Solvability Of Boundary Value Problem For Several Classes Of Nonlinear Ordinary Differential Equations

The study of multi-point boundary value problems for linear second order ordinary differential equations was initiated by Il'in and Moiseev^[17]. Gupta^[18] studied three-point boundary value problems for nonlinear ordinary differential equations. Since then, the existence of solutions for more general nonlinear multipoint boundary value problems have been studied by several authors. Lots of important results have been obtained, for example, [9 — 11] and references therein.In the first section, we consider the existence of multiple positive solutions of a class of singular nonlinear three-point boundary value problemwhere α≥ 0, β≥0, 0 < η < 1, 0 < κ < (α+β)/(αη+β), and ρ := α(1-κη)+β(1-κ) > 0, f can be singular at t = 0, t =1 and u = 0.This section establishes the existence of multiple positive solutions of singular nonlinear three-point boundary value problems (1.1.1)(1.1.2) by means of the fixed point index theorem on cones. The results presented in this paper essentially improve and generalize the known results of [1,2].For convenience, we list the following assumptions:There exists a constant σ > 0, such that 0 ≤ u ≤σ, implies q(u) ≤ Mσ, where M < [max] G(t, s) is defined by (1.2.2);(H₅) There exists a constant σ > 0, such that γσ ≤ u ≤σ , implies f(t, u)≥ mσ, where if 0 < k < 1, then , then The main results obtained in the section are the following:Theorem 1.3.1 Assume (H₁)(H₂)(H₄) hold, the problem (1.1.1)(1.1.2) have at least two positive solutions u₁,u₂, satisfying 0 < ||u₂|| < σ < ||u₂||.Theorem 1.3.2 Assume (H₁)(H₃)(H₅) hold, the problem (1.1.1)(1.1.2) have at least two positive solutions u₁,u₂, satisfying 0 < ||u₁|| < σ < ||u₂||.Remark 1.4.1 Our theorem do not require any monotonicity assumptions of f, and utilizing one kind of new method which is different of [1][2], so essentially improve and generalize the known results of [1,2].In the second section, we study the existence of positive solutions of the following nonlinear three-point boundary value problem:where 0 < η < 1.The existence of positive solutions is obtained by making use of the Kras-nosel'skii fixed point theorem of cone expansion-compressions type, the nonlinear term f in the section does not satisfy usual conditions in [9].The fundamental assumptions are: (H₃) 0 < αφ₁(η) < 1, where φ₁(t) is the unique solution of the linear boundary value problemThe main results obtained in the section are the following: Theorem 2.1.1 Assume (Hi)(H2)(H3) hold, if the following conditions are fulfilled:(i) There exist q{ —; '-?0 t 7^5 t a(ii) There exist 0 < cx < c2, 0 < r2 < 1 < t1} and hi G La([0,1], [0, +00)), /i2 e Lx([5,1 - 5], [0, +00)) such that \ (t,i) e [0,1] x [o,Cl],/(*, 0 > 92(0 - h2(t)lT\ (t, I) e[S,1-6]x [c2, +00),where 5, a are defined by (2.2.4), then the problem (2.1.1) has at least one positive solution.Theorem 2.1.2 Assume (Hi)(H2)(H3)hold, if the following conditions are fulfilled:(i) There exist q{ G C([0, +00], [0, +00)), i = 3, 4 such that, inA;(ii) There exist 0 < c4 < c3, 0 < r3 < 1 < r4 and /i3 € -^Ht0' !]> [°> +°°))> /i4 G IHI*. ! *]. [0>+°°)) such that/(U) < 93(0 + h3(t)l73, (t,l) E [0,1] x [c3,+oo), , (*,0 e[S,l-5}x [0,c4),where 5, i(t) is the unique solution of the linear boundary value problemr u"(t) + a(t)u'(t) + b(t)u(t) = 0, 0 < t < 1,\u(0) = 0, m(1) = 1.The main results obtained in the section are the following: Theorem 3.3.1 Assume (Hi)-(H4) hold, if there exist positive numbersa,b(a ^ b) such that cp(a) < aA, ip(b) > bB, then the problem (3.1.1) has atleast one positive solution u satisfyingmin{a, b} < \\u\\ < max{a, b}.Corollary. 3.3.2 Assume (Hi)-(H4) hold, suppose one of the following conditions holds:(1) (p0 < A, and ^ > B. (especially, B. (especially, <ï¿¡^ = 0 and ipQ = +oo). Then, the problem (3.1.1) has at least one positive solution u.Corollary 3.3.3 Assume (Hi)-(HA)h.o\&, suppose one of the following conditions holds:(1) /0 < A, and /^ > — (especially, max/0 = 0, and min/^ = +oo);(2) /^ < A, and /0 > ^ (especially, max/oo = 0, and min/0 = +oo). Then, the problem (3.1.1) has at least one positive solution u.Theorem 3.3.4 Assume (#i)-(#4) hold, there exist n + 1 positive numbers ai, ? ? ? ,fln+i and Qi < ?2 < ? ? â–  < On+i suppose one of the following conditions holds:(1) a2hA, 2k e {1, â–  â–  ? , n + 1};(2) ^(a2fc_x) > a2fc_i^, 2k - 1 € {1, â–  ? ? , n + 1}, (p{a2h) < a2kA, 2k e {1, ? â–  ? ,fi+l},Then, the problem (3.1.1) have at least n positive solutions Uk, and satisfying a-k < \\uk\\ < ak+i(k = 1, 2 ? ? ? n).Theorem 3.3.5 Assume {H{)-{H\) hold, suppose one of the following conditions holds: if (pQ < A, and ^o > B. (especially, <ï¿¡0 = Oand i/j0 = +oo). then the problem (3.1.1) have a sequence of positive solutions uk k = 1, 2, ? ? ?, such that \\uk\\ ->? 0.Corollary 3.3.6 Assume (Hi)-(H4) hold, suppose one of the following conditions holds: if ^ < A, and ipoo> B. (especially, f^ = 0 and V'oo = +°°)-then the problem (3.1.1) have a sequence of positive solutions uk k = 1, 2, ? ? ?, such that ||ufc|| -> +oo.Remark 3.4.1 If ?i = a, qjj = 0(z = 2, ? ? ?, m — 2). is the problem of [9].then the Theorem 3.1 of [9] become a special situation of Corollary 3.3.3 in the section, and the multiplicity of positive solutions are obtained in the section.In the fourth section, we investigate the existence of nontrivial solution for the (n-1,1) three-point boundary value problemu{n)(t) + f(t,u) = 0, 0 < t < 1,(4.1.1) (0) = au(ri), u(1)=Pu{ti), uw(0) = 0, i = 1, 2, ? ? ?, n - 2.where 0 < 77 < 1, a e R, @ e R, M = l-a-(/3- a)^'1 ^ 0, n > 3, / G C([0,1] x R,R). Under certain growth conditions on the nonlinearity /, several new existence results are obtained by using Leray-Schauder nonlinear alternative.The main results obtained in the section are the following: Theorem 4.1.1 Suppose f(t, 0) ^ 0, M ^ 0, and there exist nonnegative functions k, h G Ll[0,1] such that\f(t,u)\ < k(t)\u\ + h(t), a.e. (t,u) G [0,1] x R,Then, the problem (4.1.1) has at least one nontrivial solution u* G C[0,1].Theorem 4.1.2 Suppose f(t,O) ^ 0, M > 0 and there exist nonnegative functions k, h G I^fO, 1] such that\f(t,u)\ < k(t)\u\ + h{t), a.e. {t,u) G [0,1] x R.If one of the following conditions is fulfilled:(1) there exists p > 1 such that & h*{t)dt < [(1-Q-it-"g'where i(2) there exists p > -1 such that k(t) < ^"^K^r^l^"-^^. a-e- [0,1], and(3) there exists ,x > -2 such that k(t) < ^^f^r^f'^O- - t)?, a.e. t G [0,1], andmesH g TO 11 ? *(*) < (^ - 1)\(2 + ^)[1 - a - (fi - a)V^] _meS(i G [U, lj . /fcW < 2 a (/? a)?f-i + ^j I1 *M > ??(4) *(t) < ^1^-^, a.e. t G [0,1], andmes{t G [0,1] : k(t) 1 lJ w -a-(0- a)r)n-1 + \/3\r)n fThen, the problem (4.1.1) has at least one nontrivial solution u* € C[0,1].Theorem 4.1.3 Suppose f(t,O) ^ 0, M < 0 and there exist nonnegative functions k, h G Ll[0,1] such that\f(t,u)\ < k{t)\u\ + h{t), a.e. (t,u) e [0,1] x R.If one of the following conditions is fulfilled:(1) there exists p > 1 such that ^ W{t)dt < where ^ + ^ = 1.(2) there exists /, > -1 such that k(t) < a.e. t G [0,1], andm?{? e [0,1] = *(*) < ("-'W(3) there exists fi > -2 such that A;(t) < ^"^'gffjL^y^^""1^! - *)", a.e. t G [0,1], and(4) M0 < ^^-T^, a.e. t G [0,1], and G [0,1] : kit) < n![-*+a + (/?7Then, the problem (4.1.1) has at least one nontrivial solution u* G C[0,1].