The Paley-Wiener theorem for the Jacobi transform and the local Huygens' principle for root systems with even multiplicities

This note is a continuation of the previous paper by same authors [this issue]. Its purpose is to extend the results to the context of root systems with even multiplicities. Under the even multiplicity assumption, we prove a local Paley-Wiener theorem for the Jacobi transform and the strong Huygens' principle for the wave equation associated with the modified compact Laplace operator.     

A new proof of a theorem on optimal control of switched systems

Recently Sorin C. Bengea and Raymond A. DeCarlo [Sorin C. Bengea, Raymond A. DeCarlo, Optimal control of switching systems, Automatica J. IFAC 41 (2005) 11-27] have offered a key result that the set of trajectories of the two-switched system is dense in the set of trajectories of the embedded system. This result was proven by means of relaxed controls and the Chattering Lemma. In this paper we use the Lyapunov theorem to give a new simple proof.     

Multiplicity results for some elliptic systems near resonance with a nonprincipal eigenvalue

Some multiplicity results of solutions are obtained for a class of elliptic systems which are near resonance with a nonprincipal eigenvalue by the classical saddle point theorem and a local saddle point theorem in critical point theory.     

On a three critical points theorem

In this paper, using a recent result by J. Saint Raymond ([6]), we improve the three critical points theorem established in [5].     

On Shimura, Shintani and Eichler-Zagier correspondences

In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.

     

Existence and multiplicity of solutions for semilinear hemivariational inequalities at resonance

In this paper we study semilinear hemivariational inequalities at resonance. Assuming a partially anticoercive nonsmooth potential we prove an existence and a multiplicity theorem. The method of the proof is variational and in the multiplicity result, we employ the nonsmooth local linking theorem.     

Commutative Standard Table Algebras with at Most One Nontrivial Multiplicity

We classify commutative standard table algebras (STA) with at most one nontrivial multiplicity. The main result shows that there exists exactly one nontrivial multiplicity if and only if the table basis is the wreath product of a two-dimensional subalgebra and an abelian group. The theorem applies to adjacency algebras of commutative association schemes with exactly one primitive idempotent matrix of rank greater than one. A theorem of Seitz that characterizes finite groups with exactly one irreducible representation of degree greater than one is another corollary of the main theorem.     

Multiplicity of Positive Periodic Solutions to Second Order Singular Dynamical Systems

The existence and multiplicity of non-collision periodic solutions for second order singular dynamical systems are discussed in this paper. Using the Green’s function of linear differential equation, we consider general singularity and do not need any kind of strong force condition. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones.     

Positive periodic solutions of Hill’s equations with singular nonlinear perturbations

We study the existence and multiplicity of positive periodic solutions of Hill’s equations with singular nonlinear perturbations. The new results are applicable to the case of a strong singularity as well as the case of a weak singularity. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones. Some recent results in the literature are generalized and improved.     

Fourier-Jacobi type spherical functions for principal series representations of Sp(2, R)

In this paper, we study the Fourier-Jacobi type spherical functions on Sp(2, R) for irreducible principal series representations. We give the multiplicity theorem and an explicit formula for this function.     

POSITIVE SOLUTIONS OF m-POINT BOUNDARY-VALUE PROBLEMS

The paper is concerned with the existence and multiplicity of positive solutions for a nonlinear m-point boundary value problem. The proofs are based on a fixed-point theorem and a fixed-point index theorem in cones.     

Critical points for nondifferentiable functions in presence of splitting

A classical critical point theorem in presence of splitting established by Brézis-Nirenberg is extended to functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. The obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational-hemivariational eigenvalue problems.     

Three solutions to discrete anisotropic problems with two parameters

In this note we derive a type of a three critical point theorem which we further apply to investigate the multiplicity of solutions to discrete anisotropic problems with two parameters.     

Convergence and certain control conditions for hybrid viscosity approximation methods

Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition β_n→0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space.     

Existence and multiplicity of solutions for asymptotically linear, noncoercive elliptic equations

First we prove the existence of a nontrivial smooth solution for a p-Laplacian equation with a (p ? 1)-linear nonlinearity and a noncoercive Euler functional, under hypotheses including resonant problems with respect to the principal eigenvalue of ${(-{\it \Delta}_p,\,W^{1,p}_0(Z))}$ . Then, for the semilinear problem (i.e., p = 2), assuming nonuniform nonresonance at infinity and zero, we prove a multiplicity theorem which provides the existence of at least three nontrivial solutions, two being of opposite constant sign. Our approach combines minimax techniques with Morse theory and truncation arguments.     

Über den Verzweigungsindex bei ganzen Funktionen

Recently the author proved ([3]) for entire functions f(z) and any complex a: Each set of n a-points of f(z), which are near (with respect to the order of f(z)) to each another can be considered in view of the second fundamental theorem of Nevanlinna as an a-point of at least multiplicity two if n>1. In the present paper it will be proved: Each set of n a-points of f(z), which are very near (with respect to the order of f(z)) to each another can be considered in view of the second fundamental theorem of Nevanlinna as an a-point of at least multiplicity n–1.     

具Neumann边界条件的拟线性椭圆方程组的多解存在性

通过运用Ricceri的一个三临界点定理，得到了一类具变分结构的拟线性椭圆方程组的多解的存在性．     

Ramanujan’s Master Theorem for the Hypergeometric Fourier Transform Associated with Root Systems

Ramanujan’s Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the “Master Theorem”. In this paper we prove an analogue of Ramanujan’s Master theorem for the hypergeometric Fourier transform associated with root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.     

Existence and multiplicity of solutions for Kirchhoff type equations

In this paper, the existence and multiplicity of weak solutions are obtained for a class of Kirchhoff type problems with Dirichlet boundary value conditions by using the mountain pass theorem, the local linking theorem, the fountain theorem and the symmetric mountain pass lemma in critical point theory.     

Verzweigung an Eigenwerten unendlicher Vielfachheit-Bemerkungen zu einer Arbeit von H.-P.Heinz

This article is dealing with two theorems on nonlinear eigenvalue problems given by H.-P.Heinz in [2]: The theorem 3.6 on bifurcation at an eigenvalue of infinite multiplicity is shown to be empty. The theorem 3.7, concerning eigenvalue problems with a homogeneous nonlinearity, is considerably improved. This results follow easily from a classical theorem on eigenvectors of nonlinear completely continuous operators.