Existence of homoclinic solutions for nonlinear second-order coupled systems

Minhós, Feliz; Sousa, Robert de

doi:10.1016/j.jde.2018.07.072

Cited by 6 publications

(2 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As can be seen in the Introduction, sufficient conditions for the existence of heteroclinic solutions require strong assumptions on the nonlinearities. The goal of this paper is to weaken these conditions on the nonlinearity f , replacing them by assumptions on the inverse of the homeomorphism φ, following the ideas and methods suggested in [27,28].…”

Section: Discussionmentioning

confidence: 99%

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

Minhós

2019

Axioms

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

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

Minhós

2019

Axioms

Self Cite

View full text Add to dashboard Cite

show abstract

“…Motivated by these works and applying the techniques suggested in [16,20,21,22], we apply the fixed point theory, to obtain sufficient conditions for the existence of heteroclinic solutions of the coupled system (1), (2), assuming some adequate conditions on φ −1 , ψ −1 .…”

Section: Introductionmentioning

confidence: 99%

Heteroclinic and homoclinic solutions for nonlinear second-order coupled systems with phi-Laplacians

Sousa¹,

Minhós²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we present sufficient conditions for the existence of heteroclinic or homoclinic solutions for second order coupled systems of differential equations on the real line.We point out that it is required only conditions on the homeomorphisms and no growth or asymptotic conditions are assumed on the nonlinearities.The arguments make use of the fixed point theory, L 1 -Carathéodory functions and Schauder's fixed point theorem.An application to a family of second order nonlinear coupled systems of two degrees of freedom, shows the applicability of the main theorem.

show abstract

Lower and Upper Solutions for System of Differential Equations Involving Homeomorphism and Nonlinear Boundary Conditions

Rodríguez–López,

Szymańska-Dȩbowska,

Zima

2024

Results Math

View full text Add to dashboard Cite

We study the existence of solution to the system of differential equations $$(\phi (u'))'=f(t,u,u')$$ ( ϕ ( u ′ ) ) ′ = f ( t , u , u ′ ) with nonlinear boundary conditions $$\begin{aligned} g(u(0),u,u')=0, \quad h(u'(1),u,u')=0, \end{aligned}$$ g ( u ( 0 ) , u , u ′ ) = 0 , h ( u ′ ( 1 ) , u , u ′ ) = 0 , where $$f:[0,1]\times \mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$$ f : [ 0 , 1 ] × R n × R n → R n , $$g,h:\mathbb {R}^{n}\times C([0,1],\mathbb {R}^{n})\times C([0,1],\mathbb {R}^{n})\rightarrow \mathbb {R}^{n}$$ g , h : R n × C ( [ 0 , 1 ] , R n ) × C ( [ 0 , 1 ] , R n ) → R n are continuous, $$\phi :\prod _{i=1}^{n}(-a_i,a_i) \rightarrow \mathbb {R}^{n}$$ ϕ : ∏ i = 1 n ( - a i , a i ) → R n , $$0<a_i\le +\infty $$ 0 < a i ≤ + ∞ , $$\phi (s)=\left( \phi _1(s_1),\dots ,\phi _n(s_n)\right) $$ ϕ ( s ) = ϕ 1 ( s 1 ) , ⋯ , ϕ n ( s n ) and $$\phi _i:(-a_i,a_i)\rightarrow \mathbb {R}$$ ϕ i : ( - a i , a i ) → R is a one dimensional regular or singular homeomorphism. Our proofs are based on the concept of the lower and upper solutions.

show abstract

Existence of homoclinic solutions for nonlinear second-order coupled systems

Cited by 6 publications

References 21 publications

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

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

Heteroclinic and homoclinic solutions for nonlinear second-order coupled systems with phi-Laplacians

Lower and Upper Solutions for System of Differential Equations Involving Homeomorphism and Nonlinear Boundary Conditions

Contact Info

Product

Resources

About