EXISTENCE OF A SOLUTION FOR A GENERAL ORDER BOUNDARY VALUE PROBLEM USING THE LERAY–SCHAUDER FIXED POINT THEOREM

Introduction/purpose: This paper illustrates the existence of a generic Green’s function for a boundary value problem of arbitrary order that appears in many phenomena of heat convection, e.g. in the atmosphere, in the oceans, and on the Sun’s surface. Methods: A fixed point theorem in the Leray–Schauder form has been used to establish the existence of a fixed point in the problem. Results: The existence of a solution has been shown for an arbitrary order of the problem. Some practical examples are proposed. Conclusions: The boundary problem has a solution for an arbitrary order n.

(1) This kind of equations occurs, for instance, when studying the problem of the beginning of thermal instability in horizontal layers of fluid heated from below. This kind of phenomena could be observed in convection patterns in several situations, for instance, in the atmosphere, in the oceans, when considering the coupling with a strong electromagnetic field, or on the Sun's surface (Chandrasekhar, 1961). This work will extend the results of (Fabiano et al, 2020), (Ahmad & Ntouyas, 2012) and (Ma, 2000) to an equation of a generic order 2n.
Introduce the Green's functions G l (x, s, n) and G r (x, s, n) of problem (1) where G l (x, s, n) is defined for 0 ≤ x < s ≤ 1 and G r (x, s, n) is defined for 0 ≤ s < x ≤ 1, (x, s) → G l,r (x, s, n); (x, s) ∈ [0, 1] × [0, 1], G l,r ∈ C 2n over R such that solve the following equation: ∂ ∂x 2n G l,r (x, s, n) = δ(s − x) . ( The complete Green's function is thus obtained by the linear combination of the above two, θ is the Heaviside step function. Given the inhomogeneous problem solved by the Green's function provides solution to (4) in the integral form The functions G l,r (x, s, n) are multivariate polynomials in two variables x and s of the order 2n − 1 to be sought in the form and where the coefficient c(k, n) is clearly given in combinatoric terms, k ≤ n. Imposing boundary conditions (4) to the Green's function, we obtain that So, we could infer the following results. For G l (x, s, n) the powers of x range from n to 2n − 1, while for the powers of s we have the range from 0 to n − 1. For G r (x, s, n) we find the same situation when swapping s with x. Therefore, the coefficient c(k, n) has to be symmetric under this exchange.
We conclude that the coefficient c(k, n) of both functions G l,r (x, s, n) is given by: Notice that |c(k, n)| < 1 for all k, n. This observation will be useful in the sequel. The resulting Green's function G(x, s, n) and its x derivatives are continuous up to order 2n − 2, and present the discontinuity in −1 at order 2n − 1, because of the Dirac's δ function.
The above discussion concludes the proof of the following lemma: Lemma 1. Let x → y(x), x ∈ [0, 1] be a function of class C 2n in R, let (x, y, z) → χ(x, y, z); x ∈ [0, 1], (y, z) ∈ R 2 be a function of class C in R and let χ be a function of class C in R. Then the Green's function of the problem (4) obeying to equation (2) is given by formulas (3), (6), (7), and (10).

Solution
In this section, we will provide the main result of this work: the solution of the problem in (1) for a generic n .
Define the integral operator Υ as follows: According to Lemma 1, this operator provides a solution of problem (4) for a generic order n provided that it has a fixed point. We shall make use of the following theorem of (Bekri & Benaicha, 2018) and (Shanmugam et al, 2019), the Leray-Schauder form of the fixed point theorem appears in (Isac, 2006), (Deimling, 1985) and (Zvyagin & Baranovskii, 2010): an open bounded subset for which 0 ∈ U and let Υ : U → E be a completely continuous operator. Then only one of the following possibilities is true: 1. Υ possesses a fixed pointx ∈ U 2. there exist an element x ∈ ∂U and a real number λ > 1 such that Therefore, in order to establish the existence of a solution it is necessary to prove that our integral operator Υ possesses a fixed point. The following two theorems are devoted to this problem.
Up to this point, we have established the existence of a solution for the boundary value problem. In the following theorem, we show some parameter dependent bounds that actually lead to the existence of a solution.

There exists a constant a > 1 such that
Then problem (1) has at least one nontrivial solution x → ξ(x), x ∈ [0, 1] of class C 2n in R.
Proof. In order to prove this theorem, one has to show that the integral operator (11) has A < 1, A being defined in Theorem 2.
To prove point 1, we proceed as follows: (14) and when > −2, one has For point 2, we have In the case of point 3, we make use of Hölder inequality for which S |f (s)g(s)|ds ≤ S |f (s)| a ds 1/a S |g(s)| b ds 1/b , whenever f and g are measurable functions on the domain S and 1/a + 1/b = 1. We have
Setting α = one has to consider the inequality for all x ∈ [0, 1]. In the parameter space (α, , n), the above inequality is satisfied, for instance, for the case α = = n, whenever In this case, the existence of a nontrivial solution ξ(x) ∈ C 2n [0, 1] is guaranteed for problem (17).
Example 2 Consider the problem for a generic n ≥ 1 This problem satisfies all requirements of Theorem 2. In fact, one has For a generic α > 0, from hypothesis 2 of Theorem 3, we have that for all x ∈ [0, 1] and for all (y, z) ∈ R 2 .
Setting α = m, one has to investigate the following inequality: for all x ∈ [0, 1]. In the parameter space (α, m, n), the above inequality is satisfied, for instance, for the case α = m = n, whenever n ≥ 1.
In this case, the existence of a nontrivial solution ξ(x) ∈ C 2n [0, 1] is guaranteed for problem (20).
We have that leads to the relation Setting a = b = 2, inequality (24) becomes: where ζ(s, q) := ∞ k=0 1 (k + q) s is the Hurwitz zeta function, defined for s = 1 and (q) > 0. In our problem, it is always well defined since q > 0 and s = 1/2.
In the parameter space (α, n), setting α = n, for instance one obtains that inequality (25) is satisfied whenever Letting α = n 2 we obtain that inequality (25) is satisfied whenever For the above choice of parameters, the existence of a nontrivial solution ξ(x) ∈ C 2n [0, 1] is guaranteed for problem (23).