Order Nonlinear Functional Differential Equations Research Articles

Oscillation theorems for 𝑛th order nonlinear differential equations with deviating arguments

In this note we study the oscillatory behavior of solutions of the n n th order nonlinear functional differential equation \[ x ( n ) ( t ) + q ( t ) f ( x [ g ( t ) ] ) = 0 , n even, {x^{(n)}}(t) + q(t)f(x\left [ {g(t)} \right ]) = 0,\quad n\;{\text {even,}} \] without assuming that the deviating argument is retarded or advanced. Sufficient conditions are established for all solutions of the equation to be oscillatory.

Oscillation theorems for nth order nonlinear functional differential equations

Oscillation theorems for nth order nonlinear functional differential equations

On the positive solutions of a higher order functional differential equation with a discontinuity

The n‐th order nonlinear functional differential equation urn:x-wiley:01611712:media:ijmm769678:ijmm769678-math-0001 is considered; necessary and sufficient conditions are given for this equation to have: (i) a positive bounded solution x(t) → B > 0 as t → ∞; and (ii) all positive bounded solutions converging to 0 as t → ∞. Other results on the asymptotic behavior of solutions are also given. The conditions imposed are such that the equation with a discontinuity urn:x-wiley:01611712:media:ijmm769678:ijmm769678-math-0002 is included as a special case.

Some nonoscillation theorems for the higher order nonlinear functional differential equations

We investigate the equation $$\left( {r_{n - 1} \left( t \right)\left( {r_{n - 2} \left( t \right)\left( {...} \right)\left( {r_2 \left( t \right)\left( {r_1 \left( t \right)x'\left( t \right)} \right)'...} \right)'} \right)'} \right)' + a\left( t \right)h\left( {x\left( t \right)} \right)p\left( {x'\left( t \right)} \right) + b\left( t \right)f\left( {x\left( {g\left( t \right)} \right)} \right) = c\left( t \right)$$ and give sufficient conditions for the approach to zero of all nonoscillatory solutions or all bounded nonoscillatory solutions as t → ∞. Let us consider the following two cases. Case1. b(t) is oscillatory on [τ, ∞). Case2. b(t) is nonnegative on [τ, ∞).

A nonoscillation theorem for a second order sublinear retarded differential equation

Sufficient conditions are obtained for all solutions of a class of second order nonlinear functional differential equations to be nonoscillatory.

Second order almost linear functional differential equations—oscillation

It is shown that all solutions of certain second order nonlinear functional differential equations are oscillatory if all solutions of an associated minorizing linear equation are oscillatory.

Oscillation theorems for a second order nonlinear functional differential equation

We consider (3) to be a retarded equation with a maximum delay of p > 0. The term xt is the restriction of x(a) to the interval [t p, t] and shifted to the interval [8, 01, i.e. x~(s) = x(t + S) for s E [-/I, 01. C[/3,0] will denote all continuous curves on the interval [ @, 0] and so X$ E C[ /3, 01. We define C+[/JO] to be the set of all #(s) E C[& 0] such that #(s) is nonnegative on [ & 0] and C T [ /I, 0] to be the set of all #(s) E C+[ /3,0] such that #(s) is monotone increasing on [/I, 01. Parallel definitions are to hold for C-[/3,0] and C J [8, 01. Let