Singular nonlinear boundary value problems for second-order ordinary differential equations
The authors study the boundary value problem \((1)\quad y''+f(x,y)=0,\) \(\alpha y(0)-\beta y'(0)=0,\) \(\gamma y(1)+\delta y'(1)=0,\) where f: (0,1)\(\times (0,\infty)\to (0,\infty)\) is continuous and decreasing in y for each fixed x and integrable on [0,1] for each fixed y, \(\lim_{y\to 0^+}f(x,y)=\infty\) uniformly on compact subsets of (0,1). They establish an existence theorem for the problem (1) and state conditions under which the problem (1) has at most one positive solution. The technical arguments involve properties of solutions and the use of iterative techniques. A new fixed point theorem for cones, for decreasing mappings, is developed and used in proofs.
scientific article; zbMATH DE number 4122357
scientific article