Positive Solutions Of Nonlinear Singular Problems And Non-trivial Solution,

With the development of society and the progress of science and technology, moreand more nonlinear problems have been extensively investigated. As a result of this, aim-ing at dealing with various nonlinear problems, nonlinear functional analysis is not onlyof profound theoretical significance but also is widely applied. Based on various nonlinearproblems arising in mathematical sciences themselves and natural sciences, it provides uswith many methods for tackling all kinds of nonlinear problems, such as topological de-gree theory, cone theory, critical point theory and monotone operator theory. Naturally,the results established therein can be widely applied to various nonlinear equations, in-cluding di?erential and integral, as well as to computational mathematics, control theory,optimization theory, dynamical systems, economic mathematics, etc.The theory of boundary value problems for nonlinear ordinary and partial di?erentialequations is among the most active and fruitful fields. Such problems can find roots inapplied mathematics, physics, control theory, and other applied sciences. Therefore, theresearch of boundary value problems is not only of great theoretical significance and butalso of wide applicability. A boundary value problem for a nonlinear ordinary di?erentialequation, usually, can be transformed into an equivalent integral equation. This meansthat the study of integral equations is also important.In recent years, many papers have been published on boundary value problems forsingular ordinary di?erential equations, but with nonlinearities being merely singular atone or two endpoints of the interval [0,1] instead of being singular on the whole interval[0,1]. By using cone theory, topological degree theory and fixed point index theory, in thisthesis, we study existence of positive or nontrivial solutions for singular Sturm-Liouvilleboundary value problems, higher order singular problems with integral boundary condi-tions, and singular Hammerstein integral equations. The novelty of this thesis include twoaspects. First, our nonlinearities may be singular on the whole interval [0,1], in contrastto ones in the existing literature, which are only allowed to be singular at one or two end-points of [0,1]. Second, growth conditions of our nonlinearities, superlinear and sublinear,are described in terms of first eigenvalues of associated linear integral operators. Thismeans that our results presented here are optimal in some sense.This thesis contains four chapters.In Chapter One, by using the Krosnoselskii fixed point theorem, we study the ex- istence of positive solutions for the following singular Sturm-Liouville boundary valueproblem (?)ingular and is almost everywhere positiveon [0,1] with 01 g(t)dt > 0; a∈C1([0,1],R+),b∈C([0,1],R+),αi≥0,βi≥0,αi2 +βi2 =0(i = 1,2);α,βare increasing on [0,1] and right continuous on∈[0,1), left continuousat t = 1 withα(0) =β(0) = 0;γ0∈[0,π/2],γ1∈[0,π/2]. The novelty of this is twofold.First, our boundary conditions are expressed in terms of Riemann-Stieltjes integrals, incontrast to two-point or multipoint boundary conditions in the literature. Second, ourweight function g may be singular on the whole interval [0,1].In Chapter Two, by using topological degree theory, we first investigate the existenceof nontrivial solutions for the following higher order nonlinear singular boundary valueproblem (?)where f∈C([0,1]×R,R), and a∈L(0,1) may be singular on [0,1], with a > 0 a.e. [0,1]and 01 a(t)dt > 0.Next, by using the Guo-Krasoselskii fixed point theorem, we study the existence ofpositive solutions for the following higher order boundary value problem(?)Our Lemma 2 in this chapter corrects an error in monograph [13, section6.5.1] due to GeWeigao.In Chapter Three, by using the Krosnoselskii fixed point theorem, we first discussthe existence and multiplicity of positive solutions for the following nonlinear singular Hammerstein integral equation(?)where f∈C([0,1]×R+,R+), a∈L(0,1) may be singular and almost nonnegative every-where on [0,1] with 01 a(t)dt > 0, and k∈C([0,1]×[0,1],R+). Our main results for boththe superlinear case and the sublinear case are described in terms of first eigenvalues ofassociated linear integral operators, extending and improving the results in the existingliteratures essentially. As applications, we apply our main results to discuss the existenceand mutiplicity of positive solutions for a second-order singular Sturm-Liouville problem.Next, we study the existence of positive solutions for the following perturbed Hammersteinintegral equation(?)where f∈C([0,1]×R+,R+), and g∈L(0,1) is nonnegative and is allowed to be singularon [0,1] with 01 g(t)dt > 0; k∈C([0,1]×[0,1],R+); ?i∈C[0,1] satisfies ?i(t) > 0 forall t∈(0,1);αi is increasing on [0,1] and right continuous on∈[0,1), left continuous att = 1 withαi(0) = 0,i = 1,2,...,n.In Chapter Four, by using fixed point index theory, we study the existence of positivesolutions for the following systems of nonlinear second-order ordinary di?erential equations(?)where f1,f2∈C([0,1]×R+×R+,R+);αi is an increasing and nonconstant function on[0,1] withαi(0) = 0(i = 1,2,3,4); Hi∈C(R+,R+)(i = 1,2,3,4). Our main resultsobtained extend and improve ones in the existing literatures, and are di?erent from thecorresponding ones due to G. Infante and P. Pietramala in 2009.