On the solvability of nonlinear Sturm-Liouville problems (Q645390)

The authors study the existence of solutions for two nonlinear differential equations of Sturm-Liouville type subject to general boundary conditions. To be more precise, they consider the problems \((SL_1):\) \[ (p(t)x^{\prime }(t)) ^{\prime }+q(t)x(t)+\psi (x(t))=G(x(t)), \] \[ \alpha x(0)+\beta x^{\prime }(0)+\eta _1(x)=\phi _1(x), \] \[ \gamma x(1)+\delta x^{\prime }(1)+\eta _2(x)=\phi _2(x), \] and \((SL_2):\) \[ (p(t)x'(t))'+q(t)x(t)+\psi (x(t))=h(t), \] \[ \alpha x(0)+\beta x^{\prime }(0)+\eta _1(x)=v_1, \] \[ \gamma x(1)+\delta x^{\prime }(1)+\eta _2(x)=v_2, \] where \(\psi\in C^{1}(\mathbb{R},\mathbb{R}),\) \(p(t)>0,\) \(q(t)\) is real on \([0,1]\), \(p,p',q\) are continuous on \((0,1);\) \(h\) is a square integrable function on \([0,1],\) \(v_1,v_2\) are real values; \(G, \eta_1, \eta_2, \phi_1,\phi_2\) are nonlinear operators defined in the set \(D(\mathcal{L})\) of functions \(x\in L^2[0,1]\) such that \(x'\) is absolutely continuous and \(x''\in L^2[0,1]\); and the real coefficients of the boundary conditions satisfy \(\alpha^2+\beta^2\neq 0, ~\gamma^2+\delta^2\neq 0.\) Under suitable sufficient conditions, firstly it is proved (Theorem 3.3) that the problem \((SL_2)\) has a unique solution for each \(h\) and each \((v_1,v_2)\); next, under the same hypothesis of Theorem 3.3 and additional assumptions the authors establish in Theorem 4.2 that \((SL_1)\) has at least one solution. It is interesting to point out that the imposed conditions are related with: a) the distribution of the infinite real eigenvalues of the classical linear Sturm-Liouville boundary value problem, b) the range of \(\psi'\), and c) the size of the operator norm of \((\eta'_1,\eta'_2)\). To obtain the main results of the paper, the strategy consists in translating the problems \((SL_j), ~j=1,2,\) to appropriate operator equations in \(D(\mathcal{L})\) for which the global inverse function theorem is applied; in the case of \((SL_1)\) the Schauder fixed point theorem is also needed.