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Uniqueness, existence, and optimality for fouth-order Lipschitz equations. (English) Zbl 0642.34005

The authors of this paper consider solutions of boundary value problems for the fourth order differential equation (x) \(y^{(4)}=f(t,y,y',y'',y\prime''),\) where \(f: (a,b)\times R^ 4\to R\) is continuous and f satisfies the Lipschitz condition \[ | f(t,y_ 1,y_ 2,y_ 3,y_ 4)-f(t,z_ 1,z_ 2,z_ 3,z_ 4)| \leq \sum^{4}_{i=1}k_ i| y_ i-z_ i|, \] for \((t,y_ 1,y_ 2,y_ 3,y_ 4)\) and \((t,z_ 1,z_ 2,z_ 3,z_ 4)\in (a,b)\times R^ 4\). They characterize optimal length subintervals of (a,b) in terms of the Lipschitz constants \(k_ i\), \(i=1,2,3,4\) on which certain two, three and four point boundary value problems for (*) have unique solutions. Existence results for solutions to the boundary value problems for (*) are proved using the Pontryagin maximum principle and uniqueness.
Reviewer: N.Parhi

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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