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It is shown for any nonnegative continuous function q q on [ 0 , ∞ ) [0,\infty ) and any c > 1 c > 1 that any positive increasing solution y y of y − q 2 y = 0 y - {q^2}y = 0 satisfies y ( x ) ≥ y ( 0 ) exp ⁡ ( c ∫ 0 x q ( t ) d t ) y(x) \geq y(0)\exp (c\int _0^x {q(t)dt)} on the complement of a set of finite Lebesgue measure. It is also shown that if lim inf ( ∫ 0 x q ( t ) d t / x ) > 0 \lim \inf (\int _0^x {q(t)dt/x) > 0} then the equation has an exponentially increasing solution and an exponentially decreasing solution.

A class of integro-differential equations is described, for which the regularity of the corresponding operator is equivalent to the exponential dichotomy of solutions of the homogeneous equation.

Publisher Summary This chapter discusses a brief account of two such methods that seem particularly suited to the questions of conditional stability and of the boundedness of some solution of a differential equation. The chapter outlines in each case the general approach and indicates only in passing various old and new results that should be obtained in a unified manner. The chapter discusses the special case when the coefficient matrix is constant and has no Eigen values with zero real part in a fairly simple way, particularly under stronger assumptions on h guaranteeing the uniqueness of a solution of equation. However, the general case in which no considerations of Eigen values are available requires an entirely different approach. The chapter generalizes the situation that occurs if the coefficient matrix in equation is a constant matrix A none of whose Eigen values has zero real part. In this case, the dimension of the linear subspace that induces an exponential dichotomy equals the number of Eigen values of A having negative real part.