We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\partial u/\partial t +H(x,D_xu,u)=0$ $M\times(0,\infty)$, where Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on variable $u$. In framework of semicontinuous viscosity solutions due to Barron-Jensen, we establish comparison principle, existence theorem, representation formula as value functions for extended real-valued,...

* Correspondence: [email protected]. kr National Institute for Mathematical Sciences, KT Daeduk 2 Research Center, 463-1 Jeonmin-dong, Yuseong-gu, Daejeon 305-811, Korea Full list of author information is available at the end of the article Abstract Lex X be a normed space and Y be a Banach fuzzy space. Let D = {(x, y) Î X × X : || x|| + ||y|| ≥ d} where d > 0. We prove that the Pexiderized Jensen...

We will present a fixed point method for the stability theorems of functional equations of Jensen type as given by S.-M. Jung [11] and Wang Jian [10].

Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations those real Stieltjes measure dλ which is not necessarily using several Green functions. Utilizing these define new mean value theorems Lagrange and Cauchy types, derive Cauchy-type means.