Evolution equations of higher order in banach spaces

Miura transform is known as the transformation between Korweg de-Vries equation and modified Korweg de-Vries equation. Its formal similarity to the Cole-Hopf transform has been noticed. This fact sheds light on the logarithmic type transformations as an origin of a certain kind of nonlinearity in the soliton equations. In this article, based on the logarithmic representation of operators in infinite-dimensional Banach spaces, a structure common to both Miura and Cole-Hopf transforms is discussed. In conclusion, the Miura transform is generalized as the transform in abstract Banach spaces, and it is applied to the higher order abstract evolution equations.

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

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0219-8 ◽

2014 ◽

Vol 17 (4) ◽

Cited By ~ 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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Existence and optimal controls for nonlocal fractional evolution equations of order (1,2) in Banach spaces

Advances in Difference Equations ◽

10.1186/s13662-021-03430-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Denghao Pang ◽

Wei Jiang ◽

Azmat Ullah Khan Niazi ◽

Jiale Sheng

Keyword(s):

Banach Spaces ◽

Mild Solution ◽

Measure Of Noncompactness ◽

Evolution Equations ◽

Well Posedness ◽

Optimal Controls ◽

Lagrange Problem ◽

Fractional Differential ◽

Fractional Evolution Systems ◽

Definition Of

AbstractIn this paper, we mainly investigate the existence, continuous dependence, and the optimal control for nonlocal fractional differential evolution equations of order (1,2) in Banach spaces. We define a competent definition of a mild solution. On this basis, we verify the well-posedness of the mild solution. Meanwhile, with a construction of Lagrange problem, we elaborate the existence of optimal pairs of the fractional evolution systems. The main tools are the fractional calculus, cosine family, multivalued analysis, measure of noncompactness method, and fixed point theorem. Finally, an example is propounded to illustrate the validity of our main results.

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