Some New Nonlinear Integral Inequalities And Their Applications

Differential and integral equations have got more and more attention in recent years.As those eq uations are hard,or even cannot get analytic solutions,some attention has been focused on special integral inequalities to analyze their solutions.There have been some excellent results on the study of integral inequalities,and a large number of generalizations based on the most famous Gronwall-Bellman-type integral inequalities have been made.This paper builds some new inequalities on the foundation of what have been achieved and we have got some new results.According to the context,this pap er is divided into four chapters.Chapter 1 Preference,we introduce the main contents and the back-ground of this paper.Chapter 2 Inspired in the results which Abdeldaim and Wang have established,we made further generalizations of retarded Gronwall-Bellman-Pachpatte-type integral inequalities and studied u(t)≤u0+k[∫t0α(t)f(s)φ(u(s))ds]3 +∫t0α(t)f(s)φ(u(s))[φ(u(s))+k(∫t0α(s)f(λ)φ(i(λ))dλ)2]ds. φ1(u(t))≤u0+∫t0α(t)σ1(s)φ12(u(s))ds +∫t0α(t)σ2(s)[φ2(u(s))+∫0sσ3(λ)φ2(u(λ))dλ]ds. u(t)≤u0+k[∫t0α(t)f(s)up(s)[(u(s))+m(∫t0α(s)f(λ)up(λ)dλ2]pds. and other new inequalities.We studied the bounds of integral-differential inequalities by them.Chapter 3 Inspired in the Gronwall-Bellman integral inequalities with two independent variables which Abdeldaim and Cheung have es-tablished,we studied ψ(u(x,y))≤a(x,y)+b(x,y)∫γ(x0)γ(x)∫δ(y0)δ(y)σ1(x,y,s,t)∫0sσ3(τ,t)φ(u(τ,t))dτdtds, ψ(u(x,y))≤a(x,y)+b(x,y)∫γ(x0)γ(x)∫δ(y0)δ(y)σ1(x,y,s,t),u(s,t)dtds +c(x,t)∫α(x0)α(x)∫β(y0)β(y)σ2(x,y,t)[∫0sσ3(τ,t)u(τ,t) [φ(u(τ,t))η(u(τ,t))+∫0τσ4(ζ,t)m(u(ζ,t))dζ]dτ]dtds. and other new inequalities.we studied the bounds of nonlinear integral inequalities with two independent variables by them.Chapter 4 Under the definition of the modified Riemann-Liouville fractional derivative.we established u(t)≤h(t)+[1/Γ(α)∫0t(t-s)α-1h1/3(s)u(s)g(s)ds]3+1/Γ(α)∫0t(t-s)α-1 g(s)u(s)[u(s)+(1/Γ(α)∫0s(s-λ)α-1g(λ)u(λ)h1/2(λ)dλ)2]ds (t)≤h(t)+[1/Γ(α)∫0t(t-s)α-1g(s)ur(s)h-r(s) [u(s)h1-r/r(s)+(1/Γ(α)∫0s(s-λ)α-1g(λ)ur(λ)h1/2r2/2r(λ)dλ)2]rds, ur+1(t)≤hr+1(t)+[1/Γ(α)∫0t(t-s)α-1g(s)ur-1(s)h4/2r/3(s)ds]3 1/Γ(α)∫0t(t-s)α-1g(s)ur-1(s)h2-r(s) [ur(s)+1/Γ(α)∫0s(s-λ)α-1g(λ)ur-1(λ)h2-r/2(λ)dλ)2]ds. new inequalities.We have studied their properties and extended results of chapter two, and we studied the bounds of fractional integral inequal-ities by them.