Continuous Dependence of the Singular Nonlinear Van der Pol Equation Solutions with Respect to the Boundary Conditions: Elements of p-regularity Theory

This paper studies the problem of the continuous dependence of Van der Pol equation solutions with respect to the boundary conditions. We provide a new approach for the existence of such solutions via p-regularity theory. Several existence theorems about continuous solutions are established.


Introduction
The continuous dependence of the solutions of differential equations with respect to the boundary conditions is a difficult topic in mathematical sciences. Usually such a problem is given as a nonlinear equation F(x) = 0, where F is a sufficiently smooth map between Banach spaces X and Y . The mapping F is regular at the point x * , if the operator F (x * ) is invertible, i.e. when F (x * ) is surjective. The p-regularity theory (see, for example [4,5,11]) deals with nonregular (degenerate, singular) cases, i.e. with the problems, when the operator F (x * ) is not surjective. In this paper, we consider a nonlinear boundary value problem of the following type: (1.1) where μ = (ν, ρ) are small parameters and μ * = (0, 0). If ν = 0 and ρ = 0, then this problem is the well-known nonlinear Van der Pol equation from the theory of oscillations, and its solutions represent essential interest in different fields of physics and mathematics. This problem always has the trivial solution x * (t) = 0. We will consider the singular case of this equation, namely, with the singular parameter σ = 0 or σ = ± √ 3, since only for these σ the operator F σ (x * ) is singular. For simplicity, further all our constructions and investigations will be carried for σ = 0, i.e. for the equation x + x + x p = 0, x(0) = 0, x(2π) = 0, (1.2) because the case σ = ± √ 3 can be studied similarly. We will solve the problem of the existing continuous solution of the perturbed Van der Pol boundary value problem where ν and ρ are small parameters, which we denote as μ(ν, ρ). We consider the case p = 2, although the case of p > 2 is treated similarly, and based on p-regularity theory, we prove the existence of continuous dependent solutions of (1.3) with respect to boundary conditions μ (of course, small parameter μ) and, more importantly, provide the formula for these solutions.
In degenerate problems, it is typical to study the existence of solutions. Usually, there is not just one solution but many solutions. This is an important problem. However, the question about the existence of continuous solutions is equally interesting because there are no results in this area. Note that in the regular case, the classical Implicit Function Theorem is often used to address such problems. In the nonregular case, this theorem cannot be applied. The results obtained in the paper are based on the generalization of the Implicit Function Theorem for nonregular mappings, which were presented, for example, in [1,2,9,10]. We will provide a strong definition of the constructions of p-regularity theory and adapt it to the issue under consideration.
Here we'll make a note about multivalued operators. Let X and Y be the Banach spaces. By the mapping : X → 2 Y we mean a multivalued mapping (multimapping) from X to the set of all subsets of a space Y. Let ρ(x, y) = x − y be the distance between elements x and y in a Banach space and let ρ( For a linear surjective operator : X → Y we denote by −1 its right inverse, that is −1 : Y → 2 X which takes an element y ∈ Y to its complete inverse image of the mapping , −1 y = {x ∈ X : x = y}, and of course −1 = I Y . We set −1 y = inf{ x : x ∈ −1 y}. By the "norm" of such right inverse operator we mean the number By the Banach theorem on surjective linear operator, we have −1 < ∞. Note, that if is one-to-one, then −1 is the usual norm of the operator −1 . In our considerations, by −1 we shall mean just right inverse multivalued operator with the norm defined by (1.4). An important element of our study is Theorem 3.1. It is an analogue of the Lyusternik Theorem on the tangent cone, which concerns the existence of continuous solutions of the In its proof, we apply the Michael Selection Theorem (see [8]), which we provide in modified form: there exists a continuous mapping M : Y → X , such that AM(y) = y and M(y) ≤ c y , where c > 0 is a constant independent of y.
To the end of our article we formulate two other existence theorems about continuous solutions. We can prove them analogously.
The illustration of our problem is the following singular Van der Pol nonlinear boundary value problem (1.3) where ν and ρ are small parameters from U (ν * , ρ * ) = U (0, 0), ν * = 0, ρ * = 0. We point out that (1.5) always has a trivial solution x * (t) = 0 if μ = 0. We will consider the case p = 2 and parameters μ = (ν, ρ). Based on p-regularity theory and our theorems, in this paper, we prove that a continuous solution of Eq. (1.5), dependent on parameter μ, exists for all μ sufficiently small.

P-factor Lyusternik Theorem and some Elements of p-regularity Theory
The apparatus of p-regularity is an important tool for studying nonlinear problems. In this section, we present some definitions, notations and theorems of p-regularity theory to be used in what follows (see [4][5][6][7]10,11]). We are interested in the following nonlinear problem: where the mapping F : X × M → Z and X , M and Z are Banach spaces. Assume that for some point ( where Z 1 = cl(ImF (x * , μ * )) and W 1 = Z . For W 2 , we use one of the closed complement of Z 1 in Z (if such one there exists). Let P W 2 : Z → W 2 be the projector onto W 2 along Z 1 . By Z 2 , we denote the closure of the linear span of the image of the quadratic mapping P W 2 F (x * , μ * )[·] 2 . Then, inductively, where W i is a choice of closed complement of Z 1 ⊕ · · · ⊕ Z i−1 , i = 2, . . . , p with respect to Z , and P W i : Z → W i is a projector onto W i along Z 1 ⊕ · · · ⊕ Z i−1 , i = 2, . . . , p with respect to Z . Finally, Z p = W p . The order p is the minimal number (if it exists) for which the decomposition (2.2) holds.
In what follows, we will be denote ϕ (0) = ϕ for any mapping ϕ. Define the following mappings: where P Z i : Z → Z i is the projection operator onto Z i along Z 1 ⊕ · · · ⊕ Z i−1 ⊕ Z i+1 ⊕ · · · ⊕ Z p . Then, the mapping F can be represented as such that for any z = (x, μ) is called a p-factor operator depending of h or shortly a p-factor operator if it is clear from the context.

Remark 2.5
For each mapping f i , we have in the completely degenerate case This means that f i−1 are i-factor operators corresponding to completely degenerate mappings f i up to order i. Therefore, the general degeneration of F can be reduced to the study of completely degenerated mappings f i , i = 1, . . . , p and their compositions.
Let's introduce the nonlinear operator p [·] p such that

Definition 2.6
The p-kernel of the operator p is a set Note that the following relation holds: The p-kernel of the operator F ( p) (x * , μ * ) in the completely degenerate case is a set Define the solution set for the mapping F as the set and let T (x * ,μ * ) S denote the tangent cone to the set S at the point (x * , μ * ), i.e., The following theorems describe the tangent cone to the solution set of Eq. (2.1) in the p-regular case.

Theorem 2.11 Let X , M and Z be the Banach spaces, and let the mapping F
Theorem 2.12 (Generalized Lyusternik Theorem, [4]) Let X , M and Z be the Banach spaces, and let the mapping F ∈ C p (X × M, Z ) be p-regular at (x * , μ * ) ∈ X × M. Then, (2.14) The following Lemma (see [7]) will be used in the proof of Theorem 3.1.

Lemma 2.13
Let F : X × M → Z , where X , M, Z are Banach spaces, z = z 1 + · · · + z p , z i ∈ Z i , i = 1, . . . , p, h = 1 and Then, The following Lemma will be important in the study of the surjectivity of p-factor operators in our Example.
This Lemma is a consequence of the following.
The proof is obvious. Lemma 2.14 follows from Lemma 2.15 if we put A 1 = A and A 2 = P 2 B.
Some generalizations of the Implicit Function Theorem to the p-order Implicit Function Theorem for nonregular mappings and the p-order Implicit Function Theorem for the nontrivial kernel are in [1].
The Multivalued Contraction Mapping Theorem will be used in the proof of Theorem 3.1. Its content is available in [3].

Generalization of p-factor Lyusternik Theorem and p-order Implicit Function Theorem
In this section, we prove the Theorem, which is an analogue and generalization of the Lyusternik Theorem on the tangent cone, and discuss the existence of the continuous solution of the equation F(x, μ) = 0. Let V ε (μ * ) will be the neighborhood of μ * and C(V ε (μ * )) will be the continuous mapping from M to X on neighborhood V ε (μ * ).
Therefore, let us note that By the Taylor formula, we have According to (2.9), the mapping f i.e., Then, according to the property of a norm, we obtain where k 2 = max{cd 1 , cd 2 }, t ∈ (0, δ), δ > 0 is sufficiently small and θ p = 2k p t.
This ends the first part of the proof (we proved the existence of solutions).
The last element of the proof, i.e., the continuity of x(μ), follows from the modified form of the Michael Selection Theorem 1.1. Therefore, the multimapping : C(V ε (0)) × V ε (0) → 2 X ×M , which we defined in (3.4) by the formula gives us the continuity selector, i.e., we can choose the continuity solutions (x(tμ),μ(tμ)) of F.
From the continuity of the function x(tμ), the continuity of the function x(μ) follows. This finishes the proof of the Theorem.

Remark 3.3
If we assume that the spaces X × M and Z are finite-dimensional, we can prove the existence of the continuous function x(μ) by the consideration of the following contraction process: Such a process will converge to the continuity mapping x(tμ).

Application
The illustration of our problem is the singular Van der Pol nonlinear boundary value problem where ν and ρ are small parameters from U (ν * , ρ * ) = U (0, 0), ν * = 0, ρ * = 0. Note that (4.1) always has the trivial solution x * (t) = 0. We will consider the case p = 2. We show that for any h ν , h ρ such that h ν = h ρ , the mapping F is 2-regular along element H = [0, h ν , h ρ ], i.e., based on Theorem 3.1, there exists a continuous solution of (4.1) (with p = 2), dependent on parameters μ = (ν, ρ), for h ν = h ρ ; for h ν = h ρ , the mapping F is 2-regular along element H = [sin t, h ν , h ρ ]. All of this means that based on our Theorems, a continuous solution of Eq. (4.1), dependent on parameter μ, exists for all sufficiently small μ.
In the case of a mapping x(t, ν, ρ) that is nonsingular at the solution point x * = x(t, 0, 0) (in what follows, such types of mappings will be considered specifically), from the Implicit Function Theorem, it implies that for sufficiently small (ν, ρ), there exists ν , ρ and a function x μ such that and In this case, we can transfer the problem (4.2) tö and substitute We obtain the problem (4.5) in the form whereF : X → C[0, 2π], X = {C 2 [0, 2π] × R 2 |x(0) = 0,x(2π) = 0} and We will call the mappingF the canonical form of the mapping F.

Example 4.1
Consider the following Van der Pol boundary value problem: where ν and ρ are small parameters from U (ν * , ρ * ) = U (0, 0), ν * = 0, ρ * = 0. Substitute We obtain the following equation: Taking this into account, we consider the following problem equivalent to (4.2): be the canonical form of Eq. (4.8). Then, (4.14) Let us evaluate the second derivative of the mappingF Note that Introducing a scalar product in the form we describe the space Z 1 = ImF x (0, 0, 0), such that Z = Z 1 ⊕ W 2 , W 2 = Ker F x (0, 0, 0). We have It is easy to show that the projection onto W 2 has the form Therefore, i.e., P W 2 = P Z 2 . Now, we define the 2-factor operator hxh ρ sin τ dτ and 2-kernel of the 2-factor operator 2 (h) Taking into account the equations 2π 0 sin 3 τ dτ = 0, 2π 0 sin 2 τ dτ = π (4.25) and the fact that hx = q sin τ (since hx ∈ KerF x (0, 0, 0)), we solve the following equation with unknown q, h ν , h ρ : Using Lemma 2.14, it suffices to take any element z = z 2 = a sin t ∈ Z 2 = W 2 = KerF x (0, 0, 0) = span{sin t}. We are looking for an elementhx = b sin t ∈ KerF x (0, 0, 0). After substitution to the obtained form of the 2-factor operator, we obtain Let Using Lemma 2.14, we take any element z = z 2 = a sin t ∈ Z 2 = W 2 = KerF x (0, 0, 0) = span{sin t}. We look for elementhx = b sin t ∈ KerF x (0, 0, 0). After substituting to the designated 2-factor operator's form, we obtain Hence, Then,F is 2-regular at the point (0, 0, 0) with respect to element H = [0, h ν , h ρ ], such that h ν = h ρ . Therefore, based on our Theorem 3.1, we can formulate the following result for our Example 4.1. By applying the p-regularity theory, we formulate and prove an analogue of the Lyusternik Theorem on the tangent cone, i.e., Theorem 3.1, which consists of the problem regarding the existence of continuous solutions of singular equations of the form F(x * , μ * ) = 0, where F(x, μ) ∈ C p+1 (X × M), F : X × M → Z , M is finite-dimensional space, and X and Z are Banach spaces. We illustrate this problem by a second-order Van der Pol differential Eq. (4.8) with boundary conditions.
Here, we provide the general result of our paper. Consider the following differential equation: k (x * , μ * ). Then, there exists the continuous mapping x = x(t, μ) ∈ U ε (x * (t)), μ ∈ V ε (μ * ), t ∈ [a, b], x(t, μ) ∈ C(V ε (μ * )), for sufficiently small ε > 0, such that F(x(t, μ), μ) = 0, x(a) = ν, x(b) = ρ (5. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.