In this paper, we consider an operator $D_α$ which maps a polynomial $P(z)$ in to $D_αP(z):= np(z)+(α−z)P′(z)$, where $α ∈ \mathcal{C}$ and obtain some $L^{\gamma}$ inequalities for lucanary polynomials having zeros in $|z|≤k≤1$. Our results yields several generalizations and refinements of many known results and also provide an alternative proof of a result due to Dewan et al. [7], which is independent of Laguerre’s theorem.

In this paper we study shadowing property for sequences of mappings on compact metric spaces, i.e., nonautonomous discrete dynamical systems. We investigate the relations of various expansivity properties with shadowing and $h$-shadowing property.

Using the method of construction, with the help of inequalities, we research the Muntz rational approximation of two kinds of special function classes, and give the corresponding estimates of approximation rates of these classes under wide conditions. Because the Orlicz Spaces is bigger than continuous function space and the Lp space, the results of this paper has a certain expansion significance.

We give an existence result of the obstacle parabolic equations $$\frac{\partial b(x,u)}{\partial t} -div(a(x,t,u,\nabla u))+div(\phi(x,t,u)) =f\quad \text{in}\ \ Q_T,$$ where $b(x,u)$ is bounded function of $u$, the term $- {\rm div}(a(x,t,u,\nabla u))$ is a Leray-Lions type operator and the function $\phi$ is a nonlinear lower order and satisfy only the growth condition. The second term $f$  belongs to $L^{1}(Q_T)$. The proof of an existence solution is based on the penalization methods.

In this work, we will prove the existence of bounded solutions in $W_{0}^{1,p}(\Omega) \cap L^{\infty}(\Omega)$ for nonlinear elliptic equations $-\mbox{ div}(a(x,u,\nabla u)) + g(x,u,\nabla u) + H(x,\nabla u) = f$, where $a$, $g$ and $H$ are Carathéodory functions which satisfy some conditions, and the right hand side $f$ belongs to $W^{-1,q}(\Omega)$.

In this article, we study on the existence of solution for a singularities of a system of nonlinear fractional  differential equations (FDE). We construct a formal power series solution for our considering FDE and prove convergence of formal solutions under conditions. We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function.

Suppose that a continuous $2\pi$-periodic function $f$ on the real axis changes its monotonicity at points $y_i: -\pi\le y_{2s}< y_{2s-1}< \cdots< y_1<\pi,\ s\in\Bbb N$. In this paper, for each $n\ge N,$ a trigonometric polynomial $P_n$ of order $cn$ is found such that: $P_n$ has the same monotonicity as $f,$ everywhere except, perhaps, the small intervals$$(y_i-\pi/n,y_i+\pi/n)$$and$$\|f-P_n\|\le c(s)\omega_3(f,\pi/n),$$where $N$ is a constant depending only on $\min\limits_{i=1,\cdots,2s}\{y_i-y_{i+1}\},\ c,\ c(s)$ are constants depending only on $s,\ \omega_3(f,\cdot)$ is the modulus of smoothness of the $3$-rd order of the function $f,$ and $\|\cdot\|$ is the max-norm.