53,760 research outputs found
Existence of the solution to a nonlocal-in-time evolutional problem
This work is devoted to the study of a nonlocal-in-time evolutional problem
for the first order differential equation in Banach space. Our primary
approach, although stems from the convenient technique based on the reduction
of a nonlocal problem to its classical initial value analogue, uses more
advanced analysis. That is a validation of the correctness in definition of the
general solution representation via the Dunford-Cauchy formula. Such approach
allows us to reduce the given existence problem to the problem of locating
zeros of a certain entire function. It results in the necessary and sufficient
conditions for the existence of a generalized (mild) solution to the given
nonlocal problem. Aside of that we also present new sufficient conditions which
in the majority of cases generalize existing results.Comment: This article is an extended translation of the part of Dmytro
Sytnyk's PhD Thesi
Uniformization and an Index Theorem for Elliptic Operators Associated with Diffeomorphisms of a Manifold
We consider the index problem for a wide class of nonlocal elliptic operators
on a smooth closed manifold, namely differential operators with shifts induced
by the action of an isometric diffeomorphism. The key to the solution is the
method of uniformization: We assign to the nonlocal problem a
pseudodifferential operator with the same index, acting in sections of an
infinite-dimensional vector bundle on a compact manifold. We then determine the
index in terms of topological invariants of the symbol, using the Atiyah-Singer
index theorem.Comment: 16 pages, no figure
Global existence for a nonlocal model for adhesive contact
In this paper we address the analytical investigation of a model for adhesive
contact, which includes nonlocal sources of damage on the contact surface, such
as the elongation. The resulting PDE system features various nonlinearities
rendering the unilateral contact conditions, the physical constraints on the
internal variables, as well as the integral contributions related to the
nonlocal forces. For the associated initial-boundary value problem we obtain a
global-in-time existence result by proving the existence of a local solution
via a suitable approximation procedure and then by extending the local solution
to a global one by a nonstandard prolongation argument
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