On a max-type and a min-type difference equation

This note shows that every positive solution to the following third order non–autonomous max-type difference equation

when is a three-periodic sequence of positive numbers, is periodic with period three. The same result was proved for the following min-type difference equation

     

Weighted estimates for a class of multilinear fractional type operators

In this paper, a class of multiple fractional type weights is defined as

     

GLOBAL WELL-POSEDNESS FOR A FIFTH-ORDER SHALLOW WATER EQUATION ON THE CIRCLE

The periodic initial value problem of a fifth-order shallow water equation t u 2 x t u + 3 x u 5 x u + 3u x u 2 x u 2 x u u 3 x u = 0 is shown to be globally well-posed in Sobolev spaces˙ H s (T) for s > 2/3 by I-method. For this equation lacks scaling invariance, we first reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data, and then prove the almost conservation law, and finally obtain the global well-posedness by an iteration process.     

Duality techniques for certain integral transforms to be starlike

For β<1, let denote the class of all normalized analytic functions f such that

     

Estimates for eigenvalues of fourth-order weighted polynomial operator

In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △~2u-a△u+bu=Λρu,inΩR~n,u|Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper bounds of the (k+1)-th eigenvalueΛ_k+1 in terms of the first k eigenvalues.Moreover,these results contain some results for the biharmonic operator.     

GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH NONLINEAR DAMPING AND SOURCE TERMS

The initial boundary value problem for a viscoelastic equation | u t | ρ u tt △u-△u tt + t 0 g(ts)△u(s)ds + | u t | m u t = | u | p u in a bounded domain is considered, where ρ, m, p > 0 and g is a nonnegative and decaying function. The general uniform decay of solution energy is discussed under some conditions on the relaxation function g and the initial data by adopting the method of [14, 15, 19]. This work generalizes and improves earlier results in the literature.