Periodic-integrable Boundary Value Problem For Differential Equations

It is well known that the boundary value problem of differential equations has always been an important content of the field of differential equation re-search,and a series of important results have been obtained successively.See references[2,11,23,28,39,46,57,63,76].The existence and uniqueness of the solution of this kind of problem,as a basic problem in the study,has caused in-terest of many mathematics researchers.In the course of the study,many kinds of research methods have been developed and established,and profound re-search results have been obtained.See references[12,41,62,87,88,89,90,93].The main content of this paper is to study the existence and uniqueness of solutions for several types of periodic-integral boundary value problems for differential equations by using Schauder's fixed point theorem,nonlinear functional analysis theory and so on.In the first part,we first consider the periodic-integrable boundary value problem of the second order differential equations?denoted as PIBVP for short?x"=f?t,x,x'?,where f:[0,27?]×R2—?R is continuous.We will give a sufficient condition which ensure the existence of solution.?A1?There exist two continuous functions a?t?and b?t?,and a nonnegative constant M1,such that 0?a?t??f?t,x,y?/x?b?t?,for any x with |x|?M1 and?t,y??[0,2?]×R;?A2?There exist two nonnegative constants M2 and M3,such that|f?t,x,y?/y|?M2,for any?t,x??[0,2?]×R whereas |y|?M3.Theorem 0.0.1 If Assumptions?A1?and?A2?hold,then the PIBVP?0.0.1?has at least one solution.We also will give a sufficient condition which ensure the existence and uniqueness of solution,as f is differential.Assume that:?A3?There exist two continuous functions a?t?and ??t?,such that0???t??fx?t,x,y????t?,for any?t,x,y?E[0,2?]×R2;?A4?There exists a positive number M,such that?A4?There exists a positive number M,such that for all t ?0,27r]and?x,y?? R2.Theorem 0.0.2 If Assumptions?A3?and?A4?hold,then the PIBVP?0.0.1?has unique solution.In the second part,we deal with the existence and uniqueness of solution of the high order periodic-integrable boundary value problem.Consider the n order periodic-integrable boundary value problem x?n?=f?t,x,x',...,x?n-1??.where f:[0,2?]× Rn—? is continuous.?H1?There exist two nonnegative continuous functions b?t?and c?t?,b?t?? 0,when |xn-2|>a0,one has b?t??f?t,x0,x1,...,xn-1?xn-2?c?t?.for all?t,x0,x1,...,xn-3,xn-1??[0,2?]×Rn-1,where a0 is a positive constant.?H2?There exist two nonnegative constants a1 and cn-1,when |xn-1|?a1,for any?t,x0,x1,...,xn-2??[0,2?]×Rn-1,one has|f?t,x0,x1,...,xn-1?/xn-1|?cn-1?H3?The function f?t,x0,x1,...,xn-1?satisfies,for all?t,x0,x1,...,xn-1??[0,27r]×Rn-2×[-a0,a0]×[-a1,a1],|f?t,x0,x1,...,xn-1?|?M,where M is a positive constant.Theorem 0.0.3 If Assumptions?H1?,?H2?and?H3?hold,then the PIB-VP?0.0.2?has at least one solution.To obtain the uniqueness of solutions of the problems we consider the periodic-integrable boundary value problem as followsx?n?=an-1?t?x?n-1?+a?n-2??t?x?n-2?+h?t,x,x',...,x?n-1??.We assume throughout that possesses?H4?h is continuous bounded function on[0,2?]×Rn,an-1?t?and an-2?t?are continuous bounded function on[0,2?],and an₂?t??0,an-2?t??0.?H5?f is continuous function on[0,2?]× Rn,and f?t,x0,x1,...,xn-1?ex-ist continuous partial derivative with respect to xn-2 and xn-1,there exist nonnegative constant ?0,such that|f?t,x0,x1,...,xn-3,0,0?|??0,for any?t,x0,x1,...,xn-3??[0,2?]× Rn-2.?H6?There exist nonnegative constants ?1,?2,?3,such that|fx?n-1?|??3,?1?|fx?n-2?|??2.on[0,2?]× Rn.Theorem 0.0.4 If Assumption?H4?holds,then the PIBVP?0.0.3?has an unique solution.Theorem 0.0.5 If Assumptions?H5?,?H6?hold,then the PIBVP?0.0.3?has an unique solution.In the third part,we consider odd order differential equation with periodic-integrable boundary value problem where ?k,k=0,1,2,...,n-1,are constants,and f:[0,2?]×R—?R is continuous.We assume throughout that?S1?f is continuous,and possesses continuous partial derivative in x.There exist two constants ?,?>0,such that??|fx?t,x?|??,????t,x??[0,2?]× R.?S2?f is continuous.There exist constants M*>0 and/3,?,?>0,satisfying that??|f?t,x?/x|??with |x| ? M*,for any t ?[0.2?].Theorem 0.0.6 If Assumption?S1?holds,then the problem?0.0.4?has unique solution.Theorem 0.0.7 If Assumption?S2?holds,then the problem?0.0.4?has at least one solution.We next consider vector differential equation with periodic-integrable bound-ary value problemwhere X is function,X:[0,2?]?Rm,F:[0,2?]×Rm—?Rm,?k,k=0,1,2,...,n-1,are constans.Theorem 0.0.8 If F is a continuous vector function,and Fx is a con-tinuous Jacobi matrix in X,there exist two constants ?,?>0,satisfying that ??U?2?<U,Fx?t,X?U>???U?2,???U?Rm\{0}.for any t ?[0,2?],X ? Rm or -??U?2?<U,Fx?T,x?u>?-??U?2,???U ? Rm\{0}.for any t ?[0,2?],X ? Rm,then the PIBVP?0.0.5?has unique vector value solution.