Maximal Regularity

We address the study of well posedness on Lebesgue spaces of sequences for the following fractional semidiscrete model with finite delay where T is a bounded linear operator defined on a Banach space X (typically a space of functions like L p (Ω), 1 < p < ∞) and ∆ α corresponds to the time discretization of the continuous Riemann-Liouville fractional derivative by means of the Poisson distribution. We characterize the existence and uniqueness of solutions in vector-valued Lebesgue spaces of sequences of the model (0.1) in terms of boundedness of the operator-valued symbol whenever 0 < α ≤ 1 and X satisfies a geometrical condition. For this purpose, we use methods from operator-valued Fourier multipliers and resolvent operator families associated to the homogeneous problem. We apply this result to show a practical and computational criterion in the context of Hilbert spaces.

The orthoglide is a 3-DOF parallel mechanism designed at IRCCyN for machining applications. It features three fixed parallel linear joints which are mounted orthogonally and a mobile platform which moves in the Cartesian x-y-z space with fixed orientation. The orthoglide has been designed as function of a prescribed Cartesian workspace with prescribed kinetostatic performances. The interesting features of the orthoglide are a regular Cartesian workspace shape, uniform performances in all directions and good compactness. A small-scale prototype of the orthoglide under development is presented at the end of this paper.

We use Fourier multiplier theorems to establish maximal regularity results for a class of integro-differential equations with infinite delay in Banach spaces. Concrete equations of this type arise in viscoelasticity theory. Results are obtained for periodic solutions in the vector valued Lebesgue and Besov spaces. An application to semilinear equations is considered.

This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, and providing a copy to your institution's administrator.

Let us denote by Φ(λ, µ) the statement that B(λ) = D(λ) ω , i.e. the Baire space of weight λ, has a coloring with µ colors such that every homeomorphic copy of the Cantor set C in B(λ) picks up all the µ colors. We call a space X π-regular if it is Hausdorff and for every nonempty open set U in X there is a non-empty open set V such that V ⊂ U . We recall that a space X is called feebly compact if every locally finite collection of open sets in X is finite. A Tychonov space is pseudocompact iff it is feebly compact. The main result of this paper is the following. Theorem. Let X be a crowded feebly compact π-regular space and µ be a fixed (finite or infinite) cardinal. If Φ(λ, µ) holds for all λ < c(X) then X is µ-resolvable, i.e. contains µ pairwise disjoint dense subsets. (Here c(X) is the smallest cardinal κ such that X does not contain κ many pairwise disjoint open sets.) This significantly improves earlier results of van Mill [6], resp. Ortiz-Castillo and Tomita .

We improve some results of Pavlov and of Filatova, respectively, concerning a problem of Malychin by showing that every regular space X that satisfies ∆(X) > e(X) is ω-resolvable. Here ∆(X), the dispersion character of X, is the smallest size of a non-empty open set in X and e(X), the extent of X, is the supremum of the sizes of all closed-and-discrete subsets of X. In particular, regular Lindelöf spaces of uncountable dispersion character are ω-resolvable. We also prove that any regular Lindelöf space X with |X| = ∆(X) = ω 1 is even ω 1 -resolvable. The question if regular Lindelöf spaces of uncountable dispersion character are maximally resolvable remains wide open.

We consider some non-autonomous second order Cauchy problems of the form u + B(t)u + A(t)u = f (t ∈ [0, T ]), u(0) =u(0) = 0. We assume that the first order probleṁ u + B(t)u = f (t ∈ [0, T ]), u(0) = 0, has L p-maximal regularity. Then we establish L p-maximal regularity of the second order problem in situations when the domains of B(t 1) and A(t 2) always coincide, or when A(t) = κB(t).

In this paper, we prove some new results on operational second order dierential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is performed when the second member belongs to a Sobolev space. Existence, uniqueness and optimal regularity of the classical solution are proved using interpolation theory and results on the class of operators with bounded imaginary powers. We also give an example to which our theory applies. This paper improves naturally the ones studied in the commutative case by M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri: in fact, introducing some operational commutator, we generalize the representation formula of the solution given in the commutative case and prove that this representation has the desired regularity. Keywords: second-order elliptic dierential equations; Robin boundary conditions in non commutative cases; analytic semigroup; maximal regularity. Dedicated to Professor Angelo Favini on his 70th birthday.

We consider a non-autonomous evolutionary probleṁ u(t) + A(t)u(t) = f (t), u(0) = u0 where the operator A(t) : V → V ′ is associated with a form a(t, ., .) : V × V → R and u0 ∈ V. Our main concern is to prove well-posedness with maximal regularity which means the following. Given a Hilbert space H such that V is continuously and densely embedded into H and given f ∈ L 2 (0, T ; H) we are interested in solutions u ∈ H 1 (0, T ; H) ∩ L 2 (0, T ; V). We do prove well-posedness in this sense whenever the form is piecewise Lipschitz-continuous and symmetric. Moreover, we show that each solution is in C([0, T ]; V). We apply the results to non-autonomous Robinboundary conditions and also use maximal regularity to solve a quasilinear problem.

We show that the incompressible 3D Navier-Stokes system in a C 1,1 bounded domain or a bounded convex domain Ω with a non penetration condition ν • u = 0 at the boundary ∂Ω together with a time-dependent Robin boundary condition of the type ν × curl u = β(t)u on ∂Ω admits a solution with enough regularity provided the initial condition is small enough in an appropriate functional space.

We consider non-autonomous wave equations ü(t) + B(t)u(t) + A(t)u(t) = f (t) t-a.e. u(0) = u0,u(0) = u1. where the operators A(t) and B(t) are associated with time-dependent sesquilinear forms a(t, ., .) and b defined on a Hilbert space H with the same domain V. The initial values satisfy u0 ∈ V and u1 ∈ H. We prove well-posedness and maximal regularity for the solution both in the spaces V ′ and H. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem.

A well known result by Rubio de Francia asserts that for every finite family of disjoint intervals {I k } in R, and p in the range 2 ≤ p < ∞, there exists Cp > 0 such that X k r k S I k f L p L p ([0,1]) (R) ≤ Cp f L p (R) , where the r k 's are the Rademacher functions. In this note we prove that, given a compact connected abelian group G with dual group Γ and p in the range 2 ≤ p < ∞, there is a constant Cp, independent of G and the particular ordering on Γ, such that for every sequence {I k } of disjoint intervals in Γ, we have X k r k S I k f L p L p ([0,1]) (G) ≤ Cp f L p (G). We obtain the result by a transference approach that can be used for functions taking values in Banach spaces.

Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes are often used in this context. However, up to our knowledge, the family of MPRK has been only developed up to third order of accuracy. In this work, we propose a method to solve PDS problems, but using the Deferred Correction (DeC) process as a time integration method. Applying the modified Patankar approach to the DeC scheme results in provable conservative and positivity preserving methods. Furthermore, we demonstrate that these modified Patankar DeC schemes can be constructed up to arbitrarily high order. Finally, we validate our theoretical analysis through numerical simulations.

Lifting idempotents modulo ideals is an important tool in studying the structure of rings. This paper lays out the consequences of lifting other properties modulo ideals, including lifting of von Neumann regular elements, lifting isomorphic idempotents, and lifting conjugate idempotents. Applications are given for IC rings, perspective rings, and Dedekind-finite rings, which improve multiple results in the literature. We give a new characterization of the class of exchange rings; they are rings where regular elements lift modulo all left ideals. We also uncover some hidden connections between these lifting properties. For instance, if regular elements lift modulo an ideal, then so do isomorphic idempotents. The converse is true when units lift. The logical relationships between these and several other important lifting properties are completely characterized. Along the way, multiple examples are developed that illustrate limitations to the theory.

This paper is devoted to the numerical analysis of abstract elliptic differential equations in L p ([0, T ]; E) spaces. The presentation uses general approximation scheme and is based on C 0-semigroup theory and a functional analysis approach. For the solutions of difference scheme of the second order accuracy, the almost coercive inequality in L p τn ([0, T ]; E n) spaces with the factor min{| ln 1 τn |, 1 + | ln B n B(En) |} is obtained. In the case of UMD space E n we establish a coercive inequality for the same scheme in L p τn ([0, T ]; E n) under the condition of R-boundedness.

Parti Foundations 1 1 Why equations with Levy noise? 3 1.1 Discrete-time dynamical systems 3 1.2 Deterministic continuous-time systems ' ' ' 5 1.3 Stochastic continuous-time systems 6 1.4 Courrege's theorem 8 1.5 Ito's approach 9 1.6 Infinite-dimensional case 12 2 Analytic preliminaries 13 2.1 Notation ' 13 2.2 Sobolev and Holder spaces 13 2.3 L p-and C p-spaces " ' ' 15 2.4 Lipschitz functions and composition operators 16 2.5 Differential operators 17 3 Probabilistic preliminaries 20 "3.1 Basic definitions ~ 20 3.2 Kolmogorov existence theorem 22 3.3 Random elements in Banacti spaces 23 3.4 Stochastic processes in Banach spaces 25 3.5 Gaussian measures on Hilbert spaces 28 3.6 Gaussian 1 measures on topological spaces 30 ' 3.7 Submartingales , ; •. 31 3.8 Semimartingales • 36 3.9 Burkholder-Davies-Guhdy inequalities .

We consider the Schrödinger evolution on graphs, i.e., solutions to the equation $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, where $\mathcal {A}$ is the set of vertices of the graph and the matrix $(L(\alpha ,\beta ))_{\alpha ,\beta \in \mathcal {A}}$ describes interaction between the vertices, in particular two vertices α and β are connected if $L(\alpha ,\beta )\neq 0$. We assume that the graph has a “web-like” structure, i.e., it consists of an inner part, formed by a finite number of vertices, and some threads attach to it.We prove that such a solution $u(t,\alpha )$ cannot decay too fast along one thread at two different times, unless it vanishes at this thread.We also give a characterization of the dimension of the vector space formed by all the solutions of $\partial _t u(t,\alpha ) = i\sum _{\beta \in \mathcal {A}}L(\alpha ,\beta )u(t,\beta )$, when $\mathcal {A}$ is a finite set, in terms of the number of the different eigenvalu...

We investigate a quasilinear initial-boundary value problem for Kuznetsov's equation with non-homogeneous Dirichlet boundary conditions. This is a model in nonlinear acoustics which describes the propagation of sound in fluidic media with applications in medical ultrasound. We prove that there exists a unique global solution which depends continuously on the sufficiently small data and that the solution and its temporal derivatives converge at an exponential rate as time tends to infinity. Compared to the analysis of Kaltenbacher & Lasiecka, we require optimal regularity conditions on the data and give simplified proofs which are based on maximal Lp-regularity for parabolic equations and the implicit function theorem.

We study continuity points of functions with values in generalized metric spaces. We define the generalized oscillation, which is a useful tool in our study. Let X be a topological space and Y be a weakly developable space. Let f : X → Y be a function. Then the set C(f) of continuity points of f is a G δ-set in X. Some results concerning continuity points of separately continuous functions as well as functions with closed graphs are also given.

Two new cardinal functions defined in the class of n-Hausdorff and n-Urysohn spaces that extend pseudocharacter and closed pseudocharacter respectively are introduced. Through these new functions bounds on the cardinality of n-Urysohn spaces that represent variations of known results are given. Also properties of n-Urysohn n-H-closed spaces are proved.

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: "When is the Laplace-Beltrami operator ∆ : H k+1 (M) ∩ H 1 0 (M) → H k−1 (M), k ∈ N 0 , invertible?" We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nach. 2001). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let ∂ D M ⊂ ∂M be an open and closed subset of the boundary of M. We say that (M, ∂ D M) has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance dist(x, ∂ D M) from a point x ∈ M to ∂ D M ⊂ ∂M is bounded uniformly in x (and hence, in particular, ∂ D M intersects all connected components of M). For manifolds (M, ∂ D M) with finite width, we prove a Poincaré inequality for functions vanishing on ∂ D M , thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of ∆ with domain given by mixed boundary conditions, in particular, ∆ is invertible for manifolds (M, ∂ D M) with finite width. The bounded geometry assumption then allows us to prove the well-posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces H s (M), s ≥ 0. Contents B. AMMANN, N. GROSSE, AND V. NISTOR 4.2. Higher regularity 26 4.3. Mixed boundary conditions 28 References 29

By using Fourier multiplier theorems, the maximal B-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations.

The main purpose of this paper is to identify topologies on the closed subsets C(X) of a Hausdorff space X that are sequentially equivalent to classical Kuratowski-Painlevé convergence K. This reduces to a study of upper topologies sequentially equivalent to upper Kuratowski-Painlevé convergence K+, where we are of course led to consider the sequential modification of upper Kuratowski-Painlevé convergence. We characterize those miss topologies induced by a cobase of closed sets that are sequentially equivalent to K+, with special attention given to X first countable. Separately in the final section, we revisit Mrowka&#39;s theorem on the compactness of Kuratowski-Painlevé convergence.

This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein's constraints equations. We establish existence theorems for the Hamiltonian and the momentum constraints with constant mean curvature and with a background metric that satisfies very low regularity assumptions. These results extend the regularity results of Holst, Nagy and Tsogtgerel about the constraint equations on compact manifolds in the Besov space B s p,p [23], to asymptotically flat manifolds. We also consider the Brill-Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds [15, 29], as well as they enable us to construct the initial data for the Einstein-Euler system.

In this paper we are concerned with nonlinear Schrödinger equations with random potentials. Our class includes continuum and discrete potentials. Conditions on the potential Vω are found for existence of solutions almost sure ω. We study probabilistic properties like central limit theorem and law of larger numbers for the obtained solutions by independent ensembles. We also give estimates on the expected value for the L ∞-norm of the solution showing how it depends on the size of the potential.

We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.

We study the convergence rates of solutions to drift-diffusion systems (arising from plasma, semiconductors and electrolytes theories) to their self-similar or steady states. This analysis involves entropy- type Lyapunov functionals and logarithmic Sobolev inequalities.

In this paper the existence and uniqueness of different types of solutions to a class of semilinear retarded differential equations with nonlocal history conditions are obtained by a fixed point argument. Also finite dimensional approximation of these solutions in a Hilbert space is established.

In this paper, we present some new results about the Alexandroff Duplicate Space. We prove that if a space X has the property P , then its Alexandroff Duplicate space A(X) may not have P , where P is one of the following properties: extremally disconnected, weakly extremally disconnected, quasi-normal, pseudocompact. We prove that if X is αnormal, epinormal, or has property wD, then so is A(X). We prove almost normality is preserved by A(X) under special conditions.

In this paper we study the behaviour of selective separability properties in the class of Frechét-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frechét-Urysohn (hence R-separable and selectively separable) space which is not H-separable; assuming p = c, we construct such an example which is also zero-dimensional and α 4. Also, motivated by a result of Barman and Dow stating that the product of two countable Frechét-Urysohn spaces is Mseparable under PFA, we show that the MA is not sufficient here. In the last section we prove that in the Laver model, the product of any two H-separable spaces is mH-separable.

Initial boundary value problems with a time-nonlocal condition for a subdiffusion equation with the Riemann-Liouville time-fractional derivatives are considered. The elliptical part of the equation is the Laplace operator, defined in an arbitrary N−dimensional domain Ω with a sufficiently smooth boundary ∂Ω. The existence and uniqueness of the solution to the considered problems are proved. Inverse problems are studied for determining the right-hand side of the equation and a function in a time-nonlocal condition. The main research tool is the Fourier method, so the obtained results can be extended to subdiffusion equations with a more general elliptic operator.

In this note we will generalize the results deduced in Figalli and Glaudo (Arch Ration Mech Anal 237(1):201–258, 2020) and Deng et al. (Sharp quantitative estimates of Struwe’s Decomposition. Preprint http://arxiv.org/abs/2103.15360, 2021) to fractional Sobolev spaces. In particular we will show that for $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$n&gt;2s$$ n &gt; 2 s and $$\nu \in \mathbb {N}$$ ν ∈ N there exists constants $$\delta = \delta (n,s,\nu )&gt;0$$ δ = δ ( n , s , ν ) &gt; 0 and $$C=C(n,s,\nu )&gt;0$$ C = C ( n , s , ν ) &gt; 0 such that for any function $$u\in \dot{H}^s(\mathbb {R}^n)$$ u ∈ H ˙ s ( R n ) satisfying, $$\begin{aligned} \left\| u-\sum _{i=1}^{\nu } \tilde{U}_{i}\right\| _{\dot{H}^s} \le \delta \end{aligned}$$ u - ∑ i = 1 ν U ~ i H ˙ s ≤ δ where $$\tilde{U}_{1}, \tilde{U}_{2},\ldots \tilde{U}_{\nu }$$ U ~ 1 , U ~ 2 , … U ~ ν is a $$\delta $$ δ -interacting family of Talenti bubbles, there exists a family of Talenti bubbles $$U_{1}, U_{2},\ldots U_{\nu }$$ U 1 , U 2 , …...

We prove the exact multiplicity of flat and compact support stable solutions of an autonomous non-Lipschitz semilinear elliptic equation of eigenvalue type according to the dimension N and the two exponents, 0 &lt;α&lt;β&lt; 1, of the involved nonlinearites. Suitable assumptions are made on the spatial domainΩwhere the problem is formulated in order to avoid a possible continuum of those solutions and, on the contrary, to ensure the exact number of solutions according to the nature of the domainΩ. Our results also clarify some previous works in the literature. The main techniques of proof are a Pohozhaev’s type identity and some fibering type arguments in the variational approach.

For 0 < ω ≤ π and 0 < ω we consider the open sector and the open strip S ω := {z ∈ C \ {0} | |arg z| < ω}, St ω := {z ∈ C | |Im z| < ω }. So, S ω is symmetric about the positive real axis with opening angle 2ω. For ω = π /2 we have Sπ /2 = C +. (Note that 2ω > π is allowed here.) The strip St ω extends horizontally, symmetric about the real axis. We also define S 0 := (0, ∞) and St 0 := R and consider them as a degenerate sector and strip, respectively. For 0 ≤ ω < π, the exponential function is a biholomorphic mapping (conformal equivalence) e z : St ω → S ω with inverse log z : S ω → St ω , the (principal branch of the) logarithm. Note that the additive group R acts by translations on each strip, whereas the multiplicative group (0, ∞) acts by multiplication on each sector. An operator A on a Banach space X is called sectorial of angle ω ∈ [0, π) if σ(A) ⊆ S ω and for each α ∈ (ω, π) M (A, α) := sup{ λR(λ, A) | λ ∈ C \ S α } < ∞. 1 An operator A is simply called sectorial if it is sectorial of angle ω for some ω ∈ [0, π). In this case, 160 9 The Holomorphic Functional Calculus for Sectorial Operators ω se (A) := min{ω ∈ [0, π) | A sectorial of angle ω} is called the sectoriality angle of A. Analogously to the half-plane case, we say that a set A of operators is uniformly sectorial of angle ω < π if sup A∈A M (A, α) < ∞ for all α ∈ (ω, π). Examples 9.1. (See Exercise 9.1). 1) Let Ω be a semi-finite measure space and a : Ω → C a measurable mapping. The multiplication operator M a on L p (Ω), 1 ≤ p ≤ ∞, is sectorial of angle ω if and only if a ∈ S ω almost everywhere. 2) Let H be a Hilbert space and A a closed operator on H with numerical range W(A) ⊆ S ω and such that ran(I + A) = H. Then A is sectorial of angle ω. In particular, a positive, self-adjoint operator is sectorial of angle 0. * (∂S δ) ω se (A) < δ < ω < ∞. Proof. The proof of a)-c) is analogous to the proof of Theorem 8.3.

In this paper, we study the existence of positive solutions of a nonlinear m-point p-Laplacian dynamic equation (ϕp(x ∆ (t))) ∇ + w(t)f (t, x(t), x ∆ (t)) = 0, t1 < t < tm, subject to one of the following boundary conditions x(t1) − B0 (m−1 ∑ i=2 aix ∆ (ti)) = 0, x ∆ (tm) = 0, or x ∆ (t1) = 0, x(tm) + B1 (m−1 ∑ i=2 bix ∆ (ti)) = 0, where ϕp(s) =| s | p−2 s, p > 1. Sufficient conditions for the existence of at least three positive solutions of the problem are obtained by using a fixed point theorem. The interesting point is the nonlinear term f is involved with the first order derivative explicitly. As an application, an example is given to illustrate the result.

We study separating function sets. We find some necessary and sufficient conditions for C p (X) or C 2 p (X) to have a point-separating subspace that is a metric space with certain nice properties. One of the corollaries to our discussion is that for a zero-dimensional X, C p (X) has a discrete pointseparating space if and only if C 2 p (X) does.

We characterize the given extent in finite powers of X in terms of the topology of C p (X). It is shown that many properties of C p (X) are determined by dense subsets of C p (X). We introduce the density tightness and establish that for compact spaces of π-weight ω 1 countable density tightness implies countable π-character.

In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brézis-Gallouët-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but also the wider class of Hölder continuous functions.

Professor Florent Perek has a PhD in English and General Linguistics (University of Freiburg) and is a Lecturer in Cognitive Linguistics at the Department of English Language and Applied Linguistics at University of Birmingham, UK. Professor Perek is the author of several articles in international peer-reviewed journals and has, among his most important publications, the 2015 book, Argument structure in usage-based construction grammar: experimental and corpus-based perspectives, edited by John Benjamins.------------------------------------------------------------------------------------ENTREVISTA COM FLORENT PEREKFlorent Perek é Doutor em Inglês e Linguística Geral (Universidade de Freiburg, Alemanha) e Professor da área de Linguística Cognitiva do Departamento de Língua Inglesa e Linguística Aplicada na Universidade de Birmingham, no Reino Unido. Perek é autor de uma série de artigos em revistas renomadas internacionalmente e tem, entre suas importantes publicações, seu livro de 2...

We study the well-posedness of a linear inverse problem for a multidimensional mixed-type equation including the classical equations of elliptic, hyperbolic, and parabolic types as special cases. For this problem, using the "ε-regularization," a priori estimate, and successive approximation methods, we prove the existence and uniqueness theorems for the solution in some function class.

We study the well-posedness of a linear inverse problem for a multidimensional mixed-type equation including the classical equations of elliptic, hyperbolic, and parabolic types as special cases. For this problem, using the "ε-regularization," a priori estimate, and successive approximation methods, we prove the existence and uniqueness theorems for the solution in some function class.

Scaling functions that generate a multiresolution analysis (MRA) satisfy, among other conditions, the so-called 'two-scale relation' (TSR). In this paper we discuss a number of properties that follow from the TSR alone, independently of any MRA: position of zeros (mainly for continuous scaling functions), existence theorems (using fixed point and eigenvalue arguments) and orthogonality relation between integer translates.

In this paper, we consider a class of singular fractional differential equations with two different orders of derivation, such that its right hand side has an arbitrary singularity on a certain interval of the real axis. We obtain new results on the existence and uniqueness of solutions. Some existence results are also discussed.

We introduce a class of Morrey-type spaces M λ p,q, , which includes the classical Morrey spaces and discuss their properties. We prove a Marcinkiewicz-type interpolation theorem for such spaces. This theorem is then applied to obtaining an analogue of O'Neil's inequality for convolutions and to proving the boundedness in the introduced Morrey-type spaces of the Riesz potential and singular integral operators.