Boundary Value Problems for a Super-Sublinear Asymmetric Oscillator: The Exact Number of Solutions

Properties of asymmetric oscillator described by the equation x󸀠󸀠 = −λ(x)p + μ(x)q (i), where p ≥ 1 and 0 < q ≤ 1, are studied. A set of (λ, μ) such that the problem (i), x(0) = 0 = x(1) (ii), and |x󸀠(0)| = α (iii) have a nontrivial solution, is called α-spectrum. We give full description of α-spectra in terms of solution sets and solution surfaces.The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.


Introduction
Asymmetric oscillators were studied intensively starting from the works by Kufner and Fu č ík; see [1] and references therein.Simple equations like (2) given with the boundary conditions allow for complete investigation of spectra.It is known that the spectrum of the problem (2), ( 4) is a set of hyperbola looking curves in the (, )-plane.On the other hand, there is a plenty of works devoted to one-parameter case of equations   + () = 0 given together with the two-point boundary conditions.Due to nonlinearity of  one should consider solutions (; ) with different  =   (0).Bifurcation diagrams in terms of  and , or ‖‖ and , can serve then to evaluate the number of solutions [2][3][4].
The aim of our study in this paper is to describe properties of the spectrum of the problem (0) = 0,  (1) = 0.
For this we study first the time maps for the related functions (Section 2), then we give the analytical description of the spectrum (Section 3), and formulate the properties of the spectrum (Section 4), including the asymptotics.In Section 5 we consider the solution sets and solution surfaces which bear information on multiplicity of solutions to the problem.Analysis of properties of solution surfaces (Section 6) can give us estimations of the number of solutions to the problem (Section 7).These estimations contain also information of properties of solutions such as the number of zeros and evaluations of   (0).

Time Maps
Consider the Cauchy problem where  > 0. Solutions of this problem for  and  positive have a zero.This zero will be denoted   (, ) and called time map function; more on time maps can be found in [9,10].
(2) The function   (, ) for the problem (5) is where The function   (, ) for fixed  and  is strictly decreasing function of  and possesses the properties (4) The function   (, ) for fixed  and  is decreasing function of .
(5) The function   (, ) for fixed  and  is increasing function of .
Proof.By standard computations.
Proof.We prove the theorem only for solutions which have exactly one zero in (0, 1) and satisfy the initial condition |  (0)| =  > 0.
Proof.Follows from the relations ( 11) and Proposition 7.

Solution Sets and Solution Surfaces
Definition 11.A solution set of the problems (3) and ( 4) is a set  of all triples (, , ) ( ≥ 0,  ≥ 0,  > 0) such that there exists a nontrivial solution of the problem.
Let us distinguish between solutions of the problems (3) and ( 4) with different number of zeros in the interval (0, 1).

Description and Properties of a Solution
Set.We will identify the cross section of a solution surface    with the plane  =  0 > 0 ( 0 is fixed) with its projection to the (, )plane.This projection is in fact the respective branch    ( 0 ) of the spectrum of the problem (3), (4), and (9).
For given  ̸ =  and  > 0 set  = (/).Suppose (, , ) ∈  Remark 14.Since solution surfaces  ± 2−1 and  ± 2−1 ( ̸ = ) are centro-affine equivalent, they have similar shape.Therefore it is enough to study properties of the solution surface  ± 1 , in order to know properties of other odd-numbered solution surfaces.The same is true for  − 2−1 .Irrespective of the choice of  no oscillatory (with at least one zero in (0, 1)) solution of the problems (3), (4), and (9) exists for (, ) in the "dead zone" below the envelope (see Figure 1).

Cross-Sections of Solution
The analytical description of envelopes, corresponding to branches of spectra, follows.

The Number of Solutions by Geometrical Analysis of Solution Surfaces
We can detect the precise number of solutions to the problem for given positive (, ).We can evaluate the initial values    (0) for solutions on a basis of geometrical analysis of solution surfaces and the respective envelopes.The nodal structure of solutions can be described also.
Proposition 17.There exists a unique parameter  0 > 0 such that the line  =  * > 0 Proof.(a) Consider equation   () =  2    () +    () = 0, which turns to It can be found from the above that is the only root of the equation   () = 0. Hence at the point ( 0 ,  0 ) = (( 0 ), ( 0 )) the curve G  has a unique tangent line parallel to the  axis.
(b) Since the given parametrization of the curve G  is regular, then   ( 0 ) ̸ = 0 and in some neighborhood of the point ( 0 ,  0 ) the curve G  can be represented as the graph of the function  = ().Now find +  2 ( − 1) The routine calculations show that the expression in parentheses is equal to that if a parameter  > 0 goes from 0 to +∞, then a point (, ) ∈ G  goes from the point (+∞, 0+) to the point (+∞, +∞).
(d) It follows from the above argument that if the parameter  > 0 goes from 0 to +∞ then a point (, ) ∈ G  goes from (+∞, 0+) to ( 0 ,  0 ), then turns to the right, and goes to (+∞, +∞).Since the parametrization of the curve G  is without self-intersection points, one can deduce that the curve G  is a union of two branches (i.e., graphs of the functions  =  1 () and  =  2 () , where ( 0 <  < +∞) which do not intersect and are continuously "glued" at the point ( 0 ,  0 ).
Before presenting "the exact number of solutions" result we make the following conventions: (1) solution means a nontrivial solution of the problems (3) and ( 4), (2) E   means the envelope, where  ∈ N and  is either + or − or ±, Proof.It can be verified analytically that ( 0 ,  0 ) ∈ E   , where and  0 =  2 ( 0 ).Recall that the curve G  is the projection of the 3D curve F| = 2  to the (, )-plane.Taking in mind Proposition 17 we can assert that there exists a unique parameter  0 > 0, see (53), such that the line parallel to the  axis in the (, , )-space going through the point (, ), where  > 0 and  =  2 , and the curve F| = 2  (hence the solution surface F also) (i) does not intersect if  <    (), (ii) intersects only once if  =    (), (iii) intersects exactly at two points if  >    (), (see Figure 3).
Depending on the value of  one can detect the number of -solutions as the theorem states.For example, in the case (d): if  is odd,  >    () and  = ±, then the mentioned above line intersects the solution surface  +  exactly twice.Since  −  =  +  the mentioned above line intersects the solution surface  −  exactly twice also.Hence the problem (3), ( 4) has exactly four solutions with  zeros in (0; 1).Now we are able to prove the main theorem.In this theorem we suppose that  − 0 () = 0 and  + 0 () = 0 for all  > 0 and will use auxiliary envelopes E − 0 :  =  − 0 () and E + 0 :  =  + 0 ().
(  Proof.First notice that if  > 0 and  > 0 then there exists exactly one ±-solution without zeros in (0, 1), in fact, accordingly to conventions, two solutions without zeros in (0, 1), and of opposite sign values of   (0).We consider only the case  − 1 > 0. Similarly two other possible cases  − 1 < 0 and  − 1 = 0 can be treated.
It follows from Proposition 15 that the envelopes are ordered as (ii) if  = 2, then there are two solutions without zeros, two ±-solutions with 1 zero (that is, four solutions with 1 zero), two "+"-solution with 2 zeros and two "−"solutions with 2 zeros; totally there are 10 = 8 ⋅ 2 − 6 (nontrivial) solutions; (iii) if  = 3, then in addition to 10 solutions mentioned in the previous step, there are also two ±-solutions with 3 zeros (that is, four solutions with 3 zeros), two "+"-solutions with 4 zeros and two "−"-solution with 4 zeros; totally there are 18 = 8 ⋅ 3 − 6 (nontrivial) solutions and so on; (iv) if  ∈ N, then totally there are 8 − 6 (nontrivial) solutions.
The other cases can be considered analogously.