Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 04, pp. 1–19. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CONVERGENCE OF APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FRACTIONAL DIFFERENCE EQUATIONS WITH BOUNDARY CONDITIONS XIANG LIU, BAOGUO JIA, SCOTT GENSLER, LYNN ERBE, ALLAN PETERSON Abstract. This article studies a boundary value problem for a nonlinear Ca- puto nabla fractional difference equation. We obtain quadratic convergence results for this equation using the generalized quasi-linearization method. Fur- ther, we obtain the convergence of the sequences is potentially improved by the Gauss-Seidel method. A numerical example illustrates our main results. 1. Introduction It is well-known that fractional difference equations can be used to model many problems, such as population models, tumor growth model, and so on. In many situations, fractional difference equations have proved to be better than their coun- terpart with integer difference. Therefore, research in the theory of fractional differ- ence equations has become very important. Previous studies have mainly focused on the theory of integer-order difference equations, and classical results have been established; see for example the monographs [2, 17]. Recently, there has been a great deal of interest in fractional difference equations. The basic theory of the linear and nonlinear fractional difference equations can be found in [8, 9, 13, 14, 15, 16]. How- ever, we note that the qualitative theory of nonlinear fractional difference equations is not complete and the convergence of approximate solutions plays an important role in the development of qualitative theory. The generalized quasi-linearization method can be used to construct approximate solutions of nonlinear problems. There are many applications of this method. In [3, 4, 5, 6, 10, 11, 12, 18, 22, 21], the authors used this method to obtain the convergence of the sequences for different types of differential equations. Further, in [7, 20], the authors accelerated the convergence of the sequences by the Gauss- Seidel method. However, there are few applicable results of the above methods to nonlinear fractional difference equations. In [8, 19], the authors only discussed the existence and convergence of solutions for nonlinear fractional difference equations with initial conditions. In this article, we study the convergence of solutions for a nonlinear Caputo nabla fractional difference equation with boundary conditions. We obtain quadratic convergence by the generalized quasi-linearization method when the forcing function 2010 Mathematics Subject Classification. 39A12, 39A70. Key words and phrases. Caputo nabla fractional difference equation; upper and lower solution; generalized quasi-linearization method; Gauss-Seidel method. c©2020 Texas State University. Submitted January 11, 2019. Published January 10, 2020. 1 2 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 is the sum of convex and concave functions. Furthermore, we obtain the convergence of the sequences is potentially improved by the Gauss-Seidel method. Finally, a numerical example illustrates the obtained results. 2. Preliminaries For the convenience of readers, we will list some relevant results here. We use the notation Na := {a, a+ 1, a+ 2, . . . }, Za := {. . . , a− 2, a− 1, a, a+ 1, a+ 2, . . . }, where a is a real number. For the function f : Za → R, the backward difference or nabla operator is defined as ∇f(t) = f(t) − f(t − 1) for t ∈ Za and the higher order differences are defined recursively by ∇nf(t) = ∇(∇n−1f(t)) for t ∈ Za. In addition, we take ∇0 as the identity operator. We define the definite nabla integral of f : Za → R for b ∈ Za by∫ b a f(s)∇s = {∑b s=a+1 f(s), a < b, 0, a ≥ b. (2.1) Definition 2.1 (See [15, Definition 3.4]). The (generalized) rising function is de- fined in terms of the gamma function by tr := Γ(t+ r) Γ(t) (2.2) for those values of t and r so that the right-hand side of (2.2) is well defined. Also, we use the convention that if t is a nonpositive integer, or t+ r is not a nonpositive integer, then tr = 0. We then define the ν-th order Taylor monomials based at a (see [15, Definition 3.56]) by Hν(t, a) := (t− a)ν Γ(ν + 1) = Γ(t− a+ ν) Γ(t− a)Γ(ν + 1) for ν 6= −1,−2, . . . , and t ∈ Na. In this article, we extend the ν-th order Taylor monomials by defining H̃ν(t, a) := lim ε→0 Γ(t− a+ ν + ε) Γ(t− a+ ε)Γ(ν + 1 + ε) for all ν ∈ R, and all t ∈ Za. Note that when t > a and ν > −1, H̃ν(t, a) = Hν(t, a). The extended Taylor monomials will be used in Lemmas 2.9 and 2.10. For some important formulas for these Taylor fractional monomials see [15, The- orem 3.57 and Theorem 3.93]. Definition 2.2 (Nabla Fractional Sum [15, Definition 3.58]). Let f : Na → R, ν > 0 be given. Then (∇−νa f)(t) = ∫ t a Hν−1(t, ρ(s))f(s)∇s, t ∈ Na, (2.3) where ρ(t) := t− 1. Note, by (2.1), ∇−νa f(a) = 0. Definition 2.3 (Nabla Fractional Difference [15, Definition 3.61]). Let f : Na → R, ν > 0 be given, and let N := dνe, where d·e is the ceiling function. Then we define the νth-order nabla fractional difference (∇νaf)(t) by (∇νaf)(t) = (∇N (∇−(N−ν) a f))(t), t ∈ Na+N . (2.4) EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 3 Definition 2.4 (Caputo Nabla Fractional Difference [15, Definition 3.117]). Let f : Na−N+1 → R, ν > 0 be given, and let N := dνe. Then we define the νth-order Caputo nabla fractional difference (∇νa∗f)(t) by (∇νa∗f)(t) = (∇−(N−ν) a (∇Nf))(t), t ∈ Na+1. (2.5) Since ν > 0 ⇒ N ≥ 1, it follows from this definition that ∇νa∗c = 0 for any constant c. Lemma 2.5 (See [15, Definition 3.61 and Theorem 3.62]). Assume f : Na → R, ν > 0, ν /∈ N1, and choose N ∈ N1 such that N − 1 < ν < N . Then (∇νaf)(t) = ∫ t a H−ν−1(t, ρ(s))f(s)∇s, t ∈ Na+N , (2.6) where ρ(t) := t− 1. Note, by (2.1), ∇νaf(a) = 0. Lemma 2.6 (Nabla Leibniz Formula [15, Corollary 3.41]). Assume that f : Na × Na+1 → R. Then for t ∈ Na+1, ∇ (∫ t a f(t, τ)∇τ ) = ∫ t a ∇tf(t, τ)∇τ + f(ρ(t), t). (2.7) Also, ∇ (∫ t a f(t, τ)∇τ ) = ∫ t−1 a ∇tf(t, τ)∇τ + f(t, t). (2.8) The following corollary appears in Goodrich et al [15, Corollary 3.122, Corollary 3.167]. Corollary 2.7. For ν > 0, N = dνe, and h : Na+1 → R, we have (∇−(N−ν) a (∇N−νa h))(t) = h(t), t ∈ Na+1. Lemma 2.8. The nabla Taylor monomials satisfy the following: (i) H̃0(t, a) = 1, t ∈ Za. (ii) ∇H̃r(t, a) = { H̃r−1(t, a), 0 6= r ∈ R, t ∈ Za, 0, r = 0, t ∈ Za. Proof. (i) From the definition of H̃r(t, a), we have H̃0(t, a) = lim ε→0 Γ(t− a+ ε) Γ(t− a+ ε)Γ(ε+ 1) = 1. (ii) For r = 0, using (i), we have ∇H̃0(t, a) = 0. For r 6= 0, by the definition of H̃r(t, a), we obtain ∇H̃r(t, a) = lim ε→0 Γ(t− a+ r + ε) Γ(t− a+ ε)Γ(r + ε+ 1) − lim ε→0 Γ(t− 1− a+ r + ε) Γ(t− 1− a+ ε)Γ(r + ε+ 1) = lim ε→0 [ t− 1− a+ r + ε t− 1− a+ ε − 1 ] Γ(t− 1− a+ r + ε) Γ(t− 1− a+ ε)Γ(r + ε+ 1) = lim ε→0 rΓ(t− 1− a+ r + ε) (r + ε)Γ(t− a+ ε)Γ(r + ε) = lim ε→0 r (r + ε) · lim ε→0 Γ(t− a+ r − 1 + ε) Γ(t− a+ ε)Γ(r + ε) 4 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 = H̃r−1(t, a). The proof is complete. � For convenience, in this paper, we define the operator La : Da−N+1 → Da+1 by (Lax)(t) := ∇[(∇νa∗x)(t+ 1)], t ∈ Na+1, where x ∈ Da−N+1 := {x : Na−N+1 → R}. Lemma 2.9. Assume ν > 0 and N is a positive integer such that N − 1 < ν ≤ N . Then a general solution of the fractional difference equation (Lax)(t) = 0, t ∈ Na+1 is given by x(t) = c0H̃0(t, a) + c1H̃1(t, a) + c2H̃2(t, a) + · · ·+ cN−1H̃N−1(t, a) + cHν(t, a) for t ∈ Na−N+1. Proof. Let xk(t) := H̃k(t, a), 0 ≤ k ≤ N − 1, by Lemma 2.8, we have (Laxk)(t) = ∇[∇−(N−ν) a ∇N H̃k(t+ 1, a)] = ∇[∇−(N−ν) a ∇N−k∇kH̃k(t+ 1, a)] = ∇[∇−(N−ν) a ∇N−kH̃0(t+ 1, a)] = ∇[∇−(N−ν) a 0] = 0 for t ∈ Na+1. Let x̄(t) := Hν(t, a), by Corollary 2.7, we have (Lax̄)(t) = ∇[∇−(N−ν) a ∇NHν(t+ 1, a)] = ∇[∇−(N−ν) a ∇N∇−νa 1(t+ 1)] = ∇[∇−(N−ν) a ∇N−νa 1(t+ 1)] = ∇[χ(a,∞)(t+ 1)] = ∇[χ[a,∞)(t)] = 0 for t ∈ Na+1, where χ[a,∞)(t) = { 1, t ∈ Na, 0, t /∈ Na. . Next, we show that these solutions are linearly independent. We want to show that if c0H̃0(t, a) + c1H̃1(t, a) + c2H̃2(t, a) + · · ·+ cN−1H̃N−1(t, a) + cHν(t, a) = 0 (2.9) for all t ∈ Na−N+1, then c0 = c1 = · · · = cN−1 = c = 0. Taking t = a, a− 1, . . . , a− (N − 1), and a+ 1 in (2.9), we obtain H̃0(a, a) . . . H̃N−1(a, a) Hν(a, a) H̃0(a− 1, a) . . . H̃N−1(a− 1, a) Hν(a− 1, a) ... . . . ... ... H̃0(a− (N − 1), a) . . . H̃N−1(a− (N − 1), a) Hν(a− (N − 1), a) H̃0(a+ 1, a) . . . H̃N−1(a+ 1, a) Hν(a+ 1, a)   c0 c1 ... cN−1 c  =  0 0 ... 0 0  ; EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 5 that is,  1 0 . . . 0 0 ∗ −1 . . . 0 0 ... ... . . . ... ... ∗ ∗ . . . (−1)N−1 0 ∗ ∗ . . . ∗ 1   c0 c1 ... cN−1 c  =  0 0 ... 0 0  . So, we arrive at c0 = c1 = · · · = cN−1 = c = 0. Therefore, these solutions are linearly independent. The proof is complete. � Lemma 2.10 (See [15, Theorem 3.175]). Assume 0 < ν ≤ 1, a, b ∈ R, and b− a ∈ N2. Then the Green function for the BVP (Lax)(t) = 0, t ∈ Nb−1 a+1, x(a) = 0, x(b) = 0 (2.10) is G(t, s) = { u(t, s), (t, s) ∈ Nsa × Nbt , v(t, s), (t, s) ∈ Nbs × Nta, where u(t, s) = − (b− s)ν(t− a)ν Γ(ν + 1)(b− a)ν , v(t, s) = u(t, s) + (t− s)ν Γ(ν + 1) = u(t, s) + x(t, s). Lemma 2.11 (See [15, Theorem 3.177]). Assume 0 < ν ≤ 1, a, b ∈ R, and b− a ∈ N2. Then the Green function for the BVP (Lax)(t) = 0, t ∈ Nb−1 a+1, x(a) = 0, x(b) = 0 satisfies the following inequalities (i) G(t, s) ≤ 0, (ii) G(t, s) ≥ − ( b−a 4 )( Γ(b−a+1) Γ(ν+1)Γ(b−a+ν) ) , (iii) ∫ b a |G(t, s)|∇s ≤ (b−a)2 4Γ(ν+2) , for t ∈ Nba, and (iv) ∫ b a |∇tG(t, s)|∇s ≤ b−a ν+1 , for t ∈ Nba+1. The next corollary is an immediate consequence of [15, Theorem 3.173]. It relates to the nonhomogeneous BVP with homogeneous boundary conditions, (Lay)(t) = h(t), t ∈ Nb−1 a+1, y(a) = 0, y(b) = 0. (2.11) Corollary 2.12 (See [15, Theorem 3.173]). Assume 0 < ν ≤ 1, a, b ∈ R, and b− a ∈ N2. The unique solution of (2.11) is y(t) = ∫ b a G(t, s)h(s)∇s = b∑ s=a+1 G(t, s)h(s), t ∈ Nba, where G(t, s) is the Green function of (2.10). 6 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 Proof. This is a special case of [15, Theorem 3.173], where α = γ = 1 and β = δ = 0 in equation (3.115) of that theorem. Note that, by [15, Theorem 3.170], α = γ = 1 and β = δ = 0 ensure the hypotheses of [15, Theorem 3.173] are satisfied. � Lemma 2.13. Assume 0 < ν ≤ 1. Then the solution of the BVP (Laz)(t) = 0, t ∈ Nb−1 a+1, z(a) = A, z(b) = B (2.12) is z(t) = A+ (B −A) Hν(t, a) Hν(b, a) , t ∈ Nba. Proof. Let z be a solution of the fractional difference equation (Laz)(t) = 0. It follows from Lemma 2.9 that z(t) = c0H̃0(t, a) + cHν(t, a) = c0 + cHν(t, a). Using the first boundary condition z(a) = A, we obtain A = c0 + cHν(a, a) = c0, which implies c0 = A. Using the second boundary condition z(b) = B, we obtain B = A+ cHν(b, a). Solving for c, we obtain c = (B −A) Hν(b, a) . Thus, we have z(t) = c0 + cHν(t, a) = A+ (B −A) Hν(t, a) Hν(b, a) . The proof is complete. � Lemma 2.14. Assume 0 < ν ≤ 1, and h : Nb−1 a+1 → R. Then the solution of the nonhomogeneous BVP (Lax)(t) = h(t), t ∈ Nb−1 a+1, x(a) = A, x(b) = B (2.13) is x(t) = z(t) + b∑ s=a+1 G(t, s)h(s), t ∈ Nba, where G(t, s) is the Green’s function of the BVP (2.10) and z(t) is the unique solution of (2.12). Proof. Let y(t) = b∑ s=a+1 G(t, s)h(s), t ∈ Nba. By Corollary 2.12, y(t) is the solution of (2.11) on Nba. Let z(t) be as in the statement of this theorem. Then x(a) = z(a) + y(a) = A+ 0 = A, x(b) = z(b) + y(b) = B + 0 = B. EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 7 Finally, (Lax)(t) = (Laz)(t) + (Lay)(t) = 0 + h(t) = h(t) for t ∈ Nb−1 a+1. The proof is complete. � Lemma 2.15. Assume 0 < ν ≤ 1, x : Nba → R, and put M := max{x(t) : t ∈ Nba}. If x(t0) = M for some t0 ∈ Nb−1 a+1, then (Lax)(t0) ≤ 0. Proof. For ν = 1, according to the assumptions, we have (Lax)(t0) = (∇2x)(t)|t=t0+1 = x(t0 + 1)− 2x(t0) + x(t0 − 1) ≤M − 2M +M = 0 For the case when 0 < ν < 1, we will use the fact that for any t ∈ Na+1, (Lax)(t) = x(t+ 1)−H−ν−1(t+ 1, a)x(a) + t∑ s=a+1 H−ν−2(t+ 1, ρ(s))x(s). (2.14) To see this note that (Lax)(t) = ∇[(∇νa∗x)(t+ 1)] = ∇[∇ν−1 a ∇x(t+ 1)] = ∇ [ t+1∑ s=a+1 H−ν(t+ 1, ρ(s))∇x(s) ] = ∇ [ t+1∑ s=a+1 H−ν(t+ 1, ρ(s))x(s)− t+1∑ s=a+1 H−ν(t+ 1, ρ(s))x(s− 1) ] = ∇ [ t+1∑ s=a+1 H−ν(t+ 1, ρ(s))x(s)− t∑ s=a H−ν(t+ 1, s)x(s) ] = ∇ [ t∑ s=a+1 H−ν(t+ 1, ρ(s))x(s) +H−ν(t+ 1, ρ(t+ 1))x(t+ 1) − t∑ s=a+1 H−ν(t+ 1, s)x(s)−H−ν(t+ 1, a)x(a) ] = ∇ [ t∑ s=a+1 H−ν−1(t+ 1, ρ(s))x(s) +H−ν−1(t+ 1, ρ(t+ 1))x(t+ 1) −H−ν(t+ 1, a)x(a) ] = ∇ [ t+1∑ s=a+1 H−ν−1(t+ 1, ρ(s))x(s)−H−ν(t+ 1, a)x(a) ] = [ t+1∑ s=a+1 H−ν−1(t+ 1, ρ(s))x(s)−H−ν(t+ 1, a)x(a) ] − [ t∑ s=a+1 H−ν−1(t, ρ(s))x(s) +H−ν(t, a)x(a) ] 8 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 = x(t+ 1) + t∑ s=a+1 H−ν−2(t+ 1, ρ(s))x(s)−H−ν−1(t+ 1, a)x(a). Note that for 0 < ν ≤ 1, −H−ν−1(t0 + 1, a) is positive whenever t0 ∈ Na+1 and H−ν−2(t0 + 1, ρ(s)) is positive whenever s ∈ Nt0−1 a . Thus, we obtain (Lax)(t0) = x(t0 + 1)−H−ν−1(t0 + 1, a)x(a) + t0∑ s=a+1 H−ν−2(t0 + 1, ρ(s))x(s) = x(t0 + 1)−H−ν−1(t0 + 1, a)x(a) + t0−1∑ s=a+1 H−ν−2(t0 + 1, ρ(s))x(s) +H−ν−2(t0 + 1, ρ(t0))x(t0) ≤M −H−ν−1(t0 + 1, a)M +M t0−1∑ s=a+1 H−ν−2(t0 + 1, ρ(s)) +H−ν−2(t0 + 1, ρ(t0))M = M −H−ν−1(t0 + 1, a)M +M t0∑ s=a+1 H−ν−2(t0 + 1, ρ(s)) = M [1−H−ν−1(t0 + 1, a)] +M [(−H−ν−1(t0 + 1, t0) +H−ν−1(t0 + 1, a)] = 0. The proof is complete. � 3. Existence and comparison results Consider the following BVP for a nonlinear Caputo nabla fractional difference equation (Lax)(t) = f(t, x(t)), t ∈ Nb−1 a+1, x(a) = A, x(b) = B, (3.1) where f : Nb−1 a+1×R→ R is continuous with respect to x, x : Nba → R, and 0 < ν ≤ 1. In this article, we define the norm of x on Nba by ‖x‖ = maxs∈Nb a |x(s)|. Through- out this paper, we use the notation f (k)(t, x) := ∂kf(t,x) ∂kx (k = 0, 1, 2 . . .). For con- venience, when α0(t) and β0(t) are two functions such that α0(t) ≤ β0(t) on Nba, we use the following sets: Ω = Ω(α0, β0) := {(t, x) : α0(t) ≤ x ≤ β0(t), for all t ∈ Nb−1 a+1}, S = S(α0, β0) := {x : α0(t) ≤ x(t) ≤ β0(t), for all t ∈ Nba}. We will simply refer to the sets Ω and S when it is clear from context what α0 and β0 are. Definition 3.1. A function α0(t) is said to be a lower solution of (3.1), if (Laα0)(t) ≥ f(t, α0(t)), t ∈ Nb−1 a+1, α0(a) ≤ A, α0(b) ≤ B. (3.2) If all three inequalities in (3.2) are reversed, we have an upper solution. Now we present an existence result relative to BVP (3.1), which we will use in our main EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 9 results. Since the proof is a standard application of Schauder’s fixed point theorem we will omit the proof of this lemma. Lemma 3.2. Assume that (H3.1) the function f : Nb−1 a+1 × R → R is continuous with respect to x, and for M ≥ 0, define C := max{|f(t, x)| : t ∈ Nb−1 a+1, |x| ≤ 2M}. Then the nonlinear BVP (3.1) has a solution provided there is some M ≥ ‖z‖ = max{|A|, |B|}, where z is the unique solution of (2.12), such that C(M) > 0 and (b− a)2 ≤ 4MΓ(ν + 2) C . In particular, if f 6≡ 0 and is bounded, then (3.1) has a solution. Lemma 3.3. Assume that (H3.2) the function f : Nb−1 a+1 × R→ R is nondecreasing with respect to x for each t. (H3.3) the functions α0, β0 : Nba → R are lower and upper solutions respectively of (3.1). Then α0(t) ≤ β0(t) on Nba. Proof. Let us first prove the lemma for strict inequality. That is, suppose (Laα0)(t) > f(t, α0(t)), t ∈ Nb−1 a+1, α0(a) ≤ A, α0(b) ≤ B. and (Laβ0)(t) ≤ f(t, β0(t)), t ∈ Nb−1 a+1, β0(a) ≥ A, β0(b) ≥ B. The boundary conditions give us α0(a) ≤ A ≤ β0(a) and α0(b) ≤ B ≤ β0(b). Next, we will show that α0(t) < β0(t) for t ∈ Nb−1 a+1. If this is not true, then there exists t0 ∈ Nb−1 a+1 such that x(t) := α0(t)− β0(t) has a nonnegative maximum at t0. That is, for some t0 ∈ Nb−1 a+1, x(t0) = α0(t0)− β0(t0) = max{α0(t)− β0(t), t ∈ Nb−1 a+1} ≥ 0, But then, x(t0) = maxt∈Nb a x(t), since x(a) = α0(a) − β0(a) ≤ 0 and x(b) = α0(b)− β0(b) ≤ 0. Therefore, by Lemma 2.15, we obtain (Lax)(t0) = (Laα0)(t0)− (Laβ0)(t0) ≤ 0. So, we have f(t0, α0(t0)) < (Laα0)(t0) ≤ (Laβ0)(t0) ≤ f(t0, β0(t0)), On the other hand, our assumption that x(t0) = α0(t0) − β0(t0) ≥ 0 ⇒ β0(t0) ≤ α0(t0). By (H3.2), f(t, x) is nondecreasing with respect to x for each t, so this, in turn, implies f(t0, β0(t0)) ≤ f(t0, α0(t0)) which is a contradiction. Hence, α0(t) < β0(t) on Nb−1 a+1. Now, we define α̃0(t) := α0(t) + ε(Hν+1(t, a) −Hν+1(b, a)), where ε > 0. Then α̃0(t) < α0(t) for t ∈ Nb−1 a+1. Using condition (H3.2), we obtain (Laα̃0)(t) = (Laα0)(t) + εLa[Hν+1(·, a)−Hν+1(b, a)](t) 10 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 = (Laα0)(t) + ε ≥ f(t, α0(t)) + ε ≥ f(t, α̃0(t)) + ε > f(t, α̃0(t)). Thus α̃0(t) is a lower solution for which strict inequality holds. It therefore follows from the previous argument that α̃0(t) < β0(t) for all t ∈ Nb−1 a+1. Note this holds for all ε > 0. Letting ε → 0, we obtain α0(t) ≤ β0(t) for all t ∈ Nb−1 a+1. Thus, we have α0(t) ≤ β0(t) on Nba. The proof is complete. � Corollary 3.4. Assume that conditions (H3.1), (H3.2) hold. Then BVP (3.1) has a unique solution on Nba. Proof. According to Lemma 3.2, we obtain that BVP (3.1) has a solution on Nba. Now, we suppose x(t) and x̄(t) are two solutions of (3.1). Since any solution is both a lower and an upper solution, by Lemma 3.3, we have x(t) ≤ x̄(t) ≤ x(t)⇒ x(t) = x̄(t) on Nba. That is, the solution of (3.1) is unique on Nba. The proof is complete. � Next, we consider BVP (3.1) in the special case where f(t, x) = C(t)x and A = B = 0. That is, we consider the BVP (Lax)(t) = C(t)x(t) for t ∈ Nb−1 a+1, x(a) = 0, x(b) = 0. (3.3) Corollary 3.5. Assume that (H3.4) the function C(t) ≥ 0 for t ∈ Nb−1 a+1. If x(t) is a lower solution of (3.3), then x(t) ≤ 0 on Nba. If y(t) is an upper solution of (3.3), then y(t) ≥ 0 on Nba. Proof. By hypothesis, x(t) is a lower solution of (3.3) and, by inspection, z(t) ≡ 0 is a (upper) solution to (3.3). So Lemma 3.3 guarantees x(t) ≤ 0. Similarly, y(t) is an upper solution of (3.3) and z(t) ≡ 0 is a (lower) solution to (3.3). So Lemma 3.3 guarantees y(t) ≥ 0. The proof is complete. � Lemma 3.6. Assume that (H3.5) the functions α0, β0 : Nba → R are lower and upper solutions respectively of (3.1) such that α0(t) ≤ β0(t) on Nba. (H3.6) the function f : Ω → R is continuous in its second variable and f 6≡ 0 on Ω. Then there exists a solution x(t) of (3.1) satisfying α0(t) ≤ x(t) ≤ β0(t) on Nba. Proof. Let P : Nb−1 a+1 × R → R be defined by P (t, x) = max { α0(t),min{x, β0(t)} } . Then f(t, P (t, x)) defines a extension of f to Nb−1 a+1 × R, which is continuous in its second variable, f 6≡ 0 and bounded. Therefore, by Lemma 3.2, the BVP (Lax)(t) = f̄(t, x) := f(t, P (t, x)) for t ∈ Nb−1 a+1, x(a) = A, x(b) = B. (3.4) has a solution x(t) on Nba. To complete the proof, we just need to show that α0(t) ≤ x(t) ≤ β0(t) on Nba. Doing so will mean that f̄(t, x(t)) = f(t, x(t)), which will imply x(t) is not only a solution of (3.4) but also actually a solution of (3.1). Toward this end, we first show that α0(t) ≤ x(t) on Nba. The boundary conditions immediately give us that EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 11 α0(a) ≤ x(a) and α0(b) ≤ x(b). To see that α0(t) ≤ x(t) on Nb−1 a+1, as in the proof of Lemma 3.3, for ε > 0 define α̃ε(t) := α0(t) + ε[Hν+1(t, a) − Hν+1(b, a)]. Then α̃ε(t) < α0(t) for t ∈ Nb−1 a+1, α̃ε(a) < α0(a) ≤ x(a), and α̃ε(b) = α0(b) ≤ x(b). If we can show α̃ε(t) < x(t) on Nb−1 a+1, then letting ε → 0, we obtain limε→0 α̃ε(t) = α0(t) ≤ x(t) for t ∈ Nb−1 a+1 and we will be done. So, toward contradiction, assume that for some fixed ε > 0 this is not true. That is, assume that for some fixed ε > 0 there exists t1 ∈ Nb−1 a+1 such that d(t) := α̃ε(t) − x(t) has a nonnegative maximum at t1, i.e. for this t1 ∈ Nb−1 a+1, d(t1) = α̃ε(t1)− x(t1) = max{α̃ε(t)− x(t), t ∈ Nb−1 a+1} ≥ 0. But then, d(t1) = maxt∈Nb a d(t), since d(a) = α̃ε(a)− x(a) ≤ 0 and d(b) = α̃ε(b)− x(b) ≤ 0. And so, by Lemma 2.15, we must have (Lad)(t1) ≤ 0. (3.5) On the other hand, our assumption that α̃ε(t1)−x(t1) ≥ 0 implies x(t1) ≤ α̃ε(t1) ≤ α0(t1) and so we have P (t1, x(t1)) = α0(t1). Thus, (Laα0)(t1) ≥ f(t1, α0(t1)) = f(t1, P (t1, x(t1))) = (Lax)(t1) which implies (Laα0)(t1)− (Lax)(t1) ≥ 0 which, in turn, implies, (Lad)(t1) = (Laα̃ε)(t1)− (Lax)(t1) = (Laα0)(t1) + εLa[Hν+1(·, a)−Hν+1(b, a)](t1)− (Lax)(t1) = (Laα0)(t1) + ε(1− 0)− (Lax)(t1) = (Laα0)(t1)− (Lax)(t1) + ε > 0. which is in contradiction to (3.5). Thus, for all ε > 0, α̃ε(t) < x(t) on Nb−1 a+1. And so, as argued above, letting ε→ 0 we obtain our result that α0(t) ≤ x(t) on Nb−1 a+1. Similarly, we can show that x(t) ≤ β0(t) on Nb−1 a+1. Therefore, we conclude that α0(t) ≤ x(t) ≤ β0(t), t ∈ Nba. The proof is complete. � Lemma 3.7. Assume that the conditions (H3.2), (H3.5) and (H3.6) hold. Then (3.1) has a unique solution x(t). Furthermore, α0(t) ≤ x(t) ≤ β0(t) on Nba. Proof. By Lemma 3.6, there exists a solution of (3.1) which lies in S. By Corollary 3.4 this solution is unique. The proof is complete. � 4. Main results In this section, we consider BVP (3.1) in the special case where f(t, x) is in- creasing with respect to x and f(t, x) = f1(t, x) + f2(t, x), where f1(t, x) is concave up with respect to x and f2(t, x) is concave down with respect to x. We obtain se- quences of successive approximations by applying the generalized quasi-linearization method to our nonlinear Caputo nabla fractional difference equation and show that the sequences so obtained converge quadratically to the solution. Furthermore, we use the Gauss-Seidel method to improve the rate of convergence. Theorem 4.1. Assume that condition (H3.5) holds, and (A4.1) f 6≡ 0 and f = f1 + f2 where the functions f1, f2 : Ω→ R are such that: 12 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 (i) f (i) 1 (t, x), f (i) 2 (t, x) (i = 0, 1, 2) exist and are continuous in the second variable, (ii) f (1) 1 (t, x) + f (1) 2 (t, y) ≥ 0 for all (t, x), (t, y) ∈ Ω, and (iii) f (2) 1 (t, x) ≥ 0 and f (2) 2 (t, x) ≤ 0 on Ω. Then there exist two sequences {αn(t)} and {βn(t)}, which converge monotonically to a function x in S(α0, β0) which is a solution of (3.1). Furthermore, the conver- gence is quadratic. As in Agarwal et al [3, 3. Main Results, Theorem 1], we define the quadratic convergence of two sequences {αn} and {βn} which both converge to a common point x, by the condition that there exist constants C1, C2, C3, and C4 such that for all n ≥ 0, ‖x− αn+1‖ ≤ C1‖x− αn‖2 + C2‖βn − x‖2, ‖βn+1 − x‖ ≤ C3‖βn − x‖2 + C4‖x− αn‖2. Proof. From condition (A4.1)(iii) and the mean value theorem, we obtain, for all (t, x), (t, y) ∈ Ω, f1(t, y) ≥ f1(t, x) + f (1) 1 (t, x)(y − x), (4.1) f2(t, y) ≥ f2(t, x) + f (1) 2 (t, y)(y − x). (4.2) Next, consider the following two BVPs (Lay)(t) = f1(t, α) + f (1) 1 (t, β)(y − α) + f2(t, α) + f (1) 2 (t, α)(y − α) ≡ F (t, α, β; y), t ∈ Nb−1 a+1, y(a) = A, y(b) = B, (4.3) and (Laz)(t) = f1(t, β) + f (1) 1 (t, β)(z − β) + f2(t, β) + f (1) 2 (t, α)(z − β) ≡ G(t, α, β; z), t ∈ Nb−1 a+1, z(a) = A, z(b) = B. (4.4) Letting α = α0, β = β0 in BVPs (4.3), (4.4). We first prove that α0(t) and β0(t) are lower and upper solutions of (4.3), respectively. In fact, from condition (H3.5), we have (Laα0)(t) ≥ f1(t, α0) + f2(t, α0) = F (t, α0, β0;α0), t ∈ Nb−1 a+1, α0(a) ≤ A, α0(b) ≤ B, and by using inequalities (4.1), (4.2), it follows that (Laβ0)(t) ≤ f1(t, β0) + f2(t, β0) ≤ f1(t, α0) + f (1) 1 (t, β0)(β0 − α0) + f2(t, α0) + f (1) 2 (t, α0)(β0 − α0) = F (t, α0, β0;β0), t ∈ Nb−1 a+1, β0(a) ≥ A, β0(b) ≥ B. These show that α0(t) and β0(t) are lower and upper solutions of (4.3). Further- more, note that F : Nb−1 a+1 × R → R is continuous and nondecreasing with respect to y. Thus, by Lemma 3.7, it follows that there exists a unique solution α1(t) of (4.3) such that α0(t) ≤ α1(t) ≤ β0(t) on Nba. EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 13 Similarly, using condition (H3.5), and inequalities (4.1), (4.2), we obtain (Laα0)(t) ≥ f1(t, α0) + f2(t, α0) ≥ f1(t, β0) + f (1) 1 (t, β0)(α0 − β0) + f2(t, β0) + f (1) 2 (t, α0)(α0 − β0) = G(t, α0, β0;α0), t ∈ Nb−1 a+1, α0(a) ≤ A, α0(b) ≤ B, and (Laβ0)(t) ≤ f1(t, β0) + f2(t, β0) = G(t, α0, β0;β0), t ∈ Nb−1 a+1, β0(a) ≥ A, β0(b) ≥ B. These show that α0(t) and β0(t) are lower and upper solutions of (4.4). Further- more, note that G : Nb−1 a+1 × R → R is continuous and nondecreasing with respect to z. Thus, by Lemma 3.7, it follows that there exists a unique solution β1(t) of (4.4) such that α0(t) ≤ β1(t) ≤ β0(t) on Nba. Next, we show that α1(t) and β1(t) are lower and upper solutions of the original BVP (3.1). Toward this end, using the fact that α1(t) is the unique solution of (4.3), condition (A4.1)(iii), and the inequalities (4.1), (4.2), we have (Laα1)(t) = f1(t, α0) + f (1) 1 (t, β0)(α1 − α0) + f2(t, α0) + f (1) 2 (t, α0)(α1 − α0) ≥ f1(t, α0) + f (1) 1 (t, α1)(α1 − α0) + f2(t, α0) + f (1) 2 (t, α0)(α1 − α0) ≥ f1(t, α1) + f2(t, α1), t ∈ Nb−1 a+1, α1(a) = A, α1(b) = B, which proves α1(t) is a lower solution of (3.1). Similar arguments show that (Laβ1)(t) ≤ f1(t, β1) + f2(t, β1), t ∈ Nb−1 a+1, β1(a) = A, β1(b) = B, which shows that β1(t) is an upper solution of (3.1). So by Lemma 3.3, we have α1(t) ≤ β1(t) on Nba. Thus, we obtain α0(t) ≤ α1(t) ≤ β1(t) ≤ β0(t) on Nba. Thus, by iteration, we obtain α0(t) ≤ α1(t) ≤ · · · ≤ αn(t) ≤ βn(t) ≤ · · · ≤ β1(t) ≤ β0(t) on Nba where we obtain the functions αn+1 and βn+1 by the iterative schemes: (Laαn+1)(t) = f1(t, αn) + f (1) 1 (t, βn)(αn+1 − αn) + f2(t, αn) + f (1) 2 (t, αn)(αn+1 − αn) = F (t, αn, βn;αn+1), t ∈ Nb−1 a+1, αn+1(a) = A, αn+1(b) = B, (4.5) and (Laβn+1)(t) = f1(t, βn) + f (1) 1 (t, βn)(βn+1 − βn) + f2(t, βn) + f (1) 2 (t, αn)(βn+1 − βn) = G(t, αn, βn;βn+1), t ∈ Nb−1 a+1, βn+1(a) = A, βn+1(b) = B. (4.6) 14 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 In addition, using the fact that αn(t), βn(t) are lower and upper solutions of (3.1) with αn(t) ≤ βn(t), and the conditions of Lemma 3.7 are satisfied, we conclude that there exists a unique solution x(t) of (3.1) such that αn(t) ≤ x(t) ≤ βn(t) on Nba. Thus α0(t) ≤ α1(t) ≤ · · · ≤ αn(t) ≤ x(t) ≤ βn(t) ≤ · · · ≤ β1(t) ≤ β0(t) on Nba For any fixed t ∈ Nba, the monotone sequences {αn(t)} and {βn(t)} are bounded by {α0(t)} and {β0(t)}. As such, they converge pointwise to some limit functions, ρ and r. That is, the functions ρ, r : Nba → R and satisfy lim n→∞ αn(t) = ρ(t) ≤ x(t) ≤ r(t) = lim n→∞ βn(t). By taking the limit as n → ∞ of the difference equation in (4.5) and (4.6) we can show that ρ(t) and r(t) are solutions of (3.1). Since ρ(t) and r(t) also lie in S(α0, β0), it must be the case that ρ(t) = x(t) = r(t) on Nba. Hence αn(t) and βn(t) both converge monotonically to x(t). Finally, we show that the convergence of the sequences {αn(t)} and {βn(t)} is quadratic. For this purpose, set pn+1(t) = x(t)− αn+1(t) ≥ 0, and qn+1(t) = βn+1(t)− x(t) ≥ 0, t ∈ Nba. From condition (A4.1), the mean value theorem, and Lemma 2.11, we obtain pn+1(t) = ∫ b a G(t, s)f(s, x(s))∇s− ∫ b a G(t, s)F (s, αn(s), βn(s);αn+1(s))∇s = ∫ b a G(t, s) [ f1(s, x) + f2(s, x) ] ∇s− ∫ b a G(t, s)[f1(s, αn) + f (1) 1 (s, βn)(αn+1 − αn) + f2(s, αn) + f (1) 2 (s, αn)(αn+1 − αn)]∇s = ∫ b a G(t, s) [( f (1) 1 (s, βn) + f (1) 2 (s, αn) ) pn+1 + ( f (1) 1 (s, ξ3)− f (1) 1 (s, βn) ) pn + ( f (1) 2 (s, ξ4)− f (1) 2 (s, αn) ) pn ] ∇s ≤ ∫ b a G(t, s) [ f (2) 1 (s, η1)(ξ3 − βn)pn + f (2) 2 (s, η2)(ξ4 − αn)pn ] ∇s ≤ ∫ b a |G(t, s)| [ A1‖pn + qn‖‖pn‖+B1‖pn‖‖pn‖ ] ∇s ≤MA1‖pn‖(‖pn‖+ ‖qn‖) +MB1‖pn‖2 = (3 2 MA1 +MB1 ) ‖pn‖2 + 1 2 MA1‖qn‖2, where M := max t∈Nb a b∑ s=a+1 |G(t, s)| = (b− a)2 4Γ(ν + 2) , αn(t) ≤ ξ3(t), ξ4(t) ≤ x(t), ξ3(t) ≤ η1(t) ≤ βn(t), αn(t) ≤ η2(t) ≤ ξ4(t), |f (2) 1 (t, x)| ≤ A1, |f (2) 2 (t, x)| ≤ B1 EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 15 for t ∈ Nba. So, we have ‖pn+1‖ ≤ (3 2 MA1 +MB1 ) ‖pn‖2 + 1 2 MA1‖qn‖2, Similarly, we have ‖qn+1‖ ≤ ( MA1 + 3 2 MB1 ) ‖qn‖2 + 1 2 MB1‖pn‖2. The proof is complete. � Next, we apply the Gauss-Seidel method to possibly improve upon the conver- gence rate of the iterative scheme described in Theorem 4.1. Theorem 4.2. Let the hypotheses of Theorem 4.1 hold. Consider the iterative schemes given by (Laα ∗ n+1)(t) = f1(t, α∗n) + f (1) 1 (t, β∗n)(α∗n+1 − α∗n) + f2(t, α∗n) + f (1) 2 (t, α∗n)(α∗n+1 − α∗n) ≡ F (t, α∗n, β ∗ n;α∗n+1), t ∈ Nb−1 a+1, α∗n+1(a) = A, α∗n+1(b) = B, (4.7) and (Laβ ∗ n+1)(t) = f1(t, β∗n) + f (1) 1 (t, β∗n)(β∗n+1 − β∗n) + f2(t, β∗n) + f (1) 2 (t, α∗n+1)(β∗n+1 − β∗n) ≡ G(t, α∗n+1, β ∗ n;β∗n+1), t ∈ Nb−1 a+1, β∗n+1(a) = A, β∗n+1(b) = B. (4.8) starting with α∗0 = α0 on Nba. The two sequences obtained via this iterative scheme {α∗n(t)} and {β∗n(t)}, n ≥ 0 converge monotonically to the x(t), the solution of (3.1) that lies between α0 and β0 and the convergence is at least quadratic. Proof. Initially, we compute α1(t) using the following BVPs (Laα1)(t) = f1(t, α0) + f (1) 1 (t, β0)(α1 − α0) + f2(t, α0) + f (1) 2 (t, α0)(α1 − α0) ≡ F (t, α0, β0;α1), t ∈ Nb−1 a+1, α1(a) = A, α1(b) = B. (4.9) Relabel α1(t) as α∗0(t). Now, we compute β1(t) using β0(t) and α∗0(t), that is, β1(t) is a solution of (Laβ1)(t) = f1(t, β0) + f (1) 1 (t, β0)(β1 − β0) + f2(t, β0) + f (1) 2 (t, α∗0)(β1 − β0) ≡ G(t, α∗0, β0;β1), t ∈ Nb−1 a+1, β1(a) = A, β1(b) = B. (4.10) It is clear that α0(t) ≤ α1(t) ≤ α∗0(t) and β1(t) ≤ β0(t) on Nba. Put p(t) := β∗0(t)− β1(t). Then p(a) = p(b) = 0. Also, we have (Lap)(t) = [f1(t, β0) + f (1) 1 (t, β0)(β∗0 − β0) + f2(t, β0) + f (1) 2 (t, α∗0)(β∗0 − β0)] − [f1(t, β0) + f (1) 1 (t, β0)(β1 − β0) + f2(t, β0) + f (1) 2 (t, α0)(β1 − β0)] 16 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 ≥ f (1) 1 (t, β0)(β∗0 − β1) + f (1) 2 (t, α∗0)(β∗0 − β1) = [f (1) 1 (t, β0) + f (1) 2 (t, α∗0)]p. So p is a lower solution to a BVP of (3.3). Thus, by Corollary 3.5, we know p(t) ≤ 0 on Nba. That is, β∗0(t) ≤ β1(t) on Nba. Continuing the process, we will be able to show that the sequences {α∗n(t)} and {β∗n(t)} must converge at least as fast as the sequences {αn(t)} and {βn(t)} that were computed using the iterative scheme described in Theorem 4.1. The proof is complete. � Remark 4.3. When the function f(t, x) is the sum of (n − 1)-hyperconvex and (n−1)-hyperconcave functions (i.e. f(t, x) = f1(t, x)+f2(t, x), where f (n) 1 (t, x) ≥ 0, and f (n) 2 (t, x) ≤ 0), we can obtain two monotone sequences {αn(t)} and {βn(t)}, whose convergence is of order n (n ≥ 2). The proof is similar to that of Theorem 4.1, so we omit it. 5. Examples Table 1. Three α, β-iterates of (5.1). t α0(t) α1(t) α2(t) α3(t) β3(t) β2(t) β1(t) β0(t) 0 0 0 0 0 0 0 0 1 1 0 0.007355 0.021545 0.027694 0.028628 0.067598 0.354038 1 2 0 0.025743 0.058914 0.069876 0.071137 0.115480 0.405802 1 3 0 0.087341 0.152969 0.167201 0.168294 0.205791 0.454208 1 4 0 0.295581 0.391591 0.403170 0.403700 0.424171 0.582092 1 5 0 1 1 1 1 1 1 1 Table 2. Three α∗, β∗-iterates of (5.1). t α∗0(t) α∗1(t) α∗2(t) α∗3(t) β∗3(t) β∗2(t) β∗1(t) β∗0(t) 0 0 0 0 0 0 0 0 1 1 0 0.007355 0.021572 0.028417 0.028532 0.066598 0.353610 1 2 0 0.025743 0.058982 0.070027 0.070946 0.112921 0.404336 1 3 0 0.087341 0.153110 0.167290 0.168026 0.200221 0.449635 1 4 0 0.295581 0.391971 0.403220 0.403523 0.418531 0.570861 1 5 0 1 1 1 1 1 1 1 Now, we give an example to illustrate the results established in the previous section. Example 5.1. Consider the BVP (Lax)(t) = −1 3 x3(t) + 1 2 x2(t) + x(t), t ∈ N4 1, x(0) = 0, x(5) = 1, (5.1) where a = 0, ν = 1/2. Taking α0(t) ≡ 0, β0(t) ≡ 1, it is quick to verify that α0(t), β0(t) are lower and upper solutions of (5.1), respectively. Let f(t, x(t)) denote the right-hand side of EJDE-2020/04 APPROXIMATE SOLUTIONS TO NONLINEAR CAPUTO NABLA FDES 17 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t x Figure 1. α, β-iterates 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t x Figure 2. α∗, β∗-iterates (5.1), and split it into two functions as f(t, x(t)) = f1(t, x(t)) + f2(t, x(t)), where f1(t, x(t)) = 1 2x 2(t) + x(t), f2(t, x(t)) = − 1 3x 3(t). Note that f (1) 1 (t, x) = x+ 1 > 0, f (2) 1 (t, x) = 1 > 0 on Ω = N4 1 × [0, 1], f (1) 2 (t, x) = −x2 ≤ 0, f (2) 2 (t, x) = −2x(t) ≤ 0 on Ω. 18 X. LIU, B. G. JIA, S. GENSLER, L. ERBE, A. PETERSON EJDE-2020/04 Now, we apply the iteration scheme of Theorem 4.1. After three iterations we find the α, β-iterates given in Table 1. The graph in Figure 1 shows the α-iterates (with broken line) and the β-iterates (with unbroken line). 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Peterson; Existence and rapid convergence results for nonlinear Caputo nabla fractional difference equations, Electron. J. Qual. Theory Differ. Equ., 51 (2017), 1-16. [20] M. Sowmya, A. S. Vatsala; Generalized iterative methods for Caputo fractional differential equations via coupled lower and upper solutions with superlinear convergence, Nonlinear Dyn. Syst. Theory, 15 (2) (2015), 198-208. [21] P. G. Wang, H. X. Wu, Y. H. Wu; Higher even-order convergence and coupled solutions for second-order boundary value problems on time scales, Compu. Math. Appl., 55 (2008), 1693-1705. [22] A. Yakar; Initial time difference quasi-linearization for Caputo fractional differential equa- tions, Adv. Difference Equ., 2012, 2012: 92, 9 pages. Xiang Liu School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China Email address: liux256@126.com Baoguo Jia (corresponding author) School of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, China Email address: mcsjbg@mail.sysu.edu.cn Scott Gensler Department of Mathematics, University of Nebraska-Kearney, Kearney, NE 68849, USA Email address: scott.gensler@gmail.com Lynn Erbe Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA Email address: lerbe@unl.edu Allan Peterson Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA Email address: apeterson1@math.unl.edu 1. Introduction 2. Preliminaries 3. Existence and comparison results 4. Main results 5. Examples Conclusions Acknowledgements References