Positivity and hierarchical structure of Green's functions of 2-point boundary value problems for bending of a beam (Q5944171)

The authors study the following boundary value problems for fourth-order ordinary differential equations on a finite interval denoted by \((m_0,m_1;n_0,n_1)\) \[ L u(x) \equiv u^{(4)}(x)-p u''(x) +q u(x)=f(x),\quad x\in (0,L); \] \[ u^{(m_i)}(0)=\alpha_i, \quad u^{(n_i)}(L)=\beta_i, \quad i=0,1. \] Here, \(f\) is a given function, \(\alpha_0\), \(\alpha_1\), \(\beta_0\) and \(\beta_1\) are given constants, and the coefficients \(p\) and \(q\) are positive constants such \((p/2)^2>q\). The authors focus their attention on the boundary value problems \((0,1;0,1)\), \((0,1;0,2)\), \((0,1;\) \(1,3)\), \((0,2;0,2)\), \((0,2;1,3)\) and \((1,3;1,3)\). Thus, existence and uniqueness results are proved and an expression of Green's functions is given that depend on the values of the unique solution to the Cauchy problem \(Lu(x) =0\); \(x \in (0,L)\); \(u(0)=u'(0)=u''(0)=0\), \(u'''(0)=1\). Finally, the authors prove some order relations among the values of the different Green functions in \([0,T] \times [0,T]\) depending on the parameters \(m_i\) and \(n_i\), and in this last case, problems \((0,2;0,1)\), \((1,3;0,1)\) and \((1,3;0,2)\) are also included.