Existence and uniqueness results for third-order nonlinear differential systems (Q426280)

The author consideres the third order nonlinear differential system NEWLINE\[NEWLINEx''' = f(t,x,x',x''),\quad 0 < t < 1, \tag{1}NEWLINE\]NEWLINE together with the nonlinear boundary conditions NEWLINE\[NEWLINE x(0) = A,\quad g(x'(0),x''(0)) = 0,\quad h(x'(1),x''(1)) = 0,\tag{2}NEWLINE\]NEWLINE where \(x \in \mathbb R^n,\;A = (a_1 ,a_2 ,\dots,a_n )^T \in \mathbb R^n,\;f \in C([0,1]\times \mathbb R ^{3n},\mathbb R^n),\;g,h \in C^1(\mathbb R^{2n},\mathbb R^n).\)NEWLINENEWLINEUsing the method of upper and lower solutions and the method of descent, the author discusses the existence of solutions of (1), (2). By constructing appropriate upper and lower solutions, existence and uniqueness results for (1), (2) are obtained.