The Existence Of Solutions For Several Kinds Of Boundary Value Problems Of Nonlinear Differential Equations

The world is nonlinear in essence. Because nonlinear phenomena is studied by nonlinear theories and methods, every field becomes nonlinear and then nonlinear mechanics, nonlinear optics and nonlinear mathematics appear. Since the development of physics and applied mathematics calls for the global and high level development of the mathematics ability of analyzing and controling objective phenomena, nonlinear functional analysis which is one of the most important research fields in modern mathematics is formed by the continuously accumulation of nonlinear results. Until 1950's, nonlinear functional analysis has initially formed a theory system. In recent years, because nonlinear functional analysis has been an important tool for studying the nonlinear problem in mathematics, physics, aerospace engineering, biology engineering, it is greatly significant in the theory and application to study nonlinear functional analysis and its application.Since the 20th century, the development of nonlinear functional analysis has achieved the great breakthrough. L. E. J. Brouwer had established the conception of topological degree for finite dimensional space in 1912. Then J. Leray and J. Schauder extended the conception to completely continuous field of Banach space in 1934, afterward E. Rothe, M. A. Krasnosel'skii, P. H. Rabinowitz, H. Amann, and K. Deimling carried on embedded research on topological degree and cone theory. Many well known mathematicians in China, for example, Guo Dajun, Zhang Gongqing, Chen Wenyuan, Ding Guanggui, Sun Jingxian etc., had proud works in various fields of nonlinear functional analysis(See [1-12]).The method to research nonlinear problems mainly has topological degree method, critical point theory, partial order method, lower and upper solution method, fixed point theory, coincidence degree theory, monotone iterative technique, topological transversal degree and so on. The main questions to research are the existence of solution for nonlinear operator equation, uniqueness o solution, multi-solution, structure of solution, approximate solution, divergent theory of solution, iteration arithmetic, nonlinear operator theory as well as the application for partial differential equation, differential equation, integral equation and differential-integral equation. All these problems are among the most active domain in analyzing mathematics at present. Among them, firstly, singular boundary value problem of nonlinear differential equations. It has resulted from the applied disciplines of nuclear physics, hydromechanics, boundary layer theory, nonlinear optics and so on. It is an important research field of differential equations fields. Because it plays a very extensive and important role in the fields of physics, mathematics, aerospace engineering, biology engineering and so on, it has received high attention of numerous mathematicians. By applying the theories and methods of nonlinear functional analysis, the numerous famous mathematicians in the world have deeply studied the existence, uniqueness and multiplicity of solutions of singular boundary value problems and obtained lots of new results. However, because there are lots of difficulties in studying singular ordinary differential equations, at present it is still the advance orientation in the study of nonlinear analysis. Secondly, nonlocal boundary value problems for ordinary differential equations. The meaning of the nonlocal problems is that the definite condition of definite problem of ordinary differential equations not only depends on the value of solution in the end of interval, but also depends on the value of solution in some points of the interior of interval. Although lots of problems in theory and application can be reduced to nonlocal boundary value problems for ordinary differential equations, people started to fairly late study the nonlocal problems for the difficulties of nonlocal problems itself. Kiguradze, Lomtatidze(1984), Il'in and Moiseev(1987) began to discuss the existence of solutions of nonlinear multi-point boundary value problems for ordinary differential equations. Within the following ten years, the study on nonlocal boundary value problems for ordinary differential equations has been made great progress. However, it is not good enough and it is also a research topic to have a strong interest and maybe obtain some new significant achievements. Thirdly, system of nonlinear ordinary differential equations. Since lots of higher order differential-integral equations and implicit form equations can be reduced to the system of differential-integral equations by the appropriate variable substitution, the research of the system of equations plays a very important role in studying those equations.The present paper mainly investigates the existence of solutions for several kinds of boundary value problems of nonlinear differential equations and its application by using topological degree theory and cone theory. We obtain many new results which have the vital significance whether in the theory or in nature. The main contents are as follows:Chapter 1 gives some preliminary definitions and properties of nonlinear functional analysis, and gives several lemmas on the existence of fixed point, which play an important role in the next chapters.Chapter 2 considers the following singular nonlocal boundary value problems for systems of nonlinear second order ordinary differential equations:where a_i∈C((0,1),[0,+∞)),a_i(t) are allowed to be singular at t = 0 or t = 1; f_i:[0,1]×[0,+∞)×[0,+∞)→[0,+∞),g_i:[0,+∞)×[0,+∞)→[0,+∞)are continuous and a₁(t)f₁(t,0,0) or a₂(t)f₂(t,0,0) is not identical to zero on (0,1);α_i≥0,β_i≥0,γ_i≥O,δ_i≥0, andρ_i=α_iγ_i+α_iδ_i+β_iγ_i>0; (?)u(s)dφ_i(s),and (?) v(s)dφ_i(s) denote the Riemann-Stieltjes integrals, i= 1,2. We shall givean integral expression of the solution for the above problem. Then by using the Leggett-Williams fixed point theorem, we obtain the existence of at least three positive solutions for the above systems if the nonlinear terms f_i and g_i(i = 1,2) satisfy growth conditions.Chapter 3 considers the existence of positive solutions for the following nonlinear fourth-order differential systemswhere f₁,f₂:I×R⁺×R⁺×R^-×R^-→R⁺ are continuous functions, and the u", v" in f₁ and f₂ are the bending moment terms which represent bending effect. I=[0,1],R⁺=[0,+∞),R^-=(-∞,0]. Applying the product formula for the fixed point index on product cone and the fixed point index theory, we study the existence of positive solutions for the above systems in the case that the nonlinear terms have the different features. Our results improve and extend some known results.Chapter 4 investigates the following singular third-order nonhomogeneous boundary value problems with nonlocal boundary conditionswhereλ∈(O,∞) is a parameter, a∈C((0,1),[0,+∞)) and may be singular att=0 or t=1;f:[0,1]×[0,+∞)×[0,+∞)→+[0,+∞),g₁,g₂:[0,+∞)→[0,+∞) are continuous; (?)u(s)dα(s), and (?)u'(s)dβ(s) denote the Riemann-Stieltjes integrals,α,βare increasing nonconstant functions defined on [0,1] withα(0)=β(0)=0. Then by using the fixed point theorem of cone expansion and compression, we obtain various results on the existence and nonexistence of positive solutions for the above problem if the parameterλ∈(0,∞).Chapter 5 considers the following 2p-order and 2q-order systems of singular boundary value problems with integral boundary conditions whereλ>0,μ>0,p,q∈N,a_i∈C((0,1),[0,+∞)),a_i(t) are allowed to be singular at t = 0 or t = 1,i=1,2;f,g:[0,1]×(R⁺)^p×(R⁺)^q→R⁺(R⁺= [0,+∞)) are continuous; a_η≥0,b_η≥0,c_η≥0,d_η≥0,ρ_η=a_ηc_η+a_ηd_η+b_ηc_η> 0, 0≤η≤p-1; andα_θ≥0,β_θ≥0,γ_θ≥0,δ_θ≥0,ρ_θ=α_θγ_θ+α_θδ_θ+β_θγ_θ>0, 0≤θ≤q-1. This chapter will give a simpler expression of Green's function for the above problem, and discuss it's properties. Then by using fixed-point theorem of cone expansion and compression type due to Krasnosel'skill, this chapter discusses the existence and multiplicity of positive solutions for the above systems ifλandμare in some intervals . The results obtained extend and complement some known results.Chapter 6 establishs various results on the existence of positive solutions to the following fourth-order singular boundary value problems with integral bound-ary conditions in abstract spaces under various weaker conditions by establishing a specially constructed cone and using the fixed point theory in cone for a strict set contraction operatorwhere a,b∈C([0,1],[0,+∞));f,g:[0,1]×P→P are continuous; a_i≥0,b_i≥0, c_i≥0,d_i≥0 andρ_i=a_ic_i+a_id_i+b_ic_i>0;m_i,n_i∈L¹[0,1] are nonnegative, i = 1,2.