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Positive solutions of nabla fractional boundary value problem

Author
Gopal, N. S.

Jonnalagadda, J. M.

Full text
https://cubo.ufro.cl/ojs/index.php/cubo/article/view/3210
10.56754/0719-0646.2403.0467
Abstract
In this article, we consider the following two-point discrete fractional boundary value problem with constant coefficient associated with Dirichlet boundary conditions. \begin{align*}\begin{cases} -\big{(}\nabla^{\nu}_{\rho(a)}u\big{)}(t) + \lambda u(t) = f(t, u(t)), \quad t \in \mathbb{N}^{b}_{a + 2}, \\u(a) = u(b) = 0, \end{cases} \end{align*} where \(1 < \nu < 2\), \(a,b \in \mathbb{R}\) with \(b-a\in\mathbb{N}_{3}\), \(\mathbb{N}^b_{a+2} = \{a+2,a+3, . . . ,b\}\), \(|\lambda| < 1\), \(\nabla^{\nu}_{\rho(a)}u\) denotes the \(\nu^{\text{th}}\)-order Riemann–Liouville nabla difference of \(u\) based at \(\rho(a)=a-1\), and \(f : \mathbb{N}^{b}_{a + 2} \times \mathbb{R} \rightarrow \mathbb{R}^{+}\). We make use of Guo–Krasnosels'kiÄ­ and Leggett–Williams fixed-point theorems on suitable cones and under appropriate conditions on the non-linear part of the difference equation. We establish sufficient requirements for at least one, at least two, and at least three positive solutions of the considered boundary value problem. We also provide an example to demonstrate the applicability of established results.
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Artes, Arquitectura y UrbanismoCiencias Agrarias, Forestales y VeterinariasCiencias Exactas y NaturalesCiencias SocialesDerechoEconomía y AdministraciónFilosofía y HumanidadesIngenieríaMedicinaMultidisciplinarias
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Universidad de ChileUniversidad Católica de ChileUniversidad de Santiago de ChileUniversidad de ConcepciónUniversidad Austral de ChileUniversidad Católica de ValparaísoUniversidad del Bio BioUniversidad de ValparaísoUniversidad Católica del Nortemore

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