## Differential and Integral Equations

- Differential Integral Equations
- Volume 24, Number 7/8 (2011), 619-644.

### Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators

Giovanni Cupini, Cristina Marcelli, and Francesca Papalini

#### Abstract

We discuss the solvability of the following strongly nonlinear non-autonomous boundary-value problem: \[ (P) \quad \left \{ \begin{array}{ll} (a(x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \ \ \mbox{a.e. } t\in {\mathbb R} & \\ x(-\infty)=\nu^- ,\ \ x(+\infty)= \nu^+ & \end{array} \right . \] with $\nu^-< \nu^+$, where $\Phi:{\mathbb R} \to {\mathbb R}$ is a general increasing homeomorphism, with $\Phi(0)=0$, $a$ is a positive, continuous function and $f$ is a Caratheódory nonlinear function. We provide some sufficient conditions for the solvability of $(P)$ which turn out to be optimal for a large class of problems. In particular, we highlight the role played by the behavior of $f(t,x,\cdot)$ and $\Phi(\cdot)$ as $y\to 0$ related to that of $f(\cdot,x,y)$ as $|t|\to +\infty$. We also show that the dependence on $x$, both of the differential operator and of the right-hand side, does not influence in any way the existence or non-existence of solutions.

#### Article information

**Source**

Differential Integral Equations, Volume 24, Number 7/8 (2011), 619-644.

**Dates**

First available in Project Euclid: 27 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356628826

**Mathematical Reviews number (MathSciNet)**

MR2830312

**Zentralblatt MATH identifier**

1249.34100

**Subjects**

Primary: 34B40: Boundary value problems on infinite intervals 34C37: Homoclinic and heteroclinic solutions 34B15: Nonlinear boundary value problems 34L30: Nonlinear ordinary differential operators

#### Citation

Cupini, Giovanni; Marcelli, Cristina; Papalini, Francesca. Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators. Differential Integral Equations 24 (2011), no. 7/8, 619--644. https://projecteuclid.org/euclid.die/1356628826