Oscillatory properties of the second order nonlinear equation \[ (r(t)x^{\prime })^{\prime }+q(t)f(x)=0 \] are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too.
In this note we consider the third order linear difference equations of neutral type (E) ∆ 3 [x(n) − p(n)x(σ(n))] + δq(n)x(τ (n)) = 0, n ∈ N(n0), where δ = ±1, p, q : N(n0) → ℝ+; σ, τ : N(n0) → ℕ, lim n→∞ σ(n) = lim n→∞ τ (n) = ∞. We examine the following two cases: {0 < p(n) ≤ 1, σ(n) = n + k, τ (n) = n + l}, {p(n) > 1, σ(n) = n − k, τ (n) = n − l}, where k, l are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.
Consider the difference equation ∆x(n) +∑m i=1 pi(n)x(τi(n)) = 0, n ≥ 0 [ ∇x(n) − ∑m i=1 pi(n)x(σi(n)) = 0, n ≥ 1 ] , where (pi(n)), 1 6 i 6 m are sequences of nonnegative real numbers, τi(n) [σi(n)], 1 6 i 6 m are general retarded (advanced) arguments and ∆ [∇] denotes the forward (backward) difference operator ∆x(n) = x(n + 1) − x(n) [∇x(n) = x(n) − x(n − 1)]. New oscillation criteria are established when the well-known oscillation conditions lim sup n→∞ ∑m i=1 ∑n j=τ(n) pi(j) > 1 [ lim sup n→∞ ∑m i=1 σ∑ (n) j=n pi(j) > 1 ] and lim inf n→∞ ∑m i=1 n∑−1 j=τi(n) pi(j) > 1⁄e [ lim inf n→∞ ∑m i=1 σ∑i(n) j=n+1 pi(j) > 1⁄e ] are not satisfied. Here τ (n) = max 1≤i≤m τi(n) [σ(n) = min 1≤i≤m σi(n)]. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given.
This paper is concerned with the oscillatory behavior of first-order nonlinear difference equations with variable deviating arguments. The corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.