Plot On The Infinite Interval - Differential Equations, Existence And Uniqueness

Nonlinear functional analysis is an important branch of mordern analysis mathmatics, because it can explain all kinds of natural phenomenal, more and more mathematicans are devoting their time to it. Among them, the problem for the existence of nonlinear impulsive equations comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analyse mathematics. The present paper employs the cone theory and fixed point index theory and so on, to investigate the existence of positive solutions of several classes of impulsive differential equations on unbounded domains. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The paper is divided into four chapters according to contents. In the first chapter, the sufficient and necessary condition are given for relative compactness of a class of abstract continuous function groups. Applying it, we prove the existence of solutions of terminal value problems for second order differential equation. We state the main results sa follows:引理1.2.4 Let E be a Banach space and u_n(t) C[J, E] n = 1, 2, ...., if there exists ρ(t) ∈ L[0, ∞) such thatThen α({u_n(t) : n = 1,2, ..... }) is integrable on J, andRemark 1.2.1. Lemma 1.2.4 generalizes lemma 1.2.3 in infinite interval, and it plays an important role in researching the differential equations ininfinite interval.Remark 1.2.2. The proof of lemma 1.2.4 is not seen in the relevant papers.Theorem 1.3.1 H C Cq[J,E] is relative compactness if and only if (i) For all b > a, the functons of H' — {x' : x e H} are equiconstiuous on [a, b], and for all t € J, H'{t) — {x'(t) : x G #} is relative compactness in E;(ii) There exists t0 G J, such that H(t0) is a relative compactness subset of E;(in) When t -> oo, x(t) —> 9, x'(t) —> 6 are satisfied uniform for all x ￡ H.Theorem 1.3.2 H C C?[J, E) is relative compactness if and only if (a) For all b > a, the functions of H^ are equiconstiuous on [a,b], and for all t e J, H^m)(t) - {x'(t) : x G H} is relative compactness in E;(6) For all k(k = 0,1,--- ,m — 1, there exists tk G J, such that H^k\tk) is a relative compactness subset of E;(c) When t —)■ oo, x(t) —> 9, x'(t) —t 6, ? ? ?, >? ^ are satisfied uniform for all x G #.Remark 1.3.1 Theorem 1.1.2 is generalized to the infinite interval in Theorem 1.3.1 and 1.3.2.Applying to the terminal value problem of differntial equation in a abstract spacex" = f(t,x,x'),teR+;(1.4.1) x(oo) = x'(oo) = 0;where / G C[R+ x E x E,E], R+ = [0, oo), E is a Banach space, x(oo) = lim x(t), x'(oo) = lim x'(t).t—>oo i—>ooUsing the following conditions:(Hi) There exists nonnegative a(t),P(t),j(t),ta(t),t^(t),tj(t) Gsuch that for all t€R+,x(t),y{t)e C[R+, E],\\f(t,x,y)\\ < a(t) {fi{t) + y{t))dt < 1, / t(p(t) + 7(t))dt < 1;Jo(H2) For all bounded sets BUB2C E, f(t, Bu B2) is relative. We have one following result:Theorem 1.4.1 Assume that (Hi) and (H2) are satisfied, then there exists solutions of Eq. (1.4.1) in C2[R+,E].Remark 1.4.1Using the results of [22], the problem isn't solved, so ve improv the results of [22]and verify the important role of 1.3.1 in differential equations.In the second chapter, by means of the monotone iterative technique and a comparison result, we obtains the existence of minimal and maximal solutions of an initial value problem in an infinite interval for first order impulsive integro-differential equations in Banach spacesx'= f{t,x,Tx), teJ,t?tk,Ax\t=tk=Ik(x(tk)), k = l,2,---, (2.1.1)^ x(t0) = x0,Using the following lemmas and conditions:Lemma 2.2.1 Assume that m(t) € BPC[J,E]{\Cl[J',E] satisfiesm'{t) < -M{t)m{t) - N(t)(Tm)(t), t e J, Am|t=t4 <-Lkm[tk), fc=l,2,--- , m(0) < 6,(2.2.1)where M(t),N(t) are bounded integral nonnegative functions, Lt < l(z = 1,2,---) and ￡￡0￡i, Il^iC1 - U) are convergent. Then m(t) < 9 on J provided one of the following two conditions holds :(b) M* + N*K* < -Jruyp-*, ^ where/OO />00 ( /"OO "jM(t)dt, N* = N{t)dt, K* = sup { k(t,s)ds\. (2.2.2) Jo teJ Uo JLemma 2.2.2 Let a,r) e BPC[J,E], suppose M* + N*K* < 1. Then the IVP for linear impulsive integro-differential equation' x'(t) = -M(t)x{t) - N{t){Tx)(t) + a(t),teJ,t?tk, Ax\t=tk = Ik(v(tk)) - Lk[x(tk) - V(tk)], k = 1,2, ? ? ? , (2.2.9)x(0) = x0,has a solution x e BPC[J, E) f|Cl[J', E).(Hx) There exists u0, v0 G BPC[J, E] f) Cl[J', E) satisfying uQ(t) < vo(t){t J) andu'o < f{t,uo,Tuo), t ￡ J, t^tk,Auo\t=tk < h(uo(tk)), k = 1,2, ■ ? ? , uo(O) < Xo,v'o > f{t, vQ, Tvq), teJ, t^tk,i.e., u0 and v0 are lower and upper solutions of IVP(2.1.1), respectively;(H2) There exists bounded integrable nonnegative functions M(t) and N[t) satisfying one of the conditions (a), (b) of Lemma 2.2.1, such thatf(t,x,Tx) - f(t,y,Ty) > -M(t)(x - y) - N(t)(Tx - Ty),where te J,uo-Lk{x-y),00 00 whenever uq < y < x < Vo{k = 1,2, ? ? ■) and ￡ Li, Y\ (1 — Li) are convergence;i=0 i=0(^4) For any t G J and any bounded equicontinuous on each Jk monotone sequence B = {un} C [uo^o], there exists nonnegative constants c, c* and ck{k = 1, 2, ? ? ?) such thata(f(t,B(t),(TB)(t)) < ca(B(t))+c'a((TB)(t))tanda(Ik(B(t)))￡] of IVP(2.1.1) in [uo,vo} respectively. That is, if x e BPC[J, E] f| Cl[J', E) is any solution of IVP(2.1.1) satisfying x G [u0, v0], then for all t G J, we haveU0(t) < Ui(t) < < Un{t) < < X.(t)< x(t) < x*{t) <■■ < vn(t) (f\ — p-t _ p-7t (f\ _ -5t / .(Hij./^j,,.JOAx\t=tk = , , ou , QMtk), k = 1,2, ? ?;(n + 2)(n + 3)x(0) = Xq, where tk — t\-- Evidently, x(t) = 0 is not a solution of IVP (2.4.1).(2.4.1)Conclusion IVP (2.4.1) admits minimal and maximal solutions which are continuously differentiable on J' and satisfy0 < x(t) ]■t\ l tcf i1Jfc + i'—' 1 + 1t,t e [l.oo).Remark2.4.1 Using the theory of [28], we can not solve the example 2.4.1, so our conclusion can't follow from the main theorem in [28]. This shows that this paper improves and generalizes the related results in [28].In the third chapter, by applying Schauder fixed point theorem, in a Banach space we prove the existence of solutions of infinite boundary value systems for first order differential equations' x' = f(t,x,y),y' = g{t,x,y), te[0,oo), (3.1.1)x(oo) =/3x{0), y{oo) = 5y(0),where ft, 8 > 1.Using the following cnditions:(d) f(t,x,y),g{t,x,y) € C[J x P x P,P]t andwhere at(t), bi(t) (i = 1,2,3) are integrable on J;(C2) /(￡, Bi, B2) and g(t, ￡?i, B2) are relative compactness, where Bi C E (i = 1, 2) are bounded;We have the following main results:Theorem 3.3.1 If conditions (Ci),(C2) are satisfied, andwhere a* = Jo°° ai{t)dt,b* = f?bi(t)dt,i = 1,2,3. Then (3.1.1) has solutions onl.Remark 3.3.1 By applying Schauder fixed point theorem, the existence of solutions of the infinite boundary value systems for first order differential equations is solved in Theorem 3.3.1, and isn't solved using others.