We introduce function spaces for the treatment of non-linear parabolic partial differential equations with variable log–Hölder continuous exponents that only incorporate information symmetric part a gradient. As an analogue Korn's inequality these functions is not available, construction appropriate smoothing method proves to be difficult. Using point-wise Poincaré near boundary bounded Lipschi...

Much has been written on variable exponent spaces in recent years. Most of the literature deals with normed space structure such spaces. However, because variability exponent, underlying modular these is radically different from that induced by norm. In this article, we focus our attention progress made toward study sequence Lebesgue exponents. particular, present a survey state art regarding g...

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (Lp(·)). Both existence, uniqueness, and stability sense Ulam–Hyers are established. Our results obtained using two fixed-point theorems, then illustrating with comprehensive example.

By applying the vector-valued inequalities for the Littlewood-Paley operators and their commutators on Lebesgue spaces with variable exponent, the boundedness of the Littlewood-Paley operators, including the Lusin area integrals, the Littlewood-Paley g-functions and g μ *-functions, and their commutators generated by BMO functions, is obtained on the Morrey spaces with variable exponent.

‎By considering a degenerate $p(x)-$Laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent Sobolev space is presented‎. ‎Multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding Nehari manifold‎.