Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale
In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales: \begin{align*} (\psi(t)y^\Delta (t))^\nabla + \lambda_1 g(t, \,y(t)) + \lambda_2 h(t,\,y(t)) = 0, \,t \in [\rho(c), \,\sigma(d)]_\mathbb{T}, \end{align*}...
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| Format: | Article |
| Language: | English |
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Universidad de La Frontera
2022-12-01
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| Series: | Cubo |
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| Online Access: | https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3207/2258 |
| _version_ | 1797678795038130176 |
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| author | Saroj Panigrahi Sandip Rout |
| author_facet | Saroj Panigrahi Sandip Rout |
| author_sort | Saroj Panigrahi |
| collection | DOAJ |
| description | In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales:
\begin{align*} (\psi(t)y^\Delta (t))^\nabla + \lambda_1 g(t, \,y(t)) + \lambda_2 h(t,\,y(t)) = 0, \,t \in [\rho(c), \,\sigma(d)]_\mathbb{T}, \end{align*}
with mixed boundary conditions
\begin{align*} \alpha y(\rho(c))-\beta \psi(\rho(c)) y^\Delta(\rho(c))=0,\\ \gamma y(\sigma(d))+\delta \psi(d) y^\Delta(d)=0, \end{align*}
where \(\psi:C[\rho(c),\, \sigma(d)]_\mathbb{T}\), \(\psi(t)>0\) for all \(t \in [\rho(c),\,\sigma(d)]_\mathbb{T}\); both \(g\) and \(h : [\rho(c),\,\sigma(d)]_\mathbb{T} \times [0,\,\infty) \to \mathbb{R}\) are continuous and semipositone. We have established the existence of at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a Banach space, when \(g\) and \(h\) are both superlinear or sublinear or one is superlinear and the other is sublinear for \(\lambda_i>0;\,i=1,\,2\) are sufficiently small. |
| first_indexed | 2024-03-11T23:05:07Z |
| format | Article |
| id | doaj.art-0474bee2d02f4e758312d8c397d6951c |
| institution | Directory Open Access Journal |
| issn | 0716-7776 0719-0646 |
| language | English |
| last_indexed | 2024-03-11T23:05:07Z |
| publishDate | 2022-12-01 |
| publisher | Universidad de La Frontera |
| record_format | Article |
| series | Cubo |
| spelling | doaj.art-0474bee2d02f4e758312d8c397d6951c2023-09-21T13:13:23ZengUniversidad de La FronteraCubo0716-77760719-06462022-12-0124341343710.56754/0719-0646.2403.0413Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scaleSaroj Panigrahi0https://orcid.org/0000-0003-4704-5102Sandip Rout1https://orcid.org/0000-0002-6575-9910School of Mathematics and Statistics, University of Hyderabad, Hyderabad, 500 046, India.School of Mathematics and Statistics, University of Hyderabad, Hyderabad, 500 046, India.In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales: \begin{align*} (\psi(t)y^\Delta (t))^\nabla + \lambda_1 g(t, \,y(t)) + \lambda_2 h(t,\,y(t)) = 0, \,t \in [\rho(c), \,\sigma(d)]_\mathbb{T}, \end{align*} with mixed boundary conditions \begin{align*} \alpha y(\rho(c))-\beta \psi(\rho(c)) y^\Delta(\rho(c))=0,\\ \gamma y(\sigma(d))+\delta \psi(d) y^\Delta(d)=0, \end{align*} where \(\psi:C[\rho(c),\, \sigma(d)]_\mathbb{T}\), \(\psi(t)>0\) for all \(t \in [\rho(c),\,\sigma(d)]_\mathbb{T}\); both \(g\) and \(h : [\rho(c),\,\sigma(d)]_\mathbb{T} \times [0,\,\infty) \to \mathbb{R}\) are continuous and semipositone. We have established the existence of at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a Banach space, when \(g\) and \(h\) are both superlinear or sublinear or one is superlinear and the other is sublinear for \(\lambda_i>0;\,i=1,\,2\) are sufficiently small.https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3207/2258positive solutionsboundary value problemsfixed point theoremconetime scales |
| spellingShingle | Saroj Panigrahi Sandip Rout Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale Cubo positive solutions boundary value problems fixed point theorem cone time scales |
| title | Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale |
| title_full | Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale |
| title_fullStr | Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale |
| title_full_unstemmed | Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale |
| title_short | Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale |
| title_sort | existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale |
| topic | positive solutions boundary value problems fixed point theorem cone time scales |
| url | https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3207/2258 |
| work_keys_str_mv | AT sarojpanigrahi existenceofpositivesolutionsforanonlinearsemipositoneboundaryvalueproblemsonatimescale AT sandiprout existenceofpositivesolutionsforanonlinearsemipositoneboundaryvalueproblemsonatimescale |