The purpose of this paper is to show that the set of weak solutions of the initial-boundary value problem for the linear wave equation is nonempty, connected, and compact.

We introduce some concepts of generalized invexity for the continuous-time multiobjective programming problems, namely, the concepts of Karush-Kuhn-Tucker invexity and Karush-Kuhn-Tucker pseudoinvexity. Using the concept of Karush-Kuhn-Tucker invexity, we study the relationship of the multiobjective problems with some related scalar problems. Further, we show that Karush-Kuhn-Tucker pseudoinvexity is a necessary and suffcient condition for a vector Karush-Kuhn-Tucker solution to be a weakly efficient solution.

Linear elliptic differential equations with periodic coefficients in one-dimensional domains are considered. The approximation properties of the homogenized system are investigated. For $H^{-1}$-data, it turns out that the order of approximation is strongly related to the decay of the Fourier coefficients of the $L^2$-functions involved.

We will show that under suitable conditions on $f$ and $h$, there exists a positive number ${\lambda }^{\ast }$ such that the nonhomogeneous elliptic equation $-\Delta u+u=\lambda \left(f\left(x,u\right)+h\left(x\right)\right)$ in $\Omega$, $u\in H_0^1\left(\Omega \right)$, $N\ge 2$, has at least two positive solutions if $\lambda \in \left(0,{\lambda }^{\ast }\right)$, a unique positive solution if $\lambda ={\lambda }^{\ast }$, and no positive solution if $\lambda >{\lambda }^{\ast }$, where $\Omega$ is the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.

We prove distributional inequalities that imply the comparability of the $L^p$ norms of the multiplicative square function of $u$ and the nontangential maximal function of $logu$, where $u$ is a positive solution of a nondivergence elliptic equation. We also give criteria for singularity and mutual absolute continuity with respect to harmonic measure of any Borel measure defined on a Lipschitz domain based on these distributional inequalities. This extends recent work of M. González and A. Nicolau where the term multiplicative square functions is introduced and where the case when $u$ is a harmonic function is considered.

We consider the elliptic problem $-\Delta u+u=b\left(x\right){\left|u\right|}^{p-2}u+h\left(x\right)$ in $\Omega$, $u\in H_0^1\left(\Omega \right)$, where $2<p<(2N/(N-2\left)\right) (N\ge 3), 2<p<\infty \left(N=2\right), \Omega$ is a smooth unbounded domain in ${ℝ}^N, b\left(x\right)\in C\left(\Omega \right)$, and $h\left(x\right)\in H^{-1}\left(\Omega \right)$. We use the shape of domain $\Omega$ to prove that the above elliptic problem has a ground-state solution if the coefficient $b\left(x\right)$ satisfies $b\left(x\right)\to b^{\infty }>0$ as $\left|x\right|\to \infty$ and $b\left(x\right)\ge c$ for some suitable constants $c\in \left(0,b^{\infty }\right)$, and $h\left(x\right)\equiv 0$. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient $b\left(x\right)$ also satisfies the above conditions, $h\left(x\right)\ge 0$ and $0<{\Vert h\Vert }_{H^{-1}}<\left(p-2\right){\left(1/(p-1)\right)}^{(p-1)/(p-2)}{\left[b_{sup}{S}^p\left(\Omega \right)\right]}^{1/(2-p)}$, where $S\left(\Omega \right)$ is the best Sobolev constant of subcritical operator in $H_0^1\left(\Omega \right)$ and $b_{sup}={sup}_{x\in \Omega }b\left(x\right)$.

By means of an expression with a kind of integral operators, some properties of the weighted Hadamard-type singular integrals are revealed. As applications, the solution for certain strongly singular integral equations is discussed and illustrated.

We prove the generalized stability of $C^{*}$-ternary quadratic mappings in $C^{*}$-ternary rings for the quadratic functional equation $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)+2f\left(y\right)$.

A fixed point theorem for nonlinear contraction in the modular space is proved. Moreover, a fixed point theorem for asymptotic contraction in this space is studied.

We introduce new model equations to describe the dynamics of the electric polarization in a ferroelectric material. We consider a thin cylinder representing the material with thickness $\varepsilon$ and discuss the asymptotic behavior of the time harmonic solutions to the model when $\varepsilon$ tends to $0$. We obtain a reduced model settled in the cross-section of the cylinder describing the dynamics of the plane components of the polarization and electric fields.

The differential equation $u\text{'}\left(t\right)+\mathbf{A}u\left(t\right)=f\left(t\right)\left(-\infty <t<\infty \right)$ in a general Banach space $\mathbf{E}$ with the strongly positive operator $\mathbf{A}$ is ill-posed in the Banach space $C\left(\mathbf{E}\right)=C\left(ℝ,\mathbf{E}\right)$ with norm ${\Vert \varphi \Vert }_{C\left(\mathbf{E}\right)}={sup}_{-\infty <t<\infty }{\Vert \varphi \left(t\right)\Vert }_{\mathbf{E}}$. In the present paper, the well-posedness of this equation in the Hölder space $C^{\alpha }\left(\mathbf{E}\right)=C^{\alpha }\left(ℝ,\mathbf{E}\right)$ with norm ${\Vert \varphi \Vert }_{C^{\alpha }\left(\mathbf{E}\right)}={sup}_{-\infty <t<\infty }{\Vert \varphi \left(t\right)\Vert }_{\mathbf{E}}+{sup}_{-\infty <t<t+s<\infty }{(\Vert \varphi \left(t+s\right)-\varphi \left(t\right)\Vert }_{\mathbf{E}}/s^{\alpha }$), $0<\alpha <1$, is established. The almost coercivity inequality for solutions of the Rothe difference scheme in $C\left({ℝ}_{\tau },\mathbf{E}\right)$ spaces is proved. The well-posedness of this difference scheme in $C^{\alpha }\left({ℝ}_{\tau },\mathbf{E}\right)$ spaces is obtained.

Using the Hyers-Ulam-Rassias stability method of functional equations, we investigate homomorphisms in $C^{*}$-algebras, Lie $C^{*}$-algebras, and $J{C}^{*}$-algebras, and derivations on $C^{*}$-algebras, Lie $C^{*}$-algebras, and $J{C}^{*}$-algebras associated with the following Apollonius-type additive functional equation $f\left(z-x\right)+f\left(z-y\right)+\left(1/2\right)f\left(x+y\right)=2f\left(z-\left(x+y\right)/4\right)$.

The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valued $L_p$ spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

We classify the equilibrium solutions of the Smoluchowski equation for dipolar (extended) rigid nematic polymers under imposed elongational flow. The Smoluchowski equation couples the Maier-Saupe short-range interaction, dipole-dipole interaction, and an external elongational flow. We show that all stable equilibria of rigid, dipolar rod dispersions under imposed uniaxial elongational flow field are axisymmetric. This finding of axisymmetry significantly simplifies any procedure of obtaining experimentally observable equilibria.

We will study meromorphic functions that share a small function, and prove the following result: let $f\left(z\right)$ and $g\left(z\right)$ be two transcendental meromorphic functions in the complex plane and let $n\ge 11$ be a positive integer. Assume that $a\left(z\right)\left(\not\equiv 0\right)$ is a common small function with respect to $f\left(z\right)$ and $g\left(z\right)$. If $f^n{f}^{\prime }$ and $g^n{g}^{\prime }$ share $a\left(z\right)$ CM, then either $f^n\left(z\right)f^{\prime }\left(z\right)g^n\left(z\right)g^{\prime }\left(z\right)\equiv a^2\left(z\right)$, or $f\left(z\right)\equiv tg\left(z\right)$ for a constant satisfying $t^{n+1}=1$. As applications, we give several examples.

We investigate the generalized Hyers-Ulam stability of the functional inequalities associated with Cauchy-Jensen additive mappings. As a result, we obtain that if a mapping satisfies the functional inequalities with perturbation which satisfies certain conditions, then there exists a Cauchy-Jensen additive mapping near the mapping.

We study second-order nonlinear periodic systems driven by the vector $p$-Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).

Here we study the polyharmonic nonlinear elliptic boundary value problem on the unit ball $B$ in ${ℝ}^n\left(n\ge 2\right)$ ${\left(-△\right)}^{m}u+g\left(\cdot ,u\right)=0$, in $B$ (in the sense of distributions) ${lim}_{x\to \xi \in \partial B}(u\left(x\right)/{\left(1-{\left|x\right|}^2\right)}^{m-1})=0\left(\xi \right)$. Under appropriate conditions related to a Kato class on the nonlinearity $g\left(x,t\right)$, we give some existence results. Our approach is based on estimates for the polyharmonic Green function on $B$ with zero Dirichlet boundary conditions, including a 3G-theorem, which leeds to some useful properties on functions belonging to the Kato class.

We study Navier-Stokes equations perturbed with a maximal monotone operator, in a bounded domain, in 2D and 3D. Using the theory of nonlinear semigroups, we prove existence results for strong and weak solutions. Examples are also provided.

The concept of linking was developed to produce Palais-Smale (PS) sequences $G\left(u_k\right)\to a$, $G\text{'}\left(u_k\right)\to 0$ for $C^1$functionals $G$ that separate linking sets. These sequences produce critical points if they have convergent subsequences (i.e., if $G$ satisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations produce more useful sequences. They not only produce PS sequences, but also Cerami sequences satisfying $G\left(u_k\right)\to a$, $(1+|\left|u_k\right|\left|\right)G\text{'}\left(u_k\right)\to 0$ as well. A Cerami sequence can produce a critical point even when a PS sequence does not. In this situation, it is no longer necessary to show that $G$ satisfies the PS condition, but only that it satisfies the easier Cerami condition (i.e., that Cerami sequences have convergent subsequences). We provide examples and applications. We also give generalizations to situations when the separating criterion is violated.

Let ${\mathbb{D}}^n$ be the unit polydisc of ${ℂ}^n$, $\varphi \left(z\right)=\left({\varphi }_1\left(z\right),\dots ,{\varphi }_n\left(z\right)\right)$ be a holomorphic self-map of ${\mathbb{D}}^n$, and $\psi \left(z\right)$ a holomorphic function on ${\mathbb{D}}^n$. Let $H\left({\mathbb{D}}^n\right)$ denote the space of all holomorphic functions with domain ${\mathbb{D}}^n$, $H^{\infty }\left({\mathbb{D}}^n\right)$ the space of all bounded holomorphic functions on ${\mathbb{D}}^n$, and $\mathfrak{B}\left({\mathbb{D}}^n\right)$ the Bloch space, that is, $\mathfrak{B}\left({\mathbb{D}}^n\right)=\left\{f\in H\left({\mathbb{D}}^n\right){|\Vert f\Vert }_{\mathfrak{B}}=\left|f\left(0\right)\right|+{sup}_{z\in {\mathbb{D}}^n}{\displaystyle \sum _{k=1}^n\left|(\partial f/\partial z_k)\left(z\right)\right|}\left(1-{\left|z_k\right|}^2\right)<+\infty \right\}$. We give necessary and sufficient conditions for the weighted composition operator $\psi C_{\varphi }$ induced by $\varphi \left(z\right)$ and $\psi \left(z\right)$ to be bounded and compact from $H^{\infty }\left({\mathbb{D}}^n\right)$ to the Bloch space $\mathfrak{B}\left({\mathbb{D}}^n\right)$.

The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. Via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. As an extensive example, we consider the $q$-difference equations case.

We show that the recently introduced $L^1$TV functional can be used to explicitly compute the flat norm for codimension one boundaries. Furthermore, using $L^1$TV, we also obtain the flat norm decomposition. Conversely, using the flat norm as the precise generalization of $L^1$TV functional, we obtain a method for denoising nonboundary or higher codimension sets. The flat norm decomposition of differences can made to depend on scale using the flat norm with scale which we define in direct analogy to the $L^1$TV functional. We illustrate the results and implications with examples and figures.

We investigate isomorphisms between $C^{*}$-algebras, Lie $C^{*}$-algebras, and $J{C}^{*}$-algebras, and derivations on $C^{*}$-algebras, Lie $C^{*}$-algebras, and $J{C}^{*}$-algebras associated withthe Cauchy–Jensen functional equation $2f\left(\left(x+y/2\right)+z\right)=f\left(x\right)+f\left(y\right)+2f\left(z\right)$.

The existence and uniqueness of the strong solution for a multitime evolution equation with nonlocal initial conditions are proved. The proof is essentially based on a priori estimates and on the density of the range of the operator generated by the considered problem.

This paper presents several characterizations of a local $\alpha$-times integrated $C$-semigroup $\left\{T\left(t\right);0\le t<\tau \right\}$ by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem. We also discuss properties concerning the nondegeneracy of $T(\cdot )$, the injectivity of $C$, the closability of subgenerators, the commutativity of $T(\cdot )$, and extension of solutions of the associated abstract Cauchy problem.

We revisit a key function arised in studies of nematic liqorder crystal polymers. Previously, it was conjectured that the function is strictly decreasing and the conjecture was numerically confirmed. Here we prove the conjecture analytically. More specifically, we write the derivative of the function into two parts and prove that each part is strictly negative.

Two upper and lower bounds for Mathieu's series are established, which refine to a certain extent a sharp double inequality obtained by Alzer-Brenner-Ruehr in 1998. Moreover, the very closer lower and upper bounds for $\zeta \left(3\right)$ are deduced.

We consider the set $𝕊$ of sequences of positive real numbers and show that some subclasses of $𝕊$ have certain nice selection and game theoretic properties.

We are interested in the first prolongational limit set of the boundary of parallelizable regions of a given flow of the plane which has no fixed points. We prove that for every point from the boundary of a maximal parallelizable region, there exists exactly one orbit contained in this region which is a subset of the first prolongational limit set of the point. Using these uniquely determined orbits, we study the structure of maximal parallelizable regions.

We show that for each minimal norm $N(\cdot )$ on the algebra ${ℳ}_n$ of all $n\times n$ complex matrices, there exist norms ${\Vert \cdot \Vert }_1$ and ${\Vert \cdot \Vert }_2$ on ${ℂ}^n$ such that $N\left(A\right)=max\left\{{\Vert Ax\Vert }_2:{\Vert x\Vert }_1=1,x\in {ℂ}^n\right\}$ for all $A\in {ℳ}_n$. This may be regarded as an extension of a known result on characterization of minimal algebra norms.

We prove a Tauberian theorem to recover moderate oscillation of a real sequence $u=\left(u_n\right)$ out of Abel limitability of the sequence $\left(V_n^{\left(1\right)}\left(\Delta u\right)\right)$ and some additional condition on the general control modulo of oscillatory behavior of integer order of $u=\left(u_n\right)$.

We are concerned with the existence and nonexistence of positive solutions for the nonlinear fractional boundary value problem: $D_{0+}^{\alpha }u\left(t\right)+\lambda a\left(t\right)\mathrm{\hspace{0.17em}f}\left(u\left(t\right)\right)=0,\hspace{0.17em}$ $0<t<1,$ $u\left(0\right)=u^{\prime }\left(0\right)=u^{\prime }\left(1\right)=0,$ where $2<\alpha <3$ is a real number and $D_{0+}^{\alpha }$ is the standard Riemann-Liouville fractional derivative. Our analysis relies on Krasnoselskiis fixed point theorem of cone preserving operators. An example is also given to illustrate the main results.

We give some sufficient and necessary conditions for an analytic function $f$ on the unit ball $B$ with Hadamard gaps, that is, for $f\left(z\right)={\displaystyle {\sum }_{k=1}^{\infty }{P}_{n_k}}\left(z\right)$ (the homogeneous polynomial expansion of $f$) satisfying $n_{k+1}/n_k\ge \lambda >1$ for all $k\in \mathbb{N}$, to belong to the space ${ℬ}_p^{\alpha }\left(B\right)=\{f|{sup}_{0<r<1}{\left(1-r^2\right)}^{\alpha }\backslash |ℛf_r{\mathrm{\backslash |}}_p<\infty ,f\in H\left(B\right)\}$, $p=1,2,\mathrm{\infty }$ as well as to the corresponding little space. A remark on analytic functions with Hadamard gaps on mixed norm space on the unit disk is also given.

Let $f$ and $g$ be distributions and let $g_n=(g*{\delta }_n)(x)$, where ${\delta }_n(x)$ is a certain sequence converging to the Dirac-delta function $\delta (x)$. The noncommutative neutrix product $f\circ g$ of $f$ and $g$ is defined to be the neutrix limit of the sequence $\left\{f{g}_n\right\}$, provided the limit $h$ exists in the sense that $\text{N‐}{lim}_{n\to \infty }\langle f\left(x\right)g_n\left(x\right),\phi \left(x\right)\rangle =\langle h\left(x\right),\phi \left(x\right)\rangle$, for all test functions in $𝒟$. In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products $x_{+}^{r}ln{x}_{+}\circ x_{-}^{-r-1}ln{x}_{-}$ and $x_{-}^{-r-1}ln{x}_{-}\circ x_{+}^{r}ln{x}_{+}$ are proved to exist and are evaluated for $r=1,2,\dots$. It is consequently seen that these two products are in fact equal.

We solve the nonhomogeneous Legendre's differential equation and apply this result to obtaining a partial solution to the Hyers-Ulam stability problem for the Legendre's equation.

We study some spectral problems for a second-order differential operator with periodic potential. Notice that the given potential is a sum of zero- and first-order generalized functions. It is shown that the spectrum of the investigated operator consists of infinite number of gaps whose length limit unlike the classic case tends to nonzero constant in some place and to infinity in other place.

We obtain the general solutions of the cubic functional equation $\text{3}\left[g\left(x+y\right)+g\left(x-y\right)+6g\left(x\right)\right]=2g\left(2x+y\right)+2g\left(2x-y\right)+g\left(-x-y\right)+g\left(-x+y\right)+6g\left(-x\right)$ and the Jensen-quadratic functional equation $f\left(\left(x+y\right)/2,z+w\right)+f(\left(x+y\right)/2,z-w)=f\left(x,z\right)+f\left(x,w\right)+f\left(y,z\right)+f\left(y,w\right)$.

A reaction-diffusion system modelling a predator-prey system in a periodic environment is considered. We are concerned in stabilization to zero of one of the components of the solution, via an internal control acting on a small subdomain, and in the preservation of the nonnegativity of both components.

The concept of unbounded Fredholm operators on Hilbert $C^{*}$-modules over an arbitrary $C^{*}$-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert $C^{*}$-modules over $C^{*}$-algebras of compact operators. In the framework of Hilbert $C^{*}$-modules over $C^{*}$-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same.

By using the strong monotonicity of the perturbed fixed-point map and the normal map associated with cocoercive variational inequalities, we establish two new global bounds measuring the distance between any point and the solution set for cocoercive variational inequalities.

In 2006, W. G. Park and J. H. Bae investigated the Hyers-Ulam stability of a Cauchy-Jensen functional equation. In this paper, we improve their results and obtain better results for a Cauchy-Jensen functional equation. Also, we establish new theorems for the generalized Hyers-Ulam stability of a Cauchy-Jensen functional equation.