Three-Point Boundary Value Problems For Differential Equations With Singularity On A Half Line

Second-order boundary value problems (BVPs) on infinite intervals , arising from the study of radially symmetric solutions of nonlinear elliptic equation and models of gas pressure in a semi-infinite porous medium, have received much attention, see and the references therein.Multi-point boundary value problems of second-order linear differential equations on a finite interval were initiated by V.A. Il'in and E.I. Moiseev and three-point, BVPs of nonlinear differential equations were studied by C.P. Gupta.Since then, more general nonlinear multi-point BVPs on finite intervals have been discussed extensively . The methods therein mainly depend on the Leray-Schaudcr continuation theorem, coincidence degree theory . However, few works are done for second-order multi-point BVPs on an infinite interval and these results are mostly on f without singularity.Few papers study existence of positive solutions for three-point boundary value problems on infinite intervals when non-linearity depends on x′and may change sign and may be singular. To fill the gap in this area, we discuss the existence of positive solutions for three-point boundary value problems on half-line, using the fixed point index theory. This paper is divided into two chapters.In chapter 1, we study the existence of positive solutions forthree-point boundary value problems for differential equations on half-line as followswhere 0 <α< 1,η∈(0, +∞),f may change sign and may be singular at·x = 0 and x′= 0.In [20] Hairong Lian, Weigao Ge studied the following second-order three-point BVP on a half-linewhereα≠1,η∈(0,+∞). With the help of the established Green function and the Leray-Schauder continuation theorem suitable conditions imposed on f(where |f(t,u,v)|≤p(t)|u| + q(t)|v| + r(t))are presented for the existence of solutions. In this papar, we consider the case that f may be change sign and singular. Using the fixed point index theory on a cone, we discuss the existence of positive solutions with less conditions imposed on f. So it's used more widely than the passage above and it is the improvement of theory for existence of positive solutions for three-point boundary value problems.It has a great role in theory and practice.Some ideas come from [20,21,24,25]and references therein.The paper is organized as follows. Firstly, we study the nonexistence of positive solutions of (1), then we discuss the existence of positive solutions when f is singular at x = 0 but not at x′= 0 and singular both at x = 0 and x′=0.The method we used is firstly constructing proper integral operators and use some new conditions to overcome the singularity and sign changing/Finally, using the fixed point index theory on a cone, we consider the set of the approximate solutions and obtain a convergent subsequence. The limit is a positive solution for equation (1).In chapter 2, we establish some simple criterions for the existence of single and two positive solutions of the three-point BVP for differential equations on half linewhere 0 <α< 1,η∈(0, +∞) .In[19], Bing Liu studied the existence of single and multiple pos itive solutions to the three-point boundary value problem as followsIn this papar, we discuss the case in an infinite interval.In order to overcome the difficulty from the unboundedness of the interval, we constructed the limit funtions f₀,f_∞and f⁰,f^∞,and then we use some new conditions to deal with the singularity caused by the nonlinearity. It is worth noting that the provement of lemma 2.1.4 provided an important theoretical conditions for the following discussion .Firstly, we discuss the existence of single positive solution for the BVP(2) under f⁰ = 0, f_∞, or f₀ =∞, f^∞= 0. Then, we establish the existence conditions of two positive solutions for the BVP(2) under f₀ = f_∞=∞or f⁰ = f^∞= 0.After that, we obtain some existence results for positive solutions of the BVP(2) under f₀, f_∞, f⁰, f^∞(?) {0,∞}. Finally, we give some examples to illustrate our results. When we study the above problems,we mainly used the Krasnosel-skii's fixed point theorem in. a cone.