July/August 2011 Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators
Giovanni Cupini, Cristina Marcelli, Francesca Papalini
Differential Integral Equations 24(7/8): 619-644 (July/August 2011). DOI: 10.57262/die/1356628826

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.

Citation

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Giovanni Cupini. Cristina Marcelli. Francesca Papalini. "Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators." Differential Integral Equations 24 (7/8) 619 - 644, July/August 2011. https://doi.org/10.57262/die/1356628826

Information

Published: July/August 2011
First available in Project Euclid: 27 December 2012

zbMATH: 1249.34100
MathSciNet: MR2830312
Digital Object Identifier: 10.57262/die/1356628826

Subjects:
Primary: 34B15 , 34B40 , 34C37 , 34L30

Rights: Copyright © 2011 Khayyam Publishing, Inc.

Vol.24 • No. 7/8 • July/August 2011
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