In this paper, we study the existence of standing waves of the coupled nonlinear Schrödinger equations. The proofs of which rely on the Lyapunov-Schmidt methods and contraction mapping principle are due to F. Weinstein in [1].

Let $\mathcal{G}$ be an Abelian group and let $\rho:\mathcal{G} \times \mathcal{G} \rightarrow [0, \infty)$ be a metric on $\mathcal{G}$. Let $\varepsilon$ be a normed space. We prove that under some conditions if $f:\mathcal{G}\to\varepsilon$ is an odd function and $C_x:\mathcal{G}\to\varepsilon$ defined by $C_x(y):=2f(x+y)+2f(x-y)+12f(x)-$ $f(2x+y)-f(2x-y)$ is a cubic function for all $x\in \mathcal{G},$ then there exists a cubic function $C:\mathcal{G}\to\varepsilon$ such that $f-C$ is Lipschitz. Moreover, we investigate the stability of cubic functional equation $2f(x+y)+2f(x-y)+12f(x)-f(2x+y)$ $-f(2x-y)=0$ on Lipschitz spaces.

Let $\mathcal{L}=-\Delta+V$ be the Schrödinger operator on $\mathbb{R}^d$, where $\Delta$ is the Laplacian on $\mathbb{R}^{d}$ and $V\ne0$ is a nonnegative function satisfying the reverse Hölder's inequality. The authors prove that Riesz potential $\mathcal{J}_{\beta}$ and its commutator $[b,\mathcal{J}_{\beta}]$ associated with $\mathcal{L}$ map from $M_{\alpha,v}^{p,q}$ into $M_{\alpha,v}^{p_1,q_1}$.

In this paper, by making use of the Hadamard products, we obtain some subordination results for certain family of meromorphic functions defined by using a new linear operator.

Let $\mathbb{P}_n$ be the class of polynomials of degree at most $n$. Rather and Shah [15] proved that if $P\in \mathbb{P}_n$ and  $P(z)\neq 0$ in $|z| < 1$, then for every $R  > 0$ and 0 $\leq q < \infty, $ $$| B[P(Rz)]|_q \leq  \frac{| R^{n}B[z^n] +\lambda_0 |_{q}}{| 1+z^n|_q} | P(z)|_q,$$where $B$ is a $ B_{n}$-operator.
In this paper, we prove some generalization of this result which in particular yields some known polynomial inequalities as special. We also consider an operator $D_{\alpha}$ which maps a polynomial $P(z)$ into $D_{\alpha} P(z) := n P(z) + ( \alpha - z ) P' (z)$ and obtain extensions and generalizations of a number of well-known $L_{q}$ inequalities.

In this paper we give three characterizations of $VMO(\mathbb{R}^n)$ space, which are of John-Nirenberg type, Uchiyama-type and Miyachi-type, respectively.

In this paper,  we give some results on the  simultaneous proximinal subset and simultaneous Chebyshev in the uniformly convex Banach space. Also we give relation between fixed point theory and simultaneous proximity.

Della Vecchia et al. (see [2]) introduced a kind of modified Bernstein operators which can be used to approximate functions with singularities at endpoints on $[0,1].$ In the present paper, we obtain a kind of pointwise Stechkin-type inequalities for weighted approximation by the modified Bernstein operators.

We give a detailed description of nonconstant harmonic functions on the level 3 Sierpinski gasket. Then we extend the method on $\beta$-set with $1/3< \beta < 1/2$.

Let $P(z)$ be a polynomial of degree $n,$ having all its zeros in $|z|\leq 1.$ In this paper, we estimate $kth$ polar derivative of $P(z)$ on $|z|=1$ and thereby obtain compact generalizations of some known results which among other things yields a refinement of a result due to Paul Turán.