We prove an existence and multiplicity result for solutions of a nonlinear Urysohn type equation (2.14) by use of the Nielsen and degree theory in an annulus in the function space.

Let H²(bΩ) be the Hardy space of a bounded weakly pseudoconvex domain in ${ℂ}^n$. The natural resolution of this space, provided by the tangential Cauchy-Riemann complex, is used to show that H²(bΩ) has the important localization property known as Bishop’s property (β). The paper is accompanied by some applications, previously known only for Bergman spaces.

The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can...

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...