With the idea taken from the King type operators which preserve some test functions, we introduce here some Durrmeyer variants of Bernstein operators based on Beta functions. Some direct approximation theorems are provided of this introduced sequence of operators. We also proved Voronovkaja type theorem. Furthermore, graphical and numerical examples are also given with the help of MATLAB.

The aim of this paper is to study the model problem:

We give a Bézier variant of Baskakov-Durrmeyer-type hybrid operators in the present article. First, we obtain the rate of convergence by using Ditzian-Totik modulus of smoothness and also for a class of Lipschitz function. Then, weighted modulus of continuity is investigated too. We study the rate of point-wise convergence for the functions having a derivative of bounded variation. Furthermore, we establish the quantitative Voronovskaja-type formula in terms of Ditzian-Totik modulus of smoothness at the end.

We prove an existence result for solutions to a class of unilateral problems for the nonlinear elliptic equation whose prototype is -div(vertical bar del u vertical bar(p-2) del u) + b(x)vertical bar del u vertical bar(lambda) = f - divF in Omega, where Omega is a bounded open set of R-N, N >= 2, 1 < p < N, 0 <= lambda <= p -1, b(x) belongs to the Lorentz space L-N,L-1 (Omega), f is an element of L-1(Omega) and F is an element of (L-p' (Omega))(N), p' = p/(p - 1).

Abstract This paper deals with the existence and regularity of some unilateral problem associated to a nonlinear equation of type - div ⁡ ( a ⁢ ( x , u , ∇ ⁡ u ) ) + H ⁢ ( x , u , ∇ ⁡ u ) = f {-\operatorname{div}(a(x,u,\nabla u))+H(x,u,\nabla u)=f} .

We prove the existence and the regularity of minima for a functional defined on a suitable Sobolev space.

In this paper, we show, in the framework of Orlicz Sobolev spaces, the existence of renormalized and entropy solutions of some nonlinear equations and the existence and uniqueness of a unilateral problem. The existence of weak and renormalized solution of a nonlinear equation is also presented in the framework of weighted Sobolev spaces.

In this paper, we prove the existence and uniqueness of solutions of unilateral problems of the type −diva(x,∇u)=f in the framework of Orlicz spaces. The main contribution of our work is proving the existence of a solution without any restriction on the N-function M of the Orlicz spaces.

An existence result is proved for a variational degenerated unilateral problems associated to the following equationsAu + y(x, u, del u) = f,where A is a Leray-Lions operator acting from the weighted Sobolev space W(0)(1,p)(Omega, w) into its dual W(-1,p')(Omega, w*), while g(x, s, xi) is a nonlinear term winch has a growth condition with respect to xi and a sign condition on s, i.e. g(x, s, xi).s >= 0 for every s is an element of R and for every x and xi in their respective domains. The source term f is supposed to belong to W(-1,p')(Omega, w*) .

In this paper, we study the problem:—div a(x, u, ∆u) — div Φ(u) + g(x; u) = f in the framework of Orlicz spaces. The main contribution of our work is to prove the existence of a renormalized solution without any restriction on the N-function of the Orlicz space.

We prove the existence results in the setting of Orlicz spaces for the unilateral problem associated to the following equation, Au+g(x,u,∇u)=f, where A is a Leray–Lions operator acting from its domain D(A)⊂W01LM(Ω) into its dual, while g(x,u,∇u) is a nonlinear term having a growth conditions with respect to ∇u and no growth with respect to u, but does not satisfy any sign condition. The right-hand side f belongs to L1(Ω), and the obstacle is a measurable function.

We prove the existence of a solution for a strongly nonlinear degenerated problem associated to the equation Au + g(x,u,∇u) = f, where A is a Leray–Lions operator from the weighted Sobolev space W_0^{1,p}(Ω, w) into its dual W^{ −1,p'}(Ω, w^*) . While g(x,s,ξ) is a nonlinear term having natural growth with respect to ξ and no growth with respect to s , it satisfies a sign condition on s , i.e., g(x,s,ξ) · s ≥ 0 for every s∈ℝ . The right-hand side f belongs to L^1(Ω) .

This article is concerned with the existence result of the unilateral problem associated to the equations of the typeAu - div phi(u) = f is an element of L-1(Omega),where A is a Leray-Lions operator having a growth not necessarily of polynomial type and phi is an element of C-0(R, R-N).

An existence result for the strongly nonlinear unilateral problems associated to the equation, div(a(x, u, ∇u))+ g(x, u, ∇u) = f2 L 1 (Ω), is proved without any regularity condition on the obstacle.

In this paper, we prove the existence of solutions to unilateral problems involving nonlinear operators of the form Au + H(x, u, del u) = f where A is a Leray Lions operator from W-0(1,p)(Omega) into its dual W--1,W-p'(Omega) and H(x, u, del u) is a nonlinearity which satisfies the following growth condition \H(x, s, xi)\ <= gamma (x)+g(s)\xi\(p) with gamma is an element of L-1(Omega) and g is an element of L-1(R), and without assuming any sign condition on H(x, s, xi). The right hand side f belongs to L-1(Omega).

SUMMARY

This paper is concerned with the existence and uniqueness result of a solution for some degenerated bilateral problem by using the penalization methods

We will be concerned with the existence result of unilateral problem associated to the equations of the form A u + g ( x , u , ∇ u ) = f , where A is a Leray‐Lions operator from its domain into . On the nonlinear lower order term g ( x , u , ∇ u ), we assume that it is a Carathéodory function having natural growth with respect to |∇ u |, and satisfies the sign condition. The right‐hand side f belongs to .

In this paper, we study the existence of weak solutions for the initial-boundary value problems of the nonlinear degenerated parabolic equa- tion