The purpose of this paper is to prove the existence of a renormalized solution of perturbed elliptic problems$ -operatorname{div}Big(a(x,u,nabla u)+Phi(u) Big)+ g(x,u,nabla u) = mumbox{ in }Omega,  $ in the framework of Orlicz-Sobolev spaces without any restriction on the $M$ N-function of the Orlicz spaces, where $-operatorname{div}Big(a(x,u,nabla u)Big)$ is a Leray-Lions operator defined f...

Suppose that for an arbitrary function $f(x,y)$ of two discrete variables, we have the formal expansions. [f(x,y)=sumlimits_{m,n=0}^{infty }a_{m,n},P_{m}(x)P_{n}(y),] $$‎ ‎x^{m}P_{j}(x)=sumlimits_{n=0}^{2m}a_{m,,n}(j)P_{j+m-n}(x)‎,$$ ‎we find the coefficients $b_{i,j}^{(p,q,ell‎ ,‎,r)}$ in the expansion‎ $$‎ ‎x^{ell }y^{r},nabla _{x}^{p}nabla _{y}^{q},f(x,y)=x^{ell‎ ‎}y^{r}f^{(p,q)}(x,y) =sumli...

and Applied Analysis 3 Definition 2.2. Let ρ t t − 1 be the backward jump operator. Then i the nabla left fractional sum of order α > 0 starting from a is defined by ∇−α a f t 1 Γ α t ∑ s a 1 ( t − ρ s α−1f s , t ∈ Na 1 2.4 ii the nabla right fractional sum of order α > 0 ending at b is defined by b∇−αf t 1 Γ α b−1 ∑ s t ( s − ρ t α−1f s 1 Γ α b−1 ∑ s t σ s − t α−1f s , t ∈b−1 N. 2.5 We want to...

Recently, new developments of the theory and applications of dynamic derivatives on time scales were made. The study provides an unification and an extension of traditional differential and difference equations and, in the same time, it is a unification of the discrete theory with the continuous theory, from the scientific point of view. Moreover, it is a crucial tool in many computational and ...

where F,G ∈ M(n × n;F), Yk ∈ M(n × 1;F) (i.e., the algebra of square matrices with elements in the field F), Xk ∈ M(m× 1;F), B ∈ M(n× l;F) and C ∈ M(m×n;F). For the sake of simplicity, we set Mn = M(n× n;F) and Mnm = M(n×m;F). We assume that the system (1) is singular, i.e. the matrix F is singular and that the input vector Vk changes only at equally space sampling instants. Many authors have s...

We present the concept of a disjunctive basis as a generic framework for normal forms in modal logic based on coalgebra. Disjunctive bases were defined in previous work on completeness for modal fixpoint logics, where they played a central role in the proof of a generic completeness theorem for coalgebraic mu-calculi. Believing the concept has a much wider significance, here we investigate it m...

In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local conditions. First, derive associated Green’s function and some of its properties. Using Guo–Krasnoselkii fixed point theorem on suitable cone under appropriate conditions non-linear part equation, establish sufficient requirements least one two positive solutions ...