A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations

Abstract

The nonlocal boundary value problem 𝑑²𝑢(𝑡)/𝑑𝑡²+𝐴𝑢(𝑡)=𝑓(𝑡)(0≤𝑡≤1), 𝑖(𝑑𝑢(𝑡)/𝑑𝑡)+𝐴𝑢(𝑡)=𝑔(𝑡)(−1≤𝑡≤0), 𝑢(0⁺)=𝑢(0⁻), 𝑢_𝑡(0⁺)=𝑢_𝑡(0⁻), 𝐴𝑢(−1)=𝛼𝑢(𝜇)+𝜑, 0<𝜇≤1, for hyperbolic Schrödinger equations in a Hilbert space 𝐻 with the self-adjoint positive definite operator 𝐴 is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for solutions of the mixed-type boundary value problems for hyperbolic Schrödinger equations are obtained.

1. Introduction

Methods of solutions of nonlocal boundary value problems for partial differential equations and partial differential equations of mixed type have been studied extensively by many researches (see, e.g., [1–12] and the references given therein).

In the present paper, the nonlocal boundary value problem𝑑²𝑢(𝑡)𝑑𝑡²(𝑡)(𝑡),𝑖(𝑡)+𝐴𝑢=𝑓)(0≤𝑡≤1𝑑𝑢(𝑡)(𝑡),𝑢0𝑑𝑡+𝐴𝑢=𝑔)(−1≤𝑡≤0₊=𝑢(0⁻),𝑢_𝑡0⁺=𝑢_𝑡(0⁻),𝐴𝑢(−1)=𝛼𝑢𝜇+𝜑,0<𝜇≤1 (1.1) for differential equations of hyperbolic Schrödinger type in a Hilbert space 𝐻 with self-adjoint positive definite operator 𝐴 is considered.

It is known that various nonlocal boundary value problems for the hyperbolic Schrödinger equations can be reduced to problem (1.1).

A function 𝑢(𝑡) is called a solution of the problem (1.1) if the following conditions are satisfied.(i)𝑢(𝑡) is twice continuously differentiable on the interval (0,1] and continuously differentiable on the segment [−1,1]. 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∈[−1,1], and the function 𝐴𝑢(𝑡) is continuous on the segment [−1,1].(iii)𝑢(𝑡) satisfies the equations and nonlocal boundary condition (1.1).

In the present paper, the stability estimates for the solution of the problem (1.1) for the hyperbolic Schrödinger equation are established. In applications, the stability estimates for the solutions of the mixed-type boundary value problems for hyperbolic Schrödinger equations are obtained.

Finally note that hyperbolic Schrödinger equations play important role in physics and engineering (see, e.g., [13–16] and the references given therein).

Furthermore, the investigation of the numerical solution of initial value problems and Schrödinger equations is the subject of extensive research activity during the last decade (indicatively [17–25] and the references given therein).

2. The Main Theorem

Let 𝐻 be a Hilbert space, and let 𝐴 be a positive definite self-adjoint operator with 𝐴≥𝛿𝐼, where 𝛿>𝛿₀>0. Throughout this paper, {𝑐(𝑡),𝑡≥0} is a strongly continuous cosine operator function defined by 𝑐(𝑡)=𝑒^{𝑖𝑡𝐴1/2}+𝑒^{−𝑖𝑡𝐴1/2}2. (2.1)

Then, from the definition of the sine operator function 𝑠(𝑡)𝑠(𝑡)𝑢=^𝑡₀𝑐(𝑠)𝑢𝑑𝑠, (2.2)

it follows that𝑠(𝑡)=𝐴^−1/2𝑒^{𝑖𝑡𝐴1/2}−𝑒^{−𝑖𝑡𝐴1/2}2𝑖. (2.3)

For the theory of cosine operator function, we refer to Fattorini [26] and Piskarev and Shaw [27].

We begin with two lemmas that will be needed as follows.

Lemma 2.1. The following estimates hold: ‖𝑐(𝑡)‖_𝐻→𝐻≤1,𝐴^1/2𝑠(𝑡)_𝐻→𝐻𝑒≤1,𝑡≥0,_{±𝑖𝑡𝐴𝐻→𝐻}≤1,𝑡≥0. (2.4)

Lemma 2.2. Let |𝛼|<𝛿√.1+𝛿 (2.5) Then, the operator 𝐼−𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴 (2.6) has an inverse 𝑇=𝐼−𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴−1, (2.7) and the estimate ‖𝑇‖_𝐻→𝐻≤𝑀 (2.8) holds, where 𝑀 does not depend on 𝛼 and 𝜇.

Proof. Actually, the proof of estimate (2.8) is based on the following estimate: −𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴_𝐻→𝐻<1. (2.9) Using the definitions of cosine and sine operator functions, 𝐴≥𝛿𝐼, 𝛿>0 (positivity), and 𝐴=𝐴^∗ (self-adjointness property), we obtain −𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴_𝐻→𝐻≤sup_{𝛿≤𝜌<∞}1−𝛼𝜌cos√+𝑖𝜌𝜇√𝜌sin√𝜌𝜇𝑒^𝑖𝜌≤sup_{𝛿≤𝜌<∞}|𝛼1|𝜌cos√+𝑖𝜌𝜇√𝜌sin√𝜌𝜇𝑒^𝑖𝜌≤|𝛼|sup_{𝛿≤𝜌<∞}1𝜌²cos²√+1𝜌𝜇𝜌sin²√≤|𝛼|𝜌𝜇1+𝜌𝜌. (2.10) Since 1+𝜌𝜌≤√1+𝛿𝛿, (2.11) we have that −𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴_𝐻→𝐻<𝛿√⋅√1+𝛿1+𝛿𝛿=1. (2.12) Hence, Lemma 2.2 is proved.

Now, we will obtain the formula for solution of problem (1.1). It is known that for smooth data of initial value problems𝑑²𝑢(𝑡)𝑑𝑡²(𝑡)(𝑡),𝑢(0)+𝐴𝑢=𝑓)(0≤𝑡≤1=𝑢₀,𝑢^′(0)=𝑢^′₀,𝑖𝑑𝑢(𝑡)(𝑡)(𝑡),𝑢()𝑑𝑡+𝐴𝑢=𝑔)(−1≤𝑡≤0−1=𝑢₋₁, (2.13)

there are unique solutions of problems (2.13), and following formulas hold:𝑢(𝑡)=𝑐(𝑡)𝑢(0)+𝑠(𝑡)𝑢^′(0)+^𝑡₀𝑠𝑡−𝑦𝑓𝑦𝑑𝑦,0≤𝑡≤1, (2.14)𝑢(𝑡)=𝑒^{𝑖(𝑡+1)𝐴}𝑢₋₁−𝑖^𝑡₋₁𝑒^{𝑖(𝑡−𝑦)𝐴}𝑔𝑦𝑑𝑦,−1≤𝑡≤0. (2.15) Using (2.14), (2.15), and (1.1), we can write𝑢(𝑡)=[𝑐(𝑡)+𝑖𝐴𝑠(𝑡)]𝑒^𝑖𝐴𝑢₋₁−𝑖⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔𝑦(𝑡)𝑔(0)+𝑑𝑦−𝑖𝑠^𝑡₀𝑠𝑡−𝑦𝑓𝑦𝑑𝑦. (2.16)

Now, using the nonlocal boundary condition𝐴𝑢(−1)=𝛼𝑢𝜇+𝜑, (2.17)

we obtain the operator equation:𝐼−𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴𝑢₋₁=𝛼−𝑖𝐴⁻¹𝑐𝜇⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔𝑦𝜇𝑑𝑦−𝑠𝑖𝐴⁻¹𝑔(0)−⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔𝑦𝑑𝑦+𝐴^−1𝜇₀𝑠𝜇−𝑦𝑓𝑦𝑑𝑦+𝐴⁻¹𝜑. (2.18)

Since the operator𝐼−𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴 (2.19)

has an inverse𝑇=𝐼−𝛼𝐴⁻¹𝑐𝜇+𝑖𝑠𝜇𝑒^𝑖𝐴−1, (2.20)

for the solution of the operator equation (2.18), we have the formula𝑢₋₁=𝑇𝛼−𝑖𝐴⁻¹𝑐𝜇⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔𝑦𝜇𝑑𝑦−𝑠𝑖𝐴⁻¹𝑔(0)−⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔𝑦𝑑𝑦+𝐴^−1𝜇₀𝑠𝜇−𝑦𝑓𝑦𝑑𝑦+𝐴⁻¹𝜑 (2.21)

Thus, for the solution of the nonlocal boundary value problem (1.1) we obtain (2.15), (2.16), and (2.21).

Theorem 2.3. Suppose that 𝜑∈𝐷(𝐴^1/2),𝑓(0)∈𝐷(𝐴^1/2), and 𝑔(0)∈𝐷(𝐴^1/2). Let 𝑓(𝑡) be continuously differentiable on [0,1] and let 𝑔(𝑡) be twice continuously differentiable on [−1,0] functions. Then, there is a unique solution of the problem (1.1) and the following stability inequalities max_{−1≤𝑡≤1}‖𝑢(𝑡)‖_𝐻≤𝑀𝐴^−1/2𝜑_𝐻+𝐴^−1/2𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴⁻¹𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝐴^−1/2𝑓(𝑡)_𝐻,max_{−1≤𝑡≤1}𝑑𝑢(𝑡)𝑑𝑡_𝐻+max_{−1≤𝑡≤1}𝐴^1/2𝑢(𝑡)_𝐻≤𝑀𝜑_𝐻+𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴^−1/2𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝑓(𝑡)_𝐻,max_{−1≤𝑡≤0}𝑑𝑢(𝑡)𝑑𝑡_𝐻+max_0≤𝑡≤1𝑑²𝑢(𝑡)𝑑𝑡²_𝐻+max_{−1≤𝑡≤1}‖𝐴𝑢(𝑡)‖_𝐻≤𝑀𝐴^1/2𝑓^′(𝑡)𝐴^1/2𝜑_𝐻+𝐴^1/2𝑔(0)_𝐻+𝑔^′(0)_𝐻+max_{−1≤𝑡≤0}𝑔^′′(𝑡)_𝐻+𝐴^1/2𝑓(0)_𝐻+max_0≤𝑡≤1𝐴^1/2𝑓^′(𝑡)_𝐻 (2.22)(2.23)(2.24) hold, where 𝑀 is independent of 𝑓(𝑡),𝑡∈[0,1],𝑔(𝑡),𝑡∈[−1,0], and 𝜑.

Note that there are three inequalities in Theorem 2.3 on the stability of solution, stability of first derivative of solution and stability of second derivative of solution. That means the solution of problem (1.1) 𝑢(𝑡) and its first and second derivatives are continuously dependent on 𝑓(𝑡),𝑔(𝑡) and 𝜑.

Proof. First, estimate (2.22) will be obtained. Using formula (2.21) and integration by parts, we obtain 𝑢₋₁=𝑇𝛼−𝐴⁻²𝑐𝜇𝑔(0)−𝑒^𝑖𝐴𝑔(−1)−⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔^′𝑦𝑑𝑦+𝑖𝐴⁻¹𝑠𝜇𝑒^𝑖𝐴𝑔(−1)+⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔^′𝑦𝑑𝑦+𝐴^−1𝜇₀𝑠𝜇−𝑦𝑓𝑦𝑑𝑦+𝐴⁻¹𝜑. (2.25) Using estimates (2.4), and (2.8), we get 𝑢_−1𝐻≤𝑀𝐴^−1/2𝜑_𝐻+𝐴⁻¹𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴⁻¹𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝐴⁻¹𝑓(𝑡)_𝐻. (2.26) Applying 𝐴^1/2 to the formula (2.25) and using estimates (2.4) and (2.8), we can write 𝐴^1/2𝑢_−1𝐻≤𝑀𝜑_𝐻+𝐴^−1/2𝑔(0)_𝐻+𝐴^−1/2𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝐴^−1/2𝑓(𝑡)_𝐻. (2.27) Using formulas (2.15) and (2.16) and integration by parts, we obtain 𝑢(𝑡)=𝑒^{𝑖(𝑡+1)𝐴}𝑢₋₁+𝐴⁻¹𝑔(𝑡)−𝑒^{𝑖(𝑡+1)𝐴}𝑔(−1)−^𝑡₋₁𝑒^{𝑖(𝑡−𝑦)𝐴}𝑔^′𝑦𝑢(𝑡)=[𝑐(𝑡)(𝑡𝑒𝑑𝑦,−1≤𝑡≤0,+𝑖𝐴𝑠)]^𝑖𝐴𝑢₋₁+𝐴⁻¹𝑔(0)−𝑒^𝑖𝐴𝑔(−1)−⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔^′𝑦(𝑡)𝑔(0)+𝑑𝑦−𝑖𝑠^𝑡₀𝑠𝑡−𝑦𝑓𝑦𝑑𝑦,0≤𝑡≤1. (2.28) Using estimates (2.4) we get ‖𝑢(𝑡)‖_𝐻≤𝑀𝑢_−1𝐻+𝐴⁻¹𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴⁻¹𝑔^′(𝑡)_𝐻‖𝑢(𝑡,−1≤𝑡≤0,)‖_𝐻≤𝑀𝐴^1/2𝑢_−1𝐻+𝐴^−1/2𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴⁻¹𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝐴^−1/2𝑓(𝑡)_𝐻,1≤𝑡≤1. (2.29) Then, from estimates (2.26), (2.27), and (2.29) it follows (2.22).
Second, (2.23) will be obtained. Applying 𝐴^1/2 to the formula (2.25) and using estimates (2.4),and (2.8), we obtain𝐴^1/2𝑢_−1𝐻≤𝑀𝜑_𝐻+𝐴^−1/2𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴^−1/2𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝐴^−1/2𝑓(𝑡)_𝐻. (2.30) Applying 𝐴 to the formula (2.25) and using estimates (2.4), (2.8), we get 𝐴𝑢_−1𝐻≤𝑀𝐴^1/2𝜑_𝐻+𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝑓(𝑡)_𝐻. (2.31) Applying 𝐴^1/2 to the formulas (2.28), and using estimates (2.4) we can write 𝐴^1/2𝑢(𝑡)_𝐻≤𝑀𝐴^1/2𝑢_−1𝐻+𝐴^−1/2𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴^−1/2𝑔^′(𝑡)_𝐻𝐴,−1≤𝑡≤0,_1/2𝑢(𝑡)_𝐻≤𝑀𝐴𝑢_−1𝐻+𝑔(0)_𝐻+max_{−1≤𝑡≤0}𝐴^−1/2𝑔^′(𝑡)_𝐻+max_0≤𝑡≤1𝑓(𝑡)_𝐻,0≤𝑡≤1. (2.32) Combining estimates (2.30), (2.31),and (2.32), we get estimate (2.23).
Third, estimate (2.24) will be obtained. Using formula (2.25) and integration by parts, we obtain𝑢₋₁=𝛼−𝐴⁻²𝑐𝜇𝑔(0)−𝑒^−𝑖𝐴𝑔(−1)−𝑖𝐴⁻¹×𝑔^′(0)−𝑒^𝑖𝐴𝑔^′(−1)−⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔^′′𝑦𝑑𝑦+𝑖𝐴⁻¹𝑠𝜇×𝑒^𝑖𝐴𝑔(−1)+𝑖𝐴⁻¹𝑔^′(0)−𝑒^𝑖𝐴𝑔^′(−1)−⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔^′′𝑦𝑑𝑦−𝐴⁻²𝑓𝜇+𝑐𝜇𝑓(0)−^𝜇₀𝑐𝜇−𝑦𝑓^′𝑦𝑑𝑦+𝐴⁻¹𝜑 (2.33) Applying 𝐴 to formula (2.33) and using estimates (2.4) and (2.8), we get 𝐴𝑢_−1𝐻≤𝑀𝐴^1/2𝜑_𝐻+𝑔(0)_𝐻+𝐴^−1/2𝑔^′(0)_𝐻+max_{−1≤𝑡≤0}𝐴^−1/2𝑔^′′(𝑡)_𝐻+𝑓(0)_𝐻+max_0≤𝑡≤1𝐴^−1/2𝑓^′(𝑡)_𝐻. (2.34) Applying 𝐴^3/2 to formula (2.33) and using estimates (2.4), and (2.8) we can write 𝐴^3/2𝑢_−1𝐻≤𝑀𝐴𝜑_𝐻+𝐴^1/2𝑔(0)_𝐻+𝑔^′(0)_𝐻+max_{−1≤𝑡≤0}𝑔^′′(𝑡)_𝐻+𝐴^1/2𝑓(0)_𝐻+max_0≤𝑡≤1𝑓^′(𝑡)_𝐻. (2.35) Using formulas (2.28), and integration by parts, we obtain 𝑢(𝑡)=𝑒^{𝑖(𝑡+1)𝐴}𝑢₋₁+𝐴⁻¹𝑔(𝑡)−𝑒^{𝑖(𝑡+1)𝐴}𝑔(−1)−𝑖𝐴⁻¹𝑔^′(𝑡)−𝑒^{𝑖(𝑡+1)𝐴}𝑔^′(−1)−^𝑡₋₁𝑒^{𝑖(𝑡−𝑦)𝐴}𝑔^′′𝑦𝑢(𝑡)=𝑐(𝑡)(𝑡𝑒𝑑𝑦,−1≤𝑡≤0,+𝑖𝐴𝑠)^𝑖𝐴𝑢₋₁+𝐴⁻¹𝑔(0)−𝑒^𝑖𝐴𝑔(−1)−𝑖𝐴⁻¹𝑔^′(0)−𝑒^𝑖𝐴𝑔^′(−1)−⁰₋₁𝑒^{−𝑖𝐴𝑦}𝑔^′′𝑦(𝑡)𝑔(0)𝑑𝑦−𝑖𝑠−𝐴⁻¹𝑓(𝑡)−𝑐(𝑡)𝑓(0)−^𝑡₀𝑐𝑡−𝑦𝑓^′𝑦𝑑𝑦,0≤𝑡≤1. (2.36) Applying 𝐴 to the formulas (2.36), and using estimates (2.4), we get ‖𝐴𝑢(𝑡)‖_𝐻≤𝑀𝐴𝑢_−1𝐻+𝐴^1/2𝑔(0)_𝐻+𝑔^′(0)_𝐻+max_{−1≤𝑡≤0}𝑔^′′(𝑡)_𝐻‖(𝑡,−1≤𝑡≤0,𝐴𝑢)‖_𝐻≤𝑀𝐴^3/2𝑢_−1𝐻+𝐴^1/2𝑔(0)_𝐻+𝑔^′(0)_𝐻+max_{−1≤𝑡≤0}𝑔^′′(𝑡)_𝐻+𝐴^1/2𝑓(0)_𝐻+max_0≤𝑡≤1𝐴^1/2𝑓^′(𝑡)_𝐻,0≤𝑡≤1. (2.37) From (2.34) and (2.35) and estimates (2.37) it follows (2.24). This completes the proof of Theorem 2.3.

Remark 2.4. We can obtain the same stability results for the solution of the following multipoint nonlocal boundary value problem: 𝑑²𝑢(𝑡)𝑑𝑡²(𝑡)(𝑡),𝑖𝑑(𝑡)+𝐴𝑢=𝑓)(0≤𝑡≤1(𝑡)(𝑡),()=𝑑𝑡+𝐴𝑢=𝑔)(−1≤𝑡≤0𝐴𝑢−1^𝑁_𝑗=1𝛼_𝑗𝑢𝜇_𝑗+𝜑,0<𝜇_𝑗≤1,1≤𝑗≤𝑁, (2.38) for differential equations of mixed type in a Hilbert space 𝐻 with self-adjoint positive definite operator 𝐴.

3. Applications

Initially, the mixed problem for the hyperbolic Schrödinger equation𝑣_𝑦𝑦−𝑎(𝑥)𝑣_𝑥𝑥+𝛿𝑣=𝑓𝑦,𝑥,0<𝑦<1,0<𝑥<1,𝑖𝑣_𝑦−𝑎(𝑥)𝑣_𝑥𝑥−𝑎(𝑥)𝑣+𝛿𝑣=𝑔𝑦,𝑥,−1<𝑦<0,0<𝑥<1,_𝑥(−1,𝑥)_𝑥()()(𝑥)𝑣+𝛿𝑣−1,𝑥=𝛼𝑣1,𝑥+𝜑,0≤𝑥≤1,𝑦,0=𝑣𝑦,1,𝑣_𝑥𝑦,0=𝑣_𝑥𝑣0𝑦,1,−1≤𝑦≤1,₊,𝑥=𝑣(0⁻,𝑥),𝑣_𝑦0⁺,𝑥=𝑣_𝑦(0⁻)|𝛼|<𝛿,𝑥,0≤𝑥≤1,√1+𝛿 (3.1) is considered, where 𝛿=const>0. The problem (3.1) has a unique smooth solution 𝑣(𝑦,𝑥) for smooth 𝑎(𝑥)≥𝑎>0(𝑥∈(0,1)), 𝜑(𝑥)(𝑥∈[0,1]), 𝑓(𝑦,𝑥)(𝑦∈[0,1],𝑥∈[0,1]), and 𝑔(𝑦,𝑥)(𝑦∈[−1,0],𝑥∈[0,1]) functions.

We introduce the Hilbert space 𝐿₂[0,1] of all the square integrable functions defined on [0,1] and Hilbert spaces 𝑊¹₂[0,1] and 𝑊²₂[0,1] equipped with norms𝜑_𝑊12[0,1]=¹₀𝜑(𝑥)²𝑑𝑥^1/2+¹₀𝜑_𝑥(𝑥)²𝑑𝑥^1/2,𝜑_𝑊22[0,1]=¹₀𝜑(𝑥)²𝑑𝑥^1/2+¹₀𝜑_𝑥(𝑥)²𝑑𝑥^1/2+¹₀𝜑_𝑥𝑥(𝑥)²𝑑𝑥^1/2, (3.2)

respectively. This allows us to reduce the mixed problem (3.1) to the nonlocal boundary value problem (1.1) in Hilbert space 𝐻 with a self-adjoint positive definite operator 𝐴 defined by problem (3.1).

Theorem 3.1. The solutions of the nonlocal boundary value problem (3.1) satisfy the following stability estimates: max_{−1≤𝑦≤1}𝑣_𝑦𝑦,⋅_𝐿2[0,1]+max_{−1≤𝑦≤1}𝑣𝑦,⋅_𝑊12[0,1]≤𝑀𝜑_𝐿2[0,1]+𝑔(0,⋅)_𝐿2[0,1]+max_{−1≤𝑦≤0}𝑔_𝑦𝑦,⋅_𝐿2[0,1]+max_0≤𝑦≤1𝑓𝑦,⋅_𝐿2[0,1],max_{−1≤𝑦≤1}𝑣𝑦,⋅_𝑊22[0,1]+max_{−1≤𝑦≤0}𝑣_𝑦𝑦,⋅_𝐿2[0,1]+max_0≤y≤1𝑣_𝑦𝑦𝑦,⋅_𝐿2[0,1]≤𝑀𝜑_𝑊12[0,1]+𝑔(0,⋅)_𝐿2[0,1]+𝑔_𝑦(0,⋅)_𝐿2[0,1]+max_{−1≤𝑦≤0}𝑔_𝑦𝑦𝑦,⋅_𝐿2[0,1]+𝑓(0,⋅)_𝑊12[0,1]+max_0≤𝑦≤1𝑓_𝑦𝑦,⋅_𝑊12[0,1], (3.3) where 𝑀 does not depend on not only 𝑓(𝑦,𝑥)(𝑦∈[0,1],𝑥∈[0,1]) and 𝑔(𝑦,𝑥)(𝑦∈[−1,0],𝑥∈[0,1]) but also 𝜑(𝑥)(𝑥∈[0,1]).

The proof of Theorem 3.1 is based on the abstract Theorem 2.3 and symmetry properties of the space operator defined by problem (3.1).

Next, we consider the mixed nonlocal boundary value problem for the multidimensional hyperbolic Schrödinger equation:𝑣_𝑦𝑦−^𝑚_𝑟=1𝑎_𝑟(𝑥)𝑣_{𝑥𝑟𝑥𝑟}𝑥=𝑓𝑦,𝑥,0≤𝑦≤1,𝑥=₁,…,𝑥_𝑚∈Ω,𝑖𝑣_𝑦−^𝑚_𝑟=1𝑎_𝑟(𝑥)𝑣_{𝑥𝑟𝑥𝑟}𝑥=𝑔𝑦,𝑥,−1≤𝑦≤0,𝑥=₁,…,𝑥_𝑚−∈Ω,_𝑛𝑟=1𝑎_𝑟(𝑥)𝑣_𝑥𝑟(−1,𝑥)_𝑥𝑟=𝑣(1,𝑥)+𝜑(𝑥),𝑥∈𝑢Ω,𝑦,𝑥=0,𝑥∈𝑆,−1≤𝑦≤1, (3.4) where Ω is the unit open cube in the 𝑚-dimensional Euclidean space ℝ^𝑚:𝑥∶𝑥=𝑥₁,…,𝑥_𝑚,0<𝑥_𝑘<1,1≤𝑘≤𝑚 (3.5)

with boundary 𝑆 and Ω=Ω∪𝑆. Here, 𝑎_𝑟(𝑥)(𝑥∈Ω),𝜑(𝑥)(𝑥∈Ω), and 𝑓(𝑦,𝑥)(𝑦∈(0,1),𝑥∈Ω),𝑔(𝑦,𝑥)(𝑦∈(−1,0),𝑥∈Ω) are given smooth functions in [0,1]×Ω and 𝑎_𝑟(𝑥)≥𝑎>0.

We introduce the Hilbert space 𝐿₂(Ω) of all square integrable functions defined on Ω, equipped with the norm 𝑓_𝐿2(Ω)=⋯_𝑥∈Ω𝑓(𝑥)²𝑑𝑥₁⋯𝑑𝑥_𝑛^1/2 (3.6)

and Hilbert spaces 𝑊¹₂(Ω) and 𝑊²₂(Ω) defined on Ω, equipped with norms 𝜑_𝑊12(Ω)=𝜑_𝐿2(Ω)+⋯_𝑥∈Ω^𝑛_𝑟=1|𝜑_𝑥𝑟|²𝑑𝑥₁⋯𝑑𝑥_𝑛^1/2,𝜑_𝑊22(Ω)=𝜑_𝐿2(Ω)+⋯_𝑥∈Ω^𝑛_𝑟=1|𝜑_𝑥𝑟|²𝑑𝑥₁⋯𝑑𝑥_𝑛^1/2+⋯_𝑥∈Ω^𝑛_𝑟=1|𝜑_{𝑥𝑟𝑥𝑟}|²𝑑𝑥₁⋯𝑑𝑥_𝑛^1/2, (3.7)

respectively. The problem (3.4) has a unique smooth solution 𝑣(𝑦,𝑥) for smooth 𝑎_𝑟(𝑥),𝑓(𝑦,𝑥), and 𝑔(𝑦,𝑥) functions. This allows us to reduce the mixed problem (3.4) to the nonlocal boundary value problem (1.1) in Hilbert space 𝐻 with a self-adjoint positive definite operator 𝐴 defined by problem (3.4).

Theorem 3.2. The following stability inequalities for solutions of the nonlocal boundary value problem (3.4) max_{−1≤𝑦≤1}𝑣_𝑦𝑦,⋅_𝐿2(Ω)+max_{−1≤𝑦≤1}𝑣𝑦,⋅_𝑊12(Ω)≤𝑀𝑔(0,⋅)_𝐿2(Ω)+max_{−1≤𝑦≤0}𝑔_𝑦𝑦,⋅_𝐿2(Ω)+max_0≤𝑦≤1𝑓(𝑦,⋅)_𝐿2(Ω)+𝜑_𝐿2(Ω),max_{−1≤𝑦≤1}𝑣(𝑦,⋅)_𝑊22(Ω)+max_{−1≤𝑦≤0}𝑣_𝑦𝑦,⋅_𝐿2(Ω)+max_0≤𝑦≤1𝑣_𝑦𝑦(𝑦,⋅)_𝐿2(Ω)≤𝑀𝜑_𝑊12(Ω)+𝑔(0,⋅)_𝐿2(Ω)+𝑔_𝑦(0,⋅)_𝐿2(Ω)+max_{−1≤𝑦≤0}𝑔_𝑦𝑦𝑦,⋅_𝐿2(Ω)+𝑓(0,⋅)_𝑊12(Ω)+max_0≤𝑦≤1𝑓_𝑦𝑦,⋅_𝑊12(Ω) (3.8) hold. Here, 𝑀 is independent of 𝑓(𝑦,𝑥)(𝑦∈[0,1],𝑥∈[0,1]),𝑔(𝑦,𝑥)(𝑦∈[−1,0],𝑥∈[0,1]), and 𝜑(𝑥)(𝑥∈[0,1]).

The proof of Theorem 3.2 is based on the abstract Theorem 2.3, symmetry properties of the space operator defined by problem (3.4), and the following theorem on the coercivity inequality for the solution of the elliptic differential problem in 𝐿₂(Ω) in Sobolevskii [28].

Theorem 3.3. For the solutions of the elliptic differential problem −^𝑚_𝑟=1𝑎_𝑟(𝑥)𝑢_{𝑥𝑟𝑥𝑟}(𝑥)𝑢(𝑥)=𝜔,𝑥∈Ω,=0,𝑥∈𝑆, (3.9) the following coercivity inequality holds: ^𝑚_𝑟=1𝑢_{𝑥𝑟𝑥𝑟𝐿2(Ω)}≤𝑀‖𝜔‖_𝐿2(Ω). (3.10)

Acknowledgment

The authors would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey) for his helpful suggestions to the improvement of this paper.