African Diaspora Journal of Mathematics

Lower and Upper Solutions Method for Nonlinear Second-order Differential Equations Involving a $\Phi-$Laplacian Operator

D. A. Behi, A. Adje, and K. C. Goli

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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.

Article information

Source
Afr. Diaspora J. Math. (N.S.), Volume 22, Number 1 (2019), 22-41.

Dates
First available in Project Euclid: 20 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.adjm/1566266463

Subjects
Primary: 34B15: Nonlinear boundary value problems

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

Citation

Behi, D. A.; Adje, A.; Goli, K. C. Lower and Upper Solutions Method for Nonlinear Second-order Differential Equations Involving a $\Phi-$Laplacian Operator. Afr. Diaspora J. Math. (N.S.) 22 (2019), no. 1, 22--41. https://projecteuclid.org/euclid.adjm/1566266463


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