Periodicity in a Nonlinear Predator-prey System with State Dependent Delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear state dependent delays predator-prey system where a_i(t),c_j(t),d_i(t) are continuous positive periodic functions with periodic ω>0, b_1(t),b_2(t) are continuous periodic functions with periodic ωand ∫_0~ωbi(t)dt>0. T_i,σ_j, p_i (i=1,2,…,n, j=1, 2,…,m) are continuous and ω-periodic with respect to their first arguments, respectively, α_i, β_j,γ_i(i=1,2,…,n, j=1,2, …, m) are positive constants.     

Existence of Positive Solutions for a Class of Sigular Boundary Value Problem of Fourth Order

Some results concerning existence of positive solutions for the singular boundary value problems u^(4)(t)=f(t,u(t)) t∈（0,1) u(0)=u(1)=0 u‘(0)=u‘(1)=0 have been given, where f(t, x) may be singular at t = 0, 1.     

Linear Volterra Integral Equations as the Limit of Discrete Systems

We consider the multidimensional abstract linear integral equation of Volterra type x（t） （*）∫Rtα（s)x(s)ds=f(t),t∈R,（1）as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Bauach space-valued defined on a compact interval R of R^n Rt is a subinterval of R depending on t ∈ R and (*) f denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems, The Henstock integral setting enables us to consider highly oscillating functions.     

Some Exponential Integral Functionals of BM(μ) and BES(3)

In the present paper, we derive the Laplace transforms of the integral functionals

On a periodic prey-predator system with infinite delays

§1　IntroductionAnvarovandLarinov[1]introducedthefollowingprey-predatorsystem:x(t)=x(t)[α-γy(t)-γ∫∞0K1(s)y(t-s)ds-∫∞0∫∞0R1(s,θ)y(t-s)y(t-θ)dθds],y(t)=y(t)[-β μx(t) μ∫∞0K2(s)x(t-s)ds ∫∞0∫∞0R2(s,θ)x(t-θ)x(t-s)dθds],(1)whereα,γ,βandμarepositiveconstants,Ki∈C([0,∞),(0,∞))andRi∈C([0,∞)×[0,∞),(0,∞)),i=1,2.Fortheecologicalsenseofsystem(1),wereferto[1,2]andrefer-encescitedtherein.Sincerealisticmodelsrequiretheinclusionoftheeffectofchangingen-vironment,itmot…     

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Nonoscillation for a Second Order Linear Delay Differential Equation with Impulses

A group of necessary and sufficient conditions for the nonoscillation of a second order linear delayequation with impulses(r(t)u')'=-p(t)u(t-τ)are obtained in this paper,where p(t)=sum from ∞to n=1 a_n δ(t-t_n),δ(t) is a Dirac δ-unction,and for each n∈N,a_n>0,t_n→∞as n→∞.Furthermore,the boundedness of the solutions is also investigated if the equationis nonoscillatory.An example is given to illustrate the use of the main theorems.     

Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument

We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x（t） = a（t） ＋ （Tx）（t） ∫0^σ（t） u（t, s, x（s）, x（λs））ds 0 〈 λ 〈 1 which can be considered in connection with the following Cauchy problem x＇（t） = u（t, s, x（t）, x（λt））, t ∈ [0, 1], 0 〈 λ 〈 1 x（0） = u0.     

Some integral inequalities for the polar derivative of a polynomial

If P(z) is a polynomial of degree n which does not vanish in |z| 1,then it is recently proved by Rather [Jour.Ineq.Pure and Appl.Math.,9 (2008),Issue 4,Art.103] that for every γ 0 and every real or complex number α with |α|≥ 1,{∫02π |D α P(e iθ)| γ dθ}1/γ≤ n(|α| + 1)C γ{∫02π|P(eiθ)| γ dθ}1/γ,C γ ={1/2π∫0 2π|1+eiβ|γdβ}-1/γ,where D α P(z) denotes the polar derivative of P(z) with respect to α.In this paper we prove a result which not only provides a refinement of the above inequality but also gives a result of Aziz and Dawood [J.Approx.Theory,54 (1988),306-313] as a special case.     

Periodic Solutions for a Class of Forced Liénard-type Equations

Abstract   By applying the topological degree theory, we establish some sufficient conditions for the existence on T-periodic solutions for the Liénard-type equation

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

On Approximation by Reciprocals of Spherical Harmonics in L p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.     

Oscillation Theorems for Nonlinear Second Order Elliptic Equations

Some oscillation theorems are given for the nonlinear second order elliptic equationsum from i,j=1 to N D_i[a_(ij)(x)Ψ(y)||▽y||~(p-2)D_(jy)] c(x)f(y)=0.The results are extensions of modified Riccati techniques and include recent results of Usami.     

Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means

We indicate criteria for the coincidence of the Knopp kernels K(f) K(A _f), and K (R _f) of bounded functions f(t); here,

Extremum problems for Golubev sums

Suppose thatG is a finitely connected domain with rectifiable boundary γ, ∞εG, the domainsD ₁,...,D _s are the complements ofG, the subsetsF _j ⊂D _j are infinite and compact,n _j ≥1,j=1,...,s, are integers, λ₀ is a complex-valued measure on γ, and

A sharp form of an embedding into multiple exponential spaces

Let Θ be a bounded open set in ℝ ⁿ , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that

Generalizations of Jensen-Steffensen and related integral inequalities for superquadratic functions

We present integral versions of some recently proved results which improve the Jensen-Steffensen and related inequalities for superquadratic functions. For superquadratic functions which are not convex we get inequalities analogous to the integral Jensen-Steffensen inequality for convex functions. Therefore, we get refinements of all the results which use only the convexity of these functions. One of the inequalities that we obtain for a superquadratic function φ is

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.