We mainly study the robust stability for linear time-varying systems within the framework of nest algebra. We consider the robust stability when the system and controller are subject to independent uncertainties measured by the gap metric, and a sufficient condition is obtained by using the trigonometric structure of the graphs about the plant and the controller. Furthermore, we also obtain some sufficient conditions for the simultaneously robust stability of several linear time-varying systems. The numerical example shows that our conclusion is effective.
In this paper, we consider the periodic Cauchy problem of an integrable two-component Camassa-Holm system, which can be regarded as a two-component extension of the modified Camassa-Holm equation. First, the periodic peakons are obtained in explicit formulas. Then the precise blow-up scenarios of strong solutions and several conditions on the initial data that produce blow-up of the induced solutions are described in detail.
By making use of the Nevanlinna theory and difference Nevanlinna theory of several complex variables, we investigate several Fermat type systems of partial differential difference equations, and obtain a series of results about the existence and the forms of entire solutions of such systems, which are some improvements and generalization of the previous results given by Cao, Gao, Liu et al. We also give some examples to show that there exists significant differences in the forms of transcendental entire solutions with finite order of the systems of the equations with between several complex variables and single complex variable.
We introduce a class of order-theoretic approaches for studying the fractional differential variational inequalities. By using the order-theoretic fixed point the-orems, we prove the existence of maximal solution and minimal solution to fractional differential variational inequalities on Hilbert lattices, and obtain some new results. Our order-theoretic approaches adopted for this kind of problems are fundamentally different from the recent literatures, in which the main tools are the topological fixed point theorems and discrete approximation methods. These order-theoretic methods can effectively weaken the continuity of the involved mappings.
We mainly determine the compatible left-symmetric algebra structures on the deformed bms_{3} algebra with some natural grading conditions by classified the compatible left-symmetric algebra structures on the deformed bms_{3} algebra.
In this paper, using the minimal faithful representation of Q_{n}, we characterize some subgroups of automorphism groups of Q_{n}, including inner automorphism groups, central automorphism groups, involution automorphism groups and outer automorphism groups.
By using Moser's twist theorem, under some smoothness conditions, we prove the existence of infinitely many invariant tori and so the Lagrange stability for the sublinear asymmetric Duffing equations x"+a(x+)1/3-b(x^{-})1/3+φ(x)=p(t), where the perturbation term φ(x) is bounded, while the forced term p(t) is periodic in t.
In this paper, the method (or technique) of bi-spaces contractive semigroup method(see Theorem 2.10) is presented in a general setting as an alternative approach to the study the asymptotic behavior of nonlinear evolutionary equation. As an application, we consider reaction-diffusion equations with fading memory, and prove the asymptotic compactness of the semigroup on H_{0}^{1}(Ω)×L_{μ}^{2}(R; H_{0}^{1}(Ω)) with initial data in L^{2}(Ω)×L_{μ}^{2}(R; H_{0}^{1}(Ω)). Thus the bi-spaces global attractor A is confirmed. Furthermore, by using the new decomposition technique, we demonstrate the asymptotic regularity of the solution to obtain the contractive function. It is noteworthy that the nonlinearity f satisfies the polynomial growth of arbitrary order and bi-spaces global attractor A⊂D(A)×L_{μ}^{2}(R; D(A)).
We provide a method for solving inverse Sturm-Liouville problem on the double loop graph. We deduce asymptotic of eigenvalues for double loop graph with the standard matching condition at the contact vertex, and then reconstruct the unknown potential by the Riesz basis constructed from the subspectrum, and finally present the uniqueness theorem and reconsbruction algorithm.
The researches on the alternating direction method of multiplier method (ADMM) for solving two-block optimization have been gradually mature and perfect. However, the studies on ADMM for solving nonconvex multi-block optimization are relatively few. In this paper, we first propose a partially symmetric regularized ADMM for nonconvex multi-block optimization with linear constraints. Second, under appropriate assumptions including the region of the two parameters in the updating formulas for the multiplier, the global convergence of the proposed method is proved. Third, when the augmented Lagrangian function satisfies the Kurdyka-Lojasiewicz (KL) property, the strong convergence of the method is proved. Furthermore, when the associated KL property function has a special structure, the sublinear and linear convergence rate of the method are obtained. Finally, some preliminary numerical experiments are carried out, and this shows that the proposed method is numerically effective.
We translate a recurrence relation into a generating function and expand the product of two bivariate Eulerian polynomials in terms of the complete homogeneous symmetric functions.
In this paper, the new concept of tripled coincidence point and weakly compatible for a pair of mappings F:X×X×X→X, g:X→X in cone bmetric spaces are introduced. Under not necessary normal conditions of cone, some tripled coincidence for contractive mappings and tripled common fixed point problems of weakly compatible mappings are studied. The obtained results generalize some coupled common fixed point theorems in corresponding literatures. Finally, an example is given to illustrate our main results.
An automorphism α of a group G is called class preserving if α(g) ∈ g^{G} for all g ∈ G, where g^{G} denotes the conjugacy class of g in G. These automorphisms form a normal subgroup of the automorphism group Aut(G) of G which we denote by Aut_{c}(G). In this paper, an upper bound of the order of Aut_{c}(G) of an extension of a finite cyclic group by a finite nilpotent group is determined. Meanwhile, an upper bound of the order of Aut_{c}(G) of an extension of a finite p-group by a finite nilpotent group is given.