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

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Rui Xie ◽  
Chuanyi Zhang
Keyword(s):  
Cauchy Problem ◽  
Function Space ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Type Function ◽  
Inclusion Relations ◽  
Asymptotically Periodic ◽  
New Space ◽  
Study Inclusion

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.

scholarly journals Stability of mild solutions of the fractional nonlinear abstract Cauchy problem

2021 ◽  
Vol 30 (1) ◽  
pp. 272-288
Author(s):  
J. Vanterler da C. Sousa ◽  
◽  
Kishor D. Kucche ◽  
E. Capelas de Oliveira ◽  
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Banach Fixed Point Theorem ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

<abstract><p>Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.</p></abstract>

On mild solutions of gradient systems in Hilbert spaces

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Andrzej Rozkosz
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Backward Stochastic Differential Equations ◽  
Stochastic Representation ◽  
Infinite Dimensional ◽  
The Cauchy Problem

AbstractWe consider the Cauchy problem for an infinite-dimensional Ornstein-Uhlenbeck equation perturbed by gradient of a potential. We prove some results on existence and uniqueness of mild solutions of the problem. We also provide stochastic representation of mild solutions in terms of linear backward stochastic differential equations determined by the Ornstein-Uhlenbeck operator and the potential.

scholarly journals Periodic Solutions andS-Asymptotically Periodic Solutions to Fractional Evolution Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jia Mu ◽  
Yong Zhou ◽  
Li Peng
Keyword(s):  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Asymptotically Periodic ◽  
Liouville Fractional Derivative ◽  
Asymptotically Periodic Solutions

This paper deals with the existence and uniqueness of periodic solutions,S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.

scholarly journals New interpolation spaces and strict Hölder regularity for fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Md Mansur Alam ◽  
Shruti Dubey ◽  
Dumitru Baleanu
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Analytic Semigroup ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Interpolation Spaces ◽  
Holder Regularity

AbstractWe know that interpolation spaces in terms of analytic semigroup have a significant role into the study of strict Hölder regularity of solutions of classical abstract Cauchy problem (ACP). In this paper, we first construct interpolation spaces in terms of solution operators in fractional calculus and characterize these spaces. Then we establish strict Hölder regularity of mild solutions of fractional order ACP.

scholarly journals Nonuniqueness and wellposedness of abstract Cauchy problems in a Fréchet space

2001 ◽  
Vol 63 (1) ◽  
pp. 123-131 ◽  
Author(s):  
Peer Christian Kunstmann
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Mild Solutions ◽  
Abstract Cauchy Problem ◽  
Continuous Selection ◽  
Fréchet Space ◽  
Inhomogeneous Equation ◽  
Frechet Space ◽  
Closed Linear Operator ◽  
Cauchy Problems

Suppose that A is a closed linear operator in a Fréchet space X. We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F, and the part Az of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels.Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

scholarly journals Existence of Solutions of a Partial Integrodifferential Equation with Thermostat and Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Carlo Bianca ◽  
Luca Guerrini ◽  
Annie Lemarchand
Keyword(s):  
Cauchy Problem ◽  
Time Delay ◽  
Integrodifferential Equation ◽  
Mathematical Analysis ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Mild Solutions ◽  
Kinetic Equations ◽  
Successive Approximations ◽  
Partial Integrodifferential Equation

This paper deals with the mathematical analysis of a retarded partial integrodifferential equation that belongs to the class of thermostatted kinetic equations with time delay. Specifically, the paper is devoted to the proof of the existence and uniqueness of mild solutions of the related Cauchy problem. The main result is obtained by employing integration along the characteristic curves and successive approximations sequence arguments. Applications and perspective are also discussed within the paper.

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