Nonlinear evolution equations in an arbitrary Banach space

1977 ◽  
Vol 26 (1) ◽  
pp. 1-42 ◽  
Author(s):  
L. C. Evans
Keyword(s):  
Banach Space ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Arbitrary Banach Space

scholarly journals Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

2020 ◽  
Vol 20 (1) ◽  
pp. 89-108 ◽  
Author(s):  
André Eikmeier ◽  
Etienne Emmrich ◽  
Hans-Christian Kreusler
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Inner Product ◽  
Initial Value ◽  
Time Discretisation ◽  
Volterra Integral Operator ◽  
Stability Results

AbstractThe initial value problem for an evolution equation of type {v^{\prime}+Av+BKv=f} is studied, where {A:V_{A}\to V_{A}^{\prime}} is a monotone, coercive operator and where {B:V_{B}\to V_{B}^{\prime}} induces an inner product. The Banach space {V_{A}} is not required to be embedded in {V_{B}} or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

INTEGRAL SOLUTIONS TO A CLASS OF NONLOCAL EVOLUTION EQUATIONS

2010 ◽  
Vol 12 (06) ◽  
pp. 1031-1054 ◽  
Author(s):  
JESÚS GARCÍA-FALSET ◽  
SIMEON REICH
Keyword(s):  
Banach Space ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Sufficient Conditions ◽  
Accretive Operator ◽  
Integral Solution ◽  
Nonlinear Evolution Equations ◽  
Nonlocal Evolution Equations ◽  
Integral Solutions

We study the existence of integral solutions to a class of nonlinear evolution equations of the form [Formula: see text] where A : D(A) ⊆ X → 2X is an m-accretive operator on a Banach space X, and f : [0, T] × X → X and [Formula: see text] are given functions. We obtain sufficient conditions for this problem to have a unique integral solution.

scholarly journals Estimates of Solutions to Nonlinear Evolution Equations

2018 ◽  
Vol 14 (2) ◽  
pp. 7812-7817
Author(s):  
Alexander G. Ramm
Keyword(s):  
Banach Space ◽  
Global Solution ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Unique Global Solution ◽  
Estimates Of Solutions ◽  
Suitable Function

Consider the equation                  u’(t) = A (t, u (t)),   u(0)= U0 ;   u' := du/dt     (1).   Under some assumptions on the nonlinear operator A(t,u) it is proved that problem (1) has a unique global solution and this solution satisfies the following estimate                                               ||u (t)|| < µ (t) -1     for every t belongs to R+ = [0,infinity). Here µ(t) > 0,   µ belongs to  C1 (R+), is a suitable function and the norm ||u || is the norm in a Banach space X with the property ||u (t) ||’   <=  ||u’ (t) ||.

scholarly journals Nonlinear evolution equations on Banach space

1991 ◽  
Vol 4 (3) ◽  
pp. 187-202 ◽  
Author(s):  
N. U. Ahmed
Keyword(s):  
Banach Space ◽  
Fixed Point Theorems ◽  
Stochastic Systems ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Semigroup Theory ◽  
Nonlinear Evolution Equations ◽  
Stochastic Evolution Equations ◽  
Uniqueness Of Solutions ◽  
Semilinear Problems

In this paper we consider the questions of existence and uniqueness of solutions of certain semilinear and quasilinear evolution equations on Banach space. We consider both deterministic and stochastic systems. The approach is based on semigroup theory and fixed point theorems. Our results allow the nonlinear perturbations in all the semilinear problems to be bounded or unbounded with reference to the base space, thereby increasing the scope for applications to partial differential equations. Further, quasilinear stochastic evolution equations seemingly have never been considered in the literature.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

Sign in / Sign up

Export Citation Format

Share Document