Multiplicity results of positive solutions for fourth-order nonlinear differential equation with singularity (Q2795292)

The paper is devoted to the existence of positive \(\omega\)-periodic solutions of the differential equation NEWLINE\[NEWLINEx^{(4)}(t)+ax'''(t)+bx''(t)+cx'(t)+dx(t)=f(t,x(t))+e(t),NEWLINE\]NEWLINE where \(a,b,c,d\in \mathbb{R}\), \(e(t)\in L^1(\mathbb{R})\) is a \(\omega\)-periodic, \(f:\mathbb{R}\times(0,\infty)\to \mathbb{R}\) is a \(L^2\)-Carathéodory, \(\omega\)-periodic with respect to \(t\) function, which may be singular at the origin, that is, \(\lim_{t\to0^+}f(t,x)=+\infty\) (or \(\lim_{t\to0^+}f(t,x)=-\infty\)) uniformly on \(t\).NEWLINENEWLINEThe authors give firstly the Green's function for the equation NEWLINE\[NEWLINEx^{(4)}(t)+ax'''(t)+bx''(t)+cx'(t)+dx(t)=h(t),NEWLINE\]NEWLINE where \(h:\mathbb{R}\to(0,+\infty)\) is continuous and \(\omega\)-periodic. Next, using it and applying a Leray-Schauder alternative principle or the Schauder fixed point theorem, they establish a variety of results guaranteeing at least one solution to the considered problem.