Abstract
We are concerned with the higher-order nonlinear three-point boundary value problems: with the three point boundary conditions ; where is continuous, are continuous, and are arbitrary given constants. The existence and uniqueness results are obtained by using the method of upper and lower solutions together with Leray-Schauder degree theory. We give two examples to demonstrate our result.
Highlights
Higher-order boundary value problems were discussed in many papers in recent years; for instance, see 1–22 and references therein
We say that f t, x0, x1, . . . , xn−1 satisfies the Nagumo condition on E if there exists a continuous function φ : 0, ∞ → 0, ∞ such that f t, x0, x1, . . . , xn−1 ≤ φ |xn−1|, t, x0, x1, . . . , xn−1 ∈ E
X0, . . . , xn−1 ∈ b, c × Rn : φi t ≤ xi ≤ ψi t, i 0, . . . , n − 2, Dac t, x0, . . . , xn−1 ∈ a, c × Rn : φi t ≤ xi ≤ ψi t, i 0, . . . , n − 2 ; iii g x0, x1, . . . , xn−1 is continuous on Rn, and −1 n−ig x0, x1, . . . , xn−1 is nonincreasing in xi i
Summary
Higher-order boundary value problems were discussed in many papers in recent years; for instance, see 1–22 and references therein. Xn−1 satisfies the Nagumo condition on E if there exists a continuous function φ : 0, ∞ → 0, ∞ such that f t, x0, x1, . Let f : a, c × Rn → R be a continuous function satisfying the Nagumo condition on
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