In this paper, we present some theorems on weighted approximation by two dimensional nonlinear singular integral operators in the following form: T λ ( f ; x , y ) = ∬ R 2 ( t − x , s − y , f ( t , s ) ) d s d t , ( x , y ) ∈ R 2 , λ ∈ Λ , $$T_{\lambda }(f;x,y)=\underset{{ℝ}^2}{\int \int }(t-x,s-y,f(t,s))dsdt,(x,y)\in {ℝ}^2,\lambda \in \Lambda ,$$ where Λ is a set of non-negative numbers with accumulation point λ0.

Applying a global bifurcation theorem for convex-valued completely continuous mappings we prove some existence theorems for convex-valued differential inclusions of the form x'∈ F(t,x), where x satisfies the Nicoletti boundary conditions.

A new characteristic property of the Mittag-Leffler function $E_{\alpha }\left(a{t}^{\alpha }\right)$ with 1 < α < 2 is deduced. Motivated by this property, a new notion, named α-order cosine function, is developed. It is proved that an α-order cosine function is associated with a solution operator of an α-order abstract Cauchy problem. Consequently, an α-order abstract Cauchy problem is well-posed if and only if its coefficient operator generates a unique α-order cosine function.

Let $\mathscr{H}$ be a separable infinite dimensional complex Hilbert space, and let $ℒ\left(\mathscr{H}\right)$ denote the algebra of all bounded linear operators on $\mathscr{H}$ into itself. Let $A=(A_1,A_2,\cdots ,A_n)$, $B=(B_1,B_2,\cdots ,B_n)$ be $n$-tuples of operators in $ℒ\left(\mathscr{H}\right)$; we define the elementary operators ${\Delta }_{A,B}ℒ\left(\mathscr{H}\right)\mapsto ℒ\left(\mathscr{H}\right)$ by ${\Delta }_{A,B}\left(X\right)={\sum }_{i=1}^n{A}_{i}X{B}_i-X.$ In this paper, we characterize the class of pairs of operators $A,B\in ℒ\left(\mathscr{H}\right)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A,B\in ℒ\left(\mathscr{H}\right)$ such that ${\sum }_{i=1}^n{B}_{i}T{A}_i=T$ implies ${\sum }_{i=1}^n{A}_i^{*}T{B}_i^{*}=T$ for all $T\in {𝒞}_1\left(\mathscr{H}\right)$ (trace class operators). The main result is the equivalence between this property and the fact that...

In this paper a new class of self-mappings on metric spaces, which satisfy the nonexpensive type condition (3) below is introduced and investigated. The main result is that such mappings have a unique fixed point. Also, a remetrization theorem, which is converse to Banach contraction principle is given.