Existence of positive solutions to fourth order quasilinear boundary value problems (Q882755)

The authors study the fourth order boundary value problem \[ x^{(4)}=f(t,x,x',x'',x'''),\quad t\in[0,1],\qquad x(0)=x(1)=x''(0)=x''(1)=0. \tag{1} \] where \(f:[0,1]\times{\mathbb R}^4\to {\mathbb R}\) is a continuous function. They use fixed point theory in cones. First they state and prove an abstract theorem: let \(P\) be a cone in a Banach space, \(T:P\to P\) completely continuous; if \(\Omega_1\subset \Omega_2\) are constructed as the intersection of sublevel sets of a finite number of certain functionals, then the behaviour of \(T\) on the boundary of these sets allows to prove the existence of a fixed point of \(T\) in \((\bar \Omega_2\setminus \Omega_1)\cap P\). In the application to (1), they work in the cone of positive, concave functions and obtain one such solution if, for \(L_i>r_i>0\), \(i=1,2,3,4\), \(f\) satisfies \[ f(t, x_1,x_2,x_3,x_4)\geq\max_i\frac{r_i}{M_i}\text{ in } [0,1]\times[0,r_1]\times[-r_2,r_2]\times[-r_3,0]\times[-r_4,r_4] \] and \[ f(t, x_1,x_2,x_3,x_4)\leq\min_i\frac{L_i}{M_i}\text{ in }[0,1]\times[0,L_1]\times[-L_2,L_2]\times[-L_3,0]\times[-L_4,L_4]. \] Here, the \(M_i\) are constants related to a Green's function.