Existence of solutions for higher order \(\phi-\)Laplacian BVPs on the half-line using a one-sided Nagumo condition with nonordered upper and lower solutions

A. Zerki; K. Bachouche; K. Ait-Mahiout

doi:10.56754/0719-0646.2502.173

DOI:

https://doi.org/10.56754/0719-0646.2502.173

Abstract

In this paper, we consider the following \((n+1)\)st order bvp on the half line with a \(\phi-\)Laplacian operator \[ \begin{cases} (\phi(u^{(n)}))'(t) = f(t,u(t),\ldots,u^{(n)}(t)), & \text{a.e.},\, t\in [0,+\infty), \\ n \in \mathbb{N}\setminus\{0\}, \\ \\ u^{(i)}(0) = A_{i}, \, i=0,\ldots,n-2, \\ u^{(n-1)}(0) + au^{(n)}(0) = B, \\ u^{(n)}(+\infty) = C. \end{cases} \]

The existence of solutions is obtained by applying Schaefer's fixed point theorem under a one-sided Nagumo condition with nonordered lower and upper solutions method where \(f\) is a \(L^{1}\)-Carathéodory function.

Keywords

Boundary value problem , One-sided Nagumo condition , Lower and upper solutions , A priori estimates

Mathematics Subject Classification:

34B10 , 34B15 , 34B40

A. Zerki Laboratoire “Théorie du point fixe et Applications”, École Normale Supérieure, BP 92, Kouba, 16006, Algiers, Algeria. https://orcid.org/0000-0002-2057-3370
K. Bachouche Departement of Mathematics, Faculty of Sciences, Algiers University 1, Algiers, Algeria and Laboratory of Mathematical Analysis and Applications, Algiers, Algeria. https://orcid.org/0000-0002-4619-9698
K. Ait-Mahiout Laboratoire “Théorie du point fixe et Applications”, École Normale Supérieure, BP 92, Kouba, 16006, Algiers, Algeria. https://orcid.org/0000-0002-5991-0442

Pages: 173–193
Date Published: 2023-07-19
Vol. 25 No. 2 (2023)

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