Positive solutions for one-dimensional third-order p-Laplacian boundary value problems
Abstract In this paper, we give improved results on the existence of positive solutions for the following one-dimensional p-Laplacian equation with nonlinear boundary conditions: { ( ϕ p ( y ″ ) ) ′ + b ( t ) g ( t , y ( t ) ) = 0 , 0 < t < 1 , λ 1 ϕ p ( y ( 0 ) ) − β 1 ϕ p ( y ′ ( 0 ) ) = 0 ,...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-03-01
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| Series: | Advances in Difference Equations |
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| Online Access: | http://link.springer.com/article/10.1186/s13662-017-1153-y |
| _version_ | 1819144527723626496 |
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| author | Yan Sun |
| author_facet | Yan Sun |
| author_sort | Yan Sun |
| collection | DOAJ |
| description | Abstract In this paper, we give improved results on the existence of positive solutions for the following one-dimensional p-Laplacian equation with nonlinear boundary conditions: { ( ϕ p ( y ″ ) ) ′ + b ( t ) g ( t , y ( t ) ) = 0 , 0 < t < 1 , λ 1 ϕ p ( y ( 0 ) ) − β 1 ϕ p ( y ′ ( 0 ) ) = 0 , λ 2 ϕ p ( y ( 1 ) ) + β 2 ϕ p ( y ′ ( 1 ) ) = 0 , y ″ ( 0 ) = 0 , $$ \textstyle\begin{cases} (\phi_{p} ( y'' )) ' + b ( t ) g ( t, y ( t ) ) = 0, \quad 0 < t < 1, \\ \lambda_{1}\phi_{p} ( y ( 0 ) ) - \beta_{1} \phi_{p} ( y' ( 0 ) ) = 0, \\ \lambda_{2}\phi_{p} ( y ( 1 ) ) + \beta_{2} \phi_{p} ( y' ( 1 ) ) = 0,\qquad y'' ( 0 ) = 0, \end{cases} $$ where ϕ p ( s ) = | s | p − 2 s $\phi_{p} ( s ) = | s | ^{ p-2 } s$ , p > 1 $p >1 $ . Constructing an available integral operator and combining fixed point index theory, we establish some optimal criteria for the existence of bounded positive solutions. The interesting point of the results is that the term b ( t ) $b ( t ) $ may be singular at t = 0 $t=0$ and/or t = 1 $t=1$ . Moreover, the nonlinear term g ( t , y ) $g(t, y)$ is also allowed to have singularity at y = 0 $y=0$ . In particular, our results extend and unify some known results. |
| first_indexed | 2024-12-22T12:43:33Z |
| format | Article |
| id | doaj.art-1f1694f6c86b4f1ebccece4913606beb |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-22T12:43:33Z |
| publishDate | 2017-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-1f1694f6c86b4f1ebccece4913606beb2022-12-21T18:25:22ZengSpringerOpenAdvances in Difference Equations1687-18472017-03-012017112410.1186/s13662-017-1153-yPositive solutions for one-dimensional third-order p-Laplacian boundary value problemsYan Sun0Department of Mathematics, Shanghai Normal UniversityAbstract In this paper, we give improved results on the existence of positive solutions for the following one-dimensional p-Laplacian equation with nonlinear boundary conditions: { ( ϕ p ( y ″ ) ) ′ + b ( t ) g ( t , y ( t ) ) = 0 , 0 < t < 1 , λ 1 ϕ p ( y ( 0 ) ) − β 1 ϕ p ( y ′ ( 0 ) ) = 0 , λ 2 ϕ p ( y ( 1 ) ) + β 2 ϕ p ( y ′ ( 1 ) ) = 0 , y ″ ( 0 ) = 0 , $$ \textstyle\begin{cases} (\phi_{p} ( y'' )) ' + b ( t ) g ( t, y ( t ) ) = 0, \quad 0 < t < 1, \\ \lambda_{1}\phi_{p} ( y ( 0 ) ) - \beta_{1} \phi_{p} ( y' ( 0 ) ) = 0, \\ \lambda_{2}\phi_{p} ( y ( 1 ) ) + \beta_{2} \phi_{p} ( y' ( 1 ) ) = 0,\qquad y'' ( 0 ) = 0, \end{cases} $$ where ϕ p ( s ) = | s | p − 2 s $\phi_{p} ( s ) = | s | ^{ p-2 } s$ , p > 1 $p >1 $ . Constructing an available integral operator and combining fixed point index theory, we establish some optimal criteria for the existence of bounded positive solutions. The interesting point of the results is that the term b ( t ) $b ( t ) $ may be singular at t = 0 $t=0$ and/or t = 1 $t=1$ . Moreover, the nonlinear term g ( t , y ) $g(t, y)$ is also allowed to have singularity at y = 0 $y=0$ . In particular, our results extend and unify some known results.http://link.springer.com/article/10.1186/s13662-017-1153-yconeexistencepositive solutionmaximum principle |
| spellingShingle | Yan Sun Positive solutions for one-dimensional third-order p-Laplacian boundary value problems Advances in Difference Equations cone existence positive solution maximum principle |
| title | Positive solutions for one-dimensional third-order p-Laplacian boundary value problems |
| title_full | Positive solutions for one-dimensional third-order p-Laplacian boundary value problems |
| title_fullStr | Positive solutions for one-dimensional third-order p-Laplacian boundary value problems |
| title_full_unstemmed | Positive solutions for one-dimensional third-order p-Laplacian boundary value problems |
| title_short | Positive solutions for one-dimensional third-order p-Laplacian boundary value problems |
| title_sort | positive solutions for one dimensional third order p laplacian boundary value problems |
| topic | cone existence positive solution maximum principle |
| url | http://link.springer.com/article/10.1186/s13662-017-1153-y |
| work_keys_str_mv | AT yansun positivesolutionsforonedimensionalthirdorderplaplacianboundaryvalueproblems |