Advanced Topics in Difference Equations

Here, first we shall set together various basic statements on the periodicity of the solutions of first order linear difference equations, then we shall define various discrete functions which are in a sense connected with the concept of periodicity. Finally, we will formulate a decomposition theorem for the solutions of first order linear difference equations with periodic coefficients.

Let $$\bar T = \left\{ {{t_0},{t_1}, \cdots } \right\}$$ denote the set of increasing time instances, and $$x:\bar T \to {\mathbb{R}^n}$$ with x(k) = (x1, x2,..., xn)(t _k ). Later in several sections we shall study a variety of problems for the difference system 2.1 $$x\left( {k + 1} \right) = {f_k}\left( {x\left( 0 \right),x\left( 1 \right), \cdots x\left( k \right)} \right),\,k \in N$$ where f _k : ℝn(k+1) → ℝn, with the dependence of f _k at the time t _k annotated in the subscript.

Here, we shall develop monotone iterative methods for the construction of quasi-solutions of first order discrete systems satisfying periodic boundary conditions. For this, necessary comparison results are established, some of which are of negative nature.

Here, we shall extend some of the results of the previous section to second order systems. For this, in addition to the notations used there, for the function x : T → ℝn we shall need the central differences, which are defined as δ2x(k) = x(k + 1) − 2x(k) + x(k − 1), 1 ≤ k ≤ J − 1.

Let 1≤p, q∈N, and let I _p , I _q and I _pq be the sets {k: k∈N, 0≤k≤ p — 1}, {ℓ: ℓ∈N, 0≤ℓ≤q — 1} and {k: k∈N, 0≤k≤p — 1}x{ℓ: ℓ∈ N, 0 ≤ ℓ ≤ q — 1}, respectively.

Variation of parameter method is a technique which provides the solutions of the perturbed problems in terms of the solutions of the unperturbed problems.

Let I(k _j 0) = {k _j 0, k _j 0 + 1, ⋯}, k _j 0 ∈ ℝ₊ ∪ {0}, j = 1,2,3 and the product I(k₁0) × I(k₂0) × I(k₃0) be denoted by I3(k₁0, k₂0, k₃0), or simply as I3(k0). An arbitrary point (k₁, k₂, k₃) in I3(k0) is represented as k. For all ℓ, k ∈ I3(k0), ℓ ≤ k means that ℓ _j ≤ k _j , j = 1,2,3. Further, we define operators E _j : I3(k0) → I3(k0), j = 1,2,3 as E₁k = (k₁ + 1, k₂, k₃), E₂k = (k₁, k₂ + 1, k₃) and E₃k = (k₁, k₂, k₃ + 1). Let E _pq = E _p E _q and E _pqr = E _p E _q E _r , p, q, r ∈ {1, 2, 3}. For a given function x(k): I3(k0) → ℝn we define the first-order difference with respect to the variable k _p as Δ _p x(k) = x(E _p k) − x(k), and define the second order difference with respect to the variables k _p , k _q as Δ _pq x(k) = Δ _p Δ _q x(k). The higher order differences are defined analogously.

Here, we shall consider linear as well as nonlinear perturbations of first order difference systems with constant coefficients having infinitely many equilibria. We shall provide sufficient conditions for the asymptotic constancy of the solutions of the perturbed systems. Finally, as a consequence of our main result, we shall obtain sufficient conditions for systems of higher order difference equations to have asymptotic equilibrium.

Here, we shall investigate qualitative behavior of solutions of the linear systems of the type (2.1), namely, 9.1 $$x\left( {k + 1} \right) = \sum\limits_{i = 0}^k {{A_k}} \left( i \right)x\left( i \right),\,k \in N$$

Here, we shall investigate several notions of stability of the general difference system (2.1) using the second method of Lyapunov. For this, first we recall some definitions of stability.

Here, we shall employ prolific Lyapunov’s second method [179–181,342] to investigate the oscillatory behavior of solutions of second order nonlinear difference equation 11.1 $$\Delta \left( {a\left( k \right)\Delta y\left( k \right)} \right) + f\left( {k,y\left( {k + 1} \right),\Delta y\left( k \right)} \right) = 0,k \in N.$$

Here, we shall use difference inequalities to study the oscillatory behavior of solutions of the difference equation 12.1 $$\Delta \left( {a\left( k \right){{\left( {\Delta y\left( k \right)} \right)}^\sigma }} \right) + q\left( {k + 1} \right)f\left( {y\left( {k + 1} \right)} \right) = r\left( k \right),k \in N$$ where σ is a positive quotient of odd integers (odd/odd), or even over odd integers (even/odd). The technique employed here is an extension of the methods in the work of Graef and Spikes [119], Kwong and Wong [170] for differential equations, and Thandapani et. al. [288,295,297] for difference equations.

Here, we shall employ the summation averaging technique to establish necessary conditions for the existence of a nonoscillatory solution of the second order nonlinear difference equation 13.1 $${\Delta ^2}y\left( k \right) + A\left( {k,y\left( k \right)} \right) = B\left( {k,y\left( k \right),\Delta y\left( k \right)} \right),{\mkern 1mu} k \in N\left( {{k_0}} \right).$$

Here, we shall offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation 14.1 $$\Delta \left( {a\left( {k - 1} \right){{\left( {\Delta y\left( {k - 1} \right)} \right)}^\sigma }} \right) + A\left( {k,y\left( k \right)} \right) = B\left( {k,y\left( k \right),\Delta y\left( k \right)} \right),k \in N\left( 1 \right)$$ where as in equation (12.1), σ is a positive quotient of odd integers (odd/odd), or even over odd integers (even/odd).

Here, we shall consider the perturbed quasilinear difference equation 15.1 $$\Delta \left( {a\left( {k - 1} \right)|\Delta y\left( {k - 1} \right){|^{\sigma - 1}}\Delta y\left( {k - 1} \right)} \right) + A\left( {k,y\left( k \right)} \right) = B\left( {k,y\left( k \right),\Delta y\left( k \right)} \right),k \in N\left( 1 \right)$$ where σ > 0, and as in equation (14.1), the function a(k) is eventually positive, A : N(1) × ℝ → ℝ, B : N(1) × ℝ × ℝ → ℝ; and that there exist functions α(k), β(k), and f satisfying (12.2), (12.3) and (13.2). For the difference equation (15.1) we shall discuss results analogous to those presented in Section 14.

Here, we shall consider the half-linear difference equation 16.1 $$\Delta [|\Delta y(k){|^{\sigma - 1}}\Delta y(k)] = \sum\limits_{i = 1}^n {{p_i}(k)|y({g_i}(k)){|^{\sigma - 1}}y({g_i}(k)),k \in N(a)} $$ where σ > 0. For each 1 ≤ i ≤ n we shall assume that (I)p _i (k) ≥ 0, max _k∈N(J)p _i (k) > 0 for any a ≤ J ∈ N, and(II)g _i : N(a) → Z is such that Δg _i (k) > 0 eventually, and lim _k→∞g _i (k) = ∞. For the difference equation (16.1) we shall provide sufficient conditions for the oscillation of all solutions, as well as necessary and sufficient conditions for the existence of both bounded and unbounded nonoscillatory solutions.

Here, we shall consider the following damped difference equation 17.1 $$\Delta (a(k){(\Delta y(k))^\sigma }) + b(k){(\Delta y(k))^\sigma } + H(k,y(k),\Delta y(k)) = 0,k \in N$$ where σ is a positive quotient of odd integers (odd/odd), the function a(k) is eventually positive, and H : N × ℝ × ℝ → ℝ. For the equation (17.1) we shall establish existence theorems for positive monotone solutions. An oscillation theorem for (17.1) will also be derived.

Here, we shall provide sufficient conditions for the oscillation of all solutions of the nth order perturbed difference equation 18.1 $$|{\Delta ^n}y(k){|^{\sigma - 1}}{\Delta ^n}y(k) + Q(k,y(k - {\eta _k}),\Delta y(k - {\eta _k}), \cdots ,{\Delta ^{n - 2}}(k - {\eta _k})) = P(k,y(k - {\eta _k}),\Delta y(k - {\eta _k}), \cdots ,{\Delta ^{n - 1}}y(k - {\eta _k})),k \in N({k_0})$$ where σ > 0, and η _k ∈ Z = {⋯, −1, 0, 1, ⋯} with lim _k→∞(k − η _k ) = ∞.

Here, we shall classify the difference equation 19.1 $${\Delta ^\alpha }y\left( k \right) + \delta p\left( k \right)f\left( {y\left( {g\left( k \right)} \right)} \right) = 0,\,k \in N$$ into four cases according to α is odd or even and δ is 1 or −1. In each case we shall provide comparison theorems for the oscillation of the difference equation. In what follows, we shall assume that p : N(K) → ℝ₊ for some K ∈ N, g ∈ G = {g : N(K) → N for some K ∈ N : lim _k→∞g(k) = ∞}, and f : ℝ → ℝ is continuous satisfying (12.2) and non-decreasing.

Here, we shall investigate the oscillatory behavior of solutions of second order nonlinear neutral delay difference equations of the type 20.1 $$\Delta \left( {p\left( k \right)\Delta \left( {y\left( k \right) + h\left( k \right)y\left( {k - \tau } \right)} \right)} \right) + q\left( {k + 1} \right)f\left( {y\left( {k + 1 - \sigma } \right)} \right) = 0,\;k \in N$$ where τ, σ are fixed non-negative integers, functions p, h, q are defined on N, p(k) > 0 for all k ∈ N and q(k) is not identically zero for large k, and the continuous function f : ℝ → ℝ satisfies (12.2), (12.3).

In the previous section, to establish oscillatory behavior of solutions of (20.1), we have provided sufficient conditions which ensure that the classes M+, M− and WOS are empty. The purpose of this section is to prove the existence of solutions of (20.1) in these classes.

Here, we shall study the oscillatory behavior of solutions of n(≥ 1)th order nonlinear neutral delay difference equations of the following form 22.1 $${\Delta ^n}\left( {y\left( k \right) + p\left( k \right)y\left( {k - \tau } \right)} \right) + q\left( k \right)f\left( {y\left( {k - \sigma } \right)} \right) = 0,{\mkern 1mu} k \in N$$ where τ, σ are fixed non-negative integers, functions p, q are defined on N, q(k) ≥ 0, k ∈ N,and the continuous function f: ℝ →> ℝ satisfies (12.2).

Here, we shall provide sufficient conditions for the oscillation of all solutions of the partial difference equation 23.1 $$u(k + 1,\ell ) + \beta (k,\ell )u(k,\ell + 1) - \delta (k,\ell )u(k,\ell ) + P(k,\ell ,u(k - \tau ,\ell - \upsilon )) = Q(k,\ell ,u(k - \tau ,\ell - \upsilon )),k \in N({k_0}),\ell \in N({\ell _0})$$ where τ, ν are non-negative integers, and β(k, ℓ), δ(k, ℓ) are functions such that for all large k and ℓ $$\beta \left( {k,l} \right) \geqslant \beta > 0\,and\,\delta \left( {k,l} \right) \leqslant \delta \left( { > 0} \right).$$ We note that δ(k, ℓ) is allowed to be negative. Functions P and Q are defined on N(k₀) × N(ℓ₀) × ℝ.

Here, we shall obtain sufficient conditions for the oscillation of all solutions of the partial difference equation 24.1 $$u\left( {k + 1,\ell } \right) + \beta \left( {k,\ell } \right)u\left( {k,\ell + 1} \right) - \delta \left( {k,\ell } \right)u\left( {k,\ell } \right) + \sum\limits_{i = 1}^\sigma {{P_i}} \left( {k,\ell ,u\left( {k - {\tau _i},\ell - {v_i}} \right)} \right) = \sum\limits_{i = 1}^\sigma {{Q_i}} \left( {k,\ell ,u\left( {k - {\tau _i},\ell - {v_i}} \right)} \right),k \in N\left( {{k_0}} \right),\ell \in N\left( {{\ell _0}} \right)$$ where τ _i , v _i , 1 ≤ i ≤ σ are non-negative integers, functions β(k, ℓ), δ(k, ℓ) satisfy the same conditions as in the previous section, and P _i , Q _i , 1 ≤ i ≤ σ are defined on N(k₀) × N(ℓ₀) × ℝ.

Here, we shall offer sufficient conditions for the oscillation of all solutions of the partial difference equations 25.1 $$u\left( {k - 1,\ell } \right) + \beta \left( {k,\ell } \right)u\left( {k,\ell - 1} \right) - \delta \left( {k,\ell } \right)u\left( {k,\ell } \right) + P\left( {k,\ell ,u\left( {k + \tau ,\ell + v} \right)} \right) = Q\left( {k,\ell ,u\left( {k + \tau ,\ell + v} \right)} \right),k \in N\left( {{k_0}} \right),\ell \in N\left( {{\ell _0}} \right)$$ and 25.2 $$u\left( {k - 1,\ell } \right) + \beta \left( {k,\ell } \right)u\left( {k,\ell - 1} \right) - \delta \left( {k,\ell } \right)u\left( {k,\ell } \right) + \sum\limits_{i = 1}^\sigma {{P_i}} \left( {k,\ell ,u\left( {k + {\tau _i},\ell + {v_i}} \right)} \right) = \sum\limits_{i = 1}^\sigma {{Q_i}} \left( {k,\ell ,u\left( {k + {\tau _i},\ell + {v_i}} \right)} \right),k \in N\left( {{k_0}} \right),\ell \in N\left( {{\ell _0}} \right)$$ where τ, v, τ _i , v _i , 1 ≤ i ≤ σ are non-negative integers, functions β(k, ℓ), δ(k, ℓ) satisfy the same conditions as in Section 23, and the functions P, Q, P _i , Q _i , 1 ≤ i ≤ σ are defined on N(k₀) × N(ℓ₀) × ℝ.

Here, we shall develop criteria for the non-existence of eventually positive (negative) and non-decreasing (non-increasing) solutions of the partial difference equations 26.1 $${\nabla _k}{\nabla _\ell }u(k,\,\ell )\, + \,P(k,\ell ,u(k + \,\tau ,\,\ell + v))\, = \,Q(k,\ell ,u(k + \,\tau ,\,\ell + v))\,$$ and 26.2 $${\nabla _k}{\nabla _\ell }u(k,\,\ell )\, + \,\sum\limits_{i = 1}^\sigma {{P_i}} (k,\ell ,u(k + {\tau _i},\ell + {\nu _i}))\, = \,\sum\limits_{i = 1}^\sigma {{Q_i}} (k,\ell ,u(k + {\tau _i},\ell + {\nu _i}))\,,k\, \in \,N({k_0}),\,\ell \in N({\ell _0})$$ where as in the previous sections τ, ν, τ_i, ν_i, 1 ≤ i ≤ σ are non-negative integers, and the functions P,Q, P _i ,Q _i , 1 ≤ i ≤ σ are defined on N(k₀) × N(ℓ₀) × ℝ. By a non-decreasing (non-increasing) solution u(k, ℓ) of (26.1) or (26.2) we mean ∇ _k u(k, ℓ) ≥ (≤) 0 and ∇ _ℓ u(k, ℓ) ≥ (≤) 0.

Here, we shall provide sufficient conditions for the existence and uniqueness of the solutions of the following three—point boundary value problem 27.1 $${\Delta ^2}y\left( k \right) = f\left( {k,y\left( k \right),\Delta y\left( k \right)} \right) + e\left( k \right),\,k \in N\left( {0,J - 1} \right)\,y\left( 0 \right) = 0,\,y\left( {J + 1} \right) = \alpha y\left( \eta \right) + b$$ where η ∈ N(1, J — 1) is a fixed integer, α, b are given finite constants and e(k) is defined for k ∈ N(0, J + 1). Throughout, in what follows the function f: N(0, J + 1) × ℝ2 → ℝ is assumed to be defined and continuous.

Here, we shall offer sufficient conditions for the existence of solutions for the nth order difference equation 28.1 $${\Delta ^n}y\left( k \right) + f\left( {k,y\left( k \right),\Delta y\left( k \right), \cdots {\Delta ^{n - 2}}y\left( k \right)} \right) = 0,n2,k \in N\left( {0,J - 1} \right)$$ satisfying the boundary conditions 28.2 $${\Delta ^i}y\left( 0 \right) = 0,0in - 3$$ 28.3 $$\alpha {\Delta ^{n - 2}}y\left( 0 \right) - \beta {\Delta ^{n - 1}}y\left( 0 \right) = 0$$ 28.4 $$\gamma {\Delta ^{n - 2}}y\left( J \right) + \delta {\Delta ^{n - 1}}y\left( J \right) = 0$$ where α, β, γ and δ are constants such that 28.5 $$\rho = \alpha \gamma J + \alpha \delta + \beta \gamma succ0$$ and 28.6 $$\alpha \succ 0,\gamma \succ 0,\beta 0,\delta \gamma $$

Here, we shall consider the n(≥ 2)th order difference equation 29.1 $${\Delta ^n}y(k) + \lambda Q(k,y(k)), \cdots ,{\Delta ^{n - 2}}y(k))\, = \,\lambda P(k,y(k),\Delta y(k), \cdots ,{\Delta ^{n - 1}}y(k)),\,k \in N(0,J - 1)$$ together with the boundary conditions (28.2) – (28.4), where the constants α, β, γ and δ satisfy the conditions (28.5) and (28.6). In (29.1), λ > 0, Q: N(0, J − 1) × ℝn−1 → ℝ, and P : N(0, J − 1) × ℝn → ℝ.

Following the notations of the previous section, here we shall offer criteria for the existence of positive solutions of the boundary value problem (29.1), (28.2) – (28.4) with λ = 1.

Here, we shall obtain results similar to those in Sections 29 and 30 for the difference equation (29.1) satisfying the (n, p) boundary conditions 31.1 $$\begin{array}{*{20}{c}} {{\Delta ^i}y\left( 0 \right) = 0,0 \leqslant i \leqslant n - 2} \\ {{\Delta ^p}y\left( {J + n - p - 1} \right) = 0} \end{array}$$ where λ > 0, n ≥ 2 and 0 ≤ p ≤ n − 1 is fixed. We begin with the characterization of λ so that the boundary value problem (29.1), (31.1) has positive solutions. By a positive solution y of (29.1), (31.1), we mean a non-trivial y : N(0, J +n − 1) → [0, ∞) satisfying (29.1) and (31.1).

For the discrete boundary value problems arising in transport processes we shall provide comparison results. These results are used to develop monotone iterative methods for the construction of the maximal and minimal solutions in a sector. The advantage of this technique is that the successive approximations are the solutions of the initial and terminal value problems. Numerical illustration showing the sharpness as well as the importance of these results is also presented.

Here, first we shall find connections between the solutions of the initial value problem (6.1), and its perturbed system (6.2) satisfying, instead of the same initial condition y(k₀) = x₀, the m-point boundary conditions 33.1 $${A_1}y({k_1}) + {A_2}y({k_2}) + \cdots + {A_m}y({k_m}) = \gamma ,$$ where k₀ ≤ k₁ ≤ k₂ ⋯ ≤ k _m ≤ k₀ + J, k _i ∈ I_0,J = {k₀, k₀ + 1, ⋯ , k₀ + J}, A _i ∈ ℝn×n, i = 1, 2, ⋯ , m are constant matrices, and γ ∈ ℝn is a constant vector. Then, these connections will be used to study the existence and uniqueness of the solutions of the boundary value problem (6.2), (33.1) in ‘generalized (vector) normed spaces’. An iterative scheme which can be used to compute approximate solutions of (6.2), (33.1) will also be provided. In what follows, it is sufficient to assume that the systems (6.1) and (6.2) are defined only on I_0,J . We begin with the following:

Here, we shall consider the difference system (2.1) for k ∈ N (0, J − 1), together with the boundary conditions 34.1 $$Ax\left( 0 \right) + Bx\left( J \right) = \alpha ,$$ where A, B are n × n known matrices and a is a known n × 1 vector. In (34.1), for the cases B = 0, and A = 0, we respectively have the initial and terminal value problems, and when A = −B = I, the identity matrix, and α = 0, we have the periodic boundary conditions (3.1). Further, it is easy to see that the boundary value problem describing transport phenomena (32.1), (32.2) is also included in (2.1), (34.1).

Let B(I_J) be the space of all real n-vector functions defined on I_J = N(0, J). Let h be an operator mapping B(I_J) into ℝn.

Here, we shall consider the general discrete system 36.1 $$x(k + 1) = \sum\limits_{i = 0}^k {{A_k}(i)x(i) + b(k) + {f_k}(x(0),x(1), \cdots ,x(k),)} ,k \in {I_{J - 1}}$$ together with the boundary conditions 36.2 $$\sum\limits_{s = 1}^K {{L_s}x({k_s}) = \ell + h(x({k_1}), \cdots ,x({k_K}),)} ,$$ where each A _k (i) is a constant n × n matrix with elements a _k p,q(i), 1 ≤ p, q ≤ n; b(k) is an n-vector with components bp(k), 1 ≤ p ≤ n; є ∈ (−є₀, є₀), є₀ is a positive number, f _k : ℝn(k+1) × (−є₀, є₀) → ℝn, 0 = k₁ < k₂ < ⋯ < k _K = J < ∞, L _s , 1 ≤ s ≤ K are n × n matrices; ℓ ∈ Rn, and h: ℝnK × (−є₀, є₀) → ℝn.

Here, we shall present existence principles for the second order discrete boundary value problem where the values of the solution lie in a Banach space E, which is not necessarily finite dimensional. Our approach is based on fixed point methods (in particular continuation methods). The existence of the solutions is proved by showing that no solutions of an appropriate family of problems lie on the boundary of a suitable open set.

In this section we shall consider the optimal control problem for the stochastic difference equation 38.1 $${x_{k + 1}} = \sum\limits_{i = 0}^k {{A_{k - i}}} {x_i} + {D_k}\eta + {B_k}{u_k} + {\sigma _k}{\xi _{k + 1}}$$ and the cost functional 38.2 $$J\left( u \right) = E\left[ {{{x'}_J}F{x_J} + \sum\limits_{i = 0}^{J - 1} {\left( {{{u'}_i}{G_i}{u_i} + {{x'}_i}{H_i}{x_i}} \right)} } \right]$$

Here, we shall study symmetries of difference systems of the type (2.1) via the so-called Lie symmetry vectorfields. This line of investigation was initiated by Maeda [201], see also chapter 10 of the monograph [280], for the simpler systems with f _k = f(x(k)). The main theorem of Maeda is that in the presence of such Lie symmetry vectorfields, a change of variable is possible such that the function f takes a simpler form with respect to a decomposition of the components of the new variable, and that in the case n = 1, the single equation can be linearized. We shall show that there is a similar decomposition in the case of the system (2.1). However, in the corresponding case n = 1, our result does not lead to linearization. Nevertheless, we shall demonstrate by two examples that linearization can occur for systems of a special nature.

The importance of polar coordinates abound in mathematical sciences. However, its use so far has been confined to the continuous case. Here, we shall provide the explicit transformation formulae between discrete Cartesian coordinates, by which we mean n-tuples of integers, and what we shall call discrete polar coordinates, which will also be having integral coordinates. Finally, as an application we will develop a new discrete inequality of “integro-differential” type.