Existence Of Solutions For Singular Differential Equations In Banach Spaces

As we know, singular initial and boundary value problems arise in the fields of gasdynamics ?uid mechanics, nuclear physics, the theory of boundary layer, nonlinear opticsand so on. Some mathematical models deriving from some realistic problems is defined onfinite interval or infinite interval, for example, some problems in quantum mechanics andthe optimal cybernetics are considered on infinite interval. In addition, the function orvariable itself of these mathematical models is may be singular at the end point. Therefore,singular initial and boundary value problem is one of the important problems attractingthe attention of mathematicians and other technicians. The existence and uniqueness ofsolutions for singular di?erential equations have been considered extensively in the lasttwenty years([1]-[5],[8]-[38]). The methods are mainly topological degree and cone theoryor method of approximation. This paper also discusses such problems more generally onthe basis of above references.Chapter 1 investigates initial value problems of first order integro-di?erential equa-tions on half-line. In paper [1], professor Dajun Guo has discussed the existence of so-lutions of initial problems for impulsive integro-di?erential equations on finite intervalby iterative monotone technique in a Banach space. Paper [2] studied the existence ofinitial problem of mixed type on infinite interval by fixed point index theory. First, weconsider initial value problems for first order nonlinear (singular) finite impulsive integro-di?erential equations of mixed type on half-line in a Banach space and obtain the existenceof solutions. Our goal is to improve and extend the results of paper [1] and [2]. The ap-proach we used here are kuratowskii noncompact measure, Mo¨nch fixed point theorem.Secondly, we discuss initial value problems for a class of nonlinear singular infiniteimpulse integro-di?erential equations of mixed type on half-line in a Banach space, espe-cially the case that the nonlinear term is unbounded. By using kuratowskii noncompactmeasure and Sadovskii fixed point theorem under suitable conditions, we get the existenceof general solution (including unbounded solution) and uniqueness result.Chapter 2 investigates higher order singular boundary value problem (SBVP, forshort) on finite interval. In recent years, the positive solutions of SBVP to higherorder nonlinear di?erential equations have been studied extensively ([16],[19],[23],[24],[27] ? [33]). In superlinear case, paper [24] and [27] obtained a necessary and su?cientcondition for the existence of C2[0,1] or C3[0,1] positive solutions of a class of di?erentialequations under some given conditions. In sublinear case, paper [28] gave a necessaryand su?cient condition for the existence of C2[0,1] positive solutions as well as C3[0,1]positive solutions by means of the method of lower and upper solutions with the max-imum principle for SBVP. In [33], Zhang discussed a necessary and su?cient conditionfor the existence of C4n?2[0,1] or C4n?1[0,1] positive solution of a class of a higher ordersingular homogeneous boundary value problem by using the method of lower and uppersolutions. Our goal is to improve and extend the results in ([24] [27] [28] [33],etc). Firstly,by constructing some special cones and using fixed point theorem of cone, some necessaryand su?cient conditions for the existence of C4n?2 positive solution of general boundaryvalue problem. Secondly, we discuss some necessary and su?cient conditions of C4n?1positive solution.