The strong type and weak type estimates of parameterized Littlewood-Paley operators on the weighted Herz spaces $\dot{K}^{α,p}_q (ω_1 ,ω_2)$ are considered. The boundedness of the commutators generated by BMO functions and parameterized Littlewood-Paley operators are also obtained.

In this paper, we have studied the Lacunary type of polynomials and proved a result which generalizes as well as refines some well-known polynomial inequalities regarding the growth of polynomials not vanishing inside a circle. Further, the paper corrects the proofs of some already known results.

In this paper, we use the complex method to obtain all meromorphic solutions of the complex Zakharov-Kuznetsov modified equal width equation, then find the exact traveling wave solutions of the Zakharov-Kuznetsov modified equal width equation. At last, we give some computer simulations to illustrate our main results.

In the present paper a numerical method is developed to approximate the solution of two-dimensional Nonlinear Schrödinger equation in the presence of a singular potential. The method leads to generalized Lyapunov-Sylvester algebraic operators that are shown to be invertible using original topological and differential calculus issued methods. The numerical scheme is proved to be consistent, convergent and stable using the Lyapunov criterion, lax equivalence theorem and the properties of the generalized Lyapunov-Sylvester operators.

In this paper some novel integrals associated with the product of classical Hermite’s polynomials $$\int^{+∞}_{−∞} (x^2)^m exp(−x^2)\{H_r(x)\}^2dx,\ \int^∞_0 exp(−x^2)H_{2k} (x)H_{2s+1}(x)dx,$$ $$\int^∞_0 exp(−x^2 )H_{2k}(x)H_{2s}(x)dx  \ \text{and}\ \int^∞_0 exp(−x^2)H_{2k+1}(x)H_{2s+1}(x)dx,$$ are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of special functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite’s polynomials by suitable simplifications of arbitrary parameters.

We use the two mappings satisfying II-expansive conditions on complex valued metric spaces to construct the convergent sequences and prove that the unique limit of the sequences is the point of coincidence or common fixed point of the two mappings. Also, we discuss the uniqueness of points of coincidence or common fixed points and give the existence theorems of unique fixed points. The obtained results generalize and improve the corresponding conclusions in references.

In the present paper, a new difference matrix via difference operator $D$ is introduced. Let $x = (x_k)$ be a sequence of real numbers, then the difference operator $D$ is defined by $D(x)_n = ∑^n_{k=0} (−1)^k\Bigg(\begin{matrix}n \\ n−k \end{matrix}\Bigg)x_k$, where $n = 0,1,2,3,···$. Several interesting properties of the new operator $D$ are discussed.

Let $f$ be an $H$−periodic Hölder continuous function of two real variables. The error $||f −N_n(p;f)||$ is estimated in the uniform norm and in the Hölder norm, where $p=(p_k)^∞_{k=0}$ is a nonincreasing sequence of positive numbers and $N_n(p; f)$ is the $n\rm{th}$ Nörlund mean of hexagonal Fourier series of $f$ with respect to $p=(p_k)^∞_{k=0}$.

On a homogeneous hierarchical gasket we give a linear extension method to compute the energy measures of harmonic functions with respect to the standard energy.