Existence of a solution for a general order boundary value problem using the Leray-Schauder fixed point theorem

Существование решения краевой задачи общего порядка с использованием теоремы Лере-Шаудера о неподвижной точке

Решење проблема граничне вредности општег реда које користи теорему непокретне тачке типа Leray-Schauder

Nicola Fabianoa nicola.fabiano@gmail.com

Independent researcher, Italy

Vahid Parvanehb zam.dalahoo@gmail.com

Islamic Azad University, Islamic Republic of Iran

This work is licensed under Creative Commons Attribution 4.0 International.

We consider the generic differential equation of order 2n, n ≥ 1, with the boundary conditions:

(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 ∈ C²ⁿ over such that solve the following equation:

(2)

The complete Green’s function is thus obtained by the linear combination of the above two,

(3)

θ is the Heaviside step function. Given the inhomogeneous problem solved by x → f(x), x ∈ [0, 1], f ∈ C over,

(4)

the Green’s function provides solution to (4) in the integral form

(5)

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

(6)

and

(7)

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

(8)

and

(9)

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:

(10)

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 C²ⁿ in , let (x, y, z) → χ(x, y, z); x ∈ [0, 1],(y, z) ∈ ² be a function of class C in and let χ be a function of class C in . Then the Green’s function of the problem (4) obeying to equation (2) is given by formulas (3), (6), (7), and (10).

Another property of these Green’s functions is their homogeneity. In fact, under the scaling transformation (x, s) → (αx, αs) for α > 0 one has

that is, G_l,r(x, s, n) is homogeneous of degree 2n − 1.

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 as follows:

(11)

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):

Theorem 1. Let (E, || · ||) be a Banach space, let U ⊂ E be an open bounded subset for which 0 ∈ U and let be a completely continuous operator. Then only one of the following possibilities is true:

1. possesses a fixed point

2. there exist an element and a real number λ > 1 such that x = λx.

Therefore, in order to establish the existence of a solution it is necessary to prove that our integral operator possesses a fixed point. The following two theorems are devoted to this problem.

Theorem 2. Let x → y(x), x ∈ [0, 1] be a function of class C²ⁿ in , let (x, y, z) → χ(x, y, z); x ∈ [0, 1],(y, z) ∈ ² be a function of the class C in and |χ(x, 0, 0)| $\ne$ 0. Suppose that there exist three nonnegative functions x → (u(x), v(x), w(x)) ∈ L¹ [0, 1] such that

Define the kernel

and suppose that

Then the problem (1) has at least one nontrivial solution x → ξ(x), x ∈ [0, 1] of class C²ⁿ in .

Proof. Define the constant

From our hypothesis A < 1 and w(s) ≥ 0. Observe that for all s ∈ [0, 1]. As |χ(x, y, z)| ≤ u(x)|y| + v(x)|z| + w(x), for all x ∈ [0, 1] and (y, z) ∈ ² and according to the fact that χ(x, 0, 0) $\ne$ 0 for all x ∈ [0, 1], there exist an interval [a, b] ⊂ [0, 1] such that max_x∈[a,b] |χ(x, 0, 0)| > 0. Therefore, |χ(x, 0, 0)| > 0 and also w(x) > 0, for some x ∈ [a, b] ⊂ [0, 1]. This implies the inequality We conclude that A < 1 and B > 0.

Define L := B(1 − A)⁻¹ which is positive by construction, and the set U = {y ∈ E : ||y|| < L}. Assume that y ∈ ∂U and λ > 1. As y = λy, then λL = λ||y|| = ||y|| = max_x∈[0,1] |(y)(x)|. Adopting the simplified notation dµ = |χ(s, y(s), y''(s)|ds we have:

(12a)

(12b)

from our hypothesis, |χ(x, 0, 0)| has an upper bound for all x ∈ [0, 1]. So, one has

(13)

Using equation (12ab), we obtain the bound λL ≤ AL + B which implies that

which contradicts the hypothesis for which λ > 1, that is point (2) of Theorem 2.1 is ruled out, while point (1) is fulfilled. Therefore, we conclude that there exists at least a nontrivial solution ξ(x) of problem (1).

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.

Theorem 3. Let (x, y, z) → χ(x, y, z); x ∈ [0, 1],(y, z) ∈ ² , χ is of class C in and |χ(x, 0, 0)| $\ne$ 0. Suppose that there exist three nonnegative functions x → (u(x), v(x), w(x)) ∈ L^-1 [0, 1] such that

Theorem 3. Let .x, y, z) ›→ .(x, y, z); . ∈ [0, 1], (y, z) ∈ R2, χ is of the class C in . and |.(x, 0, 0)| ƒ= 0. Suppose that there exist three nonnegative functions x ›→ (.(.), v(.), w(.)) ∈ ..[0, 1] such that

Define

and suppose that either one of the following conditions holds:

1. There exists a constant > −2 such that

where

2. There exists a constant m > −1 such that

where

3. here exists a constant a > 1 such that

and

Then problem (1) has at least one nontrivial solution x → ξ(x), x ∈ [0, 1] of class C²ⁿ in .

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.

(14)

and when > −2, one has

For point 2, we have

(15)

and when m > −1, one has

In the case of point 3, we make use of Hölder inequality for which , whenever f and g are measurable functions on the domain S and 1/a + 1/b = 1. We have

(16)

FIELD: Mathematics

ARTICLE TYPE: Original scientific paper

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