Existence and quasilinearization methods in Hilbert spaces

Mohamed El-Gebeily; Donal O'Regan

doi:10.1016/j.jmaa.2005.11.077

Existence and quasilinearization methods in Hilbert spaces

Mohamed El-Gebeily^*, Donal O'Regan

^*Corresponding author for this work

King Fahd University of Petroleum & Minerals

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

In this paper we discuss some existence results and the application of quasilinearization methods, developed so far for differential equations, to the solution of the abstract problem over(L, ̂) u = F u in a Hilbert space H. Under fairly general assumptions on over(L, ̂), F and H, we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions. If the assumptions are strengthened, we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations.

Original language	English
Pages (from-to)	344-357
Number of pages	14
Journal	Journal of Mathematical Analysis and Applications
Volume	324
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2005.11.077
State	Published - 1 Dec 2006

Bibliographical note

Funding Information:
* Corresponding author. Fax: +966 3 8602340. E-mail address: [email protected] (M. El-Gebeily). 1 Research of the author has been funded by King Fahd University of Petroleum and Minerals under Project number MS/Singular ODE/274.

Keywords

Existence
Nonlinear operators
Quasilinearization methods
Resonance
Self adjoint operators

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2005.11.077

Cite this

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title = "Existence and quasilinearization methods in Hilbert spaces",

abstract = "In this paper we discuss some existence results and the application of quasilinearization methods, developed so far for differential equations, to the solution of the abstract problem over(L,{\^ }) u = F u in a Hilbert space H. Under fairly general assumptions on over(L,{\^ }), F and H, we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions. If the assumptions are strengthened, we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations.",

keywords = "Existence, Nonlinear operators, Quasilinearization methods, Resonance, Self adjoint operators",

author = "Mohamed El-Gebeily and Donal O'Regan",

note = "Funding Information: * Corresponding author. Fax: +966 3 8602340. E-mail address: [email protected] (M. El-Gebeily). 1 Research of the author has been funded by King Fahd University of Petroleum and Minerals under Project number MS/Singular ODE/274.",

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AU - O'Regan, Donal

N1 - Funding Information: * Corresponding author. Fax: +966 3 8602340. E-mail address: [email protected] (M. El-Gebeily). 1 Research of the author has been funded by King Fahd University of Petroleum and Minerals under Project number MS/Singular ODE/274.

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N2 - In this paper we discuss some existence results and the application of quasilinearization methods, developed so far for differential equations, to the solution of the abstract problem over(L, ̂) u = F u in a Hilbert space H. Under fairly general assumptions on over(L, ̂), F and H, we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions. If the assumptions are strengthened, we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations.

AB - In this paper we discuss some existence results and the application of quasilinearization methods, developed so far for differential equations, to the solution of the abstract problem over(L, ̂) u = F u in a Hilbert space H. Under fairly general assumptions on over(L, ̂), F and H, we show that this problem has a solution that can be obtained as the limit of a quadratically convergent nondecreasing sequence of approximate solutions. If the assumptions are strengthened, we show that the abstract problem has a solution which is quadratically bracketed between two monotone sequences of approximate solutions of certain related linear equations.

KW - Existence

KW - Nonlinear operators

KW - Quasilinearization methods

KW - Resonance

KW - Self adjoint operators

UR - https://www.scopus.com/pages/publications/33748889370

U2 - 10.1016/j.jmaa.2005.11.077

DO - 10.1016/j.jmaa.2005.11.077

M3 - Article

AN - SCOPUS:33748889370

SN - 0022-247X

VL - 324

SP - 344

EP - 357

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

Existence and quasilinearization methods in Hilbert spaces

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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