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We introduce a weakened notion of norm attainment for bounded linear operators between Banach spaces which we call \emph{quasi norm attaining operators}. An operator $T\colon X \longrightarrow Y$ between the Banach spaces $X$ and $Y$ is quasi norm attaining if there is a sequence $(x_n)$ of norm one elements in $X$ such that $(Tx_n)$ converges to some $u\in Y$ with $\|u\|=\|T\|$. Norm attaining operators in the usual sense (i.e.\ operators for which there is a point in the unit ball where the norm of its image equals the norm of the operator) and compact operators satisfy this definition. The main result of the paper is that strong Radon-Nikod\'ym operators such as weakly compact operators can be approximated by quasi norm attaining operators (even by a stronger version of the definition), which does not hold for norm attaining operators. This allows us to give characterizations of the Radon-Nikod\'ym property in term of the denseness of quasi norm attaining operators for both domain spaces and range spaces, extending previous results by Bourgain and Huff. We also present positive and negative results on the denseness of quasi norm attaining operators, characterize both finite dimensionality and reflexivity in terms of quasi norm attaining operators, discuss conditions to obtain that quasi norm attaining operators are actually norm attaining, study the relationship with the norm attainment of the adjoint operator, and present some stability properties. We finish the paper with some open questions., Comment: 26 pages

A function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R is: almost continuous in the sense of Stallings, $$f\in \textrm{AC}$$ f ∈ AC , if each open set $$G\subset \mathbb {R}^2$$ G ⊂ R 2 containing the graph of f contains also the graph of a continuous function $$g:\mathbb {R}\rightarrow \mathbb {R}$$ g : R → R ; Sierpiński–Zygmund, $$f\in \textrm{SZ}$$ f ∈ SZ (or, more generally, $$f\in \textrm{SZ}(\textrm{Bor})$$ f ∈ SZ ( Bor ) ), provided its restriction $$f\restriction M$$ f ↾ M is discontinuous (not Borel, respectively) for any $$M\subset \mathbb {R}$$ M ⊂ R of cardinality continuum. It is known that an example of a Sierpiński–Zygmund almost continuous function $$f:\mathbb {R}\rightarrow \mathbb {R}$$ f : R → R (i.e., an $$f\in \textrm{SZ}\cap \textrm{AC}$$ f ∈ SZ ∩ AC ) cannot be constructed in ZFC; however, an $$f\in \textrm{SZ}\cap \textrm{AC}$$ f ∈ SZ ∩ AC exists under the additional set-theoretical assumption $${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$ cov ( M ) = c , that is, that $$\mathbb {R}$$ R cannot be covered by less than $$\mathfrak {c}$$ c -many meager sets. The primary purpose of this paper is to show that the existence of an $$f\in \textrm{SZ}\cap \textrm{AC}$$ f ∈ SZ ∩ AC is also consistent with ZFC plus the negation of $${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$ cov ( M ) = c . More precisely, we show that it is consistent with ZFC+$${{\,\textrm{cov}\,}}(\mathcal {M}) cov ( M ) < c (follows from the assumption that $${{\,\textrm{non}\,}}(\mathcal {N}) non ( N ) < cov ( N ) = c ) that there is an $$f\in \textrm{SZ}(\textrm{Bor})\cap \textrm{AC}$$ f ∈ SZ ( Bor ) ∩ AC and that such a map may have even stronger properties expressed in the language of Darboux-like functions. We also examine, assuming either $${{\,\textrm{cov}\,}}(\mathcal {M})=\mathfrak {c}$$ cov ( M ) = c or $${{\,\textrm{non}\,}}(\mathcal {N}) non ( N ) < cov ( N ) = c , the lineability and the additivity coefficient of the class of all almost continuous Sierpiński–Zygmund functions. Several open problems are also stated.

The paper deals with systems of linear differential equations with coefficients depending on the Markov process. Equations for particular density and the moment equations for given systems are derived and used in the investigation of solvability of initial problems and stability. Results are illustrated by examples. The paper deals with systems of linear differential equations with coefficients depending on the Markov process. Equations for particular density and the moment equations for given systems are derived and used in the investigation of solvability of initial problems and stability. Results are illustrated by examples.

In this paper, we show new results and improvements of the non-Archimedean counterpart of classical analysis in the theory of lineability. Besides analyzing the algebraic genericity of sets of functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability and analyticity, we also study the lineability of sets of sequences having properties concerning boundedness and convergence. In particular we show (among several other results) the algebraic genericity of: (i) functions that do not satisfy Liouville’s theorem, (ii) sequences that do not satisfy the classical theorem of Cèsaro, or (iii) functionals that do not satisfy the classical Hahn–Banach theorem.

In this paper it is shown that for the ordinary Dirichlet series, $$\sum _{j=0}^{\infty }\frac{\alpha _{j}}{(j+1)^{s}}$$ ∑ j = 0 ∞ α j ( j + 1 ) s , $$\alpha _{0}=1$$ α 0 = 1 , of a class, say $${\mathcal {P}}$$ P , that contains in particular the series that define the Riemann zeta and the Dirichlet eta functions, there exists $$\lim _{n\rightarrow \infty }\rho _{n}/n$$ lim n → ∞ ρ n / n , where the $$\rho _{n}$$ ρ n ’s are the Henry lower bounds of the partial sums of the given Dirichlet series, $$P_{n}(s)=\sum _{j=0}^{n-1}\frac{\alpha _{j}}{(j+1)^{s}}$$ P n ( s ) = ∑ j = 0 n - 1 α j ( j + 1 ) s , $$n>2$$ n > 2 . Likewise it is given an estimate of the above limit. For the series of $${\mathcal {P}}$$ P having positive coefficients it is shown the existence of the $$\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n$$ lim n → ∞ a P n ( s ) / n , where the $$a_{P_{n}(s)}$$ a P n ( s ) ’s are the lowest bounds of the real parts of the zeros of the partial sums. Furthermore it has been proved that $$\lim _{n\rightarrow \infty }a_{P_{n}(s)}/n=\lim _{n\rightarrow \infty }\rho _{n}/n$$ lim n → ∞ a P n ( s ) / n = lim n → ∞ ρ n / n .

In this paper, we focus on the development and study of the finite difference/pseudo-spectral method to obtain an approximate solution for the time-fractional diffusion-wave equation in a reproducing kernel Hilbert space. Moreover, we make use of the theory of reproducing kernels to establish certain reproducing kernel functions in the aforementioned reproducing kernel Hilbert space. Furthermore, we give an approximation to the time-fractional derivative term by applying the finite difference scheme by our proposed method. Over and above, we present an appropriate technique to derive the numerical solution of the given equation by utilizing a pseudo-spectral method based on the reproducing kernel. Then, we provide two numerical examples to support the accuracy and efficiency of our proposed method. Finally, we apply numerical experiments to calculate the quality of our approximation by employing discrete error norms.

In this paper we first extend from normed spaces to locally convex spaces some characterizations of denting points in convex sets. On the other hand, we also prove that in an infrabarreled locally convex space a point in a convex set is denting if and only if it is a point of continuity and an extreme point of the closure of such a convex set under the strong topology in the second dual. The version for normed spaces of the former equivalence is new and contains, as particular cases, some known and remarkable results. We also extend from normed spaces to locally convex spaces some known characterizations of the weak property (π) of cones. Besides, we provide some new results regarding the angle property of cones and related. We also state that the class of cones in normed spaces having a pointed completion is the largest one for which the vertex is a denting point if and only if it is a point of continuity. Finally we analyse and answer several problems in the literature concerning geometric properties of cones which are related with density problems into vector optimization. The authors Fernando García-Castaño and M. A. Melguizo Padial have been supported by MINECO and FEDER (MTM2017-86182-P).

In this paper we have given conditions on exponential polynomials $$P_{n}(s)$$ of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds, $$a_{P_{n}(s)}$$ , $$\rho _{N}$$ and $$b_{P_{n}(s)}$$ , $$\rho _{0}$$ associated with $$P_{n}(s)$$ . The reciprocal question has been also treated. The bounds $$a_{P_{n}(s)}$$ , $$b_{P_{n}(s)}$$ are defined as the end-points of the minimal closed and bounded real interval $$I= [ a_{P_{n}(s)},b_{P_{n}(s)} ] $$ such that all the zeros of $$P_{n}(s)$$ are contained in the strip $$I\times {\mathbb {R}}$$ of the complex plane $${\mathbb {C}}$$ . The bounds $$\rho _{N}$$ , $$\rho _{0}$$ are defined as the unique real solutions of Henry equations of $$P_{n}(s)$$ . Some applications to the partial sums of the Riemann zeta function have been also showed.

Based on Bohr’s equivalence relation for general Dirichlet series, in this paper we connect the families of equivalent exponential polynomials with a geometrical point of view related to lines in crystal-like structures. In particular we characterize this equivalence relation, and give an alternative proof of Bochner’s property referring to these functions, through this new geometrical perspective. The first author’s research was partially supported by PGC2018-097960-B-C22 (MCIU/AEI/ERDF, UE).

This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely formulated in the case of existence of an integral basis for the set of exponents of the associated Dirichlet series. J.M. Sepulcre research was partially supported by MICIU of Spain under Project Number PGC2018-097960-B-C22.

In this paper, we study flag codes on the vector space $${{\mathbb {F}}}_q^n$$ F q n , being q a prime power and $${{\mathbb {F}}}_q$$ F q the finite field of q elements. More precisely, we focus on flag codes that attain the maximum possible distance (optimum distance flag codes) and can be obtained from a spread of $${{\mathbb {F}}}_q^n$$ F q n . We characterize the set of admissible type vectors for this family of flag codes and also provide a construction of them based on well-known results about perfect matchings in graphs. This construction attains both the maximum distance for its type vector and the largest possible cardinality for that distance.

The performance of turbo codes at the error floor region is largely determined by the effective free distance, which corresponds to the minimum Hamming weight among all codeword sequences generated by input sequences of weight two. In this paper, we study turbo codes of dimension one obtained from the concatenation of two equal codes and present an upper bound on the effective free distance of a turbo code with these parameters defined over any finite field. We do that making use of the so-called (A, B, C, D) state-space representations of convolutional codes and restrict to the case where A is invertible. A particular construction, from a linear systems point of view, of a recursive systematic convolutional code of rate 1/n so that the effective free distance of the corresponding turbo code attains this upper bound is also presented. D. Napp was partially supported by the the Universitat d’Alacant (Grant No. VIGROB-287) and Generalitat Valenciana (Grant No. AICO/2017/128). V. Herranz and C. Perea were supported by the Ministerio de Economa, Industria y Competitividad within project TIN2016-80565-R.

In this paper it is shown that the complex functional equation $$F(z)+F(2z)=0$$ , $$z\in \varOmega :=\mathbb {C}\setminus ( -\infty ,0 ] $$ , is stable in the strong sense of Ulam. It means that given an analytic and bounded function f(z) on $$\varOmega $$ satisfying $$\left| f(z)+f(2z)\right| 0$$ , there exists an analytic solution F(z) of the above functional equation such that $$\left| f(z)-F(z)\right| 0$$ , not necessarily bounded on $$\varOmega $$ .

We shall consider a result of Fel’dman, where a sharp Baker-type lower bound is obtained for linear forms in the values of some E-functions. Fel’dman’s proof is based on an explicit construction of Padé approximations of the first kind for these functions. In the present paper, we introduce Padé approximations of the second kind for the same functions and use these to obtain a slightly improved version of Fel’dman’s result.

For any commutative semigroup S and any positive integer m, the power function f:S→S defined by f(x)=xm is an endomorphism of S. In this paper we characterize finite cyclic semigroups as those finite commutative semigroups whose endomorphisms are power functions. We also prove that if S is a finite commutative semigroup with 1≠0, then every endomorphism of S preserving 1 and 0 is equal to a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. Immediate consequences of the results are, on the one hand, a characterization of commutative rings whose multiplicative endomorphisms are power functions given by Greg Oman in the paper (Semigroup Forum, 86 (2013), 272–278), and on the other hand, a partial solution of Problem 1 posed by Oman in the same paper.

Let M be a shift invariant subspace in the vector-valued Hardy space \({H_{E}^{2}(\mathbb{D})}\). The Beurling–Lax–Halmos theorem says that M can be completely characterized by \({\mathcal{B}(E)}\)-valued inner function \({\Theta}\). When \({E = H^{2}(\mathbb{D}),\,H_{E}^{2}(\mathbb{D})}\) is the Hardy space on the bidisk \({H^{2}(\mathbb{D}^2)}\). Recently, Qin and Yang (Proc Am Math Soc, 2013) determines the operator valued inner function \({\Theta(z)}\) for two well-known invariant subspaces in \({H^{2}(\mathbb{D}^{2})}\). This paper generalizes the \({\Theta(z)}\) by Qin and Yang (Proc Am Math Soc, 2013) and deal with the structure of \({M = {\Theta}(z)H^{2}(\mathbb{D}^{2})}\) when M is an invariant subspace in \({H^{2}(\mathbb{D}^{2})}\). Unitary equivalence, spectrum of the compression operator and core operator are studied in this paper.

We prove that there is a factor of the Muchnik lattice that captures intuitionistic propositional logic. This complements a now clas- sic result of Skvortsova for the Medvedev lattice. 1. Introduction Among the structures arising from computability theory, the lattices intro- duced by Medvedev and Muchnik stand out for several distinguished features and a broad range of applications. In particular these lattices have additional structure that makes them suitable as models of certain propositional cal- culi. The structure of the Medvedev lattice as a Brouwer algebra, and thus as a model for propositional logics, has been extensively studied in several papers, see e.g. (10), (15), (17), (20), (22). Originally motivated in (10) as a formalization of Kolmogorov's calculus of problems (7), the Medvedev lattice fails to provide an exact interpretation of the intuitionistic propositional cal- culus IPC; however, as shown by Skvortsova (15), there are initial segments of the Medvedev lattice that model exactly IPC. On the other hand, little is known about the structure of the Muchnik lattice, and of its dual, as Brouwer algebras. The goal of this paper is to show that there are initial segments (equivalently: factors obtained dividing the lattice by principal filters) of the Muchnik lattice, in which the set of valid propositional sentences coincides with IPC. This shows that the analogue of Skvortsova's theorem also holds for the Muchnik lattice. From this, it readily follows that the propositional sentences that are valid in the Muchnik lattice are exactly the sentences of the so-called logic of the weak law of the excluded middle ((17)). Similar results (as announced, with outlined proofs, in (18)) hold of the dual of the Muchnik lattice: detailed proofs are provided in Section 5. For all unexplained notions from computability theory, the reader is re- ferred to Rogers (14); our main source for Brouwer algebras and the algebraic semantics of propositional calculi is Rasiowa-Sikorski (13). A comprehensive survey on the Medvedev and Muchnik lattices, and their mutual relation- ships, can be found in (19). Throughout the paper we use the symbols + and × to denote the join and meet operations, respectively, in any lattice. 1.1. The Medvedev and the Muchnik lattices. Although our main ob- ject of study is the Muchnik lattice, reference to the Medvedev lattice will be sometimes useful. Therefore, we start by reviewing some basic definitions and facts concerning both lattices. Following Medvedev (10), a mass problem is a set of functions from the set of natural numbers ω to ω. There are two natural ways to extend Turing reducibility to mass problems: following (10)

In a recent paper, Sun et al. (2007) studied the existence of positive solutions of nonhomogeneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem.

In this paper, we investigate the structure of reachable sets from a given point q 0 for a class of analytic control affine systems characterized, among other things, by possessing two singular trajectories initiating at q 0. The aim of the paper is to establish the connection between the minimal number of analytic functions needed for describing reachable sets and the number of geometrically optimal singular trajectories. The paper is written in a language of the sub-Lorentzian geometry. Also, the sub-Lorentzian geometry methods are used to prove theorems.

By defining a new terminology, scatter degree, as the supremum of graininess functional value, this paper studies the existence of solutions for a nonlinear two-point dynamic boundary value problem on time scales. We do not need any growth restrictions on nonlinear term of dynamic equation besides a barrier strips condition. The main tool in this paper is the induction principle on time scales.

In this paper we study extensions between Cohen–Macaulay modules for algebras arising in the categorifications of Grassmannian cluster algebras. We prove that rank 1 modules are periodic, and we give explicit formulas for the computation of the period based solely on the rim of the rank 1 module in question. We determine \(\mathrm{Ext}^i(L_I, L_J)\) for arbitrary rank 1 modules \(L_I\) and \(L_J\). An explicit combinatorial algorithm is given for the computation of \(\mathrm{Ext}^i(L_I, L_J)\) when i is odd, and when i even, we show that \(\mathrm{Ext}^i(L_I, L_J)\) is cyclic over the centre, and we give an explicit formula for its computation. At the end of the paper we give a vanishing condition of \(\mathrm{Ext}^i(L_I, L_J)\) for any \(i>0\).

This paper is concerned with the distributions of zeros of solutions to first order delay dynamic equations on time scales. The results are obtained using iterative sequences.

In this paper, we consider the biharmonic problem of a partial differential inclusion with Dirichlet boundary conditions. We prove existence theorems for related partial differential inclusions with convex and nonconvex multivalued perturbations, and obtain an existence theorem on extremal solutions, and a strong relaxation theorem. Also we prove that the solution set is compact $R_{\delta}$ if the perturbation term of the related partial differential inclusion is convex, and its solution set is path-connected if the perturbation term is nonconvex.

In this paper, we study the existence of nontrivial solution for a class of quasilinear Schrodinger equations in ${\mathbb{R}}$ with the nonlinearity asymptotically linear and, furthermore, the potential indefinite in sign. The tool used in this paper is the direct variation method.

Calvert and Gupta’s results concerning the perturbations on the ranges of m-accretive mappings have been employed widely in the discussion of the existence of solutions of nonlinear elliptic differential equation with Neumann boundary. In this paper, we shall focus our attention on certain hyperbolic differential equation with mixed boundaries. By defining some suitable nonlinear mappings, we shall demonstrate that Calvert and Gupta’s results can be applied to hyperbolic equations, in addition to its wide usage in elliptic equations. Due to the differences between hyperbolic and elliptic equations, some new techniques have been developed in this paper, which can be regarded as the complement and extension of the previous work.

In this paper, we are interested in the eigenvalues and its algebraic multiplicities of a fractional linear boundary value problem with mixed set of Neumann and Dirichlet boundary conditions. The research results are then applied to consider the sign-changing solutions of the corresponding nonlinear problem by fixed point index and Leray-Schauder degree. To date, no paper has appeared in the literature which discusses sign-changing solutions of fractional boundary value problems. This paper attempts to fill this gap in the literature.

The present paper is devoted to the investigation of a parabolic equation with moving boundaries arising in ductal carcinoma in situ (DCIS) model. Approximation solution of this problem is implemented by Ritz-Galerkin, which is a first attempt at tackling such problem. In process of dealing with this moving boundary condition, we use a trick of introducing two transformations to convert moving boundary to nonclassical boundary that can be handled with Ritz-Galerkin method. Also, existence and uniqueness are proved. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper.

This paper discusses an autonomous competitive Lotka-Volterra model in random environments. The contributions of this paper are as follows. (a) Some sufficient conditions for partial permanence and extinction on this system are established; (b) By using some novel techniques, the conditions imposed on permanence and extinction of one-species are weakened. Finally, a numerical experiment is conducted to validate the theoretical findings.

This paper is concerned with the existence of universal attractors in $H_{+}^{i}$ ( $i=1,2$ ) for one-dimensional compressible and radiative magnetohydrodynamics equations in a bounded domain $\Omega:=(0,1)$ . In this paper, the author extends the results in (Qin et al. in J. Differ. Equ. 253:1439-1488, 2012).

In this paper, we are concerned with nonlinear elliptic equations of the $p(x)$ -Laplace type operators $$\textstyle\begin{cases} -\operatorname{div}(a(x,\nabla u))+\vert u\vert ^{p(x)-2}u=\lambda f(x,u) &\mbox{in } \Omega, \\ a(x,\nabla u)\frac{\partial u}{\partial n} = \lambda\theta g(x,u) &\mbox{on } \partial\Omega, \end{cases} $$ which are subject to nonlinear Neumann boundary conditions. Here the function $a(x,v)$ is of type $\vert v\vert ^{p(x)-2}v$ with a continuous function $p: \overline{\Omega} \to(1,\infty)$ and the functions $f, g$ satisfy a Caratheodory condition. The main purpose of this paper is to establish the existence of at least three weak solutions of the above problem by applying an abstract three critical points theorem which is inspired by the work of Ricceri (Nonlinear Anal. 74:7446-7454, 2011) Furthermore, we determine two intervals of λ’s precisely such that the first is where the given problem admits only the trivial solution, and the second is where the given problem has at least two nontrivial solutions as considering the positive principal eigenvalue for the $p(x)$ -Laplacian Neumann problems and an estimate of the Sobolev trace embedding’s constant.

In this paper, we obtain a maximum principle for controlled fractional Fokker-Planck equations. We prove the well-posedness of a stochastic differential equation driven by an α-stable process. We give some estimates of the solutions by fractional calculus. A linear-quadratic example is given at the end of the paper.

This paper is concerned with fuzzy bidirectional associative memory (BAM) Cohen-Grossberg neural networks with mixed delays and impulses. By constructing an appropriate Lyapunov function and a new differential inequality, we obtain some sufficient conditions which ensure the existence and global exponential stability of a periodic solution of the model. The results in this paper extend and complement the previous publications. An example is given to illustrate the effectiveness of our obtained results.

Based on a specific decomposition of discrete singular systems, in this paper, we study the problem of state tracking control by using PD-type algorithm of iterative learning control. The convergence conditions and theoretical analysis of the PD-type algorithm are presented in detail. An illustrative example supporting the theoretical results and the effectiveness of the PD-type iterative learning control algorithm for discrete singular systems is shown at the end of the paper.

Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Gr\"{o}nwall inequality. Examples illustrating the significance of the results are also given., Comment: The paper was originally open access. The purpose of this publication is correct the original mistake that occurred in Theorems 6 and 10 (in particular, the variables under the integral in (2.11) and (2.27)) in the published paper

In this paper, we introduce a new class of multivalued contractions with respect to b-generalized pseudodistances and prove a best proximity point theorem for this class of non-self mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.

In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects. To the authors’ knowledge, there is no paper about periodic solution of impulses delay differential systems via critical point theory. Our results are completely new.

This paper deals with the global existence and blow-up of the solution for a class of nonlinear reaction diffusion problems. The purpose of this paper is to establish conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified. Finally, as applications of the obtained results, some examples are presented.

The paper addresses the asymptotic properties of Camassa-Holm equation on the half-line. That is, using the method of asymptotic density, under the assumption that it is unique, the paper proves that the positive momentum density of the Camassa-Holm equation is a combination of Dirac measures supported on the positive axis. This means that as time goes to infinity, the momentum density concentrates in small intervals moving right with different constant speeds.

Consider an evolutionary equation related to the $p(x)$ -Laplacian: ${u_{t}}= \operatorname{div} ({\rho^{\alpha}}{ \vert {\nabla u} \vert ^{p(x) - 2}}\nabla u)+ {\frac{{\partial{b_{i}}(u,x,t)}}{{\partial {x_{i}}}}}$ , $(x,t) \in{Q_{T}} = \Omega \times(0,T)$ , which arises from electrorheological fluid mechanics. Since $\rho(x) = \operatorname{dist} (x,\partial\Omega)$ , the equation is degenerate on the boundary, one may expect that there is not flux across the boundary. The paper shows that the facts may be unexpected. The paper reviews Fichera-Oleinik theory, then uses the theory to discuss the boundary value condition related to the equation. If $p^{-}>2$ , the existence and the uniqueness of the solutions are researched. Finally, if $b_{i}\equiv0$ , the behavior of the solutions near the boundary is studied by the comparison theorem.

The paper studies the equation $$ {u_{t}}= \operatorname{div} \bigl(a(x)\vert {\nabla u} \vert ^{p(x) - 2}\nabla u \bigr), $$ with the boundary degeneracy coming from $a(x)\vert_{x\in \partial \Omega }=0$ . The paper introduces a new kind of weak solutions of the equation. One can study the stability of the new kind of weak solutions without any boundary value condition.

We complete the stability results of the paper Bourles et al. (SIAM J Control Optim 53:2725---2761, 2015), and for this purpose use the linear time-varying (LTV) discrete-time behaviors and the exponential stability (e.s.) of this paper. In the main theorem we characterize the e.s. of an autonomous LTV system by standard spectral properties of a complex matrix connected with the system. We extend the theory of discrete-time LTV behaviors, developed in the quoted publication, from the coefficient field of rational functions to that of locally convergent Laurent series or even of Puiseux series. The stability test can and has to be applied in connection with the construction of stabilizing compensators.

This paper is concerned with a thermoelastic suspension bridge equations with memory effects. For the suspension bridge equations without memory, there are many classical results. However, the suspension bridge equations with both viscoelastic and thermal memories were not studied before. The object of the present paper is to provide a result on the global attractor to a thermoelastic suspension bridge equation with past history.

This paper is devoted to the study of the sign of the Green’s function related to a general linear nth-order operator, depending on a real parameter, $T_{n}[M]$ , coupled with the $(k,n-k)$ boundary value conditions. If the operator $T_{n}[\bar{M}]$ is disconjugate for a given M, we describe the interval of values on the real parameter M for which the Green’s function has constant sign. One of the extremes of the interval is given by the first eigenvalue of the operator $T_{n}[\bar{M}]$ satisfying $(k,n-k)$ conditions. The other extreme is related to the minimum (maximum) of the first eigenvalues of $(k-1,n-k+1)$ and $(k+1,n-k-1)$ problems. Moreover, if $n-k$ is even (odd) the Green’s function cannot be nonpositive (nonnegative). To illustrate the applicability of the obtained results, we calculate the parameter intervals of constant sign Green’s functions for particular operators. Our method avoids the necessity of calculating the expression of the Green’s function. We finalize the paper by presenting a particular equation in which it is shown that the disconjugation hypothesis on operator $T_{n}[\bar{M}]$ for a given M cannot be eliminated.

In this paper, some easily verifiable conditions are derived for the existence of almost periodic solution of an impulsive multispecies logarithmic population model in terms of the continuous theorem of coincidence degree theory, which is rarely applied to studying the existence of almost periodic solution of an impulsive differential equation. Our results generalize previous results by Alzabut, Stamov and Sermutlu. Besides, our technique used in this paper can be applied to study the existence of almost periodic solution of an impulsive differential equation with linear impulsive perturbations.

In this paper, we propose a novel stabilized mixed finite element method for the approximation of optimal control problems governed by reaction-diffusion equations. Compared with the classical mixed finite element methods, the main contributions of this paper are as follows. First, the novel method uses only one stabilization parameter which is not mesh-dependent, and, the new mixed bilinear formulation is coercive and continuous. Second, the novel method is easy to be implemented on a computer using the standard Lagrange finite element. Third, the solutions of the novel method to the optimal control problems require low regularities. Fourth, the Ladyzhenkaya-Babuska-Brezzi (LBB) or the inf-sup condition for the mixed element spaces is unnecessary. Based on the novel method, we derive both continuous and discrete optimality systems for the corresponding constrained optimal control problems, and then a priori error analysis in a weighted norm is discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the novel stabilized method.

This paper deals with the vectorial Schrodinger operators with δ-interactions generated by $L_{X,A,Q}:=-\frac{d^{2}}{dx^{2}} +Q(x)+\sum_{k=1}^{\infty}A_{k}\delta(x-x_{k})$ , $x\in[ 0,+\infty)$ . First, we obtain an embedding inequality. Then using standard form methods, we prove that the operator $\mathbf{H}_{X,A,Q}$ given in this paper is self-adjoint. Finally, a sufficient condition and a necessary condition are given for the spectrum of the operator $\mathbf {H}_{X,A,Q}$ to be discrete. By giving additional restrictions on the symmetric potential matrix $Q(x)$ and $A_{k}$ , we also give a necessary and sufficient condition for a special case. The conditions are analogous to Molchanov’s discreteness criteria.

In this paper, we deal with a fractional differential equation of order $\delta_{1}\in(3,4]$ with initial and boundary conditions, $\mathcal{D}^{\delta_{1}}\psi(x)=-\mathcal{H}(x,\psi(x))$ , $\mathcal{D}^{\alpha_{1}} \psi(1)=0=\mathcal{I}^{3-\delta_{1}}\psi(0)= \mathcal{I}^{4-\delta_{1}}\psi(0)$ , $\psi(1) = \frac{\Gamma(\delta_{1}-\alpha_{1})}{\Gamma(\nu_{1})}\mathcal{I}^{\delta _{1}-\alpha_{1}} \mathcal{H}(x,\psi(x))(1)$ , where $x\in[0,1]$ , $\alpha_{1} \in(1,2]$ , addressing the existence of a positive solution (EPS), where the fractional derivatives $\mathcal{D}^{\delta_{1}}$ , $\mathcal{D}^{\alpha_{1}}$ are in the Riemann-Liouville sense of the order $\delta_{1}$ , $\alpha_{1}$ , respectively. The function $\mathcal{H}\in C([0,1]\times{R} , {R})$ and $\mathcal{I}^{\delta_{1}-\alpha_{1}}\mathcal{H}(x,\psi(x))(1)=\frac {1}{\Gamma(\delta_{1}-\alpha_{1})} \int_{0}^{1}(1-z)^{\delta_{1}-\alpha_{1}-1}\mathcal{H}(z,\psi(z))\,dz$ . To this aim, we establish an equivalent integral form of the problem with the help of a Green’s function. We also investigate the properties of the Green’s function in the paper which we utilize in our main result for the EPS of the problem. Results for the existence of solutions are obtained with the help of some classical results.

This paper is devoted to the analysis of an approximately controllable system perturbed by a certain Lipschitz-continuous nonlinearity. By assuming that the intensity of the nonlinear influence is ‘weak’, we prove that the perturbed system is still approximately controllable. Using this perturbation theory, we prove that a third-order semilinear dispersion equation on a finite interval is approximately controllable whenever the nonlinearity effect is ‘weak’. The result in the paper is a complement of the results obtained in (Zhou in SIAM J. Control Optim. 21:551-565, 1983), in which the control operator (resp. infinitesimal generator of the dynamics) of the unperturbed linear system is assumed to be bounded (resp. generated a differentiable semigroup on the state space).

In this paper, the parabolic evolution equation $u'(t)+A(t)u(t)=f(t)$ in a reflexive real Banach space is considered. Assuming strong monotonicity, pseudo almost automorphy and other appropriate conditions of the operators $A(t)$ and Stepanov-like pseudo almost automorphy of the forced term $f(t)$ , we obtain the Stepanov-like pseudo almost automorphy of the solution to the evolution equation by using the almost automorphic component equation method. This paper extends a known result in the case where $A(\cdot)$ and f are almost automorphic in certain senses. Finally, a concrete example is given to illustrate our results.

In this paper, we consider the existence of periodic solutions for a class of nonautonomous second-order discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz’s saddle point theorem.