2-Regularity and Branching Theorems

Journal of Mathematical Sciences, Vol. 104, No. 1, 2001

2-REGULARITY AND BRANCHING THEOREMS A. F. Izmailov

UDC 517.972.8; 517.988.67

1. Introduction. A Level Set of a 2-Regular Mapping Branching theorems are theorems on the local structure of the set of solutions to a linear operator equation in which a part of the variables plays the role of parameters in the neighborhood of singular points of this set that are specifically understood. The present paper obtains the branching theorems by using general results on the structure of level sets of a 2-regular nonlinear mapping. All linear spaces that are considered below are assumed to be real. In the standard way, we define the distance ρX (x, M) = inf{x − ξ | ξ ∈ M} from a point x to a set M in a linear normed space X. By Tx M, we denote the tangent cone (the set of tangent vectors [16, p. 40]) to the set M at the point x ∈ M: Tx M = {h ∈ X | ρX (x + th, M) = o(t), t ∈ R+ }. For arbitrary sets X and Y and for a mapping F : X → Y , we denote by F −1 : Y → 2X the mapping that sets in correspondence to each y ∈ Y its full inverse image under the mapping F ; if F is bijective, then we assume that F −1 : Y → X is an ordinary (not multivalued) mapping. If X and Y are linear normed spaces and if Λ : X → Y is a linear operator, then Λ−1  = sup{ρX (0, Λ−1 y) | y ∈ Y, y = 1} is the “norm” of the mapping Λ−1 [16, p. 45]; in the case of bijective Λ it is in fact an operator norm in the space L(Y, X). If X1 , X2 , and Y are linear normed spaces and if W is a neighborhood of the point (x1 , x2 ) in X1 × X2 , then, for the mapping F : W → Y , we denote by ∂i F (x1 , x2 ) the partial derivative with respect to the ith variable; the second partial derivative with respect to the ith and jth variables at the point (x1 , x2 ), i, j = 1, 2, is denoted by ∂ij F (x1 , x2 ). Let us specify some other notions and notation (for the origin and meaning of these notions, see [5–7, 17, 18, 25]). Let X and Y be linear normed spaces, let V be a neighborhood of the point x∗ in X, and let the mapping F : V → Y be twice Frech´et differentiable at the point x∗ . We will assume that the subspace Y1 = Im F  (x∗ ) is closed in Y . Then the quotient space Y /Y1 can be assumed to be normed in the standard way, and the canonical projection π : Y → Y /Y1 is continuous [2, p. 119]. Consider the family of linear operators Ψ2 (h) ∈ L(X, Y1 × (Y /Y1)), Ψ2 (h)x = (F  (x∗ )x, πF  (x∗ )[h, x]), the cone




h ∈ X,





T2 = h ∈ Ker F  (x∗ )  F  (x∗ )[h, h] ∈ Im F  (x∗ ) ,

and the family of sets


 T2β = h ∈ X  F  (x∗ )h ≤ β, πF  (x∗ )[h, h] ≤ β ,

β ∈ R+ .

Definition 1. A mapping F is called 2-regular at a point x∗ on an element h ∈ X if Im Ψ2 (h) = Y . Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 65, Pontryagin Conference–4. Optimal Control, 1999.

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c 2001 Plenum Publishing Corporation 1072–3374/01/1041–0830 $ 25.00 

Remark 1. Let Y2 be the orthogonal complement of Y1 in Y , and let P be the projection on Y2 along Y1 in Y . We introduce the family of linear operators Ω2 (P, h) : X1 → Y2 ,

Ω2 (P, h)x = P F  (x∗ )[h, x],

h ∈ X,

where X1 = Ker F  (x∗ ). It is easy to see that the mapping F is 2-regular at the point x∗ on an element h ∈ X iff Im Ω2 (P, h) = Y2 . The convenience of such an interpretation lies in the fact that it not infrequently allows one to reduce the infinite-dimensional case to the finite-dimensional one when examining the 2-regularity. We note that the 2-regularity property is invariant with respect to the choice of the orthogonal complement Y2 of the subspace Y1 in Y , i.e., to the choice of the projection P which, in turn, can always be chosen in the most appropriate way. Definition 2. A mapping F is called 2-regular at a point x∗ if it is 2-regular at this point on any element h ∈ T2 \ {0}. Remark 2. In the notation of Remark 1, the mapping F is 2-regular at the point x∗ iff the quadratic mapping Ω(P, ·) : X1 → Y2 , Ω(P, x) = P F  (x∗ )[x, x], is 2-regular at zero. This follows from the relation T2 = {h ∈ X1 | Ω(P, h) = 0}, Remark 1, the relation ∂2 Ω(P, h) = Ω2 (P, h) ∀ h ∈ X1 , and from the obvious fact that the mapping Ω(P, ·) is 2-regular at zero on the element h ∈ X1 if and only if it is regular at the point h. Definition 3. A mapping F is called strongly 2-regular at a point x∗ if there exists a number β > 0 such that sup{(Ψ2 (h))−1  | h ∈ T2β , h = 1} < ∞. Obviously, if the mapping F is regular at the point x∗ , i.e., if Im F  (x∗ ) = Y, then it is 2-regular at this point on any element h ∈ X and is strongly 2-regular at this point. The open mapping theorem implies that the strong 2-regularity is equivalent to the 2-regularity of the mapping at a point in the case of a finite-dimensional X. The strong 2-regularity implies the 2-regularity in the general case (since T2 = T20 ⊂ T2β ∀ β ≥ 0), but the converse is not true; the corresponding examples can be found in [7, 17, 18]. Nevertheless, sometimes even the infinite-dimensionality of X does not prevent the equivalence of the strong 2-regularity and the 2-regularity of a mapping at a point. The following will be required below. Proposition 1. Let X and Y be Banach spaces, let V be a neighborhood of a point x∗ in X, and let a mapping F : V → X be twice Frech´et differentiable at the point x∗ ; moreover, let Im F  (x∗ ) be closed in Y , and dim Ker F  (x∗ ) < ∞.

(1)

Then the mapping F is strongly 2-regular at the point x∗ if and only if it is 2-regular at this point. Proof. The Hahn–Banach theorem and condition (1) imply that the closed subspace X1 = Ker F  (x∗ ) has the closed orthogonal complement X2 in X (see [23, p. 27]). Let S be the (continuous) projection on X1 along X2 in X, and let T = IX − S be the complementary to S projection (on X2 along X1 in X). We set Γ = F  (x∗ )|X2 . The Banach theorem on the inverse operator and the closedness of Im F  (x∗ ) imply that the linear operator Γ is continuously invertible on its image, i.e., there exists a number c > 0 such that Γx ≥ cx ∀ x ∈ X2 .

(2) 831

We consider an arbitrary sequence {hk } ⊂ X such that hk  = 1 ∀ k ∈ N and F (x∗ )hk  → 0 (k → ∞). The following is true for this sequence: ΓT hk  = F  (x∗ )Shk + F  (x∗ )T hk  = F (x∗ )hk  → 0 (k → ∞), i.e., the following holds by (2): hk − Shk  = T hk  → 0 (k → ∞).

(3)

But {Shk } ⊂ X1 , where, according to (1), dim X1 < ∞. In addition, Shk  ≤ S ∀ k ∈ N, i.e., the sequence {Shk } is bounded. Therefore, without loss of generality, we can assume that this sequence converges to some element h ∈ X. But then, due to (3), the sequence {hk } also converges to this very element h; what was required is easily obtained from this by using the open mapping theorem. The proposition is proved. The following two theorems play a crucial role in the subsequent constructions. Theorem 1. Let X and Y be Banach spaces, and let V be a neighborhood of a point x∗ in X. Let a mapping F : V → Y be twice Frech´et differentiable at the point x∗ ; moreover, let Im F  (x∗ ) be closed in Y . Then (a) Tx∗ F −1 (F (x∗ )) ⊂ T2 and (b) if the mapping F is 2-regular at the point x∗ on an element h ∈ T2 , then h ∈ Tx∗ F −1 (F (x∗ )). In particular, if the mapping F is 2-regular at the point x∗ , then Tx∗ F −1 (F (x∗ )) = T2 . For the first time, Theorem 1 (the generalized Lyusternik theorem) was obtained in [25] (in the case of F  (x∗ ) = 0) and in [5] (in a full generality). The development of the 2-regularity theory began precisely from this result. If the mapping F is regular at the point x∗ , then T2 = Ker F  (x∗ ); thus, Theorem 1 indeed generalizes the classical theorem by Lyusternik on the tangent subspace [16, p. 41]. The following result, which was proved in [7] (see also [17]), is a refinement of Theorem 1. Theorem 2. Let X and Y be Hilbert spaces; moreover, let X be separable, and let V be a neighborhood of a point x∗ in X. Let the following conditions hold for a mapping F ∈ C 3 (V, Y ): Im F  (x∗ ) is closed in Y and F is strongly 2-regular at the point x∗ . Then there exist a neighborhood U of the point 0 in X and a mapping ϕ : U → X such that ϕ(0) = x∗ , ϕ(U) ⊂ V is the neighborhood of the point x∗ in X, and ϕ is a diffeomorphism of U onto ϕ(U); moreover, ϕ (0) = IX and ϕ(T2 ∩ U) = F −1 (F (x∗ )) ∩ ϕ(U). Thus, if the conditions of Theorem 2 hold, the level set M = F −1 (F (x∗ )) of the mapping F containing its strongly 2-regular point x∗ is locally diffeomorphic to the cone Tx∗ M = T2 , which is tangent to it. In the general case, it is not possible to replace the condition of strong 2-regularity by the condition of 2-regularity here (see [17]). Perhaps the requirement of the separability of X in Theorem 2 is superfluous, but the method of proof that is proposed in [7] substantially uses this condition. Let us pass to the consideration of families of the nonlinear operator equations that are parametrized by a scalar parameter. Namely, let a smooth mapping F : V × V → Y be given, where V is an open set in R, V is an open set in the Banach space X, and Y is a Banach space. In what follows, we will study the structure of the set of solutions to the equation F (σ, x) = 0,

(4)

M = F −1 (0) ⊂ V × V

(5)

i.e., the level sets

832

of the mapping F . For this set, a point (σ∗ , x∗ ) ∈ M is called singular if the mapping F (σ∗ , ·) is irregular at the point x∗ , i.e., if Im ∂2 F (σ∗ , x∗ ) = Y. The existence of singular points in the set M is by no means a pathology (see, e.g., [4, p. 96]). Moreover, it is the structure of M in a neighborhood of the singular points that is of most interest, since such points are “characteristic” ones that determine the structure of the whole set M. 2. Regular Case Let X and Y be two Banach spaces, let V be a neighborhood of a point σ∗ in R, let V be a neighborhood of the point x∗ in X, and let a mapping F : V × V → Y be Frech´et differentiable at the point (σ∗ , x∗ ); moreover, let F (σ∗ , x∗ ) = 0. First of all, we discuss the question on the structure of the set M, which was introduced in (5) in a neighborhood of the point (σ∗ , x∗ ) in the framework of assumptions that ensure the regularity of the mapping F at this point, i.e., fulfillment of the condition Im F  (σ∗ , x∗ ) = Y.

(6)

Everywhere we will assume that the operator ∂2 F (σ∗ , x∗ ) is normally solvable, that is, Im ∂2 F (σ∗ , x∗ ) = {y ∈ Y | y ∗, y = 0 ∀ y ∗ ∈ (Ker(∂2 F (σ∗ , x∗ ))∗ }.

(7)

dim Ker(∂2 F (σ∗ , x∗ ))∗ = 1,

(8)

∂1 F (σ∗ , x∗ ) ∈ Im ∂2 F (σ∗ , x∗ ).

(9)

Ker(∂2 F (σ∗ , x∗ ))∗ = lin{ϕ∗ },

(10)

In addition, let

According to (8), where ϕ∗ ∈ Y ∗ \ {0} is some functional. Then condition (7) can be rewritten in the following form: Im ∂2 F (σ∗ , x∗ ) = {y ∈ Y | ϕ∗, y = 0},

(11)

and condition (9) can be written in the form ϕ∗ , ∂1 F (σ∗ , x∗ ) =  0.

(12)

Lemma. Let X and Y be two Banach spaces, let V be a neighborhood of a point σ∗ in R, let V be a neighborhood of a point x∗ in X, and let a mapping F : V × V → Y be Frech´et differentiable at the point (σ∗ , x∗ ); moreover, let conditions (7)–(9) hold. Then the mapping F is regular at the point (σ∗ , x∗ ), i.e., (6) is satisfied. Moreover, Ker F  (σ∗ , x∗ ) = {0} × Ker ∂2 F (σ∗ , x∗ ).

(13)

Proof. The differential of the mapping F at the point (σ∗ , x∗ ) has the form F  (σ∗ , x∗ )ζ = λ∂1 F (σ∗ , x∗ ) + ∂2 F (σ∗ , x∗ )ξ,

ζ = (λ, ξ) ∈ R × X.

(14)

We fix an arbitrary element y ∈ Y . Obviously, y ∈ Im F  (σ∗ , x∗ ) iff there exists λ = λ(y) ∈ R such that y − λ∂1 F (σ∗ , x∗ ) ∈ Im ∂2 F (σ∗ , x∗ ). By virtue of (11), the latter inclusion is equivalent to the relation ϕ∗ , y − λ∂1 F (σ∗ , x∗ ) = 0. But then, according to (12), the desired element λ is given by the following explicit formula: λ(y) =

ϕ∗ , y . ϕ∗ , ∂1 F (σ∗ , x∗ )

Relation (13) is obvious. The lemma is proved. 833

Using this lemma, the classical Lyusternik theorem, and the result on rectification (see, e.g., [17, Corollary 1]), which refines this theorem in the enhanced requirements of smoothness, we obtain the following two propositions. Proposition 2. Let X and Y be two Banach spaces, let V be a neighborhood of a point σ∗ in R, and let V be a neighborhood of a point x∗ in X. Let a mapping F : V ×V → Y be strictly differentiable at the point (σ∗ , x∗ ); moreover, let conditions (7)–(9) hold, and let F (σ∗ , x∗ ) = 0. Then T(σ∗ ,x∗ ) M = {0} × Ker ∂2 F (σ∗ , x∗ ). Proposition 3. Let X and Y be two Banach spaces, let V be a neighborhood of a point σ∗ in R, and let V be a neighborhood of a point x∗ in X. Let conditions (7)–(9) hold for a mapping F ∈ C r (V × V, Y ) (r ∈ N), let Ker ∂2 F (σ∗ , x∗ ) have the closed orthogonal complement in X, and let F (σ∗ , x∗ ) = 0. Then there exist a neighborhood U of the point 0 in R, a neighborhood U of the point 0 in X, and a mapping ρ : U ×U → R×X such that ρ(0, 0) = (σ∗ , x∗ ), ρ(U × U) ⊂ V × V is the neighborhood of the point (σ∗ , x∗ ) in R × X, ρ is a C r -diffeomorphism of U × U onto ρ(U × U), and, moreover, ρ (0, 0) = I and M ∩ ρ(U × U) = ρ({0} × (Ker ∂2 F (σ∗ , x∗ ) ∩ U)). A subspace L in R × X will be called vertical if L ⊂ {0} × X. Under the conditions of Proposition 2, T(σ∗ ,x∗ ) M is a vertical subspace. If the conditions of Proposition 3 and the condition dim Ker ∂2 F (σ∗ , x∗ ) < ∞

(15)

hold, then the intersections of the set M with some neighborhood of the point (σ∗ , x∗ ) is a C -submanifold of dimension dim Ker ∂2 F (σ∗ , x∗ ) in the space R × X (we stress the fact that the case r = 1 is not excluded here). r

Remark 3. Of course, if the point (σ∗ , x∗ ) is not a singular point of the set M, then, if (15) is satisfied, the intersection of this set with some neighborhood of the point (σ∗ , x∗ ) is also a smooth submanifold of dimension dim Ker ∂2 F (σ∗ , x∗ ) + 1 in the space R × X. In this case, the tangent space to such a submanifold at the point (σ∗ , x∗ ) is not vertical in advance. If, under the conditions of Proposition 3, dim Ker ∂2 F (σ∗ , x∗ ) = 1,

(16)

then an additional requirement of 2-regularity of the mapping F (σ∗ , ·) at the point x∗ on elements of the set Ker ∂2 F (σ∗ , x∗ ) \ {0} guarantees that the one-dimensional submanifold M lies entirely to one side of the hyperplane {σ∗ } × X in R × X that passes through the point (σ∗ , x∗ ), i.e., the latter is a turning point of this submanifold. More precisely, the following proposition holds. Proposition 4. Let r ≥ 2, the conditions of Proposition 3 hold, and, moreover, let (16) hold and ¯ h] ¯ ∈ Im ∂2 F (σ∗ , x∗ ), ∂22 F (σ∗ , x∗ )[h,

(17)

¯ ∈ Ker ∂2 F (σ∗ , x∗ ) \ {0} is some element. Then there exist a neighborhood U of the point σ∗ in R h neighborhood U of the point x∗ in Rn such that the set M ∩ (U × U) \ {(σ∗ , x∗ )} is contained in one two half-spaces {(σ, x) ∈ R × X | σ < σ∗ } or {(σ, x) ∈ R × X | σ > σ∗ } in R × X. ¯ ∈ Ker ∂2 F (σ∗ , x∗ ) \ {0}. Proposition 3 and relation (16) imply the existence of Proof. We fix an element h a number δ > 0 and two mappings ρ1 ∈ C 2 ((−δ, δ), R) and ρ2 ∈ C 2 ((−δ, δ), X) such that the intersection of ¯ 2 (t)) | t ∈ (−δ, δ)}; the set M with some neighborhood of the point (σ∗ , x∗ ) has the form {(σ∗ +ρ1 (t), x∗ +th+ρ   moreover, ρ1 (0) = 0, ρ1 (0) = 0, ρ2 (0) = 0, and ρ2 (0) = 0. But then ¯ + ρ2 (t)) 0 = F (σ∗ + ρ1 (t), x∗ + th where and a of the

¯ + ρ2 (t)) = ρ1 (t)∂1 F (σ∗ , x∗ ) + ∂2 F (σ∗ , x∗ )(th 1 ¯ th] ¯ + ω(t) ∀ t ∈ (−δ, δ), + ∂22 F (σ∗ , x∗ )[th, 2 where the mapping ω : (−δ, δ) → Rn is such that ω(t) = o(t2 ); this implies ¯ h] ¯ ∈ Im ∂2 F (σ∗ , x∗ ). ρ (0)∂1 F (σ∗ , x∗ ) + ∂22 F (σ∗ , x∗ )[h, 1

834

According to (17), the latter relation holds only in the case where ρ1 (0) =  0; this yields the required result. p The mapping F : R × R → R, F (σ, x) = σ + χx , χ ∈ R, p ∈ N, p ≥ 2, at the point (0, 0) can serve as a model example for Propositions 2–4 (Proposition 4 describes the case where χ = 0, p = 2). Example 1. Let X be a Hilbert space and let Λ : X → X be a completely continuous self-adjoint linear operator. For this operator, we consider the spectral problem as the parametric relation (4) with the operator F : R × X → X,

F (σ, x) = (Λ − σIX )x.

Let σ∗ ∈ R \ {0} be a simple eigenvalue of the operator Λ, and let x∗ ∈ X \ {0} be the corresponding eigenvector. Then, according the S. M. Nikol’skii theorem [24, p. 233], the operator ∂2 F (σ∗ , x∗ ) = Λ − σ∗ IX is a Fredholm operator. In particular, relation (7) holds; moreover, due to the simplicity of the eigenvalue σ∗ and the self-adjointness of the operator Λ, we have Ker ∂2 F (σ∗ , x∗ ) = Ker(∂2 F (σ∗ , x∗ ))∗ = lin{x∗ }; this implies that Ker ∂2 F (σ∗ , x∗ ) has a closed orthogonal complement in X and Im ∂2 F (σ∗ , x∗ ) = {x ∈ X | x∗ , x = 0}. Thus, (8) holds; moreover, ∂1 F (σ∗ , x∗ ) = −x∗ ∈ Im ∂2 F (σ∗ , x∗ ), i.e., (9) also holds. Applying Proposition 3, we obtain that only one smooth curve of solutions to Eq. (4) passes through the point (σ∗ , x∗ ); moreover, the tangent straight line to this curve at the point (σ∗ , x∗ ) has the form {0} × lin{x∗ }. At the same time, ∂22 F (σ∗ , x∗ ) = 0, i.e., Proposition 4 is not applicable, and this fits the situation well, since, in this case, the curve of solutions is obtained by a translation of the tangent straight line, i.e., it has the form {(σ∗ , θx∗ ) | θ ∈ R}. We note that in the case of a finite-dimensional X, all these considerations also hold for σ∗ = 0 whenever dim Ker Λ = 1. In the infinite-dimensional case, this is not so: for a completely continuous operator Λ, the subspace Im Λ is closed in the infinite-dimensional X iff this subspace is finite-dimensional [24, p. 225]; this easily implies that the conditions of normal solvability of ∂2 F (σ∗ , x∗ ) and the finiteness of the corank of this operator cannot hold simultaneously. Example 2. The traditional example of a nonlinear integral equation is the so-called Chandrasekhar H-equation  1 x(τ ) σ x(t) = 1 + tx(t) dτ, t ∈ [0, 1], (18) 2 0 t+τ where σ ∈ R+ is a parameter (see [8, 9, 11, 15, 19, 20, 22]); this equation appears in the theory of radiative heat transfer. Of greatest practical interest are the values of the parameter σ ∈ [0, 1]; this will be clarified in what follows. It is known [15, 20] that for σ = σ∗ = 1 Eq. (18) has a unique solution x∗ (·) ∈ C[0, 1]; moreover, obviously, x∗ (t) ≥ 1 ∀ t ∈ [0, 1]. Furthermore, this solution satisfies the normalizing condition 1 1 x∗ (t) dt = 1 2 0 (see [20]), whence the relation  1 1 τ x∗ (τ ) x∗ (t) dτ = 1 ∀ t ∈ [0, 1] 2 t+τ 0

(19)

(20)

(21) 835

easily follows. We set X = C[0, 1] and introduce the mapping F : R × X → X,

(F (σ, x))(t) = x(t) −

 1 σ x(τ ) tx(t) dτ − 1. 2 0 t+τ

This mapping has continuous derivatives of any order on X; moreover,  1 1 x∗ (τ ) (∂1 F (σ∗ , x∗ ))(t) = − tx∗ (t) dτ = 1 − x∗ (t), 2 0 t+τ  1  1 1 x∗ (τ ) ξ(τ ) 1 dτ − tx∗ (t) dτ (∂2 F (σ∗ , x∗ )ξ)(t) = ξ(t) − tξ(t) 2 2 0 t+τ 0 t+τ  1 ξ(t) 1 ξ(τ ) = − tx∗ (t) dτ, ξ ∈ X x∗ (t) 2 0 t+τ

(22)

(23)

(here (19) is taken into account),  1 1 1 1  1 ξ 2 (τ ) ξ (τ ) 1 2 (∂22 F (σ∗ , x∗ )[ξ , ξ ])(t) = − tξ (t) dτ − tξ (t) dτ, 2 2 0 t+τ 0 t+τ 1

2

ξ 1 , ξ 2 ∈ X.

(24)

Using (21) and (23), we can show that the operator ∂2 F (σ∗ , x∗ ) is Fredholm; moreover, Ker(∂2 F (σ∗ , x∗ ))∗ = lin{ϕ∗ },

Ker ∂2 F (σ∗ , x∗ ) = lin{ϕ} and where ϕ ∈ X,

ϕ(t) = tx∗ (t),

∗

∗

ϕ ∈X ,

∗

and ϕ , x =



1

x(t) dt 0

(see [15]). Thus, we have shown that conditions (7), (8), and (16) hold. In addition, due to (20) and (22), we have ϕ∗ , ∂1 F (σ∗ , x∗ ) =



0

1

(1 − x∗ (t)) dt = −1 = 0,

i.e., (12) holds; therefore, (9) also holds. Finally, by (21) and (24) we have ∗

ϕ , ∂22 F (σ∗ , x∗ )[ϕ, ϕ] = −

 0

1

 2

t x∗ (t)

0

1

τ x∗ (τ ) dτ t+τ



dt = −2

 0

1

2 t2 dt = − =  0, 3

i.e., (17) holds as well. Thus, Propositions 3 and 4 are applicable here; according to these propositions, the set of solutions to Eq. (18) in a neighborhood of the point (1, x∗ ) in R × C[0, 1] has the form {(1 + ρ1 (θ), x∗ (t) + θtx∗ (t) + ρ2 (θ, t)) | θ ∈ (−δ, δ)}, where δ > 0 is some number, the function ρ1 : (−δ, δ) → R is such that ρ1 (θ) = o(θ), ρ1 (θ) has one sign ∀ θ ∈ (−δ, δ), and the function ρ2 : (−δ, δ) × [0, 1] → R satisfies the following conditions: ρ2 (θ, ·) ∈ C[0, 1] ∀ θ ∈ (−δ, δ); moreover, ρ2 (θ, ·)C = o(θ). One can show that ρ1 (θ) < 0 ∀ θ ∈ (−δ, δ) if δ is sufficiently small. A special result on the reduction of singularities of the type under consideration to some canonical form should be a final point of the local analysis that is proposed. If one searches for an analogy to the constructions for potential mappings in [4], then one should expect that if (7), (8), and (15) hold and if the condition of the 2-regularity of the mapping F (σ∗ , ·) at the point x∗ on the elements of the set Ker ∂2 F (σ∗ , x∗ ) \ {0} is satisfied, then a germ at the point x∗ of the latter mapping is differentiably rightequivalent to a germ of a certain linear-quadratic mapping. But this is not the case even if (16) holds and even if the case is finite-dimensional. For example, for the mapping G : R2 → R2 , G(x) = (x1 , x22 + x31 ), in a neighborhood of the point 0, there is no change of variables that eliminates the term of the third order, although all other requirements are satisfied. Thus, it only remains to use the general results on the representation of a nonlinear mapping on the cone of the 2-regularity obtained in [17]. 836

3. Branching Theorems Let again X and Y be two Banach spaces, V be a neighborhood of a point σ∗ in R, V be a neighborhood of a point x∗ in X, and let a mapping F : V × V → Y be twice Frech´et differentiable at the point (σ∗ , x∗ ); moreover, let F (σ∗ , x∗ ) = 0. Here we consider the case where the set M \ {(σ∗ , x∗ )}, which is introduced in correspondence with (5), is a (disconnected) one-dimensional submanifold in the space Z = R × X in a neighborhood of the point (σ∗ , x∗ ). Each connected (smooth) one-dimensional submanifold N in Z such that N ⊂ M, (σ∗ , x∗ ) ∈ N , is called a (smooth) branch of solutions to Eq. (4) passing through the point (σ∗ , x∗ ). We assume that the operator ∂2 F (σ∗ , x∗ ) is Fredholm. Thus, let the condition of normal solvability (7) hold, and, in addition, let dim Ker ∂2 F (σ∗ , x∗ ) = dim Ker(∂2 F (σ∗ , x∗ ))∗ = s,

(25)

where s ∈ N is some number. According to what was said in Sec. 2, if s = 1 and if (9) holds, then the mapping F is regular at the point (σ∗ , x∗ ), as in the case where (σ∗ , x∗ ) is not a singular point of the set M; hence there is no real branching of this set at such points: for a sufficiently smooth mapping F , only one smooth branch of solutions passes through the point (σ∗ , x∗ ). If s ≥ 2, or if (9) does not hold, then, as is easily seen, the mapping F is nonregular at the point (σ∗ , x∗ ). But then it is natural to try to apply Theorems 1 and 2 for a description of the level set M of this mapping at the neighborhood of the point (σ∗ , x∗ ) ∈ M. In this case, the question of the 2-regularity of the mapping F at the point (σ∗ , x∗ ) is a crucial one. Formula (14) obviously implies that for the subspace Y1 = Im F  (σ∗ , x∗ ) in Y we have Y1⊥ = {y ∗ ∈ Ker(∂2 F (x∗ ))∗ | y ∗, ∂1 F (x∗ ) = 0}. But then, due to (25), dim Y1⊥ < ∞ (more precisely, dim Y1⊥ = s − 1 if (9) holds and dim Y1⊥ = s if (9) does not hold); this implies that Y1 has the closed orthogonal complement Y2 in Y ; moreover, dim Y2 = codim Y1 = dim Y1⊥ . Let P be the projection on Y2 along Y1 in Y . Then, according to Remark 2, the question on the 2-regularity of the mapping F at the point (σ∗ , x∗ ) is reduced to the question on the 2-regularity of the quadratic mapping Q : Z1 → Y2 ,

Q(z) = P F  (σ∗ , x∗ )[z, z],

(26)

where Z1 = Ker F  (σ∗ , x∗ ). The same formula (14) and condition (25) imply dim Ker F  (σ∗ , x∗ ) < ∞ (more precisely, if (9) holds, then dim Ker F  (σ∗ , x∗ ) = s; if (9) does not hold, then dim Ker F  (σ∗ , x∗ ) = s + 1). Thus, the quadratic mapping Q maps from the l-dimensional subspace Z1 of the space Z (where l = s or l = s + 1) into the (l − 1)-dimensional subspace Y2 of the space Y . The set of quadratic equations Q(ζ) = 0,

ζ2 = 1

(27)

(which is naturally considered on Z1 ) is called the branching equation. The normalizing conditions can be of another form if it is more convenient. The sense of the branching equation is clarified in the following two propositions. Being applied to the case under consideration, Theorem 1 takes the following form. Theorem 1 . Let X and Y be two Banach spaces, V be a neighborhood of a point σ∗ in R, and let V be a neighborhood of a point x∗ in X. Let a mapping F : V × V → Y be twice Frech´et differentiable at the point (σ∗ , x∗ ); moreover, let conditions (7) and (25) hold, the quadratic mapping Q that was introduced in (26) be 2-regular at zero, and let F (σ∗ , x∗ ) = 0. Then T(σ∗ ,x∗ ) M = {ζ ∈ Z1 | Q(ζ) = 0}. Theorem 2 (with due regard for Proposition 1) implies the following refinement of Theorem 1 . 837

Theorem 2 . Let X and Y be two Hilbert spaces; moreover, let X be separable, V be a neighborhood of a point σ∗ in R, and let V be a neighborhood of a point x∗ in X. Let conditions (7) and (25) hold for a mapping F ∈ C 3 (V × V, Y ), the quadratic mapping Q introduced in (26) be 2-regular at zero, and let F (σ∗ , x∗ ) = 0. Then there exist a neighborhood U of the point 0 in R, a neighborhood U of the point 0 in X, and a mapping ρ : U × U → R × X such that ρ(0, 0) = (σ∗ , x∗ ), ρ(U × U) ⊂ V × V is a neighborhood of the point (σ∗ , x∗ ) in R × X, and ρ is a diffeomorphism of U × U onto ρ(U × U); moreover, ρ (0, 0) = I and M ∩ ρ(U × U) = ρ({ζ ∈ Z1 | Q(ζ) = 0} ∩ (U × U)). We note that for quadratic mappings, the property of 2-regularity at zero is generic (see [1]). In the case of a 2-regular mapping Q at zero, the set of solutions to the branching equation (27) consists of finite (possibly, zero) number of pairs of opposite points. Therefore, under the conditions of Theorem 2 , the set M in neighborhood of the point (σ∗ , x∗ ) is the union of a finite (possibly, zero) number of smooth branches of solutions that intersect at the point (σ∗ , x∗ ); moreover, the tangent line to each of these branches at the point (σ∗ , x∗ ) passes through the corresponding pair of solutions to Eq. (27). Furthermore, using the degree theory of smooth mappings given on oriented compact manifolds [10, p. 275], we can show that the number of such pairs of solutions and hence the number of branches is even. The fulfillment of this fact is a consequence of the following two statements: (a) the degree of a smooth mapping of an oriented compact manifold into a noncompact manifold of the same dimension (for example, into the arithmetical space) is equal to zero [21, p. 257] and (b) the tangent mapping of the restriction to the unit sphere of a 2-regular at zero quadratic mapping that maps from Rl to Rl−1 simultaneously changes or does not change the orientation at the opposite zeros of such a restriction. It is rather difficult to say something more specific about the structure of the set of solutions to the branching equation (27) and hence about the local structure of the set M under such general assumptions (although the application of more refined topological methods will possibly bring about some interesting general results in the future). The study of conditions that ensure the 2-regularity of the mapping Q and also the study of the structure of the set of solutions to a branching equation in one or another more specific cases merely lead to different particular statements of branching theorems. Solely to make the computations less cumbersome, we assume that X = Y and that X = Im ∂2 F (σ∗ , x∗ ) ⊕ Ker ∂2 F (σ∗ , x∗ ).

(28)

It can be easily verified that with due regard for (25), the latter condition is equivalent to the nonsingularity of the s × s-matrix with entries ϕ∗i , ϕj , i, j = 1, . . . , s, where ϕ1 , . . . , ϕs is some basis in Ker ∂2 F (σ∗ , x∗ ) and ϕ∗1 , . . . , ϕ∗s is some basis in Ker(∂2 F (σ∗ , x∗ ))∗ . Moreover, these bases can be chosen in such a way that they are biorthogonal, i.e., the relation 

ϕ∗i , ϕj 

=

1, if i = j, 0, if i = j,

i, j = 1, . . . , s,

holds [26, p. 339]. Then the projection Π on the subspace Ker ∂2 F (σ∗ , x∗ ) along Im ∂2 F (σ∗ , x∗ ) in X is given by Πx =

s 

ϕ∗i , xϕi ,

x ∈ X.

(29)

i=1

Note that condition (7) can be rewritten in the form Im ∂2 F (σ∗ , x∗ ) = {y ∈ Y | ϕ∗i , y = 0, i = 1, . . . , s}.

(30)

1. Without imposing any constraints on the number s (except for its finiteness!), we consider first the case where ∂1 F (σ∗ , x∗ ) ∈ Im ∂2 F (σ∗ , x∗ ). 838

(31)

Then, due to (14), Y1 = Im ∂2 F (σ∗ , x∗ ), and, according to (28), we can set Y2 = Ker ∂2 F (σ∗ , x∗ ),

P = Π.

(32)

We define an element ψ ∈ X by ∂2 F (σ∗ , x∗ )ψ = −∂1 F (σ∗ , x∗ ) and ψ ∈ Im ∂2 F (σ∗ , x∗ ) (from (28) and from (31), it follows that there exists a unique element ψ ∈ X, which satisfies these relations). Then, using (14) again, we have Z1 = {ζ = (λ, λψ + ξ) | λ ∈ R, ξ ∈ Ker ∂2 F (σ∗ , x∗ )}.

(33)

The second differential of the mapping F at the point (σ∗ , x∗ ) has the form F  (σ∗ , x∗ )[ζ, ζ] = λ2 ∂11 F (σ∗ , x∗ ) + 2λ∂12 F (σ∗ , x∗ )ξ + ∂22 F (σ∗ , x∗ )[ξ, ξ],

ζ = (λ, ξ) ∈ Z.

(34)

From here, taking into account (29), (32), and (33), we obtain that the mapping Q, which is introduced in accordance with (26), is given by Q(ζ) =

s 





ϕ∗i , λ2 ∂11 F (σ∗ , x∗ ) + 2λ∂12 F (σ∗ , x∗ ) λψ +

i=1

s 

ai λ2

+ 2λ

i=1

s 

ζ = λ, λψ +

s 

s 

bji ξj +

j=1



ξj ϕj 

 s s   + ξj ϕj , λψ + ξj ϕj  ϕi j=1





j=1



+ ∂22 F (σ∗ , x∗ ) λψ =

s 



j=1

 i cjk i ξj ξk ϕ ,

j,k=1



ξj ϕj  ,

λ, ξj ∈ R, j = 1, . . . , s,

j=1

where

ai = ϕ∗i , ∂11 F (σ∗ , x∗ ) + 2∂12 F (σ∗ , x∗ )ψ + ∂22 F (σ∗ , x∗ )[ψ, ψ], bji = ϕ∗i , ∂12 F (σ∗ , x∗ )ϕj + ∂22 F (σ∗ , x∗ )[ψ, ϕj ], ∗ j k cjk i = ϕi , ∂22 F (σ∗ , x∗ )[ϕ , ϕ ],

i, j, k = 1, . . . , s.

Thus, for the chosen biorthogonal bases in the subspaces Ker ∂2 F (σ∗ , x∗ ) and Ker(∂2 F (σ∗ , x∗ ))∗ , the quadratic mapping Q can be written in an explicit form here. Note that nonsingular linear transformations of the space Y2 obviously do not influence the 2-regularity of the mapping Q and do not change the sets of zeros of this mapping. Hence, using the standard basis in Rs , we can consider the mapping ˜ : R × Rs → Rs , Q

2 ˜ (Q(γ)) i = ai λ + 2λ

s 

bji ξj +

j=1

s 

cjk i ξj ξk ,

j,k=1

γ = (λ, ξ) ∈ R × Rs ,

i = 1, . . . , s, instead of Q. The set of quadratic equations 2

ai λ + 2λ

s  j=1

bji ξj

+

s 

cjk i ξj ξk = 0,

i = 1, . . . , s,

j,k=1

λ2 +

s 

ξj2 = 1

j=1

is naturally called the algebraic branching equation. The solution ζ of the branching equation (27) uniquely corresponds to each solution (λ, ξ) of this equation; moreover, this solution is obtained by norming the   element λ, λψ +

s 

j=1

ξj ϕj in Z. In the cases where one manages to ascertain the 2-regularity of the mapping 839

Q (and hence that of Q), the application of Theorem 1 or Theorem 2 (depending on the smoothness that is available) gives a picture of branching of the set M at the point (σ∗ , x∗ ). Consider some important particular cases. ˜ Let s = 1. We denote ϕ1 by ϕ, ϕ∗1 by ϕ∗ , and a1 , b11 , and c11 1 by a, b, and c, respectively. Then Q is the following quadratic form on two variables: ˜ : R × R → R, Q(γ) ˜ (35) Q = aλ2 + 2bλξ + cξ 2 , γ = (λ, ξ) ∈ R × R. The condition of 2-regularity of this form at zero is a condition of its nondegeneracy, that is, ac − b2 = 0. Thus, Theorem 2 in this case implies that if X is a separable Hilbert space and if the mapping F is sufficiently smooth, then ˜ introduced by (35) is strictly definite, then (σ∗ , x∗ ) is an isolated point of the (1) if the quadratic form Q set M; in particular, no single branch of solutions to Eq. (4) passes through this point; ˜ is not definite, then the set M in the neighborhood of the point (σ∗ , x∗ ) is a (2) if the quadratic form Q union of two branches of solutions to Eq. (4); these branches intersect transversally at this point. The word “transversally” in this context means that the tangent lines to these branches at the point (σ∗ , x∗ ) are distinct. Such lines are uniquely determined by two pairs of opposite solutions to the algebraic branching equation (see above). ˜ introduced by (35) is definite, but not strictly definite, then it is degenerate If the quadratic form Q   and Theorems 1 or 2 are not applicable. For example, if X = R, then, under the above assumptions, (σ∗ , x∗ ) is a critical point of the function ˜ is a nondegeneracy condition for such a critical point. F and the nondegeneracy condition of the form Q Consider the case of existence of a trivial branch of solutions, which is important in applications. Namely, let F (σ, x∗ ) = 0 ∀ σ ∈ V.

(36)

Then ∂1 F (σ∗ , x∗ ) = 0 (in particular, (31) holds), ∂11 F (σ∗ , x∗ ) = 0, ψ = 0, a = 0, b = ϕ∗ , ∂12 F (σ∗ , x∗ )ϕ, ˜ introduced according c = ϕ∗ , ∂22 F (σ∗ , x∗ )[ϕ, ϕ], and the nondegeneracy condition of the quadratic form Q to (35) has the form b = 0, i.e., by (30), we have ∂12 F (σ∗ , x∗ )ϕ ∈ Im ∂2 F (σ∗ , x∗ ). ˜ consists of two straight lines lin{(1, 0)} and lin{(− c , 1)}. Moreover, the set of zeros of the form Q 2b   Applying Theorems 1 and 2 , we obtain the following proposition. Proposition 5 (simplest branching theorem). Let X be a Banach space, V be a neighborhood of a point σ∗ in R, and let V be a neighborhood of a point x∗ in X. Also let a mapping F : V × V → X be twice Frech´et differentiable at the point (σ∗ , x∗ ); moreover, let conditions (7) and (25) for s = 1, and (28) and (36) hold, and let ¯ ∈ Im ∂2 F (σ∗ , x∗ ) (37) ∂12 F (σ∗ , x∗ )h ¯ ∈ Ker ∂2 F (σ∗ , x∗ ) \ {0}. Then there exists a unique number χ such that for the element h ¯ T(σ∗ ,x∗ ) M = lin{(1, 0)} ∪ lin{(χ, h)}; ¯ moreover, χ = 0 iff the mapping F (σ∗ , ·) : V → X is 2-regular at the point x∗ on the element h. 3 If X is the separable Hilbert space and if F ∈ C (V × V, X), then there exist a neighborhood U ⊂ V of the point σ∗ in R and a neighborhood U ⊂ V of the point x∗ in X such that M ∩ (U × U) is the union of the set {(σ, x∗ ) | σ ∈ U} and a one-dimensional C 1 -submanifold in R × X that contains the point (σ∗ , x∗ ); ¯ the tangent line to this submanifold at the point (σ∗ , x∗ ) has the form lin{(χ, h)}. The result of Proposition 5 in one form or another was repeatedly presented in a wide range of literature devoted to branching theory (see, e.g., [4, p. 78], [12, 13]). We stress that if χ = 0 in this proposition, then a nontrivial branch of solutions can be safely parametrized by a natural parameter σ. The model example to Proposition 5 is the mapping F : R × R → R, F (σ, x) = (σ + χxp−1 )x, χ ∈ R, p ∈ N, p ≥ 2, at the point (0, 0). 840

Consider one more example [14, 27]. Example 3. Consider the boundary-value problem t ∈ [0, 1],

x¨ + σ(x + x2 ) = 0,

(38)

x(0) ˙ = x(1) = 0,

(39)

where σ ∈ R is a parameter. For any value of the parameter σ, this problem has the solution x∗ (·) ≡ 0. Consider the integral equation x(t) − σ



1

t ∈ [0, 1],

Γ(t, τ )(x(τ ) + (x(τ ))2 ) dτ = 0,

0

(40)

to which the problem (38), (39) is reduced by using the Green function 

Γ : [0, 1] × [0, 1] → R,

Γ(t, τ ) =

Introduce the mapping F : R × X → X,

F (σ, x) = x(·) − σ



1

1 − τ, if t ≤ τ ; 1 − t, if t ≥ τ.

Γ(·, τ )(x(τ ) + (x(τ ))2 ) dτ,

0

where X = L2 [0, 1]. This mapping has continuous derivatives of any order on X; moreover, ∂2 F (σ, x)ξ = ξ(·) − σ ∂12 F (σ, x)ξ = −





1

Γ(·, τ )(1 + 2x(τ ))ξ(τ ) dτ, 0

1

Γ(·, τ )(1 + 2x(τ ))ξ(τ ) dτ, 0

∂22 F (σ, x)[ξ 1 , ξ 2] = −2σ



1

Γ(·, τ )ξ 1(τ )ξ 2 (τ ) dτ,

0

x, ξ ∈ X, x, ξ ∈ X,

x, ξ 1 , ξ 2 ∈ X.

(41) (42) (43)

For any σ ∈ R, the operator ∂2 F (σ, 0) is a Fredholm integral operator of the second kind with a continuous kernel. We can show that the spectrum of the integral summand of this operator consists of 4 simple eigenvalues of the form 2 ; moreover, the corresponding eigensubspaces are spanned by the π (2k − 1)2   π 2 (2k − 1)2 π(2k − 1) elements cos t , k ∈ N. Thus, ∀ k ∈ N, σ∗ = σk = , and x∗ = 0, (7) and (25) hold; 2 4 ¯ = cos(√σ∗ t). In addition, by virtue of moreover, s = 1 and Ker ∂2 F (σ∗ , x∗ ) is spanned by the element h(t) symmetry of the Green function Γ, the operator ∂2 F (σ∗ , x∗ ) is self-adjoint. Therefore, we can assume that ¯ for a given σ∗ . Then condition (28) obviously the kernel Ker(∂2 F (σ∗ , x∗ ))∗ is spanned by the same element h hols. Finally, due to (41), (42), and the definition of σ∗ , we have  1  1  1 ¯ 2 h L2 2 ¯ ∂12 F (σ∗ , x∗ )h ¯ =− ¯ ¯ ) dτ dt = − 1 ¯ h, Γ(t, τ )h(τ (h(t)) dt = − = 0, h(t) σ σ 0 0 ∗ 0 ∗ which means that (37) holds. In addition, according to (43), and taking into account the symmetry of the Green function again, we obtain ¯ ∂22 F (σ∗ , x∗ )[h, ¯ ¯h] = −2σ∗ h, = −2σ∗

 0

1

2 ¯ (h(t))

 0

1



0

1

¯ h(t)



1

¯ ))2 dτ dt Γ(t, τ )(h(τ

0

¯ ) dτ dt = −2 Γ(t, τ )h(τ

 0

1

3 ¯ (h(t)) dt = 0

(the latter inequality is proved by a direct computation of the integral), i.e., the mapping F (σ∗ , ·) is 2-regular ¯ at the point x∗ on the element h. Thus, by virtue of Proposition 5, ∀ k ∈ N through the point (σk , 0), apart from the trivial branch {(σ, 0) | σ ∈ R}, a unique nontrivial smooth branch of solutions to Eq. (40) passes. This branch has the √ form {(σk + σ, χσk cos( σk t) + ρk (σ, t)) | σ ∈ (−δk , δk )}, where χk = 0 and δk > 0 are some numbers, and 841

the function ρk : (−δk , δk ) × [0, 1] → R satisfies the following conditions: ρk (σ, ·) ∈ L2 [0, 1] ∀ σ ∈ (−δk , δk ); moreover, ρk (σ, ·)L2 = o(σ). The case where s = 2 is much more complex. We restrict ourselves to the statement of the following ˜ : R3 → R2 is 2-regular at fact, which can be proved by elementary methods: if the quadratic mapping Q ˜ can be zero or can consist of two or four distinct straight lines. zero, then the cone of the set of zeros of Q 2. Now let (9) hold, and let s ≥ 2. Then, by (14), we have Y1 = lin{∂1 F (σ∗ , x∗ )} ⊕ Im ∂2 F (σ∗ , x∗ ). According to (9) and (30), without loss of generality we can assume that ϕ∗s , ∂1 F (σ∗ , x∗ ) =  0. Then, as is easily seen, we can set Y2 = lin{ϕ1 , . . . , ϕs−1 }; moreover, the corresponding projection P is given by Px =

s−1 



i=1

ϕ∗i , x

(44) 

ϕ∗i , ∂1 F (σ∗ , x∗ ) ∗ − ∗ ϕs , x ϕi , ϕs , ∂1 F (σ∗ , x∗ )

x ∈ X.

(45)

In addition, due to (9) and (14), we have Z1 = {0} × Ker ∂2 F (σ∗ , x∗ ). From here, using (34), (44) and (45), we obtain an explicit expression for the mapping Q, which is introduced according to (26); namely, we obtain     Q(ζ) =

s−1  i=1



s 

 i cjk i ξj ξk ϕ ,

j,k=1

ζ = 0,

s 

ξj ϕj  ,

ξj ∈ R, j = 1, . . . , s,

j=1

where

∗ j k cjk i = ϕi , ∂22 F (σ∗ , x∗ )[ϕ , ϕ ] ϕ∗i , ∂1 F (σ∗ , x∗ ) ∗ ϕs , ∂22 F (σ∗ , x∗ )[ϕj , ϕk ], − ∗ ϕs , ∂1 F (σ∗ , x∗ ) i = 1, . . . , s − 1, j, k = 1, . . . , s. The algebraic branching equation has the form s  j,k=1

cjk i ξj ξk = 0,

i = 1, . . . , s − 1,

s 

ξj2 = 1.

j=1

The further analysis is carried out in full analogy with the previous case. We note only that the existence of the trivial branch of solutions is not possible here. A model example for this case is the mapping F : R × R2 → R2 , F (σ, x) = (σ + χ1 x21 , σ + χ2 x22 ), χ1 , χ2 ∈ R, at the point (0, 0). Example 4. Consider the integral equation  2π 1 1  2π 1 x(t) = √ σ cos t + cos(t − τ )x(τ ) dτ + √ sin t sin τ cos τ (x(τ ))2 dτ, π π 0 π 0 where σ ∈ R is a parameter. For σ = σ∗ = 0, Eq. (46) has the solution x∗ (·) ≡ 0. Introduce the mapping 1 F : R × X → X, (F (σ, x))(t) = x(t) − √ σ cos t π  2π  2π 1 1 − cos(t − τ )x(τ ) dτ − √ sin t sin τ cos τ (x(τ ))2 dτ, π 0 π 0 where X = L2 [0, 1]. This mapping is infinitely differentiable on X; moreover, 1 (∂1 F (σ∗ , x∗ ))(t) = − √ cos t, π 842

t ∈ [0, 2π], (46)

(∂2 F (σ∗ , x∗ )ξ)(t) = ξ(t) −

1 π

2π 0 2π

cos(t − τ )ξ(τ ) dτ,

ξ ∈ X,

 2 sin τ cos τ ξ 1 (τ )ξ 2 (τ ) dτ, ξ 1 , ξ 2 ∈ X. (∂22 F (σ∗ , x∗ )[ξ 1 , ξ 2])(t) = − √ sin t π 0 The operator ∂2 F (σ∗ , x∗ ) is a Fredholm integral operator of the second kind with a continuous kernel, i.e., in particular, (7) holds. One can show that

Ker ∂2 F (σ∗ , x∗ ) = lin{ϕ1 , ϕ2 }, where

1 1 ϕ1 (t) = √ sin t, ϕ2 (t) = √ cos t. π π Thus, (25) holds for s = 2; moreover, due to the symmetry of the kernel of the integral operator ∂2 F (σ∗ , x∗ ), we assume that Ker(∂2 F (σ∗ , x∗ ))∗ is spanned by the same elements ϕ1 and ϕ2 . Note that the system {ϕ1 , ϕ2 } is orthonormal. Further, we have ∂1 F (σ∗ , x∗ ) = −ϕ2 ; hence ϕ1 , ∂1 F (σ∗ , x∗ ) = 0 and ϕ2 , ∂1 F (σ∗ , x∗ ) = −1. The coefficients of the branching equation are given by ϕ1 , ϕ2 ∈ X,

1 j k cjk = cjk 1 = ϕ , ∂22 F (σ∗ , x∗ )[ϕ , ϕ ],

j, k = 1, 2,

i.e., we easily see that c11 = c22 = 0,

1 c12 = c21 = − . 2

The quadratic form ˜ : R2 → R, Q(ξ) ˜ Q = −ξ1 ξ2 corresponding to these coefficients, is not definite, and the coordinate axes in R2 are zeros of this form. Thus, two smooth branches of solutions to Eq. (46) with vertical tangents {0}×lin{ϕ1 } and {0}×lin{ϕ2 } at the point (σ∗ , x∗ ), respectively, pass through the point (σ∗ , x∗ ). Example 5. Consider the integral equation (see [26, p. 254])  π 2π cos(t − τ )x(τ ) dτ + sin(t − τ )(x(τ ))2 dτ, t ∈ [0, π], (47) x(t) = σ sin t + π 0 0 where σ ∈ R is a parameter. For σ = σ∗ = 0, Eq. (47) has the solution x∗ (·) ≡ 0. This example is similar to the previous one. As a result, we can show that (σ∗ , x∗ ) is an isolated solution to Eq. (47). 4. On Branching of Solutions to Boundary-Value Problems for Ordinary Differential Equations Let us make some remarks on the application of the obtained results to the problems of differential equations. In the classical operator statements of such problems, as a rule, X = Y ; moreover, X and Y are not Hilbert spaces. However, first, the constructions of Sec. 3 can also be easily extended to the case X = Y by using pairs of biorthogonal bases [26, p. 337]. Second, for example, the boundary-value problems for ordinary differential equations admit a reduction to finite-dimensional statements if one makes use of the classical theorems on the differentiable dependence of solutions on the initial conditions and parameters [3, p. 56] and the obtained results on branching can be applied to such finite-dimensional statements without any qualifications. We consider the parametric boundary-value problem x˙ = f (t, σ, x),

t ∈ [0, 1],

g(σ, x(0), x(1)) = 0,

(48) (49) 843

where f : R × R × Rn → Rn and g : R × Rn × Rn → Rn are sufficiently smooth mappings, n ∈ N, and σ ∈ R plays the role of a parameter. Let, for σ = σ∗ ∈ R, problem (48), (49) have a solution ϕ∗ (·) ∈ C 1 ([0, 1], Rn), x∗ = ϕ∗ (0) ∈ Rn . Then there exist a neighborhood V of the point σ∗ in R and a neighborhood V of the point x∗ in Rn for which there exists a unique smooth mapping ϕ : [0, 1] × V × V → Rn such that ∀ t ∈ [0, 1], ϕ(t, σ∗ , x∗ ) = ϕ∗ (t), 

t

f (τ, σ, ϕ(τ, σ, x)) dτ

ϕ(t, σ, x) = x + 0

(50) ∀ σ ∈ V, ∀ x ∈ V.

(51)

We introduce the mapping F : V × V → Rn ,

F (σ, x) = g(σ, x, ϕ(1, σ, x)),

(52)

and consider Eq. (4) with such an operator. According to (50), F (σ∗ , x∗ ) = 0; moreover, due to (51) and (52), the structure of the set of solutions to Eq. (4) in a neighborhood of the point (σ∗ , x∗ ) in R × Rn completely determines the structure of the set of solutions to the initial parametric problem in a neighborhood of the point (σ∗ , ϕ∗ ) in R × C 1 ([0, 1], Rn ). Indeed, the solution ϕ(·, σ, x) to the boundary-value problem (48) and (49) bijectively corresponds to each point (σ, x) ∈ M. Furthermore, sufficient smoothness of the mappings f and g ensures any necessary smoothness of mappings ϕ and F ; therefore, Eq. (4) with the introduced operator F can be studied by the methods proposed above. The main difficulty here is related to the fact that the mapping ϕ (and hence also F ) is not given explicitly; ϕ is defined as a solution to some (nonlinear) initial problem. The following consideration are helpful here. For our analysis, not the mapping F itself is needed, but its derivatives, which are expressed through the corresponding derivatives of mappings g and ϕ. For example, according to (52), we have ∂1 F (σ∗ , x∗ ) = ∂1 g(σ∗ , x∗ , ϕ∗ (1)) + ∂3 g(σ∗ , x∗ , ϕ∗ (1))∂2 ϕ(1, σ∗ , x∗ ), ∂2 F (σ∗ , x∗ ) = ∂2 g(σ∗ , x∗ , ϕ∗ (1)) + ∂3 g(σ∗ , x∗ , ϕ∗ (1))∂3 ϕ(1, σ∗ , x∗ ).

(53)

But the problems of determination of the derivatives of ϕ are obtained by differentiation of the left- and right-hand sides of relation (51) and are linear initial-value problems. For example, ∂2 ϕ(·, σ∗ , x∗ ) is defined as a solution to the initial-value problem ψ˙ = ∂2 f (t, σ∗ , ϕ∗ (t)) + ∂3 f (t, σ∗ , ϕ∗ (t))ψ, t ∈ [0, 1], ψ(0) = 0, and ∂3 ϕ(·, σ∗ , x∗ ) is defined as a solution to the matrix initial-value problem ˙ = ∂3 f (t, σ∗ , ϕ∗ (t))Ψ, t ∈ [0, 1], Ψ Ψ(0) = En ,

(54) (55)

where En is the identity n × n-matrix. In the cases where one manages to solve such initial problems explicitly, we obtain the explicit formulas for the derivatives of the mapping F . Example 6. Consider the boundary-value problem √ x¨ + σg(t)x 1 − x˙ 2 = 0,

t ∈ [0, 1],

x(0) = x(1) = 0,

(56) (57)

which describes a small bend of a rectilinear rod under the action of a constant load [26, p. 490]. Here g(·) ∈ C[0, 1] is the function of the rigidity of the rod, g(t) > 0 ∀ t ∈ [0, 1], and σ ∈ R+ is the parameter of the load. For any value of the parameter σ, this problem has a trivial solution. Problem (56), (57) is reduced to the form (48), (49) for n = 2: 



f (t, σ, x) = x2 , −σg(t)x1 1 − g(σ, x0 , x1 ) = (x01 , x11 ), 844



x22

,

t ∈ [0, 1], σ ∈ R+ , x ∈ R2 ,

σ ∈ R+ , x0 , x1 ∈ R2 .

Of course, for any value of the parameter σ, problem (48), (49) has the trivial solution ϕ∗ (·) ≡ 0. If the rigidity of the rod is constant, i.e., if g(·) ≡ g > 0, then the solution to problem (54), (55) for an arbitrary σ∗ > 0 can be written in the explicit form   √ √ √ 1 sin( σ∗ gt) cos( σ∗ gt) σ∗ g ∂3 ϕ(t, σ∗ , 0) = . √ √ √ − σ∗ g sin( σ∗ gt) cos( σ∗ gt) Then, due to (53), for the mapping F , which is introduced according to (52), we have 

∂2 F (σ∗ , 0) =

1 √ cos σ∗ g

0 √ √ 1 sin σ∗ g σ∗ g



. 2 2

Thus, the point (σ∗ , 0) is a singular point of the set M iff σ∗ = σk = π gk , k ∈ N. It is easy to verify that all the conditions of Proposition 5 are satisfied at such points (to this end, it is necessary to compute ∂12 F (σk , 0), k ∈ N, which can also be done explicitly). Therefore, ∀ k ∈ N, through the point (σk , 0), apart from the trivial branch, the nontrivial smooth branch of solutions to Eq. (4) passes. The ¯ where h ¯ = (0, 1); this follows from the relation tangent line to this branch is spanned by the element (0, h), ∂22 F (σk , 0) = 0, which can easily be verified; the tangent line to the corresponding branch of solutions to ¯ = (0, ( 1 sin(πkt), cos(πkt))) the parametric problem (48), (49) is spanned by the element (0, ∂3 ϕ(t, σk , 0)h) πk in R × C 1 ([0, 1], R2 ); finally, the tangent line to the corresponding branch of solutions to the parametric problem (56), (57) is spanned by the element (0, sin(πkt)) in R × C 2 ([0, 1], R2). Finally, we note that, in contrast to the analytical branching theory [26], in the present paper, we have stated the conditions under which the picture of branching is completely determined by the linear-quadratic “part” of the operator of the equation; this fact is in line with the ideology of the entire 2-regularity theory. In each of the examples given above, we can add to the operator F under consideration any summands that do not influence the first two derivatives of F at the point (σ∗ , x∗ ), and nothing will change substantially. Acknowledgments. This work was supported by the Russian Foundation for Basic Research, project Nos. 96–01–00288 and 97–01–00188. REFERENCES 1. A. A. Agrachev, “Topology of quadratic mappings and Hessians of smooth mappings,” In: Progress in Sciences and Technology, Series on Algebra, Topology, and Geometry, Vol. 26 [in Russian], All-Union Institute for Scientific and Technical Information, Akad. Nauk SSSR, Moscow (1983), pp. 85–124. 2. V. M. Alekseev, V. M Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979). 3. V. I. Arnol’d, Ordinary Differential Equations [in Russian], Nauka, Moscow (1971). 4. J.-P. Aubine and I. Ekeland, Applied Nonlinear Analysis [Russian translation], Mir, Moscow (1988). 5. E. R. Avakov, “Extremum conditions for smooth problems with equality-type constraints,” Zh. Vychisl. Mat. Mat. Fiz., 25, No. 5, 680–693 (1985). 6. E. R. Avakov, “Theorems on estimates in a neighborhood of a singular point of a mapping,” Mat. Zametki, 47, No. 5, 3–13 (1990). 7. E. R. Avakov, A. A Agrachev, and A. V. Arutyunov, “Level set of a smooth mapping in a neighborhood of a singular point and zeros of a quadratic mapping,” Mat. Sb., 182, No. 8, 1091–1104 (1991). 8. C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford (1977). 9. G. R. Bond and C. E. Siewert, “On the nonconservative equation of transfer for a combination of Rayleigh and isotropic scattering,” Astrophys. J., 164, 97–110 (1971). 10. Yu. G. Borisovich, N. M. Bliznyakov, Ya. A. Izrailevich, and T. N. Fomenko, Introduction to Topology [in Russian], Nauka, Moscow (1995). 11. S. Chandrasekhar, Radiative Transfer, Dover, New York (1960). 12. M. G. Crandall, “An introduction to constructive aspects of bifurcation and the implicit function theorem,” In; Applications of Bifurcation Theory, Academic Press, New York (1977) pp. 1–35. 13. D. W. Decker and H. B. Keller, “Path following near bifurcation,” Commun. Pure Appl. Math., 32, No. 2, 149–175 (1981). 845

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