Boundedness And Asymptotic Behavior Of A Class Of Differential Equations

It is well known that integral inequalities play a fundamental role in the development of the theory of differential and difference equations. During the past few years,many new inequalities have been discovered, which are motivated by certain applications.In the paper,some new nonlear inequalities are established,which generalize several types of integral inequali-ties.Using our results,the boundedness of some integro-differential equations are sudied.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 mathematical model.The paper is divided into three chapters according to contents.The first chapter contains some new results being based on precursors briefly.In the second chapter,nonlinear integral are obtained which related to some generalizations of Ou-Iang integral inequality given by Yang Enhao in^[1].And also some integral inequalities are obtained which genreralise some recent results of B.G.Pachpattet^[2-3].We state the main results as follows:Theorem 2.2.1 Let u(t),a(t) ∈ C(I,R₊),b(s,t) ∈ C(I²,R₊), for to ≤ s ≤ t < ∞, α∈ C¹(I, I) be nondecreasing with α(t) ≤ t on I, let c ≥ 0 be a real constant and φ ∈ C(R₊,R₊) be strictly increasing function with φ(∞) =∞ and g ∈ C(R₊, R₊). Iffor t ∈ I, thenfor t₀ ≤ t ≤ t₁, whereG^-1 is the inverse function of G, and t e I is chosen so that for all t lying in the interval [t₀,t₁].Corollary 2.2.1 Let c ≥ 0, p > 0, q > 0 be constants, and p ≥ q, u(t), a(t), b(s,t) are as in Theorem 2.2.1. IfRemark 2.2.1 If we take p = q =1, then the inequalities established in Corollary 2.2.1 reduce to the inequalities established by pachpatte in [2, Theorem 2.1 (a₁)].Remark 2.2.2 If we take b(s, t) = 0, then the inequalities established in Theorem 2.2.1 reduce to the inequalities established by Yang Enhao in [1, Theorem 2 (A)].Theorem 2.2.2 Let u(x, y), a(x, y) ∈ C(△, R₊), b(x, y, s, t) ∈ C(△², R₊), for x₀≤s ≤ x ≤ X, y₀ ≤ t ≤ y ≤ Y, α ∈ C¹(J₁, J₁), β∈C¹(J₂, J₂) be nondecreasing with α(x) ≤ x on J₁, β(y) ≤ y on J₂. And let c ≥ 0 be aconstant. Let tp, g be as in Theorem 2.2.1. Ifpa(x) i>P(y) rV>(u{x,y)) 0, p > 0, q > 0 be constants, and p > q, u(x,y), a(x,y), b(x,y,s,t), a(x), )3(y) are as in Theorem 2.2.2. Ifup(x,y) _iI l-S- P P⁹c p + (1-----)A(x,i/) , p > q.{ L p JRemark 2.2.3 If we take p = q = 1, then the inequalities established in Corollary 2.2.2 reduce to the inequalities established by pachpatte in [2, Theorem 2.2 (&i)l.Theorem 2.2.3 Let u, /, g, h, k be real-valued nonnegative continuous functions defined on R+, and c be a nonnegative real constant. If/<*(*) r u(s) \f(s)u(s)+g(s)^ L (2-2.17)h(s) ( u(s) + / k(r)u(T)dT J Ids, Ja(t0) ) Jfor t € /, thenu(t) < cexpra(t)/ f(8)d8Ja(t0) ) [g(s)f(r)drds,(2.2.18) for t G /, wherefor t G /.Theorem 2.2.4 Letu(z,y), f(x,y), g{x,y), h(x,y), k(x,y) be real-valued nonnegative continuous functions defined on A, and c be a nonnegative real constant. a(x), fi(y) are as in Theorem 2.2.2. Ifu2{x, y) R+ = [0, oo), (p(t) is a differentiable continuous function defined on R+ —>■ R+ with (p(t) < t, ip'(t) > 0, (p(t) > 0 eventually .2° fi(t,s)(i = 1,2,- ? -,m), 9i(t,s)(i = 1,2,- ? -,n) , /*(*,*)(* = 1,2,- ? ?,n) are continuous functions which be nondecreasing with respect to t for keeping s defined on R+ x R+ —> R+.3° Ifu{t) rⁱ jo ., '-1 (3.1.14)exp^2 Gi{t)hi(t,s)pikPi¹ds, t>0where k > 0 is any constant. = E /(3.1.15)n -v-i(O)ai(t) = a(t) + J2 Gi(t)^(f,s)[u(V(a))]?ds> (3.1.16)Gi(t) = / ft(t, s)ds. (i = 1,2, ? ? -, n) (3.1.17)ip^x{t) is the inverse function of cp(t).Theorem 3.1.2 Let1°u(t), a(t), fi(t)(i = 1,2, ■■■,m), gj{t){j = 1,2, ■■■,n),hw{t),mw(t){w = 1,2,- ? ■, I) are positive continuous functions defined on [a, oo).2° R with 0, (p(t) > a eventually.3° Ift=l 7fl 7=1 ?/fl (3.1.23) / / / rnw(r)[u((p(r))]Pwdrds, te[a,oo)where n € (0,1], (1 < i < m), qj € (0,1], (1 < j < n), pw € (0,1], (1 < w < I) are real constants, thenu(t) < ai(t) V / 9j{s)qjkq' ds￡>/T"n X I /-f l/lTD ( ClTl k*"1" /7C V" ^ 1/7 (*V^1(3.1.24)where k > 0 is any constant.Ids E/₁ [ Hw{ip{t))mw{s)\{l -pw)k^ +pwkp?-1al{ 0 is any constant,^i Jv-ai(t) = a(t) + ^T ft(?)[ j=\ J"Remark 3.1.1 If we take hw(t) = 0 or mw(t) = 0(w = 1,2, ?? -,/), then Theorem 3.1.2 contains Corollary 3.1.1.In the second section ,we study the asymptotic behavior of a class of the second order integro-differential equation. Some sufficiet conditions for the boundedness of the solutions are obtained. And the aim of this section is to improve a class of Bihari integral inequality with deviating argument from [10]. Then we study asymptotic behavior of solutions of a class of Integro-differential equation with deviating argument using this inequality(r(t)x'(t))' + f (t, x(t), J g(t, s, x(y(s)))ds) = 0. (3.2.3)It improves the results of [8 — 9],we state the main results as follows:Theorem 3.2.1 Besides the hypotheses above for (3.2.3), if(i) lim^oo R(t) < oo ;(ii) f* bi(s)R{t, s)ds, (1 < i < m+2) are bounded on R+, f*I*$ G(t) ki(s)ds are bounded on R+, Jo°° e(t)dt < oo. Where G(t) = f0 bm+i(s)R(t, s) ds , and ip^l(t) is the inverse function of bm+1(s) + bm+2(s)ds < oo, /0°°e(S)dS < oo, /0°° hi{8)[R(8)Y< ds < oo, (1 < i < m). /0°° ki{s)[R{(p{s))]rids < oo, (1 < i < n). Then anysolution x(i) of equation (3.2.3) satisfies:\x(t)\ = O(R(t)), \r(t)x'(t)\=O(l),. t—>oo (3.2.14)In the third section, the real meaning is that this section is improving a class of Bihari integral inequality with deviating argument from [12]. Then we study asymptotic behavior of solutions of a class of Integro-differential equation with deviating argument using this inequality(r(t)x'(t))' + f (t, x(t), x(bm+1(s)+bm+2(s)ds < oo, JQ°° e(s)ds < oo, /a°° b^R^pds < oo, (1 < i < m), /a°° Cj(s)[R^(S))^ds < oo, (1 < j < n), /a°° kw(s){R(Pi{t)dt = 00, l 0, (p(t) > a eventually. bi(t), Cj(t) are positive continuous functions defined on [a, oo),(i = 1,2,- ? -,ra), (j — 1,2, ?■ -,n). e(t) is a continuous function defined on R+ —> R+, r{,qj € (0,1] are real constants, if\F(t,x,y)\ < 2_^bi(t)\x\Ti + y^Cj(t)\y\qj +e(t) (t,x,y) e [a,oo) x R x R,(3.4.10) and if/t r -I Ti ?ooPnbi(r) Jn_i(r) dr < oo, / pne(t)dt < oo," im (3-4.11)PnCj-(r) Jn_i(y>(r)) dr < oo.Then any solution x(t) of equation (3.4.1) satisfies:x(t) = O(Jn-i(t)), t —> oo.Remark 3.4.1 If we take F(t,x,y) = F(t,x), then Theorem 3.4.1 contants [14,Theorem 1 and Theorem 2].Remark 3.4.2 If we take F(t,x,y) = F(t,x), n = 2, then Theorem 3.4.1 contains [15,Theorem 1 and Theorem 2].Corollary 3.4.1 Let ip(t) < t be a continuous function, bi,Cj : [a, oo) —> (0,oo), and let e(t) : R+ —> R+ be a continuous function with /a°° e(t)dt < oo, such that (3.4.10) andft rv¹(*)Coo, / cj(r) \cp{r) \9j dr < oo/t...