We consider an operator function (F(λ)) for (λ\in(σ,τ)\subseteq\mathbb R) whose values are semibounded selfadjoint operators in Hilbert space (\mathfrak H). Our main goal is to estimate the number (\mathcal N_F(α,β)) of the eigenvalues of (F(λ)) on a segment ([α,β)\Subset(σ,τ)). In particular, we prove the estimates (\mathcal N_F(α,β)\geqslant ν_F(β)-ν_F(α)) and (\mathcal N_F(α,β)= ν_F(β)-ν_F(α)) where (ν(ξ)) is the number of the negative eigenvalues of the operator (F(ξ)), (ξ\in(σ,τ)).
The obtained results are applied for the functions of ordinary differential operators on a finite interval.
In this paper, we investigate the existence and uniqueness of $(ω,ρ)$-periodic solutions for a class of the abstract impulsive integro-differential equations on Banach space.
In this note, we analyze frequently hypercyclic solutions of abstract higher-order differential equations in separable infinite-dimensional complex Banach spaces. We essentially apply results from the theory of $C$-regularized semigroups, providing several illustrative examples and possible applications.
In this paper, we investigate the inverse generator problem for abstract degenerate Volterra integro-differential equations in locally convex spaces. The classes of degenerate $(a,k)$-regularized $C$-resolvent families and degenerate mild $(a,k)$-regularized $(C_{1},C_{2})$-existence and uniqueness families play an important role in our analysis. We provide several illustrative applications of obtained theoretical results, primarily to abstract degenerate fractional differential equations with Caputo derivatives.
In this paper, we analyze the existence and uniqueness of generalized weighted pseudo-almost automorphic solutions of abstract Volterra integro-differential inclusions in Banach spaces. The main results are devoted to the study of various types of weighted pseudo-almost periodic (automorphic) properties of convolution products. We illustrate our abstract results with some examples and possible applications.
In this paper, we are primarily concerned with the study of entire and analytical solutions of abstract degenerate (multi-term) fractional differential equations with Caputo time-fractional derivatives. We also analyze systems of such equations and furnish several illustrative examples to demonstrate usage of obtained theoretical results.
In this paper we identify, for small $t$ and a fixed $T>0,$ the order $α>0$ in the abstract fractional differential equation $$\partial^αu(t)=Au(t),$$ where the time-fractional derivative $\partial^α$ is understood in the sense of Caputo and Riemann-Liouville, $A$ is a closed (possibly unbounded) linear operator in a Banach space $X,$ and $0<α<1$ or $1<α<2.$
In this paper, we first use the Bohr property to give a definition of Besicovitch almost periodic functions, and study some basic properties of Besicovitch almost periodic functions, including the equivalence of the Bohr property and the Bochner property. Then, as an application, we use the contraction principal to obtain the existence and uniqueness of Besicovitch almost periodic solutions for a class of abstract semi-linear differential equations with delay.
For an operator generating a group on $L^p$ spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction of first order differential operators and prove similar spectral multiplier estimates assuming only that the group is bounded on $L^2$ rather than $L^p$. We also prove an R-bounded Hörmander calculus result by assuming an abstract Sobolev embedding property and show that the square of a perturbed Hodge-Dirac operator has such calculus.
We prove results of existence of a solution (resp. existence and uniqness of a solution) for nonlinear differential equations of type $x'(t) +G(x,t) x(t) = F(x,t),$ in an abstract Banach subspace $X$ of the space of bounded real-valued continuous functions, satisfying some general and natural property. In our work, the functions $F$ and $G$ jointly depend on the variables $(x,t)\in X\times \R$. Several examples will be given, in various function spaces, to illustrate our results. The vector-valued framework is also considered.
In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\mathbb{H}(G):=G \times \widehat{G} \times \mathbb{T},$ where $G$ a locally compact abelian group with its dual group $\widehat{G}$. We obtain a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. As a key step in proving this we derive a trace formula for the trace class $j$-Weyl transform, $j \in \mathbb{Z}^*$ with symbols in $L^{2}(G\times \widehat{G}).$ We go on to present a characterization of t…
We provide a convenient framework for the study of the well-posedness of a variety of abstract (integro)differential equations in general Banach function spaces. It allows us to extend and complement the known theory on the maximal regularity of such equations. More precisely, by methods of harmonic analysis, we identify large classes of Banach spaces which are invariant with respect to distributional Fourier multipliers. Such classes include general vector-valued Banach function spaces $Φ$ and/or the scales of Besov and Triebel-Lizorkin spaces defined by $Φ$. We apply this result to the study…
In this paper we aim to construct an abstract model of a differential operator with a fractional integro-differential operator composition in final terms, where modeling is understood as an interpretation of concrete differential operators in terms of the infinitesimal generator of a corresponding semigroup. We study such operators as a Kipriyanov operator, Riesz potential, difference operator. Along with this, we consider transforms of m-accretive operators as a generalization, introduce an operator class $\mathfrak{G_α}$ and provide a description of its spectral properties.
In this paper, Mikhlin and Marcinkiewicz--Lizorkin type operator-valued multiplier theorems in weighted Lebesgue-Bochner spaces are studied. By using this results embedding theorems in Sobolev-Lions type spaces is obtained. Moreover, maximal regularity properties of abstract elliptic and parabolic equations are derived
The present work is devoted to the study of a boundary value problem for second order linear differential equation set on singular cylindrical domain. This problem can be regarded via a natural change of variables as an elliptic abstract differential equation with variable operators coefficients subject to some anti-periodic conditions. The complete study of this abstract version allows us to establish some interesting regularity results for our problem. The study is performed in the framework of Hölder spaces.
It is shown how the linear method of the Yosida-approximation of the derivative applies to solve possibly nonlinear abstract functional differential equations in both, the finite and infinite delay case. A generalization of the integral solution will provide regularity results. Moreover, this method applies to derive uniform convergence on the halfline, and therefore general results on boundedness and various types of asymptotic almost periodicity.
In this paper we deal with a second order multidimensional fractional differential operator. We consider a case where the leading term represented by the uniformly elliptic operator and the final term is the Kipriyanov operator of fractional differentiation. We conduct classification of such a type of operators by belonging of their resolvent to the Schatten-von Neumann class and formulate the sufficient condition for completeness of the root functions system. Finally we obtain an asymptotic formula.
Using dual perturbation theory in a non-sun-reflexive context, we establish a correspondence between 1. a class of nonlinear abstract delay differential equations (DDEs) with unbounded linear part and an unknown taking values in an arbitrary Banach space and 2. a class of abstract weak* integral equations of convolution type involving the sun-star adjoint of a translation-like strongly continuous semigroup. For this purpose we also characterize the sun dual of the underlying state space. More generally we consider bounded linear perturbations of an arbitrary strongly continuous semigroup and w…
In this short note we present several infinite dimensional theorems which generalize corresponding facts from the finite dimensional differential inclusions theory.
We first strictly expressed the basic notions and research methods of abstract operators, which systematically expounded the main results of abstract operator theory. By combining abstract operators with the Laplace transform, we can easily apply this Laplace transform to n+1 dimensional partial differential equations. Further, all the analytic solutions to an initial value problem of an arbitrary order linear partial differential equation are expressed in these abstract operators. By writing abstract operators in this class into integral forms, the solutions in operator form are represented i…
We consider the operator of taking the $2p$th derivative of a function with zero boundary conditions for the function and its first $p-1$ derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for $p=1$ and $p=3$.
In this paper, we study the separability and spectral properties of singular degenerate elliptic equations in vector valued spaces. We prove that a realization operator by this equation with some boundary conditions is separable and Fredholm in Lebesque spaces. The leading part of the associated differential operator is not self-adjoint. The sharpe estimate of the resolvent, discreetness of spectrum and completeness of root elements of this operator is obtained
Applying perturbation theory methods, the absence of the point spectrum for some nonselfadjoint integro-differential operators is investigated. The considered differential operators are of arbitrary order and act in either $\mathbf{L}_{p}(\mathbb{R}_{+})$ or $\mathbf{L}_{p}(\mathbb{R}) (1\leq p<\infty)$. As an application of general results, new spectral properties of the perturbed Hill operator are derived.
In this paper we develop generalized d'Alembert's formulas for abstract fractional integro-differential equations and fractional differential equations on Banach spaces. Some examples are given to illustrate our abstract results, and the probability interpretation of these fractional d'Alembert's formulas are also given. Moreover, we also provide d'Alembert's formulas for abstract fractional telegraph equations.
In the underlying study it is shown how the linear method of the Yosida-approximation of the derivative applies to solve possibly nonlinear and multivalued functional differential equations like: \begin{eqnarray*} u^\prime(t) &\in& A(t,u_t)u(t) +ωu(t), \ t \in \mathbb{R} \end{eqnarray*} Furthermore, in the case of finite and infinite delay we give an answer about whether the solution is bounded, periodic, almost periodic, or some kind of almost automorphy.
We provide isoperimetric Szegö-Weinberger type inequalities for the first nontrivial Neumann eigenvalue $μ_{1}(Ω)$ in Gauss space, where $Ω$ is a possibly unbounded domain of $\mathbb{R}^{N}$. Our main result consists in showing that among all sets of $\mathbb{R}^{N}$ symmetric about the origin, having prescribed Gaussian measure, $μ_{1}(Ω)$ is maximum if and only if $Ω$ is the euclidean ball centered at the origin.
Let $m\in \mathbb{N}$, $α\in[0,1]$, and $V$ be a 1-periodic complex-valued distribution in the negative Sobolev space $H^{-mα}[0,1]$. The singular non-self-adjoint eigenvalue problem $D^{2m}u+V u=λu$, $D=-i d/dx$, with semi-periodic boundary conditions is investigated. The uniform in $V$ asymptotic and non-asymptotic eigenvalue estimates are found and proved. The case of periodic boundary conditions was studied by authors earlier.
The periodic eigenvalue problem for the differential operator $(-1)^{m}d^{2m}/dx^{2m}+V$ is studied for complex-valued distribution V in the Sobolev space $H^{-mα}_{per}[-1,1]\;(m\in\mathbb{N},\; 0\leqα<1)$. The following result is shown: The periodic spectrum consists of a sequence $(λ_{k})_{k\geq0}$ of complex eigenvalues satisfying the asymptotics (for any $\varepsilon>0$) $$ λ_{2n-1},λ_{2n}=n^{2m}π^{2m}+\hat{V}(0)\pm \sqrt{\hat{V}(-2n)\hat{V}(2n)}+o(n^{m(2α-1+\varepsilon)}), $$ where $\hat{V}(k)$ denote the Fourier coefficients of V.
The paper is concerned with conditions for the existence of almost periodic solutions of the following abstract functional differential equation $ \dot u(t) = Au(t) + [{\cal B}u](t) +f(t), $ where $A$ is a closed operator in a Banach space $\X$, $\cal B$ is a general bounded linear operator in the function space of all $\X$-valued bounded and uniformly continuous functions that satisfies a so-called {\it autonomous} condition. We develop a general procedure to carry out the decomposition that does not need the well-posedness of the equations. The obtained conditions are of Massera type, which …
We adapt a technique of nonsmooth critical point theory developed by Degiovanni-Zani for a semilinear problem involving the Laplacian to the the case of the $p$-Laplacian. We suppose only coercivity conditions on the potential and impose no growth condition of the nonlinearity. The coercivity is obtained using similar nonresonance conditions to [Mawhin-Ward-Willem] and to [Landesman-Lazer] in two different results and using some comparison functions and comparison spaces in a third one. It is also shown that neither of the three theorems implies the two others.
Finiteness of the point spectrum of linear operators acting in a Banach space is investigated from point of view of perturbation theory. In the first part of the paper we present an abstract result based on analytical continuation of the resolvent function through continuous spectrum. In the second part, the abstract result is applied to differential operators which can be represented as a differential operator with periodic coefficients perturbed by an arbitrary subordinated differential operator.
We presents the study the separability properties for differential-operator equations in Morrey spaces. We prove that the corresponding differential operator is a generator of analytic semigroup in vector-valued Morrey spaces. Moreover, maximal regularity properties of corresponding parabolic equation ıs obtained. In applications, the maximal regularity properties of Wentzell-Robin type problem for elliptic equations and mixed value problem for degenerate parabolic equations in Morrey spaces are derived.
Free bondary value problem for elliptic differential-operator equations with variable coefficients is studied. The uniform maximal regularity properties and Fredholmness of this problem are obtained in vector-valued Holder spaces.
Recent work in arXiv:1901.11526 by the author about a class of abstract delay differential equations (DDEs), as well as earlier work by Diekmann and Gyllenberg on other classes of delay equations, motivates the introduction of the general notion of an admissible range and an admissible perturbation for a given $\mathcal{C}_0$-semigroup $T_0$ on a Banach space $X$ that is not assumed to be sun-reflexive with respect to $T_0$. We investigate the relationship between admissible ranges for $T_0$ and the subspace $X^{\odot\times}$ of $X^{\odot\star}$ introduced by Van Neerven. We answer two questio…
The one-dimensional harmonic oscillator wave functions are solutions to a Sturm-Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on $(-\infty,\infty)$ in terms of a system of integro-differential equations of Hartree-Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm-Liouville problem. We demon…
We consider the Cauchy problem for a second-order nonlinear evolution equation in a Hilbert space. This equation represents the abstract generalization of the Ball integro-differential equation. The general nonlinear case with respect to terms of the equation which include a square of a norm of a gradient is considered. A three-layer semi-discrete scheme is proposed in order to find an approximate solution. In this scheme, the approximation of nonlinear terms that are dependent on the gradient is carried out by using an integral mean. We show that the solution of the nonlinear discrete problem…
