Solvability of a boundary-value problem for a variable-order differential equation on a geometric graph (Q881056)

The authors investigate the following boundary problem on the graph \(\Gamma\): \[ (p_{\gamma}u''_{\gamma})''=f_{\gamma}(x),\;x\in\gamma,\;\;\gamma\in E(\Gamma_1), \] \[ -(q_{\gamma}u'_{\gamma})'=f_{\gamma}(x), x\in\gamma,\;\;\gamma\in E(\Gamma_2), \] \[ u_{\gamma}(a)=u_{\mu}(a),\;\gamma, \mu\in\Gamma(a),\;a\in J(\Gamma) \] \[ (p_{\gamma}u''_{\gamma})(a)=0,\;\gamma\in\Gamma_1(a), a\in V(\Gamma), \] \[ \sum_{\gamma\in\Gamma_1(a)}(p_{\gamma}u''_{\gamma})'(a)= \sum_{\gamma\in\Gamma_2(a)}(q_{\gamma}u'_{\gamma})(a),\;a\in J(\Gamma), \] \[ u_{\gamma}(a)=0,\;\gamma\in\Gamma(a),\;a\in\partial\Gamma, \] where \(f\in C(\Gamma^0),\;\Gamma^0=\bigcup_{i=1}^{r}\gamma_i,\) \(\gamma_i: i=1,\dots,r\) edges of the graph \(\Gamma,\) \(E(\Gamma)\) is the set of edges, \(V(\Gamma)\) the set of vertices, \(J(\Gamma),\) the set of internal vertices. The proofs are based on the conditions for degeneracy and nondegeneracy of the problem. Also the authors prove that the nondegeneracy of the problem is equivalent to the maximum principle.