Solvability for second-order nonlocal boundary value problems with a p-Laplacian at resonance on a half-line

Aijun Yang, Beijing Institute of Technology, Beijing, P. R. China
Chunmei Miao, Beijing Institute of Technology, Beijing, P. R. China
Weigao Ge, Beijing Institute of Technology, Beijing, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 19. (2009), pp. 1-15.

Abstract: This paper investigates the solvability of the second-order boundary value problems with the one-dimensional $p$-Laplacian at resonance on a half-line
$$
\left\{\begin{array}{llll}
(c(t)\phi_{p}(x'(t)))'=f(t,x(t),x'(t)),~~~~0<t<\infty,\\
x(0)=\sum\limits_{i=1}\limits^{n}\mu_ix(\xi_{i}),
~~\lim\limits_{t\rightarrow +\infty}c(t)\phi_{p}(x'(t))=0
\end{array}\right.
$$
and
$$\left\{\begin{array}{llll}
(c(t)\phi_{p}(x'(t)))'+g(t)h(t,x(t),x'(t))=0,~~~~0<t<\infty,\\
x(0)=\int_{0}^{\infty}g(s)x(s)ds,~~\lim\limits_{t\rightarrow
+\infty}c(t)\phi_{p}(x'(t))=0
\end{array}\right.
$$
with multi-point and integral boundary conditions, respectively, where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. The arguments are based upon an extension of Mawhin's continuation theorem due to Ge. And examples are given to illustrate our results.

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