Differential Equations In Banach Space, The Existence And Application

?Nonlinear functional analysis is an important branch of modern analysismathematics, because it can explain all kinds of natural phenomena, moreand more mathematicians are devoting themselves to it. Among them, theproblems of nonlinear differential equations are one of the most active fieldsthat is studied in analysis mathematics at present, especially, the impulsiveintegro-differential equation in infinite intervals. In this paper, in the formertwo sections, we devote our time to give the theorems for solutions of non-linear impulsive integro-differential equation in infinite intervals. In the thirdChapter, we study singular three-point boundary value problem by the fixedpoint theorem of Strict-Set-Contraction.?In the first Chapter, by means of a comparison result, the monotoneiterative technique and the method for upper, lower solution, we consider firstorder impulsive functional differential equation boundary value problem ininfinite interval (BVP)We obtain the existence of minimal and maximal solutions or weaker onesof (BVP)(1.1.1), and monotone iterative sequences uniformly converge to theextremal solutions. In this paper, the interval J is an unboundary domain,therefore the results in this paper improve and generalize the related resultsin [2-5]. If the impulsive item I_k is linear or g is the specific boundary valuecondition, we needn't any limited conditions on g and I_k, the results also hold.?In the second Chapter, by using the Month fixed point theorem andcomparison result, we study the existence, unique and uniformly converge ofiterative sequence of solution for two-order impulsive integro-differential initial value problem in Banach space (IVP)In this paper, the conditions are weaker than [17]. We not only weaken theregular of P, but also cancel the uniformly continuous of f on J×B_r×B_r×B_r.And the method of proof is different from [13], [17], [19]. We needn't anylimited conditions on impulsive item, when the interval J is finite and△x(t_k) =L_kx′(t_k),△x′(t_k) = L_k~*x(t_k) (L_k≥0, L_k~*≤0), the results are also true. Hence,we improve the results in [13].In the third Chapter, we discuss the existence of solution for the singulartwo-order three-point boundary value problem in Banach space (BVP)where a(t)∈C((0,1), R~+), and a(t) is singular at t = 0,1. On the conditionsof f, a, we obtain the existence of positive solution of (BVP)(3.1.1) , existenceof two positive solutions especially, by means of the fixed point theorem ofStrict-Set-Contraction.