Fixed Point Theorems And Nontrivial Solutions Of Singular Equation

Semiclosed 1-set-contractive operators are a class of new operators, and many people are interested in these operators. We mainly discuss fixed point theorems of semiclosed 1-set-contractive operators.In the second chapter of this thesis, we study fixed point theorems of semiclosed 1-set-contractive operators in Banach spaces. There are many research methods about this class of operators, but the method in this thesis is a little different from others. The conditions and relevant expressions in the literature in the past are in the form of norm, but here we replace norm in common use with some convex functional p defined on a cone. We use inequalities given by this functional to obtain some new fixed point theorems by. fixed point index theory.At the same time, the third chapter of this thesis makes some complements of the results in Shaoyuan Xu [4]. Besides discussing the existence of fixed point under the condition ofα≥1,β>0, we also discuss whether the fixed point exists under the conditions of 0<α<1,β≤0 orα<0.Recent years, nonlinear boundary value problems of singular ordinary differential equations and systems are always the focal points that are studied and discussed. People endow the boundary with various different conditions when considering the existence of solutions, then obtain many results. People are interested in the existence results of positive solutions for singular second-order second-point boundary value problems. In the third chapter of this thesis, we are concerned with nontrivial solutions for singular sublinear second-order multi-point boundary value problems.It is necessary when u≥0, f(u)≥0 in this class of papers in the past. But this condition is weakened in this thesis, that is,f can be nonnegative.Singular sublinear second-order m-point boundary value problems: where h(x) is allowed to be singular at x=0 and x=1, and f is not necessary to be nonnega-tive.In the preceding works, this problem is studied by the methods of upper and lower solutions, Schauder's fixed point theorem or the fixed point index, but we obtain the existence results. of nontrivial solution, by means of the topological degree theory under some conditions concerning the first eigenvalue corresponding to the relevant linear operator.