Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem

The existence of n and infinitely many positive solutions is proved for the nonlinear fourth-order periodic boundary value problemu(4)(t)-βu′′(t)+αu(t)=f(t,u(t)),0⩽t⩽1,u(i)(0)=u(i)(1),i=0,1,2,3,where n is an arbitrary natural number and β>-2π2,0<α<(12β+2π2)2,α/π4+β/π2+1>0. This kind of fourth-order boundary value problems usually describes the equilibrium state of an elastic beam with periodic boundary condition. The main results show that the problem may have n or infinitely many positive solutions provided the growth rates of nonlinear term f(t,l) are appropriate on some bounded subsets of its domain.