On second solutions of the sixth-order nonlinear mathematical model with measured derivatives

In the present paper the sixth-order nonlinear mathematical model with nonsmooth solutions is studied. We consider a case, when the problem is guaranteed to have one solution and investigate the question on the presence of one more. Using the pointwise approach of Yu. V. Pokorny, which has shown its effectiveness in analyzing models of the second and fourth orders, sufficient conditions of the existence of the second solution for the sixth-order model with derivatives with respect to measure are obtained.


Introduction
In the present paper a mathematical model that obviously has one known solution is studied. We investigate the question on the presence of one more solution. Without loss of generality, we will consider the known solution to be zero, since it is possible to make a functional change that makes this solution be equal to zero. Notice that a qualitative theory of equations with nonsmooth solutions began to develop rapidly after the publication in 1999 of the work of Yu. V. Pokorny [1]. The most profound results concerning this topic associated with monographs [2][3][4][5], works [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. This efficiency is explained quite simply: when we apply derivatives with respect to the measures, the equation, in contrast to theory of generalized functions, becomes defined in each point and makes it possible to use qualitative methods for analyzing of solutions. Indeed, when using the theory of Schwartz-Sobolev distributions according difficult problems are emerged. The first problem is that only weak solvability can be established. Hence equations are not suitable for applications. The second problem, which has not yet been solved, arises when the generalized function multiply by the discontinuous one. The third problem is that equations in generalized functions are the equalities of two functionals defined on the space of basic functions. Hence it is extremely difficult to apply methods of qualitative analysis to such equations.

Preliminaries
We will consider a solution of (1) in the class E of twice continuously differentiable functions u(x) At the points ξ belonging to the set of discontinuity points of µ(x), the equation in (1) is understood as the equality where ∆u(ξ) is a complete jump of the function u(x) at the point ξ.
The equation from (1) is given almost everywhere (with respect to the measure µ) on the special extension of the interval [0; ℓ]. Let S(µ) be a set of discontinuity points of the function µ(x). We introduce on J µ = [0; ℓ] \ S(µ) the metric ̺(x; y) = |µ(x) − µ(y)|. The resulting metric space (J µ ; µ) is not complete. The standard completion leads (up to isomorphism) to the set [0; ℓ] S , in which each point ξ ∈ S(µ) is replaced with a pair of elements ξ − 0, ξ + 0, which were previously limit values. By inducing ordering from the original set, we obtain the inequalities x < ξ − 0 < ξ + 0 < y for all x, y for which the inequalities x < ξ < y hold in the initial segment.
The function v(x) at the points ξ − 0 and ξ + 0 of the set [0; ℓ] S is defined by the limiting values. For a function defined in this way, we retain the previous notation. The function defined on this set becomes continuous with respect to the metric ̺(x; y).
The union of [0; ℓ] S and S(µ) gives us the set [0; ℓ] µ , in which each point ξ ∈ S(µ) is replaced by a triple of elements {ξ − 0; ξ; ξ + 0}. We suppose that the equation is given on this set.
We assume that the functions p(x), r(x), g(x) and Q(x) are µ-absolutely continuous on p(x) > 0, Q(x) does not decrease, and f (x, u) satisfies the conditions of Carathéodory, i.e., • f (x, u) is defined and continuous in u for almost all x (with respect to the µ-measure); • the function f (x, u) is measurable in x for every u; We say that the homogeneous equation is non-oscillating on [0; ℓ] if any of its non-trivial solutions has at most five zeros with respect to multiplicities. We denote by K the cone of non-negative functions on moreover, (2) possesses the non-degeneracy property (the boundary value problem for F ′ σ (x) ≡ 0 has only a trivial solution); homogeneous equation where m, M are finite positive constants belonging to [0, ℓ] for all x and s .
Proof. Let us show that the non-oscillation of the homogeneous equation implies the positive invertibility of the boundary value problem. Let . Based on the Polya-Mamman representation (the proof is similar to [5]), we have the representation (here ψ i (x) are separated from zero and can be differentiated with respect to the corresponding measure) from which it follows that for all x. The last inequality shows that the function does not increase on the segment [0; ℓ]. Therefore, it has at most one sign change. Since x (x) has at most one sign change.
Continuing the reasoning, we obtain that the functions and x (x) have no more than two sign changes; functions µ (x) have no more than three sign changes; func- x (x) have no more than four sign changes. The boundary conditions imply the equalities ( has at most two sign changes. Similarly, the function (ψ 0 u) ′ x (x) has no more than one sign change, functions ψ 0 u(x) and u(x) keep the sign on [0; ℓ]. It remains to note that u(x) can not have negative values.
To complete the proof, it remains to note that the relation G(x,s) G * (x,s) is bounded and separated from zero.
Proof. Since Lu = 0 does not oscillate on [0; ℓ] σ the integral operator transforms the cone K into a narrower set moreover, it acts and is completely continuous on C[0; ℓ]. Moreover, any fixed point (8) is a solution to the differential model (1). Thus, the question of the existence of a fixed point in K for the operator A is restricted to K( u 0 ). The solvability (6) is equivalent to the solvability of the equation λAu = u with the operator (8). If, in addition, it turns out that the last equation has a solution u 1 ∈ K( u 0 ) satisfying the inequality u 1 C ≥ R 0 for some R 0 > 0, then according to the definition of K( u 0 ) u 1 (x) ≥ R 0 u 0 (x) for all x ∈ [0; ℓ]; solution u 1 (x) satisfies (7) with R = M · R 0 . Therefore, the condition of the theorem on the absence of such solutions means that the equation λAu = u for λ ∈ (0, 1) has no solutions u 1 (x) ∈ K( u 0 ) such that u 1 ≥ R = R M . Consider on K( u 0 ) the operator A: The operator A is completely continuous on K( u 0 ) and transforms K( u 0 ) into a bounded part, that is, A leaves invariant the intersection of K( u 0 ) with a ball of some radius centered at the origin. And since this intersection is convex, bounded and closed, by virtue of the Schauder principle A has a fixed point u in K( u 0 ): u = A u.
If we assume that u C > R, then In other words, problem (6) with λ = R u C < 1 has a solution v satisfying the inequality v C > R. Thus, the inequality u C > R is impossible. Therefore, v C ≤ R. By virtue of the definition A, we obtain A u = u. The theorem is proved.
It is natural to check the conditions of the theorem for sufficiently large R. Therefore, we can specify its conditions in terms of the asymptotic properties of the function f (x, u).
The function f (x, u), which generates a superposition operator acting from C[0, ℓ] to some is asymptotically zero. In this case, we will write q(x) = f ′ ∞ (x). The simplest example of an asymptotically zero function is a function that is bounded on the entire plane. Less trivial: f (x, u) is not bounded, but has slow, for example, logarithmic, growth at infinity. Theorem 2. Let the following conditions be satisfied: for all x ∈ [0, ℓ] and u ≥ 0; f ⊕ (x, u) is asymptotically linear; 2) equation Lu = 0 does not oscillate on [0; ℓ] µ ; has no spectrum points in the unit circle.
Proof. Let us show that for a sufficiently large R and any λ ∈ (0, 1) the model has no solutions satisfying the inequality u(x) ≥ Rũ 0 (x). Suppose the opposite: for any R n → ∞ there exists a sequence of functions u n (x) ∈ K and numbers λ n ∈ (0, 1) such that u n (x) is a solution of the differential model (10) and satisfies the inequality u n (x) ≥ R n u 0 (x). The last inequality implies u n C → ∞. The function u n (x) satisfies the equation where A = GF , G, and F are defined by Thus, by the first condition of the theorem and the functions f ⊕ 0 (x, v k (x)) belong to L p,σ [0; ℓ] σ . Inequality (11) can be rewritten as Let us show that the supremum on (0, ℓ) of the last term on the right-hand side of (12) tends to zero. Indeed, the asymptotic linearity of the function f ⊕ (x, u) implies that for any ǫ > 0 there exists N = N (ǫ) such that the inequality holds for every v(x) satisfying the inequality v(x) ≥ N (ǫ). Since u n ∈ K( u 0 ) and u n C → ∞ we obtain u n (x) ≥ u n C u 0 (x), and for u n C ≥ N we will have u n C ≥ N u 0 (x). Substituting un(x) u 0 (x) in (13) instead of v(x), we have f ⊕ (x, u n (x)) − q(x)u n (x) p < ǫ u n C . Since