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Existence of positive solutions of a singular initial problem for a nonlinear system of differential equations. (English) Zbl 1080.34001
This work considers initial value problems (IVPs) defined by a diagonally implicit ODE of the form g(x)y ' =A(x)α(y)-ω(x), together with y(0 + )=0; here y n , y ' =dy/dx, g(x)=diag(g 1 (x),...,g n (x)), A(x)=(a ij (x)) is a non-singular matrix with non-vanishing diagonal entries, α(y)=(α 1 (y 1 ),...,α n (y n )), and ω(x)=(ω 1 (x),...,ω n (x)); g is continuous, and A, α, ω are C 1 . The condition g i (0 + )=0 for some i defines a singular Cauchy problem. Under componentwise positivity hypotheses on g(x), α(y), α ' (y), Ω(x)A -1 (x)ω(x) and Ω ' (x), and assuming additionally α(0 + )=Ω(0 + )=0, α i (y)Mα i ' (y), the authors first show that there exists a well-defined, positive, C 1 mapping z=φ(x)=α -1 (Ω(x)) with φ(0 + )=0. This allows them to construct a “funnel”, that is, a domain which will contain the graph of a solution to the singular IVP; specifically, under additional conditions on a ij , ω i , g i and Ω i , they prove using a topological retract method that there exists a family of positive solutions to the IVP. Some generalizations, simplification for linear cases (in which α i =id), and examples are discussed.
34A09Implicit equations, differential-algebraic equations
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions