| It is widely recognized today, that the integral inequalities are of very importance in the studying of stability, many other qualitative as well as quantitative properties of solutions of differetial equations. The well-known Grown-wall inequality, Bihari inequality and their generalization have been frequently employed in the studying of this direction. For instance, such types of inequalities are profitably used by Brauer [15] to study the asymptotic behavior of the solutions of differential systems. The linear and nonlinear generalizations of these inequalities have been obtained while studying the pointwise estimates of linear and nonlinear Volterra integral equations. It is natural to expect that some new generalizations and extensions of these inequalities would be equally important in certain new applications. Our main aim is to establish some new generalizations and extensions of these inequalities which can be used as handy tools in the study of certain new classes of ordinary differential equations and partial differential equations.The thesis is composed of three chapters.Chapter 1 is the introduction of this paper, which introduces the problem and the background of the problem we have studied.In the second chapter, we consider the generalizations of continuous retarded integral inequalities and their applications. In the first section of the second chapter, based on the results of Olivia Lipovan [2], we get the generalization of Ou-Iang inequality and Pachpatte inequality. Then we give some applications of our generalizations and corollary. Our main results are as follows:Theorem 2.1.2 Let u, f and g be nonnegative continuous functions defined on R_+ and let c be a nonnegative constant. Moreover, let φ∈ C(R_+, R_+)be an increasing function with 0 on (0, oo) and a G Cl(R+,R+) be nondecreasing with a{t) < t on R+. Ifra(t)f(u(t)) 0 if t > 0.(ii) \ip(u) < tp(^) for v > 0.Definition 2.2.2 A function ip G C(R+,R+) is said to belong to the class /C, if(i) (p is nondecreasing, (p(t) > 0 if t > 0,(ii) \ (p(^) for v > 1, and 1 is nondecreasing on I.(ii) fi(t, a), gj(t, s) G C(D, ,R+)(z = 1,2, ■? ? , m)(j = 1, 2, ? ? ? , n) are non-decreasing in t for s G / is fixed.(iii) tti, Wj e F (i = 1, 2, ? ? ? ,m)(j = 1, 2, ? ? ? ,n), zo>0.i3i(*) = ^r1i=i J° r=(i = 2,3,---,n),tG/, especially,Ft x is the inverse of Fi,fz ds Fi{z)= —rT, (i = 1,2, ? ■■,n), z > zQ > 0.The third chapter is concerned with the generalizations of discrete retarded integral inequalities and their applications. In the first and cecond section of the third chapter, we generalize the content of the first section of the secnd chapter to the discrete form. Prom this, we can discuss the properties of solutions of retarded difference equations. Moreover, we subsequently generalize them to the form of two variables, which extends the domain of applications more. The results what we get are as follows:Theorem 3.2.1 Assume that {un}, {/?}, {gn} G S(N-m,R+), c > 0 is constant, m G No. Moreover, we assume that r0 > 0, G(z)Q,'1, cp~l, G~x are the inverse of Q, <~p and G respectively. M satisfies that forany n < M, n G A^o,n-lG(￡2(c)) + Y,9s-m G DomiG'1), 3.2.3n—1 n—1) + ^gs-m] + ^2fs-m G DomlO-1). 3.2.4s=0 s=0Theorem 3.2.8 Let {u(m,n)}, {/(m,n)}, {^(m,n)} G S(N^1xN-i2,R+), h, h G No, and c > 0 is constant. Moreover, we assume that cp G C(R+, R+)is strictly increasing with (p(oo) = oo and ip 6 C(R+, R+) is nondecreasing. Ifm—1n—1* ~ ^)?(s - /i, t - k)ij}(u(s -h,t- l2))4- #(s - /i, i - /2)?(s ~h,t- h)], m,n€ No,3.2.12holds, then for any m,n ￡ Nq, m < M, n < N,m—1n—1,n) < ^-^n-^G-1s=o t=o 3 2 13m—1n—1s=0 t=0Where fi(r) = ^ ^%, r > r0 > 0, G{z) = ￡ ^^j, z > z0 > 0.V?"1, fi"1, G-1 are the inverse of tp, Q and G respectively. M, N satisfies thatfor any m,n G AT0, m < M, n < N,rn—X n—1 m—1 n—1 /(1^2)(1). 3.2.14s=0 t=0 s=0 t=0m—1 n—1 m—1 ti—1s=o t=o s=o t=o3.2.15In the third section of the third chapter, similar to the generalization of the second section of the second chapter, we generalize one signal of sum to arbitrary finite signals of sum. The main results are as follows:Theorem 3.3.5 Letm n—1 s ~ ri)tti(u(s - rt-))p n—1+ ^2J2gi(n,s)wi(u(s)), n e NQ.i=l s=0If (i) u(n) € S(N-r,R+), a(n) € S(N0,R+) and a(n) > 1 is nondecreasing on No, where r = max! |
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