On Hilbert-Pachpatte multiple integral inequalities.

Zhao, Changjian; Chen, Lian-Ying; Cheung, Wing-Sum

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Uncountably many solutions of a system of third order nonlinear differential equations

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In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations $\frac{d}{d t} \{r_{i} (t) \frac{d}{d t} [λ_{i} (t) \frac{d}{d t} (x_{i} (t) - f_{i} (t, x_{1} (t - σ_{i 1}), x_{2} (t - σ_{i 2}), x_{3} (t - σ_{i 3})))]\} + \frac{d}{d t} [r_{i} (t) \frac{d}{d t} g_{i} (t, x_{1} (p_{i 1} (t)), x_{2} (p_{i 2} (t)), x_{3} (p_{i 3} (t)))] + \frac{d}{d t} h_{i} (t, x_{1} (q_{i 1} (t)), x_{2} (q_{i 2} (t)), x_{3} (q_{i 3} (t))) = l_{i} (t, x_{1} (η_{i 1} (t)), x_{2} (η_{i 2} (t)), x_{3} (η_{i 3} (t))), t \geq t_{0}, i \in {1, 2, 3}$ in the following bounded closed and convex set $Ω (a, b) = \{x (t) = (x_{1} (t), x_{2} (t), x_{3} (t)) \in C ([t_{0}, + \infty), ℝ^{3}) : a (t) \leq x_{i} (t) \leq b (t), \forall t \geq t_{0}, i \in {1, 2, 3}\},$ where $σ_{i j} > 0$ , $r_{i}, λ_{i}, a, b \in C ([t_{0}, + \infty), ℝ^{+})$ , $f_{i}, g_{i}, h_{i}, l_{i} \in C ([t_{0}, + \infty) \times ℝ^{3}, ℝ)$ , $p_{i j}, q_{i j}, η_{i j} \in C ([t_{0}, + \infty), ℝ)$ for $i, j \in {1, 2, 3}$ . By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.