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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
Personen und Körperschaften: Yang, Chen, Zhai, Chengbo, Hao, Mengru
In: Journal of Inequalities and Applications, 2014, 2014, 1
Medientyp: E-Article
Sprache: Englisch
veröffentlicht:
Springer Science and Business Media LLC
Schlagwörter:
author_facet Yang, Chen
Zhai, Chengbo
Hao, Mengru
Yang, Chen
Zhai, Chengbo
Hao, Mengru
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>&gt;</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>&gt;</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>&gt;</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
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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>&gt;</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>&gt;</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>&gt;</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
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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>&gt;</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>&gt;</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>&gt;</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>&gt;</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>&gt;</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>&gt;</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