2019 Lower and Upper Solutions Method for Nonlinear Second-order Differential Equations Involving a $\Phi-$Laplacian Operator
D. A. Behi, A. Adje, K. C. Goli
Afr. Diaspora J. Math. (N.S.) 22(1): 22-41 (2019).

Abstract

In this paper, we consider the following nonlinear second-order differential equations: $-(\Phi(u'(t)))' = f (t, u(t), u'(t)) + \Xi (u(t)) \text { a.e on } \Omega = [0, T]$ with a discontinuous perturbation and multivalued boundary conditions. The nonlinear differential operator is not necessarily homogeneous and incorporates as a special case the one-dimensional p-Laplacian. By combining lower and upper solutions method, a fixed point theorem for multifunction and theory of monotone operators, we show the existence of solutions and existence of extremal solutions in the order interval $[\alpha, \beta]$ where $\alpha$ and $\beta$ are assumed respectively an ordered pair of lower and upper solutions of the problem. Moreover, we show that our method of proof also applies to the periodic problem.

Citation

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D. A. Behi. A. Adje. K. C. Goli. "Lower and Upper Solutions Method for Nonlinear Second-order Differential Equations Involving a $\Phi-$Laplacian Operator." Afr. Diaspora J. Math. (N.S.) 22 (1) 22 - 41, 2019.

Information

Published: 2019
First available in Project Euclid: 20 August 2019

zbMATH: 07161359
MathSciNet: MR3992764

Subjects:
Primary: 34B15

Keywords: $\Phi-$laplacian , Bernstein-Nagumo-Winter condition , extremal solutions , fixed point , lower and upper solutions , multifunctions , truncation and penalty functions

Rights: Copyright © 2019 Mathematical Research Publishers

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Vol.22 • No. 1 • 2019
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