Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

Banaś, Józef; Mursaleen, M.

doi:10.1007/978-81-322-1886-9

Cited by 165 publications

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“…Taking the supremum of |L G f (v)| over all v ∈ V and applying (3.11) gives Hence, L G ∞→∞ ≤ D + 1. Now that we have shown that L G is bounded on both ℓ 1 and ℓ ∞ , standard interpolation result over the ℓ p (V ) spaces imply that L G : ℓ p (V ) → ℓ p (V ) are uniformly bounded for all 1 < p < ∞ (see for example Exercise 12 of §2.6 from [1]). This completes the proof.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

Stability of infinite systems of coupled oscillators via random walks on weighted graphs

Bramburger

2019

Trans. Amer. Math. Soc.

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Weakly coupled oscillators are used throughout the physical sciences, particularly in mathematical neuroscience to describe the interaction of neurons in the brain. Systems of weakly coupled oscillators have a well-known decomposition to a canonical phase model which forms the basis of our investigation in this work. Particularly, our interest lies in examining the stability of synchronous (phase-locked) solutions to this phase system: solutions with phases having the same temporal frequency but differ through time-independent phase-lags. The main stability result of this work comes from adapting a series of investigations into random walks on infinite weighted graphs. We provide an interesting link between the seemingly unrelated areas of coupled oscillators and random walks to obtain algebraic decay rates of small perturbations off the phase-locked solutions under some minor technical assumptions. We also provide some interesting and motivating examples that demonstrate the stability of phase-locked solutions, particularly that of a rotating wave solution arising in a well studied paradigm in the theory of coupled oscillators.

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Section: Preliminary Definitionsmentioning

confidence: 99%

Stability of infinite systems of coupled oscillators via random walks on weighted graphs

Bramburger

2019

Trans. Amer. Math. Soc.

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“…The following definition introduces the concept of measure of non-compactness. The list of axioms (MN1) -(MN6) is based on the classical collections that can be found in [3] (page 11), [19] (pages [18][19] or [4] (page 170).…”

Section: Measure Of Non-compactness On C B (T )mentioning

confidence: 99%

A measure of non-compactness on T_3½ spaces based on Arzelà-Ascoli type theorem

Turoboś

2019

Quaestiones Mathematicae

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The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the T 3 1 2 space. Motivated by the generalized Arzelà-Ascoli theorem for Tichonoff space T via Wallman compactifiaction Wall(T ), we constuct a measure of non-compactness for the space C b (T ). We also study some of the properties of this object and give another version of Darbo-type theorem, suitable for this particular case.

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“…An upper semicontinuous map G : X → P(X) is said to be condensing if, for any subset B ⊆ X with α(B) = 0, we have α(G(B)) < α(B), where α denotes the Kuratowski measure of noncompactness [4].…”

Section: Preliminariesmentioning

confidence: 99%

On a nonlinear integrodifferential evolution inclusion with nonlocal initial conditions in Banach spaces

Zhang¹

2012

OpMath

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Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

Cited by 165 publications

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Stability of infinite systems of coupled oscillators via random walks on weighted graphs

Stability of infinite systems of coupled oscillators via random walks on weighted graphs

A measure of non-compactness on T_3½ spaces based on Arzelà-Ascoli type theorem

On a nonlinear integrodifferential evolution inclusion with nonlocal initial conditions in Banach spaces

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Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

Cited by 165 publications

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Stability of infinite systems of coupled oscillators via random walks on weighted graphs

Stability of infinite systems of coupled oscillators via random walks on weighted graphs

A measure of non-compactness on T3½ spaces based on Arzelà-Ascoli type theorem

On a nonlinear integrodifferential evolution inclusion with nonlocal initial conditions in Banach spaces

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A measure of non-compactness on T_3½ spaces based on Arzelà-Ascoli type theorem