| The impulsive phenomena are adequate apparatus for the sudden change at cer-tain moments, and have a wide range of applications, for example, in control systems, communications, life sciences, Medicine, economics, information science, etc.The ma-jority of scholarsband experts, are interested in the investigation for such phenomena, and are committed to the theory research.The differential systems with non-impulse effects have been profound understand-ing. The differential systems have impulse effects. They are many similarities, and also have a lot of different.In many literature, the paper consider the existence of solutions for the following differential equation with p(t)=±1,g(t)=0: when p(t)≠±1,g(t)≠0, we naturally consider the existence of solutions for the differential equations,First, we consider the existence of the solution for the following differential equa-tions with impulse effects: where J=[0,1],g(x,u)∈C(I×R+,R+),P(x)>0,Q(x)>0,P(x)∈C1(J),Q(x)∈C(J),-Δ(Pu')|x=xk=-P(xk)(u'(xk+)-u'(xk-)) and Ik∈E C(J,R+),0<x1<x2< …<xp=1are given.Here u'(xk+)(u'(xk-))) denotes the right limit (respectively left limit) of u'(x) at x=xk, and the following period boundary value problem where J=[0,1], f(t,x)∈C(J×R+,R+),p(t)>0,g(t)>0,p(t)∈C1(J),q(t)∈C(J),-Δ(px')|t=tk=-p(tk)(x'(tk+)-x'(tk-)), x(t)=-p(t)x'(t) and Ik∈C(J,R+),0t1<t2<…<tp=1are given. Here x'(tk+)(respectively x'(tk-)) denotes the right limit (respectively left limit) of x'(t) at t=tk.Further, we will think about the existence of solutions when the nonlinear term is singular and superlinear.The whole contents is divided into four chapters:Chapter1, as the beginning of this paper, offers the theoretical background of impulsive differential equations.Chapter2, as prior knowledge, gives some relative the lemmas and theorems, such as Alternative of Leray-Schauder,the Arzela-Ascoli Theorem, krasnoselskii fixed point theorem, etc. And the Green's Function of the following equation is offered. where where Q=I×I, Q1={(x,y)∈Q|0≤x≤y≤1}, Q2={(x, y)∈Q|0≤y≤x≤1}, and m and n are linear independent, and m, n and ω satisfy the following Lemma2.6.Chapter3, we obtained the existence positive solutions of Neumann boundary value problem for second order differential equation with impulse effects and multi-plicity of positive solutions to singularity and superlinear equations with second order Neumann boundary value problems with impulse effects. This chapter have two parts. In one part, we obtained the existence positive solutions of Neumann boundary value problem for second order differential equation with impulse effects.Consider the equation, where J=[0,1],g(x,u)∈C(I×R+,R+),P(x)>0,Q(x)>0,P(x)∈C1(J),Q(x)∈C(J),-Δ(Pu')|x=xk=-P(xk)(u'(xk+)-u'(xk-)) and Ik∈C(J,R+),0<x1<x2<xp=1are given.Here u'(xk+)(u'(xk-))) denotes the right limit (respectively left limit) of u'(x)at x=xk.First, we gives the operators(Tu)(x) in the conek with the Green's function.Using the the fixed-point theorem2.2and theorem2.3, we obtained the equation has at least two positive solutions.We take the following hypotheses:Theorem1Assume that (H1) and (H3) are satisfied. Then(7) has at least two positive solutions u1and u2satisfying Theorem2Assume that (H2) and (H4) are satisfied. Then(7)has at least two positive solutions u1and u2such thatIn two part, we obtained multiplicity of positive solutions to singularity and su-perlinear equations with second order neumann boundary value problems with impulse effects. Mainly it divided into the Positive and Semi-positive.When it is the Positive, we consider the equationwhere J=[0,1]and0<t1<t2<…<tp=1are given. Ik∈C(I, R).-Δpx'|t=tk=-p(tk)(x'(tk+)-x'(tk-)), x[1](t)=-p(t)x'(t).x'(tk+)(x'(tk-))denotes the right limit (respectively left limit) of x'(t) at t=tk. Define J'=J\{t1,t2,…,tP}. X={x:J→R|x∈C(J, R); x'|(tk,tk+1)∈C(tk, tk+1),x'(tk-)=u'(tk),(?)x'(tk+)}. and whereσ is as in (3.2.4). One may readily verify that K is a cone in X. Suppose that F:[0,1]×R→[0,∞) is a continous function. Define an operator T: X→X,for x∈X, and t∈[0,1],Using the Alternative of Leray-Schauder, we given the existence of one solution. Assume that the following conditions are satisfied: (H1) For each constant L>0, there exists a function φL>0such that f(t,x)> φL(x) for all (t,x)∈[0,1]×(0,L];(H2) there exist continous, non-negative function g(x),h(x) and γ(x) on (0,∞)上such that and g{x)>0is non-increasing, and h(x)/g(x) and γ(x) are non-decreasing in x∈(0,∞);(H3) there exists a positive number r such that where σ, B and ω(x) are as in (3.2.4).Theorem3Under assumptions (H1),(H2) and (H3) are satisfied. Then equa-tion(8)has at least one positive solution with0<‖x‖<r.Further, using the fixed point theorem, we give the existence of another solution..Theorem4Suppose that (H1)-(H3) are satisfied, Furthermore, assume that(H4) There exist continous non-negative function g1(x), h1(x) and γ1(x) on (0,∞) such that and f1(x)>0is non-increasing and h1(x)/fi(x), γ1(x) are non-decreasing in x∈(0,∞);(H5) there exist a positive number R>r such that where σ, A and ω(t) are as in (3.2.4). Then, besides the solution x constructed in Theorem3, Equation (8)has znother posi-tive solution x with r<‖x‖≤R.And we carried out a detailed example application.When it is the Semi-positive, we consider the equationwhere f(t,x) may be singular at x=0. In particular, f(t,x) is superlinear near x=+∞, i.e..And f(t, x)is positive.We give the existence of a solution with the upper and lower solutions.Assume that the following conditions are satisfied:(C1) There exists a constant M>0such that for all (t,x)∈[0,1]×[0,∞),(C2) There exist continous, non-negative g(x), h{x) and γ(x), such that where g(x) is non-increasing function and h(x)/g(x), γ(x) are non-decreasing in x∈(0,∞). where σ, B and ω(t) are as in(3.2.4).(C4) There exists constants L>M,ε>0such that(C5) There exist continous, non-negative g1(x), h1(x) and γ1(x), such that where g1(x)>0is non-increasing, h1(x)/g1(x),γ1(x) are non-decreasing in x∈(0,∞).(C6) There exists R> r such thatTheorem5Suppose that condition (C1)-(C4) are satisfied. Then problem (10)has at least one positiveUsing the fixed point theorem, we obtained the existence of another solution.Theorem6Suppose that conditions (C1)-(C6) hold. Then the equation (10) has another positive solution x∈C[0,1], x'∈C[0,1], x(t)>0,with r<‖x+Mω‖≤R.Chapter4, discuss the existence of positive solutions to second order period bound-ary value problems with impulse effects and the existence of positive solutions to sin-gularity and superlinear equations with second order period boundary value problems with impulse effects.This chapter have two parts. In one part, we obtained the existence of positive solutions to second order period boundary value problems with impulse effects.Consider the equation, wheredenotes the rightlimit (respectively left limit) of x'(t) at t=tk·Define the Banach space And the operator on X as following:for x∈X. Clearly,Φ is a completely continous operator on X Using the the fixed-point theorem, we obtained the equation has at least onepositive solution and has at least two positive solutions. Suppose that the following conditions are satisfied: Theorem7Assume that (H1) or (H2) is satisfied. Then (12)has at least apositive solution x. Theorem8For all x,xi,∈[λr,R],0<r<R,t E [0,1], we have f(t,x)≥0, Ii(x)≥0, Ji(x)≥0,1≤i≤p, And assume that one following hypothese is satisfied: Then (12) has at least two posttive solutions x1and x2such that r<‖x1‖<s‖x2‖<R.tIn the second part, we obtained the existence of positive solutions to singularity and superlinear equations with second order period boundary value problems with impulse effects. Mainly it divided into the Positive and Semi-positive.When it is the Positive, we consider the equationwhere J=[0,1], f(t,x)∈C(J×R+,R+),p(t)>0,g(t)>0,p(t)∈C1(J),q(t)∈C(J),-Δ(px')|t=tk=-p(tk)(x'(tk+)-x'(tk-)),x[1](t)=-p(t)x'(t) and Ik∈C(J,R+),0<t1<t2<…<tp=1are given. Here x'(tk+)(respectively x'(tk-)) denotes the right limit (respectively left limit) of x'(t) at t=tk. The perturbation f(t,x) may be singular at x=0and f(t,x)is positive.Using the Alternative of Leray-Schauder and the Arzela-Ascoli theorem, we given the existence of a solution.Assume the following conditions are satisfied:(Hi) There exists a constant L>0, φL>0such that f(t,x)≥φL(x) for all (t,x)∈[0,1]x(0,L];(H2) There exist continous, non-negative function g(x), h(x) and γ(x), such that and g(x)>0is non-increasing and h(x)/g(x),γ(x) are non-decreasing in x∈(0,∞);(H3) There exists r such that where M,σ are as in (4.1.7)and(4.1.8). Theorem9Assume (H1),(H2) and (H3) hold. Then the equation (14) has at least a positive solution0<‖x‖<r.Further, using the fixed point theorem, we give the existence of another solution..Theorem10Suppose that conditions (H1)-(H3) hold, in addition, it is assumed that the following two conditions are satisfied:(H4) For some continous non-negative functions g1(x), h1(x)and γ(x) with the properties that g1(x)>0is non-increasing and h1(x)/g1(x), γ (x) are non-decreasing on x G (0,∞);(H5) There exists R> r such that where σ,m, ω(t) are as in(4.1.8) and (4.1.9). Then, besides the solution x constructed in Theorem9, problem (14) has another positive solution x with r <‖x‖<R.When it is the Semi-positive, we consider the equation where J=[0,1], f(t,x)∈C(J×R+,R+),p(t)>0,g(t)>0,p(t)∈C1(J),q(t)∈C(J),-Δ(px')|=tk=-p(tk)(x'(tk+)-x'(tk-)),x[1](t)=-p(t)x'(t) and Ik∈C(J,R+),0t1<t2<…<tp=1are given. Here x'(tk+)(respectively x'(tk-)) denotes the right limit (respectively left limit) of x'(t) at t=tk..The perturbations f(t,x) may be singular at x=0and f(t,x) is superlinear near x=+∞i. e., in particular, f(t,x)is semi-positive.Using the Alternative of Leray-Schauder and the Arzela-Ascoli theorem, we ob-tained the existence of at least one solution. Theorem11Assume the following hypotheses are satisfied:(C1) For all (t,x)∈[0,f]×[0,∞), there exists a constant M>0such that(C2) There exist continous, non-negative functions g(x), h(x) and γ(x), such thatand g(x) is non-increasing function, h(x)/g(x), γ(x) are non-decreasing functions in x∈(0,∞).(C3) In x E (0,+∞), there exist continous, non-increasing function g0(x) and a constant R0(x)>0sunh that f(t,x)≥g0(x),(t,x) E [0,1]×(0, R0], where g0(x) where σ, B and ω(t) are as in (4.1.8)and(4.1.9). Then in t∈[0,1], the problem (15) has at least positive solution x∈C[0,1], x(t)>0with0<‖x+Mω‖<r.using the fixed point theorem, we give the existence of another solution.Theorem12Suppose that conditions (C1)-(C4)hold, In addition, it is assumed that the following conditions are satisfied:(C5)Some continous non-negative functions g1(x) and h1(x), γ(x) satisfied:(?)(t,x)∈I×(0,∞), where g1(x)is non-increasing,h1(x)/g1(x),γ1(x)are non-decreasing in(0,+∞).(C6) There exists R>r such that Then besides the solution x constructed in Theorem11,problem(15)has another posi-tive solution...
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