Modular uniform convexity structures and applications to boundary value problems with non-standard growth

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$. Our analysis considers a continuous and bounded exponent $p$ satisfying $1<\inf\limits_{x\in \Omega}p(x)$ and $\sup\limits_{x\in \Omega}p(x)<\infty$, and is based on the uniform convexity of the Dirichlet integral, which is highly non trivial and in the variable exponent case is not related to the uniform convexity of the Sobolev norm

Multivalued SK-contractions with respect to b-generalized pseudodistances

A new class of multivalued non-self-mappings, called SK-contractions with respect to b-generalized pseudodistances, is introduced and used to investigate the existence of best proximity points by using an appropriate geometric property. Some new fixed point results in b-metric spaces are also obtained. Examples are given to support the usability of our main result

Continuous selections of multivalued mappings

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our present survey considers more restricted and specific areas of mathematics. Note that we do not consider the theory of selectors (i.e. continuous choices of elements from subsets of topological spaces) since this topics is covered by another survey in this volume