Solutions Of Initial Value Problems And Multi-Point Boundary Value Problems For Nonliner Equations

Nonlinear functional analysis is an important branch of modern analysis mathematics, because it can explain all kinds of natural phenomenal, more and more mathematicians are devoting their time to it. Among them, the multi-point boundary value problems for ordinary differential equations arise in a variety of different applied mathematics and physics, it is at present one of the most active fields that is studied in analysis mathematics. The cone theory, fixed point theory and so on, are used in the paper. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The paper is divided into three chapters according to contents.In the first chapter, by using the partial order method and a new comparison result,we investigate the unique solution of the initial value problem for nonlinear second order integro-differential equationsu" = f(t,u,u',Tu), u(0)=x0, u'(0) = x1, (1.1.1)We require only a lower solution or an upper solution. Without any compactness type assumption,we obtain the explicitly iterative sequences of approximation solutions and the error estimate.The results obtained here improve and extend recent results. We state the main results sa follows:Lemma 1.2.1 Let m € Cl[I,Rl], such thatm'{t) > - M{t) I m(s)ds - N(t)m(t) Jo- L(t) / k2(tJs)m(s)ds, m(0) > 0, t e I, Jowhere M(t),N(t),L(t) are non-negative continuous functions on /, /c2(t, s) = J k(r, s)dr, and one of the following conditions(i) f \M{t)t + N(t) + L(t) J k2{t,s)ds\dt<\,(ii) r |m(*) / eti Jo L JoJ rt ?r/ \ i r0 J 'is required. Then m(t) > 0, Vt G /. Using the following conditions:(#i) There exist ?0 G C2 [/,ï¿¡],such that u{f < f(t,uo,u'o,Tuo), uQ(0) < x0, u'0(0) < x\.{H2) Fort G J, u,v G Q = {x ï¿¡ Cl[I,E]\x> uo,x' > u'0},u< v,u' < v',f(t, v, v', Tv) - f(t, u, u', Tu) > -M(t)(v-u)- N(t)(v' - u') - L(t)T(v - u),where M(t), N(t), L(t)ave in Lemma 1.2.1.(#3) There exist non-negative Lebesgue integrable functions Mi(t), Ni(t), Li(t), such thatf(t, v, v',Tv) - f(t, u, u',Tu) < Mx{t){v -u) + N^v' - u') + Lx{t)T{v - u),for t G /, u, v G Q, u < v, u' < v'. We have the following results:Theorem 1.3.1 Let P be a normal cone in Banach space E. Suppose (Hi) - (H3) hold, then IVP(l.l.l) has a unique solution u* in Q, and Vz0 € Q,the iterative sequence \°° rtzn(t) = Fn₁(t) + J2i-1)i / hi(t,s)Fn.1(s)da,converges uniformly to u* on /, whereFB_i(t) = [1 + t/V(0)]zo + tei + / (t- s)[f(s,zn_l(s),z'n_l(s), {Tzn-Ms)Jo+ MCs)^!^) + N(s)z'n_l(s) + L(s)(Tzn^)(s)]ds, hi(t, s) = N(s) + (t - s)[M{s) - N'(s)] + {t- r)L(r)k(r, s)dr,J shi(t,s)= hi(t,r)hi^.1(r,s)dr.JsIn addition, there exist positive integer N*, and the error estimate\\Zn - U% < βoβn\\z'o - W'ollc + A) j^lkl - "ollc, ? > N*,where 0 < ft < 1, ,9o > 0 are two Constances, wi(t) =Mt)n=1^(*) =ii + f [f(s, uo(s), u'Q(s), Tuo(s)) + M(s)uo(s) + N(s)u'0(s) Jo4- L(s)Tuq(s) — M(s)x0 — L(s)xo I k(s, r)dr]ds,JoK1{t,s)=N(s)+ f [M(r) + L(r)k2(r,s)]dr, k2(t,s)= f k(t,r)dr,J s J sKn(t,s) =N(r) + [ (M(t) + L(r)k2(r,r))drJKn-i(r,s)dr.Theorem 1.3.2 Let P be a normal cone in Banach space E. Suppose (H{)^Hl hold, then IVP(l.l.l) has a unique solution v* in Q*, and Vzq € Q*,the iterative sequenceconverges uniformly to v*, whereK-i(t) = [1 + tN(0)]xQ + tXl+ f (t- s)[f{s, zn^(s), z'^Johx{t, s) = N{s) + {t- s)[M{s) - N'{s)} + f (t - r)L{r)k{r, s)dr,J 3hi(t,s)= / hi(t,r)hi_l(r,s)dr.JsIn addition, there exist positive integer iV*, and the error estimate||^||<^n||4li + A11 "n ? 11 c —^ fu^ ||^u "U lie ' /^u-I _<sub> o Iwhere 0 < P < 1, /?o>O are two Constances,i*-^||C! n> N*.wAt) =/) + Y,{-l)n / Kn(t, 8)^(3^3, n=1 Johl(t)=xl+ [[f(s,vo(s),v'0(s),Tv0{s)) + M(s)vo(s) + N{s)v'0{s) Jo+ L(s)Tvo(s) - M(s)xq - L(s)xo / k(s, r)dr]ds,JoK^s) =N(s)+ f [M(r) + L(r)k2(r,s)}dr, k2(t,s) = f k(t,r)dr,J s J sKn(t,s) =J* fiV(r) + y (M(t) + L(r)A;2(r,r))drj Kn^{r,s)dr.In the second chapter, the existence of multiple nontrival positive solutions for a class of singular second order three point boundary value problems withsign changing non linearitiesu"(t) + q(t)f(t,u)=O, 0<* 0(^0).(H2) a > 0,8 > 0,a.+ -/3 ^ 0,0 < r? < 1,0 < k < ^±| < J, and p = a(l - krj) + p(l - k) > 0.(-ff3) g € C((0,l),[0,+oo)) such that 0 < J*(P + as)q(s)ds < +oo, 0 < f* q(s)ds < +OO.The existence of at least three nontrival positive solutions by using the Leggett-Williams fixed point theorem.Theorem 2.3.1 Suppose (Hi) — (H3) hold. In addition there exist real numbers a,b,c > 0 such that 0 < a < b < min{7, ^}c, where 7 = min \ *iifc^, kr),T]\, and / satisfies the following conditions:(#4) f(t,u) > 0, for all (t,u) G [0,1] x [b,c];(H5) f(t,u) <±, for all (t,u) e [0,1] x [0,c];(He) f(t,u) < ±, for all (t,u) 6 [0,1] x [0,o];(H7) f(t,u) > -, for all (t,u) e [v,l] x [6,6/7].Then the three-point boundary value problem (2.1.1) has at least three nontrivial positive solutions^,^2,^3, such that0 < llwiH < a, b < min U2, \\u3\\ > a with min u* < b. n 0(^ 0),(H2) q : (0,1) -> [0, +oo), / q(t)dt < +oo, q(t) ï¿¡ 0 on any subinterval of (0,1)m-2(#3) a>0, ï¿¡ > 0, p = ^ + a>0andA = p-^A^()8 + a^) > 0.i=1We have the following main results:Theorem 3.3.1 Suppose (Hi), [H2), (H3) hold. In addition there exist real numbers o, b, c such that 0 < a < b < min{7, A/A}c and / satisfies the following conditions:(HA) f(t, u) < c/A for all (t, u) € [0,1] x [0, c],(#5) f(t, u) < a/A for all (t, u) G [0,1] x [0, a],(H6) f(t, u) > b/X for all (t, u) G [a, 1 - a] x [6,6/7],(#7) /(t,?) > 0 for all (t, u) e [0,1] x [b, c].Then the m-point boundary value problem (1.1) has at least three nontrivial positive solutions u\, v.2, W3, such that0 < IIt*iSI < fl, fr < mm v>2 and IIU3II > a with min u* < b.cr...