On Solvability of Boundary Value Problem for Asymmetric Differential Equation Depending on x'

We state the conditions of geometrical nature which guarantee the existence of a solution to the boundary value problem x′′ + 2δx′ + λf(x) − μg(x−) = h(t, x, x′), x(0) = 0 = x(1) with a damping term 2δx′ and nonnegative parameters λ, μ, provided that f(x) − g(x−) is a sector-bounded nonlinearity.


Introduction
Boundary value problems for asymptotically asymmetric ordinary differential equations attract significant attention in the last decades. This partially can be explained by theoretical reasons: it is natural to study the behaviour of asymmetric oscillators versus symmetric (classical) ones. On the other hand, asymmetric equations appear in applications, such as mechanics. Especially interesting is the study of theory of suspension bridges, since possibly unknown nonlinear phenomena lie in underground of it. The interested reader may consult the works [11] and [12] for additional information and more references.
Let us trace now the development of the theoretical issues connected with the problem under consideration.
The classical harmonic differential equation x + λx = 0 (1.1) may be changed to the asymmetric equation as follows The similar question about the problem also can be answered. A set of (λ, µ) such that the problem (1.2), (1.3) has a nontrivial solution is the spectrum of the problem (1.2), (1.3). This spectrum is well known. In the first quadrant of the (λ, µ)-plane it consists of two orthogonal straight lines and a set of hyperbolas (this spectrum is called usually the Fučík spectrum). The complement of the spectrum in the first quadrant is a union of two disjoint sets, S 1 and S 2 . These sets are fully described in the work [8]. It is known [2] that if (λ, µ) ∈ S 1 then the problem (1.5) is solvable for bounded nonlinearities h. Nonlinearities can be introduced in equation (1.2) in another way. A nonlinear asymmetric oscillator in the meaning that restoring forces on the left and right sides nonlinearly depend on replacements x(t) may be associated with equation x + λf (x + ) − µg(x − ) = 0, where f and g are nonlinear nonnegative functions.
In the works [3,4,5,13] the problem together with the normalization condition was considered. The condition (1.7) is not needed in case of the problem (1.2), (1.3) since it is fulfilled automatically. The spectrum of the problem (1.6), (1.7) is a set of all pairs (λ, µ) such that the problem (1.6), (1.7) has a nontrivial solution -in the first quadrant of the (λ, µ)-plane it consists of branches: two orthogonal straight lines and a set of hyperbola looking curves. In [4] some properties of the spectrum were analyzed and was pointed out in particular that branches of the spectrum may have separate components connected at infinity and these separate components can be even bounded. In [5] the problem (1.6), (1.7) was studied provided that the functions f = g are piece-wise linear.
If we release |x (0)| from 1 to α the spectrum of the problem (1.6) is a union of solution surfaces [13] -solution surfaces are in the 3D space (λ, µ, α), where α = x (0). Knowledge of solution surfaces in case of given f and g opens the possibility to evaluate the number of solutions to BVP of the type (1.6). In the works [6,7] the Neumann problem x +λf (x + )−µg(x − ) = 0, x (a) = 0 = x (b), a < b, was considered also.
If the problem is considered with a bounded right side h provided that f = g are sector-bounded nonlinearities [1], the existence results can be obtained also [8]. This can be interpreted as description of an asymmetric oscillator with positive and negative restoring forces given respectively by f and g = f and in presence of external force given by h.
In this paper we consider the boundary value problem The equation in (1.8) describes an asymmetric oscillator with nonlinear restoring forces f and g, an external force h and in presence of damping term given by 2δx . So despite of asymmetric character of the restoring forces the resistance of medium is symmetric.

Generalization of the classical Fučík spectrum
In what follows we need to know the spectrum of the boundary value problem where k > 0, λ > 0, µ > 0. This problem is a generalization (extension) of the classical Fučík problem [2]. We wish to prove the following result. Proposition 1. The boundary value problem (2.1) has a nontrivial solution if and only if (λ, µ) is in the Fučík type spectrum Σ F (k, δ) = ( )) consisting of the branches
Proof. By constructing continuously differentiable solutions with prescribed number of zeros in the interval (0, 1). Suppose for a moment that k = 1 and consider the problem In the case of δ 2 − λ < 0 the solution of the Cauchy problem x + 2δx + λx = 0, x(0) = 0, x (0) = α > 0 has the zeros t i = i π/ √ λ − δ 2 (i = 1, 2, . . .). "Positive" solutions without zeros. It follows from (2.9) that x(t; α) is a solution of (2.8) with positive derivative (x (0) > 0) and without zeros in the interval (0, Similarly solutions of the problem (2.8) with negative derivative (x (0) < 0) and without zeros in the interval (0, 1) arise if δ and µ satisfy the relation . "Positive" solutions with exactly one zero. Consider a solution Consider also a solution with a negative derivative of the problem (2.8) with exactly one zero in the interval (0, 1). There is a smooth junction at the first zero t λ . This solution is positive in the interval (0, t λ 1 ) and negative in (t λ 1 , 1).
Summing up, we have that "Negative" solutions with exactly one zero. Similarly the case of a solution with x (0) < 0, which has exactly one zero in (0, 1), can be considered. For a given δ choose λ and µ such that The junction at the first zero point t µ The obtained solution is negative in the interval (0, t µ 1 ) and positive in (t µ 1 , 1). Hence the expression for the branch Replacing λ and µ in the above considerations by kλ and kµ respectively we obtain the expressions (2.2)-(2.7) for the branches of the spectrum of the problem (2.1). Remark 1. 1) A branch F s j (k, δ) describes all nontrivial "positive" (resp.: "negative") solutions of the problem (2.1), that is, all nontrivial solutions with x (0) > 0 (resp.: is a nontrivial solution of the problem (2.1), then the derivatives x (z) at j-th zeros z ∈ [0, 1] of the solution x(t) (from left to right) are: • for a "positive" solution • for a "negative" solution , . . . , x (1) = (−1) j+1 αe −δ .  3) The branches of the Fučík type spectrum Σ F (k, δ) can be obtained from the classical Fučík spectrum Σ F (1, 0) by composition of the translation parallel to the vector (δ 2 , δ 2 ) and the homothety with the center (0, 0) and coefficient 3 Solvability of the problem (3.1) We wish to prove the existence results for the boundary value problem where f and g are sector-bounded nonlinearities and the damping term 2δx is present. The left-hand side is called a principal part of the problem (3.1). We are looking for (sufficient) conditions on the parameters λ, µ > 0 and the damping coefficient δ which guarantee the existence of a solution to the problem (3.1). We suppose that the following conditions are fulfilled.  Remark 2. 1) The sector-bounded nonlinearity concept, see Fig. 3, was used in Aizerman's conjecture [1] and exploited in the feedback systems theory [9].
2) Roughly speaking, the problem (3.1) has a solution if a triple (k, l, δ) satisfies some requirements. In order to make this point precise, we introduce the sets D j (k, δ).

Regions D j (k, δ)
Suppose that k > 0 and consider the problem (2.1) with the spectrum Σ F (k, δ) described in Proposition 1.
Let us introduce a subset D j (k, δ) (j = 0, 1, 2, . . .) of the first quadrant in the (λ, µ)-plane in which the solutions of the initial value problems have opposite signs at t = 1 and exactly j zeros in (0, 1).

Some preparations to Main Theorem
Since it is essential for proving the main result, we discuss solvability of the problem (3.1) when f (x) = g(x) = x for all x ≥ 0 and h satisfies the condition C). Respectively, consider the problem It follows from Proposition 1 that if (λ, µ) ∈ Σ F (1, δ) then the problem (2.1) with k = 1 has only the trivial solution. This is insufficient for solvability of the problem (3.6). The solvability, however, can be guaranteed for regions D j (1, δ) (j = 0, 1, 2, . . .) -"good " regions for the problem (3.6).

The Main Theorem: statement
The next theorem states the main result.
To prove the Main Theorem, we need some comparison results.

Differential inequality
The below arguments almost one-by-one repeat the analogous considerations in the paper of the authors [8].
The following assertion is a slight modification of Theorem 14.1 in [10].

Angular functions
In what follows we restate the comparison results of [10,Ch. 15] adapted for the case under consideration.

Comparison of angular functions
Consider shortened equation and compare it to equations 14) having in mind the conditions B).
Hence the alternative form of the main theorem can be given. If there exists j ∈ {0, 1, 2, . . .} such that the damping coefficient δ and the (l, k)sector satisfy the conditions (4.1), then there exists a pair of positive parameters (λ, µ) in the region D j (k, l, δ) = D j (k, δ) ∩ D j (l, δ) such that the problem (3.1) has a solution.
• Somewhat peculiar properties of the sets D j (k, l, δ) are described in Proposition 4.