In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on $[0,1]$ are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.

In this work, using an analogue of Sadovskii's fixed point result and several important inequalities we investigate and give new existence theorems for the nonlinear operator equation $F(x)=\mu x$, $(\mu \geq 1)$ for some weakly sequentially continuous, weakly condensing and weakly $1$-set weakly contractive operators with different boundary conditions. Correspondingly, we can obtain some applicable fixed point theorems of Leray-Schauder, Altman and Furi-Pera types in the weak topology setting which generalize and improve the corresponding results of [3,15,16].

Let $w$ be a Muckenhoupt weight and $H^p_w(\mathbb R^n)$ be the weighted Hardy space. In this paper, by using the atomic decomposition of $H^p_w(\mathbb R^n)$, we will show that the Bochner-Riesz operators $T^\delta_R$ are bounded from $H^p_w(\mathbb R^n)$ to the weighted weak Hardy spaces $WH^p_w(\mathbb R^n)$ for $0 < p < 1$ and $\delta=n/p-(n+1)/2$. This result is new even in the unweighted case.

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form$$ \left\{\begin{array}{ll} \dfrac{\partial b_1(x,u_1)}{\partial t}- \mathop{div}\big(a(x,t,u_1,Du_1)\big)+\mathop{div}\big(\Phi_1(u_1)\big)+ f_1(x,u_1,u_2)= 0  &  \quad\text{in}\ \  Q, \\\dfrac{\partial b_2(x,u_2)}{\partial t}- \mathop{div}\big(a(x,t,u_2,Du_2)\big)+\mathop{div}\big(\Phi_2(u_2)\big)+ f_2(x,u_1,u_2)= 0  & \quad\text{in}\ \  Q, \\\end{array}\right.$$in the framework of weighted Sobolev spaces, where $b(x,u)$ is unbounded function on $u$, the Carathéodory function $a_i$ satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function $\phi_i$ is assumed to be continuous on $\mathbb{R}$ and not belong to $(L^1_{loc}(Q))^N$.

In this paper, we consider a countable family of surjective mappings $\{T_n\}_{n \in\mathbb{N}}$ satisfying certain quasi-contractive conditions. We also construct a convergent sequence $\{x_n\}_{n \in \mathbb{N}}$ by the quasi-contractive conditions of $\{T_n\}_{n \in\mathbb{N}}$ and the boundary condition of a given complete and closed subset of a cone metric space $X$ with convex structure, and then prove that the unique limit $x^{*}$ of $\{x_n\}_{n \in \mathbb{N}}$  is the unique common fixed point of $\{T_n\}_{n \in \mathbb{N}}$. Finally, we will give more generalized common fixed point theorem for mappings $\{T_{i,j}\}_{i,j \in \mathbb{N}}$. The main theorems in this paper generalize and improve many known common fixed point theorems for a finite or countable family of mappings with quasi-contractive conditions.

In this paper, we establish two integral inequalities for Hardy operator's conjugate operator at the endpoint on $n$-dimensional space. The operator $H^*_n$ is bounded from $L^1_{x^\alpha}(\mathbb{G}^n)$ to $L^q_{x^\beta}(\mathbb{G}^n)$ with the bound explicitly worked out and the similar result holds for $\mathcal{H}^\ast_n$.

In this note, we establish a companion result to the theorem of J. Szabados on the maximum of fundamental functions of Lagrange interpolation based on Chebyshev nodes.

Let $C$ be a closed convex weakly Cauchy subset of a normed space $X$. Then we define a new $\{a,b,c\}$ type nonexpansive and $\{a,b,c\}$ type contraction mapping $T$ from $C$ into $C$. These types of mappings will be denoted respectively by $\{a,b,c\}$-$n$type and $\{a,b,c\}$-$c$type. We proved the following:
1. If $T$ is $\{a,b,c\}$-$n$type mapping, then $\inf\{\|T(x)-x\|:x\in C\}=0$, accordingly $T$ has a unique fixed point. Moreover, any sequence $\{x_{n}\}_{n\in \mathcal{N}}$ in $C$ with $\lim_{n\to \infty}\|T(x_{n})-x_{n}\|=0$ has a subsequence strongly convergent to the unique fixed point of $T$.
2. If $T$ is $\{a,b,c\}$-$c$type mapping, then $T$ has a unique fixed point. Moreover, for any $x\in C$ the sequence of iterates $\{T^{n}(x)\}_{n\in \mathcal{N}}$ has subsequence strongly convergent to the unique fixed point of $T$.
This paper extends and generalizes some of the results given in [2,4,7] and [13].

This paper is concerning the commutators generated by the multilinear singular integral with rough kernels and BMO functions. The boundedness of the multilinear commutators $T_{\vec{b}}(\vec{f})$ is established on the Morrey-Herz space by using the John-Nirenberg inequality.