Weak and strong solutions for differential equations in Banach spaces

2003 ◽  
Vol 18 (4) ◽  
pp. 687-692 ◽  
Author(s):  
A.M. Gomaa
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Strong Solutions ◽  
Weak And Strong Solutions

scholarly journals Yamada-Watanabe Results for Stochastic Differential Equations with Jumps

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Mátyás Barczy ◽  
Zenghu Li ◽  
Gyula Pap
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Stochastic Equations ◽  
Original Method ◽  
Pathwise Uniqueness ◽  
General Version ◽  
Strong Existence ◽  
Weak And Strong Solutions

Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness implies strong existence for stochastic differential equations with jumps.

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.

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