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Existence and uniqueness of initial value problems for a class of second- order differential equations. (English) Zbl 0691.34005

The authors investigate the initial value problem for the second order differential equations of the form (1) \(x''(t)=g(x(t),x'(t),x''(t)),\) which cannot be solved for the highest order derivative. Their work is related to that of Petryshyn who used the theory of A-mappings. The authors’ investigations arise from a global bifurcation result for an A- proper operator, where some of the possible behaviour of a global bifurcation can be eliminated if the corresponding initial value problem has a unique solution. The authors claim that if g is Lipschitz in all variables with Lipschitz constant I in the third variable, then solutions of the initial value problem are not unique as it might be expected.
The authors prove in section 2 that the solution can be modified on a set of measure zero so that \(x''\) is continuous while in section 3 they give an example to illustrate the theorem proved in this section namely. If \(g(0,0,0)=0\) and the initial date is taken to be zero, then 0 is always a solution of the initial value problem \(x''=g(x,x',x''),\) \(x(0)=A=0\), \(x'(0)=B=0\) which is written as the equation \(Lx=Nx\), \(x\in X\) where \(L,N: X\to Y\) are given by \(Lx=x''\), \(Nx(t)=g(x(t),x'(t),x''(t)).\)
Reviewer: F.M.Ragab

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
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[1] Petryshyn, W.V, On the approximation solvability of equations involving A-proper, and pseudo A-proper mappings, Bull. amer. math. soc., 81, 223-312, (1975) · Zbl 0303.47038
[2] Petryshyn, W.V, Using degree theory for densely defined A-proper maps in the solvability of semilinear equations with unbounded and noninvertible linear part, Nonlinear anal. TMA, 4, 259-281, (1980) · Zbl 0444.47046
[3] Petryshyn, W.V; Yu, Z.S, Periodic solutions of nonlinear second-order differential equations which are not solvable for the highest derivative, J. math. anal. appl., 89, 462-488, (1982) · Zbl 0516.34019
[4] Webb, J.R.L, Topological degree and A-proper operators, Linear algebra appl., 84, 227-242, (1966) · Zbl 0619.47047
[5] Welsh, S.C, Global results concerning bifurcation for Fredholm maps of index zero with a transversality condition, Nonlinear anal. TMA, 12, 1137-1148, (1988) · Zbl 0672.47051
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