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# Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations

Departamento de Matemática, Escola de Ciências e Tecnologia, Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora. Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
Axioms 2019, 8(1), 22; https://doi.org/10.3390/axioms8010022
Received: 28 December 2018 / Revised: 28 January 2019 / Accepted: 11 February 2019 / Published: 15 February 2019
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
In this paper, we consider the second order discontinuous differential equation in the real line, $a t , u ϕ u ′ ′ = f t , u , u ′ , a . e . t ∈ R , u ( − ∞ ) = ν − , u ( + ∞ ) = ν + ,$ with $ϕ$ an increasing homeomorphism such that $ϕ ( 0 ) = 0$ and $ϕ ( R ) = R$ , $a ∈ C ( R 2 , R )$ with $a ( t , x ) > 0$ for $( t , x ) ∈ R 2$ , $f : R 3 → R$ a $L 1$ -Carathéodory function and $ν − , ν + ∈ R$ such that $ν − < ν +$ . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities $ϕ$ and $f$ . To the best of our knowledge, this result is even new when $ϕ ( y ) = y$ , that is for equation $a t , u ( t ) u ′ ( t ) ′ = f t , u ( t ) , u ′ ( t ) , a . e . t ∈ R$ . Moreover, these results can be applied to classical and singular $ϕ$ -Laplacian equations and to the mean curvature operator. View Full-Text
MDPI and ACS Style

Minhós, F. Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations. Axioms 2019, 8, 22.

AMA Style

Minhós F. Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations. Axioms. 2019; 8(1):22.

Chicago/Turabian Style

Minhós, Feliz. 2019. "Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations" Axioms 8, no. 1: 22.

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