Bounds for Solutions of Nonlinear Differential Systems

摘要

Bounds are established for solutions of differential systems of the form dx/dt = A(t) x +F(t, x(t)) with various kinds of nonlinearities, some that increase slower than the linear terms, A(t)x, when the norm of x becomes very large and others that decrease faster than the linear terms when the norm of x approaches zero. In each case, the bounds are derived from bounds to the general solution of the linear part assumed as known. In the case of (relatively) slowly increasing nonlinear terms, F(t, x(t)), the bounds hold for solutions with large initial values; in the case of (relatively) rapidly decreasing nonlinear terms the bounds are valid for solutions with small initial values. The bounds reflect at least the growth of the various terms involved and are rather sharp for some equations. The bounds are developed from the solution of a nonlinear integral inequality. The solution is an extension of a result for the corresponding linear integral inequality.