scholarly journals Existence Theorem for Noninstantaneous Impulsive Evolution Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-4
Author(s):  
Gul I Hina Aslam ◽  
Amjad Ali ◽  
Maimona Rafiq
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Constructive Theory ◽  
First Order ◽  
Linear Evolution ◽  
Semilinear Problems ◽  
Variational Structure ◽  
Linear Evolution Equation ◽  
Variational Form ◽  
Impulsive Evolution Equations

In this note, the variational form of the classical Lax–Milgram theorem is used for the divulgence of variational structure of the first-order noninstantaneous impulsive linear evolution equation. The existence and uniqueness of the weak solution of the problem is obtained. In future, this constructive theory can be used for the corresponding semilinear problems.

scholarly journals Fractional approximation of solutions of evolution equations

Analysis ◽  
2016 ◽  
Vol 36 (2) ◽  
Author(s):  
Anatoly N. Kochubei ◽  
Yuri G. Kondratiev
Keyword(s):  
Evolution Equation ◽  
Analytic Continuation ◽  
Evolution Equations ◽  
Order Linear ◽  
First Order ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Fractional Approximation ◽  
Fractional Equation

AbstractWe show how to approximate a solution of the first order linear evolution equation, together with its possible analytic continuation, using a solution of the time-fractional equation of order

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

scholarly journals On the stability of first order impulsive evolution equations

2014 ◽  
Vol 34 (3) ◽  
pp. 639 ◽  
Author(s):  
JinRong Wang ◽  
Michal Fečkan ◽  
Yong Zhou
Keyword(s):  
Evolution Equations ◽  
First Order ◽  
The Stability ◽  
Impulsive Evolution Equations

scholarly journals A new class of hyperbolic variational–hemivariational inequalities driven by non-linear evolution equations

2020 ◽  
Vol 32 (1) ◽  
pp. 59-88
Author(s):  
STANISŁAW MIGÓRSKI ◽  
WEIMIN HAN ◽  
SHENGDA ZENG
Keyword(s):  
Dynamical System ◽  
Evolution Equations ◽  
Frictional Contact ◽  
Maximal Monotone ◽  
Contact Boundary ◽  
Hemivariational Inequalities ◽  
Response Condition ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Non Linear

The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.

scholarly journals Analysis of a frictionless contact problem for elastic-viscoplastic materials

2012 ◽  
Vol 17 (1) ◽  
pp. 99-117
Author(s):  
Mohamed Selmani ◽  
Lynda Selmani
Keyword(s):  
Contact Problem ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Order Differential Equation ◽  
Nonlinear Evolution ◽  
Monotone Operators ◽  
Uniqueness Result ◽  
Nonlinear Evolution Equations ◽  
Frictionless Contact ◽  
First Order

We consider a dynamic frictionless contact problem for elastic-viscoplastic materials with damage. The contact is modelled with normal compliance condition. The adhesion of the contact surfaces is considered and is modelled with a surface variable, the bonding field whose evolution is described by a first order differential equation. We derive variational formulation for the model and prove an existence and uniqueness result of the weak solution. The proof is based on arguments of nonlinear evolution equations with monotone operators, a classical existence and uniqueness result on parabolic inequalities, differential equations and fixed-point arguments.

BASIC THEORY FOR A CLASS OF MODELS OF HIERARCHICALLY STRUCTURED POPULATION DYNAMICS WITH DISTRIBUTED STATES IN THE RECRUITMENT

2006 ◽  
Vol 16 (10) ◽  
pp. 1695-1722 ◽  
Author(s):  
ÀNGEL CALSINA ◽  
JOAN SALDAÑA
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Local Existence ◽  
Structured Populations ◽  
Uniqueness Of Solution ◽  
Infinite Dimensional ◽  
Linear Evolution ◽  
Local Existence And Uniqueness ◽  
Size Rank ◽  
The Individual

In this paper we present a proof of existence and uniqueness of solution for a class of PDE models of size structured populations with distributed state-at-birth and having nonlinearities defined by an infinite-dimensional environment. The latter means that each member of the population experiences an environment according to a sort of average of the population size depending on the individual size, rank or any other variable structuring the population. The proof of the local existence and uniqueness of solution as well as the continuous dependence on initial continuous is based on the general theory of quasi-linear evolution equations in nonreflexive Banach spaces, while the global existence proof is based on the integration of the local solution along characteristic curves.

scholarly journals Monotone Iterative Technique for the Initial Value Problems of Impulsive Evolution Equations in Ordered Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
He Yang
Keyword(s):  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
Uniqueness Results ◽  
Impulsive Evolution Equations

This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach spaceE:u′(t)+Au(t)=f(t,u(t),Gu(t)),t∈[0,a],t≠tk,Δu|t=tk=Ik(u(tk)),0<t1<t2<⋯<tm<a,u(0)=u0, whereA:D(A)⊂E→Eis a closed linear operator, andf:[0,a]×E×E→Eis a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearityf, some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.

scholarly journals Two-point boundary value problems for linear evolution equations

2001 ◽  
Vol 131 (3) ◽  
pp. 521-543 ◽  
Author(s):  
A. S. FOKAS ◽  
B. PELLONI
Keyword(s):  
Boundary Value Problems ◽  
Evolution Equations ◽  
Boundary Value ◽  
Point Boundary ◽  
Linear Evolution ◽  
Linear Evolution Equation ◽  
Derivatives Of ◽  
Linear Evolution Equations ◽  
Well Posed ◽  
General Method

We study boundary value problems for a linear evolution equation with spatial derivatives of arbitrary order, on the domain 0 < x < L, 0 < t < T, with L and T positive finite constants. We present a general method for identifying well-posed problems, as well as for constructing an explicit representation of the solution of such problems. This representation has explicit x and t dependence, and it consists of an integral in the k-complex plane and of a discrete sum. As illustrative examples we solve some two-point boundary value problems for the equations iqt + qxx = 0 and qt + qxxx = 0.

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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