Referring to the properties of Fourier transform, the authors find the uncertainty principle of discrete fractional Fourier transform and uncertainty principle of continuous fractional Fourier transform under Lebesgue measure, which makes the uncertainty principles of fractional Fourier transform more general.

The symmetric realizations of the product of two general regular quasi-differential expressions in Hilbert space are investigated. The two-point boundary conditions which determine symmetric operators are characterized and a sufficient and necessary condition for the product of two higher-order regular differential operators to be symmetric is obtained. The presented result contains the self-adjoint do-main characterization as a special case. Several examples of regular symmetric product operators are given.

The investigation on the numerical range of the complex Volterra operator on Hardy Hilbert space has always been a hot topic for mathematicians, which has not been solved. In this paper, we present the formulas for the numerical radius of an unilateral weighted shift operator with weights $ (h, k, j, b, a, b, a,\cdots ) $, where $ a, b, h, k, j> 0 $. In particular, we apply the above result to calculate the numerical range of the complex Volterra operator on Hardy Hilbert space. These results can not only effectively facilitate further study of the numerical ranges of weighted shift operators with disturbed periodic weights and harmonic weights, but also provide typical examples for the numerical range of bounded linear operators on Hardy Hilbert space.

In this paper, the compactness of commutator $[\vec{b},T]$ generated by the $m$-linear $\omega$-type C-Z operator and generalized Campanato space function $\vec{b}$ with variable growth conditions on generalized Morrey space is studied. A sufficient condition for the compactness of commutator $[\vec{b},T ]$ is obtained on the generalized Morrey space.

Using the Nabla integral on time scales, this paper establishes the monotonicity rules for the ratios of parametric Nabla integrals

and the ratios of the parametric Nabla integrals with variable limits

In the part of monotonicity rules for the ratios of parametric Nabla integrals, some different special cases are considered in detail, including the ratio of two polynomials on time scales and the ratio of two Nabla Laplace transforms. Using these monotonicity rules, the monotonicity of the functions $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}{u_i}(s)}{n \mathcal{J}{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{J}{v}(u_is)}{n \mathcal{J}{v}(\bar{u}s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n K_{u_i}(s)}{n K_{\bar{u}}(s)} $, $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}{u_i}(s)}{n \mathcal{Y}{\bar{u}}(s)} $ and $ s\mapsto\frac{\sum\limits_{i=1}^n \mathcal{Y}{v}(u_is)}{n \mathcal{Y}{v}(\bar{u}s)} $ is proved, where $ \bar{u}=\sum\limits_{i=1}^n u_i/n $, $ I_u(\cdot), K_u(\cdot) $ are the modified Bessel functions of the first and second kind, respectively, $ \mathcal{J}_u(s):= \big( \frac{s}{2} \big)^{-u} I_{u}(s) $ and $ \mathcal{Y}_u(s):=K_u(s)-K_0(s) $.

Most known results on the solutions of iterative functional equations are about their continuity and smoothness. However, concerning the study of conjugation and linearization in functional equations and dynamical systems, the property of homeomorphism play an important role in these theories. Although partial results were obtained, a full description is still open. In this paper, we first consider homeomorphic solutions to a general iterative functional equation by the methods of construction and approximation. Then, the obtained results are used to study the solutions of polynomial-like iterative functional equations.

To the best of the authors' knowledge, almost no literature focuses on the almost automorphic dynamics to Schrödinger or Klein-Gordon equations. This paper gives some results on almost automorphic weak solutions to a nonlocal Laplacian saturating Schrödinger-Klein-Gordon equations by employing a mix of Galerkin method, Laplace transform, Fourier series and Picard iteration. Beyond that, global exponential convergence of the equations is investigated.

In this paper, we consider the Riemann problem for a system arising in chemotaxis. The system is of mixed type and transitions from a hyperbolic to an elliptic region. It is linearly degenerated along the $v$-axis. We obtain a solvable domain in the phase plane for the existence of Riemann solutions when the initial left and right states are in different regions. In particular, when the left state is fixed in the second quadrant and any right state is in the first quadrant, the initial value problem of this mixed-type system is Riemann solvable.

This paper is mainly concerned with almost automorphy for a class of finite delay differential equations $ u'(t)=Au(t)+Lu_t+f(t,u_t),\ t\in \mathbb {R} $ on a Banach space $ X $, where $ A $ is a Hille-Yosida operator with the domain being not dense, $ L $ is a bounded linear operator, and $ f $ is a binary $ S^p$-almost automorphic function. Compared with the previous research results, we do not require the semigroup generated by the Hille-Yosida operator to be compact, and only under weaker Lipschitz hypothesis of $ f $ and $ S^p$-almost automorphy hypothesis, which is weaker than almost automorphy, of $ f $, the solution of the above delay differential equation is showed to be compact almost automorphic (stronger than almost automorphic). Moreover, the abstract results are applied to a class of partial differential equations arising in age-structured models.

In this paper, by Kato's theory of semigroup and the definition of the dissipative operator, we study the initial value problem of a modified Camassa-Holm equation with cubic nonlinearity, and establish the existence and uniqueness of its solutions in Sobolev space $ H^{s,p}(\mathbb{R}) $, $ s\ge 1 $, $ p\in (1,\infty ) $, which extend the well-posedness of its solutions in Besov space $ B_{p,r}^{s}(\mathbb{R}) $ $ (p,\ r\ge 1,\ s> \max \{2+\frac{1}{p},\frac{5}{2}\}) $ obtained by Fu et al. (J Differ Equations, 2013, 255: 1905-1938).

The existence of solutions to Dirichlet problems for a class of singular superlinear $k$-Hessian systems with parameters is studied. Based on the Krasonsel'skii type fixed point theorem in a Banach space, the existence, multiplicity and nonexistence results of nontrivial radial solutions are obtained. At the same time, the asymptotic behavior dependent on parameter is discussed.

In this paper, we consider the existence of multiple solutions of the following multi-critical nonlocal elliptic equations with magnetic field

where $\Omega$ is bounded domain with smooth boundary in $\mathbb{R}^N$, $N\geq4$, i is imaginary unit, $2^*_s=\frac{N+\alpha_s}{N-2}$ with $N-4$<$\alpha_s$<$N, s=1,2,\cdots,k$ $(k\geq2)$, $\lambda$>0 and $2\leq p$<$2^*=\frac{2N}{N-2}$. Suppose the magnetic vector potential $A(x)= (A_1(x), A_2(x),\cdots, A_N(x))$ is real and local Hölder continuous, we show by the Ljusternik-Schnirelman theory that our problem has at least ${\rm cat}_\Omega(\Omega)$ nontrivial solutions for $\lambda$ small.

In this paper, we study the multiplicity of solutions for nonhomogeneous quasilinear Schrödinger equation with coercive potential. We obtain two different solutions by applying the Mountain Pass Theorem and Ekeland's Variational Principle. The results in this paper extend and complement previously known results to the quasilinear equations.

In this paper, we study the long-time dynamical behavior of solutions for the nonclassical diffusion equation with time-dependent memory kernel in the time-dependent space $H_{0}^1(\Omega)\times L_{\mu_{t}}^2(\mathbb R^+; H_{0}^1(\Omega))$. Under the new theorical framework, the well-posedness of the solution, the existence and the regularity of the time-dependent global attractors are proved by using the delicate integral estimation method and decomposition technique.

In this paper, we establish the monotonicity and symmetry of singular solutions of semilinear mixed local-nonlocal elliptic equations by moving planes method.

In this paper, by employing the Picard operator and dynamic inequalities, we investigate Hyers-Ulam-Rassias stability of a class of first-order nonlinear dynamic equations on time scales, which is more general than the equations discussed in the references. Three examples are presented to illustrate the applications of the conclusions.

In this paper, we study the existence and orbital stability of standing waves for a class of nonhomogeneous nonlinear Schrödinger equations under mass subcritical conditions. By means of a variational principle, we discuss the compactility of minimization sequence of constrained variational problems. From this, we obtain the existence of standing waves and prove the orbital stability of standing waves.

With the help of the physics-informed neural networks (PINNs), the forward and inverse problems of extended fifth-order mKdV(emKdV) equation are tackled, and the dynamic behaviors of solitons are also analyzed and simulated in this paper. The hyperbolic tangent function $\tanh$ is selected as the activation function to solve the one, two and three-soliton solutions of the equation. Moreover, the data-driven solutions obtained by PINNs method are compared with the exact solution given by the simplified Hirota method. Specifically, the accuracy of one-soliton solution is $\mathcal{O}(10^{-4})$, and the accuracy of the two-soliton and three-soliton solutions is $\mathcal{O}(10^{-3})$. For the inverse problem, the coefficients of the equation are discovered by the data of one, two and three-soliton solutions, respectively. Meanwhile, the robustness of the PINNs algorithm is explored under different noises. The accuracy of the data-driven coefficients can reach $\mathcal{O}(10^{-3})$ or $\mathcal{O}(10^{-2})$ respectively, when 1% initial noise or observation noise is added to the training data. And the prediction accuracy can still reach $\mathcal{O}(10^{-2})$ even if 3% initial noise or observation noise is added. According to the analysis of experimental data, the impact of observation noise on PINNs model is slightly greater than the initial noise.

Based on the Bouligand tangent cone, Clarke tangent cone and their corresponding normal cones, the optimality theories of the non-negative group sparse constrained optimization problem are studied. This paper defines the Bouligand tangent cone and its normal cone and the Clarke tangent cone and its normal cone of the non-negative group sparse constraint set, and presents their equivalent characterizations. Under the assumption that the objective function is continuously differentiable, with the help of the tangent cone and the normal cone of the sparse constrained set of the non-negative group, the definitions of four types of stable points for the optimization problem are given and the relationships between these four types of stable points are discussed. Finally, the first-order and second-order optimality conditions for the optimization problem of sparse constraint of non-negative groups are established.