In this work we prove improved converse theorems of trigonometric approximation in variable exponent Lebesgue spaces with some Muckenhoupt weights.

Please click on PDF to view the abstract.

In the present work, we investigate an interval of real parameters $ \lambda $ for which the problem admits at least one nontrivial solution. Moreover we deal with the existence results of three solutions for anisotropic problems with variable exponents.

1. Introduction: There is a long-standing conjecture attributed to I. Schur that if G is a , nite group with Schur multiplier M(G) then the exponent of M(G) divides the exponent of G. It is easy to show that this is true for groups G of exponent 2 or exponent 3, but it has been known since 1974 that the conjecture fails for exponent 4. Bayes, Kautsky and Wamsley [1] give an example of a group G of order 2 with exponent 4, where M(G) has exponent 8. (Bayes, Kautsky and Wamsley are heros of the early days of computing with , nite p-groups. ) However the truth or otherwise of this conjecture has remained open up till now for groups of odd exponent, and in particular it has remained open for groups of exponent 5 and exponent 9. For a survey article on Schur's conjecture see Thomas [6]...

In this work, we study the existence of an initial boundary problem of a quasilinear parabolic problem with variable exponent  and  $ L ^{1} $-data of the type\begin{equation*}\left\{\begin{array}{ll}(b(u))_{t}-\text{div}(\left\vert \nabla u\right\vert ^{p(x)-2}\nabla u)+\lambda\left\vert u\right\vert ^{p(x)-2}u=f(x,t,u) \text{ } &\text{in}\hspace{0.5cm}Q=\Omega \times ]0,T[, \\u=0 & \text{on}\hspace{0.5cm}\Sigma =\partial \Omega \times ]0,T[, \\b(u)(t=0)=b(u_{0}) & \text{in}\hspace{0.5cm}\Omega , \end{array}\right.\end{equation*}where $ \lambda>0$ and $ T $ is positive constant. The main contribution of our work is to prove the existence of a renormalized solution. The functional setting involves Lebesgue– Sobolev spaces with variable exponents.

Some functional inequalities in variable exponent Lebesgue spaces are presented. The bi-weighted modular inequality with variable exponent p(: ) for the Hardy operator restricted to non-increasing function which is....

The present study examined the behavior of solutions for a viscoelastic double-Kirchhoff-type wave equation with nonlocal degenerate damping term and variable exponent nonlinearities. Under appropriate conditions for the data and exponents, we prove the global existence, asymptotic stability, and blow up of solutions with arbitrary initial energy.