Existence and uniqueness of solutions for third order nonlinear boundary value problems (Q1802554)

The paper deals with the third order differential equation (1) \(y'''=f(x,y,y',y'')\) and the boundary conditions (2) \(ay'(0)-by''(0)=A\), \(y(1)=B\), \(y'(1)=C\) or (3) \(y(0)=A\), \(y'(0)=B\), \(ay'(1)+by''(1)=C\), where \(A,B,C,a,b\in\mathbb{R}\), \(a,b\geq 0\), \(a+b>0\), and \(f\) is supposed to be continuous on \([0,1]\times\mathbb{R}^ 3\). The author constructs upper and lower solutions for problem (1), (2) and then he proves the existence and uniqueness for the problem provided (1) has a trivial solution and partial derivatives of \(f\) satisfy certain inequalities. Similar results are proved for problem (1), (3). Four examples are given.