On positive solutions for some semilinear periodic parabolic eigenvalue problems

For a bounded domain Ω in , N?2, satisfying a weak regularity condition, we study existence of positive and T-periodic weak solutions for the periodic parabolic problem Lu_λ=λg(x,t,u_λ) in , u_λ=0 on . We characterize the set of positive eigenvalues with positive eigenfunctions associated, under the assumptions that g is a Caratheodory function such that ξ→g(x,t,ξ)/ξ is nonincreasing in (0,∞) a.e. satisfying some integrability conditions in (x,t) and

     

On the asymptotic stability of equilibrium

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a positive definite function V which has infinitesimal upper bound such that is negative definite. In this paper we prove that if is bounded then the condition that is negative definite can be weakened and replaced by that and is negative definite.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

A generalization of Barbashin-Krasovski theorem

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a function V(t,x), a(‖x‖)?V(t,x)?b(‖x‖) for some a,b∈K, such that for some c∈K. In this paper we prove that if f(t,x) is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , t∈[−T,T], where , uniformly for (t,x)∈[−T,T]×BH.     

Strong convergence of the composite Halpern iteration

Let C be a closed convex subset of a uniformly smooth Banach space E and let T:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈C, the initial guess x₀∈C is chosen arbitrarily and given sequences , and in (0,1), the following conditions are satisfied:

(i)

;

(ii)

αn→0, βn→0 and 0<a?γn, for some a∈(0,1);

(iii)

, and . Let be a composite iteration process defined by

     

The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems

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, , .     

Regularity results for a class of obstacle problems under nonstandard growth conditions

We prove regularity results for minimizers of functionals in the class , where is a fixed function and f is quasiconvex and fulfills a growth condition of the type

L⁻¹|z|^p(x)?f(x,ξ,z)?L(1+|z|^p(x)),     

On powers of the Heaviside function for negative integers

The distributions and were defined as the neutrix limit of the sequences and respectively for , see [J.D. Nicholos, B. Fisher, The distribution composition , J. Math. Anal. Appl. 258 (2001) 131-145; B. Fisher, On defining the distribution , Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 15 (1985) 119-129]. We here consider these distributions when r=0. In other words, we define the sth powers of the Heaviside function H(x) in the distributional sense for negative integers. Further compositions are also considered.     

Systems of orthogonal polynomials arising from the modular j-function

Let be the polynomial whose zeros are the j-invariants of supersingular elliptic curves over . Generalizing a construction of Atkin described in a recent paper by Kaneko and Zagier (Computational Perspectives on Number Theory (Chicago, IL, 1995), AMS/IP 7 (1998) 97-126), we define an inner product on for every . Suppose a system of orthogonal polynomials {P_n,ψ(x)}_n=0^∞ with respect to exists. We prove that if n is sufficiently large and ψ(x)P_n,ψ(x) is p-integral, then over . Further, we obtain an interpretation of these orthogonal polynomials as a p-adic limit of polynomials associated to p-adic modular forms.     

On controlled extensions of functions

For any numerical function we give sufficient conditions for resolving the controlled extension problem for a closed subset A of a normal space X. Namely, if the functions , and satisfy the equality E(f(a),g(a))=h(a), for every a∈A, then we are interested to find the extensions f? and ? of f and g, respectively, such that , for every x∈X. We generalize earlier results concerning E(u,v)=u·v by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.     

Generalized vector valued almost periodic and ergodic distributions

Schwartz's almost periodic distributions are generalized to the case of Banach space valued distributions , and furthermore for a given arbitrary class A to for φ∈ test functions D(R,C)}. It is shown that this extension process is iteration complete, i.e. . Moreover the T from are characterized in various ways, also tempered distributions with P={X-valued functions of polynomial growth} are shown. Under suitable assumptions , , where for all h>0}, is defined with the corresponding extension of Mh. With an extension of the indefinite integral from to D^′(R,X) a distribution analogue to the Bohl-Bohr-Amerio-Kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained, also for almost automorphic, Levitan almost periodic and recurrent functions, similar for a result of Levitan concerning ergodic indefinite integrals. For many of the above results a new (Δ)-condition is needed, we show that it holds for most of the A needed in applications. Also an application to the study of asymptotic behavior of distribution solutions of neutral integro-differential-difference systems is given.     

A supplement to precise asymptotics in the law of the iterated logarithm

Let be a sequence of real-valued i.i.d. random variables with E(X)=0 and E(X²)=1, and set , n?1. This paper studies the precise asymptotics in the law of the iterated logarithm. For example, using a result on convergence rates for probabilities of moderate deviations for obtained by Li et al. [Internat. J. Math. Math. Sci. 15 (1992) 481-497], we prove that, for every b∈(−1/2,1],

     

Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E^∗ satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . Let be a sequence in [?,1−?],?∈(0,1), for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

Reflexivity and hyperreflexivity of operator spaces

In this paper, we show that if is an n-dimensional subspace of L(V) such that every nonzero transformation of has rank greater than or equal to 2n−1 then is algebraically reflexive. If is an n-dimensional subspace of B(H) such that every nonzero transformation of  has rank greater than or equal to 2n−1 then is hyperreflexive. We also consider how to construct some new hyperreflexive subspaces.     

Removable singularities for a Sobolev space

Let Ω⊂RN be an open set and F a relatively closed subset of Ω. We show that if the (N−1)-dimensional Hausdorff measure of F is finite, then the spaces and coincide, that is, F is a removable singularity for . Here is the closure of in H¹(Ω) and H¹(Ω) denotes the first order Sobolev space. We also give a relative capacity criterium for this removability. The space is important for defining realizations of the Laplacian with Neumann and with Robin boundary conditions. For example, if the boundary of Ω has finite (N−1)-dimensional Hausdorff measure, then our results show that we may replace Ω by the better set (which is regular in topology), i.e., Neumann boundary conditions (respectively Robin boundary conditions) on Ω and on coincide.