共查询到20条相似文献,搜索用时 62 毫秒
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In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α1y(b)+β1y′(b)=γ1, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C1(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L1(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined. 相似文献
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Norihide Tokushige 《Discrete Mathematics》2010,310(3):453-460
Let m(n,k,r,t) be the maximum size of satisfying |F1∩?∩Fr|≥t for all F1,…,Fr∈F. We prove that for every p∈(0,1) there is some r0 such that, for all r>r0 and all t with 1≤t≤⌊(p1−r−p)/(1−p)⌋−r, there exists n0 so that if n>n0 and p=k/n, then . The upper bound for t is tight for fixed p and r. 相似文献
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Jie Xiao 《Journal of Differential Equations》2006,224(2):277-295
Let u(t,x) be the solution of the heat equation (∂t-Δx)u(t,x)=0 on subject to u(0,x)=f(x) on Rn. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t2,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞). 相似文献
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Mohammad Javaheri 《Journal of Mathematical Analysis and Applications》2010,361(2):332-337
Let γ:[0,1]→2[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈2(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A0=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π1, π2 are projections on the axes. 相似文献
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Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with u∈Lp′(I), v∈Lq(I) and let be the Hardy-type operator given by
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Hongtao Xue 《Journal of Mathematical Analysis and Applications》2011,384(2):439-443
By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero. 相似文献
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B.V. Subramanya Bharadwaj 《Discrete Mathematics》2009,309(4):834-1274
Let G=(V,E) be a finite, simple and undirected graph. For S⊆V, let δ(S,G)={(u,v)∈E:u∈S and v∈V−S} be the edge boundary of S. Given an integer i, 1≤i≤|V|, let the edge isoperimetric value of G at i be defined as be(i,G)=minS⊆V;|S|=i|δ(S,G)|. The edge isoperimetric peak of G is defined as be(G)=max1≤j≤|V|be(j,G). Let bv(G) denote the vertex isoperimetric peak defined in a corresponding way. The problem of determining a lower bound for the vertex isoperimetric peak in complete t-ary trees was recently considered in [Y. Otachi, K. Yamazaki, A lower bound for the vertex boundary-width of complete k-ary trees, Discrete Mathematics, in press (doi:10.1016/j.disc.2007.05.014)]. In this paper we provide bounds which improve those in the above cited paper. Our results can be generalized to arbitrary (rooted) trees.The depth d of a tree is the number of nodes on the longest path starting from the root and ending at a leaf. In this paper we show that for a complete binary tree of depth d (denoted as ), and where c1, c2 are constants. For a complete t-ary tree of depth d (denoted as ) and d≥clogt where c is a constant, we show that and where c1, c2 are constants. At the heart of our proof we have the following theorem which works for an arbitrary rooted tree and not just for a complete t-ary tree. Let T=(V,E,r) be a finite, connected and rooted tree — the root being the vertex r. Define a weight function w:V→N where the weight w(u) of a vertex u is the number of its successors (including itself) and let the weight index η(T) be defined as the number of distinct weights in the tree, i.e η(T)=|{w(u):u∈V}|. For a positive integer k, let ?(k)=|{i∈N:1≤i≤|V|,be(i,G)≤k}|. We show that . 相似文献
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We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V0={p1,p2,p3} be the set of vertices of SG and the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw=uw1uw2?uwn for any sequence w=(w1,w2,…,wn)∈n{1,2,3}. The union of the images of V0 under these iterations is the set of nth stage vertices Vn of SG. Let F:Vn→R be any function. Given any numbers αw(w∈n{1,2,3}) with 0<|αw|<1, there exists a unique continuous extension of F, such that
f(uw(x))=αwf(x)+hw(x) 相似文献
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Zhijun Zhang 《Journal of Mathematical Analysis and Applications》2005,312(1):33-43
By Karamata regular variation theory and constructing comparison functions, we show the exact asymptotic behaviour of the unique classical solution near the boundary to a singular Dirichlet problem −Δu=k(x)g(u), u>0, x∈Ω, u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN; g∈C1((0,∞),(0,∞)), , for each ξ>0, for some γ>0; and for some α∈(0,1), is nonnegative on Ω, which is also singular near the boundary. 相似文献
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In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C2(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C0-semigroup of contractions on C(d[0,1]) and C2(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f. 相似文献
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Zhijun Zhang 《Journal of Mathematical Analysis and Applications》2005,308(2):532-540
By constructing the comparison functions and the perturbed method, it is showed that any solution u∈C2(Ω) to the semilinear elliptic problems Δu=k(x)g(u), x∈Ω, u|∂Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c0>0, ; g∈C1[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , . 相似文献
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We consider the sub- or supercritical Neumann elliptic problem −Δu+μu=u5+ε, u>0 in Ω; on ∂Ω, Ω being a smooth bounded domain in , μ>0 and ε≠0 a small number. Hμ denoting the regular part of the Green's function of the operator −Δ+μ in Ω with Neumann boundary conditions, and , we show that a nontrivial relative homology between the level sets ?μc and ?μb, b<c<0, induces the existence, for ε>0 small enough, of a solution to the problem, which blows up as ε goes to zero at a point a∈Ω such that b??μ(a)?c. The same result holds, for ε<0, assuming that 0<b<c. It is shown that, (resp. >0) for μ small (resp. large) enough, providing us with cases where the above assumptions are satisfied. 相似文献
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Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L2(E;m) and ((Xt)t?0,(Px)x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f2) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5). 相似文献
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Let be a function satisfying Carathéodory's conditions and (1−t)e(t)∈L1(0,1). Let ξi∈(0,1), ai∈R, i=1,…,m−2, 0<ξ1<ξ2<?<ξm−2<1 be given. This paper is concerned with the problem of existence of a C1[0,1) solution for the m-point boundary value problem
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M. Loayza 《Journal of Differential Equations》2006,229(2):509-528
We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut−Δu=a(x)uq+b(x)up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a∈Lα(Ω), b∈Lβ(Ω) and 0<q?1<p. The initial data u(0)=u0 is considered in the space Lr(Ω), r?1. In the main result (0<q<1), we assume a,b?0 a.e. in Ω and we assume that u0?γdΩ for some γ>0. We find a unique solution in the space . 相似文献
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Biagio Ricceri 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):4151-4157
If X is a real Banach space, we denote by WX the class of all functionals possessing the following property: if {un} is a sequence in X converging weakly to u∈X and lim infn→∞Φ(un)≤Φ(u), then {un} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C1 functional, belonging to WX, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X∗; a C1 functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C1 functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation
Φ′(x)=λJ′(x)+μΨ′(x) 相似文献