Existence and uniqueness of solutions of a fourth-order boundary value problem with non-homogeneous boundary conditions

Kaufmann, Eric R. and Whaley, Zach (2023) Existence and uniqueness of solutions of a fourth-order boundary value problem with non-homogeneous boundary conditions. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (59). pp. 1-11. ISSN 1417-3875

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Abstract

Let m ≥ 2 and a,b,c > 0. We consider the existence and uniqueness of solutions for the fourth order iterative boundary value problem, x(4)(t) = −f(t,x(t),x[2](t),. . ., x[m](t)), −a ≤t≤a where x[2](t) =x x(t) and for j = 3,...,m, x[j](t) = x x[j−1](t) , with solutions satisfying one of the following sets of conjugate boundary conditions: x(−a) = −a, x(−a) = −a, x′(−a) = b, x(a) = a, x′′(−a) = c, x′(a) = b, x(a) = a, x′′(a) = c. The main tool used is the Schauder fixed point theorem.

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