Mem. Differential Equations Math. Phys. 10(1997), 162–163

Ye. Galakhov

ON DIFFERENTIAL OPERATORS WITH INTEGRAL CONDITIONS

(Reported on December 2–9)

Ordinary differential equations with nonlocal conditions are studied in [1 - 4]. Such conditions are usually written in the form of the Stieltjes integral containing an atomic measure at the ends of an interval. Using methods of the theory of nonlocal elliptic problems [5], we can avoid such restrictions. First this method was employed in [6].

In the present paper, we consider the equation

(Au − λu)(t) = −a_{0}(t)u′′(t) + a_{1}(t)u′(t) +
+ a_{2}(t)u(t) − λu(t) = f_{0}(t) (t ∈ (0, 1))  (1)

with integral conditions

B_{j}u = ∫_{0}^{1} φ_{j}(t)u(t)dt = f_{j} (j = 1, 2).  (2)

Here a_{i} are real-valued functions (i = 0, 1, 2); a_{0}(t) ≥ k > 0 (0 ≤ t ≤ 1); a_{1}, a_{2} ∈ C[0, 1]; f_{0} ∈ L_{2}(0, 1) is a complex-valued function; f_{j} ∈ ℂ (j = 1, 2) are constants; λ ∈ ℂ is a spectral parameter; φ_{j} are linearly independent real-valued functions.

The functions a_{0} and φ_{j} have to satisfy some conditions. In order to formulate them, we introduce some notation:

C^{0,α}[a, b] is the Hölder space consisting of the functions u(t) such that u(t) are continuous on [a, b] and

|u(t_{1}) − u(t_{2})| ≤ c|t_{1} − t_{2}|^{α} (t_{1}, t_{2} ∈ [a, b]),

where c > 0 and 0 < α ≤ 1 do not depend on t_{1}, t_{2};

W^{1,∞}(a, b) is the space of the functions u(t) which are absolutely continuous on [a, b] and u′ ∈ L_{∞}(a, b);

C_{β}^{0,α}[0, 1] = {u ∈ L_{2}(0, 1) : u ∈ C^{0,α}[0, β] ∩ C^{0,α}[1 − β, 1]},
W_{β}^{1,∞}(0, 1) = {u ∈ C[0, 1] : u ∈ W^{1,∞}(0, β) ∩ W^{1,∞}(1 − β, 1)},

where 0 < β < 1/2;

W^{m}(0, 1) is the Sobolev space of complex-valued functions which are absolutely continuous on [0, 1] along with their derivatives up to the order (m−1) inclusively and whose m-th derivative belongs to L_{2}(0, 1). The norm is defined by

∥u∥_{W^{m}(0,1)} = { ∑_{j=0}^{m} ∫_{0}^{1} |u^{(j)}(t)|^{2}dt}^{1/2}.

1991 Mathematics Subject Classification. 34B99, 47E05.

Key words and phrases. Differential operator, integral conditions, discrete spectrum.

V (0, 1) = L_{2}(0, 1) × ℂ × ℂ; Ω_{ε,q} = {λ ∈ ℂ : | arg λ| ≥ ε, |λ| ≥ q}, where ε, q > 0. In the spaces W^{m}(0, 1) and V (0, 1), we introduce the following equivalent norms

depending on λ:

|∥u|∥_{W^{m}(0,1)} = {∥u∥_{W^{m}(0,1)}^{2} + |λ|^{m}∥u∥_{L_{2}(0,1)}^{2}}^{1/2},
|∥f|∥_{V (0,1)} = {∥f_{0}∥_{L_{2}(0,1)}^{2} + |λ|^{2}(|f_{1}|^{2} + |f_{2}|^{2})}^{1/2},

where f = (f_{0}, f_{1}, f_{2}).

We define a bounded operator L(λ) : W^{2}(0, 1) → V (0, 1) by the formula L(λ)u = (Au−λu, B_{1}u, B_{2}u). We also introduce an unbounded operator A_{B} : L_{2}(0, 1) → L_{2}(0, 1) with the domain D(A_{B}) = {u ∈ W^{2}(0, 1) : B_{j}u = 0, j = 1, 2} by the formula A_{B}u = Au (u ∈ D(A_{B})).

Suppose that a_{0} ∈ W_{β}^{1,∞}(0, 1), φ_{j} ∈ C_{β}^{0,α}[0, 1] (1/2 < α ≤ 1), and

Δ_{φ} = φ_{1}(0)φ_{2}(1) − φ_{1}(1)φ_{2}(0) ≠ 0.

Theorem 1. Let Δ_{φ} ≠ 0. Then for any ε > 0 there exists q > 1 such that for all λ ∈ Ω_{ε,q} a solution u ∈ W^{2}(0, 1) of the problem (1), (2) satisfies

|∥u|∥_{W^{2}(0,1)} ≤ cλ^{1/4}|∥f|∥_{V (0,1)},  (3)

where c > 0 does not depend on λ and f.

From Theorem 1 it follows:

Theorem 2. Let Δ_{φ} ≠ 0. Then the spectrum σ(A_{B}) consists of isolated eigenvalues. For any ε > 0, there exists q > 0 such that

σ(A_{B}) ⊂ {λ ∈ ℂ : | arg λ| < ε} ∪ {λ ∈ ℂ : |λ| < q}.

L(λ) : W^{2}(0, 1) → V (0, 1) is a Fredholm operator and ind L(λ) = 0 for every λ ∈ ℂ. For λ ∉ σ(A_{B}), the operator L(λ) has a bounded inverse L^{−1}(λ) : V (0, 1) → W^{2}(0, 1).

Under the assumptions a_{0} ∈ C^{2}[0, 1], a′(0) = a′(1) = 0, ϕ_{j} ∈ C^{1}[0, 1], these theorems were proved in [6].

The author is thankful to A. L. Skubachevskiǐ for the formulation of the problem and valuable advice and to I. T. Kiguradze for fruitful discussion.

References

1. M. Picone, Rend. Acad. Lincei 17(1908), 340–347.

2. J. D. Tamarkin, Math. Zeit. 27(1928), 1–54.

3. A. M. Krall, Rocky Mountain J. Math. 5(1975), 493–542.

4. Il’in, V. A. and Ye. I. Moiseev. Differentsial’nye Uravneniya 24(1988), 795–804; English transl. in Differential Equations 24(1988).

5. A. L. Skubachevskiǐ, Mat. Sb. 121(163)(1983), 201–210; English transl. in Math. USSR Sb. 49(1984).

6. A. L. Skubachevskiǐ and G. M. Steblov. Dokl. Akad Nauk SSSR 321(1991), 1158–1163; English transl. in Soviet Math. Dokl. 44(1992).

Author’s address: Moscow State Aviation Institute (Technical University) 4, Volokolamskoe shosse, Moscow 125871 Russia