Boundary value problems for second order differential equations with \(\varphi \)-Laplacians (Q2071780)

From the text of the paper: The paper gives a simple proof of the existence and multiplicities of solutions to the boundary value problems (BVPs) \[ \Bigl(\varphi(u')\Bigr)'=f(t,u,u')-s,\eqno(1.1) \] \[ L(u,u')=0,\eqno(1.2) \] relative to the value of \(s,\) assuming that \(\varphi:(-a,a)\to\mathbb{R}, \varphi(0)=0,\) is an increasing homeomorphism and \(L:\mathbb{R}^2\to\mathbb{R}^2\) represents various boundary conditions.'' Also from the text: ``In the proposed approach, we consider the BVP \[ u'=\varphi^{-1}(v),\;\;v'=f(t,u,\varphi^{-1}(v))-s,\eqno(1.3) \] \[ L(z(t,c))=(L_1(u(t,c_1), v(t,c_2)), L_2(u(t,c_1), v(t,c_2)))=0,\eqno(1.4) \] equivalent to the BVP (1.1), (1.2) and are looking for the initial points of the \(T\)-periodic solutions of (1.3), which reduces the problem to finding zeros of the mapping \(\Psi:\mathbb{R}^2\to\mathbb{R}^2,\) \[ \Psi(c)=L(z(t,c))=(L_1(u(t,c_1), v(t,c_2)), L_2(u(t,c_1), v(t,c_2))),\eqno(1.5) \] where \[ u(t,c_1)=\int_0^t\varphi^{-1}(v(\sigma,c_2))d\sigma+c_1, \] \[ v(t,c_2)=\int_0^t(f(\sigma,u(\sigma,c_1),\varphi^{-1}(v(\sigma, c_2)))-s)d\sigma+c_2,\eqno(1.6) \] satisfy the initial value problem (1.3), \(z(0,c)=c=(c_1, c_2).\) When the set of all zeros of \(\Psi\) is known, the solution of (1.5) is determined using a corollary to Borsuk's theorem [\textit{J. T. Schwartz}, Nonlinear functional analysis. New York-London-Paris: Gordon and Breach Science Publishers (1969; Zbl 0203.14501)]. The searching of zeros of (1.5) and their multiplicities can be solved simultaneously, not referring to the theory of the upper/lower solutions, which considerably simplifies and permits to get an extension of results of the quoted papers.