On the solvability and asymptotics of solutions to a functional-differential equation with singularity (Q2754819)

The author studies the following functional-differential equation NEWLINE\[NEWLINE\alpha(t)\dot x(t)=at+b_1x(t)+b_2x(g(t))+\varphi(t,x(t),x(g(t)),\dot x(t),\dot x(h(t))) .\tag{1}NEWLINE\]NEWLINE Here, \(a,b_1,b_2\in \mathbb{R}, \alpha(t),g(t),h(t)\in C((0,\tau ] \mapsto(+0,\infty)), g(t)\leq t, h(t)\leq t\), and \(\lim_{t\to 0}\alpha(t)=0\). The function \(\varphi(t,x,y,u,v)\) is continuous in a domain with the corner point at the origin. The author seeks for solutions to (1) from the set \(U(\rho,M)\) which consists of functions \(u(t)\in C^1((0,\rho) \mapsto \mathbb{R})\) satisfying the inequalities \(|u(t)|\leq Mt\xi(t), |\dot u(t)|\leq Mt/\alpha(t)\), with NEWLINE\[NEWLINE \xi(t):=I(t,\sigma):=\sigma \left(\int_{(1-\sigma)\tau /2}^{t} (\alpha(r))^{-1} dr\right)^{\sigma },NEWLINE\]NEWLINE and if \(\lim_{s\to 1-0}I(t,s)<\infty\), then \(\sigma =1\), else \(\sigma =-1\). Conditions are found under which equation (1) has at least one solution \(x(t)\) from \(U_1(\rho,M)\) or infinitely many solutions from such a class. The proof is based on the Schauder fixed-point theorem and Wazewski's topological principle.