scholarly journals Kneser's theorem for differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 419-434 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Differential Equations ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Solution Set ◽  
The Cauchy Problem ◽  
Uniformly Continuous

We consider the Cauchy problem x (t) = f (t,x (t)), x (O) = xO defined in a nonreflexive Banach space and with the vector field f: T × X → X being weakly uniformly continuous. Using a compactness hypothesis that involves the weak measure of noncompactness, we prove that the solution set of the above Cauchy problem is nonempty, connected and compact in .

scholarly journals Weak solutions of differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 407-418 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Continuous Vector ◽  
Weakly Continuous ◽  
Continuous Vector Field ◽  
The Cauchy Problem

We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe 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.

UNIQUE SOLUTIONS OF DISCONTINUOUS O.D.E.'S IN BANACH SPACES

2006 ◽  
Vol 04 (03) ◽  
pp. 247-262 ◽  
Author(s):  
ALBERTO BRESSAN ◽  
WEN SHEN
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Right Hand ◽  
Directional Variation ◽  
The Cauchy Problem

We consider the Cauchy problem for an ordinary differential equation with discontinuous right-hand side in an L∞ space. Under the assumptions that the vector field is directionally continuous with bounded directional variation, we prove that the O.D.E. has a unique Carathéodory solution, which depends Lipschitz continuously on the data.

scholarly journals An existence theorem for ordinary differential equations in Banach spaces

1984 ◽  
Vol 30 (3) ◽  
pp. 449-456 ◽  
Author(s):  
Bogdan Rzepecki
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Regularity Condition ◽  
Measure Of Noncompactness ◽  
Bounded Solution ◽  
Carathéodory Conditions

We prove the existence of bounded solution of the differential equation y′ = A(t)y + f(t, y) in a Banach space. The method used here is based on the concept of “admissibility” due to Massera and Schäffer when f satisfies the Caratheodory conditions and some regularity condition expressed in terms of the measure of noncompactness α.

scholarly journals Optimal time and space regularity for solutions of degenerate differential equations

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Alberto Favaron
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Optimal Time ◽  
Time And Space ◽  
Optimal Regularity ◽  
Degenerate Differential Equation ◽  
The Cauchy Problem ◽  
Degenerate Differential Equations

AbstractWe derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.

scholarly journals On Semilinear Integro-Differential Equations with Nonlocal Conditions in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-26
Author(s):  
Tran Dinh Ke ◽  
Valeri Obukhovskii ◽  
Ngai-Ching Wong ◽  
Jen-Chih Yao
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Topological Structure ◽  
Continuous Dependence ◽  
Nonlocal Conditions ◽  
Abstract Cauchy Problem ◽  
Nonlinear Perturbations ◽  
Continuous Dependence Results

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set.

scholarly journals On existence theorems for differential equations in Banach spaces

1985 ◽  
Vol 32 (1) ◽  
pp. 73-82 ◽  
Author(s):  
Józef Banaś
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Banach Spaces ◽  
Existence Theorems ◽  
The Cauchy Problem

In this paper we show that a number of existence theorems for the Cauchy problem of ordinary differential equations in Banach spaces are only apparent generalizations of the previous ones.

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