EIGENVALUES AND EIGENFUNCTIONS FOR REGULAR STURM-LIOUVILLE EQUATION WITH NON-LOCAL BOUNDARY CONDITIONS

In this paper, we study the existence and some general properties of eigenvalues and eigenfunctions of a nonlocal boundary value problem of the Sturm-Liouville differential equation. AMS Subject Classification: 34A55, 34B10, 34B15, 34B18, 34L10, 34L40, 34K10


Introduction
Boundary value problems for various differential equations with nonlocal boundary conditions, were actively investigated during the last three decades.Re-search is motivated by both their interest to pure mathematics and new applications in physics, mechanics, biochemistry, ecology (see [1]- [5]and [8], [9]).
Here we study the existence and some general properties of the eigenvalues and eigenfunctions of the two non-local boundary value problems (1) and (4).Comparison with the local boundary value problem problem of equation (1) with the local boundary value problem y ′ (0) − Hy(0) = 0, y(π) = 0 will be given.

General Properties
Here we prove some results concerning the eigenvalues and eigenfunctions of the non-local problem (1)-( 4).
Lemma 1.The eigenvalues of the non-local boundary value problem (1) and (4) are real.
Proof.Let y 0 (x) be the eigenfunction that corresponds to the eigenvalue λ 0 of the problem (1) and ( 4), then and y 0 (η) − Hy 0 (η) = y 0 (π) = 0 (6) Multiplying both sides of (5) by ȳ0 and then integrating form 0 to ξ with respect to x, we have using the boundary condition ( 6), we have From which it follows the reality of λ 2 0 .
Lemma 2. The eigenfunctions that corresponds to two different eigenvalues of the non-local boundary value problem (1) and (4) are orthogonal.
Proof.Let λ 1 = λ 2 be two different eigenvalues of the non-local boundary value problem (1) and (4).Let y 1 (x), y 2 (x) be the corresponding eigenfunctions, then y 2 (η) − Hy 2 (η) = y 2 (π) = 0 (10) Multiplying both sides of (7) by ȳ2 and integrating with respect to x, we obtain By taking the complex conjugate of ( 9) and multiply it by y 1 and integrate the resulting expression with respect to x, we have Subtracting (11) from (12) and using the boundary conditions of ( 8) and (10) we obtain which completes the proof.

The Asymptotic Formulas for the Solution
Here we study the asymptotic formulas for the solutions of problem ( 1) and ( 4).Lemma 1.1 deals with the nature the eigenvalues.Let be φ(x, λ) the solution of equation ( 1) and ( 4) satisfying the initial conditions and by ϑ(x, λ) the solution of the same equation, satisfying the initial conditions We notes that φ(x, λ) and ϑ(x, λ) are linearly independent if and only if The solution Y (x, λ) as eigenfunction must satisfy the first condition (4), we have and then, After using the condition (13), ( 14), we get αφ(π, λ) = 0, where α = 0. therefore, The characteristic equation will be Lemma 3. The solution φ(x, λ) of problem ( 1) and ( 4) satisfy the integral equations Proof.First we obtain formula (16) Indeed,with solution of the form q(x) = 0. (1) becomes becomes −y ′′ = λ 2 y by means of variation of parameter method, we have and the direct calculation of C 1 (x, s) and C 2 (x, s), we have substituting from ( 18) into (17) equation ( 16) follows.Second we show that the integral representation (16) satisfies the problem (1) and ( 13).Let ϕ(x, λ) be the solution of (1), so that We multiply both sides by sin λ(x − τ ) λ and integrating with respect to τ from η to x we obtain Integrating by parts twice and using the condition (13), we have By substituting from (20) into (19) we get the required formula (16).
Theorem 5. Let λ = σ + it and suppose that q(x) has a second order piecewise differentiable derivatives on [0, π].Then the solution φ(x, λ) of nonlocal boundary value (1) and ( 4) have the following asymptotic formula where Proof.By substituting from (21) into the integral equation ( 16), we have Integrating the last integration of (28) by parts and noticing that there exists q ′ (x) such that q ′ ∈ L 1 [0, π] substituting from ( 29) into (28) , we get where α 1 (x) is defined by (27) .In order to make φ(x, λ) more precise we repeat this procedure again by substituting from the last result (30) into the same integral equation ( 16), we have Now we estimate each term in (31).Integrating by parts twice the first term of (31), and noticing that q ′′ ∈ L 1 [0, π], we have Similarly, we have Substituting from (32) and (33) into (31), we get where α 1 (x) and α 2 (x) is defined by (27) .In order to make φ(x, λ) more precise we repeat this procedure again by substituting from the last result (34) into the same integral equation ( 16), we have Now we estimate each term in (35).Integrating by parts twice the first term of (35), and noticing that q ′′ ∈ L 1 [0, π], we have Further, and Theorem 6.Let q ∈ L 1 (0, π), then we have the following asymptotic formulas for λ n of non-local boundary value ( 1) and ( 4) where α 1 (x) defined in (27).