The Study On The Existence Of Solutions For Second-order M-point Boundary Value Problem

The boundary value problem of ordinay differential equations comes from a lot of branches of applied mathematics and physics,and it is one of the most active fields that are studied in analysis mathematics at present.The boundary value problem has been vastly applied in many fields such as gas diffusion through porous media,catalysts theory,thermal self-ignition of a chemical active mixture of gases and so on.In this paper,we first introduce the existence of solutions to multi-point boundary value problems and the bifurcation properties of the boundary value problems.We introduce the results of D.D.Hai's -u″(x)+p(x)u′(x)=f(u(x)),r＜x＜R, u(r)=0,u(R)=λ.Theorem 1.Assume the following two conditions hold: (A₁) p:[r,R]—(0.+∞) is countious. (A₂)f:[0,+∞)→[0,+∞) is local Lipschitz countious,satisfies where F(x):=integral from n=0 to x(f(u)du).Then.there exists a positiveλ^*,such that:whenλ∈(0,λ^*),there are at least two solutions to boundary value problem.Whenλ=λ^*, there is at least one solution to boundary value problem.Whenλ＞λ^*,there is no solution to boundary value problem.We introduce the results of Ma's -u″(x)=p(x)f(u(x)),0＜x＜1. u(0)=0,u(1)-αu(β)=λ, where A is a nonnegative parameter.Theorem 2.Assume the following conditions hold: (C₁)β∈(0.1) andα∈(0.1/β). (C₂)p:[0,1]→[0,+∞) is cotmtious. (C₂) f[0,+∞)→[0,+∞) is countious,and satisfies(?)f(u)/u=0.Then,there exists 0＜λ^*≤+∞,for everyλ∈(0,λ^*),there is at least one solution to three-point boundary value problems.Zhang assumed the following conditions hold (H₁)β∈(0,1) andα∈(0,1/β) are the real numbers,λ≥0. (H₂) p(x) is nonnegative measurable function defined in(0,1),and satisfies: 0≤integral from n=0 toβ(sp(d)ds)＜+∞,0＜integral from n=βto 1((1-t)p(t)dt)＜+∞. (H₃) f:[0,+∞)→[0,+∞) is countious. (H₃)^* f:[0,+∞)→[0,+∞) is local Lipschitz eountious. whereβ^*:= 1/min{β,1—β},for given[a,b],Theorem 3.Assume(H₁),(H₂),(H₃)^*,(H₅),(H₆) hold.Then.there exists a positive numberλ^*,for anyλ∈[0,λ^*),there are at least two solutions to three-point boundary value problem.Whenλ=λ^*,there is at least one solution to three-point boundary value problem.When Aλ＞λ^*.there is no solution to three-point boundary value problem.Based upon shooting method,we introduce a numerical method for approximating solutions and fold bifurcation solutions of second order muhi-point boundary value problems.We introduce the equivalence of the regularity of solutions and simple fold bifurcation solutions between the numerical method and the multi-point boundary value problems.Then,we introduce the result of a class of second order three-point boundary value problems whereλis a nonncgative parameter.β∈(0,1).Theorem 4.Assume the conditions(A1),(A2),(A3) hold (A1)λ≥0,β∈[0.1], (A2) f(t,u) is a real function defined in(0,1)×R,and satisfies:(â…°) for every fixed u∈R,f(t,u) is measurable in(0,1),(â…±) f is almost every countious with t∈(0,1),(â…²) for every fixed N≥0,there exists a function k_N(t)∈E:= {h(t)∈L_loc¹(0,1);‖h‖_E≤+∞},such that: |f(t,u)|≤k_N(t) u∈[-N,N],(A3) there exist two functions v(t),w(t),which are the upper solution and lower solution of second order three-point boundary value problem,and w(t)≤v(t),t∈[0,1],then there exists one solution to second order three-point boundary value problem,and w(t)≤u(t)≤v(t).Zhang considered the following condition. (AA):f:[0,1]×R→R is eountious and satifies: |f(t,u)|≤p(t)|u|+r(t), where p(·),r(·)∈L¹(0,1),‖p‖L¹ï¼œ1.Theorem 5.Assume the condition(AA) hold,u₁,u₂,u₃ are the strict lower solution,the strict upper solution and the strict lower solution to second order three-point boundary value problem and u₁＜u₂＜u₃,t∈[0,1] then there exists at least two different solutions u(t),v(t) to second order threepoint boundary value problem and u₁(t)＜u(t)＜u₂(t),u₁(t)＜u(t)(?)t∈[0.1], u₂(t_v)＜v(t_v)＜u₃(t_v)(?)t_v∈(0,1).Theorem 6.Assume the condition(A4) hold,uU₁,u₂,u₃ are the strict lower solution,the strict upper solution and the strict lower solution to second order three-point boundary value problem and u₁＜u₂＜u₃,t∈[0,1] then there exists at least two different solutions u(t),v(t) to second order threepoint boundary value problem and u₂(t)＜v(t)＜u₃(t).u(t)＜u₃(t)(?)t∈[0.1], u₁(t_v)＜u(t_v)＜u₂(t_v)(?)t_v∈(0,1].Theorem 7.Assume u₁,u₂,u₃,u₄ are the strict lower solution,the strict upper solution,the strict lower solution and the strict upper solution to second order three-point boundary value problem and u₁＜u₂＜u₃＜u₄,t∈[0,1] then there exists at least three different solutions u(t):u(t),w(t) to second order three-point boundary value problem and u₁(t)＜u(t)＜u₂(t),u₃(t)＜w(t)＜u₄(t), u₁(t)＜u(t)＜u₄(t),u₁＜v,(?)t∈[0,1], u₂(t_v)＜v(t_v)＜u₃(t_v),(?)t_v∈(0,1]. Finally,we give a,n example to illustrate there may be multiplicity of solutions for a class of second order three-point boundary value problem at resonance or unresonance condition...