Favardʼs theorem of piecewise continuous almost periodic functions and its application

In this paper, by constructing Bochner–Fejér polynomials for piecewise continuous almost periodic functions (PCAP, for short), the authors establish Favard?s theorem of PCAP functions, which illustrates when the primitive function of PCAP function is a PCAP function. As its application, combining coincidence degree theory, we consider the existence of PCAP solution of impulsive single population model with hereditary effects. To our best knowledge, it is the first time when coincidence degree theory is used to study the existence of PCAP solution of impulsive differential equation.     

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.     

Smooth properties for semigroups of Lévy processes and their application

Explicit smooth properties for the semigroup of Lévy processes are derived in terms of its symbol. As an application, we obtain new sufficient conditions for the strong Feller property of stochastic differential equations driven by the additive Lévy process.     

Generalized Moser–Trudinger inequality for unbounded domains and its application

We give a version of the Moser–Trudinger inequality for Orlicz–Sobolev spaces embedded into exponential and multiple exponential spaces on unbounded domains in ${\mathbb R^n, n \geq 2}$ . Applying this result and the Mountain Pass Theorem we study the existence of non-trivial weak solutions to the problem $$\begin{array}{ll}u \in W^1 L^{\Phi}(\mathbb R^n)\quad{\rm and}\\\quad -{\rm div} \left(\Phi ' (|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)+V(x)\Phi'(|u|)\frac{u}{|u|} =f(x,u)\quad{\rm in}\, \mathbb R^n,\end{array}$$ where Φ is a Young function such that the space ${W^1 L^{\Phi}(\mathbb R^n)}$ is embedded into an Orlicz space of the exponential or multiple exponential type, the nonlinearity f(x, t) has the corresponding critical growth and V(x) is a continuous potential.     

The Blok–Ferreirim theorem for normal GBL-algebras and its application

Generalized basic logic algebras (GBL-algebras for short) have been introduced in [JT02] as a generalization of Hájek’s BL-algebras, and constitute a bridge between algebraic logic and ℓ-groups. In this paper we investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum decomposition theorem. This result allows us to derive many interesting consequences, such as the decidability of the universal theory of commutative GBL-algebras, the fact that n-potent GBL-algebras are commutative, and a representation theorem for finite GBL-algebras as poset sums of GMV-algebras, a result which generalizes Di Nola and Lettieri’s [DL03] representation of finite BL-algebras. Presented by J. G. Raftery. Received May 23, 2007; accepted in final form February 20, 2008.     

Weak KAM theory for discounted Hamilton–Jacobi equations and its application

Weak KAM theory for discounted Hamilton–Jacobi equations and corresponding discounted Lagrangian/Hamiltonian dynamics is developed. Then it is applied to error estimates for viscosity solutions in the vanishing discount process. The main feature is to introduce and investigate the family of \(\alpha \)-limit points of minimizing curves, with some details in terms of minimizing measures. In error estimates, the family of \(\alpha \)-limit points is effectively exploited with properties of the corresponding dynamical systems.     

Stochastic equations and ergodicity for two-type continuous-state branching processes with immigration in Lévy random environments

This paper establishes a stochastic differential equation system with both positive and negative jumps and proves the existence and uniqueness of the strong solution and presents an equivalent condition for ergodicity of the solution. The strong solution is called two-type continuous-state branching processes with immigration in Lévy random environments. The model can be extended to any finite dimensional case.     

Existence,uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection.The existence criterion of the process is firstly considered.We prove that if the sum of the resurrection rates is finite,then there does not exist any process.An existence criterion is then established when the sum of the resurrection rates is infinite.Some equivalent criteria,possessing the advantage of being easily checked,are obtained for the latter case.The uniqueness criterion for such process is also investigated.We prove that although there exist infinitely many of them,there always exists a unique honest process for a given q-matrix.This unique honest process is then constructed.The ergodicity property of this honest process is analysed in detail.We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.     

The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes

We use Bochner’s subordination to give a representation of the genealogical structure associated with general continuous-state branching processes. We then apply this representation to connections between a branching process introduced by Neveu, and the coalescent process recently investigated by Bolthausen-Sznitman and others. Received: 25 March 1999 / Revised version: 13 September 1999 /?Published online: 11 April 2000     

A note on monotone mean–variance preferences for continuous processes

We extend a recent result of Trybuła and Zawisza (2019), who investigate a continuous-time portfolio optimization problem under monotone mean–variance preferences. Their main finding is that the optimal strategies for monotone and classical mean–variance preferences coincide in a stochastic factor model for the financial market. We generalize this result to any model for the financial market where asset prices are continuous.     

The first eigenvalue for a quasilinear Schrödinger operator and its application

In this paper, we study the first eigenvalue for a quasilinear Schrödinger operator, which is greater than the first eigenvalue of the usual laplacian operator. As an application we treat a quasilinear resonance problem involving a subcritical growth perturbation.     

Functional ergodic limits for occupation time processes of site-dependent branching Brownian motions in ℝ

We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ(·). We focus on the functional ergodic limits for the occupation time processes of the models in ?. It is proved that the limiting process has the form of λξ(·), where λ is the Lebesgue measure on ? and ξ(·) is a real-valued process which is non-degenerate if and only if σ is integrable. When ξ(·) is non-degenerate, it is strictly positive for t > 0. Moreover, ξ converges to 0 in finite-dimensional distributions if the integral of σ tends to infinity.     

An existence theorem for a generalized self-dual Chern–Simons equation and its application

In this paper, we establish an existence theorem for a generalized self-dual Chern–Simons equation over a doubly periodic domain and use the existence theorem to prove the existence of doubly periodic self-dual vortices in a Maxwell–Chern–Simons model with non-minimal coupling. We find a necessary and sufficient condition for the existence of solutions of the generalized Chern–Simons equation. We prove the existence result by using two methods, a super- and sub-solution method and a constrained minimization method. Our main contribution is that we find a general inequality-type constraint by using the second method and it maybe applied to some related problems with the similar structures.     

Wiener–Ingham type inequality for Vilenkin groups and its application to harmonic analysis

An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums \(f(x)=\sum A_ke^{in_kx}\), where \(A_k\)’s are complex numbers, \(n_{-k}=-n_k\) and \(\{n_k\}\) satisfies the small gap condition \((n_{k+1}-n_k)\ge q\ge 1\) for \(k=0,1,2,\ldots \), says that if I is any subinterval of \([-\pi ,\pi ]\) of length \(|I|=2\pi (1+\delta )/q>2\pi /q\) then \(\sum |A_k|^2\le A_{\delta }|I|^{-1}\int _I|f|^2\), \(|A_k|\le A_{\delta }|I|^{-1}\int _I|f|\), wherein \(A_{\delta }\) depends only on \(\delta \). Such an inequality is proved here in the setting of the Vilenkin groups G. The inequality is then applied to generalize the Bernstěin, Szász and Ste?hkin type results concerning the absolute convergence of Fourier series on G.     

An application of Furuta inequality and its best possibility

This paper shows the best possibility for outer exponents of some inequalities under some conditions, and a counter example is obtained.     

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund's result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.