Positive periodic solutions to the planar Lp dual Minkowski problem in the critical case

In this paper, we investigate the planar $L_p$ dual Minkowski problem$x^{″}+x=x^{p-1}{(x^2+x^{\prime 2})}^{\frac{2-q}{2}}f\left(t\right),$ where p and q are two constants, $f\in L^1(R/TZ;R^{+})$, and $T>0$. Notice that in the critical case $T=\pi$, the $L_p$ dual Minkowski problem is closely related to the half-period symmetry problem in convex geometry. By using Krasnosel'skii-Guo fixed point theorem and Schauder's fixed point theorem, we derive sufficient conditions for the existence of positive π-periodic solutions to this equation. In addition, we employ numerical bifurcation analysis to explore the dynamical behavior of positive π-periodic solutions.