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Oscillation and nonoscillation of functional differential equations

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A qualitative approach is usually concerned with the behavior of solutions of a given differential equation and usually does not seek specific explicit solutions;This dissertation is the analysis of nonoscillation of even order delay differential equations, and the oscillation of solutions of arbitrary order functional differential equations. This is done mainly in Chapters II and III. Chapter IV deals with the oscillation and nonoscillation of bounded solutions of n-th order delay differential equations. By an oscillatory solution we mean that the solution has infinitely many zeros; otherwise, it is called nonoscillatory solution;The functional differential equations under consideration are:(UNFORMATTED TABLE OR EQUATION FOLLOWS)≤ (A) [r(t)y[superscript]'(t)][superscript](2n-1) + [sigma][subscript]spi=1k P[subscript]i(t) F[subscript]i[y[subscript][tau](t),y[subscript]sp[sigma][subscript]1'(t),...,y[subscript]sp[sigma][subscript]2n-1(2n-1)&(t)] & = f(t), n ≥ 1& x[superscript](n)(t) + [sigma] [sigma][subscript]spi=1m P[subscript]i(t) x[g[subscript]i(t)] + [delta] q(t) x[h(t)]& = f(t), n ≥ 3& (B) L[subscript]n x(t) + (-1)[superscript]n-1 [sigma][subscript]spi=1m a[subscript]i(t)f[x(g[subscript]i(t)]& = b(t), n ≥ 1& (C) (TABLE/EQUATION ENDS);Equation (A) is considered in Chapter II, where we find sufficient conditions for which all solutions of equation (A) are nonnegative. Equation (B) is studied in Chapter III for the cases: [delta] = 0, [sigma] = 0 and [sigma] = [delta] = ±1 i.e., delay arguments, advanced arguments, and mixed type arguments, respectively. Finally, equation (C) is studied in Chapter IV. Sufficient conditions are found to force all bounded solutions to be oscillatory, and sufficient conditions are also found for which every bounded solution of equation (C) tends to zero as t → [infinity].