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Uniqueness of positive solutions for several classes of sum operator equations and applications
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Zeitschriftentitel: | Journal of Inequalities and Applications |
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In: | Journal of Inequalities and Applications, 2014, 2014, 1 |
Medientyp: | E-Article |
Sprache: | Englisch |
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Springer Science and Business Media LLC
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author_facet |
Yang, Chen Zhai, Chengbo Hao, Mengru Yang, Chen Zhai, Chengbo Hao, Mengru |
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author |
Yang, Chen Zhai, Chengbo Hao, Mengru |
spellingShingle |
Yang, Chen Zhai, Chengbo Hao, Mengru Journal of Inequalities and Applications Uniqueness of positive solutions for several classes of sum operator equations and applications Applied Mathematics Discrete Mathematics and Combinatorics Analysis |
author_sort |
yang, chen |
spelling |
Yang, Chen Zhai, Chengbo Hao, Mengru 1029-242X Springer Science and Business Media LLC Applied Mathematics Discrete Mathematics and Combinatorics Analysis http://dx.doi.org/10.1186/1029-242x-2014-58 <jats:title>Abstract</jats:title> <jats:p>In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:msubsup> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:math> </jats:inline-formula>, where <jats:italic>f</jats:italic> and <jats:italic>G</jats:italic> are both nonnegative, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:inline-formula>, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> in Ω, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> on <jats:italic>∂</jats:italic> Ω, where Ω is a bounded domain with smooth boundary in <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:math> </jats:inline-formula> (<jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> </jats:inline-formula>), <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> and <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:inline-formula> is allowed to be singular on <jats:italic>∂</jats:italic> Ω. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.</jats:p> <jats:p> <jats:bold>MSC:</jats:bold>47H10, 47H07, 45G15, 35J60, 35J65.</jats:p> Uniqueness of positive solutions for several classes of sum operator equations and applications Journal of Inequalities and Applications |
doi_str_mv |
10.1186/1029-242x-2014-58 |
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Springer Science and Business Media LLC, 2014 |
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Springer Science and Business Media LLC, 2014 |
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publishDateSort |
2014 |
publisher |
Springer Science and Business Media LLC |
recordtype |
ai |
record_format |
ai |
series |
Journal of Inequalities and Applications |
source_id |
49 |
title |
Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_unstemmed |
Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_full |
Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_fullStr |
Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_full_unstemmed |
Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_short |
Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_sort |
uniqueness of positive solutions for several classes of sum operator equations and applications |
topic |
Applied Mathematics Discrete Mathematics and Combinatorics Analysis |
url |
http://dx.doi.org/10.1186/1029-242x-2014-58 |
publishDate |
2014 |
physical |
|
description |
<jats:title>Abstract</jats:title>
<jats:p>In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>u</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi>λ</mml:mi>
<mml:msubsup>
<mml:mo>∫</mml:mo>
<mml:mi>a</mml:mi>
<mml:mi>b</mml:mi>
</mml:msubsup>
<mml:mi>G</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>t</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>s</mml:mi>
<mml:mo>)</mml:mo>
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>s</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>s</mml:mi>
<mml:mo>)</mml:mo>
<mml:mo>)</mml:mo>
<mml:mspace />
<mml:mi>d</mml:mi>
<mml:mi>s</mml:mi>
</mml:math>
</jats:inline-formula>, where <jats:italic>f</jats:italic> and <jats:italic>G</jats:italic> are both nonnegative, <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>λ</mml:mi>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</jats:inline-formula> is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mo>−</mml:mo>
<mml:mi>Δ</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>=</mml:mo>
<mml:mi>λ</mml:mi>
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
<mml:mo>)</mml:mo>
</mml:math>
</jats:inline-formula>, <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>u</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>)</mml:mo>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</jats:inline-formula> in Ω, <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>u</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</jats:inline-formula> on <jats:italic>∂</jats:italic> Ω, where Ω is a bounded domain with smooth boundary in <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mi>N</mml:mi>
</mml:msup>
</mml:math>
</jats:inline-formula> (<jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>N</mml:mi>
<mml:mo>≥</mml:mo>
<mml:mn>1</mml:mn>
</mml:math>
</jats:inline-formula>), <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>λ</mml:mi>
<mml:mo>></mml:mo>
<mml:mn>0</mml:mn>
</mml:math>
</jats:inline-formula> and <jats:inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>f</mml:mi>
<mml:mo>(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
<mml:mo>)</mml:mo>
</mml:math>
</jats:inline-formula> is allowed to be singular on <jats:italic>∂</jats:italic> Ω. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.</jats:p>
<jats:p>
<jats:bold>MSC:</jats:bold>47H10, 47H07, 45G15, 35J60, 35J65.</jats:p> |
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author | Yang, Chen, Zhai, Chengbo, Hao, Mengru |
author_facet | Yang, Chen, Zhai, Chengbo, Hao, Mengru, Yang, Chen, Zhai, Chengbo, Hao, Mengru |
author_sort | yang, chen |
container_issue | 1 |
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container_title | Journal of Inequalities and Applications |
container_volume | 2014 |
description | <jats:title>Abstract</jats:title> <jats:p>In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:msubsup> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:math> </jats:inline-formula>, where <jats:italic>f</jats:italic> and <jats:italic>G</jats:italic> are both nonnegative, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:inline-formula>, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> in Ω, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> on <jats:italic>∂</jats:italic> Ω, where Ω is a bounded domain with smooth boundary in <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:math> </jats:inline-formula> (<jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> </jats:inline-formula>), <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> and <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:inline-formula> is allowed to be singular on <jats:italic>∂</jats:italic> Ω. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.</jats:p> <jats:p> <jats:bold>MSC:</jats:bold>47H10, 47H07, 45G15, 35J60, 35J65.</jats:p> |
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series | Journal of Inequalities and Applications |
source_id | 49 |
spelling | Yang, Chen Zhai, Chengbo Hao, Mengru 1029-242X Springer Science and Business Media LLC Applied Mathematics Discrete Mathematics and Combinatorics Analysis http://dx.doi.org/10.1186/1029-242x-2014-58 <jats:title>Abstract</jats:title> <jats:p>In this article we study several classes of sum operator equations on ordered Banach spaces and present some new existence and uniqueness results of positive solutions, which extend the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for several classes of sum operator equations. As applications, we utilize the main results obtained in this paper to study two classes nonlinear problems; one is the integral equation <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:msubsup> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:math> </jats:inline-formula>, where <jats:italic>f</jats:italic> and <jats:italic>G</jats:italic> are both nonnegative, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> is a parameter; the other is the elliptic boundary value problem for the Lane-Emden-Fowler equation <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:inline-formula>, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> in Ω, <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> on <jats:italic>∂</jats:italic> Ω, where Ω is a bounded domain with smooth boundary in <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:math> </jats:inline-formula> (<jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> </jats:inline-formula>), <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> </jats:inline-formula> and <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> </jats:inline-formula> is allowed to be singular on <jats:italic>∂</jats:italic> Ω. The new results on the existence and uniqueness of positive solutions for these problems are given, which complement the existing results of positive solutions for these problems in the literature.</jats:p> <jats:p> <jats:bold>MSC:</jats:bold>47H10, 47H07, 45G15, 35J60, 35J65.</jats:p> Uniqueness of positive solutions for several classes of sum operator equations and applications Journal of Inequalities and Applications |
spellingShingle | Yang, Chen, Zhai, Chengbo, Hao, Mengru, Journal of Inequalities and Applications, Uniqueness of positive solutions for several classes of sum operator equations and applications, Applied Mathematics, Discrete Mathematics and Combinatorics, Analysis |
title | Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_full | Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_fullStr | Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_full_unstemmed | Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_short | Uniqueness of positive solutions for several classes of sum operator equations and applications |
title_sort | uniqueness of positive solutions for several classes of sum operator equations and applications |
title_unstemmed | Uniqueness of positive solutions for several classes of sum operator equations and applications |
topic | Applied Mathematics, Discrete Mathematics and Combinatorics, Analysis |
url | http://dx.doi.org/10.1186/1029-242x-2014-58 |