In this paper we generalize the concept of $f$-divergence to a convex function defined on a convex cone in a linear space. Some fundamental results are established. Applications for some well known divergence measures are provided as well.

This paper deals with the existence and uniqueness results for nonlinear and double perturbed BVPs for fractional differential equations with firstorder dependence derivative. Our approach is based on fixed point theorems and monotone iterative technique. Some illustrative examples are also presented.

A Carleman estimates is established to prove a unique continuation property of the solution of the Boussinesq system. We can prove that if the solution of the Boussinesq systems vanishes in an open subset, then this solution is identically equal to zero in the horizontal component of the open subset.

This paper is concerned with the nonlinear stability of periodic traveling wave solutions for the coupled Benjamin-Bona-Mahony system. We show the existence of a family of dnoidal type traveling waves. We find conditions on parameters of the waves which imply the nonlinear stability of periodic traveling waves.

We generalize the concept of biorthogonal wavelets to a local field $K$ of positive characteristic. We show that if the translates of the scaling functions of two multiresolution analyses are biorthogonal, then the associated wavelet families are also biorthogonal. Under mild assumptions on the scaling functions and the wavelets, we also show that the wavelets generate Riesz bases for $L^2(K)$.

Cellular geometries can vary significantly, how they influence signaling remains largely unknown. In this article, we describe a new model of the most extensively studied signal transduction pathways, the Janus kinase (JAK)/signal transducer and activator of transcription (STAT) pathway based on a mixed system of linear differential equations (PDEs + ODEs) coupled by Robin boundary conditions. This model was introduced to analyze the influence of the cell shape on the regulatory response to the activated pathway. In this article, we present an analysis of the wellposedness of the resulting system, i.e., the existence of a unique solution, its nonnegativity, boundedness and Lyapunov stability. As byproduct, we show the well-posedness and convergence of a suitable discretization of this model providing the basis for its reliable numerical simulation.

Let $\Omega \in L^1(S^{n-1})$ have mean value zero and satisfy the condition $$\sup_{\zeta' \in S^{n-1}} \int_{S^{n-1}} |\Omega(y')| (\ln |\zeta' \cdot y'|^{1})^{(\ln(e + \ln |\zeta' \cdot y'|^{1}))^{\beta}} \, d\sigma(y') \lt \infty \text{ for some } \beta > 0.$$ Under certain conditions on the measurable function $h$, we show that the singular integral $$Tf(x) = \text{p. v.} \int_{\mathbb{R}^n} \dfrac{h(|y|) \Omega (y')}{|y|^n} f(x y)\,dy$$ is bounded on the Triebel-Lizorkin weighted spaces $\dot{F}^{\alpha,w}_{p,q}\mathbb{R}^n$. We also study the Marcinkiewicz integral (with the same kernel $\Omega$ as above) in the $L^p$ weighted spaces.

In this paper, we study the wellposedness in scales of Hilbert spaces $E^{\alpha},\alpha\in\mathbb{R}$ defined by the noncoupled system partial differential operator of a chemotaxis model of aggregation of microglia in Alzheimer's disease for a perturbated analytic semigroup, which decays exponentially in the large time asymptotic dynamics of the problem to a finite dimensional set $K\subset \mathbb{R}^{3}$ of the spatial average solutions. Uniform bounds in $\Omega\times (0,T)$ of solutions and gradient solutions to the system of equations are proved. Thus via a bootstrap argument solutions to the problem are shown to be classical solutions. Furthermore, under natural conditions on the coupled elliptic system quasilinear differential operator, we prove the existence of a fundamental solution or evolution operator for the model equations in cited function spaces. In conclusion numerical simulation results are provided.