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Let KK be a closed convex subset of a qq-uniformly smooth separable Banach space, T:K→KT:KK a strictly pseudocontractive mapping, and f:K→Kf:KK an LL-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)t(0,1), let xtxt be the unique fixed point of tf+(1-t)Ttf+(1-t)T. We prove that if TT has a fixed point, then {xt}{xt} converges to a fixed point of TT as tt approaches to 0.  相似文献   

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This paper treats some variational principles for solutions of inhomogeneous p  -Laplacian boundary value problems on exterior regions U?RNU?RN with dimension N?3N?3. Existence-uniqueness results when p∈(1,N)p(1,N) are provided in a space E1,p(U)E1,p(U) of functions that contains W1,p(U)W1,p(U). Functions in E1,p(U)E1,p(U) are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an LpLp-imbedding theorem. Also an existence result for a general variational problem of this type is obtained.  相似文献   

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We consider the semilinear elliptic equation Δu+K(|x|)up=0Δu+K(|x|)up=0 in RNRN for N>2N>2 and p>1p>1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r−?K(r)r?K(r) with ?>−2?>2 around a positive constant is small near r=∞r= and p   is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent pcpc which is determined by N   and the order of the behavior of K(r)K(r) as r=|x|→0r=|x|0 and ∞. In order to understand how subtle the structure is on K   at p=pcp=pc, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2)p=(N+2)/(N2).  相似文献   

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Let A be an Archimedean f  -algebra and let N(A)N(A) be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→Ad:AA is a derivation if and only if d(A)⊂N(A)d(A)N(A) and d(A2)={0}d(A2)={0}, where A2A2 is the set of all products ab in A.  相似文献   

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In this paper, we consider the problem (Pε)(Pε) : Δ2u=un+4/n-4+εu,u>0Δ2u=un+4/n-4+εu,u>0 in Ω,u=Δu=0Ω,u=Δu=0 on ∂ΩΩ, where ΩΩ is a bounded and smooth domain in Rn,n>8Rn,n>8 and ε>0ε>0. We analyze the asymptotic behavior of solutions of (Pε)(Pε) which are minimizing for the Sobolev inequality as ε→0ε0 and we prove existence of solutions to (Pε)(Pε) which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for εε small, (Pε)(Pε) has at least as many solutions as the Ljusternik–Schnirelman category of ΩΩ.  相似文献   

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For almost all x>1x>1, (xn)(xn)(n=1,2,…)(n=1,2,) is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (bn)(bn) in [0,1[[0,1[ and ε>0ε>0, the x  -set such that |xn−bn|<ε|xnbn|<ε modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X][1,X] has a dimension <1, depending on ε and X. Some results are given and a question is proposed.  相似文献   

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This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)s(m,n,k) such that (m+1)(m+1) generic skew-symmetric (k+1)(k+1)-forms in (n+1)(n+1) variables as linear combinations of the same s(m,n,k)s(m,n,k) decomposable skew-symmetric (k+1)(k+1)-forms.  相似文献   

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In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of RdRd on Lp(X)Lp(X)-spaces are convergent for d?3d?3 and p>d/(d-1)p>d/(d-1).  相似文献   

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In this note we derive a maximum principle for an appropriate functional combination of u(x)u(x) and |∇u|2|u|2, where u(x)u(x) is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in RN+1RN+1.  相似文献   

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Given a rank-r   binary matroid we construct a system of O(r3)O(r3) linear equations in O(r2)O(r2) variables that has a solution over GF(2)GF(2) if and only if the matroid is graphic.  相似文献   

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In this paper we investigate the one-dimensional Schrodinger operator L(q)L(q) with complex-valued periodic potential q   when q∈L1[0,1]qL1[0,1] and qn=0qn=0 for n=0,−1,−2,...n=0,1,2,..., where qnqn are the Fourier coefficients of q   with respect to the system {ei2πnx}{ei2πnx}. We prove that the Bloch eigenvalues are (2πn+t)2(2πn+t)2 for n∈ZnZ, t∈CtC and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.  相似文献   

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In this paper the question of finding infinitely many solutions to the problem −Δu+a(x)u=|u|p−2uΔu+a(x)u=|u|p2u, in RNRN, u∈H1(RN)uH1(RN), is considered when N≥2N2, p∈(2,2N/(N−2))p(2,2N/(N2)), and the potential a(x)a(x) is a positive function which is not required to enjoy symmetry properties. Assuming that a(x)a(x) satisfies a suitable “slow decay at infinity” condition and, moreover, that its graph has some “dips”, we prove that the problem admits either infinitely many nodal solutions or infinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.  相似文献   

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