We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions of work are enumerated as follows: (i) develop an approximate inversion technique for operators, based on which we analyze differential equations; (ii) prove that sensitive dependence well-posedness classical Riemann-Liouville initial value could be resolved by adjus...

In this article, we study the steady generalized Navier-Stokes equations in a half-space in the setting of variable exponent spaces. We first establish variable exponent spaces of Clifford-valued functions in a half-space. Then, using this operator theory together with the contraction mapping principle, we obtain the existence and uniqueness of solutions to the stationary Navier-Stokes equation...

In this paper we define a Grushin-type operator with a variable exponent growth and establish existence results for an equation involving such an operator. A suitable function space setting is introduced. Regarding the tools used in proving the existence of solutions for the equation analyzed here they rely on the critical point theory combined with adequate variational techniques. 2010 Mathema...

We prove both local and global Poincaré inequalities with the variable exponent for differential forms in the John domains and s L -averaging domains, which can be considered as generalizations of the existing versions of Poincaré inequalities.

We study unilateral series in a single variable q where its exponent is an unbounded increasing function, and the coefficients are periodic. Such series converge inside the unit disk. Quadratic polynomials in the exponent correspond to partial theta series. We compute limits of those series as the variable tends radially to a root of unity. The proofs use ideas from the q-integral and are eleme...

In this paper, we have discussed about the inclusion properties of homogeneous Herz-Morrey spaces with variable exponent and weak exponent. We also studied relation between those spaces.