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1.
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 uX and lim infnΦ(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 λI, the functional ΦλJ 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   

2.
Let sR. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,), the Triebel-Lizorkin-type space with p∈(0,), q∈(0,] and τ∈[0,), the Besov-Hausdorff space with p∈(1,), q∈[1,) and and the Triebel-Lizorkin-Hausdorff space with and , where t denotes the conjugate index of t∈[1,]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces.  相似文献   

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In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all tR, and W(t,u) is of subquadratic growth as |u|→.  相似文献   

5.
In this paper, we prove existence of radially symmetric minimizersuA(x)=UA(|x|), having UA(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |u(x)| is the Euclidean norm of the gradient vector and BR is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?∗∗:R×R→[0,] is just convex lsc and and ?∗∗(s,⋅) is even; while ρ1(⋅) and ρ2(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?∗∗(s,v)= is freely allowed.  相似文献   

6.
It is shown that every almost linear bijection of a unital C-algebra A onto a unital C-algebra B is a C-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all unitaries uA, all yA, and n=0,1,2,…, and that almost linear continuous bijection of a unital C-algebra A of real rank zero onto a unital C-algebra B is a C-algebra isomorphism when h(n2uy)=h(n2u)h(y) for all , all yA, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all xX and all unitaries uA, is an A-linear isomorphism. This is applied to investigate C-algebra isomorphisms between unital C-algebras.  相似文献   

7.
If X is a Banach space and CX∗∗ a convex subset, for x∗∗∈X∗∗ and AX∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w-compact subset KX∗∗ we have if and only if φ satisfies the Δ2-condition at 0. We also prove that for every Banach space X, every nonempty convex subset CX and every w-compact subset KX∗∗ then and, if KC is w-dense in K, then .  相似文献   

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Let be a differentiable (but not necessarily C1) vector field, where σ>0 and . Denote by R(z) the real part of zC. If for some ?>0 and for all , no eigenvalue of DpX belongs to , then: (a) for all , there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [−∞,∞) (called the index of X at infinity) such that for some constant vector vR2 the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R2∪{∞} is a repellor (respectively an attractor) of the vector field X+v.  相似文献   

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The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,X×X→[0,] satisfying, for all x,y,zX, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x0X, the set Xw={xX:limλw(λ,x,x0)=0} is a metric space with metric , called a modular space. The modular w is said to be convex if (λ,x,y)?λw(λ,x,y) is also a modular on X. In this case Xw coincides with the set of all xX such that w(λ,x,x0)< for some λ=λ(x)>0 and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.  相似文献   

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Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤pq<, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v0(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.  相似文献   

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We establish the existence of a continuous family of fast positive wavefronts u(t,x)=?(x+ct), ?(−)=0, ?(+)=κ, for the non-local delayed reaction-diffusion equation . Here 0 and κ>0 are fixed points of gC2(R+,R+) and the non-negative K is such that is finite for every real λ. We also prove that the fast wavefronts are non-monotone if .  相似文献   

16.
This note is devoted to a generalization of the Strassen converse. Let gn:R→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,yR, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X1,X2,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X1,X2,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.  相似文献   

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Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any yY. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any yY and zZ and for any MN there exists xMX for which nMB(xn,y),z〉=〈B(xM,y),z〉 for all yY and zZ. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.  相似文献   

20.
Let I=[a,b]⊂R, let 1<p?q<∞, let u and v be positive functions with uLp(I), vLq(I) and let be the Hardy-type operator given by
  相似文献   

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