In this paper, we show that general nonlinear switched systems are asymptotically controllable if and only if there exist control-Lyapunov functions for their relaxation systems. If the switching signal is dependent on the time, then the control-Lyapunov functions are continuous. And if the switching signal is dependent on the state, then the control-Lyapunov functions are -smooth. We obtain the results from the viewpoint of control system theory. Our approach is based on the relaxation theorems of differential inclusions and the classic Lyapunov characterization.

The purpose of this paper is to derive a perturbation theory of evolution systems of the hyperbolic second order hyperbolic equations. We give an example of a partial functional equation as an application of the preceding result in case of the mixed problems for hyperbolic equations of second order with unbounded principal operators.

In this paper, we prove that infinite-dimensional vector spaces of -dense curves are generated by means of the functional equations f(x)+f(2x)++f(nx) = 0, with , which are related to the partial sums of the Riemann zeta function. These curves -densify a large class of compact sets of the plane for arbitrary small , extending the known result that this holds for the cases n = 2, 3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the power of the density approaches the Jordan content of the compact set which the curve densifies.

This paper is concerned with nonlinear evolution differential equations with infinite delay in Banach spaces. Using Kato's approximating approach, existence and uniqueness of strong solutions are obtained.

In this paper, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive p-Laplace equation , 1 < p < 2, in a bounded domain with . More precisely, it is shown that if q > p-1, any solution vanishes in finite time when the initial datum or the coefficient a or the Lebesgue measure of the domain is small, and if 0 < q < p-1, there exists a solution which is positive in for all t > 0. For the critical case q = p-1, whether the solutions vanish in finite time or not depends crucially on the value of , where and is the unique positive solution of the elliptic problem -div() = 1, ; =0, . This is a main difference between equations with local and nonlocal sources.

Let X be a divergence-free vector field defined on a closed, connected Riemannian manifold. In this paper, we show the equivalence between the following conditions: X is a divergence-free vector field satisfying the shadowing property. X is a divergence-free vector field satisfying the Lipschitz shadowing property. X is an expansive divergence-free vector field. X has no singularities and is Anosov.

We prove that if is large and is small enough, then every approximate zero of power series equation =0 can be approximated by a true zero within a good error bound. Further, we obtain Hyers-Ulam stability of zeros of the polynomial equation of degree n, + + + + = 0 for a given integer n > 1.

During the last decade, several papers have focused on linear q-difference equations of the form with entire or meromorphic coefficients. A tool for studying these equations is a q-difference analogue of the lemma on the logarithmic derivative, valid for meromorphic functions of finite logarithmic order . It is shown, under certain assumptions, that = max + 1. Moreover, it is illustrated that a q-Casorati determinant plays a similar role in the theory of linear q-difference equations as a Wronskian determinant in the theory of linear differential equations. As a consequence of the main results, it follows that the q-gamma function and the q-exponential functions all have logarithmic order two.

A nonlinear elliptic problem involving p-Laplacian and nonlinear boundary condition is considered in this paper. By using the method of Nehari manifold, it is proved that the system possesses two nontrivial nonnegative solutions if the parameter is small enough.

In this paper, we study the second approximate Matsumoto metric F = on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.

In this paper, a new proof of the following result is given: The product of the volumes of an origin-symmetric convex bodies K in and of its polar body is minimal if and only if K is a parallelogram.

In this paper, we investigate the error estimates of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the order k = 1 Raviart-Thomas mixed finite element and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of order in the -norm and order h in the -norm for the control variable are proved.

We show that the recurrence rates of Laurent series about continued fractions almost surely coincide with their pointwise dimensions of the Haar measure. Moreover, let be the set of points with lower and upper recurrence rates , (), we prove that all the sets , are of full Hausdorff dimension. Then the recurrence sets have constant multifractal spectra.

In this article, we introduce and study the Gorenstein cotorsion dimension of modules and rings. It is shown that this dimension has nice properties when the ring in question is left GF-closed. The relations between the Gorenstein cotorsion dimension and other homological dimensions are discussed. Finally, we give some new characterizations of weak Gorenstein global dimension of coherent rings in terms of Gorenstein cotorsion modules.

In this paper, we prove the global existence and uniqueness of the dissipative Kirchhoff equation with nonlinear boundary damping by Galerkin approximation benefited from the ideas of Zhang et al. [33]. Furthermore,we overcome some difficulties due to the presence of nonlinear terms and by introducing a new variables and we can transform the boundary value problem into an equivalent one with zero initial data by argument of compacity and monotonicity.

The aim of this paper is to prove the next result. Let n > 1 be an integer and let R be a n!-torsion free semiprime ring. Suppose that f : R R is an additive mapping satisfying the relation [f(x), ] = 0 for all . Then f is commuting on R.

We prove that a semi-symmetric 3-dimensional gradient Ricci soliton is locally isometric to a space form , , (Gaussian soliton); or a product space , , where the potential function depends only on the nullity.

Our goal is to relax a sufficient condition for the exponential almost sure stability of a certain class of stochastic differential equations. Compared to the existing theory, we prove the almost sure stability, replacing Lipschitz continuity and linear growth conditions by the existence of a strong solution of the underlying stochastic differential equation. This result is extendable for the regime-switching system. An explicit example is provided for the illustration purpose.

We enumerate black and white colored d-ary trees with no leftmost -edges, which is a generalization of hybrid binary trees. Then the multi-colored hybrid d-ary trees with the same condition is studied.

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators , where = , F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space . We also show some necessary conditions for the hyponormality of with on .

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.

An upstream scheme based on the pseudostress-velocity mixed formulation is studied to solve convection-dominated Oseen equations. Lagrange multipliers are introduced to treat the trace-free constraint and the lowest order Raviart-Thomas finite element space on rectangular mesh is used. Error analysis for several quantities of interest is given. Particularly, first-order convergence in norm for the velocity is proved. Finally, numerical experiments for various cases are presented to show the efficiency of this method.

In this paper, we prove a necessary optimality theorem for a nonsmooth optimization problem in the face of data uncertainty, which is called a robust optimization problem. Recently, the robust optimization problems have been intensively studied by many authors. Moreover, we give examples showing that the convexity of the uncertain sets and the concavity of the constraint functions are essential in the optimality theorem. We present an example illustrating that our main assumptions in the optimality theorem can be weakened.