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

Liu, Min. "Uncountably many solutions of a system of third order nonlinear differential equations." Commentationes Mathematicae Universitatis Carolinae 52.3 (2011): 369-389. <http://eudml.org/doc/246859>.

@article{Liu2011,
abstract = {In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations \[ \aligned & \frac\{d\}\{dt\}\Big \lbrace r\_i(t)\frac\{d\}\{dt\}\Big [\lambda \_i(t)\frac\{d\}\{dt\} \Big (x\_i(t)-f\_i(t,x\_1(t-\sigma \_\{i1\}),x\_2(t-\sigma \_\{i2\}), x\_3(t-\sigma \_\{i3\}))\Big )\Big ]\Big \rbrace \cr & \qquad \quad + \frac\{d\}\{dt\}\Big [r\_i(t)\frac\{d\}\{dt\}g\_i(t,x\_1(p\_\{i1\}(t)), x\_2(p\_\{i2\}(t)),x\_3(p\_\{i3\}(t)))\Big ] \cr & \qquad \quad + \frac\{d\}\{dt\}h\_i(t,x\_1(q\_\{i1\}(t)),x\_2(q\_\{i2\}(t)), x\_3(q\_\{i3\}(t))) \cr & = l\_i(t,x\_1(\eta \_\{i1\}(t)),x\_2(\eta \_\{i2\}(t)),x\_3(\eta \_\{i3\}(t))), \quad t\ge t\_0,\quad i\in \lbrace 1,2,3\rbrace \endaligned \] in the following bounded closed and convex set \[ \aligned \Omega (a,b)=\Big \lbrace x(t)=\big (x\_1(t),x\_2(t),x\_3(t)\big )\in C([t\_0,+\infty ),\mathbb \{R\}^3):a(t)\le x\_i(t)\le b(t), \qquad \forall \, t\ge t\_0, i\in \lbrace 1,2,3\rbrace \Big \rbrace , \qquad \endaligned \] where $\sigma _\{ij\}>0$, $r_i,\lambda _i,a,b\in C([t_0,+\infty ),\mathbb \{R\}^\{+\})$, $f_i,g_i,h_i,l_i\in C([t_0,+\infty )\times \mathbb \{R\}^3,\mathbb \{R\})$, $p_\{ij\},q_\{ij\},\eta _\{ij\}\in C([t_0,+\infty ),\mathbb \{R\})$ for $i,j\in \lbrace 1,2,3\rbrace $. 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.},
author = {Liu, Min},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {system of third order nonlinear neutral delay differential equations; contraction mapping; completely continuous mapping; condensing mapping; uncountably many bounded positive solutions; neutral delay differential equation; third order nonlinear system; positive solution; contraction mapping; condensing mapping},
language = {eng},
number = {3},
pages = {369-389},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Uncountably many solutions of a system of third order nonlinear differential equations},
url = {http://eudml.org/doc/246859},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Liu, Min
TI - Uncountably many solutions of a system of third order nonlinear differential equations
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 3
SP - 369
EP - 389
AB - In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations \[ \aligned & \frac{d}{dt}\Big \lbrace r_i(t)\frac{d}{dt}\Big [\lambda _i(t)\frac{d}{dt} \Big (x_i(t)-f_i(t,x_1(t-\sigma _{i1}),x_2(t-\sigma _{i2}), x_3(t-\sigma _{i3}))\Big )\Big ]\Big \rbrace \cr & \qquad \quad + \frac{d}{dt}\Big [r_i(t)\frac{d}{dt}g_i(t,x_1(p_{i1}(t)), x_2(p_{i2}(t)),x_3(p_{i3}(t)))\Big ] \cr & \qquad \quad + \frac{d}{dt}h_i(t,x_1(q_{i1}(t)),x_2(q_{i2}(t)), x_3(q_{i3}(t))) \cr & = l_i(t,x_1(\eta _{i1}(t)),x_2(\eta _{i2}(t)),x_3(\eta _{i3}(t))), \quad t\ge t_0,\quad i\in \lbrace 1,2,3\rbrace \endaligned \] in the following bounded closed and convex set \[ \aligned \Omega (a,b)=\Big \lbrace x(t)=\big (x_1(t),x_2(t),x_3(t)\big )\in C([t_0,+\infty ),\mathbb {R}^3):a(t)\le x_i(t)\le b(t), \qquad \forall \, t\ge t_0, i\in \lbrace 1,2,3\rbrace \Big \rbrace , \qquad \endaligned \] where $\sigma _{ij}>0$, $r_i,\lambda _i,a,b\in C([t_0,+\infty ),\mathbb {R}^{+})$, $f_i,g_i,h_i,l_i\in C([t_0,+\infty )\times \mathbb {R}^3,\mathbb {R})$, $p_{ij},q_{ij},\eta _{ij}\in C([t_0,+\infty ),\mathbb {R})$ for $i,j\in \lbrace 1,2,3\rbrace $. 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.
LA - eng
KW - system of third order nonlinear neutral delay differential equations; contraction mapping; completely continuous mapping; condensing mapping; uncountably many bounded positive solutions; neutral delay differential equation; third order nonlinear system; positive solution; contraction mapping; condensing mapping
UR - http://eudml.org/doc/246859
ER -