本文给出了Bihari不等式成在高维空间的一种推广形式。即证明了定理:设Ω_r表R^n中的球;S^2=sum from i=1 to n (S_i^2≤r^2),Q为R^n中有界可测集,u(s,x),f(s,x)为Ω_R×Q(R>r)下的非责有界连续函数,c≥0为常数,若 u(t,y≤c+∫f(...本文给出了Bihari不等式成在高维空间的一种推广形式。即证明了定理:设Ω_r表R^n中的球;S^2=sum from i=1 to n (S_i^2≤r^2),Q为R^n中有界可测集,u(s,x),f(s,x)为Ω_R×Q(R>r)下的非责有界连续函数,c≥0为常数,若 u(t,y≤c+∫f(s,x)φ[u(s,x)dxds] (1)对(t,y)∈Ω_r×Q(r<R)成立,其中φ(u)当0<u<ü(ü≤∞)为正的连续非减函数,又设ψ(u)=integral from n=0 to u du_1/(φu_1)(c<u<ü)这时如果 ∫Ω_r×Q~[f(s,x)dxds]<ψ(ü-0) (2)则有 supu(t,y)≤ψ^(-1)[f(s,x)dxds] (t,y)∈Ω_r×Q展开更多
This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari＇s inequality in t...This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari＇s inequality in the plane. Moreover, we also discuss the time regularity property of the solution by Kolmogorov＇s continuity criterion.展开更多
文摘本文给出了Bihari不等式成在高维空间的一种推广形式。即证明了定理:设Ω_r表R^n中的球;S^2=sum from i=1 to n (S_i^2≤r^2),Q为R^n中有界可测集,u(s,x),f(s,x)为Ω_R×Q(R>r)下的非责有界连续函数,c≥0为常数,若 u(t,y≤c+∫f(s,x)φ[u(s,x)dxds] (1)对(t,y)∈Ω_r×Q(r<R)成立,其中φ(u)当0<u<ü(ü≤∞)为正的连续非减函数,又设ψ(u)=integral from n=0 to u du_1/(φu_1)(c<u<ü)这时如果 ∫Ω_r×Q~[f(s,x)dxds]<ψ(ü-0) (2)则有 supu(t,y)≤ψ^(-1)[f(s,x)dxds] (t,y)∈Ω_r×Q
基金supported by NSF (10971076 and 11061032) of ChinaScience and Technology Research Projects of Hubei Provincial Department of Education (Q20132505)
文摘This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari＇s inequality in the plane. Moreover, we also discuss the time regularity property of the solution by Kolmogorov＇s continuity criterion.
