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Linear positive operators and their applications to differential equations. (English) Zbl 0564.34008
The main object here is to show that the operator which was introduced by E. W. Cheney and A. Sharma [Can. J. Math. 16, 241-252 (1964; Zbl 0128.290)] can be used to define recursively the solution of the initial value problem \(y'=f(x,y)\), \(y(0)=y_ 0\), \(x\in [0,a)\) (with \(a\leq 1)\) where f satisfies the Lipschitz condition \(| f(x,y_ 1)- f(x,y_ 2)| \leq \lambda | y_ 1-y_ 2|\) with \(\lambda\in [0,1)\) in the strip \(0\leq x<a\), \(-\infty <y<\infty\) and f, \(f_ x\), \(f_ y\) are continuous and bounded in the domain \(0\leq x<\infty\), \(- \infty <y<\infty\).
Reviewer: J.O.C.Ezeilo

34A30 Linear ordinary differential equations and systems
34A99 General theory for ordinary differential equations
47E05 General theory of ordinary differential operators
Zbl 0128.290
Full Text: EuDML