In this paper, we are interested in the dynamic evolution of an elastic body, acted by resistance forces depending also on the displacements. We put the mechanical problem into an abstract functional framework, involving a second order nonlinear evolution equation with initial conditions. After specifying convenient hypotheses on the data, we prove an existence and uniqueness result. The proof is based on Faedo-Galerkin method.

Let A be a closed linear operator acting in a separable Hilbert space. Denote by co(A) the closed convex hull of the spectrum of A. An estimate for the norm of f(A) is obtained under the following conditions: f is a holomorphic function in a neighbourhood of co(A), and for some integer p the operator $A^p-{(A*)}^p$ is Hilbert-Schmidt. The estimate improves one by I. Gelfand and G. Shilov.

Let $1\le q<p<\infty$ and $1/r:=1/pmax(q/2,1)$. We prove that ${ℒ}_{r,p}^{\left(c\right)}$, the ideal of operators of Geľfand type $l_{r,p}$, is contained in the ideal ${\Pi }_{p,q}$ of $(p,q)$-absolutely summing operators. For $q>2$ this generalizes a result of G. Bennett given for operators on a Hilbert space.

In this paper, a nonlinear backward heat problem with time-dependent coefficient in the unbounded domain is investigated. A modified regularization method is established to solve it. New error estimates for the regularized solution are given under some assumptions on the exact solution.

For a bounded and sectorial linear operator V in a Banach space, with spectrum in the open unit disc, we study the operator $Ṽ={\int }_0^{\infty }d\alpha V^{\alpha }$. We show, for example, that Ṽ is sectorial, and asymptotically of type 0. If V has single-point spectrum 0, then Ṽ is of type 0 with a single-point spectrum, and the operator I-Ṽ satisfies the Ritt resolvent condition. These results generalize an example of Lyubich, who studied the case where V is a classical Volterra operator.

The properties of a transformation $$f\mapsto {\tilde{f}}_h$$ by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that $$\left({\tilde{f}}_h\right)\widetilde{{}_k}={\tilde{f}}_{h+k}$$ for certain complex h and k, and that $$f\left(t\right)=\underset{h\to 0^{+}}{lim}{\tilde{f}}_h\left(t\right)$$ , where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous...

Let H(B) denote the space of all holomorphic functions on the unit ball B of ℂⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. We study the integral-type operator $C_{\phi }^{g}f\left(z\right)={\int }_0^1\Re f\left(\phi \left(tz\right)\right)g\left(tz\right)dt/t$, f ∈ H(B). The boundedness and compactness of $C_{\phi }^g$ from Privalov spaces to Bloch-type spaces and little Bloch-type spaces are studied