To the theory of two-point boundary value problems for second-order differential equations (Q2703896)

The authors study the nonlinear boundary value problem NEWLINE\[NEWLINE x^{\prime \prime } =P(x(t),x'(t))+f(t,x(t),x'(t)),\qquad 0< t<1, NEWLINE\]NEWLINE NEWLINE\[NEWLINE x'(0) =k_0x(0)+h_0(x),\qquad x'(1)=k_1x(1)+h_1(x), NEWLINE\]NEWLINE where \(P(x,y)\) is a continuous function positively homogeneous of order \(m>1\), \(P(\lambda x,\lambda y)=\lambda ^mP(x,y)\) for \(\lambda >0\) and \((x,y)\in \mathbb{R}^2\), \(k_0\) and \(k_1\) are constants, \(f(t,x,y)\) is a jointly continuous function defined for \(0\leq t\leq 1\) and \((x,y)\in \mathbb{R}^2\) and such that NEWLINE\[NEWLINE \lim_ {|x|+|y|\to 0} (|x|+|y|)^{-m}\max_{ 0\leq t\leq 1} |f(t,x,y)|=0 NEWLINE\]NEWLINE and \(h_0(x),h_1:C^1[0,1]\to \mathbb{R}^1\) are continuous functionals satisfying the conditions NEWLINE\[NEWLINE \lim_{\|x\|_{C^1}\to \infty }\|x\|_{C^1}^{-1}|h_i(x)|=0,\quad i=0,1. NEWLINE\]NEWLINE The existence of an a priori estimate on the solutions to problem (1), (2) is established in the first section of the paper. The next two sections describe the topological structure of the solutions to the problem. On the basis of the results, the authors present several tests that allow one to determine the solvability and unsolvability of the problem, including the existence of a nonzero solution to the problem.