Multiple Positive Solutions For Systems Of Conjugate Integral Equations

In this paper, the existence and multiplicity of positive solutions for the three systems of nonlinear conjugate integral equations are discussed by using topological degree theory combined with fixed point index, cone and semi-ordering methods of nonlinear functional analysis, then the new results are obtained and the previous results are improved.The paper is divided into three chapters, G is all supposed to be a bounded closed area in the Rⁿ, P={u∈E :u(x)>0,x∈G} is a cone in E=C(G)which is a real Banach space. P×P is a cone in product space E×E. B₁and B₂:E→E arelinear integral operators as follows:(B₁u)(x)=(?)k(x,y)u(y)dy, (B₂u)(x)=(?)k(y,x)u(y)dy.Then suppose(1) k₁=(?)k(x,y)dy>0, k₂=(?)k(y,x)dy>0;(2) r(B₁) and r(B₂) are spectral radiuses and positive of B₁ and B₂, are satisfied.Because of the assumptions, we easily know r(B₁) = r(B₂) which is denotedasλ₁.In chapter one, the existence and multiplicity of positive solutions for the system of Hammerstein integral equations with the same property of nonlinear termsare considered. Namely, the nonlinear terms f and g are satisfied with expansion conditions or compression conditions in zero or infinite point,where f,g∈C(G×R⁺×R⁺,R⁺).In this chapter, the nucleus functions of the system of integral equations of (1.1) are supposed as follows:If the h∈C(G) exists and is almost everywhere positive in G such that k(x, y)≥h(x)k(z,y), k(y, x)≥h(x)k(y,z), (?)x,y,z∈G.Through related property of linear integral transposed operator spectral, the criterion for the existence of single positive solution or multiple positive solutions is given by using fixed point index theory in cone.The assumptions:(H₁) There exist p,q∈C(R⁺,R⁺)such that(?)f(x,u,v)/(u+p(v))>1/λ₁, (?)g(x,u,v)/(v+q(u))>1/λ₁ uniformly with respectto x∈G. (H₂) There exist a,b≥0 such that a+b<1/(max{k₁,k₂}),and(?)f(x,u,v)/(u+v)≤a, (?)g(x,u,v)/(u+v)≤b.(H₃) There exists,t∈C(R⁺,R⁺) such that(?)f(x,u,v)/(u+s(v))>1/λ₁,(?)g(x,u,v)/(v+t(u))>1/λ₁ uni formly with respectto(x,v)∈G×R⁺ and (x,u)∈G×R⁺, respectively.(H₄) There exist c,d≥0 such that c + d<1/(max{k₁,k₂}), and(?)f(x,u,v)/(u+v)≤c,(?)g(x,u,v)/(u+v)≤d,(H₅) f(x,u,v)and g(x,u,v)are nondecreasing in u and v, and there exists N>0 such thatf(x,N,N)1), g(x,N,N)2), a.e.x∈G.Then the mainly results of chapter one are as follows:Theorem 1.2.1 Assume (H₁) and (H₂) are fulfilled, then the problem of(1.1)has at least one positive solution.Theorem 1.2.2 Assume (H₃) and (H₄) are satisfied, then the problem of(1.1) has at least one positive solution.Theorem 1.2.3 Assume(H₁),(H₃)and(H₅)are satisfied, then the problem of(1.1)has at least two positive solutions.In chapter two, the existence of positive solutions for the system of Hammerstein integral equations with the different property of nonlinear terms is considered. Namely, f₁ and f₂ res pectively satisfy superlinear condition and sublinear condition, where f_i,h_i∈C(R⁺×R⁺,R⁺),(i=1,2).In this chapter, the nucleus functions of the system of integral equations of (2.1) are supposed as follows:Suppose G₀(?)G is a bounded closed area,αandβ∈(0,1) are constants, and(â… ) k(x,y)>0,k(y,x)>0, (?)x,y∈G\(?)G.(â…¡) k(x,y)≤k(y,y),k(y,x)≤k(y,y),(?)x∈G,y∈G.(â…¢) k(x,y)≥αk(y,y),k(y,x)≥βk(y,y), (?)x∈G₀,y∈G.By computing the fixed point index in K₁×K₂, we establish the criterion for the existence of positive solution of(2.1).The assumptions:(H₃) (?)h₁(u,v)/u=0 uniformly with respect to v∈R⁺.(H₄) (?)h₂(u,v)/v=0 uniformly with respect to u∈R⁺.(?)h₂(u, v)=0 uniformly with respect to (?)M>0,v∈[0,M].Then the mainly result of chapter two is as follows:Theorem 2.2.1 Suppose(H₁)-(H₄) are satisfied, then the problem of(2.1) has at least one positive solution.In chapter three, by means of the fixed point theorem of cone expansion and compression, the existence and multiplicity of positive solutions for the system of nonlinear Hammerstein integral equations, which is rewritten as a integral equationare considered, where f,g∈C(G×R⁺,R⁺).In this chapter, the hypotheses which the nucleus functions of the system of integral equations of(3.1) are satisfied are the same as the system of integral equations of(2.1) .The assumptions:(H₁) g∈C(G×R⁺,R⁺)and g(x,0)=0,x∈G are satisfied.(H₆) There exist N>0 such that (?) f(y,u)1)-(H₃)are satisfied, then the problem of(3.1)has at least one positive solution. Theorem 3.2.2 Suppose(H₁),(H₄)and(H₅)are satisfied, then the problem of(3.1)has at least one positive solution.Theorem 3.23 Suppose(H₁),(H₃),(H₅)and(H₆) are satisfied, then the problem of(3.1) has at least two positive solutions.