The Sign-changing Solutions For The Second-order System Of Equations Two-point Boundary Value Problems

In this paper, by using the theory of Leray-Schauder degree and fixed point index in nonlinear functional analysis, combining cone and partial order methods, we study the multiplicity of sign-changing solutions to nonlinear second-order system of equations two-point boundary value problems (BVP):We overcome some difficulty caused by number of unkown functions increasing and extend some proof methods which are suitable for single equation to the system of equations and get the same results as the singleequation.Let us list some hypotheses of f and g.(H₁) f,g∈C(R²,R¹), f(0,0) = g(0,0) = 0, and such thatf(u,v)≥0 , g(u,v)≥0 for all u≥0,v≥0, particularly, when u or v is positive, we have f(u,v) > 0,g(u,v) >0; f(u,v)≤0,g(u,v)≤0 for all u≤0,v≤0,particularly,when u or v isnegative,we have f(u,v)<0,g(u,v)<0．(H₂) f and g are of continuous partial derivative at the point(0,0)．Writef'_x(0,0)=a₀,f'_y(0,0)=b₀,g'_x(0,0)=c₀,g'_y(0,0)=d₀,where a₀,b₀,c₀,d₀ are positive．There exists a positive integer m such that(λ₁+λ₂)²>4λ₁λ₂,(2m)²Ï€²1,λ₂)<(2m+1)²Ï€²,whereλ₁,λ₂ are eigenvalues of A₀=(?)．where a₁,b₁,c₁,d₁ are positive．There exists a positive integer l such that(μ₁+μ₂)²>4μ₁μ₂,(2l)²Ï€²1,μ₂)<(2l+1)²Ï€²,whereμ₁,μ₂ are eigerlvalues of A₁=(?)．(H₄)There exists a constant T>0 such that|f(u,v)|<2T,|g(u,v)|<2T,for all |u|≤T,|v|≤T．We obtain the following main results of this paper．Theorem 1．1．1 If conditions(H₁)-(H₄)hold,then the BVP(1．1．1)has atleast six different nontrivial solutions,which are two positive solutions,twonegative solutions and two sign-changing solutions．Theorem 1．1．2 If conditions(H₁)-(H₄)hold,and f,g are odd,i．e．f(-u,-v)=-f(u,v),g(-u,-v)=-g(u,v),for all u,v∈R¹,then the BVP(1.1.1) has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions.