eu STURM-LIOUVILLE DIFFERENTIAL EQUATION WITH NON-LOCAL BOUNDARY CONDITIONS

We investigate 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


Introduction
Many interesting applications of differential equations arise in recent years with nonlocal condition often appear in Mathematics, mechanics, physics, geophysics and other branches of natural sciences (see [1]- [5] and [10]- [13]).Afterwards the number of differential problems with nonlocal boundary conditions had increased.Quite new area, related to problems of this type, deals with investigation of the spectrum of Sturm-Liouville with nonlocal conditions.
Recently, in ( [6]- [8]) the authors studies the existence and some asympotic properties of the eigenvalues and eigenfunctions of the boundary value problem of the Sturm-Liouville differential equation with different kinds of nonlocal boundary conditions.Consider the nonlocal boundary value problem of the Sturm-Liouville equation (1) with the non-local conditions y ′ (0) − Hy(0) = 0, y(ξ) = 0, ξ ∈ (0, π], ( where the non-negative real function q(x) has a second piecewise integrable derivatives on (0, π), H is real and λ is spectral parameter.
Here we study the existence and some general properties of the eigenvalues and eigenfunctions of the two non-local boundary value problems ( 1) and ( 2).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.
Multiplying both sides of (3) by ȳ0 and then integrating form 0 to ξ with respect to x, we have Using the boundary condition (4), 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 (2) are orthogonal.
Proof.Let λ 1 = λ 2 be two different eigenvalues of the non-local boundary value problem (1) and (2).Let y 1 (x), y 2 (x) be the corresponding eigenfunctions, then and Multiplying both sides of (5) by ȳ2 and integrating with respect to x, we obtain by taking the complex conjugate of ( 7) and multiply it by y 1 and integrate the resulting expression with respect to x, we have Subtracting ( 9) from ( 10) and using the boundary conditions of ( 6) and ( 8) we obtain which completes the proof.
After using the condition (11), (12), we get αφ(ξ, λ) = 0, where α = 0, therefore, The characteristic equation will be Lemma 3. The solution φ(x, λ) of problem ( 1) and (2) satisfy the integral equations Proof.First we obtain formula (14) 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 ( 16) into (15) equation ( 14) follows.Second we show that the integral representation (14) satisfies the problem ( 1) and (11).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 0 to x we obtain Integrating by parts twice and using the condition (11), we have By substituting from (18) into (17) we get the required formula (14).
Proof.We show first that where the inequality is uniformly with respect to x. ( By the aid of (21) we find that From ( 14) and ( 21) it follows that, ϕ(x, λ) has the asymptotic formula (19).
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 ( 2) have the following asymptotic formula where Proof.By substituting from (19) into the integral equation ( 14), we have Integrating the last integration of (26) by parts and noticing that there exists q ′ (x) such that q ′ ∈ L 1 [0, π] Substituting ( 27) in (26), we get where α 1 (x) is defined by (25).In order to make φ(x, λ) more precise we repeat this procedure again by substituting from the last result (28) into the same integral equation ( 14), we have Now we estimate each term in (29).Integrating by parts twice the first term of (29), and noticing that q ′′ ∈ L 1 [0, π], we have Similary, we have Substituting (30) and (31) in (29), we get where α 1 (x) and α 2 (x) is defined by (25).In order to make φ(x, λ) more precise we repeat this procedure again by substituting from the last result (32) into the same integral equation ( 14), we have Now we estimate each term in (33).Integrating by parts twice the first term of (33), and noticing that q ′′ ∈ L 1 [0, π], we have x 0 sin λ(x − t) cos λt λ q(t)dt = sin λx 2λ x 0 q(t)dt Further, and Substituting from (34)-( 36) into (33) we get the required formula (24).Now inserting the values of the functions ϕ(x, λ) from the estimate (24) into the second of the boundary conditions in (2), we obtain the following equation for the determination of the eigenvalues equation ( 19) is the characteristic equation which gives roots of λ λ Then the ω(λ) has the same root of the function sinλξ (By Rouche's theorem) 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 ( 2) where α 1 (x) defined in (25).

Lemma 1 .
The eigenvalues of the nonlocal boundary value problem (1) and (2) are real.