Abstract Cauchy Problem Research Articles

Multiplicative operator functions and abstract Cauchy problems

Multiplicative operator functions and abstract Cauchy problems

Local k-convoluted c-semigroups and complete second order abstract Cauchy problems

Let C : X ? X be a bounded linear operator on a Banach space X over the field F(=R or C), and K : [0,T0)?F a locally integrable function for some 0 < T0 ? ?. Under some suitable assumptions, we deduce some relationship between the generation of a local (or an exponentially bounded) K-convoluted (C 0 0 C)-semigroup on X x X with subgenerator (resp., the generator) (0 I B A) and one of the following cases: (i) the well-posedness of a complete second-order abstract Cauchy problem ACP(A,B,f,x,y): w??(t) = Aw?(t) + Bw(t) + f (t) for a.e. t ?(0,T0) with w(0) = x and w?(0) = y; (ii) a Miyadera-Feller-Phillips-Hille- Yosida type condition; (iii) B is a subgenerator (resp., the generator) of a locally Lipschitz continuous local ?-times integrated C-cosine function on X for which A may not be bounded; (iv) A is a subgenerator (resp., the generator) of a local ?-times integrated C-semigroup on X for which B may not be bounded.

An abstract nonlocal functional-differential second order evolution Cauchy problem

An abstract nonlocal functional-differential second order evolution Cauchy problem

Rank two solutions of the abstract Cauchy problem

In this paper we discuss solutions of the Abstract Cauchy Problem Bu(α) (t) = Au(t)+ f (t) u(0) = x0 that are of the form u = u1⊗δ1+u2⊗δ2 , where u and f are functions defined on [0,a] with values in the Hilbert space l2.

Spaces of Smooth and Generalized Vectors of the Generator of an Analytic Semigroup and Their Applications

For a strongly continuous analytic semigroup {e tA }t ≥ 0 of linear operators in a Banach space B, we study some locally convex spaces of smooth and generalized vectors of its generator A, as well as the extensions and restrictions of this semigroup to these spaces. We extend the Lagrange result on the representation of a translation group in the form of exponential series to the case of these semigroups and solve the Hille problem on the description of the set of all vectors x∈𝔅 for which the limit $$ {\lim}_{n\to \infty }{\left(I+\frac{tA}{n}\right)}^nx $$ exists and coincides with e tA x. Moreover, we present a brief survey of particular problems whose solutions are required for the introduction of the above-mentioned spaces, namely, the description of all maximal dissipative (self-adjoint) extensions of a dissipative (symmetric) operator, the representation of solutions of operator-differential equations on an open interval and the investigation of their boundary values, and the existence of solutions of an abstract Cauchy problem in various classes of analytic vector-valued functions.

The Solution Of Nonhomogen Abstract Cauchy Problem by Semigroup Theory of Linear Operator

In this article we will investigate how to solve nonhomogen degenerate Cauchy problem via theory of semigroup of linear operator. The problem is formulated in Hilbert space which can be written as direct sum of subset Ker M and Ran M*. By certain assumptions the problem can be reduced to nondegenerate Cauchy problem. And then by composition between invers of operator M and the nondegenerate problem we can transform it to canonic problem, which is easier to solve than the original problem. By taking assumption that the operator A is infinitesimal generator of semigroup, the canonic problem has a unique solution. This allow to define special operator which map the solution of canonic problem to original problem. ©2016 JNSMR UIN Walisongo. All rights reserved.

A characterization of well-posedness for abstract Cauchy problems with finite delay

Let A be a closed operator defined on a Banach space X and F be a bounded operator defined on a appropriate phase space. In this paper, we characterize the well-posedness of the first order abstract Cauchy problem with finite delay,{u′(t)=Au(t)+Fut,t>0;u(0)=x;u(t)=ϕ(t),−r≤t<0, solely in terms of a strongly continuous one-parameter family {G(t)}t≥0 of bounded linear operators that satisfy the functional equationG(t+s)x=G(t)G(s)x+∫−r0G(t+m)[SG(s+⋅)x](m)dm for all t,s≥0,x∈X. In case F≡0 this property reduces to the characterization of well-posedness for the first order abstract Cauchy problem in terms of the functional equation that satisfy the C0-semigroup generated by A.

Abstract Cauchy problems for quasilinear operators whose domains are not necessarily dense or constant

The solvability of the abstract Cauchy problem for the quasilinear evolution equation u′(t)=A(u(t))u(t) for t>0 and u(0)=u0∈D is discussed. Here {A(w);w∈Y} is a family of closed linear operators in a real Banach space X such that Y⊂D(A(w))⊂Y¯ for w∈Y, Y is another Banach space which is continuously embedded in X, and D is a closed subset of Y. The existence and uniqueness of C1 solutions to the Cauchy problem is proved without assuming that Y is dense in X or D(A(w)) is independent of w. The abstract result is applied to obtain an L1-valued C1-solution to a size-structured population model.

Abstract Cauchy problem for the Bessel–Struve equation

We consider the Cauchy problem for the Bessel–Struve equation in a Banach space. A sufficient condition for the solvability of this problem is proved, and the solution operator is written in explicit form via the Bessel and Struve operator functions. A number of properties is established for the solutions.

Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions

This paper deals with exact controllability of a class of abstract nonlocal Cauchy problem with impulsive conditions in Banach spaces. By using Sadovskii fixed point theorem and Mönch fixed point theorem, exact controllability results are obtained without assuming the compactness and Lipschitz conditions for nonlocal functions. An example is given to illustrate the main results.

Local K-convoluted C-cosine functions and abstract cauchy problems

Let K : [0,T0)? F be a locally integrable function, and C : X ? X a bounded linear operator on a Banach space X over the field F(=R or C). In this paper, we will deduce some basic properties of a nondegenerate local K-convoluted C-cosine function on X and some generation theorems of local Kconvoluted C-cosine functions on X with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-cosine function on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem: U''(t)=Au(t)+f(t) for a.e. t ? (0, T0), u(0) = x, u'(0) = y when K is a kernel on [0, T0), C : X ? X an injection, and A : D(A) ? X ? X a closed linear operator in X such that CA ? AC. Here 0 &lt; T0 ? ?, x,y ? X, and f ? L1,loc([0,T0),X).

Fractional abstract Cauchy problem with order $\alpha \in (1,2)$

Fractional abstract Cauchy problem with order $\alpha \in (1,2)$

Discrete maximal regularity for abstract Cauchy problems

Maximal regularity is a fundamental concept in the theory of nonlinear partial differential equations, for example, quasilinear parabolic equations, and the NavierStokes equations. It is thus natural to ask whether the discrete analogue of this notion holds when the equation is discretized for numerical computation. In this paper, we introduce the notion of discrete maximal regularity for the finite difference method (θ-method), and show that discrete maximal regularity is roughly equivalent to (continuous) maximal regularity for bounded operators. In addition, we show that this characterization is also true for unbounded operators in the case of the backward Euler method.

Optimal Stable Approximation for the Cauchy Problem for Laplace Equation

Cauchy problem for Laplace equation in a strip is considered. The optimal error bounds between the exact solution and its regularized approximation are given, which depend on the noise level either in a Hölder continuous way or in a logarithmic continuous way. We also provide two special regularization methods, that is, the generalized Tikhonov regularization and the generalized singular value decomposition, which realize the optimal error bounds.

Abstract Cauchy problem for weakly continuous operators

We discuss the solvability of the abstract Cauchy problem u′(t)=Au(t) for t>0 and u(0)=u0∈D, where D is a weakly closed subset of a Banach space Y continuously embedded in the underlying Banach space X and A is a nonlinear operator from D into X satisfying a local weak continuity condition described by a family of functionals. We introduce a new type of weak compactness on the level sets of functionals and give a necessary and sufficient condition for the existence of a weakly continuously differentiable solution satisfying a growth condition. We apply the abstract result obtained to the Cauchy problem for a nonlinear Schrödinger equation and a mixed problem for a logarithmic wave equation.

Regularity of mild Solutions for fractional abstract Cauchy problem with order $${\alpha \in (1, 2)}$$ α ∈ ( 1 , 2 )

In this paper, we deal with the regularity of mild Solutions for fractional abstract Cauchy problem with order \({\alpha \in (1, 2)}\). Based on properties of solution operator and analytic solution operator, we obtain the sufficient condition under which a mild solution becomes a classical solution, and if the Cauchy problem has an analytic solution operator, we obtain more stronger results.

Well-posedness for coupled bulk-interface diffusion with mixed boundary conditions

Abstract In this paper, we consider a quasilinear parabolic system of equations describing coupled bulk and interface diffusion, including mixed boundary conditions. The setting naturally includes non-smooth domains Ω. We show local well-posedness using maximal Ls -regularity in dual Sobolev spaces of type W - 1 , q ( Ω ) $W^{-1,q}(\Omega )$ for the associated abstract Cauchy problem.

An Application Of The Theory Of Scale Of Banach Spaces

Abstract The abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

Space ofω-Periodic Limit Functions and Its Applications to an Abstract Cauchy Problem

We introduce a new space consisting of what we callω-periodic limit functions. We investigate some properties of the new function space. In particular, we study inclusion relations among asymptotically periodic type function spaces. Finally, we apply theω-periodic limit functions to investigate the existence and uniqueness of asymptoticallyω-periodic mild solutions of an abstract Cauchy problem.

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

We extract rates of convergence and rates of metastability (in the sense of Tao) for convergence results regarding abstract Cauchy problems generated by ϕ-accretive at zero operators A:D(A)(⊆X)→2X where X is a real Banach space, proved in [8], by proof-theoretic analysis of the proofs in [8] and having assumed a new, stronger notion of uniform accretivity at zero, yielding a new notion of modulus of accretivity. 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.