This paper mainly discusses the self-adjointness and eigenvalue dependence of a class of second-order differential operator with internal point conditions containing an eigenparameter. First, a problem-related linear operator $T$ is defined in an appropriate Hilbert space, and the study of the problem to be transformed into the research of the operator $T$ in this space, and the operator $T$ is proved to be self-adjoint according to the definition of self-adjoint operator. In addition, on the basis of self-adjoint, it is proved that the eigenvalues are not only continuously dependent but also differentiable on each parameter of the problem, and the corresponding differential expressions are given. Meanwhile, the monotonicity of the eigenvalues with respect to the part parameters of the problem is also discussed.

In this paper, we study the product operators of Volterra operator $ V $ and Toeplitz operator $ T_\varphi $ on the classical Hardy space $ L_\varphi=T_\varphi V $ and $ R_\varphi=VT_\varphi $, and we obtain some basic properties of $ L_\varphi $ and $ R_\varphi $, we also get a result of $ L_\varphi $ which is similar to the Coburn theorem of Toeplitz operator. The necessary and sufficient conditions for $ V $ and $ T_\varphi $ to be commutative are given in this paper, and the symbol of $ L_\varphi $ and $ R_\varphi $ is characterized when $ z^jH^2\ (j=1, 2, \cdots ) $ are the common invariant subspaces of $ L_\varphi $ and $ R_\varphi $.

Introducing the concept of super-homogeneous function, firstly constructing the Hilbert-type discrete inequality with super-homogeneous kernel, and then discussing discrete operator with super-homogeneous kernel by using the relationship between Hilbert-type inequality and operator with the same kernel, obtained the construction conditions for bounded discrete operators with super-homogeneous kernel in weighted normed sequence spaces, and a method for determining whether an operator is bounded or not and the estimation of operator norm are solved.

For any $ x\in (0,1)$, let $x=[ d_{1}(x), d_{2} (x), \cdots, d_{n} (x)]$ be its Lüroth expansion. Denote the maximal digits of the first $n$ digits by $L_{n}(x)=\max \left\{d_{1}(x), \cdots, d_{n}(x)\right\}.$ For any real number $0< \alpha < \beta < \infty$, we determine the Hausdorff dimension of the exceptional set

$F_{\phi}(\alpha, \beta)=\left\{x \in(0,1): \liminf _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\alpha, \limsup _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\beta\right\},$

where $ \phi (n)= n^{\gamma} (\gamma>0)$ or $ {\rm e}^{n^{\gamma}} (\gamma>0 )$. This supplements the results of [13]. Similarly, the corresponding exceptional sets of the sums of digits in Lüroth expansion are also studied.

This paper studies inverse spectral problems for the Sturm-Liouville operator on $(0,1)$ with the Robin boundary conditions and a discontinuity at $x=d\in(0,\frac{1}{2}]$. Suppose that the known data contains one subspectrum, the potential function on $(d,1)$ as well as partial parameters in the right boundary condition and the discontinuous conditions. The paper proves the local solvability and stability for the inverse problems of recovering the potential function on $(0,d)$ and the parameter in left boundary condition, where the known potential and the parameter in the right boundary condition are allowed to contain errors.

In this paper, we consider the following Kirchhoff-type equation

$\begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\ d x\right)\Delta u=\lambda u+|u|^{p-2}u \quad \mathrm{in}\ \mathbb{R}^{3},\\ \|u\|^2_{2}=\rho,\end{cases}$

where $a$, $b$, $\rho>0$ and $\lambda\in\mathbb{R}$ arises as Lagrange multiplier with respect to the mass constraint $\|u\|^2_{2}=\rho$. When $p\in\left(2,\frac{10}{3}\right)$ or $p\in\left(\frac{14}{3},6\right)$, we establish the existence of infinitely many radial $L^2$-normalized solutions by using the genus theory. Furthermore, we testify an asymptotic behavior of the above solutions with respect to the parameter $b\rightarrow 0^+$.

This paper studies the Riemann problem of the coupled Aw-Rascle-Zhang traffic model with different pressure laws on the connected roads. Using the method of characteristic analysis and theories of phase transition, we construct the Riemann solution to the coupled Aw-Rascle-Zhang model for the ommitted case $ v_- + \eta (\rho_-)^{\gamma} = v_+ $ in reference [5], and correct Riemann solution for the case $ v_+ + \eta(\rho_-)^{\gamma} < v_+ $, which complete the work of Herty, et al. Furthermore, when the parameter of the pressure term $ \mu \to \eta $, the uniqueness and stability of the Riemann solution of the coupled Aw-Rascle model are proved.

In this paper, we consider a class of $\phi$-Laplacian Rayleigh equation, where the nonlinear term is non-autonomous and has a singularity at the origin. By applications of Mawhin's continuation theorem and some analysis methods, we prove the existence of periodic solutions to the equation with a strong singularity of repulsive type (or weak and strong singularities of attractive type).

In this paper, we study the constrained variational problem of a class of nonlinear Schrödinger-Poisson equations with Dirac potentials. Under the assumptions of certain parameters and indices, we prove the existence of constrained minimizers, and the relevant conclusions is further extended in reference [2].

This paper studies the global regularity problem for the 2D micropolar Rayleigh-Bénard convection system with velocity zero dissipation, micro-rotation velocity Laplace dissipation and temperature fractional dissipation. By introducing two combined quantities and using the technique of Littlewood-Paley decomposition, this paper establishes the global regularity result of solutions to this system.

In this paper, We demonstrates that when the initial cell mass is small, the solution of the initial boundary value problem converges at an algebraic rate to the corresponding parabolic-elliptical Keller-Segel-Stokes system during the fast signal diffusion limit process by performing appropriate energy iterative estimation on the three-dimensional parabolic-parabolic Keller-Segel-Stokes system.

In this paper, the two-dimensional Mindlin-Timoshenko plate system with local damping is studied. First, the original system is transformed into an abstract Cauchy problem, and the well posedness of the system is obtained by using operator semigroup theory. With the help of the frequency domain stability results of the linear system, the uniform exponential stability of the system is obtained by introducing geometric optical conditions and multiplier techniques.

For the linearized ARZ traffic flow model, the existing literatures are usually based on the following assumptions: First, the equilibrium state of the system is exactly equal to the speed of free flow; Second, the traffic flow entering the upstream section is exactly equal to the mathematical expectation of traffic demand; Third, the boundary feedback doesn't consider the impact of time delay factors. In this paper, a PDE-PDE infinite-dimensional coupled closed-loop system with model drift term and boundary disturbance term is established without these constraints by combining the time-delay boundary control strategy. Specifically, the closed-loop system is transformed into an abstract evolution equation by using operator semigroup theory. The well-posedness of the closed-loop system is proved by combining the admissible theory of linear system solutions and control operators. The weighting ISS-Lyapunov function is constructed, and the input-to-state stability(ISS) of the closed-loop system is proved. The dissipative conditions of the feedback parameters are obtained. The effectiveness of the proposed controller and the feasibility of the parameter conditions are further verified by numerical simulation experiments.

In this paper, we study an inverse boundary value problem for fractional elliptic equation of Tricomi-Gellerstedt-Keldysh-type. For this ill-posed problem, a conditional stability result is established. Based on the ill-posedness analysis, a fractional Tikhonov regularization method was constructed to recover the continuous dependence of the solution on the measurement data. Under the a-priori and a-posteriori selection rules for regularization parameter, the corresponding convergence results of Hölder type are derived and proved, respectively. Finally, the simulation effectiveness of the fractional Tikhonov method is verified by two numerical examples. The numerical results show that the method works stably and effectively in dealing with the inverse problem in the text.

In this paper, we investigate a periodic stage structure single-population model with infinitely distributed delay and feedback control. Firstly, a sufficient criterion for the existence of a unique globally asymptotically stable periodic solution for the auxiliary system-periodic stage structure single-population model is derived. Secondly, we establish sufficiency criteria on the permanence of the model and the existence of a positive periodic solution in the form of integral. Finally, we illustrate the results with numerical simulations.

The present study focuses on robust discriminant regression models for feature extraction, which can be rephrased as minimizing matrix trace function subject to product manifold constraints. By building upon the Zhang-Hager technique, the authors develop a Riemannian nonlinear conjugate gradient method for solving a simplified version of the reconstruction problem. The method exploits the geometric properties of the product manifold, and the global convergence analysis of the proposed algorithm is provided. Empirical results demonstrate that the proposed algorithm is effective and feasible for solving the underlying problem. In terms of iteration efficiency, the proposed algorithm outperforms the existing method, other Riemannian gradient-like algorithms and Riemannian first-order and second-order algorithms available in the MATLAB toolbox Manopt.

To provide a finite characterization of feasible solution sequences for optimization problems (OP) and variational inequality problems (VIP), an augmented set value map is introduced for the solution sets of these problems. Additionally, the concepts of augmented weak sharpness with respect to feasible solution sequences are established. These novel notions extend the traditional concepts of weak sharpness and strongly non-degeneracy relative to feasible solution sequences, addressing the limitation that solution sets often lack weak sharpness or strongly non-degeneracy in many cases. When the feasible solution sets of optimization problems and variational inequality problems exhibit augmented weak sharpness, the necessary and sufficient conditions for the finite termination of feasible solution sequences are provided for each problem. These conditions extend the corresponding results found in existing literature, where solution sets are weak sharp or strongly non-degenerate. Furthermore, sufficient conditions with fewer restrictions are provided for the finite termination of various optimization algorithms.

In this paper, An inertial viscosity-type algorithm is proposed to solve proximal split feasibility problems in Hilbert spaces. In this algorithm, a non-Lipschitz stepsize rule is given, which overcomes the drawback that the stepsize tends to zero. Further, a strong convergence theorem for our proposed algorithm is established without Lipschitz continuity of the gradient operators. As theoretical applications, the split equilibrium problem is investigated. Finally, numerical experiments are provided for demonstration and comparison.

Based on a convex constrained nonlinear monotone equations equivalent to the split feasibility problem, in this paper, a novel inertial conjugate gradient projection method is proposed. The presented method does not calculate the maximum eigenvalue of the matrix ${A^\top}A$ and the complex projections of multiple times. Under mild conditions, the global convergence of the proposed method is proved, and its rate of convergence is analyzed. Numerical experiments show that the proposed method is efficient and robust.

In this paper, we present a new golden ratio primal-dual algorithm to solve the nonsmooth saddle point problems, which is full-splitting.Under some appropriate conditions, we prove the sequence generated by the algorithm iteration converges to the solution of the problem, as well as an $ O(1/N) $ ergodic convergence rate result. Finally, with comparisons to Zhu, Liu and Tran-Ding's algorithms, we give some numerical experiments to show the less iterate numbers and CPU time of the proposed method.

Based on the energy conversion mechanism of wind power street lamps, this paper constructed and analyzed a fluid model driven by the M/M/1 queueing system with set-up time and complete failure. Firstly, the driving system is described and the stationary probability distribution of the driving system is obtained by using the matrix-geometry method. Secondly, the differential and difference equations of the fluid level in steady-state conditions are obtained based on the net input rate structure of the fluid model and using the probability analysis method. Then, the expected buffer content and the probability of the empty buffer under steady-state conditions are obtained by using the Laplace-Stieltjes transform(LST) method. The cost function of the system is constructed according to the performance index. Finally, the influence of parameters changing on the performance indicators and cost function are illustrated in numerical analysis.

In this paper, we consider a sequence of non-negative dependent and not necessarily identically distributed random variables with local long-tailed marginal distributions and Bernstein copula and study the local asymptotic behavior of the tail of their partial sum and maximum. Then, under a suitable condition for local subexponentiality, we obtain the local max-sum equivalence. The result indicates that the big-jump principle of random walks remains valid in its local version under more general dependency assumptions. The numerical experimental results under different parameter settings further validate the stability and feasibility of the obtained results.