Difference Equations and Inequalities

Monotone iterative technique with respect to initial time difference for initial value problem for difference equations with maxima

“…To prove (P1) set Qα = α (1) , where α (1) is the unique solution of (9) with η = α and let m(n) = α(n) − α (1) (n), n ∈ Z[n 0 , n p+1 ] For any n ∈ p k=0 I k we obtain the inequality m(n + 1) = α(n + 1) − P (n)α (1) (n) − K(n)α (1) (n + 1) − ψ(n, α(n), α(n + 1)) ≤ P (n)(α(n) − α (1) (n)) + K(n)(α(n + 1) − α (1) (n + 1)) = P (n)m(n) + K(n)m(n + 1).…”

“…where τ (n) is given by (5) for γ n = ξ(n, A(n), A(n j )), j ∈ Z[1, p] and ζ(n) is given by (7) for γ n k +d k = ξ(n k +d k , A(n k +d k ), A(n k )), µ k = υ(k, A(n k −1)), k ∈ Z [1, p]. From (17) it follows the function A(n) is a solution of NIDE (1). Similarly, we prove the function B(n) is a solution of NIDE (1).…”

“…Similarly, we prove the function B(n) is a solution of NIDE (1). Let u ∈ S(α, β) be a solution of IVP for NIDE (1). From inequalities (16) it follows there exists a natural number p such that p ∈ N :…”

Existence and approximation of solutions of non-linear discrete systems on infinite intervals

“…for n3 N>. Note that (1.2)L, for fixed n 3 N>, determines y(i) for i'n in terms of y(m), m3N L so in fact the finite section approximations reduce to discrete equations on the finite interval N L , n 3 N> (problems of this form have received a lot of attention in the literature [1]). The technique we present when discussing approximation of solutions to (1.1) involves using the Furi-Pera Fixed Point Theorem together with the notion of strict convergence [6].…”

Explicit inverse of nonsingular Jacobi matrices

“…since, in other case, J(a, b, c) is a reducible matrix and hence the inversion 10 problem leads to the invertibility of a matrix of lower size. Moreover, the values of the coefficients for the sequences a and c at n + 1 have no influence in the analysis of the matrix (1), since these coefficients do not appear in it.…”