Boundary Harnack principle for subordinate Brownian motions

We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κ$\kappa$-fat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κ$\kappa$-fat open sets with respect to these processes with their Euclidean boundaries.     

On the asymptotic behaviour of Lévy processes,Part I: Subexponential and exponential processes

We study tail probabilities of the suprema of Lévy processes with subexponential or exponential marginal distributions over compact intervals. Several of the processes for which the asymptotics are studied here for the first time have recently become important to model financial time series. Hence our results should be important, for example, in the assessment of financial risk.     

Approximation of analytic functions by Legendre functions

We will solve the inhomogeneous Legendre’s differential equation and apply this result to estimate the error bound occurring when an analytic function is approximated by an appropriate Legendre function.     

Hopf bifurcation in symmetric configuration of predator-prey-mutualist systems

In this paper we apply the equivariant degree method to study Hopf bifurcations in a system of differential equations describing a symmetric predator-prey-mutualist model with diffusive migration between interacting communities. A topological classification (according to symmetry types), of symmetric Hopf bifurcation in configurations of populations with D₈, D₁₂, A₄ and S₄ symmetries, is presented with estimation on minimal number of bifurcating branches of periodic solutions.     

Weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type

This paper studies suitable sufficient conditions to ensure the existence and uniqueness of weighted pseudo-almost periodic solutions to a neutral delay integral equation of advanced type introduced by T.A. Burton in the literature. The abstract results are then utilized to characterize weighted pseudo-almost periodic solutions to the well-known logistic equation.     

Asymptotically strict pseudocontractive mappings in the intermediate sense

It is proved that the modified Mann iteration process: _xn+1=(1−_αn)_xn+_αnTⁿ_xn,n∈N$x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_n{T}^n{x}_n,n\in \mathbb{N}$, where {_αn}$\left\{{\alpha }_n\right\}$ is a sequence in (0, 1) with δ≤_αn≤1−κ−δ$\delta \le {\alpha }_n\le 1-\kappa -\delta$ for some δ∈(0,1)$\delta \in (0,1)$, converges weakly to a fixed point of an asymptotically κ$\kappa$-strict pseudocontractive mapping T$T$ in the intermediate sense which is not necessarily Lipschitzian. We also develop CQ method for this modified Mann iteration process which generates a strongly convergent sequence.     

Nonexistence of positive weak solutions of elliptic inequalities

In this paper we give sufficient conditions for the nonexistence of positive entire weak solutions of coercive and anticoercive elliptic inequalities, both of the p$p$-Laplacian and of the mean curvature type, depending also on u$u$ and x$x$ inside the divergence term, while a gradient factor is included on the right-hand side. In particular, to prove our theorems we use a technique developed by Mitidieri and Pohozaev in [E. Mitidieri, S.I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001) 1–362], which relies on the method of test functions without using comparison and maximum principles. Their approach is essentially based first on a priori estimates and on the derivation of an asymptotics for the a priori estimate. Finally nonexistence of a solution is proved by contradiction.