In this paper, we study initial value problems of fractional differential equations and inclusions of Hadamard type, supplemented with nonlocal conditions. Some new existence and uniqueness results are obtained by using fixed point theorems for single valued and multi-valued maps. Examples illustrating the main results are also presented.

The Bocher-Grace theorem can be stated as follows: let $p$ be a third degree complex polynomial. Then, there is a unique inscribed ellipse interpolating the midpoints of the triangle formed from the roots of $p$, and the foci of the ellipse are the critical points of $p$. Here, we prove the following generalization: let $p$ be an $n$th degree complex polynomial, and let its critical points take the form \[ \alpha +\beta \cos k\pi /n,\quad k=1,\ldots ,n-1,\ \beta \ne 0. \] Then, there is an inscribed ellipse interpolating the midpoints of the convex polygon formed by the roots of $p$, and the foci of this ellipse are the two most extreme critical points of $p$: $\alpha \pm \beta \cos \pi /n$.

We study the eigenvalue problem for a system of fractional $p$-Laplacians, that is, \begin{equation} \begin{cases} (-\Delta _p)^r u = \lambda \tfrac {\alpha }p|u|^{\alpha -2}u|v|^{\beta } &\mbox{in } \Omega ,\\ (-\Delta _p)^s v = \lambda \tfrac {\beta }p|u|^{\alpha }|v|^{\beta -2}v &\mbox{in } \Omega ,\\ u=v=0 &\mbox{in } \Omega ^c=\mathbb{R} ^N\setminus \Omega. \end{cases} \end{equation} We show that there is a first (smallest) eigenvalue that is simple and has associated eigenpairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues $\lambda _n$ such that $\lambda _n\to \infty $ as $n\to \infty $.

In addition, we study the limit as $p\to \infty $ of the first eigenvalue, $\lambda _{1,p}$, and we obtain $[\lambda _{1,p}]^{{1}/{p}}\to \Lambda _{1,\infty }$ as $p\to \infty ,$ where $$\Lambda _{1,\infty }= \inf _{(u,v)} \bigg \{\frac {\max \{ [u]_{r,\infty } ; [v]_{s,\infty } \} }{ \| |u|^{\Gamma } |v|^{1-\Gamma } \|_{L^\infty (\Omega )}}\bigg \} = \bigg [ \frac {1}{R(\Omega )} \bigg ]^{ (1-\Gamma ) s + \Gamma r }. $$ Here, $$R(\Omega ):=\max _{x\in \Omega }dist (x,\partial \Omega )\mbox { and } [w]_{t,\infty } := \!\sup _{(x,y)\in \overline {\Omega }} \frac {| w(y) - w(x)|}{|x-y|^{t}}.$$

Finally, we identify a PDE problem satisfied, in the viscosity sense, by any possible uniform limit along subsequences of the eigenpairs.

Using the degeneration technique, we study the behavior of Welschinger invariants under the blow-up and obtain some blow-up formulae of Welschinger invariants. To analyze the variation of Welschinger invariants when replacing a pair of real points in the real configuration by a pair of conjugated points, Welschinger introduced the $\theta $-invariant. In this paper, we also verify that the $\theta $-invariant is the Welschinger invariant of the blow-up of the symplectic $4$-manifold.

In this article, we study the existence of nontrivial solutions for the following $p(x)$ Kirchhoff-type problem \begin{equation} \begin{cases}\smash {\!-M\big (\textstyle \int _{\Omega }A(x,\nabla u)\,dx\big ){div}(a(x,\nabla u))} =\lambda h(x)\frac{\partial F}{\partial u} (x,u), \quad \mbox {in } \Omega \\ u=0, \quad \mbox {on } \partial \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb {R}^{n}$, $n\geq 3$, is a smooth bounded domain, $\lambda >0$, $h\in C(\Omega )$, $F:\overline {\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$ is continuously differentiable and $a, A:\Omega \times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ are continuous. The proof is based on variational arguments and the theory of variable exponent Sobolev spaces.

In this paper, we obtain numerical ranges of normal weighted composition operators on $\ell ^{2}(\mathbb {N})$.

Let $\mathcal{G} $ be a locally compact group. In continuation of our studies on the first and second duals of measure algebras by the use of the theory of generalized functions, here we study the C$^*$-subalgebra $GL_0(\mathcal{G})$ of $GL(\mathcal{G})$ as an introverted subspace of $M(\mathcal{G} )^*$. In the case where $\mathcal{G} $ is non-compact, we show that any topological left invariant mean on $GL(\mathcal{G} )$ lies in $GL_0(\mathcal{G} )^\perp $. We then endow $GL_0(\mathcal{G} )^*$ with an Arens-type product, which contains $M(\mathcal{G} )$ as a closed subalgebra and $M_a(\mathcal{G} )$ as a closed ideal, which is a solid set with respect to absolute continuity in $GL_0(\mathcal{G} )^*$. Among other things, we prove that $\mathcal{G} $ is compact if and only if $GL_0(\mathcal{G} )^*$ has a non-zero left (weakly) completely continuous element.

Let $\mathcal {A}$ be the set of all integers of the form $\gcd (n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\#(\mathcal {A} \cap [1, x])\gg x / \log x$ for all $x \geq 2$ and that $\mathcal {A}$ has zero asymptotic density. Our proofs rely upon a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.

A predator-prey model with Leslie-Gower and Holling-type II schemes with regime switching will be considered, that is, both white and color noises are taken into account. We firstly show that there exists a globally unique solution to the stochastic predator-prey model by use of the comparison theorem. Then, asymptotic properties of the system will be examined and the conditions under which the system is stochastically persistent will be given. Moreover, lastly, we analyze the optimal harvesting policy of the stochastic prey-predator model with Markovian switching.

Let $\mathcal {K}(\mathcal {H})$ be the $C^*$-algebra of compact op\-erators on a Hilbert space $\mathcal {H}$. Let $E$ be a Hilbert $\mathcal {K}(\mathcal {H})$-mod\-ule and $\mathcal {L}(E)$ the $C^*$-algebra of all adjointable maps on $E$. In this paper, we prove that, if $\varphi :\mathcal {L}(E)\to \mathcal {L}(E)$ is a unital surjective bounded linear map, which preserves quasi-isometries in both directions, then there are unitary oper\-ators $U, V\in \mathcal {L}(E)$ such that \[ \varphi (T)=UTV\quad \mbox {or}\quad \varphi (T)=UT^{tr }V \] for all $T\in \mathcal {L}(E)$, where $T^{tr }$ is the transpose of $T$ with re\-spect to an arbitrary but fixed orthonormal basis of $E$.

In this article, we formulate a version of the analytic Novikov conjecture for semigroups rather than groups and show that the descent argument from coarse geometry generalizes effectively to this new situation.

We introduce the notions of Chern-Dirac bundles and Chern-Dirac operators on Hermitian manifolds. They are analogues of classical Dirac bundles and Dirac operators, with the Levi-Civita connection replaced by the Chern connection. We then show that the tensor product of the canonical and the anticanonical spinor bundles, called the $\mathcal{V} $-spinor bundle, is a bigraded Chern-Dirac bundle with spaces of harmonic sections isomorphic to the full Dolbeault cohomology class. A similar construction establishes isomorphisms among other types of harmonic sections of the $\mathcal{V} $-spinor bundle and twisted cohomology.

In this paper, the unique existence of generalized solutions to periodic boundary value problems for a class of systems of nonlinear equations with $x$-dependent coefficients is discussed under a resonance condition. The argument presented makes use of the global inverse theorem and Galerkin's method.

In this paper, we present a definition of the tracial Rokhlin property for (cocyclic) actions of countable discrete amenable groups on simple $C^*$-algebras, which generalize Matui and Sato's definition. We show that generic examples, like Bernoulli shift on the tensor product of copies of the Jiang-Su algebra, has the weak tracial Rokhlin property, while it is shown in {Hirshberg 2014} that such an action does not have finite Rokhlin dimension. We further show that forming a crossed product from actions with the tracial Rokhlin property preserves the class of $C^*$-algebras with real rank $0$, stable rank $1$ and has strict comparison for pro\-jec\-tions, generalizing the structural results in {Osaka 2006}. We use the same idea of the proof with significant simplification. In another joint paper with Chris Phillips and Joav Orovitz, we shall show that pureness and $\mathcal{Z} $-stability could be preserved by crossed product of actions with the weak tracial Rokhlin property. The combination of these results yields an application to the classification program, which is discussed in the aforementioned paper. These results indicate that we have the correct definition of tracial Rokhlin property for actions of general countable discrete amenable groups.

In this paper, we give a negative answer to a problem presented by Bharanedhar and Ponnusamy concerning univalency of a class of harmonic mappings. More precisely, we show that for all values of the involved parameter, this class contains a non-univalent function. Moreover, several results on a new subclass of close-to-convex harmonic mappings, motivated by the work of Ponnusamy and Sairam Kaliraj, are obtained.

We show that, for contracting and regular self-similar graph actions, the shift maps on limit spaces are positively expansive local homeomorphisms. From this, we find that limit solenoids of contracting and regular self-similar graph actions are Smale spaces and that the unstable Ruelle algebras of the limit solenoids are strongly Morita equivalent to the Cuntz-Pimsner algebras by Exel and Pardo if self-similar graph actions satisfy the contracting, regular, pseudo free and $G$-transitive conditions.