Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem

This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<...
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Autor*in:	Yongxiang Li [verfasserIn] Weifeng Ma [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index

Übergeordnetes Werk:	In: Mathematics - MDPI AG, 2013, 10(2022), 17, p 3063
Übergeordnetes Werk:	volume:10 ; year:2022 ; number:17, p 3063

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/math10173063

Katalog-ID:	DOAJ01225164X

Internformat


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520			\|a This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones.
650		4	\|a fully fourth-order boundary value problem
650		4	\|a simply supported beam equation
650		4	\|a positive solution
650		4	\|a cone
650		4	\|a fixed point index
653		0	\|a Mathematics
700	0		\|a Weifeng Ma \|e verfasserin \|4 aut
773	0	8	\|i In \|t Mathematics \|d MDPI AG, 2013 \|g 10(2022), 17, p 3063 \|w (DE-627)737287764 \|w (DE-600)2704244-3 \|x 22277390 \|7 nnns
773	1	8	\|g volume:10 \|g year:2022 \|g number:17, p 3063
856	4	0	\|u https://doi.org/10.3390/math10173063 \|z kostenfrei
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856	4	0	\|u https://www.mdpi.com/2227-7390/10/17/3063 \|z kostenfrei
856	4	2	\|u https://doaj.org/toc/2227-7390 \|y Journal toc \|z kostenfrei
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allfields	10.3390/math10173063 doi (DE-627)DOAJ01225164X (DE-599)DOAJ4268a919ad0a444c83424fc4681ddfdb DE-627 ger DE-627 rakwb eng QA1-939 Yongxiang Li verfasserin aut Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones. fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index Mathematics Weifeng Ma verfasserin aut In Mathematics MDPI AG, 2013 10(2022), 17, p 3063 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:10 year:2022 number:17, p 3063 https://doi.org/10.3390/math10173063 kostenfrei https://doaj.org/article/4268a919ad0a444c83424fc4681ddfdb kostenfrei https://www.mdpi.com/2227-7390/10/17/3063 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2022 17, p 3063
spelling	10.3390/math10173063 doi (DE-627)DOAJ01225164X (DE-599)DOAJ4268a919ad0a444c83424fc4681ddfdb DE-627 ger DE-627 rakwb eng QA1-939 Yongxiang Li verfasserin aut Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones. fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index Mathematics Weifeng Ma verfasserin aut In Mathematics MDPI AG, 2013 10(2022), 17, p 3063 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:10 year:2022 number:17, p 3063 https://doi.org/10.3390/math10173063 kostenfrei https://doaj.org/article/4268a919ad0a444c83424fc4681ddfdb kostenfrei https://www.mdpi.com/2227-7390/10/17/3063 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2022 17, p 3063
allfields_unstemmed	10.3390/math10173063 doi (DE-627)DOAJ01225164X (DE-599)DOAJ4268a919ad0a444c83424fc4681ddfdb DE-627 ger DE-627 rakwb eng QA1-939 Yongxiang Li verfasserin aut Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones. fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index Mathematics Weifeng Ma verfasserin aut In Mathematics MDPI AG, 2013 10(2022), 17, p 3063 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:10 year:2022 number:17, p 3063 https://doi.org/10.3390/math10173063 kostenfrei https://doaj.org/article/4268a919ad0a444c83424fc4681ddfdb kostenfrei https://www.mdpi.com/2227-7390/10/17/3063 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2022 17, p 3063
allfieldsGer	10.3390/math10173063 doi (DE-627)DOAJ01225164X (DE-599)DOAJ4268a919ad0a444c83424fc4681ddfdb DE-627 ger DE-627 rakwb eng QA1-939 Yongxiang Li verfasserin aut Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones. fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index Mathematics Weifeng Ma verfasserin aut In Mathematics MDPI AG, 2013 10(2022), 17, p 3063 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:10 year:2022 number:17, p 3063 https://doi.org/10.3390/math10173063 kostenfrei https://doaj.org/article/4268a919ad0a444c83424fc4681ddfdb kostenfrei https://www.mdpi.com/2227-7390/10/17/3063 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2022 17, p 3063
allfieldsSound	10.3390/math10173063 doi (DE-627)DOAJ01225164X (DE-599)DOAJ4268a919ad0a444c83424fc4681ddfdb DE-627 ger DE-627 rakwb eng QA1-939 Yongxiang Li verfasserin aut Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones. fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index Mathematics Weifeng Ma verfasserin aut In Mathematics MDPI AG, 2013 10(2022), 17, p 3063 (DE-627)737287764 (DE-600)2704244-3 22277390 nnns volume:10 year:2022 number:17, p 3063 https://doi.org/10.3390/math10173063 kostenfrei https://doaj.org/article/4268a919ad0a444c83424fc4681ddfdb kostenfrei https://www.mdpi.com/2227-7390/10/17/3063 kostenfrei https://doaj.org/toc/2227-7390 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 10 2022 17, p 3063
language	English
source	In Mathematics 10(2022), 17, p 3063 volume:10 year:2022 number:17, p 3063
sourceStr	In Mathematics 10(2022), 17, p 3063 volume:10 year:2022 number:17, p 3063
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index Mathematics
isfreeaccess_bool	true
container_title	Mathematics
authorswithroles_txt_mv	Yongxiang Li @@aut@@ Weifeng Ma @@aut@@
publishDateDaySort_date	2022-01-01T00:00:00Z
hierarchy_top_id	737287764
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Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. 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callnumber-first	Q - Science
author	Yongxiang Li
spellingShingle	Yongxiang Li misc QA1-939 misc fully fourth-order boundary value problem misc simply supported beam equation misc positive solution misc cone misc fixed point index misc Mathematics Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem
authorStr	Yongxiang Li
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callnumber-label	QA1-939
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issn	22277390
topic_title	QA1-939 Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem fully fourth-order boundary value problem simply supported beam equation positive solution cone fixed point index
topic	misc QA1-939 misc fully fourth-order boundary value problem misc simply supported beam equation misc positive solution misc cone misc fixed point index misc Mathematics
topic_unstemmed	misc QA1-939 misc fully fourth-order boundary value problem misc simply supported beam equation misc positive solution misc cone misc fixed point index misc Mathematics
topic_browse	misc QA1-939 misc fully fourth-order boundary value problem misc simply supported beam equation misc positive solution misc cone misc fixed point index misc Mathematics
format_facet	Elektronische Aufsätze Aufsätze Elektronische Ressource
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title	Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem
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title_full	Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem
author_sort	Yongxiang Li
journal	Mathematics
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author_browse	Yongxiang Li Weifeng Ma
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class	QA1-939
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author-letter	Yongxiang Li
doi_str_mv	10.3390/math10173063
author2-role	verfasserin
title_sort	existence of positive solutions for a fully fourth-order boundary value problem
callnumber	QA1-939
title_auth	Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem
abstract	This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones.
abstractGer	This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones.
abstract_unstemmed	This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<msup<<mi<u</mi<<mrow<<mo<(</mo<<mn<4</mn<<mo<)</mo<</mrow<</msup<<mo<=</mo<<mi<f</mi<<mrow<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mo<′</mo<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mo<,</mo<<mspace width="0.166667em"<</mspace<<msup<<mi<u</mi<<mrow<<mo<‴</mo<</mrow<</msup<<mo<)</mo<</mrow<</mrow<</semantics<</math<</inline-formula< on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<</semantics<</math<</inline-formula< with the boundary condition <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<mi<u</mi<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<0</mn<<mo<)</mo<</mrow<<mo<=</mo<<msup<<mi<u</mi<<mrow<<mo<″</mo<</mrow<</msup<<mrow<<mo<(</mo<<mn<1</mn<<mo<)</mo<</mrow<<mo<=</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula<, which models a statically bending elastic beam whose two ends are simply supported, where <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<:</mo<<mrow<<mo<[</mo<<mn<0</mn<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mn<1</mn<<mo<]</mo<</mrow<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<×</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<−</mo<</msup<<mo<×</mo<<mi mathvariant="double-struck"<R</mi<<mo<→</mo<<msup<<mi mathvariant="double-struck"<R</mi<<mo<+</mo<</msup<</mrow<</semantics<</math<</inline-formula< is continuous. Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. Our discussion is based on the fixed point index theory in cones.
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container_issue	17, p 3063
title_short	Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem
url	https://doi.org/10.3390/math10173063 https://doaj.org/article/4268a919ad0a444c83424fc4681ddfdb https://www.mdpi.com/2227-7390/10/17/3063 https://doaj.org/toc/2227-7390
remote_bool	true
author2	Weifeng Ma
author2Str	Weifeng Ma
ppnlink	737287764
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doi_str	10.3390/math10173063
callnumber-a	QA1-939
up_date	2024-07-04T00:20:01.880Z
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Some precise inequality conditions on <i<f</i< guaranteeing the existence of positive solutions are presented. The inequality conditions allow that <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<f</mi<<mo<(</mo<<mi<t</mi<<mo<,</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<</mrow<</semantics<</math<</inline-formula< may be asymptotically linear, superlinear, or sublinear on <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mi<u</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<v</mi<<mo<,</mo<<mspace width="0.166667em"<</mspace<<mi<w</mi<</mrow<</semantics<</math<</inline-formula<, and <i<z</i< as <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mn<0</mn<</mrow<</semantics<</math<</inline-formula< and <inline-formula<<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"<<semantics<<mrow<<mo<\|</mo<<mo<(</mo<<mi<u</mi<<mo<,</mo<<mi<v</mi<<mo<,</mo<<mi<w</mi<<mo<,</mo<<mi<z</mi<<mo<)</mo<<mo<\|</mo<<mo<→</mo<<mo<∞</mo<</mrow<</semantics<</math<</inline-formula<. 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Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem

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