The Positive Solutions For Three Order Three-point Singular Boundary Value Problems For Differential Equations

The boundary value problems for ordinary differential equations are very important. With the development of science and technology, all kinds of natural and marginal prob-lems in the fields of engineering, mechanics, astronomy, economics, cybernetics, biology etc can be described into the ordinary differential equations boundary value problems. As we all know, it is quite difficult to find out the solutions of the differential equations. So, the theoretical study of the existence of the solutions and their character from the theory attracts a great attention. With the constant development of the ordinary differ-ential equations, the study of multi point boundary value problems becomes increasingly active.The ordinary differential equations with multi point boundary value problems mean that the boundary conditions not only depend on the solution in the interval endpoints, but also depend on the solution in the range of interval points. For the reason, it can describe many important physics phenomenas more clearly, on the other hand, it can put many classical two point boundary value problems into the same framework. Therefore, it has important theoretical significance and application value and gets a great attention of many domestic and external mathematics workers. The concrete examples include the density of the same cross-section of different sub-branch of cable vibration problem in engineering and lots of problems in the theory of elastic stability. It is precisely because multi point boundary value problems have broad application background, and it pos-sesses important research value.The earliest research document of multi point boundary value problems was D.Barr, T.Sherman's paper in 1973. For the second order three point boundary value problems, Gupta, O'Regan, Ma and so on published many research achievements [3,14,15,16,21]. With more broad research of the second order multi point boundary value problems, three order multi point boundary value problems are becoming the enthuse research problems. But so far, the research of third order multi point boundary value problems is less. This provides vast space for the research of third order multi point boundary value problems.Third order boundary value problems for singular differential equations also have broad actual significance. It has broad application in the turbulent medium flow hole theory of temperament, the elastic beam vibration theory, the topological transversality theory, the physical universe, the plasma problem and so on. Early in 1988, Baxley pub-lished a paper which researched the third order singular boundary value problem using the topological transversality theory in SIAM.J.Appl.Math. We can see, the research for the existence and multi-solution of third order boundary value problems for singular differential equations has important theoretical significance and application value.In 1998, Douglas Anderson studied the third order three point boundary value problem where,η∈[1/2,1),f:[0,+∞)→[0,+∞) is continuous and for any x≥0,f≥0. This paper gave f some restrict conditions using the famous Legget and Williams fixed-point theorem, and got at least three nonnegative solutions of BVP(Ⅰ). In 2003, Yao also studied the third order three point boundary value problem. He enlarges the nonlinear term f into f(t,x(t)), where f:[0,1]×[0,+∞)→[0,+∞) is continuous. Using the cone Krasnosel'skii fixed-point theorem, there exists at least one positive solution. With additional conditions, at least n positive solutions exist. Therefore, this paper puts (Ⅰ) some extend spread. However, the nonlinear term f has no singularities. In 2005, Sun studied the third order three point boundary value problem where, f:[0,1]×[0,+∞)→[0,+∞), a:(0,1)→[0,+∞),λ>0. We can see f has no singularities, but a(t) is allowed to be suitable singular at t=0, t=1. This paper also uses the cone Krasnosel'skii fixed-point theorem and lets the nonlinear term f be superlinear or sublinear. Then BVP (Ⅱ) at least exists one positive solution or two positive solutions. In order to overcome the singularity of a(t), it leadinto an(t) which has no singularities. So, when n→+∞, an(t)→a(t) compined with approximation techniques Ascoli-Arzela theorem, the corresponding conculution can be obtained.In 2009, Yao studied the third order three point singular boundary value problem for differential equations whenλ=1 in BVP(Ⅱ) and the nonlinear term f:(0,1)×(0,+∞)→[0,+∞) is continuous. We can see that f(t,x(t)) is singular at t=0, t=1 and x=0 which enlarges the application of BVP (Ⅱ). In order to overcome the singularity of f, this paper defines the operator Tn→T which is consistent. Using the completely continuous of Tn, the completely continuous of T can be obtained, where, (Tx)(t)=∫01 G(t, s)a(t)f(t,x(t))ds. Compined with the cone tension and compression fixed-point theorem, it at least exists two positive solutions and n positive solutions. This paper restrict the nonlinear term in a limited subset, with other conditions, so BVP (Ⅱ) (whenλ=1) has positive solutions. However, how to get more intuitional existed conditions? This is the problem we are going to solve.From the above-mentioned papers, we can see, the nonlinear term is nonnegative in most of them. So the correspond operator is a cone mapping. Then we can use the cone fixed-point theorem to discuss the existence of positive solutions. If the nonlinear term is sign-changing, the corresponding operator may be not a cone mapping, and then we can't directly use the fixed-point theorem to discuss the existence of the positive solutions. So, we need to treat the nonlinear term, so that we can use the fixed-point theorem indirectly. In recent years, this kind boundary value problem attract lots of scientists'attention. Of course, there are many results.In 2003, Yao studied the third order three point semi-positione boundary value problem: where,η∈(1/2,1),f:[0,1]×[0,+∞)→(-∞,+∞) is continuous and maybe sign-changing. However, there exists an M>0 such that f(t,x)+M>0. Then whenλ∈(0,λ), using the cone tension and compression fixed-point theorem, BVP (Ⅲ) at least exists one positive solution. In 2004, Yu enlarged BVP (Ⅲ), considering the nonlinear term f(t,x) being sign-changing and singular at t=0, t=1, and x=0. Also in the same above-mentioned conditions, using the fixed-point theory, there at least exists two nonzero positive solutions. From now, we can see, in order to get the solutions existence,λplays the conformity, and it belongs a special interval. At the same time, Xu studied the second order three point semi-positione differential equations boundary value problems. With some conditions, there exists many positive solutions. Some ideas in my paper originate from this paper.In recent years, there are many workers considering the nonlinear term f not only releted to x, but also depending on x', even more higher-order derivative. For example, [5][6][11][28]. However, the third order three point boundary value problem for differential equations: is not being studied. In the second part of this paper, we discuss the problem, whereη∈[1/2,1). Using the fixed-point theory and the approximation techniques, we get that when f is nonsingular and singular, the existence of positive solutions is obtained. At last, we give some examples.In the first part of this paper, we consider the third order three point boundary value problem:Also using the fixed-point theory and the approximation techniques, we get that when the nonlinear term f is singular and sign-changing, there exists positive solutions. The results are broader than [23][29][31]. At last, we give some examples to prove the appli-cations of the theory.