On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator

Abstract

A third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert space 𝐻 with self-adjoint positive definite operator 𝐴. Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered.

1. Introduction

In modeling several phenomena of physics, biology, and ecology mathematically, there often arise problems with nonlocal boundary conditions (see [1–5] and the references given therein). Nonlocal boundary value problems have been a major research area in the case when it is impossible to determine the boundary conditions of the unknown function. Over the last few decades, the study of nonlocal boundary value problems is of substantial contemporary interest (see, e.g., [6–14] and the references given therein).

We consider the nonlocal boundary value problem 𝑑²𝑢(𝑡)𝑑𝑡²(𝑡)(𝑡)𝑢(0)(1)𝑢+𝐴𝑢=𝑓,0<𝑡<1,=𝛼𝑢+𝜑,_′(0)=𝛽𝑢^′(1)+𝜓, (1) for hyperbolic equations in a Hilbert space 𝐻 with self-adjoint positive definite linear operator 𝐴 with domain 𝐷(𝐴).

A function 𝑢(𝑡) is called a solution of problem (1) if the following conditions are satisfied.(i)𝑢(𝑡) is twice continuously differentiable on the segment [0,1]. The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii)The element 𝑢(𝑡) belongs to 𝐷(𝐴) for all 𝑡∈[0,1] and the function 𝐴𝑢(𝑡) is continuous on the segment [0,1].(iii)𝑢(𝑡) satisfies the equations and the nonlocal boundary conditions (1).

Here, 𝜑(𝑥), 𝜓(𝑥)(𝑥∈[0,1]) and 𝑓(𝑡,𝑥)(𝑡,𝑥∈[0,1]) are smooth functions.

In the study of numerical methods for solving PDEs, stability is an important research area (see [6–27]). Many scientists work on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitudes of the grid steps 𝜏 and ℎ with respect to the time and space variables are connected. This particularly means that 𝜏‖𝐴_ℎ‖→0 when 𝜏→0.

We are interested in studying high order of accuracy unconditionally stable difference schemes for hyperbolic PDEs.

In the present paper, third order of accuracy difference scheme generated by integer power of 𝐴 for approximately solving nonlocal boundary value problem (1) is presented. The stability estimates for solution of the difference scheme are established.

In [8], some results of this paper, without proof, were presented.

The well posedness of nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11–14, 19–32] and the references therein).

2. Third Order of Accuracy Difference Scheme Subject to Nonlocal Conditions

In this section, we obtain stability estimates for the solution of third order of accuracy difference scheme 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+23𝐴𝑢_𝑘+16𝐴𝑢_𝑘+1+𝑢_𝑘−1+1𝜏12²𝐴²𝑢_𝑘+1=𝑓_𝑘,𝑓_𝑘=23𝑓𝑡_𝑘+16𝑓𝑡_𝑘+1+𝑓𝑡_𝑘−1−1𝜏12²−𝐴𝑓𝑡_𝑘+1+𝑓^′′𝑡_𝑘+1,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,₀=𝛼𝑢_𝑁𝜏+𝜑,𝐼+²𝜏12𝐴+⁴𝐴144²𝜏⁻¹𝑢₁−𝑢₀+𝜏2𝐴𝑢₀−𝜏𝑓_1,1𝜏=𝛽𝐼−²𝐴×127𝑢_𝑁−8𝑢_𝑁−1+𝑢_𝑁−2+𝜏6𝜏3𝑓_𝑁−𝐴𝑢_𝑁+𝜏𝐼−²𝐴𝜓12 (2) for numerical solution of nonlocal boundary value problem (1). Here, 𝑓_1,1=𝑓(0)+−𝑓(0)+𝜏𝑓^′(01)2−2𝑓^′(0)𝜏6. (3) We study the stability of solutions of difference scheme (2) under the following assumption: |𝛼|+2𝛽+2|𝛼|𝛽<1. (4) We give a lemma that will be needed in the sequel which was presented in [18]. First, let us present the following operators: 1𝑅=𝐼−3𝜏²𝐴+𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²×1𝐼+6𝜏²1𝐴+𝜏12⁴𝐴²⁻¹, (5) and its conjugate 𝑅, 𝑅₁=−5𝜏⁴𝐴144²+7𝜏⁶𝐴216³−𝑖𝜏𝐴^1/2×𝜏𝐼+²𝜏12𝐴+⁴𝐴144²1𝐼+𝜏72⁴𝐴²×−𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²×𝜏𝐼+²𝜏12𝐴+⁴𝐴144²⁻¹, (6) and its conjugate 𝑅₁, 𝑅₂=𝜏𝐼−²𝐴𝜏12𝐼+²6𝜏𝐴+⁴𝐴12²×−𝑖𝐴^1/2𝜏𝐼+²𝜏12𝐴+⁴𝐴144²1𝐼+𝜏72⁴𝐴²⁻¹,𝑅₃=𝜏𝐼+²6𝜏𝐴+⁴𝐴12²×𝜏𝐼+²𝜏12𝐴+⁴𝐴144²−𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²⁻¹,𝑅₄=𝜏𝐼+²3𝜏𝐴+⁴9𝐴²+𝜏⁶𝐴72³×−𝑖𝐴^1/21𝐼+𝜏72⁴𝐴²𝜏𝐼+²6𝜏𝐴+⁴𝐴12²×𝜏𝐼+²6𝐴⁻¹,𝑅₅=−𝜏²2𝜏𝐴−⁴𝐴12²+𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²×𝜏𝐼+²6𝜏𝐴+⁴𝐴12²⁻¹, (7) and its conjugate 𝑅₅, and 𝑅₆=1𝐼−3𝜏²𝐴+𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²×𝜏²2𝜏𝐴+⁴𝐴12²−𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²⁻¹, (8) and its conjugate 𝑅₆.

We consider the following operators: 𝑅₇=(7𝑅−𝐼)=56𝜏𝐼−𝜏12²1𝐴+𝜏72⁴𝐴²+76𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²×𝜏⁻¹1𝐼+6𝜏²1𝐴+𝜏12⁴𝐴²⁻¹, (9) and its conjugate 𝑅₇, 𝑅₇=7𝑅−𝐼=56𝜏𝐼−𝜏12²1𝐴+𝜏72⁴𝐴²−76𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²×𝜏⁻¹1𝐼+6𝜏²1𝐴+𝜏12⁴𝐴²⁻¹,𝑅₈=7𝐼−2𝜏²𝐴𝜏6𝜏𝐼+²𝐴3+𝜏⁴𝐴²9+𝜏⁶𝐴³72×𝜏⁻¹𝜏𝐼+²𝐴6⁻¹𝜏𝐼+²6𝜏𝐴+⁴𝐴12²⁻²,𝑅₉=5𝐼−3𝜏²𝜏𝐴+⁴𝐴²9𝜏𝐼+²𝐴3+𝜏⁴𝐴²9+𝜏⁶𝐴³72×𝜏⁻¹𝜏𝐼+²𝐴6⁻¹1𝐼+6𝜏²1𝐴+𝜏12⁴𝐴²⁻³,𝑅₁₀5=𝐼+𝜏144⁴𝐴²−9𝜏288⁶𝐴³+9𝜏1728⁸𝐴⁴×𝑖𝜏𝐴^1/21𝐼+𝜏72⁴𝐴²𝜏𝐼+²𝜏12𝐴+⁴𝐴144²⁻¹, (10) and its conjugate 𝑅₁₀.

Lemma 1. The following estimates hold: ‖𝑅‖_𝐻→𝐻≤1,𝑅_𝐻→𝐻𝑅≤1,_1𝐻→𝐻≤1,𝑅_1𝐻→𝐻𝐴≤1,_1/2𝑅_2𝐻→𝐻≤1,𝜏𝐴^1/2𝑅_3𝐻→𝐻𝐴≤1,_1/2𝑅_4𝐻→𝐻≤1,𝐴^−1/2𝑅_5𝐻→𝐻𝐴≤𝜏,_−1/2𝑅_5𝐻→𝐻≤𝜏,𝜏𝐴^1/2𝑅_6𝐻→𝐻≤1,𝜏𝐴^1/2𝑅_6𝐻→𝐻≤1. (11)

Now let us give, without proof, the second lemma.

Lemma 2. The following estimates hold: 𝐼+𝑖𝜏𝐴^1/2𝑅_𝐻→𝐻≤2,𝐼+𝑖𝜏𝐴^1/2𝑅_𝐻→𝐻≤2,𝜏𝑅_7𝐻→𝐻≤1,𝜏𝑅_7𝐻→𝐻1≤1,3𝜏𝐴^1/2𝑅²_𝐻→𝐻1≤1,3𝜏𝐴^1/2𝑅²_𝐻→𝐻≤1,𝜏𝑅_8𝐻→𝐻≤76,𝜏𝑅_9𝐻→𝐻𝑅≤1,₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝑅≤2,₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻≤2. (12)

Throughout the section, for simplicity, we denote 𝐵_𝜏1=𝛽2𝑅₂𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−21+𝛽2𝑅₂𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−21−𝛼2𝑅₁𝑅^𝑁−𝑅₁𝑅^𝑁1+𝛼𝛽4𝑅₁𝑅₂𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁𝑅^𝑁−21+𝛼𝛽4𝑅₁𝑅₂𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁𝑅^𝑁−21−𝛼𝛽4𝑅₁𝑅₂𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁𝑅^𝑁−21−𝛼𝛽4𝑅₁𝑅₂𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁𝑅^𝑁−2. (13)

Lemma 3. Suppose that assumption (4) holds. Then, the operator 𝐼−𝐵_𝜏 has an inverse 𝑇_𝜏=(𝐼−𝐵_𝜏)⁻¹. From symmetry and positivity properties of operator 𝐴, the following estimate is satisfied: 𝑇_{𝜏𝐻↦𝐻}≤1|𝛼|𝛽|𝛼𝛽.1−−2−2| (14)

Proof. Using the definitions of 𝐵_𝜏,𝑅,𝑅, estimates (11), and the following simple estimates, 𝜏𝐴^1/2𝜏𝐼−²𝐴𝜏12𝐼+²𝜏12𝐴+⁴𝐴144²⁻¹_𝐻→𝐻≤12,𝜏𝐴^1/21𝐼+𝜏12²1𝐴+𝜏144⁴𝐴²⁻¹_𝐻→𝐻≤12√1112+√,11 (15) and the triangle inequality, we get 𝐵_𝜏≤𝛽12𝐴^1/2𝑅_2𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻+𝛽12𝐴^1/2𝑅_2𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻1+𝛼2𝑅_1𝐻→𝐻𝑅^𝑁_𝐻→𝐻+𝑅_1𝐻→𝐻𝑅^𝑁_𝐻→𝐻+|𝛼𝛽1|4𝑅_1𝐻→𝐻𝐴^1/2𝑅_2𝐻→𝐻×𝑅^𝑁_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻+|𝛼𝛽1|4𝑅_1𝐻→𝐻𝐴^1/2𝑅_2𝐻→𝐻×𝑅^𝑁_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻+|𝛼𝛽1|4𝑅_1𝐻→𝐻𝐴^1/2𝑅_2𝐻→𝐻×𝑅^𝑁_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻+|𝛼𝛽1|4𝑅_1𝐻→𝐻𝐴^1/2𝑅_2𝐻→𝐻×𝑅^𝑁_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻≤𝑞, (16) where 𝑞=|𝛼|+2𝛽+2|𝛼|𝛽. (17) Since 𝑞<1, the operator 𝐼−𝐵_𝜏 has a bounded inverse and 𝐼−𝐵_𝜏⁻¹_𝐻→𝐻≤1=11−𝑞|𝛼|𝛽|𝛼𝛽.1−−2−2| (18) Lemma 3 is proved.

Now, let us obtain formula for the solution of problem (2). Using the results of [18], one can obtain the following formula: 𝑢₀𝑢=𝜇,₁=𝜏𝐼+²𝜏12𝐴+⁴𝐴144²⁻¹×5𝐼−𝜏12²𝜏𝐴+⁴𝐴144²𝜇𝜏+𝜏𝐼−²𝐴12𝜔+𝜏²𝑓_1,1,𝑢_𝑘=12𝑅₁₀𝑅^𝑘−𝑅₁₀𝑅^𝑘1𝜇+2𝑅^𝑘−𝑅^𝑘𝑅₂𝜔+12𝑅^𝑘−𝑅^𝑘𝑅₃𝜏²𝑓_1,1+12𝑅₄^𝑘−1_𝑠=1𝑅^𝑘−𝑠−𝑅^𝑘−𝑠𝑓_𝑠𝜏² (19) for the solution of difference scheme 𝜏⁻²𝑢_𝑘+1−2𝑢_𝑘+𝑢_𝑘−1+23𝐴𝑢_𝑘+16𝐴𝑢_𝑘+1+𝑢_𝑘−1+1𝜏12²𝐴²𝑢_𝑘+1=𝑓_𝑘,𝑓_𝑘=23𝑓𝑡_𝑘+16𝑓𝑡_𝑘+1+𝑓𝑡_𝑘−1−1𝜏12²−𝐴𝑓𝑡_𝑘+1+𝑓^′′𝑡_𝑘+1,𝑡_𝑘𝑢=𝑘𝜏,1≤𝑘≤𝑁−1,𝑁𝜏=1,₀𝜏=𝜇,𝐼+²𝜏12𝐴+⁴𝐴144²𝜏⁻¹𝑢₁−𝑢₀+𝜏2𝐴𝑢₀−𝜏𝑓_1,1=𝜏𝐼−²𝐴12𝜔. (20) Applying formula (19) and nonlocal boundary conditions 𝑢₀=𝛼𝑢_𝑁+𝜑,𝜔=𝛽7𝑢_𝑁−8𝑢_𝑁−1+𝑢_𝑁−2+𝜏6𝜏3𝑓_𝑁−𝐴𝑢_𝑁+𝜓, (21) one can write 1𝜇=𝛼2𝑅₁₀𝑅^𝑁−𝑅₁₀𝑅^𝑁1𝜇+2𝑅^𝑁−𝑅^𝑁𝑅₂𝜔+12𝑅^𝑁−𝑅^𝑁𝑅₃𝜏²𝑓_1,1+12𝑅₄^𝑁−1_𝑠=1𝑅^𝑁−𝑠−𝑅^𝑁−𝑠𝑓_𝑠𝜏²1+𝜑,𝜔=𝛽2𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅₁₀𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅₁₀𝑅^𝑁−2𝜇+12𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2𝑅₂𝜔+12𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2𝑅₃𝜏²𝑓_1,1+𝜏3𝑓_𝑁+12𝑅₈𝑓_𝑁−1𝜏²+12𝑅₉𝑓_𝑁−2𝜏²+12𝑅₄𝜏^𝑁−3_𝑠=1𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}𝑓_𝑠𝜏+𝜓. (22) Using formulas in (22), we obtain 𝜇=𝑇_𝜏𝛼12𝑅^𝑁−𝑅^𝑁𝑅₃𝜏²𝑓_1,1+12𝑅₄𝜏^𝑁−1_𝑠=1𝑅^𝑁−𝑠−𝑅^𝑁−𝑠𝑓_𝑠𝜏×1+𝜑𝐼−2𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2𝑅₂+𝛼12𝑅^𝑁−𝑅^𝑁𝑅₂×𝛽12𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2×𝑅₃𝜏²𝑓_1,1+2𝜏3𝑓_𝑁+𝑅₈𝑓_𝑁−1𝜏²+𝑅₉𝑓_𝑁−2𝜏²+𝑅₄𝜏^𝑁−3_𝑠=1𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}−𝑅₇𝑅₅−𝜏𝐴3𝑅²×𝑅^{𝑁−2−𝑠}𝑓_𝑠𝜏+𝜓,𝜔=𝑇_𝜏1𝐼−𝛼2𝑅₁₀𝑅^𝑁−𝑅₁₀𝑅^𝑁×𝛽12𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2×𝑅₃𝜏²𝑓_1,1+2𝜏3𝑓_𝑁+𝑅₈𝑓_𝑁−1𝜏²+𝑅₉𝑓_𝑁−2𝜏²+𝑅₄𝜏×^𝑁−3_𝑠=1𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}−𝑅₇𝑅₅−𝜏𝐴3𝑅²×𝑅^{𝑁−2−𝑠}𝑓_𝑠𝜏+1+𝜓2𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅₁₀𝑅^𝑁−2+𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅₁₀𝑅^𝑁−2×𝛼12𝑅^𝑁−𝑅^𝑁𝑅₃𝜏²𝑓_1,1+12𝑅₄𝜏^𝑁−1_𝑠=1𝑅^𝑁−𝑠−𝑅^𝑁−𝑠𝑓_𝑠𝜏+𝜑. (23)

So, formulas (19) and (23) give a solution of problem (2).

Unfortunately, the estimates for max_{1≤𝑘≤𝑁}‖𝑢_𝑘‖_𝐻, max_{1≤𝑘≤𝑁}‖𝐴^1/2𝑢_𝑘‖_𝐻, and max_{1≤𝑘≤𝑁}‖𝐴𝑢_𝑘‖_𝐻 cannot be obtained under the conditions max_{1≤𝑘≤𝑁}𝑢_𝑘𝐻≤𝑀^𝑁−1_𝑠=1𝐴^−1/2𝑓_𝑠𝐻𝜏+𝜑_𝐻+𝐴^−1/2𝜓_𝐻+𝜏𝐴^−1/2𝑓_1,1𝐻,max_{1≤𝑘≤𝑁}𝐴^1/2𝑢_𝑘𝐻≤𝑀^𝑁−1_𝑠=1𝑓_𝑠𝐻𝜏+𝐴^1/2𝜑_𝐻+𝜓_𝐻+𝜏𝑓_1,1𝐻,max_{1≤𝑘≤𝑁}𝐴𝑢_𝑘𝐻≤𝑀^𝑁−1_𝑠=2𝑓_𝑠−𝑓_𝑠−1𝐻+𝑓_1𝐻+𝐴𝜑_𝐻+𝐴^1/2𝜓_𝐻+𝜏𝐴^1/2𝑓_1,1𝐻. (24) Nevertheless, we have the following theorem.

Theorem 4. Suppose that assumption (4) holds and 𝜑∈𝐷(𝐴^3/2), 𝜓∈𝐷(𝐴^1/2). Then, for solution of difference scheme (2), the following stability estimates hold: max_{1≤𝑘≤𝑁}𝑢_𝑘𝐻≤𝑀^𝑁−1_𝑠=1𝐴^−1/2𝑓_𝑠𝐻𝜏+𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^−1/2𝜓_𝐻+𝜏𝐴^−1/2𝑓_1,1𝐻,max_{1≤𝑘≤𝑁}𝐴^1/2𝑢_𝑘𝐻≤𝑀^𝑁−1_𝑠=1𝑓_𝑠𝐻𝜏+𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝜓_𝐻+𝜏𝑓_1,1𝐻,max_{1≤𝑘≤𝑁}𝐴𝑢_𝑘𝐻≤𝑀^𝑁−1_𝑠=2𝑓_𝑠−𝑓_𝑠−1𝐻+𝑓_1𝐻+𝐴𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^1/2𝜓_𝐻+𝜏𝐴^1/2𝑓_1,1𝐻, (25) where 𝑀 does not depend on 𝜏, 𝜑, 𝜓, 𝑓_1,1(𝑥), and 𝑓_𝑠(𝑥), 1≤𝑠≤𝑁−1.

Proof. Using formulas in (23) and estimates (11), (12), and (14), we obtain 𝐼+𝑖𝜏𝐴^1/2𝜇_𝐻≤𝑇_{𝜏𝐻→𝐻}×𝛼|1|2𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝐴^−1/2𝑓_1,1𝐻+12𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−1_𝑠=1𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻×𝐴^−1/2𝑓_{𝑠𝐻→𝐻}𝜏+𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻×11+2𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻×𝐴^1/2𝑅_2𝐻→𝐻+|𝛼|12𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝐴^1/2𝑅_2𝐻→𝐻×𝛽12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝐴^−1/2𝑓_1,1𝐻+2𝜏3𝐴^−1/2𝑓_𝑁𝐻+𝜏𝑅_8𝐻→𝐻𝐴^−1/2𝑓_𝑁−1𝐻𝜏+𝜏𝑅_9𝐻→𝐻𝐴^−1/2𝑓_𝑁−2𝐻𝜏+𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−3_𝑠=1𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻×𝐴^−1/2𝑓_𝑠𝐻𝜏+𝐴^−1/2𝜓_𝐻≤𝑀^𝑘−1_𝑠=1𝐴^−1/2𝑓_𝑠𝐻𝜏+𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^−1/2𝜓_𝐻+𝜏𝐴^−1/2𝑓_1,1𝐻,𝐴^−1/2𝜔_𝐻≤𝑇_{𝜏𝐻→𝐻}×|𝛼|11+2𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝛽12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝐴^−1/2𝑓_1,1𝐻𝜏+23𝐴^−1/2𝑓_𝑁𝐻𝜏+𝜏𝑅_8𝐻→𝐻𝐴^−1/2𝑓_𝑁−1𝐻𝜏+𝜏𝑅_9𝐻→𝐻𝐴^−1/2𝑓_𝑁−2𝐻𝜏+𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−3_𝑠=1𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻×𝐴^−1/2𝑓_𝑠𝐻𝜏+𝐴^−1/2𝜓_𝐻+12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}𝑅^𝑁−2_𝐻→𝐻×𝛼|1|2𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝐴^−1/2𝑓_1,1𝐻+𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−1_𝑠=1𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻×𝐴^−1/2𝑓_𝑠𝐻𝜏+𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻≤𝑀^𝑘−1_𝑠=1𝐴^−1/2𝑓_𝑠𝐻𝜏+𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^−1/2𝜓_𝐻+𝜏𝐴^−1/2𝑓_1,1𝐻. (26)
Applying 𝐴^1/2 to formulas in (23), we get 𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜇_𝐻≤𝑇_{𝜏𝐻→𝐻}×𝛼|1|2𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝑓_1,1𝐻+12𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−1_𝑠=1𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻×𝑓_{𝑠𝐻→𝐻}𝜏+𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻×11+2𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻×𝐴^1/2𝑅_2𝐻→𝐻+|𝛼|12𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝐴^1/2𝑅_2𝐻→𝐻×𝛽12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝑓_1,1𝐻+2𝜏3𝑓_𝑁𝐻+𝜏𝑅_8𝐻→𝐻𝑓_𝑁−1𝐻𝜏+𝜏𝑅_9𝐻→𝐻×𝑓_𝑁−2𝐻𝜏+𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−3_𝑠=1𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻×𝑓_𝑠𝐻𝜏+𝜓_𝐻≤𝑀^𝑘−1_𝑠=1𝑓_𝑠𝐻𝜏+𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝜓_𝐻+𝜏𝑓_1,1𝐻,‖𝜔‖_𝐻≤𝑇_{𝜏𝐻→𝐻}×|𝛼|11+2𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝛽12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝑓_1,1𝐻2𝜏+3𝑓_𝑁𝐻𝜏+𝜏𝑅_8𝐻→𝐻𝑓_𝑁−1𝐻𝜏+𝜏𝑅_9𝐻→𝐻𝑓_𝑁−2𝐻𝜏+𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−3_𝑠=1𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻𝑓_𝑠𝐻𝜏+𝜓_𝐻+12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}𝑅^𝑁−2_𝐻→𝐻×𝛼|1|2𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝑓_1,1𝐻+12𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−1_𝑠=1𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻×𝑓_𝑠𝐻𝜏+𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻≤𝑀^𝑘−1_𝑠=1𝑓_𝑠𝐻𝜏+𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝜓_𝐻+𝜏𝑓_1,1𝐻. (27) Now, applying Abel’s formula to (23), we obtain the following formulas: 𝜇=𝑇_𝜏𝛼12𝑅^𝑁−𝑅^𝑁𝑅₃𝜏²𝑓_1,1+12𝑅₄𝜏^2𝑁−1_𝑠=2𝑅₆𝑅^𝑁−𝑠−𝑅₆𝑅^𝑁−𝑠×𝑓_𝑠−𝑓_𝑠−1+𝑅₆−𝑅₆𝑓_𝑁−1−𝑅₆𝑅^𝑁−1−𝑅₆𝑅^𝑁−1𝑓₁×1+𝜑𝐼−2𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2𝑅₂1+𝛼2𝑅^𝑁−𝑅^𝑁𝑅₂×𝛽12𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2×𝑅₃𝜏²𝑓_1,1+𝜏3𝑓_𝑁+12𝑅₈𝑓_𝑁−1𝜏²+12𝑅₉𝑓_𝑁−2𝜏²+𝑅₄12𝜏^2𝑁−3_𝑠=2𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}×𝑓_𝑠−𝑓_𝑠−1+𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑓_𝑁−3−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−3−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−3×𝑓₁+𝜓,𝜔=𝑇_𝜏1𝐼−𝛼2𝑅₁𝑅^𝑁−𝑅₁𝑅^𝑁×𝛽12𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−2×𝑅₃𝜏²𝑓_1,1+𝜏3𝑓_𝑁+12𝑅₈𝑓_𝑁−1𝜏²+12𝑅₉𝑓_𝑁−2𝜏²+𝑅₄12𝜏^2𝑁−3_𝑠=2𝑅₆𝑅₇𝑅₅𝜏−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^{𝑁−2−𝑠}×𝑓_𝑠−𝑓_𝑠−1+𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑓_𝑁−3−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅^𝑁−3−𝑅₆𝑅₇𝑅₅−𝜏𝐴3𝑅²×𝑅^𝑁−3𝑓₁+1+𝜓2𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅₁𝑅^𝑁−2−𝑅₇𝑅₅−𝜏𝐴3𝑅²𝑅₁𝑅^𝑁−2×𝛼12𝑅^𝑁−𝑅^𝑁𝑅₃𝜏²𝑓_1,1+12𝑅₄𝜏^2𝑁−1_𝑠=2𝑅₆𝑅^𝑁−𝑠−𝑅₆𝑅^𝑁−𝑠×𝑓_𝑠−𝑓_𝑠−1+𝑅₆−𝑅₆𝑓_𝑁−1−𝑅₆𝑅^𝑁−1−𝑅₆𝑅^𝑁−1×𝑓₁+𝜑. (28)(29)
Next, let us obtain the estimates for ‖𝐴(𝐼+𝑖𝜏𝐴^1/2)𝜇‖_𝐻 and ‖𝐴^1/2𝜔‖_𝐻. First, applying 𝐴 to formula (28) and using estimates (11), (12), and (14) and the triangle inequality, one can obtain 𝐴𝐼+𝑖𝜏𝐴^1/2𝜇_𝐻≤𝑇_{𝜏𝐻→𝐻}×𝛼|1|2𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝐴^1/2𝑓_1,1𝐻𝜏+12𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−1_𝑠=2𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−𝑠_𝐻→𝐻×𝑓_𝑠−𝑓_𝑠−1𝐻+𝜏𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝑅_6𝐻→𝐻+𝜏𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝑓_𝑁−1𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−1_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−1_𝐻→𝐻×𝑓_1𝐻+𝐴𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻×11+2𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻𝐴^1/2𝑅_2𝐻→𝐻+|𝛼|12𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻𝐴^1/2𝑅_2𝐻→𝐻×𝛽12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻×𝜏𝐴^1/2𝑓_1,1𝐻+2𝜏3𝐴^1/2𝑓_𝑁𝐻+𝜏𝑅_8𝐻→𝐻𝐴^1/2𝑓_𝑁−1𝐻𝜏+𝜏𝑅_9𝐻→𝐻𝐴^1/2𝑓_𝑁−2𝐻𝜏+𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−3_𝑠=2𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻𝑓_𝑠−𝑓_𝑠−1𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑓_𝑁−3𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−3_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−3_𝐻→𝐻𝑓_1𝐻+𝐴^1/2𝜓_𝐻≤𝑀^𝑁−1_𝑠=2𝑓_𝑠−𝑓_𝑠−1𝐻+𝑓_1𝐻+𝐴𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^1/2𝜓_𝐻+𝜏𝐴^1/2𝑓_1,1𝐻. (30)
Second, applying 𝐴^1/2 to formula (29) and using estimates (11), (12), and (14) and the triangle inequality, we get 𝐴^1/2𝜔_𝐻≤𝑇_{𝜏𝐻→𝐻}×|𝛼|11+2𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻+𝐼+𝑖𝜏𝐴^1/2−1𝑅_{10𝐻→𝐻}×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁_𝐻→𝐻×𝛽12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−2_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝐴^1/2𝑓_1,1𝐻+23𝜏𝐴^1/2𝑓_𝑁𝐻+𝜏𝑅_8𝐻→𝐻𝐴^1/2𝑓_𝑁−1𝐻𝜏+𝜏𝑅_9𝐻→𝐻𝐴^1/2𝑓_𝑁−2𝐻𝜏+𝜏𝐴^1/2𝑅_4𝐻→𝐻𝜏×^𝑁−3_𝑠=2𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^{𝑁−2−𝑠}_𝐻→𝐻𝑓_𝑠−𝑓_𝑠−1𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻×𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑓_𝑁−3𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−3_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅^𝑁−3_𝐻→𝐻𝑓_1𝐻+𝐴^1/2𝜓_𝐻+12𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−2_𝐻→𝐻+𝜏𝑅_7𝐻→𝐻𝜏⁻¹𝐴^−1/2𝑅_5𝐻→𝐻+13𝜏𝐴^1/2𝑅²_𝐻→𝐻×𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2𝑅^𝑁−2_𝐻→𝐻×𝛼|1|2𝑅^𝑁_𝐻→𝐻+𝑅^𝑁_𝐻→𝐻×𝜏𝐴^1/2𝑅_3𝐻→𝐻𝜏𝐴^1/2𝑓_1,1𝐻+𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑁−1_𝑠=2𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑁−𝑠_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑁−𝑠_𝐻→𝐻×𝑓_𝑠−𝑓_𝑠−1𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻×𝑓_{𝑁−1𝐻→𝐻}+𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑁−1_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑁−1_𝐻→𝐻×𝑓_1𝐻+𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝐴𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻≤𝑀^𝑁−1_𝑠=2𝑓_𝑠−𝑓_𝑠−1𝐻+𝑓_1𝐻+𝐴𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^1/2𝜓_𝐻+𝜏𝐴^1/2𝑓_1,1𝐻. (31)
Now, we will prove estimates (25). Using formula (19), estimates (11), (12), (26), and (27), and the triangle inequality, we obtain 𝑢_𝑘𝐻≤12𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝑅^𝑘_𝐻→𝐻×𝐼+𝑖𝜏𝐴^1/2𝜇_𝐻+12𝐴^1/2𝑅_2𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝐴^1/2𝑅_2𝐻→𝐻𝑅^𝑘_𝐻→𝐻×𝐴^−1/2𝜔_𝐻+12𝜏𝐴^1/2𝑅_3𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝜏𝐴^1/2𝑅_3𝐻→𝐻𝑅^𝑘_𝐻→𝐻𝜏𝐴^−1/2𝑓_1,1𝐻+12𝜏𝐴^1/2𝑅_4𝐻→𝐻^𝑘−1_𝑠=1𝑅^𝑘−𝑠_𝐻→𝐻+𝑅^𝑘−𝑠_𝐻→𝐻×𝐴^−1/2𝑓_𝑠𝐻𝜏≤𝑀^𝑁−1_𝑠=1𝐴^−1/2𝑓_𝑠𝐻𝜏+𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^−1/2𝜓_𝐻+𝜏𝐴^−1/2𝑓_1,1𝐻 (32) for any 𝑘≥2. Applying 𝐴^1/2 to (19), we get 𝐴^1/2𝑢_𝑘𝐻≤12𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝑅^𝑘_𝐻→𝐻×𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜇_𝐻+12𝐴^1/2𝑅_2𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝐴^1/2𝑅_2𝐻→𝐻𝑅^𝑘_𝐻→𝐻×‖𝜔‖_𝐻+12𝜏𝐴^1/2𝑅_3𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝜏𝐴^1/2𝑅_3𝐻→𝐻𝑅^𝑘_𝐻→𝐻×𝜏𝑓_1,1𝐻+12𝜏𝐴^1/2𝑅_4𝐻→𝐻^𝑘−1_𝑠=1𝑅^𝑘−𝑠_𝐻→𝐻+𝑅^𝑘−𝑠_𝐻→𝐻×𝑓_𝑠𝐻𝜏≤𝑀^𝑁−1_𝑠=1𝑓_𝑠𝐻𝜏+𝐴^1/2𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝜓_𝐻+𝜏𝑓_1,1𝐻 (33) for 𝑘≥2. Now, applying Abel’s formula to (19), we have 𝑢_𝑘=12𝑅₁𝑅^𝑘−𝑅₁𝑅^𝑘1𝜇+2𝑅^𝑘−𝑅^𝑘𝑅₂𝜔+12𝑅^𝑘−𝑅^𝑘𝑅₃𝜏²𝑓_1,1+𝜏²𝑅₄12^𝑘−1_𝑠=2𝑅₆𝑅^𝑘−𝑠−𝑅₆𝑅^𝑘−𝑠𝑓_𝑠−𝑓_𝑠−1+𝑅₆−𝑅₆𝑓_𝑘−1−𝑅₆𝑅^𝑘−1−𝑅₆𝑅^𝑘−1𝑓₁,2≤𝑘≤𝑁. (34) Applying 𝐴 to formula (34) and using estimates (11) and (12) and the triangle inequality, we obtain 𝐴𝑢_𝑘𝐻≤12𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝑅₁₀𝐼+𝑖𝜏𝐴^1/2−1_𝐻→𝐻𝑅^𝑘_𝐻→𝐻×𝐴𝐼+𝑖𝜏𝐴^1/2𝜇_𝐻+12𝐴^1/2𝑅_2𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝐴^1/2𝑅_2𝐻→𝐻𝑅^𝑘_𝐻→𝐻×𝐴^1/2𝜔_𝐻+12𝜏𝐴^−1/2𝑅_3𝐻→𝐻𝑅^𝑘_𝐻→𝐻+𝜏𝐴^−1/2𝑅_3𝐻→𝐻𝑅^𝑘_𝐻→𝐻×𝜏𝐴^1/2𝑓_1,1𝐻+12𝜏𝐴^1/2𝑅_4𝐻→𝐻×^𝑘−1_𝑠=2𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑘−𝑠_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑘−𝑠_𝐻→𝐻×𝑓_𝑠−𝑓_𝑠−1𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑓_𝑘−1𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑘−1_𝐻→𝐻+𝜏𝐴^1/2𝑅_6𝐻→𝐻𝑅^𝑘−1_𝐻→𝐻𝑓_1𝐻≤𝑀^𝑁−1_𝑠=2𝑓_𝑠−𝑓_𝑠−1𝐻+𝑓_1𝐻+𝐴𝐼+𝑖𝜏𝐴^1/2𝜑_𝐻+𝐴^1/2𝜓_𝐻+𝜏𝐴^1/2𝑓_1,1𝐻 (35) for 𝑘≥2. Theorem 4 is proved.

Note that the stability estimates obtained previously permit us to get the convergence estimate of difference scheme (2) under the smoothness property of solution (1). Actually, under the condition 𝑢(𝑡)∈𝐶([0,1],𝐻), we can obtain the third order of accuracy for the error of difference scheme (2). Since 𝑢⁽⁶⁾(𝑡)=−𝐴³𝑢(𝑡)+𝐴²𝑓(𝑡)−𝐴𝑓^′′(𝑡)+𝑓⁽⁴⁾(𝑡), this condition is satisfied under the given data 𝜑∈𝐷(𝐴³), 𝜓∈𝐷(𝐴^5/2), 𝑓^′(𝑡)∈𝐷(𝐴²), and 𝑓(0)∈𝐷(𝐴³).

Now, let us give application of this abstract result for nonlocal boundary value problem 𝑢_𝑡𝑡−𝑎(𝑥)𝑢_𝑥𝑥()𝑢()()(𝑥)𝑢+𝛿𝑢=𝑓𝑡,𝑥,0<𝑡<1,0<𝑥<1,0,𝑥=𝛼𝑢1,𝑥+𝜑,0≤𝑥≤1,_𝑡(0,𝑥)=𝛽𝑢_𝑡()(𝑥)𝑢()()1,𝑥+𝜓,0≤𝑥≤1,𝑡,0=𝑢𝑡,1,𝑢_𝑥(𝑡,0)=𝑢_𝑥(𝑡,1),0≤𝑡≤1 (36) for hyperbolic equation. Problem (36) has a unique smooth solution 𝑢(𝑡,𝑥), 𝛿>0and the smooth functions 𝑎(𝑥)≥𝑎>0(𝑎(0)=𝑎(1),𝑥∈(0,1)), 𝜑(𝑥),𝜓(𝑥)(𝑥∈[0,1]), and 𝑓(𝑡,𝑥)(𝑡,𝑥∈[0,1]). This allows us to reduce mixed problem (36) to nonlocal boundary value problem (1) in a Hilbert space 𝐻=𝐿₂[0,1] with a self-adjoint positive definite operator 𝐴^𝑥 defined by (36).

The discretization of problem (36) is carried out in two steps. In the first step, let us define the grid space [0,1]_ℎ=𝑥∶𝑥_𝑟=𝑟ℎ,0≤𝑟≤𝐾,𝐾ℎ=1. (37) We introduce Hilbert space 𝐿_2ℎ=𝐿₂([0,1]_ℎ), 𝑊¹_2ℎ=𝑊¹_2ℎ([0,1]_ℎ), and 𝑊²_2ℎ=𝑊²_2ℎ([0,1]_ℎ) of the grid functions 𝜑^ℎ(𝑥)={𝜑_𝑟}₁^𝐾−1 defined on [0,1]_ℎ, and we assign the difference operator 𝐴^𝑥_ℎ by the formula 𝐴^𝑥_ℎ𝜑^ℎ(𝑥)=−(𝑎(𝑥)𝜑_𝑥)_𝑥,𝑟+𝛿𝜑_𝑟1^𝐾−1, (38) acting in the space of grid functions 𝜑^ℎ(𝑥)={𝜑_𝑟}^𝐾₀ satisfying the conditions 𝜑₀=𝜑_𝐾, 𝜑₁−𝜑₀=𝜑_𝐾−𝜑_𝐾−1.

With the help of 𝐴^𝑥_ℎ, we arrive at the nonlocal boundary value problem 𝑑²𝑣^ℎ(𝑡,𝑥)𝑑𝑡²+𝐴^𝑥_ℎ𝑣^ℎ(𝑡,𝑥)=𝑓^ℎ(),[]𝑡,𝑥0<𝑡<1,𝑥∈0,1_ℎ,𝑣^ℎ(0,𝑥)=𝛼𝑣^ℎ(1,𝑥)+𝜑^ℎ(𝑥),𝑥∈[0,1]_ℎ,𝑣^ℎ_𝑡(0,𝑥)=𝛽𝑣^ℎ_𝑡(1,𝑥)+𝜓^ℎ(𝑥),𝑥∈[0,1]_ℎ (2) for a system of ordinary differential equations.

In the second step, we replace problem (2) with difference scheme (40) 𝜏⁻²𝑢^ℎ_𝑘+1(𝑥)−2𝑢^ℎ_𝑘(𝑥)+𝑢^ℎ_𝑘−1(𝑥+2)3𝐴^𝑥_ℎ𝑢^ℎ_𝑘(𝑥)+16𝐴^𝑥_ℎ𝑢^ℎ_𝑘+1(𝑥)+𝑢^ℎ_𝑘−1(𝑥)+1𝜏12²𝐴^𝑥_ℎ²𝑢^ℎ_𝑘+1(𝑥)=𝑓^ℎ_𝑘(𝑥),𝑓^ℎ_𝑘(𝑥)=23𝑓^ℎ𝑡_𝑘+1,𝑥6𝑓^ℎ𝑡_𝑘+1,𝑥+𝑓^ℎ𝑡_𝑘−1−1,𝑥𝜏12²−𝐴𝑓^ℎ𝑡_𝑘+1,𝑥+𝑓^ℎ_𝑡𝑡𝑡_𝑘+1,𝑥,𝑥∈[0,1]_ℎ,𝑡_𝑘𝑢=𝑘𝜏,𝑁𝜏=1,1≤𝑘≤𝑁−1,_ℎ0(𝑥)=𝛼𝑢^ℎ_𝑁(𝑥)+𝜑^ℎ(𝑥),𝑥∈[0,1]_ℎ,𝜏𝐼+²𝐴12^𝑥ℎ+𝜏⁴𝐴144^𝑥ℎ2𝜏⁻¹𝑢^ℎ₁(𝑥)−𝑢^ℎ₀(𝑥)+𝜏2𝐴^𝑥_ℎ𝜑^ℎ(𝑥)−𝜏𝑓^ℎ_1,1(𝑥)𝜏=𝛽𝐼−²𝐴12^𝑥ℎ×16𝜏7𝑢^ℎ_𝑁(𝑥)−8𝑢^ℎ_𝑁−1(𝑥)+𝑢^ℎ_𝑁−2(𝑥+𝜏)3𝑓^ℎ_𝑁(𝑥)−𝐴𝑢^ℎ_𝑁(𝑥+𝜏)𝐼−²𝐴12^𝑥ℎ𝜓^ℎ(𝑥),𝑥∈[0,1]_ℎ,𝑓^ℎ_1,1(𝑥)=12𝑓^ℎ()+𝜏0,𝑥6𝑓^ℎ_𝑡(0,𝑥). (40)

Theorem 5. Let 𝜏 and ℎ be sufficiently small numbers. Then, the solution of difference scheme (40) satisfies the following stability estimates: max_{0≤𝑘≤𝑁}𝑢^ℎ_𝑘𝐿2ℎ+max_{0≤𝑘≤𝑁}𝑢^ℎ_{𝑘𝑊12ℎ}≤𝑀₁max_{1≤𝑘≤𝑁−1}𝑓^ℎ_𝑘𝐿2ℎ+𝜓^ℎ_𝐿2ℎ+𝜑^ℎ_𝑊12ℎ+𝜏𝜑^ℎ_𝑊22ℎ+𝜏𝑓^ℎ_1,1𝐿2ℎ,max_{1≤𝑘≤𝑁−1}𝜏⁻²𝑢^ℎ_𝑘+1−2𝑢^ℎ_𝑘+𝑢^ℎ_{𝑘−1𝐿2ℎ}+max_{0≤𝑘≤𝑁}𝑢^ℎ_{𝑘𝑊22ℎ}≤𝑀₁𝑓^ℎ_1𝐿2ℎ+max_{2≤𝑘≤𝑁−1}𝜏⁻¹𝑓^ℎ_𝑘−𝑓^ℎ_{𝑘−1𝐿2ℎ}+𝜓^ℎ_𝑊12ℎ+𝜑^ℎ_𝑊22ℎ+𝜏𝜑^ℎ_𝑊32ℎ+𝜏𝑓^ℎ_1,1𝑊12ℎ. (41) Here, 𝑀₁ does not depend on 𝜏,ℎ,𝜑^ℎ(𝑥),𝜓^ℎ(𝑥),𝑓^ℎ_1,1(𝑥), and 𝑓^ℎ_𝑘(𝑥),1≤𝑘<𝑁.

The proof of Theorem 5 is based on the proof of abstract Theorem 4 and the symmetry property of operator 𝐴^𝑥_ℎ defined by (38).

Acknowledgments

The authors would like to thank Professor Pavel E. Sobolevskii and reviewers for their helpful comments.