On a Lyapunov-type inequality and the zeros of a certain Mittag–Leffler function

In this work, a Lyapunov-type inequality is obtained for the case when one is dealing with a fractional differential boundary value problem. We then use that result to obtain an interval where a certain Mittag–Leffler function has no real zeros.     

Asymptotic convergence of solutions for Laplace reaction–diffusion equations

We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their $\omega$-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.     

Existence,compactness, and connectedness of solution sets to nonlocal and impulsive Cauchy problems

In this paper, by using some well-known inequalities and Schaefer fixed point theorem, we show existence results for impulsive Cauchy problems with nonlocal conditions. The compactness of solution sets can be shown in some certain conditions. Moreover, connectedness of solution sets to impulsive Cauchy problems is shown.     

Explicit H1-Estimate for the Solution of the Lamé System with Mixed Boundary Conditions

In this paper we consider the Lamé system on a polygonal convex domain with mixed boundary conditions of Dirichlet-Neumann type. An explicit L² norm estimate for the gradient of the solution of this problem is established. This leads to an explicit bound of the H¹ norm of this solution. Note that the obtained upper-bound is not optimal.     

Asymptotic analysis of risk quantities conditional on ruin for multidimensional heavy-tailed random walks

In this paper we consider a multidimensional renewal risk model with regularly varying claims. This model may be used to describe the surplus of an insurance company possessing several lines of business where a large claim possibly puts multiple lines in a risky condition. Conditional on the occurrence of ruin, we develop asymptotic approximations for the average accumulated number of claims leading the process to a rare set, and the expected total amount of shortfalls to this set in finite and infinite horizons. Furthermore, for the continuous time case, asymptotic results regarding the total occupation time of the process in a rare set and time-integrated amount of shortfalls to a rare set are obtained.     

Harnack inequalities for SDEs driven by subordinate Brownian motions

By using regularization approximations of the underlying subordinator and a gradient estimate approach, the dimension-independent Harnack inequalities are established for the inhomogeneous semigroup associated with a class of SDEs with Lévy noise containing a subordinate Brownian motion. Our estimates in Harnack type inequalities improve the corresponding ones in the recent paper by Wang and Wang (2014) [10].     

On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension

In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold of even complex dimension satisfies the inequality , where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T ²×N, where T ² is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases.     

Bounds for the Characteristic Functions of the System of Monomials in Random Variables and of Its Trigonometric Analogue

Using a multidimensional analogue of Vinogradov's inequality for a trigonometric integral, the upper bounds are constructed for the moduli of the characteristic functions both of the system of monomials in components of a random vector with an absolutely continuous distribution in and of the system where is uniformly distributed in [0, l]^s.     

Characterization of chaotic order and its application to furuta inequality

In this note, we give a simple characterization of the chaotic order among positive invertible operators on a Hilbert space. As an application, we discuss Furuta's type operator inequality.