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The Existence of Solutions for Several Classes of Nonlinear Third-Order Differential Equations Boundary Value Problem

Author: SongWenDong
Tutor: QiShiShuo
School: Zhengzhou University
Course: Applied Mathematics
Keywords: positive solution Cone Boundary value problem Systems of ordinary differential equations The fixed point theorem in cone Green function Leray-Schauder fixed-point theorem the Leggett-Williams fixed point theorem
CLC: O175.8
Type: Master's thesis
Year: 2008
Downloads: 12
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Abstract


Nonlinear functional analysis is an important branch of modern analysis mathematics, because it can explain kinds of natural phenomena, more and more mathematiciansare devoting their time to it. Among them, the nonlinear boundary value problem comesfrom a lot of branches of applied mathematics and physics, it is at present one of the mostactive fields that is studied in analysis mathematics. The present paper employs the conetheory, fixed point theorem and so on, to investigate the existence of solutions to boundaryvalue problem of several kinds of nonlinear differential equations. The obtained results areeither new or intrinsically generalize and improve the previous relevant ones under weakerconditions.The thesis is divided into four chapters according to the contents.Chapter 1 is divided into tow sections according to the contents. In section 1, by using the fixed point theorems in cones the existence of positive solutions for the third-ordertow-point boundary value problems :(E) u’’’(t) =λf(t,u)+μg(t,u),0<t<1. Whereλ,μ>0 are parameters.f,g :[0,1]×[0,+∞)→R are continuous ,in the following any boundary value conditions:(c1) u(0) = u’(0) =u(1) = 0, (c2) u(0) = u’(0) = u’(1) = 0, (c3) u(0) = u’=u"(1) = 0,(c4) u(0) = u"(0) = u(1) = 0,(c5) u(0) = u"(Q) = u’(1) = 0, (c6) u’(0) = u"(0) =u(1)=0.Which nonlinear f,g are both semi-positive is obtained under the condition thatf,g are all super-linear (sub-linear), or one is super-linear, the other is sub-linear. Theresults obtained in this section essentially improve , generalized many well-known results.In section 2, several existence results of solution and positive solution are obtained for thesemi-linear third-order two-point boundary value problem the equivalent norm the Leray-Schauder fixed-point theorem and the system of integralequations.In chapter 2, by using the fixed point theorem in cone, the existence of multiplepositive solutions are given to singular boundary value problems of a class of third-orderthree-point differential equations :λ1∈(0,1),λ2∈(1,+∞),0<η<1,1<α<1/η. At least two positive solutions areestablished.a(t) :(0,1)→[0,+∞) is continuous and may be singular at t = 0 and/or t = 1.The chapter 3 is concerned with boundary value problems for systems of nonlinearthree-order differential equations. Under the suitable conditions, the existence and multiplicity of positive solutions are established by using the fixed-point theorem in cone.In chapter 4, the existence of multiplicity of positive solutions for the boundary valueproblem is presented by means of the property of the corresponding Green’s function andthe Leggett-Williams fixed point theorem.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Boundary Value Problems
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