Type Theorem Research Articles

A priori estimates and Liouville type theorems for semilinear equations and systems with fractional Laplacian

In this paper, we firstly obtain a priori estimates of nonnegative solutions to fractional Laplacian equations and systems with critical Sobolev exponent. Secondly, by deriving the universal estimates of solutions, we establish the connections between Liouville type theorems and local properties of nonnegative solutions to fractional Laplacian equations and fractional Lane-Emden systems with subcritical Sobolev exponent. Our main results improve the work of Chen et al. (2016) [8] and Polácik et al. (2007) [26]. Specially, for fractional Lane-Emden systems, our results seem to be the first results on universal estimates to cover the full subcritical range. Furthermore, the novelty of our method lies in a completely new blow-up analysis coupled with a truncating technique and the construction of barrier function which is sufficiently different from the blow-up method used in [8,26].

Bernstein type theorems of translating solitons of the mean curvature flow in higher codimension

Bernstein type theorems of translating solitons of the mean curvature flow in higher codimension

A theorem of Caristi-Kirk type for b-metric spaces

A theorem of Caristi-Kirk type for b-metric spaces

New Liouville type theorems for the stationary Navier–Stokes, MHD, and Hall–MHD equations

We establish new Liouville-type theorems for weak solutions of the stationary Navier–Stokes equations, stationary magnetohydrodynamics (MHD) equations and stationary Hall–MHD equations under some conditions on the growth of certain Lebesgue norms of the velocity and the magnetic field.

Lebesgue decomposition type theorems for weakly null-additive functions on D-posets

Lebesgue decomposition type theorems for weakly null-additive functions on D-posets

Existence of solution sets for Φ-Laplacian for random impulsive differential equations

In this article, we provide results on the existence and compactness of the set of solutions for a class of initial-value problem representing a first-order random impulsive differential equation. The key tool to achieve our goal is based on an alternative nonlinear Leary-Schauder type theorem in the settings of generalized Banach spaces.

Existence of sign-changing radial solutions with prescribed numbers of zeros for elliptic equations with the critical exponential growth in ℝ²

In this paper, we are concerned with the existence of sign-changing radial solutions with any prescribed numbers of zeros to the following Schrodinger equation with the critical exponential growth: { − Δ u + u = λ u e u 2 in R 2 , lim | x | → ∞ u ( x ) = 0 , \begin{equation*} \begin {cases} -\Delta u +u=\lambda ue^{u^2} \quad \quad \text {in } \quad \mathbb {R}^2,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0, \end{cases} \end{equation*} where 0 > λ > 1 0>\lambda >1 . Our proof relies on the shooting method, the Sturm’s comparison theorem and a Liouville type theorem in exterior domain of R 2 \mathbb {R}^2 . It seems to be the first existence result of sign-changing solution for Schrodinger equation with the critical exponential growth.

Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper.

Hamilton and Li–Yau type gradient estimates for a weighted nonlinear parabolic equation under a super Perelman–Ricci flow

In this paper we derive elliptic and parabolic type gradient estimates for positive smooth solutions to a class of nonlinear parabolic equations on smooth metric measure spaces where the metric and potential are time dependent and evolve under a super Perelman–Ricci flow. A number of implications, notably, a parabolic Harnack inequality, a class of Hamilton type dimension-free gradient estimates and two general Liouville type theorems along with their consequences are discussed. Some examples and special cases are presented to illustrate the results.

Internal and Observational Parametricity for Cubical Agda

Two approaches exist to incorporate parametricity into proof assistants based on dependent type theory. On the one hand, parametricity translations conveniently compute parametricity statements and their proofs solely based on individual well-typed polymorphic programs. But they do not offer internal parametricity: formal proofs that any polymorphic program of a certain type satisfies its parametricity statement. On the other hand, internally parametric type theories augment plain type theory with additional primitives out of which internal parametricity can be derived. But those type theories lack mature proof assistant implementations and deriving parametricity in them involves low-level intractable proofs. In this paper, we contribute Agda --bridges: the first practical internally parametric proof assistant. We provide the first mechanized proofs of crucial theorems for internal parametricity, like the relativity theorem. We identify a high-level sufficient condition for proving internal parametricity which we call the structure relatedness principle (SRP) by analogy with the structure identity principle (SIP) of HoTT/UF. We state and prove a general parametricity theorem for types that satisfy the SRP. Our parametricity theorem lets us obtain one-liner proofs of standard internal free theorems. We observe that the SRP is harder to prove than the SIP and provide in Agda --bridges a shallowly embedded type theory to compose types that satisfy the SRP. This type theory is an observational type theory of logical relations and our parametricity theorem ought to be one of its inference rules.

A Liouville type theorem for a scaling invariant parabolic system with no gradient structure

We prove a Liouville-type theorem for semilinear parabolic systems of the form∂tui−Δui=∑j=1mβijuirujp−r,i=1,2,...,m in the whole space RN×R, where r>0, p>r+1 and (βij) is a real (m,m) symmetric matrix with nonnegative entries, positive on the diagonal. Due to the lack of gradient structure, a nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron is introduced. We expect to see more applications of this method to other problems with no gradient structure.

Bounds for the Generalization of Baer’s Type Theorems

Bounds for the Generalization of Baer’s Type Theorems

A Finite Topological Type Theorem for Open Manifolds with Non-negative Ricci Curvature and Almost Maximal Local Rewinding Volume

Abstract In this paper, we present finite topological type theorems for open manifolds with non-negative Ricci curvature, under almost maximal local rewinding volume. Unlike previous related research, our theorems remove the constraints of sectional curvature or conjugate radius, which were crucial additional assumptions on metric regularity in prior results. Notably, our settings do not necessarily satisfy a triangle comparison of Toponogov type. In fact, the method we adopt also extends to many previous related studies.

A new variant of the modified Bernstein-Kantorovich operators defined by Özarslan and Duman

In the present article, we introduce a new function $ \zeta $ in the modified Bernstein-Kantorovich operators defined by Özarslan and Duman (http://dx.doi.org/10.1080/01630563.2015.1079219) and properties of the function $ \zeta $ is define as: this is an infinitely differentiable function in the closed interval $ [0, 1] \; \text{such that}\; \zeta(0) = 0, \; \zeta(1) = 1 $ and $ \zeta'(\varkappa) $. We discussed an approximation properties with the help of the Bohman-Korovkin's type theorem and scrutinize the rate of convergence with the aid of modulus of continuity, Lipschitz type functions. Furthermore, we study the rate of convergence by means of derivatives of bounded variation and an approximation theorem by using of the Bohman-Korovkin's type theorem in $ \mathcal{A}- $Statistical convergence.In lastly, in order to validate our theoretical results, we provide some numerical experiments that are produced by a MATHEMATICA compiler and show that a careful choice of the function $ \zeta(\varkappa) $ for the both operators and we see that the our operator gives the better approximation results as compare to the modified Bernstein-Kantorovich operators defined by Özarslan (doi.org/10.1080/01630563.2015.1079219).

Approximation by q‐Post‐Widder Operators Based on a New Parameter

The purpose of this paper is to introduce q‐Post–Widder operators based on a new parameter and study their approximation properties. The moments and central moments are investigated. And some local approximation properties of these operators by means of modulus of continuity and Peetre’s K‐functional are presented. Furthermore, the rate of convergence for these operators is obtained. Weighted approximation and the quantitative q‐Voronovskaja type theorem are discussed. Finally, numerical illustrative examples have been given to show the convergence of these newly defined operators.

Abelian and Tauberian results for the fractional Fourier cosine (sine) transform

<abstract><p>In this paper, we presented Tauberian type results that intricately link the quasi-asymptotic behavior of both even and odd distributions to the corresponding asymptotic properties of their fractional Fourier cosine and sine transforms. We also obtained a structural theorem of Abelian type for the quasi-asymptotic boundedness of even (resp. odd) distributions with respect to their fractional Fourier cosine transform (FrFCT) (resp. fractional Fourier sine transform (FrFST)). In both cases, we quantified the scaling asymptotic properties of distributions by asymptotic comparisons with Karamata regularly varying functions.</p></abstract>

Pulsed Electromagnetic Excitation of a Narrow Slot Between Two Dielectric Halfspaces

The transient electromagnetic (EM) excitation of a narrow slot in a perfectly electrically conducting (PEC) screen that separates two homogeneous dielectric halfspaces, a simplified model of a typical feeding structure of leaky lens antennas, is analyzed numerically in the time domain (TD). The problem is formulated using the TD reciprocity theorem of the time-convolution type and subsequently solved with the aid of the Cagniard-DeHoop method of moments (CdH-MoM). Numerical results are validated using a general-purpose EM-field solver.

Voronovskaja type results for the Aldaz-Kounchev-Render versions of generalized Baskakov operators

The modification of Bernstein operators given by Aldaz, Kounchev and Render has been intensively studied in the recent years. In this paper we define a corresponding modification for the general class of Baskakov-type operators. All these operators preserve the constants and j-th monomial for a given natural number j. We prove a general result of Voronovskaja type (Theorem 2.1) for positive linear operators acting on the infinite interval [0,?) and use this theorem to prove a Voronovskaja-type theorem for the whole class of the modified Baskakov-type operators.

On approximating properties of blended variant of Kantorovich-Stancu type Lupaş operators

We introduce blended variant of Kantorovich-Stancu type Lupa? operators and study convergence and q-statistical convergence properties using Korovkin theorem. We investigate rate of convergence interms of modulus of continuity, Peetre?s K-functional and Lipschitz functions. Some Direct results and Voronovskaja type theorem are established. Moreover, graphical analysis and error estimations are presented using MATLAB.

On fourth order retarded equations with functional boundary conditions: A unified approach

By means of a recent Birkhoff-Kellogg type theorem, we discuss the solvability of a fairly general class of parameter-dependent fourth order retarded differential equations subject to functional boundary conditions. We seek solutions within a translate cone of nonnegative functions. We provide an example to illustrate our theoretical results.