Positive solution of a singular nonlinear third-order periodic boundary value problem (Q5946102)

Here, the authors study the nonlinear periodic boundary value problem NEWLINE\[NEWLINEu'''+\rho^3 u=f(t,u),\;0\leq t\leq 2\pi, \quad u^{(i)}(0)= u^{(i)}(2\pi), \;i=0,1,2, \tag{A}NEWLINE\]NEWLINE where \(\rho\) is a positive constant.NEWLINENEWLINENEWLINEA function \(u(t)\) is said to be a positive solution to problem (A), if it satisfiesNEWLINENEWLINENEWLINE(1) \(u\in C^2\langle 0,2\pi \rangle\), \(u^{(i)}(0)= u^{(i)}(2\pi)\), \(i=0,1,2\);NEWLINENEWLINENEWLINE(2) \(u'''\) exists a.e. and \(u(t)>0\) on \(\langle 0,2\pi\rangle\);NEWLINENEWLINENEWLINE(3) \(u'''(t)+\rho^3 u(t)= f(t,u(t))\) a.e. on \(\langle 0,2\pi \rangle\).NEWLINENEWLINENEWLINEUnder certain (real?) hypotheses, by employing perturbation technique and Schauder fixed-point theorem, there is proved, that problem (A) has at least one positive solution if \(\rho\in (0,{1 \over\sqrt 3})\).