Electron. J. Differential Equations, Vol. 2018 (2018), No. 150, pp. 1-6.

Existence and uniqueness of solutions for a second-order iterative boundary-value problem

Eric R. Kaufmann

Abstract:
We consider the existence and uniqueness of solutions to the second-order iterative boundary-value problem
$$ x''(t) = f(t, x(t), x^{[2]}(t)), \quad a \leq t \leq b, $$
where $x^{[2]}(t) = x(x(t))$, with solutions satisfying one of the boundary conditions $x(a) = a$, $x(b) = b$ or $x(a) = b$, $x(b) = a$. The main tool employed to establish our results is the Schauder fixed point theorem.

Submitted September 20 2017. Published August 8, 2018.
Math Subject Classifications: 34B15, 34K10, 39B05.
Key Words: Iterative differential equation; Schauder fixed point theorem; contraction mapping principle.

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Eric R. Kaufmann
Department of Mathematics & Statistics
University of Arkansas at Little Rock
Little Rock, AR 72204, USA
email: erkaufmann@ualr.edu

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