# Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale

- Saroj Panigrahi panigrahi2008@gmail.com
- Sandip Rout sandiprout7@gmail.com

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https://doi.org/10.56754/0719-0646.2403.0413## Abstract

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.

## Keywords

D. R. Anderson and C. Zhai, “Positive solutions to semi-positone second-order three-point problems on time scale”, Appl. Math. Comput., vol. 215, no. 10, pp. 3713–3720, 2010.

D. R. Anderson and P. J. Y. Wong, “Positive solutions for second-order semipositone problems on time scales”, Comput. Math. Appl., vol. 58, no. 2, pp. 281–291, 2009.

R. Aris, Introduction to the analysis of chemical reactors, New Jersey: Prentice Hall, Engle- wood Cliffs, 1965.

D. Bai, and Y. Xu, “Positive solutions for semipositone BVPs of second-order difference equations”, Indian J. Pure Appl. Math., vol. 39, no.1, pp. 59–68, 2008.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Ap- plications, Boston: Birkh Ìˆauser, 2001.

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Boston: Birkhäuser, 2003.

R. Dahal, “Positive solutions for a second-order, singular semipositone dynamic boundary value problem”, Int. J. Dyn. Syst. Differ. Equ., vol. 3, no. 1–2, pp. 178–188, 2011.

L. Erbe and A. Peterson, “Positive solutions for a nonlinear differential equations on a measure chain”, Math. Comput. Modelling, vol. 32, no. 5–6, pp. 571–585, 2000.

C. Giorgi and E. Vuk, “Steady-state solutions for a suspension bridge with intermediate supports”, Bound. Value Probl., Paper No. 204, 17 pages, 2013.

C. S. Goodrich, “Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale”, Comment. Math. Univ. Carolin., vol. 54, no. 4, pp. 509–525, 2013.

S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus”, Results Math., vol. 18, no. 1–2, pp. 18–56, 1990.

G. Infante, P. Pietramala and M. Tenuta, “Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory”, Commun. Nonlinear Sci. Numer. Simul., vol. 19, no 7, pp. 2245–2251, 2014.

M. A. Krasnosel‘skii, Positive Solutions of Operator Equations, Groningen: P. Noordhoff, 1964.

E. Kreyszig, Introductory Functional Analysis With Applications, New York: John Wiley & Sons, Inc., 1978.

J. Selgrade, “Using stocking and harvesting to reverse period-doubling bifurcations in models in population biology”, J. Differ. Equations Appl., vol. 4, no. 2, pp. 163–183, 1998.

J. P. Sun and W. T. Li, “Existence of positive solutions to semipositone Dirichlet BVPs on time scales”, Dynam. Systems Appl., vol. 16, no. 3, pp. 571–578, 2007.

Y. Yang and F. Meng, “Positive solutions of the singular semipositone boundary value problem on time scales”, Math. Comput. Modelling, vol. 52, no. 3–4, pp. 481–489.

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*CUBO*, vol. 24, no. 3, pp. 413–437, Dec. 2022.