We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation T between uniformly log-concave probability measures. Let T : R → R be an optimal transportation pushing forward μ = e dx to ν = e dx. Assume that 1) the second differential quotient of V can be estimated from above by a power function, 2) modulus of convexity of W can be estimated from below by...

In this paper we present a fixed point theorem of Banach type in modular space. We give an application of this result to a nonlinear integral equation in Musielak-Orlicz space. 0. Introduction It is well known that one of the standard proofs of Banach’s fixed point theorem is based on Cantor’s theorem in complete metric spaces [3, 4]. To this end, using some convenient constants in the contract...

We investigate two greedy strategies for finding an approximation to the minimum of a convex function E defined on a Hilbert space H. We prove convergence rates for these algorithms under suitable conditions on the objective function E. These conditions involve the behavior of the modulus of smoothness and the modulus of uniform convexity of E.

In this paper, we deal with the following double phase problem $$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }\left( |\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u\right) =&{} \gamma \left( \displaystyle \frac{|u|^{p-2}u}{|x|^p}+a(x)\displaystyle \frac{|u|^{q-2}u}{|x|^q}\right) \\ &{}+f(x,u) &{} \text{ in } \Omega ,\\ u=0&{} \partial , \end{array} \right. \end{aligned}$$ where...

In this study, the mechanical and thermal behavior of the nano-reinforced polymer composite reinforced by Montmorillonite (MMT) nanoparticles is investigated. Due to low cost of computations, the 3D representative volume elements (RVE) method is utilized using ABAQUS finite element commercial software. Low density poly ethylene (LDPE) and MMT are used as matrix and nanoparticle material, respec...