The asymptotic behavior of the solutions of the Cauchy problem generated by ϕ-accretive operators

Abstract

The purpose of this paper is to study the asymptotic behavior of the solutions of certain type of differential inclusions posed in Banach spaces. In particular, we obtain the abstract result on the asymptotic behavior of the solution of the boundary value problemwhere Ω is a bounded open domain in R n with smooth boundary ∂Ω, f (t, x) is a given L 1 -function on ]0, ∞[ × Ω, γ 1 and 1 p < ∞. ∆ p represents the p-Laplacian operator, ∂ ∂η is the associated Neumann boundary operator and β a maximal monotone graph in R × R with 0 ∈ β(0).  2005 Elsevier Inc. All rights reserved.

Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators

“…In fact, this is a corollary of a general theorem in [8] about this convergence for general so-called almostorbits v(t) of the nonexpansive semigroup Analyzing the proof of the main theorem from [8] we extract an explicit computation which, in particular, eventually translates any given rate of convergence for ( * ) into a rate of convergence of the integral solution of the Cauchy problem towards the unique zero z of A. In the case of f ∈ L 1 (0, ∞, X) such a rate of convergence on ( * ) amounts to knowing a rate of convergence of ( * * )…”

“…We compute the rate of metastability for the convergence of the solution of the abstract Cauchy problem generated by a uniformly accretive at zero operator to the unique zero of A via proof mining based on a result by the first author. Finally, we apply our results to a special class of Cauchy problems considered in [8]. This work is the first application of proof mining to the theory of partial differential equations.…”

“…The operator A is assumed to satisfy the range condition, to have a zero z ∈ D(A) i.e. 0 ∈ Az, and to satisfy the condition of being φ-accretive at zero in the sense of [8]. This condition, in particular, implies that z is uniquely determined.…”

Existence results and asymptotic behavior for nonlocal abstract Cauchy problems

“…(See [21].) Let X be a Banach space, let φ : X → [0, ∞) be a continuous function such that φ(0) = 0, φ(x) > 0 for x = 0 and which satisfies the following condition:…”

“…Now, having in mind the results concerning asymptotic behavior for φ-accretive at zero operators which appear in [21], it is not difficult to obtain Proposition 17. Consider problem (10) under the assumptions (A), (C) and (D) where A ⊆ X × X satisfies condition (15) for some z ∈ X.…”