New Estimations for Sturm-Liouville Problems in Difference Equations
aa r X i v : . [ m a t h . C A ] A p r New Estimations for Sturm-Liouville Problems in DifferenceEquations
Erdal BAS, Ramazan OZARSLAN
Department of Mathematics, Faculty of Science, Firat University, Elazig,23119, Turkeye-mail: [email protected], [email protected]
Abstract
In this paper, Sturm-Liouville problem for difference equations is considered with poten-tial function q ( n ) . The representations of solutions are obtained by variation of parametersmethod. These solutions are proved, using summation by parts. Also, estimation of asymp-totic expansion of the solutions are established.
AMS:
Keywords:
Sturm-Liouville, difference equation, eigenfunction, asymptoticformula, Casoratian.
1. Introduction
Sturm-Liouville operators have been studied for a number of years. Firstly,Sturm Liouville problem developed in a number of articles published by theseauthors in 1836 and 1837. It is known that the spectral characteristics arespectra, spectral functions, scattering data, norming constants, etc. The rep-resentation of solution of Sturm-Liouville problem and asymptotic formulas foreigenfunctions have been obtained by Levitan and Sargsjan [6]. The followingproblem Ly ( t ) = − d ydt + q ( t ) y = λy,y (0) cos α + y ′ (0) sin α = 0 y (1) cos β + y ′ (1) sin β = 01s called Sturm-Liouville problem in differential equation. Differential equationsare related to difference equations closely.In general, it is known that difference equations related to recursive relationsfor a long time. Its actual development appeared by being able to compared todifferential equations. Difference equations have many application areas whichare problems of physics, mathematics and engineering, vibrating string, econ-omy, population, actueria and logistics, etc.As a natural result of comparing the difference equations to differential equa-tions, the theory of linear difference equations have begun to appear in a similarway to the theory of differential equations. Basic theory of linear difference equa-tions improved by De Moivre, Euler, Lagrange, Laplace et al [13]. Hereafter,Hartman [17], Ahlbrandt-Hooker [16], Peterson-Kelley [1], Agarwal [2], Jirari[5], Bender-Orszag [7], Goldberg [12], Elaydi [15], Lakshmikantham, Trigianteand Peterson [13], Mickens [14] and they have contributed to linear differenceequations with publications and books.Especially, in recent years, Sturm-Liouville difference equation has seen agreat interest and a lot of study has published, but there are still a lot of thingsto develop about this subject. Peterson, Kelley, Agarwal, Bender gave a placethis subject in studies, also, Jirari studied pecularly Sturm-Liouville differenceequation in his studies.Atkinson studied discrete and continuous boundary value problems in hisbook and also he considered self-adjoint second-order difference equations. Ji-rari contributed Atkinson’s study by Second Order Sturm-Liouville DifferenceEquations and Orthogonal Polynomials in his thesis.Hinton and Lewis [9], Clark [19], Shi and Wu [8] , Shi and Sun [10] , Shiand Chen [18] and Hilscher [20] investigated spectral analysis of second orderdifference equations and operators. Wang and Shi [11] , Ji and Yang [21] , Sunand Shi [22] studied eigenvalues of second order difference equations.In the most of studies for Sturm-Liouville difference equations, q ( n ) is con-sidered as a real number, solutions and spectral properties for Sturm-Liouvilledifference equation are investigated by Hamiltoni systems. The following (1 . − .
3) problem [1] , [5]∆ (∆ p ( n − x ( n − q ( n ) x ( n ) + λr ( n ) x ( n ) = 0 , n = a, ..., b (1.1) x ( a −
1) + hx ( a ) = 0 , (1.2) x ( b + 1) + kx ( b ) = 0 . (1.3)is called Sturm-Liouville problem in difference equations.Our study is organized as follows. Fundamental theorems and definitionsin Section 2, representations of solutions with two different initial conditions inSection 3 and asymptotic behavior of eigenfunctions are given in Section 4.
2. PreliminariesDefinition.2.1. [1] The matrix of Casoratian is given by w ( n ) = x ( n ) x ( n ) ... x r ( n ) x ( n + 1) x ( n + 1) ... x r ( n + 1)... ... ... x ( n + r − x ( n + r − ... x r ( n + r − where x ( n ) , x ( n ) , ..., x r ( n ) are given functions. The determinant W ( n ) = det w ( n )is called Casoratian. Note that, it is similar to Wronskian determinant. Theorem.2.1. [5] (Wronskian-Type Identity) Let y and z be a solutions of(1 . . Then, for a ≤ n ≤ bW [ y, z ] ( n ) = p ( n −
1) [ y ( n ) ∆ z ( n − − z ( n ) ∆ y ( n − − p ( n −
1) [ y ( n ) z ( n − − y ( n − z ( n )]is a constant (In particular equal to W [ y, z ] ( a )). Definition.2.2. [5] Let’s express (1 .
1) Sturm-Liouville equation as follows, Lx ( n ) = − λx ( n ) , n ∈ [ a, b ] , (2.2)3ith initial conditionscos αx ( a ) − sin α ( p ( a ) ∇ x ( a )) = 0 , (2.3)where 0 < α, β < π, ∇ is the backward difference operator, ∇ x ( n ) = x ( n ) − x ( n − , are equivalent to x ( a −
1) + (cid:18) cot αp ( a ) − (cid:19) x ( a ) = 0 , (2.4)in other words, x ( a −
1) + hx ( a ) = 0 , (2.5)where L is self-adjoint Sturm-Liouville operator and h is real number. (2 . − (2 .
5) initial value problem is called Sturm-Liouville problem.
Theorem.2.2. [1] (
Summation by parts ) If m < n , then n − X k = m x ( k ) ∆ y ( k ) = [ x ( k ) y ( k )] nm − n − X k = m ∆ x ( k ) y ( k + 1) . (2.6) Theorem.2.4 . [1] If y n is an indefinite sum of x n , then n − X k = m y ( k ) = x ( n ) − x ( m ) . (2.7) Theorem.2.5. [1] , [3] ( Annihilator method ) Suppose that x ( n ) solves fol-lowing difference equation, E is the shift operator, Ex ( n ) = x ( n + 1) , (cid:0) E t + p ( t − E t − + ... + p (0) (cid:1) x ( n ) = r ( n ) , and that r ( n ) satisfies (cid:0) E m + q ( m − E m − + ... + q (0) (cid:1) r ( n ) = 0 . Then x ( n ) satisfies (cid:0) E m + q ( m − E m − + ... + q (0) (cid:1) (cid:0) E t + p ( t − E t − + ... + p (0) (cid:1) x ( n ) = 0 .
3. Main Results
4n this paper, we are interested in the representations of solutions of theSturm-Liouville problem in difference equations with potential function q ( n ) asfollows, ∆ x ( n −
1) + q ( n ) x ( n ) + λx ( n ) = 0 , n = a, ..., b (3.1)with initial conditions, x ( a −
1) + hx ( a ) = 0 , (3.2)where a, b are finite integers with a ≥ , a ≤ b, h is a real number, ∆ isthe forward difference operator, ∆ x ( n ) = x ( n + 1) − x ( n ) , λ is the spectralparameter, q ( n ) is a real valued potential function for n ∈ [ a, b ] . In the generalliterature, potential function q ( n ) is taken as real number but we take it asa variable coefficients in a similar way in Levitan and Sargsjan’s study [6].Levitan and Sargsjan obtained the representation of solution of Sturm-Liouvilleproblem in differential equations and asymptotic formulas for eigenfunctions. Inanalogous manner, we tried to obtain the representation of solution of Sturm-Liouville problem in difference equations.A self-adjoint difference operator corresponds to equation (3 .
1) noted by, Lx ( n ) = ∆ x ( n −
1) + q ( n ) x ( n ) = − λx ( n ) . In ℓ ( a, b ) , the Hilbert space of sequences of complex numbers x ( a ) , ..., x ( b )with the inner product, < x ( n ) , y ( n ) > = b X n = a x ( n ) y ( n ) , for every x ∈ D L, let define as follows D L = (cid:8) x ( n ) ∈ ℓ ( a, b ) : Lx ( n ) ∈ ℓ ( a, b ) , x (0) = − h, x (1) = 1 (cid:9) . Hence, equation (3 .
1) can be written as follows Lx ( n ) = − λx ( n ) . At this section, we present the representation of solution of (3 . − (3 .
2) Sturm-Liouville problem by variation of parameters method.5 heorem 3.1.
Let define Sturm-Liouville problem in difference equations asfollows; Lx ( n ) = − λx ( n ) , (3.3) x (0) = − h, x (1) = 1 , (3.4)then (3 . − (3 .
4) Sturm-Liouville problem has a unique solution for x ( n ) as x ( n, λ ) = − h (cid:16) q (0) − λ + p λ ( λ − (cid:17) p λ ( λ − − λ + p λ ( λ − ! n (3.5)+ − h (cid:16) q (0) − λ − p λ ( λ − (cid:17) p λ ( λ − − λ − p λ ( λ − ! n − n X i =0 q ( i ) x ( i ) (cid:18) − λ − √ λ ( λ − (cid:19) i p λ ( λ − − λ + p λ ( λ − ! n + n X i =0 q ( i ) x ( i ) (cid:18) − λ + √ λ ( λ − (cid:19) i p λ ( λ − − λ − p λ ( λ − ! n . Where − P i =0 . = 0 . Proof. If x ( n ) and x ( n ) are linearly independent solution for homogen partof (3 . , then it is easily found by characteristic polynomial [1] x h ( n ) = c x ( n ) + c x ( n ) , (3.6) x h ( n ) = c − λ + p λ ( λ − ! n + c − λ − p λ ( λ − ! n . (3.7)Assume that | λ − | < . . By variation ofparameters method [1] , [7], we take x p ( n ) = c ( n ) x ( n ) + c ( n ) x ( n ) , (3.8)If we perform necessary processes, we find c and c as follows c ( n ) = n X i =0 q ( i ) x ( i ) x ( i ) W ( x ( i ) , x ( i )) , (3.9) c ( n ) = n X i =0 − q ( i ) x ( i ) x ( i ) W ( x ( i ) , x ( i )) . x ( n, λ ) = c − λ + p λ ( λ − ! n + c − λ − p λ ( λ − ! n + n X i =0 q ( i ) x ( i ) (cid:18) − λ − √ λ ( λ − (cid:19) i W ( x ( i ) , x ( i )) − λ + p λ ( λ − ! n + n X i =0 − q ( i ) x ( i ) (cid:18) − λ + √ λ ( λ − (cid:19) i W ( x ( i ) , x ( i )) − λ − p λ ( λ − ! n . Where, W Casoratian determinant is a constant by Theorem 2.1, W ( x ( i ) , x ( i )) = − p λ ( λ − . If we use the initial conditions (3 . x ( n, λ ) = − h (cid:16) q (0) − λ + p λ ( λ − (cid:17) p λ ( λ − − λ + p λ ( λ − ! n (3.10)+ − h (cid:16) q (0) − λ − p λ ( λ − (cid:17) p λ ( λ − − λ − p λ ( λ − ! n − n X i =0 q ( i ) x ( i ) (cid:18) − λ − √ λ ( λ − (cid:19) i p λ ( λ − − λ + p λ ( λ − ! n + n X i =0 q ( i ) x ( i ) (cid:18) − λ + √ λ ( λ − (cid:19) i p λ ( λ − − λ − p λ ( λ − ! n . Now, let’s show that (3 .
15) holds Sturm-Liouville problem (3 . − (3 . . q ( n ) x ( n ) = − ∆ x ( n − − λx ( n ) . (3.11)First, let’s take last two term in (3 .
10) and write equality (3 .
11) in place of q ( i ) x ( i ). Hence, we obtain − n X i =0 [ − ∆ x ( i − − λx ( i ) ] x ( i ) √ λ ( λ − x ( n ) = n X i =0 ∆ x ( i − x ( i ) √ λ ( λ − x ( n )+ n X i =0 λx ( i ) x ( i ) √ λ ( λ − x ( n ) , (3.12)7 X i =0 [ − ∆ x ( i − − λx ( i ) ] x ( i ) √ λ ( λ − x ( n ) = − n X i =0 ∆ x ( i − x ( i ) √ λ ( λ − x ( n ) − n X i =0 λx ( i ) x ( i ) √ λ ( λ − x ( n )(3.13)Second, let’s apply twice summation by parts method to first term at the righthand side of equation (3 .
12) and (3 .
13) by Theorem 2.2, we obtain − n X i =0 (cid:2) − ∆ x ( i − − λx ( i ) (cid:3) x ( i ) p λ ( λ − x ( n ) = [ x ( n + 1) ∆ x ( n ) − x (0) ∆ x ( − − ∆ x ( n + 1) x ( n + 1) + ∆ x (0) x (0) − ∆ x ( − x (0)] x ( n ) p λ ( λ − n X i =0 (cid:2) ∆ x ( i −
1) + λx ( i ) (cid:3) x ( i ) p λ ( λ − x ( n )and n X i =0 (cid:2) − ∆ x ( i − − λx ( i ) (cid:3) x ( i ) p λ ( λ − x ( n ) = − [ x ( n + 1) ∆ x ( n ) − x (0) ∆ x ( − − ∆ x ( n + 1) x ( n + 1) + ∆ x (0) x (0) − ∆ x ( − x (0)] x ( n ) p λ ( λ − − n X i =0 (cid:2) ∆ x ( i −
1) + λx ( i ) (cid:3) x ( i ) p λ ( λ − x ( n ) . Since x and x satisfy the homogen part of (3 . .
14) and (3 .
15) equal to zero and hence, (3 .
14) and (3 .
15) asfollows respectively, [ x ( n + 1) ∆ x ( n ) − x (0) ∆ x ( −
1) (3.16) − ∆ x ( n + 1) x ( n + 1) + ∆ x (0) x (0) − ∆ x ( − x (0)] x ( n ) p λ ( λ − , − [ x ( n + 1) ∆ x ( n ) − x (0) ∆ x ( −
1) (3.17) − ∆ x ( n + 1) x ( n + 1) + ∆ x (0) x (0) − ∆ x ( − x (0)] x ( n ) p λ ( λ − . − n X i =0 q ( i ) x ( i ) x ( i ) p λ ( λ − x ( n ) + n X i =0 q ( i ) x ( i ) x ( i ) p λ ( λ − x ( n ) = 1 p λ ( λ −
4) [ x ( n ) p λ ( λ −
4) (3.18)+ x ( n ) ( x (0) x ( − − x ( − x (0))+ x ( n ) ( x ( − x (0) − x (0) x ( − . And then, we have the following equation x ( n ) = (cid:18) − h (cid:16) q (0) − λ + √ λ ( λ − (cid:17) √ λ ( λ − (cid:19) x ( n ) + (cid:18) − h (cid:16) q (0) − λ − √ λ ( λ − (cid:17) √ λ ( λ − (cid:19) x ( n ) − n X i =0 q ( i ) x ( i ) x ( i ) p λ ( λ − x ( n ) + n X i =0 q ( i ) x ( i ) x ( i ) p λ ( λ − x ( n ) . Note that, detail of the proof will be given in published paper.
Theorem 3.2.
Let define Sturm-Liouville problem in difference equations asfollows; Ly ( n ) = − λy ( n ) , (3.19) y (0) = 1 , y (1) = 0 , (3.20)then (3 . − (3 .
32) Sturm-Liouville problem has a unique solution for y ( n ) as y ( n, λ ) = − λ + p λ ( λ − − q (0)2 p λ ( λ − ! − λ + p λ ( λ − ! n (3.21)+ − λ + p λ ( λ −
4) + 2 q (0)2 p λ ( λ − ! − λ − p λ ( λ − ! n − n X i =0 q ( i ) y ( i ) (cid:18) − λ − √ λ ( λ − (cid:19) i p λ ( λ − − λ + p λ ( λ − ! n + n X i =0 q ( i ) y ( i ) (cid:18) − λ + √ λ ( λ − (cid:19) i p λ ( λ − − λ − p λ ( λ − ! n . Proof.
This is proved similarly to the proof of Theorem 3.1.9 . Asymptotic Formulas for Sturm-Liouville Problem in DifferenceEquations
At this section, we present the asymptotic formulas for the solution of Sturm-Liouville problem. Let’s take (3 . − (3 .
4) Sturm-Liouville problem. Then wecan give the following Theorem.
Theorem 4.1. (3 . − (3 .
4) Sturm-Liouville problem has the estimate x ( n ) = O ( | h | ) . Theorem 4.2. (3 . − (3 .
20) Sturm-Liouville problem has the estimate y ( n ) = O (1) . The proofs will be given in published version of paper.
Conclusion
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