orlicz function

This paper is devoted to studying the existence of a solution to the  Dirichlet problem for a specific class of elliptical anisotropic equations  of the type\begin{eqnarray}\label{P.1}\left \{\begin{array}{rl}&A(u)+g(x,u)= f  \ \  in\\Omega\\& u=0\ \ on\ \partial \Omega,\end{array}\right.\end{eqnarray}in the anisotropic Orlicz-Sobolev spaces, where A is a Leray-Lions operator  $A(u)=\displaystyle\sum_{i=1}^{N}-\frac{\partial}{\partial x_{i}} (a_{i}(x,D^{i} u)),$   the Carathéodory function $g(x, s )$ is a non-linear lower order term that verify some natural growth and sign conditions, where the data $f$ is framed in anisotropic Orlicz-Sobolev spaces, and it is described by an Orlicz function that does not meet the $\Delta_2$-condition. Within this framework, we prove the existence of a weak solution for our strongly nonlinear elliptic problem.

In recent years, a variety of work has been done in the field of single, double and triple sequences. Study on n-tuple sequence is new in this field. The main interest of this paper is to explore the idea of n-tuple sequences x = (xi_1,i_2,...,i_n) in metric spaces. We introduce the concept of n-sequence space of interval number and discussed its arithmetic properties. Furthermore, we combined the concept of Orlicz function, statistical convergence, interval number and n-sequence to construct some new nsequence spaces and discussed their properties. Some suitable examples for these spaces have been constructed.

In this article we have introduced the sequence space $m(\phi,d)$ and $m(M,\phi,d)$ of W. L. C. Sargent type in a metric space $(X, d)$ on generalising the sequence space $m(\phi)$ and we have defined these sequence spaces using the Orlicz function $M$. We have investigated their different properties like solidness, symmetricity, monotone, sequence algebra, completeness etc. We have established some inclusion results involving the space $m(M,\phi,d)$ and some of the existing sequence spaces. We have provided suitable examples and discussed in detail, in order to justify the failure cases and the definitions we have introduced. The results established in this article generalized and unifies several existing results.

In this article, we introduce a new class of ideal convergent sequence spaces using an infinite matrix, Musielak-Orlicz function and a new generalized difference matrix in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We also give some relations related to these sequence spaces.

In this paper we introduce strongly $left[V_{2},lambda_{2},M,pright] -$summable double vsequence spaces via Orlicz function and examine some properties of the resulting these spaces. Also we give natural relationship between these spaces and $S_{lambda_{2}}-$statistical convergence.