Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Existence of positive solutions for the nonlinear fractional differential equation D ^αu = f(x,u), 0 < α < 1 has been given (S. Zhang. J. Math. Anal. Appl. 252 (2000), 804–812) where D ^α denotes Riemann–Liouville fractional derivative. In the present work we extend this analysis for n-term non autonomous fractional differential equations. We investigate existence of positive solutions for the following initial value problem

Large Time Asymptotics for the BBM–Burgers Equation

We study large time asymptotics of solutions to the BBM–Burgers equation

Second-order Linearity of Wilcoxon Statistics

The rank statistic , with R _i (t) being the rank of and e ₁ , . . . , e _n being the random sample from a distribution with a cdf F, is considered as a random process with t in the role of parameter. Under some assumptions on c _i , x _i and on the underlying distribution, it is proved that the process converges weakly to the Gaussian process. This generalizes the existing results where the one-dimensional case was considered. We believe our method of the proof can be easily modified for the signed-rank statistics of Wilcoxon type. Finally, we use our results to find the second order asymptotic distribution of the R-estimator based on the Wilcoxon scores and also to investigate the length of the confidence interval for a single parameter β _l .     

Generalised twists,stationary loops,and the Dirichlet energy over a space of measure preserving maps

Let be a bounded Lipschitz domain and consider the Dirichlet energy functional

Gelfand-Hille type theorems in ordered Banach algebras

We consider the Gelfand-Hille Theorems, specifically conditions under which an element in an ordered Banach algebra (A,C) with spectrum {1} is the identity of the algebra. In particular we show that for , where C is a closed normal algebra cone, if and x is doubly Abel bounded then x = 1. Furthermore in the case where and C is a closed proper algebra cone, then x = 1 if and only if x^L is Abel bounded and for some .      

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

We establish a new 3G-Theorem for the Green’s function for the half space We exploit this result to introduce a new class of potentials that we characterize by means of the Gauss semigroup on . Next, we define a subclass of and we study it. In particular, we prove that properly contains the classical Kato class . Finally, we study the existence of positive continuous solutions in of the following nonlinear elliptic problem

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data

This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type

Some properties of essential spectra of a positive operator, II

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ_ew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ_ef(T), then for every a ≠ 0, where T₋₁ is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if , where , then an operator on D_i is band irreducible.      

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

Convex Minimization over the Fixed Point Set of Demicontractive Mappings

This paper deals with a viscosity iteration method, in a real Hilbert space , for minimizing a convex function over the fixed point set of , a mapping in the class of demicontractive operators, including the classes of quasi-nonexpansive and strictly pseudocontractive operators. The considered algorithm is written as: x _n+1 := (1 − w) v _n + w T v _n , v _n := x _n − α _n Θ′(x _n ), where w ∈ (0,1) and , Θ′ is the Gateaux derivative of Θ. Under classical conditions on T, Θ, Θ′ and the parameters, we prove that the sequence (x _n ) generated, with an arbitrary , by this scheme converges strongly to some element in Argmin _Fix(T) Θ.      

Positive bases in spaces of polynomials

For a nonempty compact set we determine the maximal possible dimension of a subspace of polynomial functions over Ω with degree at most m which possesses a positive basis. The exact value of this maximum depends on topological features of Ω, and we will see that in many of the cases m can be achieved. Whereas only for low m or finite sets Ω it is possible that we have a subspace X with positive basis and with dim X = m + 1. Hence there is no Ω for which a positive basis exists in for all . This work was accomplished during the 2^nd author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927.     

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .     

Increasing functions and inverse Santaló inequality for unconditional functions

Let be a convex function and be its Legendre tranform. It is proved that if is invariant by changes of signs, then . This is a functional version of the inverse Santaló inequality for unconditional convex bodies due to J. Saint Raymond. The proof involves a general result on increasing functions on together with a functional form of Lozanovskii’s lemma. In the last section, we prove that for some c > 0, one has always . This generalizes a result of B. Klartag and V. Milman.      

On the Aumann–Shapley value

Let L be a D-lattice, i.e. a lattice ordered effect algebra, and let BV be the Banach space of all real-valued functions of bounded variation on L (vanishing at 0) endowed with the variation norm. We prove the existence of a continuous Aumann–Shapley value φ on NA, the subspace of BV spanned by all functions of the form , where  is a non-atomic σ-additive modular measure and is of bounded variation and continuous at 0 and at 1.      

Some Properties of Essential Spectra of a Positive Operator

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ_ew(T) of the operator T is a set , where is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum

Vectorspacelike representation of absolute planes

The pointset E of an absolute plane can be provided with a binary operation "+" such that (E, +) becomes a loop and for each a E \ {o} the line [a] through o and a is a commutative subgroup of (E, +). Two elements a, b E \ {o} are called independent if [a] ∩ [b] = {o} and the absolute plane is called vectorspacelike if for any two independent elements we have E = [a] + [b] := {x + y | x [a], y [b]}. If is singular then (E, +) is a commutative group and is vectorspacelike iff is Euclidean. If is a hyperbolic plane then is vectorspacelike and in the continous case if a, b are independent, each point p has a unique representation as a quasilinear combination p = α · a + μ · b where α · a [a]and β · b [b] are points, α, β real numbers such that λ (o, λ · a) = |λ|· λ (o, a) and λ (o, μ · b) = |μ|. λ(o, b) and λ is the distance function. This work was partially supported by the Research Project of MIUR (Italian Ministery of Education and University) “Geometria combinatoria e sue applicazioni” and by the research group GNSAGA of INDAM. Dedicated to Walter Benz on the occasion of his 75 ^th birthday, in friendship     

Strong convergence of viscosity approximation methods with strong pseudocontraction for Lipschitz pseudocontractive mappings

In this paper, for a Lipschitz pseudocontractive mapping T, we study the strong convergence of iterative schemes generated by

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.