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In this note we offer some criticism on the paper “On an identity for zeros of Bessel functions” by Baricz et al. [3] . The paper gives identities of type Stieltjes–Calogero for the sums of reciprocals of differences of fourth powers of zeros of Bessel functions. Although interesting in principle, by containing one too many sums of similar complexity the identities fail to convey the true spirit of the work of Stieltjes and Calogero. We rectify this by providing what we think is the correct type of identity for the above-said sums, in the general setup of entire functions of order

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers “Boundedness of solutions to a quasilinear parabolic–parabolic chemotaxis model with nonlinear signal production” by Tao et al. (2019) [2] and “Boundedness for a fully parabolic Keller–Segel model with sublinear segregation and superlinear aggregation” by Frassu and Viglialoro (2021) [1] . These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant M 1 and for each k > 1 there are cones with normal constant M > k . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results.

We prove oscillation and nonoscillation theorems for the second order linear differential equation (E) y″+q(t)y=0, where q(t)⩾0 and locally integrable on [t∗,∞). These results are extensions of earlier results of Huang [J. Math. Anal. Appl. 210 (1997) 712–723]. Furthermore, we show that the oscillation criterion established for Eq. (E) can be extended to the delayed differential equation y″+q(t)y(σ(t))=0, where σ(t)⩽t and limt→∞σ(t)=∞.

Using the method of interval integral average, we establish some new oscillation criteria which improve the oscillation criteria due to James S.W. Wong.

A proposal for the correction of the main result of the title's paper is made. Some accompanying comments on other points are given. So, a simpler central result is derived. An explicit representation of the set of stable delays is achieved.

We point out errors and oversights in a paper by Giordano, Laforgia, and Pecaric [3] on inequalities involving Bessel functions.

Lemma 4.1 in [Comm. Math. Phys.91(1983)] by Ron Diperna pertaining to the vacuum case for an existence proof of polytropic gas dynamics using compensated compactness is incomplete as given. Here we give a quick fix of the lemma plus some generalization.

In this paper we generalize two results of Khan and Majumdar on the weak sequential convergence of functions in the Lebesgue-Bochner space L X 1 (Ω) and on a Fatou-like convergence theorem.

and proved the following assertions: Suppose that f~ C ((0,2n) x R, R) is decreasing in u for each fixed t E (0,27r) and that X, p E C2 ((0,27r)) are lower and upper solutions for (1). Then (1) if c( d p on (0, 2n), then (1) admits a solution and the solution set of (1) in the sector (a, /I) is an Rg; (2) if c( 2 /I on (0, 2rc), then (a) (1) admits a solution; (b) there exist positive constants sr , s? such that the solution set of (1) in the sector (LX-S,, fl+s2) is an R,. The aim of this paper is to improve these assertions.

If ƒ(z) is an entire function of order larger than one that satisfies certain additional conditions, characterizations of the growth of its Fourier transform are given, in terms of the growth of ƒ(z) . The results obtained are extensions of theorems of Paley and Wiener, and are a refinement of prior work of Gel'fand and Silov.

The theory of Varley and Cumberbatch [l] giving the intensity of discontinuities in the normal derivatives of the dependent variables at a wave front can be deduced from the more general results of Prasad which give the complete history of a disturbance not only at the wave front but also within a short distance behind the wave front. In what follows we omit the index M in Eq. (2.25) of Prasad [2].

In this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions.

In this short paper, we investigate blow-up rate of solutions of reaction–diffusion equations with localized reactions. We prove that the solutions have a global blow-up and the rate of blow-up is uniform in all compact subsets of the domain.

In this short paper the core of the direct method for proving stability of functional equations is described in a clear way and in a quite general form.

While there have been extensive studies regarding the theory of composition operators in standard Bergman spaces, there have not been many results pertaining to large Bergman spaces due to a lack of useful tools. In this paper, we give the characterizations of the compact differences of composition operators in Bergman spaces with the exponential type weight using a newly defined Riemannian distance. Furthermore, we give a sufficient condition for the question when two composition operators lie in the same component., This paper is the corrected version of the published paper

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system { u t = ∇ ⋅ [ ( u + α ) m 1 − 1 ∇ u − χ u ( u + α ) m 2 − 2 ∇ v ] in Ω × ( 0 , T ) , v t = Δ v − v + u in Ω × ( 0 , T ) under Neumann boundary conditions and initial conditions, where Ω is a general bounded domain in R n with smooth boundary, α > 0 , χ > 0 , m 1 , m 2 ∈ R and T > 0 . Recently, Anderson–Deng [1] gave a lower bound for the blow-up time in the case that m 1 = 1 and Ω is a convex bounded domain. The purpose of this paper is to generalize the result in [1] to the case that m 1 ≠ 1 and Ω is a non-convex bounded domain. The key to the proof is to make a sharp estimate by using the Gagliardo–Nirenberg inequality and an inequality for boundary integrals. As a consequence, the main result of this paper reflects the effect of nonlinear diffusion and need not assume the convexity of Ω.

The known family of woven collections of frames have interesting properties and applications. Yet, they cannot be easily constructed. The present paper introduces new collections of frames sharing many properties with the original ones and having simpler construction. The frames { φ i } i ∈ I and { ψ i } i ∈ I for a Hilbert space H are called partition-woven (or briefly P-woven), if there exists a nontrivial subset σ of I such that the family { g i } i ∈ I represented as the triple ( { φ i } , { ψ i } , σ ) and called a P-weaven frame, is a frame defined by g i = φ i if i ∈ σ and g i = ψ i , otherwise. The paper also studies the class of CP-woven collections of frames which is a subset of the P-woven collections of frames and satisfy a stronger condition. The set of all woven frames is a strict subset of the set of all CP-woven frames which is further a strict subset of the set of all P-woven frames. Several examples and counterexamples are studied to show whether or not a pair of frames is P-woven or CP-woven.

In this paper, we study the large time behavior of a chemotaxis model with nonlinear diffusion and consumption { u t = Δ u m − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , v t − Δ v = − v u , where m > 1 . In a previous paper [5] , we have proved the existence and uniform boundedness of global weak solutions for any nonnegative initial data and any m > 1 . In this work, we show that the weak solutions strongly converge to ( 1 , 0 ) in the large time limit.

In the paper, the initial observability of abstract second-order infinite-dimensional dynamical systems is considered. Two cases of observation operator are explored. First, the observation operator is assumed to be bounded. Then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation. Using the frequency-domain method it is proved, that initial observability of second-order system under discussion can be verified by the initial observability conditions for the corresponding simplified first-order system. General results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures. Some special cases are also discussed and practical comments and remarks are given. The paper extends earlier results on initial observability for a class of infinite-dimensional second-order abstract dynamical systems.

Let $A,B\subset\mathbb{R}$. Define $$A\cdot B=\{x\cdot y:x\in A, y\in B\}.$$ In this paper, we consider the following class of self-similar sets with overlaps. Let $K$ be the attractor of the IFS $\{f_1(x)=\lambda x, f_2(x)=\lambda x+c-\lambda,f_3(x)=\lambda x+1-\lambda\}$, where $f_1(I)\cap f_2(I)\neq \emptyset, (f_1(I)\cup f_2(I))\cap f_3(I)=\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main result of this paper is $K\cdot K=[0,1]$ if and only if $(1-\lambda)^2\leq c$. Equivalently, we give a necessary and sufficient condition such that for any $u\in[0,1]$, $u=x\cdot y$, where $x,y\in K$., Comment: 12 pages

In this paper we study the priori error analysis for semidiscrete Galerkin finite element approximation of the impulsive optimal control problem governed by linear heat equation with control constraint. We first derive Pontryagin's maximum principle and construct the semidiscrete finite element approximation scheme for our impulsive optimal control problem. Then, the error estimates for optimal control u, the related state y and the adjoint state φ will be obtained. Finally, an efficient algorithm for our numerical experiment is designed, which verifies the theoretical results obtained in this paper.

In this paper we will study an option pricing problem in incomplete markets by an analytic point of view. The incompleteness is generated by the presence of a non- traded asset. The aim of this paper is to use the semigroup theory in order to prove existence and uniqueness of solutions to generalized Black-Scholes type equations that are non-linearly associated with the price of European claims written exclusively on non-traded assets. Then, we derive analytic expressions of the solutions. An approximate representation in terms of a generalized Feynman-Kac type formula is derived in cases where an explicit closed form solution is not available. Numerical examples are also given (see Appendix E ) where theoretical approximations and numerical tests reveal a remarkable agreement.

Whether or not weak solutions of classical Navier-Stokes equations decay to zero in L 2 as time tends to infinity was posed by Leray in his pioneering paper [12] , [13] and has been well solved (see [15] for instance). This paper examines the three dimensional Navier-Stokes equations with damping term | u | β − 1 u and proves that the weak solutions decay to zero in L 2 as time tends to infinity for any β ≥ 1 , by developing local-in-space estimate for weak solutions.

In this paper we consider a competitive economic equilibrium problem where preferences of consumers are expressed by means of a binary relation. The aim is to find a suitable quasi-variational inequality which characterizes the equilibria and, by using tools of variational theory, to study such equilibria. The novelty of this paper consists in the study of an economic equilibrium problem by a variational approach without the need of representing the consumer's preferences by a utility function.

In this paper we combine topological and functional analysis methods to prove that a non-locally trivial quasi-linear map defined on a C ( K ) must be nontrivial on a subspace isomorphic to c 0 . We conclude the paper with a few examples showing that the result is optimal, and providing an application to the existence of nontrivial twisted sums of l 1 and c 0 .

In our previous article, a particular function was stated to give an equality in a particular inequality, proving that inequality was sharp. In this corrigendum it is shown that the function given in the original paper does not have the stated property, but prove that another function does have the desired property, thus preserving the overall result from the original paper.

This paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems, whose unperturbed system has a center at the origin and may have invariant multiply lines. By Melnikov function method, this paper provides the upper and lower bounds for the Poincare cyclicity. In particular, the exact cyclicities of two specific cases are given using analyzing method, one of which has been discussed in the paper [7] .

Using an asymptotically additive sequence of continuous functions as a constrained condition, this paper studies the relations of several constrained ergodic averages for asymptotically additive potentials. Basic properties of constrained maximum ergodic averages are studied. In particular, if the dynamical systems satisfy the specification property, the maximal growth rate of an asymptotically additive potential on the level sets is equal to its constrained maximum ergodic averages and the maximal growth rates on the irregular sets is its unconstrained maximum ergodic averages. Finally, the applications for suspension flows are given in the end of the paper.

In this paper, we are concerned with an initial–boundary value problem for a class of non-Newtonian fluids. The viscosity term of momentum equation possesses singularity and nonlinearity, and the initial vacuum is allowed. The main feature which distinguishes this paper from other related works lies in the fact that we study the global strong solution of the compressible non-Newtonian fluids with singularity and vacuum in one-dimensional bounded intervals, and show that there exists a unique global-in-time strong solution to the problem while the initial energy is small.

In this paper, we consider the following discrete p-Laplacian nonlinear parabolic equations with q-nonlocal reaction { u t ( x , t ) = Δ p , ω u ( x , t ) + λ ∑ y ∈ S | u ( y , t ) | q − 1 u ( y , t ) , ( x , t ) ∈ S × ( 0 , ∞ ) , u ( x , t ) = 0 , ( x , t ) ∈ ∂ S × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ≥ 0 , x ∈ S ‾ . Here, S is a network with boundary ∂S. The goal of this paper is to characterize completely the parameters p > 1 and q > 0 to see when the solution blows up, vanishes, or exists globally. Indeed, the blow-up rates for the blow-up solutions are derived.

The paper is devoted to the existence of weak solutions of set-valued stochastic differential equations. Such equations were considered in the authors paper [22] , where existence and uniqueness of strong solutions have been investigated. Now, apart from the existence of weak solutions, the weak compactness in distribution of the set of all weak solutions of set-valued stochastic differential equations, is considered.