Several Types Of Nonlinear Problems, Positive Solutions And Applications

Various kinds of nonlinear problems arise in many fields of natural science, en-gineering technology and social science, such as physics, ecology and economics. Forexample, the vibrations of a guy wire of uniform cross-section and composed of Nparts of di?erent densities can be set up as a multi-point boundary value problem;many problems in the theory of elastic stability can be handled by the method ofmulti-point problems. During the development of solving such problems, nonlinearfunctional analysis has been becoming one of the most important research fields inmodern mathematics. It mainly includes partial ordering method, topological degreemethod and the variational method. Also it provides a much e?ect theoretical toolsfor solving many nonlinear problems in the fields of the science and technology. Andwhat is more, it is an important approach for studying nonlinear integral equations,di?erential equations and partial equations arising from many applied mathematics.L.E.J.Brouwer had established the concept of topological degree for finite dimensionalspace in 1912. J.Leray and J.Schauder had extended the conception to completelycontinuous field of Banach space in 1943, afterward E.Rothe, M.A.Krasnosel'skii,P.H.Rabinowitz, H.Amann, K.Deimling had carried on embedded research on topo-logical degree and cone theory. Many well known mathematicians in China, say ZhangGongging, Chen Wenyuan, Guo Dajun, Sun Jingxian etc., had profound works invarious fields of nonlinear functional analysis.For di?erential equations, many authors are particularly interested in positivesolutions, which are a class of practical solutions. The study of existence of positivesolutions to di?erential equations is often transformed into investigating the existenceof fixed points for integral operators on a cone. The theories most used for the re-search into the existence of fixed points for integral operators are that of the degree ofnonlinear functional analysis and that of fixed point index. And the theorems widelyused are Schauder fixed point theorem, Krasnosel'skii fixed point theorem, Leggett-Williams fixed point theorem and its generalization—the five functional fixed pointtheorem.Despite the fact that many authors have studied the existence of positive solutionsby these theorems and obtained many achievements, in order to use these familiar fixedpoint theorems, we need to assume that the nonlinear terms are continuous and theGreen's functions must satisfy given conditions, which makes the range of using thesetheorems restricted, so there are yet many challenging questions to be solved.This paper includes six chapters. From Chapter One to Chapter Five, we mainlyutilize the method supersolution and subsolution, fixed point index theory on a cone and fixed point theorems of expansion and compression type on a cone to study the ex-istence, uniqueness and multiplicity of positive solutions for some nonlinear problems,including boundary value problems for systems of nonlinear nth order ordinary di?er-ential equations, a system of generalized Lidstone problems, Sturm-Liouville boundaryvalue problems on the half-line, a class of singular fractional boundary value problem,a fourth order p-Laplacian boundary value problem and a system of nonlinear Ham-merstein integral equations.In Chapter One, we study the existence and multiplicity of positive solutions forthe following system of nonlinear Hammerstein integral equations:(?)where k∈C([0,1]×[0,1],R+), f,g,h∈C([0,1]×R+×R+×R+,R+)(R+ := [0,+∞)).Based on a priori estimates achieved by using Jensen's integral inequality, weobtain our main results under the assumptions on the nonlinearities, which are mostlyformulated in terms of spectral radii of associated linear integral operators. Concavefunctions are utilized to characterize coupling behaviors of f,g and h, so that theyhave different growth manners. The methodology, which uses the nonnegative concavefunctions to study the systems of di?erential equations, origins from [1]. [2,15] extendand improve the corresponding ones in [1], furthermore, our main results here extendand improve the corresponding ones in [1,2,15]. We deal with the above systems withthree equations, and the results imply that we can extend and improve the generalizedones, which the systems include n nonlinear integral equations. Our main resultscan be applied to a wide variety of systems of boundary value problems for ordinarydi?erential equations of second order and higher order. As application, we use our mainresults to establish the existence and multiplicity of positive solutions for a system ofhigh order boundary value problems for nonlinear ordinary di?erential equations.In Chapter Two, we study the existence and multiplicity of positive solutions forthe system of the generalized Lidstone boundary value problems(?) where m≥1,n≥1, f,g∈C([0,1]×Rm+ +n,R+)(R+ := [0,+∞)), a,b,c,d∈R+with ac + ad + bc > 0. In this chapter, we continue to use the method of ChapterOne, that is, concave functions are utilized to characterize coupling behaviors of thenonlinearities. Note that our study is on systems of generalized Lidstone problems andthe system contains two equations, which can be of di?erent orders. To overcome thisdi?culty, we first use the method of order reduction to transform the above systeminto an equivalent system of integro-integral equations, then prove the existence andmultiplicity of positive solutions for the resulting system under appropriate conditions,thereby establishing our main results for the above system.In Chapter Three, we consider the existence of positive solutions for Sturm-Liouville boundary value problems on the half-line:(?)where f∈C([0,+∞)×[0,+∞),[0,+∞)); h : (0,+∞)�'(0,+∞) is a Lebesgue inte-grable function and may be singular at t = 0; p∈C[0,+∞)∩C1(0,+∞) with p > 0on (0,+∞) and 0+∞p(1s)ds < +∞; a,b,c,d≥0 withρ= bc + ad + ac 0+∞p(1s)ds > 0.We also consider the problem under some conditions concerning the first eigenvaluescorresponding to the relevant linear operators, and the nonlinearity f may grow super-linearly or sublinearly. Our main results here extend and improve the correspondingones in [38–42]. At last in this chapter, we give some examples to illustrate our results.In Chapter Four, we study the existence and multiplicity of positive solutions forthe singular fractional boundary value problem(?)whereα∈(3,4] is a real number, D0α+ is the standard Riemann-Liouville derivative,f∈C([0,1]×[0,∞),[0,∞)), and h∈C(0,1)∩L(0,1) is nonnegative and may besingular at t = 0 and/or t = 1. Under some conditions concerning the first eigenval-ues corresponding to the relevant linear operators, we obtain the existence of positivesolutions for the above problems. It should be remarked that our nonlinearity f here,unlike the f in [43], may be both sublinear and superlinear. Our first theorem involvesthe existence of at least one positive solution for the above problems. with f grow-ing superlinearly, thereby complementing the results in [43]. We then establish twoexistence results of twin positive solutions for the above problems, two results thatessentially improve and extend the corresponding ones in [44]. In Chapter Five, we study the existence, multiplicity and uniqueness of positivesolutions for the fourth order p-Laplacian boundary value problem(?)Here p > 0 and f∈C([0,1]×R+,R+). Although the methodology origins from [20],our main results here extend and improve the corresponding ones in [20]. Viewing theabove problem as a perturbation of the fourth order Lidstone problem, we use fixedpoint index theory to establish our main results based on a priori estimates achievedby utilizing properties of concave functions. Our methodology and results in this paperare new.In Chapter Six, we first study the existence positive solutions for the followingsystem of singular generalized Lidstone boundary value problems(?)a1x(2i)(0) ? b1x(2i+1)(0) = c1x(2i)(1) + d1x(2i+1)(1) = 0 (i = 0,1,...,m ? 1),a2y(2j)(0) ? b2y(2j+1)(0) = c2y(2j)(1) + d2y(2j+1)(1) = 0 (j = 0,1,...,n ? 1).where m,n≥1, a(t),b(t)∈C((0,1),[0,+∞)), a(t) and b(t) are allowed to be singularat t = 0 and/or t = 1; fi∈C([0,1]×Rm+ +n,R+)(R+ := [0,+∞)); ai,bi,ci,di∈R+withρi := aici + aidi + bici > 0, i = 1,2. The features of this chapter mainly includethe following aspects. Firstly, our study is on systems of singular generalized Lidstoneproblems. Secondly, a and b are allowed to be singular at t = 0 and/or t = 1. Thirdly,the system contains two equations, which can be of di?erent orders.We second consider the existence of at least three positive solutions for the fol-lowing singular Sturm-Liouville integral boundary value problems for second-orderordinary di?erential equations.(?)where a,b,c,d≥0 with ac + ad + bc > 0; h∈C((0,1),[0,+∞)) may be singular att = 0 and/or t = 1, and 01 h(s)ds <∞; f∈C([0,1]×[0,+∞)×(-∞,+∞),[0,+∞));ξ(t) andη(t) are increasing on [0,1] and right continuous on∈[0,1), left continuousat t = 1, withξ(0) =η(0) = 0; 01 u(t)dξ(t) and 01 u(t)dη(t) denote the Riemann-Stieltjes integrals of u with respect toξandη, respectively. It is worth mentioningthat the Riemann-Stieltjes integral includes as special cases the multi-point boundary value problems and integral boundary value problems. That is why many authors areparticularly interested in Riemann-Stieltjes integral boundary value problems. At lastin this chapter, we give some examples to illustrate our results.