scholarly journals On the Behaviour of Singular Semigroups in Intermediate and Interpolation Spaces and Its Applications to Maximal Regularity for Degenerate Integro-Differential Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-37 ◽  
Author(s):  
Alberto Favaron ◽  
Angelo Favini
Keyword(s):  
Differential Evolution ◽  
Banach Spaces ◽  
Wide Class ◽  
Evolution Equations ◽  
Maximal Regularity ◽  
Linear Operators ◽  
Interpolation Spaces ◽  
Power Type ◽  
Maximal Time

For those semigroups, which may have power type singularities and whose generators are abstract multivalued linear operators, we characterize the behaviour with respect to a certain set of intermediate and interpolation spaces. The obtained results are then applied to provide maximal time regularity for the solutions to a wide class of degenerate integro- and non-integro-differential evolution equations in Banach spaces.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

scholarly journals C0-semigroups of linear operators on some ultrametric Banach spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Toka Diagana
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Crucial Role ◽  
Evolution Equations ◽  
Linear Operators ◽  
Ground Field ◽  
Semigroups Of Linear Operators ◽  
Classical Setting

C0-semigroups of linear operators play a crucial role in the solvability of evolution equations in the classical context. This paper is concerned with a brief conceptualization ofC0-semigroups on (ultrametric) free Banach spacesE. In contrast with the classical setting, the parameter of a givenC0-semigroup belongs to a clopen ballΩrof the ground fieldK. As an illustration, we will discuss the solvability of some homogeneousp-adic differential equations.

scholarly journals On Regularized Quasi-Semigroups and Evolution Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-13
Author(s):  
M. Janfada
Keyword(s):  
Banach Spaces ◽  
Infinitesimal Generator ◽  
Evolution Equations ◽  
Linear Operators ◽  
Semigroups Of Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Abstract Evolution Equations ◽  
Regularized Semigroups

We introduce the notion of regularized quasi-semigroup of bounded linear operators on Banach spaces and its infinitesimal generator, as a generalization of regularized semigroups of operators. After some examples of such quasi-semigroups, the properties of this family of operators will be studied. Also some applications of regularized quasi-semigroups in the abstract evolution equations will be considered. Next some elementary perturbation results on regularized quasi-semigroups will be discussed.

scholarly journals Approximate controllability of second-order nonlocal impulsive partial functional integro-differential evolution systems in Banach spaces

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5887-5912 ◽  
Author(s):  
Mahalingam Nagaraj ◽  
Velusamy Kavitha ◽  
Dumitru Baleanu ◽  
Mani Arjunan
Keyword(s):  
Differential Evolution ◽  
Banach Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Nonlinear Alternative ◽  
Banach Contraction

This manuscript is involved with a class of second-order impulsive partial functional integro-differential evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions ensuring the existence and approximate controllability of mild solutions are established. Theory of cosine family, Banach contraction principle and Leray-Schauder nonlinear alternative fixed point theorem are employed for achieving the required results. An example is analyzed to illustrate the effectiveness of the outcome.

Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces

2015 ◽  
Vol 15 (04) ◽  
pp. 457-475 ◽  
Author(s):  
Jin Liang ◽  
James H. Liu ◽  
Ti-Jun Xiao ◽  
Hong-Kun Xu
Keyword(s):  
Cauchy Problem ◽  
Banach Spaces ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Linear Part ◽  
Linear Operators ◽  
Autonomous Case ◽  
The Cauchy Problem ◽  
General Existence ◽  
Impulsive Delay

In this paper, we are concerned with the periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations with periodic inhomogenous terms in Banach spaces, where the operators in the linear part (possibly unbounded) depend on the time [Formula: see text] and generate an evolution family of linear operators. We first establish two new Gronwall–Bellman-type inequalities, and then prove a new and general existence theorem for periodic mild solutions to the nonautonomous impulsive delay evolution equations, which extends essentially some existing results even for the autonomous case as well as for the case when impulsive perturbations or delays are absent.

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