The Solutions Of A Kind Two-Point Boundary Value Problem Of Fourth-Order Differential Equations In Banach Space And Its Applications

All sorts of nonlinear problems resulting from mathematics , physics ,chemistry ,bi-ology , medicine , ecnomics ,cybernetics and so on have brought wide attention of people . At present , nonlinear functional analysis has been one of the most important branch of learning in modern mathematics . It provides an effect theoretical tool for sdudying many nonlinear problems , mainly including partial ordering method , topological degree method ,variational method and analytic method and so on . It plays an important role in dealing with nonlinear equations and partial differential equations arising in applied mathematics .The existence of solutions for differential equations have been considered extensively in last years. But most of these researchers only put their attentions to general space ,few papers put their attentions to the existence of solutions of fourth-order differential equations in Banach space .The author mainly studies existence of solutions of fourth-order differential equations in Banach space .This paper is divided into two chapters.In chapter 1, by means of the theory of Darbo ,the author studies the existence of solutions for the following two-point boundary value problems of fourth-order ordinary differential equations in Banach space:First we give several lemmas.lemma 1 Let/ G C[I, E],then two-point boundary value problems(x"(t) = f(t), tei,(1.2)x(o) = x(i) = e.fhas the only solution x(t) = /?* ^(i, s)/(s)ds in C^lIfE] .wheres(l-t), 0 0, / is bounded and uniformly continuous on I xTrxTr , and exist Cr > 0, C* > 0,0 < Cr + C* < 2 such thatQ(/(f,P1,Z?2) 0, .A : Br —> C72[/, E\ is a strict-set-contraction.This means that exists 0 < Kr < l.for any 5 C Br , such that ai{A{S)) < Kra2{S)The main results of this chapter are :Theory 1 : If {Hi), (H2) are satisfied , then two-point boundary problem (1.1) has at least one solution x G C4[J,E].We apply the results to the following example : Example 1.1 : consider the infinite ordinary differential equation group 0(< 0) in Q ,then maxu(i)= maxu(i) (min?(x)=min u[x)).Lemma 2.2 (the comparison theory ) Let q G C4[I, ￡J],such that(2.2)then q(t) > 0,V t G /.q(G)>e,q(\)>6,We need the following conditions in this chapter :[Hi) There exists t)0,u)0￡ C4[I, E\, such that vo[t) < wo[t),v'^[t) > ?/0'(t),V t e /, and vq,wq is the lower and upper sollutions Respectively , for the problem (2.1):■44), vtei, VtEJ,to0(0) >9,wo[l)>e,(H2) There exists monotone conditions on u, v in f(t, u, v) , which satisfy : 1) V v € [tuo(t),t#(t)],ui,u2 e fao(*),u>o(*)] ^ ?i < ?2 , which satisfy : f{t,ui,v) < f(i,u2,v),VtLet [wo,u>o] = (? € C^J.BJMt) < u(t) < u*,(*),t#(t) > u"{t) > H'(t),V t G /}.The main results of this chapter are :Theory 1 . Let E is a real Banach space ,P is a regular cone of E (see the paper [l]),and (Hi), (H2) are satisfied , then boundary value problems (2.1) has the minimal and maximal solutions v,w in [uo.u>o] (*? C*[I,E] . (That is ,if u G [uo.^o] n C4[/, ￡^ is any solution of (1) , which satisfied v(t) < u{i) < w{t),v"{t) > ?"(*) > w"(t),V t € /) And the following monotone sequences {vn(t)},{wn(t)} converge uniformly v(t) ,uJ(t)m7:vn(t)= f G(t,s)f(3,vn-1(s)X-i(s))ds, VteJ,(n = 1,2,3,...), (2.4)l/t^n-iW^^W)^, Vt €/,(? = 1,2,3,...), (2.5)'0We apply the results to the following example : Example 2.1 : consider the boundary value problems :f ?W (f) = -?"(*) + (u(t) + I)2 + aimrt - 1, t G / = [0,1],< (2.25)( u(0) = u(l) = u"(0) = u"(l) = 0.We proof: boundary value problems(2.25) has the minimal solutions C4 and the maximal solutions C4 . in 0 < u(t) < sinirt , —K2sinnt < u"(t) < 0 (i￡/=[0,1])...