We establish in this paper some Jensen’s type inequalities for functions defined by power series with nonnegative coefficients. Applications for functions of selfadjoint operators on complex Hilbert spaces are provided as well.

Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces $L₁\left({ℝ}^d\right)$ and $BV\left({ℝ}^d\right)$ are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in $L₁\left({ℝ}^d\right)$ is also shown.

We consider the existence of infinitely many solutions to the boundary value problem $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left({\frac{1}{2}}_0{D}_t^{-\beta }\left(u^{\text{'}}\left(t\right)\right)+{\frac{1}{2}}_t{D}_T^{-\beta }\left(u^{\text{'}}\left(t\right)\right)\right)+\nabla F(t,u\left(t\right))=0\text{a.e.}t\in [0,T],\\ u\left(0\right)=u\left(T\right)=0.\end{array}$$ Under more general assumptions on the nonlinearity, we obtain new criteria to guarantee that this boundary value problem has infinitely many solutions in the superquadratic, subquadratic and asymptotically quadratic cases by using the critical point theory.

2000 Mathematics Subject Classification: Primary 26A33; Secondary 35S10, 86A05Fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. An initial value problem involving a space-fractional diffusion equation is an abstract Cauchy problem, whose analytic solution can be written...

We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more difficult problem of computing the lower box dimension. We give some non-trivial bounds and provide examples to show that lower box dimension behaves much more strangely than upper box dimension, Hausdorff dimension and packing dimension.

We establish sufficient conditions for the existence of solutions of a class of fractional functional differential inclusions involving the Hadamard fractional derivative with order $\alpha \in (0,1]$. Both cases of convex and nonconvex valued right hand side are considered.

We provide a number of either necessary and sufficient or only sufficient conditions on a local homeomorphism defined on an open, connected subset of the n-space to be actually a homeomorphism onto a star-shaped set. The unifying idea is the existence of "auxiliary" scalar functions that enjoy special behaviours along the paths that result from lifting the half-lines that radiate from a point in the codomain space. In our main result this special behaviour is monotonicity, and the auxiliary function...

Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a Baire-$.5$ function between two comparable real-valued functions on the topological spaces that $F_{\sigma }$-kernel of sets are $F_{\sigma }$-sets.