Approximation of differential problems with singularities and time discontinuities Original Research Article

The paper deals with the singular mixed boundary value problem View the MathML source(tμu′(t))′+tμf(t,u(t))=0,limt→0+tμu′(t)=0,u(T)=A, Turn MathJax on where μ∈Nμ∈N, μ≥2μ≥2, [0,T]⊂R[0,T]⊂R, A∈[0,∞)A∈[0,∞). For s1,…,sr∈(0,T]s1,…,sr∈(0,T] and J=(0,T]∖{s1,…,sr}J=(0,T]∖{s1,…,sr} we assume that f(t,x)f(t,x) is continuous on the set J×(0,∞)J×(0,∞) and may have singularities at t=0t=0 and x=0x=0 and integrable discontinuities at t=sit=si, i=1,…,ri=1,…,r. We provide a new approach giving the existence of positive solutions of the above singular problem by means of a sequence of its discretizations. As an application we present new existence results for singular problems arising in the theory of shallow membrane caps