Initial and Boundary Value Problems for the Caputo Fractional Self-Adjoint Difference Equations
Kevin Ahrendt, Lydia DeWolf, Liam Mazurowski, Kelsey Mitchell, Tim Rolling, Dominic Veconi
aa r X i v : . [ m a t h . C A ] F e b Initial and Boundary Value Problems for theCaputo Fractional Self-Adjoint DifferenceEquations
Kevin AhrendtUniversity of Nebraska-LincolnDepartment of MathematicsLincoln, NE 68588-0130 USA [email protected]
Lydia DeWolfDepartment of MathematicsUnion CollegeJackson, TN, 38305 USA [email protected]
Liam MazurowskiDepartment of Mathematical SciencesCarnegie Mellon UniversityPittsburgh, PA 15213 USA [email protected]
Kelsey MitchellDepartment of MathematicsBuena Vista UniversityStorm Lake, IA 50588 USA [email protected]
Tim RollingDepartment of MathematicsUniversity of Nebraska-LincolntLincoln, NE 68588-0130 USA [email protected]
Dominic VeconiDepartment of MathematicsHamilton CollegeClinton, NY 13323 USA [email protected]
Communicated by Allan Peterson
VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Abstract
In this paper we develop the theory of initial and boundary value prob-lems for the self-adjoint nabla fractional difference equation containing aCaputo fractional nabla difference that is given by ∇ [ p ( t + 1) ∇ νa ∗ x ( t + 1)] + q ( t ) x ( t ) = h ( t ) , where 0 < ν ≤
1. We give an introduction to the nabla fractional calcu-lus with Caputo fractional differences. We investigate properties of thespecific self-adjoint nabla fractional difference equation given above. Weprove existence and uniqueness theorems for both initial and boundaryvalue problems under appropriate conditions. We introduce the definitionof a Cauchy function which allows us to give a variation of constants for-mula for solving initial value problems. We then show that this Cauchyfunction is important in finding a Green’s function for a boundary valueproblem with Sturm-Liouville type boundary conditions. Several inequal-ities concerning a certain Green’s function are derived. These resultsare important in using fixed point theorems for proving the existence ofsolutions to boundary value problems for nonlinear fractional equationsrelated to our linear self-adjoint equation.
AMS (MOS) Subject Classification:
Key words:
Caputo fractional difference, Caputo variation of constantsformula, Fractional boundary value problems
In this section we will develop the basic definitions and theorems needed for ourresults. For an introduction to whole order difference calculus, see Kelley andPeterson [6]. For more information in the fractional results using the backwardsdifference operator, see Hein et al. [5] and Ahrendt et al. [1].
Definition 1. [1] Let a ∈ R . Then the set N a is given by { a, a + 1 , a + 2 , . . . } .Furthermore, if b ∈ N a , then N ba is given by { a, a + 1 , . . . , b − , b } . Definition 2. [1] Let f : N a → R . Then the nabla difference of f is defined by ( ∇ f )( t ) := f ( t ) − f ( t − , for t ∈ N a +1 . For convenience, we will use the notation ∇ f ( t ) := ( ∇ f )( t ) . For N ∈ N , we have that the N th order fractional difference is recursively definedas ∇ N f ( t ) := ∇ ( ∇ N − f ( t )) , for t ∈ N a + N . Definition 3. [1] Let f : N a → R and let c, d ∈ N a . Then the definite nablaintegral of f from c to d is defined by Z dc f ( s ) ∇ s := (P ds = c +1 f ( s ) , c < d, , d ≤ c. VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Definition 4. [1] Let t ∈ R and let n ∈ Z + . Then the rising function is definedby t n := ( t )( t + 1) · · · ( t + n −
1) = Γ( t + n )Γ( t ) , where Γ is the gamma function. For ν ∈ R , the generalized rising function isthen defined by t ν := Γ( t + ν )Γ( t ) for t and ν such that t + ν
6∈ { . . . , − , − , } . If t is a non-positive integer and t + ν is not a non-positive integer then we take by convention t ν = 0 . Theorem 5 (Fundamental Theorem of Nabla Calculus) . [1] Assume the func-tion f : N ba → R and let F be a nabla antidifference of f on N ba , then Z ba f ( t ) ∇ t = F ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ba := F ( b ) − F ( a ) . The following definitions extend the nabla difference and nabla integral tofractional value orders.
Definition 6. [1] Let f : N a +1 → R , ν > , ν ∈ R . The ν th order nablafractional sum of f is defined as ∇ − νa f ( t ) := t X s = a +1 ( t − s + 1) ν − Γ( ν ) f ( s ) , for t ∈ N a . Definition 7. [1] Let f : N a +1 → R , ν ∈ R , ν > , and N = ⌈ ν ⌉ . Then ν th order nabla fractional difference of f is defined as ∇ νa f ( t ) := ∇ N ∇ − N − νa f ( t ) , for t ∈ N a + N . The next theorem gives results for composing nabla fractional sums anddifferences in certain cases.
Theorem 8 (Composition Rules) . [1] Let µ, ν > and let f : N a → R . Set N = ⌈ ν ⌉ . Then ∇ − νa ∇ − µa f ( t ) = ∇ − ν − µa f ( t ) , t ∈ N a , and ∇ νa ∇ − µa f ( t ) = ∇ ν − µa f ( t ) , t ∈ N a + N . While we gave the traditional definition of a nabla fractional difference above,we will focus on the Caputo nabla fractional difference for the rest of the paperand only appeal to the previous definition when needed. The following definitionhas been adapted from Anastassiou in [2].
VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Definition 9.
Let f : N a − N +1 → R , ν > , ν ∈ R , N = ⌈ ν ⌉ . The ν th orderCaputo nabla fractional difference is defined as ∇ νa ∗ f ( t ) := ∇ − ( N − ν ) a ( ∇ N f ( t )) , for t ∈ N a . Note that the Caputo difference operator is a linear operator. Theorem 10 (Discrete Whole-Order Taylor’s Formula) . [3] Fix N ∈ N andlet f : N a − N +1 → R . Then f ( t ) = N − X k =0 ∇ k f ( a ) ( t − a ) k k ! + t X s = a +1 ( t − s + 1) N − ( N − ∇ N f ( s ) , for t ∈ N a . The following theorem is adapted from Anastassiou in [2] where we use ourdefinition of the Caputo nabla fractional difference.
Theorem 11 (Caputo Discrete Taylor’s Theorem) . [2] Let ν ∈ R , ν > , N = ⌈ ν ⌉ , and f : N a − N +1 → R . Then for all t ∈ N a , the representation holds f ( t ) = N − X k =0 ( t − a ) k k ! ∇ k f ( a ) + 1Γ( ν ) t X τ = a +1 ( t − τ + 1) ν − ∇ νa ∗ f ( τ ) . Proof.
Using Theorem 8, notice that for f : N a − N +1 → R , ∇ − νa ∇ νa ∗ f ( t ) = ∇ − νa ∇ − ( N − ν ) a ∇ N f ( t )= ∇ − ν − N + νa ∇ N f ( t )= ∇ − Na ∇ N f ( t ) , for t ∈ N a . By Definition 6, ∇ − Na ∇ N f ( t ) = 1( N − Z ta ( t − s + 1) N − ∇ N f ( s ) ∇ s, Similarly, ∇ − νa ∇ νa ∗ f ( t ) = 1Γ( ν ) Z ta ( t − s + 1) ν − ∇ νa ∗ f ( s ) ∇ s. So from Theorem 10, f ( t ) = N − X k =0 ( t − a ) k k ! ∇ k f ( a ) + 1( N − t X s = a +1 ( t − s + 1) N − ∇ N f ( s )= N − X k =0 ( t − a ) k k ! ∇ k f ( a ) + 1Γ( ν ) t X s = a +1 ( t − s + 1) ν − ∇ νa ∗ f ( s ) , for t ∈ N a , proving the result. (cid:4) VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation We are interested in solutions to the nabla fractional initial value problem (IVP) ( ∇ νa ∗ f ( t ) = h ( t ) , t ∈ N a +1 , ∇ k f ( a ) = c k , ≤ k ≤ N − , (1)where a, ν ∈ R , ν > N = ⌈ ν ⌉ , c k ∈ R for 0 ≤ k ≤ N −
1, and f : N a − N +1 → R . Theorem 12.
The solution to the IVP (1) is uniquely determined by f ( t ) = N − X k =0 ( t − a ) k k ! c k + 1Γ( ν ) t X τ = a +1 ( t − τ + 1) ν − h ( τ ) , for t ∈ N a +1 .Proof. Let f : N a − N +1 → R satisfy ∇ k f ( a ) = c k , for 0 ≤ k ≤ N −
1. Note that this uniquely determines the value of f ( t ) for a − N + 1 ≤ t ≤ a . For t ∈ N a +1 , let f ( t ) satisfy ∇ N f ( t ) = h ( t ) − t − X s = a +1 ( t − s + 1) N − ν − Γ( N − ν ) ∇ N f ( s ) . This recursive definition uniquely determines f ( t + 1) from the values of f ( a − N + 1) , ..., f ( t ), so the function is uniquely defined for all t ∈ N a − N +1 . So forany t ∈ N a +1 , ∇ N f ( t ) + t − X s = a +1 ( t − s + 1) N − ν − Γ( N − ν ) ∇ N f ( s ) = h ( t ) . Equivalently, Γ( ν )Γ(1)Γ( ν ) ∇ N f ( t ) + t − X s = a +1 ( t − s + 1) N − ν − Γ( N − ν ) ∇ N f ( s )= (1) ν − Γ( ν ) ∇ N f ( t ) + t − X s = a +1 ( t − s + 1) N − ν − Γ( N − ν ) ∇ N f ( s )= t X s = a +1 ( t − s + 1) N − ν − Γ( N − ν ) ∇ N f ( s )= ∇ − ( N − ν ) a ∇ N f ( t )= ∇ νa ∗ f ( t )= h ( t ) . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation f ( t ) solves the IVP (1). Conversely, if we suppose that there is afunction f : N a − N +1 → R that satisfies the IVP, reversing the above algebraicsteps would lead to the same recursive definition. Therefore the solution to theIVP is uniquely defined. By the Caputo Discrete Taylor’s Theorem, f ( t ) mustsatisfy the representation f ( t ) = N − X k =0 ( t − a ) k k ! ∇ k f ( a ) + 1Γ( ν ) t X τ = a +1 ( t − τ + 1) ν − ∇ νa ∗ f ( τ )= N − X k =0 ( t − a ) k k ! c k + 1Γ( ν ) t X τ = a +1 ( t − τ + 1) ν − h ( τ ) , for t ∈ N a +1 . (cid:4) Example 13.
Solve the IVP ( ∇ . ∗ f ( t ) = t, t ∈ N ,f (0) = 2 . Applying the variation of constants formula yields the following expression for f ( t ) . f ( t ) = X k =0 t k k ! 2 + ∇ − . h ( t )= t X s =1 ( t − s + 1) − . Γ(0 . s + 2 After summing by parts, we have f ( t ) = t . Γ (2 .
7) + 2 . For development of these properties in the continuous setting, see Kelley andPeterson [7].Let D a := { x : N a → R } and let the self-adjoint fractional operator L a bedefined by( L a x )( t ) := ∇ [ p ( t + 1) ∇ νa ∗ x ( t + 1)] + q ( t ) x ( t ) , t ∈ N a +1 , where x ∈ D a , < ν < p : N a +1 → (0 , ∞ ) and q : N a +1 → R . Note that whilethe operator is for values of t ∈ N a +1 , the function x is defined on N a . Notethat L a is a linear operator. VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Theorem 14 (Existence and Uniqueness for Self-Adjoint IVPs) . Let
A, B ∈ R ,and let h : N a +1 → R . The IVP L a x ( t ) = h ( t ) , t ∈ N a +1 ,x ( a ) = A, ∇ x ( a + 1) = B, has a unique solution x : N a → R .Proof. Let x : N a → R satisfy the initial conditions (cid:26) x ( a ) = A,x ( a + 1) = A + B. Furthermore, for t ∈ N a +1 , let x ( t + 1) satisfy the recursive equation x ( t + 1) = x ( t ) − t X s = a +1 ( t − s + 2) − ν Γ(1 − ν ) ∇ x ( s )+ 1 p ( t + 1) " h ( t ) − q ( t ) x ( t ) + p ( t ) t X τ = a +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ ) . Note that as defined, x ( t + 1) is uniquely determined from the values of x ( a ), x ( a + 1), . . . , f ( t − f ( t ), for t ∈ N a +1 . Furthermore, ∇ x ( t + 1) + t X s = a +1 ( t − s + 2) − ν Γ(1 − ν ) ∇ x ( s )= 1 − ν Γ(1 − ν ) ∇ x ( t + 1) + t X s = a +1 ( t − s + 2) − ν Γ(1 − ν ) ∇ x ( s )= t +1 X s = a +1 ( t − s + 2) − ν Γ(1 − ν ) ∇ x ( s )= ∇ − (1 − ν ) a ∇ x ( t + 1)= ∇ νa ∗ x ( t + 1) . So we have that ∇ νa ∗ x ( t + 1) = 1 p ( t + 1) " h ( t ) − q ( t ) x ( t ) + p ( t ) t X τ = a +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ ) . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation p ( t + 1) ∇ νa ∗ x ( t + 1)= h ( t ) − q ( t ) x ( t ) + p ( t ) t X τ = a +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ )= h ( t ) − q ( t ) x ( t ) + p ( t ) ∇ − (1 − ν ) a ∇ x ( t )= h ( t ) − q ( t ) x ( t ) + p ( t ) ∇ νa ∗ x ( t ) , which implies ∇ [ p ( t + 1) ∇ νa ∗ x ( t + 1)] = h ( t ) − q ( t ) x ( t ) . Thus, by rearranging, we have that ∇ [ p ( t + 1) ∇ νa ∗ x ( t + 1)] + q ( t ) x ( t ) = h ( t ) . Therefore, for any value of t ∈ N a +1 , x ( t ) satisfies the IVP, So a solution exists.Reversing the preceding algebraic steps shows that if some function y ( t ) is asolution to the IVP, it must be the same solution as our original x ( t ). Thereforea unique solution exists. (cid:4) The following lemma shows that initial conditions behave nicely when deal-ing with the Caputa nabla fractional difference.
Lemma 15.
Let < ν < and let x : N a → R . Then ∇ νa ∗ x ( a + 1) = ∇ x ( a + 1) for t ∈ N a +1 .Proof. Let 0 < ν <
1. Then by definition of the Caputo difference, ∇ νa ∗ x ( a + 1) = ∇ − (1 − ν ) a ∇ x ( a + 1)= a +1 X τ = a +1 ( a + 1 − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ )= ∇ x ( a + 1) , for t ∈ N a +1 . (cid:4) The next theorem and corollary show that the self-adjoint fractional nabladifference equation behaves very similar to a second order difference equation.
Theorem 16 (General Solution of the Homogeneous Equation) . Suppose x , x : N a → R are linearly independent solutions to L a x ( t ) = 0 . Then the general so-lution to L a x ( t ) = 0 is given by x ( t ) = c x ( t ) + c x ( t ) , for t ∈ N a +1 , where c , c ∈ R are arbitrary constants. VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Proof.
Let x , x : N a → R be linearly independent solutions of L a x ( t ) = 0.Then there exist constants α, β, γ, δ ∈ R for which x , x are the unique solutionsto the IVPs Lx = 0 , t ∈ N a +1 ,x ( a ) = α, ∇ x ( a + 1) = β, and Lx = 0 , t ∈ N a +1 ,x ( a ) = γ, ∇ x ( a + 1) = δ. Since L a is a linear operator, we have for any c , c ∈ R L a [ c x ( t ) + c x ( t )] = c L a x ( t ) + c L a x ( t ) = 0 , so x ( t ) = c x ( t ) + c x ( t ) solves L a x ( t ) = 0. Conversely, suppose x : N a → R solves L a x ( t ) = 0. Note that x ( t ) solves the IVP Lx = 0 , t ∈ N a +1 ,x ( a ) = A, ∇ x ( a + 1) = B, for some A, B ∈ R . We show that the matrix equation (cid:20) x ( a ) x ( a ) ∇ x ( a + 1) ∇ x ( a + 1) (cid:21) (cid:20) c c (cid:21) = (cid:20) AB (cid:21) (2)has a unique solution for c , c . The above matrix equation can be equivalentlyexpressed as (cid:20) α γβ δ (cid:21) (cid:20) c c (cid:21) = (cid:20) AB (cid:21) . Suppose by way of contradiction that (cid:12)(cid:12)(cid:12)(cid:12) α γβ δ (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Then, without loss of generality, there exists a constant k ∈ R for which α = kγ and β = kδ . Then x ( a ) = α = kγ = kx ( a ), and ∇ x ( a + 1) = β = kδ = k ∇ x ( a + 1). Since kx ( t ) solves L a x ( t ) = 0, we have that x ( t ) and kx ( t )solve the same IVP. By uniqueness, x ( t ) = kx ( t ). But then x ( t ) and x ( t )are linearly dependent, which is a contradiction. Therefore, the matrix equation(2) must have a unique solution, so x ( t ) and c x ( t ) + c x ( t ) solve the sameIVP, and so by uniqueness in Theorem 14, every solution to L a x ( t ) = 0 can beuniquely expressed as a linear combination of x ( t ) and x ( t ). (cid:4) Corollary 17 (General Solution of the Nonhomogeneous Equation) . Suppose x , x : N a → R are linearly independent solutions of L a x ( t ) = 0 and y : N a → R is a particular solution to L a x ( t ) = h ( t ) for some h : N a +1 → R . Then thegeneral solution of L a x ( t ) = h ( t ) is given by x ( t ) = c x ( t ) + c x ( t ) + y ( t ) , for t ∈ N a +1 and where c , c ∈ R are arbitrary constants. VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Proof.
Since L a is a linear operator, one can show that x ( t ) = c x ( t ) + c x ( t ) + y ( t ) solves L a x ( t ) = h ( t ) for any c , c ∈ R in a similar way as inTheorem 16. Conversely, suppose x : N a → R solves L a x ( t ) = h ( t ). Again notethat x ( t ) solves the IVP Lx = h ( t ) , t ∈ N a +1 ,x ( a ) = A,x ( a + 1) = B, for some A, B ∈ R . Since y ( t ) is a particular solution of L a x ( t ) = h ( t ), thereexist unique constants c , c ∈ R for which x h ( t ) = c x ( t ) + c x ( t ) solves theIVP Lx h = 0 ,x h ( a ) = A − y ( a ) ,x h ( a + 1) = B − y ( a + 1) , for t ∈ N a +1 . Observe that x h ( t ) + y ( t ) satisfies L a x ( t ) = h ( t ). Further, x h ( a ) + y ( a ) = A − y ( a ) + y ( a ) = A, and x h ( a + 1) + y ( a + 1) = B − y ( a + 1) + y ( a + 1) = B. Therefore x ( t ) and x h ( t ) + y ( t ) solve the same IVP. Then by uniqueness ofIVPs, x ( t ) = c x ( t ) + c x ( t ) + y ( t ), and thus any solution to L a x ( t ) = h ( t )may be written in this form. (cid:4) In this section we develop techniques to solve initial value problems for thefractional self-adjoint operator involving the Caputo difference. See Brackins [4]for a similar development using the Riemann-Liouville definition of a fractionaldifference.
Definition 18.
The Cauchy function for L a x ( t ) is the function x : N s × N a → R that satisfies the IVP L s x ( t, s ) = 0 , t ∈ N s +1 ,x ( s, s ) = 0 , ∇ x ( s + 1 , s ) = p ( s +1) , (3) for any fixed s ∈ N a . Remark 19.
Note by Lemma 15, the IVP (3) is equivalent to the IVP L s x ( t, s ) = 0 , t ∈ N s +1 x ( s, s ) = 0 , ∇ νs ∗ x ( s + 1 , s ) = p ( s +1) VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation for any fixed s ∈ N a . Theorem 20 (Variation of Constants) . Let h : N a +1 → R . Then solution tothe IVP L a y ( t ) = h ( t ) , t ∈ N a +1 ,y ( a ) = 0 , ∇ y ( a + 1) = 0 , is given by y ( t ) = Z ta x ( t, s ) h ( s ) ∇ s, where x ( t, s ) is the Cauchy function for the homogenous equation and where y : N a → R .Proof. Note that y ( a ) = a X s = a +1 x ( a, s ) h ( s ) = 0 , so the first initial condition holds. By the definition of the Caputo difference, ∇ νs ∗ x ( t + 1 , s ) = ∇ − (1 − ν ) s ∇ x ( t + 1 , s ) = t +1 X τ = s +1 ( t + 1 − τ + 1) − ν Γ( − ν ) ∇ x ( τ, s ) . Then ∇ [ p ( t + 1) ∇ νs ∗ x ( t + 1 , s )]= p ( t + 1) t +1 X τ = s +1 ( t − τ + 2) − ν Γ(1 − ν ) ∇ x ( τ, s ) − p ( t ) t X τ = s +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ, s )= p ( t + 1) t +1 X τ = s +1 ( t − τ + 2) − ν Γ(1 − ν ) ∇ x ( τ, s ) − p ( t ) t +1 X τ = s +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ, s )+ p ( t ) ( t − t − − ν Γ(1 − ν ) ∇ x ( t + 1 , s )= t +1 X τ = s +1 ( t − τ + 2) − ν p ( t + 1) − ( t − τ + 1) − ν p ( t )Γ(1 − ν ) ∇ x ( τ, s ) . So we have that ∇ [ p ( t + 1) ∇ νs ∗ x ( t + 1 , s )]= t +1 X τ = s +1 ( t − τ + 2) − ν p ( t + 1) − ( t − τ + 1) − ν p ( t )Γ(1 − ν ) ∇ x ( τ, s ) . (4) VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation y ( t ). Note that ∇ y ( t ) = t X s = a +1 x ( t, s ) h ( s ) − t − X s = a +1 x ( t − , s ) h ( s )= t − X s = a +1 x ( t, s ) h ( s ) + x ( t, t ) h ( t ) − t − X s = a +1 x ( t − , s ) h ( s )= t − X s = a +1 ∇ x ( t, s ) h ( s ) . From here, observe that ∇ y ( a + 1) = a +1 − X s = a +1 x ( t, s ) h ( s ) = 0 , so the second initial condition holds. Then by the definition of the Caputodifference, ∇ νa ∗ y ( t ) = ∇ − (1 − ν ) a ∇ y ( t )= t X τ = a +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ y ( τ )= t X τ = a +1 ( t − τ + 1) − ν Γ(1 − ν ) τ − X s = a +1 ∇ x ( τ, s ) h ( s )= t X τ = a +1 τ − X s = a +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ, s ) h ( s ) . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation ∇ [ p ( t + 1) ∇ νa ∗ y ( t + 1)]= p ( t + 1) t +1 X τ = a +1 τ − X s = a +1 ( t − τ + 2) − ν Γ(1 − ν ) ∇ x ( τ, s ) h ( s ) − p ( t ) t X τ = a +1 τ − X s = a +1 ( t − τ + 1) − ν Γ(1 − ν ) ∇ x ( τ, s ) h ( s )= t +1 X τ = a +1 τ − X s = a +1 ( t − τ + 2) − ν p ( t + 1) − ( t − τ + 1) − ν p ( t )Γ(1 − ν ) ∇ x ( τ, s ) h ( s )= t X s = a +1 t +1 X τ = s +1 ( t − τ + 2) − ν p ( t + 1) − ( t − τ + 1) − ν p ( t )Γ(1 − ν ) ∇ x ( τ, s ) h ( s )= p ( t + 1) ∇ x ( t + 1 , t ) h ( t )+ t − X s = a +1 t +1 X τ = s +1 ( t − τ + 2) − ν p ( t + 1) − ( t − τ + 1) − ν p ( t )Γ(1 − ν ) ∇ x ( τ, s ) h ( s )= p ( t + 1) h ( t ) p ( t + 1) + t − X s = a +1 ∇ [ p ( t + 1) ∇ νs ∗ x ( t + 1 , s )] h ( s )= h ( t ) + t − X s = a +1 ∇ [ p ( t + 1) ∇ νs ∗ x ( t + 1 , s )] h ( s ) . Therefore L a y ( t ) = ∇ [ p ( t + 1) ∇ νa ∗ y ( t + 1)] + q ( t ) y ( t )= h ( t ) + t − X s = a +1 ∇ [ p ( t + 1) ∇ νs ∗ x ( t + 1 , s )] h ( s )+ t − X s = a +1 q ( t ) x ( t, s ) h ( s ) + q ( t ) x ( t, t ) h ( t )= h ( t ) + t − X s = a +1 (cid:20) ∇ [ p ( t + 1) ∇ νs ∗ x ( t + 1 , s )] + q ( t ) x ( t, s ) (cid:21) h ( s )= h ( t ) + t − X s = a +1 L s x ( t, s ) h ( s )= h ( t ) . Thus y : N a → R solves the IVP for t ∈ N a +1 . (cid:4) Theorem 21 (Variation of Constants with Non-Zero Initial Conditions) . The
VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation solution to the IVP L a y ( t ) = h ( t ) , t ∈ N a +1 ,y ( a ) = A, ∇ y ( a + 1) = B, where A, B ∈ R are arbitrary constants, is given by y ( t ) = y ( t ) + Z ta x ( t, s ) h ( s ) ∇ s, where y ( t ) solves the IVP L a y ( t ) = 0 , t ∈ N a +1 ,y ( a ) = A, ∇ y ( a + 1) = B. Proof.
The proof follows from Theorem 20 by linearity. (cid:4)
Example 22.
Find the Cauchy function for ∇ [ p ( t + 1) ∇ νa ∗ y ( t + 1)] = 0 , t ∈ N a +1 . Consider ∇ [ p ( t + 1) ∇ νs ∗ x ( t + 1 , s )] = 0 . Integrating both sides from s to t andapplying the Fundamental Theorem of Nabla Calculus along with the secondinitial condition from Remark 19 yields p ( t + 1) ∇ νs ∗ x ( t + 1 , s ) − p ( s + 1) ∇ νs ∗ x ( s + 1 , s ) = 0 p ( t + 1) ∇ νs ∗ x ( t + 1 , s ) − ∇ νs ∗ x ( t + 1 , s ) = 1 p ( t + 1) . By the definition of the Caputo difference, this is equivalent to ∇ − (1 − ν ) s ∇ x ( t + 1 , s ) = 1 p ( t + 1) ∇ − νs ∇ − (1 − ν ) s ∇ x ( t + 1 , s ) = ∇ − νs p ( t + 1) ∇ x ( t + 1 , s ) = ∇ − νs p ( t + 1) , after applying a composition rule from Theorem 8. Replacing t + 1 with t yields ∇ x ( t, s ) = ∇ − νs p ( t ) . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Applying the Fundamental Theorem after integrating both sides from s to t andapplying the first initial condition from Remark 19 yields x ( t, s ) − x ( s, s ) = Z ts ∇ − νs p ( τ ) ∇ τx ( t, s ) = Z ts ∇∇ − νs p ( τ ) ∇ τ = (cid:20) ∇ − νs p ( τ ) (cid:21) τ = tτ = s = ∇ − νs p ( t ) − ∇ − νs p ( s )= ∇ − νs p ( t ) . Therefore the Cauchy function is x ( t, s ) = ∇ − νs p ( t ) = t X τ = s +1 ( t − τ + 1) ν − Γ( ν ) (cid:18) p ( τ ) (cid:19) . Example 23.
Find the Cauchy function for ∇∇ νa ∗ y ( t + 1) = 0 , t ∈ N a +1 . Notice that this is a particular case of the previous example, where p ( t ) ≡ .Then the Cauchy function is x ( t, s ) = ( ∇ − νs t )= Z ts ( t − τ + 1) ν − Γ( ν ) ∇ τ = − ( t − τ ) ν Γ( ν + 1) (cid:12)(cid:12)(cid:12)(cid:12) τ = tτ = s = ( t − s ) ν Γ( ν + 1) . Also note that if you take ν = 1 as in the whole order self-adjoint case, theCauchy function simplifies to x ( t, s ) = t − s . Example 24.
Find the solution to the IVP ∇ [ p ( t + 1) ∇ νa ∗ y ( t + 1)] = h ( t ) , t ∈ N a +1 ,y ( a ) = 0 , ∇ y ( a + 1) = 0 . From Theorem 20, we know that the solution is given by y ( t ) = Z ta x ( t, s ) h ( s ) ∇ s. VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation By Example 22, we know the Cauchy function for the above difference equationis x ( t, s ) = ∇ − νs p ( t ) . Then the solution is given by y ( t ) = Z ta ∇ − νs p ( t ) h ( s ) ∇ s. Example 25.
Solve the IVP ∇∇ . ∗ x ( t + 1) = t, t ∈ N ,x (0) = 0 , ∇ x (1) = 0 . This is a particular case of Example 24 where h ( t ) = t , a = 0 , and ν = 0 . .Then y ( t ) = Z t x ( t, s ) s ∇ s = t X s =1 .
6) ( t − s ) . s, and after summing by parts and applying Theorem 5, we get that the solution is y ( t ) = ( t − . Γ(3 . In this section we develop techniques to solve boundary value problems for thefractional self-adjoint operator involving the Caputo difference. See Brackins [4]for a similar development using the Riemann-Liouville definition of a fractionaldifference.We are interested in the boundary value problems (BVPs) L a x ( t ) = 0 , t ∈ N b − a +1 ,αx ( a ) − β ∇ x ( a + 1) = 0 ,γx ( b ) + δ ∇ x ( b ) = 0 , (5)and L a x ( t ) = h ( t ) , t ∈ N b − a +1 ,αx ( a ) − β ∇ x ( a + 1) = A,γx ( b ) + δ ∇ x ( b ) = B, (6) VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation h : N b − a +1 → R and α, β, γ, δ, A, B ∈ R for which α + β > γ + δ >
0. Note that despite the fact that the difference equations abovehold for t ∈ N b − a +1 , the solution x ( t ) for each BVP is defined on the domain of N ba . We are primarily interested in cases where the BVP (5) has only the trivialsolution. Theorem 26.
Assume (5) has only the trivial solution. Then (6) has a uniquesolution.Proof.
Let x , x : N a → R be linearly independent solutions to L a x ( t ) = 0.By Theorem 13, a general solution to L a x ( t ) = 0 is given by x ( t ) = c x ( t ) + c x ( t ) , where c , c ∈ R are arbitrary constants. If x ( t ) solves the boundary conditionsin (5), then x ( t ) is the trivial solution, which is true if and only if c = c = 0.This is true if and only if the system of equations (cid:26) α [ c x ( a ) + c x ( a )] − β ∇ νa ∗ [ c x ( a + 1) + c x ( a + 1)] = 0 ,γ [ c x ( b ) + c x ( b )] + δ ∇ νa ∗ [ c x ( b ) + c x ( b )] = 0 , or equivalently, (cid:26) c [ αx ( a ) − β ∇ νa ∗ x ( a + 1)] + c [ αx ( a ) − β ∇ νa ∗ x ( a + 1)] = 0 ,c [ γx ( b ) + δ ∇ νa ∗ x ( b )] + c [ γx ( b ) + δ ∇ νa ∗ x ( b )] = 0 , has only the trivial solution where c = c = 0. In other words, x ( t ) solves (5)if and only if D := (cid:12)(cid:12)(cid:12)(cid:12) αx ( a ) − β ∇ νa ∗ x ( a + 1) αx ( a ) − β ∇ νa ∗ x ( a + 1) γx ( b ) + δ ∇ νa ∗ x ( b ) γx ( b ) + δ ∇ νa ∗ x ( b ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Now consider (6). By Corollary 17, a general solution to L a y ( t ) = h ( t ) is y ( t ) = a x ( t ) + a x ( t ) + y ( t ) , where a , a ∈ R are arbitrary constants and y : N a → R is a particular solutionof L a y ( t ) = h ( t ). Consider the system of equations a [ αx ( a ) − β ∇ νa ∗ x ( a + 1)] + a [ αx ( a ) − β ∇ νa ∗ x ( a + 1)]= A − αy ( a ) + β ∇ νa ∗ y ( a + 1) ,a [ γx ( b ) + δ ∇ νa ∗ x ( b )] + a [ γx ( b ) + δ ∇ νa ∗ x ( b )] = B − γy ( b ) − δ ∇ νa ∗ y ( b ) , for arbitrary A, B ∈ R as in (6). Since D = 0, this system has a unique solutionfor a , a . It may be shown algebraically that this system is equivalent to α [ a x ( a ) + a x ( a ) + y ( a )] − β ∇ νa ∗ [ a x ( a + 1) + a x ( a + 1) + y ( a + 1)]= A,γ [ a x ( b ) + a x ( b ) + y ( b )] + δ ∇ νa ∗ [ a x ( b ) + a x ( b ) + y ( b )] = B, so y ( t ) satisfies the boundary conditions for (6). Therefore for any A, B ∈ R ,(6) has a unique solution. (cid:4) VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Theorem 27.
Let ρ := αγ ∇ − νa p ( b ) + αδp ( b ) + βγp ( a + 1) . Then the BVP ∇ [ p ( t + 1) ∇ νa ∗ x ( t + 1)] = 0 , t ∈ N b − a +1 ,αx ( a ) − β ∇ νa ∗ x ( a + 1) = 0 ,γx ( b ) + δ ∇ νa ∗ x ( b ) = 0 , has only the trivial solution if and only if ρ = 0 .Proof. Note that x ( t ) = 1 , x ( t ) = ∇ − νa p ( t ) are linearly independent solutionsto ∇ [ p ( t + 1) ∇ νa ∗ x ( t + 1)] = 0 . Then a general solution of the difference equation is given by x ( t ) = c x ( t ) + c x ( t ) = c + c ∇ − νa p ( t ) . Consider the boundary conditions αx ( a ) − β ∇ νa ∗ x ( a + 1) = 0 , and γx ( b ) + δ ∇ νa ∗ x ( b ) = 0. These boundaries give us c α + c (cid:20) − βp ( a + 1) (cid:21) = 0 ,c γ + c (cid:18) δp ( b ) + γ ∇ − νa p ( b ) (cid:19) = 0 . Converting this into a linear system yields " α − βp ( a +1) γ δp ( b ) + γ ∇ − νa p ( b ) c c (cid:21) = (cid:20) (cid:21) . Consider the determinant of the coefficient matrix, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α − βρ ( a +1) γ δp ( b ) + γ ∇ − νa p ( b ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = αγ ∇ − νa p ( b ) + αδp ( b ) + βγp ( a + 1) = ρ. By properties of invertible matrices, the BVP has only the trivial solution ifand only if ρ = 0. (cid:4) Definition 28.
Assume that (5) has only the trivial solution. Then we definethe Green’s function for the homogeneous BVP (5), G ( t, s ) , by G ( t, s ) := (cid:26) u ( t, s ) , a ≤ t ≤ s ≤ b,v ( t, s ) , a ≤ s ≤ t ≤ b, VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation where u ( t, s ) solves the BVP L a u ( t ) = 0 , t ∈ N b − a +1 ,αu ( a, s ) − β ∇ u ( a + 1 , s ) = 0 ,γu ( b, s ) + δ ∇ u ( b, s ) = − [ γx ( b, s ) + δ ∇ x ( b, s )] , for each fixed s ∈ N ba and where x ( t, s ) is the Cauchy function for L a x ( t ) . Thenwe define v ( t, s ) := u ( t, s ) + x ( t, s ) . Theorem 29 (Green’s Function Theorem) . If (5) has only the trivial solution,then the solution to (6) where A = B = 0 is given by y ( t ) = Z ba G ( t, s ) h ( s ) ∇ s, where the G ( t, s ) is the Green’s function for the homogeneous BVP (5).Proof. First note that by Theorem 26, u ( t, s ) for each fixed s ∈ N ba is well-defined. Let y ( t ) = Z ba G ( t, s ) h ( s ) ∇ s = Z ta G ( t, s ) h ( s ) ∇ s + Z bt G ( t, s ) h ( s ) ∇ s = Z ta v ( t, s ) h ( s ) ∇ s + Z bt u ( t, s ) h ( s ) ∇ s = Z ta [ u ( t, s ) + x ( t, s )] h ( s ) ∇ s + Z bt u ( t, s ) h ( s ) ∇ s = Z ba u ( t, s ) h ( s ) ∇ s + Z ta x ( t, s ) h ( s ) ∇ s = Z ba u ( t, s ) h ( s ) ∇ s + z ( t ) , where z ( t ) := R ta x ( t, s ) h ( s ) ∇ s . Since x ( t, s ) is the Cauchy function for L a x ( t ) =0, by Theorem 20, z ( t ) solves the IVP L a z ( t ) = h ( t ) , t ∈ N b − a +1 ,z ( a ) = 0 , ∇ z ( a + 1) = 0 . Then, L a y ( t ) = Z ba L a u ( t, s ) h ( s ) ∇ s + L a z ( t )= 0 + h ( t ) = h ( t ) , VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation t ∈ N b − a +1 , so the difference equation is satisfied. Now we check the boundaryconditions. At t = a , we have αy ( a ) − β ∇ y ( a +1) = Z ba [ αu ( a, s ) − β ∇ u ( a +1 , s )] h ( s ) ∇ s +[ αz ( a ) − β ∇ z ( a +1)] = 0 , and at t = b , we have γy ( b ) + δ ∇ y ( b )= γz ( b ) + Z ba γu ( b, s ) h ( s ) ∇ s + δ ∇ z ( b ) + Z ba δ ∇ u ( b, s ) h ( s ) ∇ s = γ Z ba x ( b, s ) h ( s ) ∇ s + δ ∇ Z ba x ( b, s ) h ( s ) ∇ s + Z ba [ γu ( b, s ) + δ ∇ u ( b, s )] h ( s ) ∇ s = − Z ba [ γx ( b, s ) + δ ∇ x ( b, s )] h ( s ) ∇ s + Z ba [ γx ( b, s ) + δ ∇ x ( b, s )] h ( s ) ∇ s = 0 . (cid:4) Corollary 30. If (5) has only the trivial solution, then the solution to (6) with A, B ∈ R is given by y ( t ) = z ( t ) + Z ba G ( t, s ) h ( s ) ∇ s, where z : N ba → R is the unique solution to L a z ( t ) = 0 ,αz ( a ) − β ∇ z ( a + 1) = A,γz ( b ) + δ ∇ z ( b ) = B. Proof.
This corollary follows directly from Theorem 29 by linearity. (cid:4)
Example 31.
Find the Green’s function for the boundary value problem ∇ [ ∇ νa ∗ y ( t + 1)] = 0 , t ∈ N b − a +1 ,y ( a ) = 0 ,y ( b ) = 0 . The Green’s function is given by G ( t, s ) = ( u ( t, s ) , a ≤ t ≤ s ≤ b,v ( t, s ) , a ≤ s ≤ t ≤ b, where u ( t, s ) , for each fixed s ∈ N ba , solves the BVP ∇ [ ∇ νa ∗ u ( t + 1 , s )] = 0 , t ∈ N b − a +1 ,u ( a, s ) = 0 ,u ( b, s ) = − x ( b, s ) , VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation and v ( t, s ) = u ( t, s ) + x ( t, s ) . By inspection, we find that x ( t ) = 1 is a solutionof ∇ [ ∇ νa ∗ y ( t + 1)] = 0 , for t ∈ N a +1 . Let x ( t ) = ( ∇ − νa t ) . Consider ∇ [ ∇ νa ∗ x ( t + 1)] = ∇ [ ∇ νa ∗ ∇ − νa (1)]= ∇ [ ∇ − ( N − ν ) a ∇ N ∇ − νa (1)]= ∇ [ ∇ − ( N − ν ) a ∇ ( N − ν ) a (1)]= ∇ [1]= 0 , using Theorem 8. So we have that x ( t ) solves ∇ [ ∇ νa ∗ y ( t + 1)] = 0 . Since x ( t ) and x ( t ) are linearly independent, by Theorem 16, the general solution is givenby y ( t ) = c + c ( ∇ − νa t ) = c + c ( t − a ) ν Γ(1 + ν ) , and it follows that u ( t, s ) = c ( s ) + c ( s ) ( t − a ) ν Γ(1 + ν ) . The boundary condition u ( a, s ) = 0 implies that c ( s ) = 0 . The boundarycondition u ( b, s ) = − x ( b, s ) then yields − x ( b, s ) = u ( b, s ) = c ( s ) ( b − a ) ν Γ(1 + ν ) . From Example 23, we know that x ( b, s ) = ( ∇ − νs b ) = ( b − s ) ν Γ(1 + ν ) , and thus c ( s ) = − ( b − s ) ν ( b − a ) ν . Hence the Green’s function is given by G ( t, s ) = − ( b − s ) ν ( t − a ) ν Γ(1 + ν )( b − a ) ν , a ≤ t ≤ s ≤ b, − ( b − s ) ν ( t − a ) ν Γ(1 + ν )( b − a ) ν + ( t − s ) ν Γ(1 + ν ) , a ≤ s ≤ t ≤ b. Remark 32.
Note that in the continuous and whole-order discrete cases, theGreen’s function is symmetric for the equivalent BVP in Example 31. This
VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation is not necessarily true in the fractional case. By way of counterexample, take a = 0 , b = 5 , and ν = 0 . . Then computing we find that G (2 ,
3) = u (2 ,
3) = − (2) . (2) . Γ(1 . . = − , but G (3 ,
2) = v (3 ,
2) = − (3) . (3) . Γ(1 . . + (1) . Γ(1 .
5) = − . Thus for this particular BVP, unlike in the continuous and whole-order discretecases, is not symmetric.
Theorem 33.
The Green’s function for the BVP ( ∇∇ νa ∗ x ( t + 1) = 0 ,x ( a ) = x ( b ) = 0 , for t ∈ N b − a +1 , given by G ( t, s ) = − ( b − s ) ν ( t − a ) ν Γ(1 + ν )( b − a ) ν , a ≤ t ≤ s ≤ b, − ( b − s ) ν ( t − a ) ν Γ(1 + ν )( b − a ) ν + ( t − s ) ν Γ(1 + ν ) , a ≤ s ≤ t ≤ b, satisfies the inequalities1. G ( t, s ) ≤ , G ( t, s ) ≥ − (cid:18) b − a (cid:19)(cid:18) Γ( b − a + 1)Γ( ν + 1)Γ( b − a + ν ) (cid:19) , Z ba | G ( t, s ) |∇ s ≤ ( b − a ) ν + 2) , for t ∈ N ba , and4. Z ba |∇ G ( t, s ) |∇ s ≤ b − aν + 1 , for t ∈ N ba +1 .Proof. (1) Let a ≤ t ≤ s ≤ b . Then G ( t, s ) = u ( t, s ) = − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ≤ , for each fixed s ∈ N ba . Now let a ≤ s < t ≤ b . Then G ( t, s ) = v ( t, s ), so we wishto show that v ( t, s ) is non-positive. First, we show that v ( t, s ) is increasing.Taking the nabla difference with respect to t yields ∇ t (cid:20) − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν + ( t − s ) ν Γ(1 + ν ) (cid:21) = − ( t − a ) ν − ( b − s ) ν Γ( ν )( b − a ) ν + ( t − s ) ν − Γ( ν ) . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation t − a ) ν − ( b − s ) ν Γ( ν )( b − a ) ν ≤ ( t − s ) ν − Γ( ν ) . Since t − s is positive, this happens if( t − a ) ν − ( b − s ) ν ( b − a ) ν ( t − s ) ν − ≤ . Now, by definition of the rising function,( t − a ) ν − ( b − s ) ν ( b − a ) ν ( t − s ) ν − = Γ( t − a + ν − t − a ) Γ( b − s + ν )Γ( b − s ) Γ( b − a )Γ( b − a + ν ) Γ( t − s )Γ( t − s + ν − t − s + ν − t − s + ν ) · · · ( t − a + ν − t − s )( t − s + 1) · · · ( t − a − · ( b − s )( b − s + 1) · · · ( b − a − b − s + ν )( b − s + ν + 1) · · · ( b − a + ν − t − s + ν − t − s ) ( t − s + ν )( t − s + 1) · · · ( t − a + ν − t − a −
1) ( b − s )( b − s + ν ) ( b − s + 1)( b − s + ν + 1) · · ·· ( b − a − b − a + ν − ≤ (1)(1) · · · (1)(1)(1) · · · (1) = 1 . Next, we check v ( t, s ) at the right endpoint, t = b , v ( b, s ) = − ( b − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν + ( b − s ) ν Γ( ν + 1) = 0 . Thus, v ( t, s ) is nonpositive for a ≤ s < t ≤ b . Also note that v ( t, s ) = u ( t, s ) for t = s . Therefore, for t ∈ N ba , G ( t, s ) is nonpositive.(2) Since we know that v ( t, s ) is always increasing for a ≤ s ≤ t ≤ b andthat for s = t , v ( t, s ) = u ( t, s ), it suffices to show that u ( t, s ) ≥ − (cid:18) b − a (cid:19)(cid:18) Γ( b − a + 1)Γ( ν + 1)Γ( b − a + ν ) (cid:19) . Let a ≤ t ≤ s ≤ b . Then G ( t, s ) = u ( t, s ) = − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ≥ − ( s − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν . Note that for α ∈ N and 0 < ν < α ν = Γ( α + ν )Γ( α ) ≤ Γ( α + 1)Γ( α ) = α . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation − ( s − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ≥ − ( s − a ) ( b − s ) Γ( ν + 1)( b − a ) ν ≥ − ( a + b − a )( b − a + b )Γ( ν + 1)( b − a ) ν = − ( b − a )( b − a )Γ( b − a )4Γ( ν + 1)Γ( b − a + ν )= − ( b − a )Γ( b − a + 1)4Γ( ν + 1)Γ( b − a + ν )= − (cid:18) b − a (cid:19)(cid:18) Γ( b − a + 1)Γ( ν + 1)Γ( b − a + ν ) (cid:19) . Therefore, G ( t, s ) ≥ − (cid:18) b − a (cid:19)(cid:18) Γ( b − a + 1)Γ( ν + 1)Γ( b − a + ν ) (cid:19) . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Z ba | G ( t, s ) |∇ s = Z ta | v ( t, s ) |∇ s + Z bt | u ( t, s ) |∇ s = Z ta (cid:12)(cid:12)(cid:12)(cid:12) − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν + ( t − s ) ν Γ(1 + ν ) (cid:12)(cid:12)(cid:12)(cid:12) ∇ s + Z bt ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s = Z ta − (cid:20) − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν + ( t − s ) ν Γ(1 + ν ) (cid:21) ∇ s + Z bt ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s = Z ba ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s − Z ta ( t − s ) ν Γ( ν + 1) ∇ s = − ( t − a ) ν ( b − s − ν +1 Γ( ν + 2)( b − a ) ν (cid:12)(cid:12)(cid:12)(cid:12) s = bs = a + ( t − s − ν +1 Γ( ν + 2) (cid:12)(cid:12)(cid:12)(cid:12) s = ts = a = ( t − a ) ν ( b − a − ν +1 Γ( ν + 2)( b − a ) ν − ( t − a − ν +1 Γ( ν + 2)= ( t − a ) ν ( b − a − b − a ) ν Γ( ν + 2)( b − a ) ν − ( t − a − t − a ) ν Γ( ν + 2)= ( t − a ) ν Γ( ν + 2) [ b − a − − ( t − a − t − a ) ν ( b − t )Γ( ν + 2) ≤ ( t − a )( b − t )Γ( ν + 2) ≤ ( a + b − a )( b − a + b )Γ( ν + 2)= ( b − a )( b − a )4Γ( ν + 2)= ( b − a ) ν + 2) . (4) We can assume that b − a > b − a = 1, it would be an initial valueproblem and not a boundary value problem. Taking the difference of u ( t, s )with respect to t , we have ∇ t u ( t, s ) = ∇ t − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν = − ν ( t − a ) ν − ( b − s ) ν Γ( ν + 1)( b − a ) ν . Since all the factors in the above equation are individually positive, we get that
VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation ∇ u ( t, s ) is nonpositive. Let t ∈ N a . We therefore know Z ba |∇ t G ( t, s ) | ∇ s = Z t − a |∇ t G ( t, s ) | ∇ s + Z bt − |∇ t G ( t, s ) | ∇ s = Z t − a |∇ t v ( t, s ) | ∇ s + Z bt − |∇ t u ( t, s ) | ∇ s = Z t − a (cid:20) ∇ t − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν + ∇ t ( t − s ) ν Γ( ν + 1) (cid:21) ∇ s + Z bt − ∇ t ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s = Z t − a ∇ t − ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s + Z t − a ∇ t ( t − s ) ν Γ( ν + 1) ∇ s + Z bt − ∇ t ( t − a ) ν ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s = Z t − a − ν ( t − a ) ν − ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s + Z t − a ν ( t − s ) ν − Γ( ν + 1) ∇ s + Z bt − ν ( t − a ) ν − ( b − s ) ν Γ( ν + 1)( b − a ) ν ∇ s = − ν ( t − a ) ν − Γ( ν + 1)( b − a ) ν (cid:20) − ν + 1 ( b − s − ν +1 (cid:21) s = t − s = a + ν Γ( ν + 1) (cid:20) − ν ( t − s − ν (cid:21) s = t − s = a + ν ( t − a ) ν − Γ( ν + 1)( b − a ) ν (cid:20) − ν + 1 ( b − s − ν +1 (cid:21) s = bs = t − = ν ( t − a ) ν − Γ( ν + 2)( b − a ) ν h ( b − t ) ν +1 − ( b − a − ν +1 i − ν + 1) (cid:2) ( t − t + 1 − ν − ( t − a − ν (cid:3) + − ν ( t − a ) ν − Γ( ν + 2)( b − a ) ν h ( b − b − ν +1 − ( b − t ) ν +1 i = 2 ν ( t − a ) ν − ( b − t ) ν +1 Γ( ν + 2)( b − a ) ν + ( t − a − ν Γ( ν + 1) − ν ( t − a ) ν − ( b − a − ν + 2) . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation t = b . This would imply that Z ba |∇ t G ( t, s ) | ∇ s = 2 ν ( b − a ) ν − (0) ν +1 Γ( ν + 2)( b − a ) ν + ( b − a − ν Γ( ν + 1) − ν ( b − a ) ν − ( b − a − ν + 2)= ( ν + 1)( b − a − ν Γ( ν + 2) − ν ( b − a ) ν − ( b − a − ν + 2) . For t = b and b − a = 2, this becomes Z ba |∇ t G ( t, s ) | ∇ s = ( ν + 1)(1) ν Γ( ν + 2) − ν (2) ν − (1)Γ( ν + 2)= ( ν + 1)Γ( ν + 1)Γ( ν + 2) − ν Γ( ν + 1)Γ( ν + 2)= 1 − νν + 1= 1 ν + 1 ≤ ν + 1= b − aν + 1 . For t = b and b − a = 3, we have Z ba |∇ t G ( t, s ) | ∇ s = ( ν + 1)(2 ν )Γ( ν + 2) − ν (3 ν − )(2)Γ( ν + 2)= ( ν + 1)Γ( ν + 2)Γ( ν + 2) − ν Γ( ν + 2)Γ( ν + 2)Γ(3)= ν + 1 − ν = 1= 33 ≤ b − aν + 1 . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation t = b and b − a ≥
4, the result holds since Z ba |∇ t G ( t, s ) | ∇ s = ( ν + 1)( b − a − ν Γ( ν + 2) − ν ( b − a ) ν − ( b − a − ν + 2)= ( ν + 1)( b − a − ν ) · · · (2 + ν )Γ( b − a − − ν Γ( b − a − ν )( b − a − ν )Γ( b − a )= ( ν + 1)( b − a − ν ) · · · (2 + ν )Γ( b − a − − ν Γ( b − a − ν )Γ(2 + ν )Γ( b − a − ν + 1)( b − a − ν ) · · · (2 + ν )( b − a − − ν ( b − a − ν ) · · · (2 + ν )( b − a − b − a − ν ) · · · (2 + ν )( b − a − b − a − b − a − ν ) · · · (2 + ν )( b − a − ≤ ( b − a − b − a − · · · (3)( b − a − ( b − a − b − a − b − a − b − a − ≤ b − aν + 1 . So the result holds if t = b generally. Now, assume t < b . If t = a + 1, then we VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation Z ba |∇ t G ( t, s ) | ∇ s = 2 ν (1 ν − )( b − a − ν +1 + ( ν + 1)(0 ν )( b − a ) ν Γ( ν + 2)( b − a ) ν − ν (1 ν − )( b − a − ν +1 Γ( ν + 2)( b − a ) ν = 2 ν Γ( ν )( b − a − b − a ) ν − ν Γ( ν )( b − a − b − a ) ν Γ( ν + 2)( b − a ) ν = 2 ν Γ( ν )( b − a − − ν Γ( ν )( b − a − ν + 2)= Γ( ν + 1)( b − a − ν + 2)= Γ( ν + 1)( b − a − ν + 1)Γ( ν + 1)= b − a − ν + 1 ≤ b − aν + 1 . If t = a + 2, then Z ba |∇ t G ( t, s ) | ∇ s = 2 ν (2 ν − )( b − a − ν +1 + ( ν + 1)(1 ν )( b − a ) ν − ν (2 ν − )( b − a − ν +1 Γ( ν + 2)( b − a ) ν = 2 ν Γ( ν + 1)( b − a − ν +1 Γ( ν + 2)( b − a ) ν + ( ν + 1)( ν )Γ( ν )( b − a ) ν Γ( ν + 2)( b − a ) ν − ν Γ( ν + 1)( b − a − ν +1 Γ( ν + 2)( b − a ) ν = 2 ν Γ( ν + 1)( b − a − b − a − ν + 1)( b − a − ν ) + 1 − ν ( b − a − ν + 1 ≤ ν ( b − a − ν + 1 + 1 − ν ( b − a − ν + 1= 2 ν ( b − a − ν + 1 + ν + 1 ν + 1 − ν ( b − a − ν + 1= ν ( b − a −
2) + 1 ν + 1 ≤ b − a − ν + 1 ≤ b − aν + 1 . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation t = a + 3, then Z ba |∇ t G ( t, s ) | ∇ s = 2 ν (3 ν − )( b − a − ν +1 + ( ν + 1)(2 ν )( b − a ) ν − ν (3 ν − )( b − a − ν +1 Γ( ν + 2)( b − a ) ν = 2 ν Γ(2 + ν )Γ( b − a − ν )Γ( b − a )Γ(3)Γ( b − a + ν )Γ( ν + 2)Γ( b − a −
3) + ( ν + 1) − ν Γ( ν + 2)( b − a − ν + 2)Γ(3)= ν ( b − a − b − a − b − a − b − a − ν )( b − a − ν ) + ( ν + 1) − ν ( b − a − ≤ ν ( b − a −
3) + ν + 1 − ν ( b − a − νb − νa − ν + 2 ν + 2 − νb + νa + ν ν ( b − a −
3) + 22 ≤ b − a − b − a − ≤ b − aν + 1 . VPs and BVPs for a Self-Adjoint Caputo Nabla Fractional Difference Equation t = a + k , where k ∈ N b − a − . Then Z ba |∇ t G ( t, s ) | ∇ s = 2 ν ( k ) ν − ( b − a − k ) ν +1 ( b − a ) ν Γ( ν + 2) + ( ν + 1)( k − ν Γ( ν + 2) − ν ( k ) ν − ( b − a − ν + 2)= 2 ν Γ( k + ν − b − a − k + ν + 1)Γ( b − a )Γ( k )Γ( b − a + ν )Γ( ν + 2)Γ( b − a − k ) + ( ν + 1)Γ( k − ν )Γ( ν + 2)Γ( k − − ν Γ( k + ν − b − a − ν + 2)Γ( k )= 2 ν ( ν + 2) . . . ( ν + k − b − a − . . . ( b − a − k )( k − b − a − ν ) . . . ( b − a − ( k −
1) + ν ) + ( ν + 1) . . . ( ν + k − k − − ν ( ν + 2) . . . ( ν + k − b − a − k − ≤ ν ( ν + 2) . . . ( ν + k − b − a − k )( k − k − ν + 1) . . . ( ν + k − k − − ν ( ν + 2) . . . ( ν + k − b − a − k − ν ( ν + 2) . . . ( ν + k − b − a − k − b + a + 1)( k − k − ν + 1) . . . ( ν + k − k − ν ( ν + 2) . . . ( ν + k − b − a + 1 − k ) + ( k − ν + 1) . . . ( ν + k − k − ≤ (1)(3)(4) . . . ( k − b − a + 1 − k ) + ( k − . . . ( k − k − ( k − b − a + 1 − k ) + ( k − k − k − b − a + 1 − k ) + 2( k − b − a − ≤ b − aν + 1 . (cid:4) These properties of the Green’s function are important in proving the exis-tence and uniqueness of solutions of BVPs for nonlinear difference equations.
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