Necessary and Sufficient Conditions for Oscillation of Certain Higher Order Partial Difference Equations

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1 International Journal of Difference Equations ISSN , Volume 4, Number 2, pp ( Necessary and Sufficient Conditions for Oscillation of Certain Higher Order Partial Difference Equations Figen Özpınar Afyon Kocatepe University Bolvadin Vocational School Bolvadin, Afyonkarahisar, Turkey Ömer Akın TOBB University of Economics and Technology Department of Mathematics Söğütözü, Ankara, Turkey Abstract In this paper, some necessary and sufficient conditions for the oscillation of the higher order partial difference equation of the form h n r ma m,n + ( 1 h+r+1 pa m k,n l = 0, (m, n N 2 0 are established, where k, l N 0, h, r N 1, p is a nonnegative real number AMS Subject Classifications: 39A11, 34K11, 34C10 Keywords: Partial difference equation, higher order PDEs, oscillation, oscillatory 1 Introduction Partial difference equations are difference equations that involve functions with two or more independent variables Recently, there are many papers that devoted to the development of qualitative theory of difference equations [5 7, 9 Their significance is illustrated in applications involving random walk problems, molecular structure problems and numerical difference approximation problems etc Received June 11, 2008; Accepted July 26, 2009 Communicated by Gusein Guseinov

2 212 Özpınar and Akın In this paper, we consider the higher order partial difference equations of the form h n r ma m,n + ( 1 h+r+1 pa m k,n l = 0, (11 where m, n, k, l N 0, r, h N 1, N a = {a, a + 1, a + 2, }, p is a nonnegative real number The forward partial differences m and n are defined as usual, ie, m A m,n = A m+1,n A m,n and n A m,n = A m,n+1 A m,n The higher order partial differences for any positive integers r and h are defined as r ma m,n = m ( r 1 m A m,n, 0 ma m,n = A m,n, h na m,n = n ( h 1 n A m,n and 0 na m,n = A m,n There are several works about qualitative theory of higher order partial difference equations B G Zhang and S T Liu [7,8 studied the oscillatory behaviour of solutions of the partial difference equations of the forms A m+k,n+l + s 1 s 2 i=1 j=1 q i,j A m+k i,n+l j = 0 and A m,n = u p i A m ki,n l i + i=1 ν q j A m+τj,n+σ j B G Zhang, Y Zhou and Y Q Huang [10 investigated existence of positive solutions for nonlinear higher order neutral partial difference equations of the form h n r m(a m,n ca m k,n l + ( 1 h+r+1 p m,n f(a m τ,n σ = 0 In 2007, C F Li and Y Zhou [2 studied existence of bounded and unbounded nonoscillatory solutions for partial difference equations of the form h n r m(a m,n + c m,n A m k,n l + p m,n A m σ1,n ρ 1 q m,n A m σ2,n ρ 2 = 0 Ch G Philos and Y G Sficas [3 studied the oscillation of the ordinary difference equations of the form ( 1 m+1 m A n + j=1 p k A n lk = 0 By a solution of (11, we mean a nontrivial double sequence {A m,n } which is defined for m k and n l and satisfies (11 for m 0, n 0 A solution {A m,n } of (11 is said to be eventually positive (or negative if A m,n > 0 (or A m,n < 0 for all large m and n It is said to be oscillatory if it is neither eventually positive nor eventually negative A solution {A i,j } of (11 is called to be proper, if there exist positive numbers M, α and β such that A m,n Mα m β n, (12 k=0

3 Oscillation of Higher Order PDEs 213 for all large m and n The set Ω = N k N l /N 1 N 1 is called the initial domain A function φ i,j defined on Ω is called the initial function It is easy to construct by induction a double sequence {A i,j } which equals φ i,j on Ω and satisfies (11 on N 0 N 0 It is not difficult to prove that if the initial data satisfy φ m,n M 1 α m β n, (m, n Ω, (13 for some positive numbers M 1, α and β, then the corresponding solution is proper We look for the solutions of the form A m,n = λ m µ n, (14 where λ and µ are complex numbers Substituting (14 into (11, we obtain the characteristic equation or Φ(λ, µ = Φ(λ, µ = (λ 1 r (µ 1 h + ( 1 h+r+1 pλ k µ l = 0 (15 r h j=0 ( 1 i+j ( r i( h j λ r i µ h j + ( 1 h+r+1 pλ k µ l = 0 (λ, µ is said to be positive root of equation (15, if it satisfies Equation (15; moreover, λ > 0 and µ > 0 2 Some Auxiliary Lemmas Lemma 21 (See [9 Assume that there exist positive constants M 1, M, and N such that A m,n M 1 r1 m r2 n, m M, n N Then the z-transform of {A m,n }, which is defined by Z (A m,n = exists in the region z 1 > r 1 and z 2 > r 2 of m, 2, In the following, we always assume that A m,n = 0 for m < 0 and n < 0 in the series m=p n=q 2 By direct calculations, we can prove the following lemma Lemma 22 (See [9 The following formulas are true:

4 214 Özpınar and Akın (i (ii (iii (iv (v where F (k + i, z 2 = Z(A m k,n l = z k 1 z l 2 F (z 1, z 2 F (k + i, z 2 z i 1 = z k 1 ( A k+i,n z n 2 F (z 1, z 2 k 1 ( k 1 A k+i,n z1 i z2 n = z1 k A m,n z1 m z2 n where F (z 1, n = A m,n z1 m z2 n = Z(A m+k,n+l =z k 1z l ( k 1 F (z 1, z 2 k 1 2 F (m, z 2 z m 1 F (z 1, iz i 2, F (m, z 2 z m 1, A m,n z1 m z2 n F (z 1, nz n 2 3 Main Results Theorem 31 Every proper solution {A m,n } of (11 is oscillatory if and only if its characteristic equation (15 has no positive roots Proof Necessity Otherwise, let (λ 0, µ 0 be a positive root of (15 Then it is easy to find that {A m,n } with A m,n = λ m 0 µ n 0 is a positive proper solution of (11, a contradiction Sufficiency Assume that (15 has no positive roots Let {A m,n } be a positive proper solution of (11 with the initial data φ m,n such that φ m,n < c Then, by induction, it is easy to find that there exist b > 0 such that A m,n < bc m+n, (m, n N 2 0 (31

5 Oscillation of Higher Order PDEs 215 Thus, by Lemma 21, for z i > c, i = 1, 2, the z-transform of {A m,n } Z (A m,n = A m,n z1 m z2 n = F (z 1, z 2 (32 m, exists By taking the z-transform of both sides of (11, we obtain z i > c, i = 1, 2, where and Φ (z 1, z 2 = Ψ (z 1, z 2 = r r Φ (z 1, z 2 F (z 1 z 2 = Ψ (z 1, z 2, (33 h j=0 h j=0 ( We write (33 in the form Set r i 1 ( 1 i+j ( r h i( j z r i 1 z h j 2 + ( 1 h+r+1 pz1 k z2 l ( 1 i+j ( r h i( j z r i 1 z h j 2 r i 1 h j A m,n z1 m z2 n h j 1 2 Φ (1/z 1, 1/z 2 F (1/z 1, 1/z 2 = Ψ (1/z 1, 1/z 2 (34 ω (z 1, z 2 = F (1/z 1, 1/z 2 = m, A m,n z m 1 z n 2 (35 Equation (35 has radius of convergence ρ i, i = 1, 2 That is, (34 holds for z i < ρ i, i = 1, 2 Equivalently, (33 holds for z i > 1/ρ i, i = 1, 2 It is known that a power series with positive coefficients having radius of convergence ρ i, i = 1, 2 has a singularity at z i = ρ i, i = 1, 2 [1 By condition Φ (z 1, z 2 0 for (z 1, z 2 (0, (0, Thus Φ (1/ρ 1, 1/ρ 2 0, and hence, ω (z 1, z 2 = Ψ (1/z 1, 1/z 2 Φ (1/z 1, 1/z 2 is analytic in the region z 1 ρ 1 < d 1 and z 2 ρ 2 < d 2, which contradicts the singularity of ω (z 1, z 2 at z i = ρ i, i = 1, 2 Therefore we must have ρ i =, i = 1, 2, ie, (33 holds for z i > 0, i = 1, 2, which leads to A m,n = 0 for all large m and n Otherwise, the left-hand side of (33 does not equal the right-hand side This contradiction finishes the proof

6 216 Özpınar and Akın From Theorem 31, we can derive some sufficient and necessary conditions for oscillation of (11 Theorem 32 Assume that p > 0 Then every proper solution of (11 oscillates if and only if p (k + rk+r (l + h l+h > 1 (36 Proof Necessity It is sufficient to prove that if (36 does not hold, then (11 has a positive solution (i Let r + h be odd Obviously, if (36 does not hold, then in view of (15, we get and Φ (1, l/(l + h > 0 r r h h Φ (k/(k + r, l/(l + h = (k + r r (l + h h [ 1 + p (k + rk+r (l + h l+h 0 (ii Let r + h be even If (36 does not hold, then we get and Φ (1, l/(l + h < 0 r r h h Φ (k/(k + r, l/(l + h = (k + r r (l + h h [ 1 p (k + rk+r (l + h l+h 0 Since Φ(λ, µ is continuous, there exist λ 0 [k/(k + r, 1 and µ 0 = l/(l + h such that Φ (λ 0, µ 0 = 0 By Theorem 31, (11 has a positive solution This is a contradiction Sufficiency It is sufficient to prove that under condition (36, the characteristic equation (15 has no positive roots (i If r+h is odd, then (15 has no positive roots for the case that (λ 1 r (µ 1 h 0 For the case that (λ 1 r (µ 1 h < 0, we write Φ(λ, µ in the form Φ (λ, µ = (λ 1 r (µ 1 [ 1 h p + λ k µ l (λ 1 r (µ 1 h

7 Oscillation of Higher Order PDEs 217 Set f (λ, µ = λ k µ l (λ 1 r (µ 1 h It is easy to find that f (λ, µ reaches its maximum value at λ 0 µ 0 = l/(l + h Hence, = k/(k + r, max f (λ, µ = f (λ 0, µ 0 = λ,µ (0,1 Thus, for λ, µ (0, 1, we have Φ (λ, µ (λ 1 r (µ 1 h [ which implies that (15 has no positive roots (k + r k+r (l + h l+h 1 + p (k + rk+r (l + h l+h > 0, (ii If r+h is even, then (15 has no positive roots for the case that (λ 1 r (µ 1 h 0 For the case that (λ 1 r (µ 1 h > 0, we write Φ(λ, µ in the form Φ (λ, µ = (λ 1 r (µ 1 [1 h p λ k µ l (λ 1 r (µ 1 h Set f (λ, µ = λ k µ l (λ 1 r (µ 1 h Then it is not difficult to find that f (λ, µ reaches its maximum value at λ 0 = k/(k + r, µ 0 = l/(l + h Hence, max f (λ, µ = f (λ 0, µ 0 = λ,µ (0,1 or λ,µ>1 Thus, for λ, µ (0, 1 or λ, µ > 1, we have Φ (λ, µ (λ 1 r (µ 1 h [ which implies that (15 has no positive roots By Theorem 31, every proper solution of (11 oscillates (k + r k+r (l + h l+h 1 p (k + rk+r (l + h l+h < 0, References [1 R P Gilbert Function Theoretic Methods in Partial Differential Equations Academic Press, New York, 1969

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