Second order nonlinear multivalued boundary problems in Hilbert spaces

In this paper we consider second order differential inclusions in real Hilbert space, namely p(t)⋅x^″(t)+r(t)⋅x^′(t)∈Ax(t)+F(t,x(t)), a.e. on [0,T], under the nonlinear boundary conditions. Using techniques from multivalued analysis and the theory of operators of monotone type, we prove the existence of solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present a special case of interest, which fit into our framework, illustrating the generality of our results.     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.     

On approximately additive functions

Let C be a convex symmetric subset of a real Banach space F and K be a subgroup of the group (F,+). Let E be a real linear space, h:E→F, and h(x+y)−h(x)−h(y)∈K+C for x,y∈E. We prove that under some additional assumptions h can be represented in the form: h=A+γ+κ with an additive (or linear) A:E→F and some γ:E→C, κ:E→K.     

Selections of bounded variation under the excess restrictions

Let X be a metric space with metric d, c(X) denote the family of all nonempty compact subsets of X and, given F,G∈c(X), let e(F,G)=supx∈Finfy∈Gd(x,y) be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval [a,b]. The main result of the paper is the following selection theorem: If,V₊(F,[a,b])<∞,t₀∈[a,b]andx₀∈F(t₀), then there exists a single-valued functionof bounded variation such thatf(t)∈F(t)for allt∈[a,b],f(t₀)=x₀,V(f,[a,t₀))?V₊(F,[a,t₀))andV(f,[t₀,b])?V₊(F,[t₀,b]). We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1-82]. In contrast to this, a multifunction F satisfying e(F(s),F(t))?C(t−s) for some constant C?0 and all s,t∈[a,b] with s?t (Lipschitz continuity with respect to e(⋅,⋅)) admits a Lipschitz selection with a Lipschitz constant not exceeding C if t₀=a and may have only discontinuous selections of bounded variation if a<t₀?b. The same situation holds for continuous selections of when it is excess continuous in the sense that e(F(s),F(t))→0 as s→t−0 for all t∈(a,b] and e(F(t),F(s))→0 as s→t+0 for all t∈[a,b) simultaneously.     

Two-batch liar games on a general bounded channel

We consider an extension of the 2-person Rényi-Ulam liar game in which lies are governed by a channel C, a set of allowable lie strings of maximum length k. Carole selects x∈[n], and Paul makes t-ary queries to uniquely determine x. In each of q rounds, Paul weakly partitions [n]=A₀∪?∪At−1 and asks for a such that x∈Aa. Carole responds with some b, and if a≠b, then x accumulates a lie (a,b). Carole's string of lies for x must be in the channel C. Paul wins if he determines x within q rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space [n] for which Paul can guarantee finding the distinguished element is as q→∞, where Ek(C) is the number of lie strings in C of maximum length k. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel C.     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

Davenport constant with weights and some related questions, II

Let G be a finite abelian group of order n and let A⊆Z be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x₁,…,xk) with xi∈G, there exists a non-empty subsequence (xj₁,…,xjl) and a₁,…,al∈A such that . Similarly, for any such set A, EA(G) is defined to be the least t∈N such that for all sequences (x₁,…,xt) with xi∈G, there exist indices j₁,…,jn∈N,1?j₁<?<jn?t, and ?₁,…,?n∈A with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case.     

Strong solutions for differential equations in abstract spaces

Let (E,F) be a locally convex space. We denote the bounded elements of E by . In this paper, we prove that if B_Eb is relatively compact with respect to the F topology and f:I×E_b→E_b is a measurable family of F-continuous maps then for each x₀∈E_b there exists a norm-differentiable, (i.e. differentiable with respect to the ∥·∥_F norm) local solution to the initial valued problem u_t(t)=f(t,u(t)), u(t₀)=x₀. All of this machinery is developed to study the Lipschitz stability of a nonlinear differential equation involving the Hardy-Littlewood maximal operator.     

Improved algorithms for the continuous tree edge-partition problems and a note on ratio and sorted matrices searches

Let p≥2 be an integer and T be an edge-weighted tree. A cut on an edge of T is a splitting of the edge at some point on it. A p-edge-partition of T is a set of p subtrees induced by p−1 cuts. Given p and T, the max-min continuous tree edge-partition problem is to find a p-edge-partition that maximizes the length of the smallest subtree; and the min-max continuous tree edge-partition problem is to find a p-edge-partition that minimizes the length of the largest subtree. In this paper, O(n²)-time algorithms are proposed for these two problems, improving the previous upper bounds by a factor of log (min{p,n}). Along the way, we solve a problem, named the ratio search problem. Given a positive integer m, a (non-ordered) set B of n non-negative real numbers, a real valued non-increasing function F, and a real number t, the problem is to find the largest number z in {b/a|a∈[1,m],b∈B} such that F(z)≥t. We give an O(n+t_F×(logn+logm))-time algorithm for this problem, where t_F is the time required to evaluate the function value F(z) for any real number z.     

An even-order three-point boundary value problem on time scales

We study the even-order dynamic equation (−1)ⁿx^(Δ∇)n(t)=λh(t)f(x(t)), t∈[a,c] satisfying the boundary conditions x^(Δ∇)i(a)=0 and x^(Δ∇)i(c)=βx^(Δ∇)i(b) for 0?i?n−1. The three points a,b,c are from a time scale , where 0<β(b−a)<c−a for b∈(a,c), β>0, f is a positive function, and h is a nonnegative function that is allowed to vanish on some subintervals of [a,c] of the time scale.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

On multiplicative perturbation of integral resolvent families

In this paper we study multiplicative perturbations for the generator of a strongly continuous integral resolvent family of bounded linear operators defined on a Banach space X. Assuming that a(t) is a creep function which satisfies a(⁰⁺)>0, we prove that if (A,a) generates an integral resolvent, then (A(I+B),a) also generates an integral resolvent for all B∈B(X,Z), where Z belongs to a class of admissible Banach spaces. In special instances of a(t) the space Z is proved to be characterized by an extended class of Favard spaces.     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

The heat equation with singular nonlinearity and singular initial data

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 .     

On the existence of mild solutions of semilinear evolution differential inclusions

In this paper we deal with a Cauchy problem governed by the following semilinear evolution differential inclusion:

x^′(t)∈A(t)x(t)+F(t,x(t))     

Properties of the density for a three-dimensional stochastic wave equation

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let pt,x(y) be the density of the law of the solution u(t,x) of such an equation at points (t,x)∈]0,T]×R³. We prove that the mapping (t,x)?pt,x(y) owns the same regularity as the sample paths of the process {u(t,x),(t,x)∈]0,T]×R³} established in [R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press]. The proof relies on Malliavin calculus and more explicitly, the integration by parts formula of [S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./Springer-Verlag, Bombay, 1984] and estimates derived from it.     

Maximal regularity for non-autonomous Schrödinger type equations

In this paper we study the maximal regularity property for non-autonomous evolution equations t_∂u(t)+A(t)u(t)=f(t), u(0)=0. If the equation is considered on a Hilbert space H and the operators A(t) are defined by sesquilinear forms a(t,⋅,⋅) we prove the maximal regularity under a Hölder continuity assumption of t→a(t,⋅,⋅). In the non-Hilbert space situation we focus on Schrödinger type operators A(t):=−Δ+m(t,⋅) and prove Lp−Lq estimates for a wide class of time and space dependent potentials m.     

A bilinear Orlicz-Pettis Theorem

Suppose E,F,G are locally convex spaces, is a bilinear operator and λ is a scalar sequence space. A series j_∑xj in E is λ b multiplier convergent if for every t={tj}∈λ there exists xt∈E such that for every y∈F. Under continuity assumptions on the linear operators b(x,⋅), we establish several versions of the Orlicz-Pettis Theorem for multiplier convergent series. Applications to spaces of continuous linear operators are given.     

Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences

In this work we study mappings f from an open subset A of a Banach space E into another Banach space F such that, once a∈A is fixed, for mixed (s;q)-summable sequences of elements of a fixed neighborhood of 0 in E, the sequence is absolutely p-summable in F. In this case we say that f is (p;m(s;q))-summing at a. Since for s=q the mixed (s;q)-summable sequences are the weakly absolutely q-summable sequences, the (p;m(q;q))-summing mappings at a are absolutely (p;q)-summing mappings at a. The nonlinear absolutely summing mappings were studied by Matos (see [Math. Nachr. 258 (2003) 71-89]) in a recent paper, where one can also find the historical background for the theory of these mappings. When s=+∞, the mixed (∞,q)-summable sequences are the absolutely q-summable sequences. Hence the (p;m(∞;q))-summing mappings at a are the regularly (p;q)-summing mappings at a. These mappings were also studied in [Math. Nachr. 258 (2003) 71-89] and they were important to give a nice characterization of the absolutely (p;q)-summing mappings at a. We show that for q<s<+∞ the space of the (p;m(s;q))-summing mappings at a are different from the spaces of the absolutely (p;q)-summing mappings at a and different from the spaces of regularly (p;q)-summing mappings at a. We prove a version of the Dvoretzky-Rogers theorem for n-homogeneous polynomials that are (p;m(s;q))-summing at each point of E. We also show that the sequence of the spaces of such n-homogeneous polynomials, n∈N, gives a holomorphy type in the sense of Nachbin. For linear mappings we prove a theorem that gives another characterization of (s;q)-mixing operators in terms of quotients of certain operators ideals.