A Bolzano's theorem in the new millennium

Morales, Claudio H.

doi:10.1016/s0362-546x(01)00851-3

Cited by 11 publications

(19 citation statements)

References 14 publications

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“…□ As a consequence of Proposition 2.6 obtain a multivalued version of Corollary 2 in [17] x ∈ F (x) for x ∈ ∂B(0, r).…”

Section: Proposition 26 Let X Be a Banach Space And Let F : B(0 R)mentioning

confidence: 77%

“…For related boundary conditions, we mention the work of Brézis et al [3], Kachurovskii [12], Leray and Lions [14], and Rockafellar [19]. Also we remind the work of Morales [17], in which he gave a brief description of how the original problem has evolved in time passing through various generalizations for the last 30 years. However Morales [17] only studied such works for singlevalued mappings.…”

Section: Then There Exists a Solution X 0 ∈ [−R R] Of The Equation Fmentioning

confidence: 99%

“…Also we remind the work of Morales [17], in which he gave a brief description of how the original problem has evolved in time passing through various generalizations for the last 30 years. However Morales [17] only studied such works for singlevalued mappings.…”

Section: Then There Exists a Solution X 0 ∈ [−R R] Of The Equation Fmentioning

confidence: 99%

“…Domain invariance theorems for singlevalued strongly-monotone mappings were studied by many authors (see [4,15,17,18]). By using Corollary 3.2 or Proposition 3.2, we will obtain an invariance of domain theorem for locally strongly-monotone multivalued mappings.…”

Section: Then the Equation 0 ∈ F (X) Has A Solution In Gmentioning

confidence: 99%

“…Recently Morales [17] established a close connection between a classical Bolzano's theorem from real analysis and recent works about monotone operator theory on reflexive Banach spaces. The Bolzano's theorem is a kind of intermediate value theorem stated as in Dauben [7].…”

Section: Introductionmentioning

confidence: 96%

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Multivalued Versions of a Bolzano's Theorem

Bae¹,

Cho²

2011

Journal of the Korean Mathematical Society

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“…□ As a consequence of Proposition 2.6 obtain a multivalued version of Corollary 2 in [17] x ∈ F (x) for x ∈ ∂B(0, r).…”

Section: Proposition 26 Let X Be a Banach Space And Let F : B(0 R)mentioning

confidence: 77%

Section: Then There Exists a Solution X 0 ∈ [−R R] Of The Equation Fmentioning

confidence: 99%

Section: Then There Exists a Solution X 0 ∈ [−R R] Of The Equation Fmentioning

confidence: 99%

Section: Then the Equation 0 ∈ F (X) Has A Solution In Gmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

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Multivalued Versions of a Bolzano's Theorem

Bae¹,

Cho²

2011

Journal of the Korean Mathematical Society

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Remarks on the infinite‐dimensional counterparts of the Darboux theorem

Juszczyk

Kryszewski

2022

Math Methods in App Sciences

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The Darboux theorem, one of the fundamental results in analysis, states that the derivative of a real (not necessarily continuously) differentiable function defined on a compact interval has the intermediate value property, that is, attains each value between the derivatives at the endpoints. The Bolzano intermediate value theorem, which implies Darboux's theorem when the derivative is continuous, states that a continuous real‐valued function f$$ f $$ defined on false[−1,1false]$$ \left[-1,1\right] $$ satisfying ffalse(−1false)<0$$ f\left(-1\right)<0 $$ and ffalse(1false)>0$$ f(1)>0 $$, has a zero, that is, ffalse(xfalse)=0$$ f(x)&#x0003D;0 $$ for at least one number −1

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Wave propagation and strain localization in a fully saturated softening porous medium under the non‐isothermal conditions

Sun

2016

Int. J. Numer. Anal. Meth. Geomech.

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Summary The (THM) coupling effects on the dynamic wave propagation and strain localization in a fully saturated softening porous medium are analyzed. The characteristic polynomial corresponding to the governing equations of the THM system is derived, and the stability analysis is conducted to determine the necessary conditions for stability in both non‐isothermal and adiabatic cases. The result from the dispersion analysis based on the Abel–Ruffini theorem reveals that the roots of the characteristic polynomial for the THM problem cannot be expressed algebraically. Meanwhile, the dispersion analysis on the adiabatic case leads to a new analytical expression of the internal length scale. Our limit analysis on the phase velocity for the non‐isothermal case indicates that the internal length scale for the non‐isothermal THM system may vanish at the short wavelength limit. This result leads to the conclusion that the rate‐dependence introduced by multiphysical coupling may not regularize the THM governing equations when softening occurs. Numerical experiments are used to verify the results from the stability and dispersion analyses. Copyright © 2016 John Wiley & Sons, Ltd.

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A Bolzano's theorem in the new millennium

Cited by 11 publications

References 14 publications

Multivalued Versions of a Bolzano's Theorem

Multivalued Versions of a Bolzano's Theorem

Remarks on the infinite‐dimensional counterparts of the Darboux theorem

Wave propagation and strain localization in a fully saturated softening porous medium under the non‐isothermal conditions

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