Existence of solutions in the alpha-norm for partial functional differential equations with infinite delay

Ezzinbi, Khalil; Rebey, Amor

doi:10.14232/ejqtde.2012.1.59

Cited by 7 publications

(9 citation statements)

References 14 publications

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“…Previous studies on integral and/or partial differential equations involving infinite delays have found criteria for the existence, uniqueness, and stability of traveling wave solutions [ 1 , 18 ]. Following the successful extrapolation approach for infinite delays in integral and/or partial differential equations based on finite delays [ 12 , 18 ] and recalling the close relationship of partial differential equations and integro-differential equations of the type discussed here [ 25 ], we assume in the following that mathematical analysis based on finite delays are applicable. An additional discussion of this approximation may appear necessary, but is neglected since it would exceed the major aim of the present work.…”

Section: Methodsmentioning

confidence: 99%

Distributed Nonlocal Feedback Delays May Destabilize Fronts in Neural Fields, Distributed Transmission Delays Do Not

Hutt

Zhang

2013

The Journal of Mathematical Neuroscience

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The spread of activity in neural populations is a well-known phenomenon. To understand the propagation speed and the stability of stationary fronts in neural populations, the present work considers a neural field model that involves intracortical and cortico-cortical synaptic interactions. This includes distributions of axonal transmission speeds and nonlocal feedback delays as well as general classes of synaptic interactions. The work proves the spectral stability of standing and traveling fronts subject to general transmission speeds for large classes of spatial interactions and derives conditions for the front instabilities subjected to nonlocal feedback delays. Moreover, it turns out that the uniqueness of the stationary traveling fronts guarantees its exponential stability for vanishing feedback delay. Numerical simulations complement the analytical findings.

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Section: Methodsmentioning

confidence: 99%

Distributed Nonlocal Feedback Delays May Destabilize Fronts in Neural Fields, Distributed Transmission Delays Do Not

Hutt

Zhang

2013

The Journal of Mathematical Neuroscience

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“…Since the theory of differential equations with delay has extensive physical background and realistic mathematical model, it has been considerably developed and numerous properties of solutions have been studied, see [14][15][16] and references therein. Therefore, the nonlocal Cauchy problems for evolution equations with delay are an important area of investigation and have been studied extensively in recent years.…”

Section: Introductionmentioning

confidence: 99%

Existence of solutions for delay evolution equations with nonlocal conditions

Zhang

2017

Open Mathematics

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Abstract:In this paper, we are devoted to study the existence of mild solutions for delay evolution equations with nonlocal conditions. By using tools involving the Kuratowski measure of noncompactness and fixed point theory, we establish some existence results of mild solutions without the assumption of compactness on the associated semigroup. Our results improve and generalize some related conclusions on this issue. Moreover, we present an example to illustrate the application of the main results.

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“…Fu and ix x Introduction Ezzinbi [119] considered the existence of mild and classical solutions for a class of neutral partial functional differential equations with nonlocal conditions. Various classes of partial functional and neutral functional differential equations with infinite delay are studied by Adimy et al [10][11][12], Belmekki et al [52], and Ezzinbi [108]. Henriquez [136] and Hernandez [137,138] studied the existence and regularity of solutions to functional and neutral functional differential equations with unbounded delay.…”

Section: Introductionmentioning

confidence: 99%