-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.

In this article, by using Moser's twist theorem, we prove that all solutions of the equation $x'' + ax^{+} - bx^{-} + \varphi(x) = p(t)$ are bounded in the critical situation, where $p$ is a smooth periodic function, and $\varphi$ is bounded one.

In this paper, we investigate the $L^{2}$ boundedness of Fourier integral operator $T_{\phi,a}$ with rough symbol $a \in L^{\infty} S^{m}_{\rho}$ and rough phase $\phi \in L^{\infty} \Phi^{2}$ which satisfies $\big| \{ x: |\nabla_{\xi} \phi(x,\xi) - y| \leq r \} \big| \leq C(r^{n-1}+r^{n})$ for any $\xi,y \in \mathbb{R}^{n}$ and $r \gt 0$. We obtain that $T_{\phi,a}$ is bounded on $L^2$ if $m \lt \rho(n-1)/2 - n/2$ when $0 \leq \rho \leq 1/2$ or $m \lt -(n+1)/4$ when $1/2 \leq \rho \leq 1$. When $\rho = 0$ or $n = 1$, the condition of $m$ is sharp. Moreover, the maximal wave operator is a special class of $T_{\phi,a}$ which is studied in this paper. Thus, our main theorem substantially extends and improves some known results about the maximal wave operator.

In this paper we will consider various kinds of chaotic points of multifunctions and show their application to the theory of infinite topological games.

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order $n$; (2) the maximum independence number of an almost self-centered graph of order $n$ and radius $r$; (3) the minimum order of a $k$-regular almost self-centered graph; (4) the maximum size of an almost peripheral graph of order $n$; (5) possible maximum degrees of an almost peripheral graph of order $n$ and (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order $n$ with maximum degree $n-4$ which is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them.

RK-RK and REK-RK methods are the two latest while very effective randomized iteration solvers for factorized linear system $UV \beta = y$ by interlacing randomized Kaczmarz (RK) and randomized extended Kaczmarz (REK) updates. This paper considers two latest randomized iterative methods for solving large-scale linear systems and linear least-squares problems—greedy randomized Kaczmarz (GRK) method and greedy randomized Gauss–Seidel (GRGS) method. By introducing a relaxation parameter $\omega$ into the iterates of GRK and GRGS, we construct relaxed GRK and GRGS methods, respectively. In addition, by interlacing their updates, we propose relaxed GRK-GRK and GRGS-GRK methods to solve consistent and inconsistent factorized linear systems, respectively. We prove the exponential convergence of these two interlaced methods and show that relaxed GRK-GRK and GRGS-GRK can be more efficient than RK-RK and REK-RK, respectively, if the relaxation parameters are chosen appropriately.

The uniform stabilization of a semilinear wave equation with variable coefficients and nonlinear boundary conditions is considered. The uniform decay rate is established by the Riemannian geometry method.

In this paper, we find all the solutions of Diophantine equation written in our title, where $n$, $m$, $x$ and $y$ are positive integers and $F_k$ denotes the $k$th term of the Fibonacci sequence.

We study the local-in-time well-posedness of Vlasov–Poisson equation in Besov space for the large initial data. To accomplish it, we establish commutator estimates in Besov space which are quite useful in dealing with the electronic term $\nabla_{x} \phi$. Also, the $L^p$-$L^q$ type estimates for the electronic term $\nabla_{x} \phi$ are established, which are not only useful in the estimate for Poisson equation, but also play a fundamentally important role in commutator estimates involving the electronic term $\nabla_{x} \phi$.

In the paper, we study an inverse problem of recovering a time-dependent convection coefficient from the measured data at an interior/boundary point in a one-dimensional nonlinear subdiffusion model with non-homogeneous boundary conditions. Due to the nonlinearity and non-homogeneous boundary conditions of the system, such an inverse problem is novel and important. We first investigate the unique existence and some regularities of the solution to forward problem by using the transposition method and the fixed point theorem. Then a conditional stability of the inverse problem is obtained based on the regularity of solution for the direct problem and some generalized Gronwall's inequalities. Finally, we transform the inverse problem into a variational problem. The existence and convergence of the regularization solution for the variational problem are proved and we use a modified Levenberg–Marquardt method to find an approximate convection coefficient function. The efficiency and accuracy of the algorithm are illustrated with two numerical examples.

We consider how distribution of eigenvalues depends on boundary conditions of a discrete Laplacian operator on lattices. We study the Laplacian with boundary conditions given by a linear combination of Dirichlet and Neumann conditions. In particular, we derive a secular equation and investigate the Laplacian operator's eigenvalues with different boundary conditions, including the interlacing property, the first eigenvalue gaps, and the monotonicity property.

In this paper, we have obtained upper bounds for the total chromatic number of some classes of Cayley graphs, odd graphs and mock threshold graphs.

We prove Sobolev-type inequalities for modified Riesz potentials of functions in Morrey spaces of an integral form over non-doubling metric measure spaces. Our results are new even for the doubling metric measure setting. In particular, our results extend the previous results in Morrey spaces of an integral form in the Euclidean case.

In this paper we are concerned with the existence of invariant curves of quasi-periodic reversible mappings with higher order degeneracy of the twist condition under the Brjuno–Rüssmann's non-resonant condition. In the proof we use a new variant of the KAM theory, containing an artificial parameter $q$, $0 \lt q \lt 1$, which makes the steps of the KAM iteration infinitely small in the speed of function $q^n \varepsilon$, rather than super exponential function.

In this paper, we study an initial boundary value problem for a $p$-Laplacian hyperbolic equation with logarithmic nonlinearity. By combining the modified potential well method with the Galerkin method, the existence of the global weak solution is studied, and the polynomial and exponential decay estimation under certain conditions are also given. Moreover, by using the concavity method and other techniques, we obtain the blow up results at finite time.

We are concerned with divergences on the Cartan–Hadamard Riemannian manifold of symmetric cones, self-dual homogeneous cones in Euclidean spaces, and related optimization problems. We introduce a parameterized version of fidelity on symmetric cones, namely sandwiched quasi-relative entropies, and construct a one-parameter family of divergences based on these entropies. We consider the median minimization problem of finite points over these divergences and establish existence and uniqueness of minimizer. The global linear rate convergence of a gradient projection algorithm for solving the median minimization problem is analyzed based on the derived upper bound of the condition number of the Hessian function.

A class of conformable fractional differential equations \[ D_x^{\alpha} D_x^{\alpha} u(x) + \omega(x) f(u(x)) = 0 \quad \textrm{on $(0,1)$} \] is considered. We first give a sufficient condition for the existence of sign-changing solutions with the prescribed number of zeros to this problem. On the basis of this result, we turn to a specific case of the above problem and give a uniqueness theorem related to the function $\omega$. Essentially, the main methods using in this work are properties of conformable fractional calculus, the scaling argument and Prüfer-type substitutions.

We investigate Bergman type operators on the complex unit ball, which are singular integral operators induced by the modified Bergman kernel. We consider the $L^p$-$L^q$ boundedness and compactness of Bergman type operators. The results of boundedness can be viewed as the Hardy–Littlewood–Sobolev (HLS) type theorem in the case unit ball. We also give some sharp norm estimates of Bergman type operators which in fact gives the upper bounds of the optimal constants in the HLS type inequality on the unit ball. Moreover, a trace formula is given.