Asymptotic Behavior Of Solutions Of Certain Higher Differential Equations And Higher Difference Equations

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.Difference equations appear as natural descriptions of observed evolution phenomena because most measurements of time evolving variables are discrete and as such these equations are in their own right important mathematical models. More importantly, difference equations also appear in the study of discretization methods for differential equations. Several results in the theory of difference equations have been obtained as more or less natural discrete analogues of corresponding results of differential equations. Nonetheless, the theory of difference equations is a lot richer than the corresponding theory of differential equations. Consequently, the theory of difference equations is interesting in itself and it is easy to see that it will assume greater importance in the near future.The thesis is divided into four sections according to contents.In chapter 1, Preface.we introduce the importance of the main contents of this paper.In chapter 2, some generalizations of nonlinear Bihari integral inequalityare studied, and they are applied in the study of the asymptotic behavior of solutions of several classes of higher order nonlinear integro-differential equations with deviating arguments.We state the main results as follows:Lemma 2.2 Let the following conditions be satisfied :1. the functions u{t), a(t) and #,(ï¿¡) are continuous on R+ —> R+, i — 1,2,...,/, a(t) is nondecreasing on R+ with a(t) > 1;2. the functions fi(t, s)(i = 1,2,..., m), hi(t, s), ki(t, s) (i = 1, 2,..., n) : D = {(t, s) : 0 < s < t < +00} —>? R+ are continuous, and are nondecreasing in t for s E R+ fixed;3. the function cp(t) is continuously differentiable, ip(t) < t, ip'(t) > 0, and ip(t) is eventually positive, t G R+.Suppose moreover that Vi E R+,(2.5)in piu(t)] / hi(t,s) / ^j(s, m)[u(m)]p'c/m ds, ,_i >/o L-/0 Jwhere n e (0,1] (1 < i < m), p{ € (0,1] (1 < i < n), qz e (0,1] (1 < i < I) areconstants.Then we havez=lwherei-lfi(t,s)dsexpL A;=lptfi(t,s)ds -70(2.7),7-i = l(i = 1,2, .Here it is supposed thatooo J[Ek{t) = i,l[Fk{t) = l,l[Gk(t) = l, t e fl+.fc=l Jt=l fc=lRemark 2.1 The inequality in (2.6) generelizes the inequalities used in [1 - 6].Theorem 2.1 Suppose(i) Vi € R+, ui, u2,..., u2n, u2n+i € R,2nwhere ei, 6j(z = 1,2,..., 2n + 1) : R+ —>? i?+ are continuous, r, € (0, l](i = 1,2,..., 2n) are constants;where e2,e3 : R+ —> R+ are continuous, ki : R\ —$ R+ is continuous, and nondecreasing in t for s G i?+ fixed, Pi G (0, l](i = 1,2,..., n) are constants;(iii) the functions &<(*)=irn+ii-1), i = 1,2,... ,n and\k=\areall belonged to the class Li(0,oo);(iv) the integral/t n EmmPi\t)ez{r))dr ) ds,andare bounded as t —> oo;Then for any initial condition 0(t),t G [7,0], the solution y(t) of the equation(2.1), which satisfies y(t) = 9(t), t G [7,0], is defined on [7, 0]Ui?+, and satisfies(2.4).Remark 2.2 If f(t,uu ... ,u2n,u2n+i) = f(t,uu ... ,un,u2n+i),cn(t) = 0, then the main results in [1] are included in our Theorem 2.1.Remark 2.3 If n = 2, then the result in [2] is included in our Theorem 2.1.Remark 2.4 If n = 2, f{t,ui,U2,U3,u4,u5) = f(t,ui,u2,u5), then the main results in [5] are included in our Theorem 2.1.Remark 2.5 If n — 2, f{t,Ui,u2,U3,u4,u5) = f(t, uu w2,u3,u4), then the main results in [6] are included in our Theorem 2.1.In chapter 3, we appliy some generalizations of nonlinear Bihari integral inequality in the study of the asymptotic behavior of solutions of several classes of higher order nonlinear integro-differential equations with deviating arguments.We state the'main results as follows:Theorem 3.1 Suppose that."OO/ Pi{t)dt = oo, 1 < i < n - 1, (3.6)g(t) < t and there are nonnegative continuous functions qi(t),0 < i < n and on [a, oo) and Wi(u) €.J-,Q{t)w0 (j *?Lj > for (*. ?o) e [a, oo) x i?. (3.8)Suppose moreover that/?ooj (\f(t)\ + qn(t))dtooqi(t)Jn-i-l{t)dt(t))dt < oo. (3.10)Then every solution x(i) of (3.1) satisfieslim TLkX? .=beR,0oo J(i)Remark 3.1 If H(t,uo, ui,... ,un-i,un) = i?(t, uo, tti,..., un_i), then [11, Theorem 1] is included in our Theorem 3.1.Remark 3.2 If H(t,uo,U\,..., un_i, un) = H(t,u0), then [15, Theorem:-1 and 2] are included in our Theorem 3.1.Theorem 3.2 Suppose that (3.6) holds, g(t) < t, and there are nonneg-ative continuous functions <&(ï¿¡), 0 < i < n and ip(t) on [a,oo) and Wi(u) ï¿¡ T, 0 < i < n — 1, such that (3.7) and (3.8) hold. Suppose moreover that/oo />oo - />oo /-oo /"ooPl(*l) / Vi{Si) I â– â– â–  Pn-l(sB-i) / ^(3.16)< oo, 1 < i < n — 1,x> />oo roo pooPi(si) / P2(S2) / ??? / [?o(s) + '0(s)]dsdsn_i... ds\ < oo, (3.17)?'Si â– 'S2 JSn-l/oo /-oo />oo roo /-coPl(Sl) / P2(?2) / ?â– â–  / Pn-l(Sn-l) / [l/(s)|and/?ooJ sn-l[\f{s)\+qn{s)]ds 1 and f(k) > 1 be nondecreasing on N(k0), w(u) ï¿¡ T. If the discrete inequalityx(k) 0, W^l(u) is the inverse function ofRemark 4.1 The continuous analogue of Lemma 4.1 is due to Meng [23]. Remark 4.2 The inequality in Lemma 4.1 generalizes and improves the Bihari's inequality.Theorem 4.1 Suppose that00(A;) = oo, l0 e iV(A;0) x Rn. (4.8)Suppose moreover thatfc=fcoandqi{k)Jn-iⁱ{k) < oo, 0 < i < n - 1. (4.10)A;=/fc0Then every solution x(k) of (4.1) satisfiesLix(k) = O(Jn_ii(A;)) (A -?? oo), 0 < i < n - 1. (4.11)Remark 4.3 The continuous analogue of Theorem 4.1 is due to [11, Theorem 1].Remark 4.4 If H(k, x(k),Lix(k),..., Ln_xx{k)) = H(k, x(k)), then the continuous analogue of Theorem 4.1 is due to [16, Theorem].Remark 4.5 If n — 2, f(k) = 0, then Theorem 4.1 generalizes and improves the results of [23, Theorem 2 ].Theorem 4.2 Suppose that (4.7) holds, and there are real valued non-negative functions <&(&) (0 < i < n) on N(ko) and Wi(tt) €T(0i-!(s) < OOSl=k0 S2=S\ S3=S2 Sn_l=Sn₂ 5=Sn-i0l=Sn₂ ' S = Sn-\(4.19) hold, then every oscillatory solution x(k) of (4.1) satisfieslim Lix(k) = 0,0 < i < n- 1.fc—k?Remark 4.7 The continuous analogue of Theorem 4.2 is due to [11, Theorem 2].Theorem 4.3 Suppose that (4.7) holds, and there are real valued non-negative functions qi(k) (0 < i < n) on N[ko) and Wi(u) ï¿¡ T (0 < % < n — I), such that (4.8) holds. Suppose moreover thatt(k) < 0, or pl{k) < 0, 1 < i < n - 1,"-1ft(s)sn-8-1 < oo, 0 < i < n - 1, (4.24)s=koandooYsn-Il\f(s)\ + Qn(s)}