Solutions Of Several Classes Of Integral-differential Equations And Its Application

Nonlinear functional analysis is an important branch of mordern analysis mathmatics, because it can explain all kinds of natural phenomenas, more and more mathematicans are devoting theirselves to it. Among them, the nonlinear integrable-differential initial (boundary) value problems in the abstract (Banach) space is still a worthwhile study. Recently, the existence of nonlinear impulsive integrable-deifferential equations. In this paper, in Section 1. we devote our time to giving the theormes for the solutions of discontinuous impulsive integral-differential initial value problems, our results obtianed are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions. In the latter two sections, we discusse the solutions of two classes of boundary value problems for nonlinear integral-differential equations. We apply upper(lower) solution, comparable theorems and fixed point theory.The paper is divided into three chapters according to contents. In the first chapter, by giving a new comparable result, under only a lower solution or upper solution with the generalized upper control condition, we get the existence of an unique solution for the discontinuous impulsive integrable-differential euqation initial valueThe main results are as follows:Lemma 1.2.1 Assume that p G PC[J, E] DA[J', E] satisfiesp'(t) < -M(t)p(t) - N(t) (Tp) (t), a.e. teJ,t?tu Aplt^^-LMU)), 1 = 1,2,...,?, (1-2.1)P(*o) < 0,where M(t), N(t)are nonnegative Lesbesgue integrable functions , 0 < l(z = 1, 2, ? ? ? , m) and satisfy one of the following inequalities:(a) M. + N,koa < f JIVLJJt0(b) M. > 0, M^NMe 1) < rJt0m(c) if Mt > 0, {eM*a - M*a - l)N.ko + ï¿¡ U < Ml, if Mt = 0,^(d) !}°+a \M(t) + N(t) H k(t, s)dsdtwherer M* = sup M(t), A^? = supiV(t). i/ien \f t € J, p{t) < 9. teJ teJRemark 1.2.1 Let r = max{ti+i - U : 0 < i < m}. Assume p € PC[J,E] n A[J',E] satisfies (1.2.2), M, > 0, ^* > 0, 0 < U < l(i = 1,2, ??? ,m), if(e) n(1"L<)(f) M, > 0, M;lNMeM'a l)r < "H1"^ ,iE nai)then we can know that the conditions (a) and (b)are satisfied. Hence Lemma 1.2.1 extends the comparable results in [3,7].For convenience, we give the following assumptions:(Hi) there exist uQ G PC[J, E] fl A[J', E] satisfyingu'o < f(t,uo,Tuo,Suo), t G [t0, to + o],t ^ tj, Auolteti < Ii(v>0{ti)), i = 1,2, ? â–  ? , m, Wo (to) < ï¿¡o-(ff2) V?ï¿¡ PC[J,ï¿¡],Ft;(t) = /(*,?(*), (Tu)(?), (Su)(t)) is strongly measurable, and Fuo, G L[J, ï¿¡](H3) there exist bounded Lebesgue integrable functions M(t), N(t) which satisfy Lemma 1.2.1, such that V t G J, u > u > Uo, we havef(t,u,Tu,Su) - f(t,u,Tu,Su) > -M(t)(u-u) - N(t)(Tu-Tu).(HA) there exist nonnegative Lebesgue integrable functions a(t), fi(t), such that Vt G J, u > u > u0, we havef(t,u,Tu,Su) - f(t,u,Tu,Su) < a(t)(u-u) + /3(t)T(u-u), t G J.(H5) there exist 0 < Li < 1, (z = 1, 2, â–  â–  ? , m), such that if u(ti) > u(tz) > fxo(tj), we haveIi(u) - Ii(u) > -Li(u-u)(i = 1,2, ??? ,m).(H6) there exist C{ > 0(i — 1, 2, ? ? ? , m), such that u(U) > u(ti) > Uq(U), we haveIi(u)-Ii(u) 0,wheres = f(t, uQ, Tu0, Su0), t e [t0, tQ + a],t ^ U, olfet, > Ii(uo{ti)), i = 1, 2, ? ? ? , m,{H'2) V u e PC[J, E], Fv(t) = f(t, v(t), (Tu){i), (Su)(t)) strongly measurable, and Fvo, G LP0[J, E].(H'3) there exist bounded Lebesgue integrable functions M(t), N(t) which satisfy Lemma 1.2.1, such that V t G J, vq > v > v, we havef(t,v,Tv,Sv) -f(t,v,Tv,Sv) > -M(t)(v-v) - N(t)T{v - v), te J.(i/4) there exist nonnegative Lebesgue integrable functions a(t), /3(t),j(t), such that Vt G J, Vo > v > v, we havef(t,v,Tv,Sv) - f(t,v,Tv,Sv) v(ti) > v(ti), we haveh(v) - Ii(v) > -Li{v - v){% = 1,2, ? ? â–  , m).(H'6) there exist d > 0(i = 1,2, â–  ? ? ,m), such that if vo{ti) > v(U) > v(ti), we haveThen IVP(l.l.l) has a unique solutiont€J0,teJi,t G Jmfor any z0 G PC[J,E] n A[J',E], by the definition of the following iterativesZn(t) = (Ayn-x)(t) =, ï¿¡ G JnZ(n-i)i(t), t eZ(n-l)m{t), t G Jm.uniformly convergents to ui(t) in E, and for any s > 0,\zn -oj\\pc =wheres = <50, t G Jo,51, t G Ji,5m, < G Jm. 0 < Si < 1, i = 1,2, ? ? ? , ra.Theorem 1.3.3 ylssume i/iai the conditions (Hi) — (H3), (H5) and (H[) — (H'3), (H'5) are satisfied. If P is regular or E is weakly sequence P,then we have the maximal and minimum solutions in [uo,uo]-Remark 1.3.1 If I* G C[E,E], f(t,u,Tu,Su) satisfy weak C- condition and the else conditions in theorem, then there must be sequences {yn}, {zv} in [wo^ojwith respectively uniformly convergents to uj(t).Remark 1.3.2 In [3] the authors studied IVP(l.l.l), f(t,u,Tu,Su) satisfies M, N increasing conditions about u, Tu;However in this paper the theo-rme only satisfies M(t), N(t) increasing conditions, we get the similar results. Hence the main results in this paper improve and generalize the main results in [3]. Furthermore, in this paper, we get the unique solution of IVP(l.l.l) only by useing a supersolution or subsolution, and we also get the estimate of the approximate solutions for iterative sequences.Remark 1.3.3 In [4] the nonlinear term / and implusive term U are continuous, and in the condition (G3), Lj, Cj satisfy the following condition4=1However in the proof of theorems, we omit the above confmition in the condition (G3), and here / and I{ are discontinuous, then the applictions is more generalized.Remark 1.3.4 In this paper, we needn't the noncompact condition, weakly sequence space or regular cone and others. Hence the Theorem 1.3.1 and 1.3.2 not only extend the mail results in [1-3,5,7] but also improve and generalize Theroem 2 and 3 in [4] when we choose /;= 0.In the second chapter, by means of the fixed point theorem and a comparison result, we obtains the existence of minimal and maximal solutions of periodc boundary value problem for first order discontinuous impulsive integro-differential equations in Banach spacesx'{t)^f(t,x(t),(Tx){t),{Sx)(t)), te[0,a],t?ti, Ax\t=ti=Ii(x(ti))t i=l,2,---,m, (2.1.1)x(Q)=x(a).We will use the following conditions and lemmasLemma 2.2.1 Assume p € PC[J, E] n A[J', ï¿¡] satisfiesp'(t) < -Mp{t) - iVi (Tp) (t) - N2 (Sp) (t), a.e. t e J, t ? U,k ), i = l,2,...,m, (2.2.1)p(O) 0, iVx > 0, N2>0, 0 < L^ < 1(? = 1, 2, ? ? ? , m) and satisfies ont of the following conditions(n) M^l(N.k.. I N.h..)(r2Ma 11 < iP°Z'^}[a) M (Miko + iv2no){e L) < /?nJ f(t, vo,Tv0, Sv0), t G [0,o], t^U, =ti < Ii(vo(U)), i = 1, 2, ? ? ? , m, > vc(a).(H2) VuG PC[J,E],Fu(t) = f(t,u(t),(Tu)(t), {Su)(t)) is strongly measurable.(H3) There exist the constants M > 0, Nx > 0, N2 > 0, such that if V t G J,uo{t) —M(u — u) — Ni(v — v) — N2(w — w).(H4) There exist the constants 0 < Lj < 1, such that uo(ti) < u(U) < uiU) < vo{U) Bt,Ii(u) - Ii(u) > -Li(x - x), i = 1, 2, â–  ? ? ,m. {H5)FuQ,FvQeL[J,E}. The main results are as followsTheorem 2.3.1 Assume E is preweakly compact partial space, P is a normal cone in E, and the conditions (Hi) — (H5) are satisfied. If the inequalities (2.2.17) and one of (a) — (c) hold, then PBVP (2.1.1) must have a solution in D, and must have the maximal and minimal solutions.Remark 2.3.1 The proof method is different with the one in [17], and we also weaken the condition which the cone need satisfy, hence generalize the main results in [17]. And we improve and generalize the corresponding theorems in [10-11], furthermore the main results also generalize the result in [12], here the nonlinearity / and /j only need satisfy M-increasing condition.In order to get the approximate solution, we give the following assumption (H'2) For a.e. t ï¿¡ J, f(t, -,-,?) in E x E x E continuous, and for any (x,y,z) € E x E x E, f(t, x, y, z) is strongly measurable in J.Theorem 2.3.2 Assume E is preweakly compact partial space, P is 0 normal cone in E, and the conditions {Hi), {H'^) and (Hz) — (H5) are satisfied. If one of(a)-(c) holds, then there exists monotone sequences {un}, {vn} CPC[J, E]nA[J', E] respectively conformly convergents to the maximal and minimal solutions in I.Remark 2.3.2 Compared with Theorem 4 in [17], our theorem weaken the conditions of P, such that the applications are wide. We know that Pis normal if P is regular, but otherwize, the result doesn't hold. Hence the paper improve and generalize the corresponding theorems in [10 — 11]-The main results as followsTheorem 3.3.1 Let E be a real Banach space, P is a normal cone in E, u0 G C2[I, E] and G = {u G C[I,E]\u > u0}. Assume f(t,u,Tu,Su) satisfy the following conditions(H\) For any u G D, the abstract operator Fu = f(t, u, Tu, Su) maps the continuous function u into the strongly measurable function and Fu0 G L[I, E\.(H2) u0 is the lower solution of BVP(3.1.1), i.e.,— Uo < f(t, U0, TUq, Suq),uo(0) < xQ,(H3) There exist nonnegative constant L, M and N such that ^ ^- < 1, and for any u,v G G, v > u, we have6 < f{t, v,Tv, Sv) - f{t, u,Tu, Su) < L{v - u) + MT{v -u) + NS{v - u),where fc0 = maxMeï¿¡) k(t, s), h0 = maxMe/)ujn(t) -> u)*(t) uniformly in t G / with the norm and for any A G [^-L+Nh0 fthere exist no G N such that we have the following error estimate\\ujn-uj*\\ < Xn\\oj0 - uo\\ +------— f|ui — uo||, n>n0. (3.3.4)1 — ABy the similar method we may obtain the following theorem.Theorem 3.3.2 Let E be a real Banach space, P is a normal cone in E, v0 6 C2[I,E] and G' = {u G C[I,E]\u < v0}. Assume f(t,u,Tu,Su) satisfy the following conditions(Hi)' For any u G G', the abstract operator Fu = f(t,u,Tu,Su) maps the continuous function u into the strongly measurable function.(if2)' ^o is the supper solution of BVP(S.l.l), i.e.,-v'o > f(t,vo,Tvo,Svo), > x0, vo(l) > xi,andFvoï¿¡(H3)' There exist nonnegative constant L, M and N such that ï¿¡ m?- < 1, and for any u,v G G', v > u, we have6 < f(t, v,Tv, Sv) - f{t, u,Tu, Su) < L(v -u) + MT(v -u) + NS{v - u).Then BVP(S.l.l) has a unique solution u* in G' and for any ujq G G', the explicitly iterative sequenceujn(t) =xo + t(x[-xo) + t dr /(s,a;n-i(s),Tcjn₁(s),S'a;n₁(s))ds .Jo Jo- drJo Jo(3.3.16)uniformly in t e / with the norm and for any A e [L+^/lt) + ^ff*1, l), i/iere n0 (E. N such that we have the following error estimate^ n>n0. (3.3.17)\1 — ARemark 3.3.1 In [24], the following strict compactness type conditions are required(H) There exist nonnegative constant Ci > 0 (i=l, 2, 3) such that for any bounded set V{ G E (i=l, 2, 3),a(/(t, Vi, F2, F3)) < Cia(Vi) + C2a(V2) + C3a(V3).Obviously, our conditions are much wider than those in [24]. So our results generalizes and improves the main result in [24].Remark 3.3.2 If we let E be R, BVP(3.1.1) changes to the general ordinary differential equations. In [21, 22, 27], the nonlinear term f(t,u) need satisfy the continuity condition and the authors obtained the existence of one or more positive solutions of BVP(3.1.1), however, in this paper, we not only get the unique solution of BVP(3.1.1) but also give the explicitly iterative approximation of the solution and the error estimate of iterative sequence.Remark 3.3.3 In the special case, where BVP(3.1.1) do not include Tu and Su. Our results can reduce to the main results in [27].