Nonlinear Boundary Value Solution And The Existence Of Multiple Solutions For

Nonlinear functional analysis is a new branch of modern analysis math-matics. Many problems arise from models of chemical reators, neutron transport, population biology, contagious disease, economics, and other systems. We need to discuss the existence of nonnegative solutions with certain desired qualitative properties. The paper employs the cone theory, fixed point index theory, and Leggett-Williams fixed point theorem and so on, to investigate the existence of positive solutions of several classes of boundary value problem for differential equations. The paper is divided into four sections.In the first section, by using the topological degree theory, we prove the existence of solutions between lower and upper solutions for periodic boundary value problems of second order differential equations, where the lower and upper solution α and p satisfy relations: β < α.whereTheorem 1.2.1 Suppose that (1.1) has the low and upper solutions α, β satisfying β <α, and the following conditions hold:(A1) f: I x Rx R—> R satisfies Caratheodory condition , and for all A > 0, there exists hA G L1(I) such that for all t ∈ I and (u, v) G R2 with |u| ≤ A, |v|≤ A, we have |f(t,u,v)| ≤ hA(t);(A2) (?)g G L1(I) such that k(t, s) ≤ g(s), (t, s) G I xI;(A3) (?)M > 0, for all fixed u G R, we havef(t, u, v2) - f(t, u,v1) > M(v2 — v1), 0 ≤ v1 < v2Then PBVP(l.l) has a solution u(t) satisfying β(t) < u(t) < α(t).(A4) Suppose that u, u, v, v satisfy β(t) ≤u≤u≤ α(t), Tβ(t) ≤v < v < Ta(t), andf(t, u, v) - f(t, % v) < M(u -u) + N(v -v) a.ete I hold, where M, N > 0 satisfy 4tt2(M + 2ttA^A;o) < 1 and2nNk0) < 1.Theorem 1.3.1 Suppose that(Ax) - (A4) hold, there exist the monotone sequences {pn(t)}, {an(t)}, where /?0 = f3(t), a0 = a(t), such thatlim pn(t) = pit), lim an(t) = v{t)uninformly in t G /, and p, v are the maximal and minimal solutions of PBVP(l.l) between a and p.Remark 1.3.1 In this paper, we extend the periodic boundary value problem of first order of [24-26] to PBVP of second order.Remark 1.3.2 Many authors get the maximal and minimal solutions only under the condition: a < f3, but we give the solutions under the condition: (3 < a.In the second section, by the two-point expasion condition of fixed point for the mixed monotone operator, the existence of positive solution to a class of singular second-order boundary value problems:au(0) - (3u'(0) = 0, ju(l) + 6u'(l) = 0is first established.Suppose that the following conditions hold :(Bi) a, b : (0,1) —* [0,+00) is continuous, and a, b is singular at t = 0,1;(￡2) There exists t0 G (0,1), such that a(t0) > 0, b(t0) > 0 and Jo G{s, s)(a(s) + b(s))ds < 00;(B3) /, g : [0, +00) —> [0, +00) is continuous;(￡4) a, 7 > 0, 0, 6>Q, A = 7/? + cry + a5 > 0.Theorem 2.3.1 Suppose that the following conditions hold:(C\) f : [0, +00) —? [0, +00) is continuous, strictly increasing and ^+oo f(u) = +00;(C2) g : [0, +00) —y [0, +00) is continuous and decreasing; (C3) 3m, M > 0, such thatwhere ￡ = sup{6 > 0, ￡~￡G(t,s) > b ￡ G{t,s)ds}, then BVP(2.1) has a positive solution .Remark 2.4.1 We translate BVP(2.1) to a mixed monotone operator. In the present results they all require Uo < vo, so the BVP cannot be proved by the had method under the condition: Uo ^ ^o- But we get it.Remark 2.4.2 Prom theorem 2.2.1 and theorem 2.2.2 and proof of them, we know A(u, v) can only be a strictly increasing about u operator.Remark 2.4.3 If A(u, v) is a only strictly decreasing about v operator, we can get the same result by getting rid of (Ci), and suppose the same conditions hold to g , then theorem 2.3.1 hold .In the third section, by using the fixed point theorems of cone expansion and cone compression, we prove a necessary and sufficient condition for the existence of positive solution to a class of singular fourth-order boundary value problems:), 0 < * < 1,u(0) = u(l) = 0, (3-1)au"(0) - 6u"(0) = 0, cu"(l) - du"{\) = 0,Where p, q e C((0,1), [0,+oo)), and p, q are singular at t = 0, t = 1,/, p € C([0, +oo), [0, +oo)) satisfying:(￡>i) a>0, 6>0, c>0, rf>0, a + b > 0, c + d > 0, p =ac + ad + bc> 0;(D2) There exist contants Ai, /Hi(0 < Ai < /^ < 1), A2, /J2(0 < A2 < H2 < +00), co(0 < c0 < 1), such that for all t G (0,1), u G C(0, +00), we have(D2)' There exist contants Ai, >Lti(O < Ai < nx < 1), A2, /x2(0 < A2 < Hi < +00), Co(0 < Co < 1), such that for all t G (0,1), u e C(0,+oo), we haved?f{u) < f(cou) < cj?/(u);(D3) There exists Jo = [a, yS] C (0,1), and we mark mint€j0 p(t)f(l) = n > 0, minteJo q{t)g{l) = r2 > 0.Remark 3.1.1 If Co > 1, from (3.2) we know that for all t e (0,1), u G C(0, +00), we haveIn fact, if Co > 1, from (3.2) we get 0 < ￡ < 1.Remark 3.1.2 From {D2), we obtain that / is sublinear and g is super linear.Theorem 3.2.1 Suppose that (Di), (D2), (D3) or (Dx), (D2)', (D3) hold, and0 < f\p(t)f(l) + q(t)g(l))dt < -i-, (3.4)Jo Mothen the necessary and sufficient condition that the singular boundary value problem (3.1) has a solution in C3[0,1] is0 < / \p(t)f(G(s, s)) + q{t)g(G(s, s))]dt < 00,whereMo = max G(t,s),0