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.

中文翻译：

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具有 $$\phi $$ ϕ -Laplacians 的非线性二阶耦合系统的异宿和同宿解

在本文中，我们给出了实线上二阶耦合微分方程组异宿解或同宿解存在的充分条件。我们指出只需要同胚上的条件，非线性上不假设增长或渐近条件。这些论证利用了不动点理论、\(L^{1}\) -Carathéodory 函数和 Schauder 不动点定理。对一族二自由度二阶非线性耦合系统的应用表明了主定理的适用性。