The Boundary Problems Of Nonlinear Differential Equation

Nonlinear functional analysis has been one of the most important branches of learning in modern mathematics at present. It provides an effect theoretical tool for studying many nonlinear problems. It plays an important role in dealing with nonlinear equations and differential equations arising in applied mathematics.Much attention is attracted by questions of existence of positive solutions for BVP of differential equation on measure chains,but most papers considered the existence of positive solutions about non-singular differential equations. Little attention is on questions of existence of positive solutions for BVP of singular differential equation on measure chains and the relevant papers are few.For the demand of practical problems , the far research on questions of existence of positive solutions for BVP of differential equation on measure chains has certain value.The existence of positive solutions for second order m-point boundary value problems has been studied by many authors,but in most cases the nonlinear term is superlinear or sublinear.In this paper the existence and multiplicity theorem are estabished bu using the first evigenvalue of the corresponding linear problem.Many results on convex operators or concave operators are obtained,but they are changed to α concave operators or —α convex operators and little research is on α convex operators or —α concave operators because α convexoperators or — a concave operators describe superlinear problems.Recently literature^,23,24] had studied these operators.In this paper we consider the system of second-order boundary value problems. Using cone theorem,fixed point theorem Krasnosel'skii's fixed point theorem andupper and lower solutions method, we establish the existence of one or more solutions for the system.The paper is divided into four chapters according to content.The first hapters is concerned with singular differential equations on measure chains.The boundary vaiue problem' uAA+m(t)f(u(a(t))) = 0, te[0,l] (1.1.1)u(0) = 0 = u{a(l)) (1.1.2)u(0) = 0 = uA(a(l)) (1.1.3)is considered under some conditions concerning the first eigenvalue of the relevant linear operator. The existence of positive solutions is obtained by means of fixed index theory. Main Results:Theorem 1.3.1 If (Hi) — (H3) and the following conditions hold,lim inf^- > Xi (1.3.1)u>+0 Ulim sup^- < Xi (1.3.2)u-^+oo uwhereAiis first eigenvalue of T,then BVP(l.l.l) and BVP (1.1.2) have at least one positive solution.Theorem 1.3.2 If (Hi) — (H3) and the following conditions hold,lim sup^- < Ai (1.3.6)>+0 uulim inf^ > Ax (1.3.7)whereA^s first eigenvalue of T,then BVP(l.l.l) and BVP (1.1.2) have at least one positive solution.Theorem 1.4.1 If (Hi) — (H3) and the following conditions hold,lim inf-^- >Ulim sup-----< Aiu—>+oo UwhereAiis first eigenvalue of T.then BVP(l.l.l) and BVP (1.1.3) have at least one positive solution.Theorem 1.4.2 If (Hi) — (H3) and the following conditions hold,f(u) lim sup—— Xxu-?+oo UwhereAiis first eigenvalue of T,then BVP(l.l.l) and BVP (1.1.3) have at least one positive solution.In the second chapter, using Krasnonel'skii's fixed point theorem and Leray-Schauder degree, we studied the existence of positive solutions for second order m-point boundary value problemsf(t,u(t)) = o te[o,i]m-2where & G (0,1) satisfing 0 < fi < ï¿¡2 < ? ? ? < ï¿¡m-2 < 1m-2a< G [0, +oo) (i = 1, 2, â–  ? ? m - 2) and 0 < ^ a, < 1;i=l/ec([0,l] x [0,+oo),[0,+oo))Main result:Theorem 2.3.1 : If one of the following conditions holds, (I) lim sup max —'— < Ai and lim inf min —-— > Aiu->+0 te[0,l] U - u-y-t-oo t€[0,l] 14(//) lim sup max —-—- < Ai and lim inf min —-— >u>+oo te[o,i] u u->+o te[o,i] u)[] []then BVP (2.1.1) has at least one positive solution.Corollary 2.3.1 : If one of the following conditions holds,(I) lim sup max —'— = 0 and lim inf min —■— = +oo ?>+o te[o,i] w u->+oo te[o,i] u(II) lim sup max ' = 0 and lim inf min ———- = +oou>+oo tï¿¡[0,l] U u->+0 te[O,l] U[] []then BVP (2.1.1) has at least one positive solution.If we note <ï¿¡(/) = max{f(t, c) : 0 < t < 1, 0 < c < I},(2.1.1)y?(Z) = min{f(t, c) : n < t < is, jl 2 < \x < v < 1 is a constant.\ -1A =i r1/, B =7m-2A"obviously , 0 < A < B.Then we have the following theorem:Theorem 2.4.1 If lim inf min ^ ' > Ai, lim inf min l±l—L > Aiu->+0 ï¿¡ï¿¡[0,1] U u-++oo t€[0,l] Uand there is a a > 0 satifing (a) < a A, then BVP (2.1.1) has at least two positive solution.Theorem2.4.2 If lim sup max ' - < Ai, lim sup max ——— < Ai u-n-o te[o,i] u u->+oo te[o,i] uand there is a b > 0 satifing ip(b) > bB, then BVP (2.1.1) has at least two positive solution.In the third chapter,some existence result of positive fixed point for —a concave operator is given by means of fixed point theorem of cone expansion and compression. And then,they are used to superlinear integral equations.Let E be real Banach space where 9 denote zero element in E and P is a cone in E , P+ = P/9. Every cone P in E defines a partial ordering in E given by x < y if y — x G P.Main result:Theorem3.2.1 let(Hi) Ei(i — 1, 2) be ordering Banach space , Pi is a cone in E{ .P1 is normal , and the maximal element in unit ball in E\(H2) F : Pi —*? P2 is strict decreasing —a concave operator where (a > 0), and when x ^ #,we have Fx > 9.(H3) B : P% —? Pi is a linear operator , when x ^ 6 we haveBx > 9. and there are e € Px+ and real number e0 > 0 satisfingBx > e0 || x || e, Vx € P2+ (3.2.1)(#4) A = BFis a completely continuous oprator . then A have positive fixed point.Theorem3.2.2 Let E be realBanach space , P is a normal cone in E,A : P —y Pis —a concave completely continuous oprator ,(a > 0) and|| : ||x| = 1} < 00, in/{||^lx|| : ||a;| = 1} > 0then A have at least one positive fixed point in P.Corollary 3.3.1 Let(H[) Ei(i — 1, 2) be ordering Banach space , Pi is a cone in 2% . Pi is a solid cone and the maximal element in unit ball in E\(H2) F : Pi —>â–  Pi is decreasing continous —a concave operator where (a > 0), and when x ^ 0,we have Fx > 9.(H'3) B : P2 —> Pi is a linear compeletly continous oprator , when x ^ #,we have Bx > 9.and there are e € P* and real number Co > 0 satifing ( 3.2.1 ). then^l = BF have at least one positive fixed point .Theorem3.4.1If there is a r\ > 0 and a e(x) > 0 where e(x) is continous and not zero every satisfingVe{x)e(y) < K{x, y) < e{x), Vx,y G G (3.4.2)di(x) is non-negative measurable in G , a, > 0, i — 1,2 ? ? ? n,essential submume(x) 2^ai(x)dx < +oothen integral equation ( 3.4.1 ) have continous positive solution.In the fourth chapter ,by using iterative techniques theorem of fixed point of nonlinear operator equations is established. And then,they are used to integral equations in Banach spaces.Main result:theorem 4.2.1 Let E be a real Banach space,P is a normal cone in E, u0 e E, D = {ue E\u > uQ},A : D —> E, if there exists a bounded linear positive operator T,L : E -> E such that-T(x2 - xi) < Ax2 - Axi < L(^2 - xi), (4.2.1)every 21,22 € D and xi < 22,if the following conditions hold(i) u0 < Auo(ii) (J + r)j>^ieP(iii) TL=LT and the spectral radius of linear operater T, L satisfingr{T) < 1, r(L) + r(T) < inf{\\\ : A G a(I + T)}then A have only one fixed point u* in D,and for every x0 ï¿¡ D we havexn —> n* (n —>â–  oo),wehre xn = (/ + ^"^Axn-i + Txn_i), (n = 1, 2 ? â–  ?), and for every 5,inf{\X\:\ecx(I there exists nosuch that when n > no- ?*Theorem4.2.2 Let E be a real Banach space,P is a normal cone in E, voe E, D = {ue E\u < v0},A : D —> E, if there exists a bounded linear positive operator T, L : E —> E such that-T(x2 - xi) < Ax2 - Axi < L(x2 -xi),every Xi,x2 G D and xi < X2,if the following conditions hold (i) AvQ < vQ(ii) (7 + T)x > 6 => x G P(iii) TL=LT and the spectral radius of linear operater T, L satisfingr{T) < 1, r{L) + r{T) < inf{\\\ : A G a {I + T)}then A have only one fixed point w*,and for every Xq G D we have :v* (n -> oo),where xn = (7 + T)¹(Ar7i_i + Txn_i), (n = 1,2 ? ? ?), and for every 6,_<sub><sub><sub><sub>^) + r(T)_<sub><sub><sub><sub><sub><5<1m/{|A| : AG noNSn\\xn v"\\ < N5n\\x0 - vo\ y Corollary4.2.1 If we change ( 4.2.1 ) to—M(x2 — X\) < Ax2 — Axi < L(x2everya;i,X2 G D and X\ < x2,where M > Ois a conatant, L : E —>? E'is a bounded linear operator , and r(T) < 1 then theorem 4.2.1 , theorem 4.2.2 hold ,,Corollary4.2.2 If we change r(T) < 1 in 1 to ||L|| < 1 then the result holds also.