Bihari Inequality And Differential Equation,

The integral inequality is a useful tool when we study the stability theory of differential equation, especially when we study the stability, the estimate of solution and the boundedness of differential equation. In the past few years, many scholars take on the research of this field, they have achieved many good results .The theory of boundedness is one of important branches of differential equations.In the field of modern applied mathematics,it has made considerable headway in recent years,because all the structure of its emergence has deep physical background and realistic mathematicalmodel.In the first chapter,we mainly introduce the background knowledge of the whole article.In the second chapter,some generalizations of nonlinear Bahari integral inequality are studied. We state the main results as follows:lemma 2.1.1 [1] Assume that p ≥ 1, a ≥ 0, thenfor any k > 0.lemma 2.1.2 Assume that p ≥ q > 0, a ≥ 0, thenfor any k > 0.Theorem 2.1.1 let a, b ∈ C(I,R_+), α∈ C~1(I,I), be nondecreasing with α(t) ≤ t on I and u ∈ C(I, R_+), φ ∈ C(I, R_+), p≥q>0, p≥r>O,p,q,r, are constantsfor t ∈I, thenfor t G /, whereG(t)= [ [a{s)^-^-ki + a{s)-k^(s)]ds [*J P Pto P P Ja(t0) P+b(s)—k p (f)(s)\ds, Prl a a^￡ fa{t) r z-nA1(t) = a(s)-kpds Bi{t)= b(s)-kpds, (2.1.2)Jto P Ja(to) Pfor t G /.Remark 2.1.1 If we take (t) = k where k > 0 is a constant, q — r = 1, then the inequalities established in Theorem 2.1.1 improve the inequalities established by pachpatte in [2 Theorem 1 (03)]. vTheorem 2.1.2 let a, b G C(A, R+) and a G CX{JU Jx), f3 G Cl(J2,J2), be nondecreasing with a(x) < x on Jx, /3(y) < y on J2 , (j)(x,y) G C(A,i?+), u(x, y) G C(A, J?+) and p > q > 0, p > r > 0, p, q, r, are constants. If(u(x,y))p < 4>{x,y) 4- / / a(s,t)uq(s,t)dtds+I I b{s,t)ur(s,t)dtds, (2.1.7)for (x,y) G A, thenfor (x,y) G A, wherefx fy , ,? a^￡ j4(x, y)= / / a(s,t)-k p rftds,B(x,y)= f f * b{s,t)-kr-^dtds, (2.1.9)px py ~f(x,y)= / / (a(s,i)—— fe*+oOJ Xn Jyn y/ (b{s,t)------kp +b(a,t)(f>(s,t)-k p )dtds,for (x,y) G A.Remark 2.1.2 If we take 0 is a constant, q — r = 1 then the inequalities established in Theorem2.1.2 improve the inequalities established by pachpatte in [2,Theorem 3 (C3)].Theorem 2.1.4 let u(x, y), a(x, y), b(x, y), c(x, y),d(x, y), e(x, y) be nonneg-ative continuous functions defined for x,y € i?+, a(x) € C1(R+,R+) be non-decreasing with a(x) < x on R+, /3(y) € C1(jR+,jR+) be nondecreasing with P{y) >yon R+, and p > q > 0,p > r > 0, p, q, r are constants. If(u(x,y)Y 0, x, y G /?+, where70 7/3(y) P P(2.1.26)((,)) ( P Pfor x, y € Z?+.Remark 2.1.3 If we take 9 = 1, d(x,y) = 0 or r = 1, c(x,y) = 0, and oj(x) = a;, y0(y) = y then the inequalities established in Theorem 2.1.4 and Theorem 2.1.5 reduce to the inequalities established in [1,Theorem 1 and 2].Remark 2.1.4 If we take q — 1, d(x,y) = 0 or r = 1, c(x,y) = 0, and a(;r) = 2:, /3(y) = y, p = 1, e(x,T/) = 0, then the inequalities establishedin Theorem 2.1.4 and Theorem 2.1.5 reduce to the inequalities established in [3,Theorem 2.2 (a1)anrf(a2)].Theorem 2.1.6 let u(x,y),a(x,y),b(x,y),c(x,y),d(x,y),e(x,y) be nonneg-ative continuous functions defined for x,y E R+, a(x,y) be nondecreaing in x e R+,a(x) € Cl{R+,R+) be nondecreasing with a(x) < x on R+, j3(y) € Cl(R+,R+) be nondecreasing with f3(y) > y on R+, and p > q > 0,p > r > 0, p, g, r are constants.If(u(x,y))poo _n(j;,y) < (B(x,y))p[a(x,y) + F(x,y) x exp( / / -fcVc(3,t)Jo Jp(y)P (2.1.41))fi (for any A; > 0, x,y E R+, whererxB(x,y) = exp( b(s,y)ds), (2.1.42)Jooc(x)+(M)) + M) P P (2.1.43)))((,)) ( P Pfor x,y e R+.Remark 2.1.5 If we take g = 1, d(x,y) = 0 or r = 1, c(x, y) = 0, and a(x) = z, P(y) = y then the inequalities established in Theorem 2.1.6 and Theorem 2.1.7 reduce to the inequalities established in [1,Theorem 3 and 4].Remark 2.1.6 If we take q = I, d(x,y) = 0 or r = 1, c(x,y) = 0, and a(x) = x, j3(y) — y, p = 1, e(x,y) = 0, then the inequalities established in Theorem 2.1.6 and Theorem 2.1.7 reduce to the inequalities established in [3,Theorem 2.3 ].Theorem 2.2.1 let u(m, n), a(m, n), b(m, n), c(m, n), d(m, n), e(m, n) be non-negative functions defined for m,n e No and p > q > 0,p > r > 0, p, q, r are constants.IfTO — 1 OO(u(m, n))p < 0(771, n) + b(m, n) ]T ]P [c(s, t){u{s, t))Q+s=0 t-n+1d(s,t)(u{s,t))r + e{s,t)}, (2.2.1)for m,n ￡ No, thenm—1 00u(?,n)< [a(m,n) + b{m,n)f(m,n) \\{l + ^ (-k^c{s,t)b(s,t)s=o t=n+i p (2.2.2)+ -k^ d(s,t)b(s,t)))]r. for any A; > 0, m,n E No, wherem— 1 00s=0 t=n+lkr~{^^kr+a(s,t)-kr~^)+e(s,t)], (2.2.3)p pfor m,n € A^o, A; > 0.Remark 2.2.1 If we take q = 1, d(m, n) = 0 or r = 1, c(m, n) = 0, then the inequalities established in Theorem 2.2.2 and Theorem 2.2.1 reduce to the inequalities established in [5,Theorem 1 and 2].Remark 2.2.2 If we take q = 1, d(m, n) = 0 or r = 1, c(m, n) = 0, and p = 1, e(m,n) = 0, then the inequalities established in Theorem 2.2.2 and Theorem 2.2.1reduce to the inequalities established in [4,Theorem 2.6 (pi)and(p2)\.Theorem 2.2.3 let u(m, n), a(m, n), b(m, n), c(m, n), d(m, n), e(m, n) be non-negative functions defined for m,n € Nq. Assume that a(m,n) be nondecreaing in m G iV0, and p > q > 0,p > r > 0, p, q, r are constants.Ifm—\ m—\ 00(u(m,n))p 0, m,n e No, wherem—1 oos=0 i=n+l ^ P{r(s, t))Lp(^^krp + a(s, t)-^) + e(s, t)], (2.2.24)m-lr(m, n) = JJ [1 +6(s, n)], " (2.2.25)for m,n ￡ No.Remark 2.2.3 If we take q = 1, d(m,n) = 0 or r = 1, c(m,n) = 0, then the inequalities established in Theorem 2.2.4 and Theorem 2.2.3 reduce to the inequalities established in [5,Theorem 3 and 4].Remark 2.2.4 If we take q = 1, d(m,n) = 0 or r = 1, c(m,n) — 0, and p = 1, e(m,n) = 0, then the inequalities established in Theorem 2.2.4 and Theorem 2.2.3 reduce to the inequalities in [4,Theorem 2.7 ].In the third chapter,we study the boundedness and asymptotic behavior of a class of higher order nonlinear differential equation .This chapter is divided into two sections.In the first section ,we study the boundedness of solutions of a class of higher order functional differential equations .The aim of this section is to generalize the integral inequalities in [7]. The inequalities given here can be used as tools in studying boundedness of solutions of a class of higher order functional differential equations :Lnx(t) + F(t,x(t),x((t))) = 0we state the main results as follows:Theorem 3.1.1 Let fv(t), gj(t)be nonnegative continuous functions defined for [a, oo) , v = 1, 2, ■ ? ? , I, j = 1, 2, ? ? ? , k. (t) are continuous functions and satisfy 4>(t) < t, (f>'(t) > 0, 4>(t)is eventually positive, rv, qj ￡ (0,1] are constants, andI k\F(t,x,y)\ < ^/yWI^T" +^2gj(t)\y\9j (t,x,y) ￡ [a, oo) x R x R, (3.1.18)v=l j=\suppose moreoverlim In(t,a;pu-■■ ,pn-UpnprQ"fv) -1{t),a;pir-- ,pn-i,PnP903 {4>)9j) < oo, j = 1, 2, ? ? ? , k. (3.1.20)i—>oothen,for every solution x(t) of (3.1.2),D°(a;,po)(*)is bounded.Theorem 3.1.2 Let F(t, x, y)be as in Theorem3.1.1,If in addtion to (3.1.19), (3.1.20),lim ln{t, a;pnpovjv,pn-i, ■ ■ ■ ,pi) < oo, v = 1,2, ? ? ? ,1, (3.1.22)t—yoolim In(t, a; pngjplJ{(f))),pn-.i,-■? ,pi) < oo, j = 1,2, ??-,&. (3.1.23)then,for every solution x(i) of (3.1.2) ,I?o(a;]9o)(i) tends to a finite limit as ￡ —>■ oo. In particular, for every oscillatory solution x(t) of (3.1.2), D°(x;po)(t) tends to zero as t —? oo.Theorem 3.1.3 Suppose that there is a function F(t,x, y) such that\F(t,x,y)\oo(in) lim In(t,a;pnpof,pn-i,- ?■ ,pi) < oo,t-+oo(iu) lim In{t,a;pniThen all solutions of (3.1.2) are nonoscillatory .In the second section ,the asymptotic behavior of solutions of higher order nonlinear integro-differential equations with deviating argument is studied. Our technique depends on an integral inequality containing a deviating argument. From this we obtain some sufficient conditions under which all solutions of Eq.(3.2.1) have some asymptotic behavior. It improves the results of [8 — 9], we state the main results as follows:Theorem 3.2.1 In addition to the previous assumptions on (1) and (2),we assume further that(i) for t G R+, ?i, ■ ? ? ,un, vu--- ,vn, vn+l G R we haven ni,--- ,un,vi,--- ,vn,vn+1)\ R+ are continuous,^, pi G (0,1] are constants, (ii) for t, s G R+, u\, ■ ■ ■ , un G R, we haven\g(t,s,ui,--- ,un)\ < e2(t) +e3(s) + ^ h(t, s)\ui\Qi,where e-i and e$ : R+ —> R+ are continuous functions ,kt : R2+ —> R+ are continuous and nondecreasing in t for s G R+ fixed, qt G (0,1] are constants;(iii) the functions &^)(ELi l^(<)l)(E?=i l^1)(^)i)rN i = 1, 2, ? ? ? , n, and (t)\)(T.U |2JW)(#))I)W. ? = 1,2,--- ,n, and are m tne ciass ^i(0)°°);(iv) the following integrals are bounded as t —> oo,/and...