| In this paper, we apply theory of ordering, expansion and compression theory, to study some nonlinear BVPs. This paper is composed of five chapters.In chapter 1, is the introduction of this paper, which introduces the main contents of this paper.In chapter 2, we deals with he BVP for the one-dimensional p-Laplacianwhere φp(s) = |s|p-2s,p > 1, which is singular at u = 0, t = 0,1. Throughout this paper, we assume that:(H1) q ∈ e(0,1), for all t∈ (0,1), with q>0, and (H2) f : [0,1] × (0, ∞) →R is continuous. θ : R→ R is continuous and nondecreasing, and θ(0) = 0,(H3) There exists a nonincreasing sequence {pn} and satisfies linin→∞ = 0, and for all , where n = 3,4, ... ,(H4) There exists a function a ∈ c[0,1]for all t∈[0,1), with a > 0,-,u) + p(a'(t)Y > 0 (t,u) e (0, -) xth it(Hb) For all n = 3,4, ?■?, there exists a sequence of functions {/??} 6Wn{i)) < 0, 0( lim at [0,l]/S?(<) > Pn\[-,1), q(t)f(i,pn(t)) + Wn(t))' ≤ 0,(//6) sup{maxt6[o,i]/3n(<)|n = 3,4, ???} < +oo,(H7) ^(/o1 q(s)g(a(s))ds) < +oo,We have the main result:Thorem 2.2.2 Suppose (Hi) - (H7) hold, then the problem(2.1.1) has a solution u e c[0,1] Dcl(0? iJi^pOO e cl(°. 1), and for all t G [0,1], with?(/) > a(t).In chapter 3, we discuss the singular boundary problem for the p-laplacianax(0) - /3x'(0) = 0, 7.t(1) + Sx'(l) = 0 .where p(s) = \s\p~2s,p > 1, a, 7 > 0, /5, S > 0, / e C([0,oo)), fand a(t) have infinitely singularities, V 0. , a(t) = +00 , V i = 1, 2, ? ? ?0< J a(s)ds<+00. (3.2.0).and in every interval of[0,l] ,witha(<) 7^ 0.We have the main result:Thorem 3.2.2 Assume that condition (H)holds,exist,s{//,A.}^° ,, such that {tk+uh), k = 1,2,- ■? , let {/?*}￡!, and {/?*}￡,,such thatRk+l < μ^ik) < r(k) < A]rk < Rk, A: = 1,2-- - ,where Ai G (y,+oo). Furthermore, for each natural number k, assume that satisfies:(H2) f{x) < (A2/?Jt)p-1, V x e [0,7?,], where 0 < A2 < {(f + JiVjde)}-1,. Then the problem(3.1.1)havc {xj^^and satisfies r{ < \\xt\\ < Ru V i =In chapter 4, we discuss the boundary problem of the classes of p-laplacian equationsI (rhJr.')V 4- fit. x y) = 0 t G (0 1)(4.1.1) ,x,y)=0, te(0,l).= 0,7i0P(x(l)) + 6l(f>p(x'(l)) =0, () ' = 0.the existence of the positive solutions.where cf)p(s) = \s\p~2s, p > 1, a{ > 0, & > 0(z = 1,2) % > 0 5i > 0 (i =l,2)/,5ec([0,l]*[0,oc]*[0,cx),[0,oc)),and the following conditions hold:(H) Suppose x(t),y(t) 6 A', then .x(^ + y(0 > C(||.r|| + \\y\\) t € [(,1-Q where ( e (0, |) is const.Then we have the main result:Theorem 4.2.1 Suppose th,-r:> exists two different consts A and 77, such thatf(tt x(t), y(t)) < p{mx\) , 0 < * < 1 , o< x, y < ~, (4.2.1)g{t, x(t), y[t)) < 4>P{m7\) , 0 < i < 1 , 0 < x,y < -, (4.2.2)andf{t,x{t),y{t))>P{lri) , #<*(j>p(lT]) , e 0, such that(4.2.1) (4.2.2)hold, and one of the following conditions hold:(#3) fo(t) > MLe) , for 0 < * < (1 - 9), or ^0(<) > 0p(|), for 0 < i <(H2) foo(t) >Then the boundary value problem (4.1.1)-(4.1.2) have two positive solutions.Theorem 4.3.2 Suppose 377 > 0,such that one of(4.2.3)(4.2.4)hold, and one of the following conditions hold:<< |
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