| Abstract: |
| We study the nonlocal nonlinear Neumann boundary value problem of the following form
$$
x``=f(t,x,x`),\qquad x(0)=a,\qquad x`(1)=N(x`),
$$
where $a\in\mathbb R^n$ is fixed, $t\in[0,1]$, $f:[0,1]\times \mathbb R^n\times\mathbb R^n\to\mathbb R^n$ is continuous,
$N:C([0,1],\mathbb R^n)\to\mathbb R^n$ is a continuous and not necessarily linear application. The case of linear boundary conditions are also considered, i.e.
$$
N(y):=\int_0^1 y(s)\, dg(s),
$$
where $g=\mbox{diag}(g_1,\ldots,g_n)$ with $g_i:[0,1]\to\mathbb R$, $i=1,\ldots,n$. Using different topological methods, we show that the problem under consideration has at least one solution.
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\noindent
{\bf \small Bibliography:}\
[1] J. Mawhin, K. Szyma\`nska-D\c ebowska, Convex sets and second order systems with nonlocal boundary conditions at resonance, Proc. Amer. Math. Soc., 145 (2017), 2023--2032.\
[2] K. Szyma\`nska-D\c ebowska, The solvability of a nonlocal boundary value problem, Math. Slovaca, 65 (2015), 1027--1034.\ |
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