Solvability of nonlinear equations in case of branching solutions

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems are illustrated on the nonlinear integral equations.


Introduction
The progress in the methods of analysis development and the new nonlinear problems of applied mathematics enable the nonlinear functional analysis novel concepts formulation. In the seminal paper of L.A. Lusternik [1] the main directions of this field paved the anevue for the branching theory of nolinear equations development. The classic works of A.M. Lyapunov, A.I. Nekrasov, J.H. Poincaré, M.A. Krasnoselsky [2], J. Toland and others contributed to this field. In their studies, reductions of the given nonlinear models to the finite-dimensional systems with parameters were used. The finite-dimensional equivalent system is now known as Lyapunov-Schmidt branching system and the corresponding method is known as Lyapunov-Schmidt (LS) method. In the review [3] and monograph of M.M. Vainberg and V.A. Trenogin [4] the basement of the analytical theory of branching solutions in Banach spaces with applications is given. These works contributed to the modern functional analysis development with many applications [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].
Studies of the integral equations in mechanics played the principal role not only in LS method development, but also iniciated the way for functional analysis new chapters construction. Studies of integral equations' bifurcation points are necessary in various mathematical models in the various fields. For example, for some parameter values there the loss of stability may occur and another series of solutions branches off from that bifurcation point.
Such problems include both classical problems of the critical load of the rod and the formation, the emergence of new forms of equilibrium of rotating fluids, a branch at a critical wind speed from the horizontal surface of the waves, and a number of novel challenging bifurcation problems in biochemistry, plasma physics, electrical engineering and many other applied fields. Over the past decade, the branching theory of solutions of nonlinear equations with parameters and its applications have received enormous development and practical applications.
Despite the abundance of literature in the last 20 to 30 years and interesting results focused on the theory of branching solutions, the formulation and proof of the general existence theorems in nonlinear non-standard models with parameters is still an open problem. The problem of approximate methods development in the neighborhood of critical points ia still open. The clarity of the methods and results presentation using the elementary methods is also important. The objective of this article to fill the gap between abstract theory development and concrete problems solution.
It is to be noted that only some part of the total set of results in this field we discuss due to the limited size of the article. Applications and many other outstanding results including cosymmetry by Yudovich, projective-iterative techniques, center manifold reduction, global existence theorems, have remained beyond its scope.
The remainder of this paper is structured as follows. Sec. 2 demonstrates the construction of the main part of the branching Lypunov-Schmidt equation and its analysis. The existence theorems of bifurcation points and bifurcation manifolds of real solutions are proved. These theorems generalizes the numbder of well-known theorems on bifurcation points. Examples of solving integral equations with bifurcation points and points of enhanced bifurcation are given.
Methods for parameterizing the branches of solutions of nonlinear equations in a neighborhood of branch points are described in Sec. 3 and Sec. 4. Iterative methods for constructing branches with the choice of a uniformization parameter are provided that ensure uniform convergence of iterative schemes in the neighborhood of the critical parameter values. Concluding remarks, regularization and generalizations for interwined equations are provided in Sec. 5.

Existence Theorem of Bifurcation Points and Manifolds of Nonlinear Equations
Let X, Y are real Banach spaces, Λ is real normed space. We consider the equation where B : D ⊂ X → Y is closed Fredholm operator with dense domain D, λ ∈ Λ. Nonlinear operator R(x, λ) with values in Y is defined, continuous, and continuosly differentiable in Fréchet sense wrt x in the neighborhood We assume that equation (1) has trivial solution x = 0 for all λ and

Theorem 1.
If order for the point λ = 0 to be a bifurcation point, it is necessary that homogenius linear equation Bx = 0 has nontrivial solution.
Proof. If equation Bx = 0 has only trivial solution, then Fredholm operator B has bounded inverse B −1 and equation (1) can be reduced to equation which meets the condition of the contraction mapping principal in the small neighborhood of pair x = 0, λ = 0. Therefore, equation enjoys unique solution in that neighborhood. Because of improsed conditions R(0, λ) = 0 equation (2) has only trivial solution in the small neighborhood of the point x = 0, λ = 0. Theorem is proved.
Let us now focus on the sufficient conditions of bifurcation points existence. We introduce the basis Then from the Schmidt -Trenogin Lemma (see Lemma 2.1.1. [? ]) it follows that operatorB = B + ∑ n i=1 ·, γ i z i is contunuosly invertible. Let Γ =B −1 . Then Γz i = φ i , i = 1, 2, . . . , n. Let us introduce the projectors . . , n}. Let us rewrite eq. (1) as following systemB Equation (3) by multiplication with operator Γ can be reduced to Using change eq. (5) can be reduced to the following equation For sake of clarity let us assume ∑ n s=1 ξ s φ s = ξφ. For arbitrary ξ, λ from small neighborhood of origin due to contraction mapping principal the sequence u m = ΓR(ξφ + u m−1 , λ), u 0 = 0 converges to unique solution u(ξ, λ) of equation (7). In that case u(0, λ) = 0. Because of then, taking into account function u continuity wrt ξ, λ and equality R x (0, 0) = 0 in the small neighborhood of origin, we get the following equality This formula can be presented as follows as ||R x (ξφ + u, λ)|| ≤ q < 1. Therefore, using the Taylor formula, the desired function u(ξ, λ) in the problem of bifurcation point search in the neighborhood of point ξ = 0 can be represented as following series where ||r(ξ, λ)|| = o(||ξ||). Taking into account (4), (6), (7), the following finite-dimentional branching system of Lyapunov-Schmidt (LS) can be derived Taking into account equality Γ * γ k = ψ k , branching system (8) can be presented as follows or, briefly, in the matrix form Here Let us employ matrix M(λ) to get the sufficient conditions for point λ = 0 to be such a bifurcation. We introduce the set {λ ∈ Λ : det M(λ) = 0} which contain the possible bifurcation point λ = 0.
Let us introduce the condition Condition 1. Let in the neighborhood of point λ = 0 there exists set S which is Jordan continuum, and S = S + ∪ S − and 0 ∈ ∂S + ∩ ∂S − . There let exists continuous mapping λ(t) as t ∈ [−1, 1] with values in S such as λ :

Corollary 3.
Let condition I is fulfilled for ∀λ ∈ Ω 0 . Then Ω 0 will be bifurcation set of eq. (1). If in that case Ω 0 is connected set and each points belongs to the neighborhood of homeomorphic some set of space R n then Ω 0 will be n-dimentional manyfolds of bifurcation of eq. (1).
From Theorem 1 it follows the known streithern of the known Theorem of M.A. Krasnoselsky on bifurcation point of odd mutiplicity.

Definition 3. System of branching equations of Lyapunov
is symmetric for ∀(ξ, λ) from neighborhood of null. Let us outline that Then we have the following lemma on potentiality of eq. (9) . . are symmetric in the neighborhood of (0, 0).
Proof. In conditions of the symmetric operators B and R x the equalities φ i = ψ i , i = 1, . . . , n are valid and for m = 1, 2, . . . . Therefore, for arbitrary ξ, λ from zero neighnborhood there following equalities are valid and LS is potential in sense of definition 3.

Remark 1.
In some special cases for λ ∈ R 1 we have provided the analytical proofs of this lemma using the Rolle theorem, Morse lemma and local coordinates. Using Lemma 2 there following theorem on bifurcation points existence is valid. Theorem 3. Let LS eq.(9) is potential and let condition II be fulfilled for ν 1 = ν 2 . Then λ = 0 is bifurcation point of equation (1). If in such a conditions x = 0 is isolated solution of equation Bx = R(x, 0), then λ = 0 will be the strong bifurcation point of eq. (1).
is positive defined and symmetric matrix, and for λ ∈ (−ε, 0) is negative defined and symmetric matrix. Then λ = 0 is bifurcation point of equation (1).
1st case: a(λ 0 ) = 1, a (λ 0 ) = 0. Using theorem 2 λ 0 is bifurcation point. Moreover, branching equation here is following and exists two small real solutions Let t 0 ta(t) dt = 0. Using Theorems 2 and 3 we can conclude that λ = 0 is bifurcation point. All the solutions of this equation can be presented as follows Then there are two cases: 1st case. Let λ = 0. Then c 1 = 0, c 2 = Then λ = 0 is unique bifurcation point. 2nd case. Let λ = 0. Then x(t) = 3tc 1 + a(t)c 2 , and c 2 = 3c 2 1 + c 2 2 1 0 a 2 (s) ds. Hence in the second case equation has two c-parametric solutions Let us consider one more model from mechanics. Operator F(x, λ) is differentiable wrt x in sense of Fréchet and Theorems 2 and 3 can be applied. Here cos nt cos ns x(s) ds. For construction of parametric solutions in other simple cases it is usefull to use the following result.

Proof.
First of all let us notice that BΓu = u if u, φ i = 0, i = 1, . . . , n. Then, taking into account conditions of the Lemma, we get the following equation to find u The latter equation for sufficiently small |c i |, i = 1, . . . , n using the implicit operator theorem will enjoy unique continuous solution u(c) → 0 and this solution can be found using successive approximations

Example 4.
Let us consider the equation where all the function are continuous,

Solutions Parametrization and Iterations in Branch Points Neighborhood
The objective of this section is to describe the iteration scheme with uniformization parameter selection and initial approximations of branches of solution of eq. (1). It is to be noted that in sec. 3 condition R(0, λ) = 0 can be unsatisfied.
An important role of power geometry [38] and Newton diagram is well known in asymptotic analysis of finite-dimentional systems when implicite theorem's conditions are not fulfilled. Solution of operator equation (1) reduces to solution of such type finite-dimentional LS system.
The main stages of this approach we describe below. Similar with sec. 1 let us consider the equation is Fredholm operator. We have to construct the solution x → 0 as λ → 0.
Proof follows from Implicit Function Theorem due to conditions 1 and 2. Using substitution of determined ξ i (ε), i = 1, . . . , n and λ(ε) into (17) and taking into account (2) we get the desired pair x(ε), λ(ε) satisfies equation (13). Then the following Theorem takes place System (20) can contain several solutions and choice of vectors α, θ is not unique in general case, the eq. (13) can contain several solutions.
As result, the following statement can be formulated concerning the existence and construction of the analytical solution of eq. (13).

Remarks, regularization and generalizations
The right hand side of the iteration scheme (27) contains operator Γ introduced by V.A. Trenogin [39] and negative powers of the small parameter ε. But this singularity is resolvable. Indeed, in case of polynomial nonlinearity wrt negative powers of ε one can eliminate the corresponding powers of parameter ε. For more details readers may refer to [9,30,31]. Then, taking into account boundness of operator Γ and its regularizing properties [23,39] convergence of proposed N-steps method of successive approximations will be uniform in the branch point's neighborhood. If it is not possible to perform explicite eliminations, then for sake of stable computations in case of negative powers of ε one can employ the change of ε onto ε + signε δ ν , where 0 < ν < 1/2p, p = max 1≤j≤n (θ j − r), where δ is maximal absolute error of computations. Then proposed iteration scheme can be classified as Tikhonov-Lavrentiev regularisation algorithm.
Finally, let us outline that in number of applications Condition 2 for branching system is not satisfied. Analysis of corresponding branching solutions depending on free parameters linked with model's symmetry requires methods from [23,25,26,31]. Usually, in such cases it is assumed the existence of linear bounded operators S ∈ L(X → X) and K ∈ L(Y → Y) such as BS = KB, R(SX, λ) = KR(x, λ) for ∀x, λ ∈ Ω.
Operators S, K can be projectors. If problem G-invariant then S, K can be parametric representations of G-group. In that case we say that eq. (13) is (S, K)-interwined. In papers [24,26] the iterative approach is implemented and developed using ideas of analytical method of Lyapunov-Schmidt in case of (S, K)-interwined equations. In this case it is allowed to change the parameter of uniformization of solutions branches.

Conclusion
Results of this paper enable applications of the existence theorems for bifurcation points of nonlinear BVP problems and make it possible to construct an appropriate solutions. Our method has been also applied for solution of degenerate operator-differential and integral equations [17,18,21,23,32], [37].
Problem of optimal uniformization parameters selection needs to take into account an insight of the problems and it is not yet solved in algebraic form. As footnote, let us outline the formulation and proof of the nonlocal theorems of existence of branching solutions in nonstandart models remains an important problem. For solution of these problems the Trenogin's nonlocal theorems from [6,7] can be employed.