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We study the existence and positivity of solutions of a highly nonlinear periodic differential equation. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ a modification of Krasnoselskii’s fixed point theorem introduced by T. A. Burton ([4], Theorem 3) to show the existence and positivity of solutions of the equation.

Existence and uniqueness of a positive solution for a third-order three-point boundary-value problem.

Palamides, Alex P., Stavrakakis, Nikolaos M. (2010)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and uniqueness of nontrivial solutions for nonlinear higher-order three-point eigenvalue problems on time scales.

Han, Wei, Kao, Yonggui (2008)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and uniqueness of positive solution for a singular nonlinear second-order $m$ -point boundary value problem.

Lv, Xuezhe, Pei, Minghe (2010)

Boundary Value Problems [electronic only]

Existence and uniqueness of positive solutions for a BVP with a $p$ -Laplacian on the half-line.

Tian, Yu, Ge, Weigao (2009)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and uniqueness of positive solutions for a third-order three-point problem on time scales.

Xu, Huiye (2010)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Existence and uniqueness of positive solutions for Neumann problems of second order impulsive differential equations.

Zhai, Chengbo, Song, Ruipeng (2010)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Existence and uniqueness of smooth positive solutions to a class of singular $m$ -point boundary value problems.

Du, Xinsheng, Zhao, Zengqin (2009)

Boundary Value Problems [electronic only]

Existence and uniqueness results for perturbed Neumann boundary value problems.

Zhang, Jieming, Zhai, Chengbo (2010)

Boundary Value Problems [electronic only]

Existence, multiplicity and infinite solvability of positive solutions for $p$ -Laplacian dynamic equations on time scales.

Wang, Da-Bin (2006)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale

Christopher S. Goodrich (2013)

Commentationes Mathematicae Universitatis Carolinae

We consider the existence of at least one positive solution to the dynamic boundary value problem $\begin{matrix} - y^{Δ Δ} (t) & = λ f (t, y (t)), t \in {[0, T]}_{𝕋} y (0) & = \int_{τ_{1}}^{τ_{2}} F_{1} (s, y (s)) Δ s y (σ^{2} (T)) & = \int_{τ_{3}}^{τ_{4}} F_{2} (s, y (s)) Δ s, \end{matrix}$ where $𝕋$ is an arbitrary time scale with $0 < τ_{1} < τ_{2} < σ^{2} (T)$ and $0 < τ_{3} < τ_{4} < σ^{2} (T)$ satisfying $τ_{1}$ , $τ_{2}$ , $τ_{3}$ , $τ_{4} \in 𝕋$ , and where the boundary conditions at $t = 0$ and $t = σ^{2} (T)$ can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples.