Uniqueness Theorem Of A Kind Of Systems Of Nonlinear Equations And Its Applications

In this paper,a uniqueness theorem of a kind of systems of nonlinear equations is given and using this result uniqueness theorems in three aspects are easily deduced. It is divided into four parts.The first part is a uniqueness theorem of a kind of systems of nonlinear equa-tions.Theorem 1 Let all functions Fk(x1,...,xn), k=1,...,n,and their partial deriva-tives (?)Fk(x1,...,xn)/(?)xj,k,j=1,...,n,be continuous on a connected D.Assume that for each index m, 1≤m≤n,there is a set of numbers{∈m,1,...,∈m,m} with│∈m,k│=1,k=1,...,m,such that for each given set of values {ym+1,...,yn},the system of epuations with m unknowns x1,...,xm Then for each index m,1≤m≤n,and for each given set of values {ym+1,...,yn}, the sysstem of epuations (1) has a unique solution.In particular,thesystem of epuations (1)with m=n has a unique solution.The second part is a uniqueness theorems for power orthogonal polynomial.Theorem 2 Let dμbe a measure on (a,b) satisfyingμ∈C(a,b).Assume, further, thatμ∈C1(a,b)μ'(x)>0,x∈(a,b),if min1≤k≤n mk=1.Then there exists a unique vectorΧ∈X satisfyingΦn(x)=infy∈ΧΦn(y).We give a simple proof for the theorem by using Theorem 1. The third part is a uniqueness theorem for a Gaussian quadrature formula of an extended Chebyshev system.In this section and the forth part, we denote the considered vector: Denote by Ak,j(x)∈PN the fundamental functions for the Hermite and Birkhoff interpolation, satisfyingTheorem 3 Let U be an (N+1)-dimensional extended Chebyshev space on [a, b] and let a measure dμsuported on [a, b] satisfy thatμ∈C[a, b]. When min1≤k≤n mk=1 assume further thatμ∈C1[a, b] andμ'(x)>0 in (a, b). Then for a given vectorΧ2∈X2, there exists a unique vectorΧ1∈X1 such that forΧ= (Χ1,Χ2), relation ck,mk-1(Χ)=0, k=1,...,m (3) is exact for everyμ∈U.Again we can give a simple proof for the theorem by applying the Theorem 1. The forth part is a uniqueness theorem for Gaussian Birkhoff quadrature for-mula.Firstly, basis of the results the former got, we obtain two lemmas. Under the following lemmas, we will get two main theorems easily. These lemmas themselves are important results, too.Lemma 1 For fixed indices k,j,r with ek,j=1, ek,j€E, and 1≤r≤n, we haveLemma 2 Let dμbe a measure satisfyingμ∈C[a, b] and let an n×(N+1) Polya matrix E contain no odd non-Hermitian sequences. Suppose that each of the rows 1≤k≤m is Hermitian. Let 1≤k, r≤m. Then Moreover, if for a given vectorΧ2∈X2, a vectorΧ1∈X1 is a solution of the system of normal equations holds. Here Gk(Χ)=∫ab Ak,mk-1(Χ;x)S(Χ;x)dμ(x), k=1,...,m. Next,we obtain two main results of this paper.Theorem 4 Let dμbe a measure satisfyingμ∈C[a,b]and let an n×(N+1) Polya matrix E contain no odd non-Hermitian sequences.Then for a given vectorΧ2∈X2,there exists a vectorΧ1∈X1 such that forΧ=(Χ1,Χ2),the GGBQF Moreover,forΧ∈X,the following statements are equivalent: (a)the vectorΧsatisfies the GGBQF(5)with the property(3); (b)the vectorΧsatisfies the system of normal equations(4); (c)the vectorΧsatisfies the orthogonal relation: here V(Χ)={P∈PN:p(j)(x k)=0, e_k,j*,=1,e_k,j*,J∈E*},Χ∈X.Let E* be obtained from E by omitting a 1 position(k,mk-1)in each row k,1≤k≤m and S(Χ;x) be from(2).Theorem 5 Let the assumptions of Theorem A prevail.When min{m1,....mn} =1 assume further thatμis strictly monotone increasing.Suppose that each of the rows 1≤k≤m is Hermitian.Then for a given vectorΧ2∈X2,there exists a unique vectorΧ1∈X1 such that forΧ:(Χ1,Χ2),the GGBQF(5)with the property (3)holds.In comparison with Nikolov,this paper simplifies the heavy and complicated restrictive conditions of and gives simple proofs for Theorem 4 and Theorem 5.