EXISTENCE OF A SOLUTION FOR A GENERAL ORDER BOUNDARY VALUE PROBLEM USING THE LERAY-SCHAUDER FIXED POINT THEOREM Текст научной статьи по специальности «Математика»

ORIGINAL SCIENTIFIC PAPERS

ОРИГИНАЛНИ НАУЧНИ РАДОВИ ОРИГИНАЛЬНЫЕ НАУЧНЫЕ СТАТЬИ £

СП I

со <м сп s± CL

е"

ф о

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

323

о cL

"О

<u

X

<u

"О

o CO

Nicola Fabianoa, Vahid Parvanehb

<0

a Independent researcher, Rome, Italy, e-mail: nicola.fabiano@gmail.com, corresponding author, ORCID iD: ©https://orcid.org/0000-0003-1645-2071

b Islamic Azad University, Department of Mathematics, Gilan-E-Gharb Branch, Gilan-E-Gharb, Islamic Republic of Iran e-mail: zam.dalahoo@gmail.com, ORCID iD: https://orcid.org/0000-0002-3820-3351 o

pr e

DOI: 10.5937/vojtehg69-29703; https://doi.org/10.5937/vojtehg69-29703 FIELD: Mathematics

ARTICLE TYPE: Original scientific paper Abstract:

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.

ro

>

о

J2

<U "O

<U СЯ

<U О

<u

(Я

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. «>

"S

Key words: fixed point, boundary value problem, Leray-Schauder fixed point theorem.

ro

The problem

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

y(2n)(x) =x(x,y(x),/(x)) y(0)(0) =y(1)(0) = ... = y(n)(0) = 0 (1)

y(n+i) (1) =y(n+2) (i) = ... = y(2n-1) (1) = 0.

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¿(x, s, n) and Gr(x, s, n) of problem (1) where G¿(x, s, n) is defined for 0 < x < s < 1 and Gr(x, s, n) is defined for 0 < s < x < 1, (x,s) ^ G¡,r(x,s,n);(x,s) e [0,1] x [0,1],G¡,r e C2n overR such that solve the following equation:

d \2n

— ) Gi,r(x,s,n) = 5(s - x). (2)

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

G(x, s, n) = 6(s — x)G¡(x, s, n) + 6(x — s)Gr(x, s, n) , (3)

6 is the Heaviside step function. Given the inhomogeneous problem solved

by x ^ f (x),x e [0,1], f e C over R,

y(2n)(x) =f (x) y(0)(0) =y(1)(0) = ... = y(n)(0) = 0 (4)

y(n+l)(1) =y(n+2)(1) = ... = y(2n-1)(1) = 0 ,

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

y(x) = G(x,s,n)f (s)ds 0

The functions Gl>r(x,s,n) are multivariate polynomials in two variables

x and s of the order 2n - 1 to be sought in the form

Gi(x,s,n) = ^2/ c(k,n)xk 1s

k-1 „2 n-k

k=1

and

Gr(x,s,n) = ^2/ c(k,n)sk 1

k—1x2n-k

(6)

(7)

k=1

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

Gi(0, s, n) = (Jlx ) Gi(x,s,n) and

x=0

. = Gi (x,s,n)

0 (8)

x=0

cn cn I

cn <M m !± CL

e"

<D O

<u

o

CP

~o

<u

X

<u

"O

0 OT

1

> ro

dx I

n+1

Gr(x, s, n)

x=1

2n 1

dix) Gr(x,s,n)

0. (9)

x=1

So, we could infer the following results.

For Gl(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 Gr(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 Gl>r (x, s, n) is given by:

c(k, n) =

2n — 1

(-1)k

(2n - 1)! V k - 1 ) (k - 1)!(2n - k)! '

(-1)k

(10)

Notice that lc(k,n)l < 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 5 function.

The above discussion concludes the proof of the following lemma:

Lemma 1. Let x ^ y(x),x e [0,1] be a function of class C2n in R, let

(x, y, z) ^ x(x, y,z); x e [0,1], (y, z) e R2 be a function of class C in R and

E

.q o

<U

CO

>

o

.Q

<U "D

<U CS CO

O

o

CO

ro

<u o d <u co

X LU

"03 <u

CO J2

ro

let x 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).

Another property of these Green's functions is their homogeneity. In fact, under the scaling transformation (x, s) ^ (ax, as) for a > 0 one has

Gi>r (ax, as,n) = a2n-1Gi,r (x,s,n),

that is, (x, s, n) is homogeneous of degree 2n - 1.

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 Y as follows:

r x r 1

Yy(x) := / Gr(x,s,n)f(s)ds + / Gi(x, s, n)f(s)ds. (11) ./0 Jx

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 & Bara-novskii, 2010):

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

1. Y possesses a fixed point x e U

2. there exist an element x e dU and a real number A > 1 such that

Yx = Ax.

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

Theorem 2. Let x ^ y(x),x e [0,1] be a function of class C2n in R, let

(x, y, z) ^ x(x, y, z); x e [0,1], (y, z) e R2 be a function of the class C in R

and |x(x, 0,0)| = 0. Suppose that there exist three nonnegative functions

x ^ (u(x), v(x), w(x)) e L1[0,1] such that

lx(x,y,z)| < u(x)|y| + v(x)|z| + w(x).

Define the kernel

and suppose that

2n-1

Ms) := E s' fc=i

A := / Kn(s)[u(s)+ v(s)]ds < 1. Jo

Then the problem (1) has at least one nontrivial solution x ^ £(x), x e [0,1] of class C2n in R.

cn cn I

cn <M CI S± Œ

e" çu

o cu

o

CL

~o

cu

X

cu

"O

0 OT

1

> co

Proof.Define the constant

B = Kn(s)w(s)ds. o

From our hypothesis A < 1 and w(s) > 0. Observe that Kn(s) > 0 for all s e [0,1]. As |x(x,y, z)| < u(x)|y| + v(x)|z| + w(x), for all x e [0,1] and (y, z) e R2 and according to the fact that x(x, 0,0) = 0 for all x e [0,1], there exist an interval [a, b] c [0,1] such that maxxera,b] |x(x, 0,0)| > 0. Therefore, |x(x, 0,0)| > 0 and also w(x) > 0, for some x e [a, b] c [0,1]. This implies the inequality /J K„(s)w(s)ds > Jb K„(s)w(s)ds > 0. 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 E : ||y|| < L}. Assume that y e dU and A > 1. As Yy = Ay, then AL = A||y|| = ||Yy|| = maxx£[0)1] |(Yy)(x)|. Adopting the simplified notation d^ = |x(s,y(s),y"(s)|ds we have:

AL = max |(Ty)(x)| << Gr(x,s,n)d^ + / G^(x,s,n)d^ xe[o,i] LJo Jx

ra ( fX fi

/ sfc-1x2ra-fcd^ + / xk-is2ra-fc<

fc=1

327

E

.Q

o œ cu

CO

>

o

.Q

CD "O

CU

cs

CO

o

o

CO CO

cu o d cu

-I—'

CO x LU

"03

-I—'

cu

J2 CO

n ( rx /•! ^

Ylc(k,n) sk-1x2n-kdi + J xk-1s2n-kdpj =

^ c(k, n)^ £ sk-1di + jf1 s2n-kd^ =

^ c(k, n)^ J\sk-1 + s2n-k)djj <

n f r 1 ^ 2n-1 r 1 r 1

/ (sk-1 + s2n-k)dA = V / skd/1 = l fcn(s)dii. (12)

k=1 u° J k=1 7o 7o

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

r 1 r 1

Kn(s)lx(s, 0,0)lds < / Kn(s) [u(s)|y(s)| + v(s)ly"(s)l + w(s)] ds <

/0 Jo

r 1

Kn(s)

u(s) max |y(s)| + v(s) max |y (s)| + w(s) se[o,1] se[o,1]

ds

f Kn(s) [u(s)|y(s)|TO + v(s)ly"(s)l^ + w(s)] ds < Jo

i Kn(s)[u(s)IM + v(s)M + w(s)] ds = Jo

[ Kn(s)[u(s) + v(s)] 11y||ds + j Kn(s)w(s)ds = oo

A||y|| + B = AL + B. (13) Using equation (12), we obtain the bound XL < AL + B which implies that

BB X < A + B = A + B(1 -A)-1 =1,

which contradicts the hypothesis for which X > 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 £(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.

o

Theorem 3. Let (x,y, z) ^ x(x,y, z); x e [0,1], (y,z) e R2, x is of class k

C in R and |x(x, 0,0)| = 0. Suppose that there exist three nonnegative m

functions x ^ (u(x), v(x), w(x)) e L1[0,1] such that "

Cp

|x(x, y, z)| < u(x)|y| + v(x)|z| + w(x). |

Define

Kn(s) := £

2n-1

and suppose that either one of the following conditions holds: 1. There exists a constant £ > - 2 such that

s^

u(s) + v(s) < —---r-—---, 0 < s < 1 ,

w w ^(2n + £ + 1) - ^(£ + 2)' < <

where

d f™ ^(z) := — [logr(z)] and r(z) := / e-iiz-1di. dz 0

, r(m)r(2n + 1)

£(m, 2n + 1) = ^ 7 v-f.

' 7 r(2n + m + 1)

^ (u(s) + v(s))"ds 0

329

o

_fc ^ s "O

<1J

k=1 £

<1J "D

O

OT I

> ro

0 2

CT

2. There exists a constant m > -1 such that ®

2 >

XX XX (^i \ (^ , X _

u(s)+ v(s) < 1 - (m2 + m)<9(m, 2n +1) '(1 - *>" 0 < s < 1 0

.Q

where

<1J

cs ro

3. There exists a constant a > 1 such that o

1 11,

<-;-ri , - + 1 = 1

v-^2ra-W 1 ^ b a b

Z^fc=1 [ fcb+1

<u o

and a

f K„(s)[u(s)+ v(s)]ds < 1.

Jo ro

Then problem (1) has at least one nontrivial solution x ^ £(x), x e [0,1] of class C2n in R.

<1J

<0 J2

ro

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:

f Kn(s)[u(s) + v(s)]ds < ^2n + £ + 1) - W + 2) lo Kn(s)s"ds =

1 2n-1 r 1

^(2n + £ + 1) - ^(£ + 2) k= Jo

J- jf s(k+£)ds

1 2n-1 1

E

^(2n + £ + 1) - £ + 2) k= k + £ + 1

№(2n +£ + D - W + 2)] • + £ + 1) - w +2) = 1 < (14) and when £ > -2, one has

For point 2, we have

n <! - (m2 am»,+l) n ^M-»"*

2 2n-1 1

m + m ^ sk(1 - s)mds =

1 - (m2 + m)P(m, 2n + 1) ^^ Jo

m2 + m 2n-1

Y-(m2+mm+/3{m,2nT1) ^ ^(k,m

m2 + m

1 - (m2 + m)P(m, 2n + 1) and when m > -1, one has

k=1

1

m2 + m

2

m2 + m

1 - (m2 + m)P(m, 2n + 1)

- @(m, 2n + 1)

> 0.

1 , (15)

In the case of point 3, we make use of Holder inequality for which

Is f (s)g(s)d < (fs f (sTds)1/a (fs |g(s)|bds)1/b, whenever f and g are measurable functions on the domain S and 1/a + 1/b = 1. We have

Kn(s)[u(s) + v(s)]ds <

(u(s) + v(s))a ds

L-/Q

(u(s) + v(s))ads

2n-1

£

fc=1

1 2n-1

'£

k=1 i

1 \ i

(sk)bds

Q

kb + 1

2n— 1

e:

2ra-1/ 1 \b

fc=1 I fcb+1

iE

fc=1

<

1 \ b

kb +1

1 . (16)

□

cn cn I

cn <M CO S± CP

£E cu

o cu

o

CP "D

cu

X

cu

"D

Examples

0 OT

1

> co

Example 1

Consider the problem for a generic n > 1

y(2n)(x) = Xry2e-y2 + xa [WT^ sin y" + e-¥ y(Q)(0) =y(1)(0) = ... y(n) (0) = 0 ,y(n+1)(1) =y(n+2)(1) = ...y(2n-1)(1) = 0.

(17)

This problem satisfies all requirements of Theorem 2. In fact, one has

x(x,y,-) = T y2e-y2 + -

sin - + e 2

together with

u(x) = —, v(x) = —, w(x) = e-^ .

For a generic a > 0, following Theorem 3, hypothesis 1, we have that (u(x), v(x), w(x)) e L1[0,1] are nonnegative functions. Moreover,

|x(x,y,z)| < u(x)|y| + v(x)|z| + w(x)

for all x e [0,1] and for all (y, -) e R2.

E

.q o cl cu

CO

>

o

J2

cu "D

CU CS CO

o

o

CO CO

cu o c cu co X LU

"03

-I—'

cu

J2 CO

1

1

1

Q

1

Q

1

2

Setting a = £ one has to consider the inequality

1

^(2n + £ + 1) - ^(£ + 2)

1

X > 0 (18)

for all x e [0,1]. In the parameter space (a,£,n), the above inequality is satisfied, for instance, for the case a = £ = n, whenever

1 < n < 12. (19)

In this case, the existence of a nontrivial solution £(x) e C2n[0,1] is guaranteed for problem (17).

Example 2

Consider the problem for a generic n > 1

11

y(2n)(x) =^(1 - x)ay cosy + —(1 - x)ay" tanhy" + coshx

y(o)(0)=y(1)(0) = ... = y(n){0) = 0 (20)

^y(n+1)(1) =y(n+2)(1) = ... = y(2n-1)(1) = 0. This problem satisfies all requirements of Theorem 2. In fact, one has

x(x, y, z) = ^(1 - x)ay cosy + 7^(1 - x)az tanh z + coshx

together with

(1 - x)a

u(x) =-ö-, v(x) =-ö-, w(x) = coshx.

For a generic a > 0, from hypothesis 2 of Theorem 3, we have that (u(x),v(x),w(x)) e L1[0,1] are nonnegative functions. Moreover,

lx(x,y,z)l < u(x)lyl + v(x)lzl + w(x) for all x e [0,1] and for all (y, z) e R2.

Setting a = m, one has to investigate the following inequality:

m2 + m

1 — (m2 + m)ß(m, 2n + 1)

1

■ (1 — x)m > 0

(21)

for all x e [0,1]. In the parameter space (a, m, n), the above inequality is satisfied, for instance, for the case a = m = n, whenever

n > 1.

(22)

In this case, the existence of a nontrivial solution £(x) e C2n[0,1] is guaranteed for problem (20).

Example 3

Consider the problem for a generic n > 1

cn cn I

cn <M m

CP CP

E~ £ o cu

o

CP

~o

cu

X

cu

"D

0 OT

1

> co

' y(2n)(x) = ^^^tanhy + xa(y//)2e-(y")2 + ex +3

a y3

2 y^+ 4 01 4 y(Q)(0) =y(1) (0) = ... y(n)(0) = 0

^y(n+1)(1) =y(n+2)(1) = ... y(2n-1)(1) = 0.

(23)

E ju .q o cl cu

CO

>

This problem satisfies all requirements of Theorem 2. In this example,

xa y3 xa 2

x(x, y, -) = — y4+-tanh y + — -2e-z + ex +3

together with

xa xa

u(x) = —, v(x) = —, w(x) = ex + 3.

For a generic a > 0, following assumption 3 of Theorem 3, we have that (u(x), v(x), w(x)) e L1[0,1] are nonnegative functions. Moreover,

|x(x, y, -)| < u(x)|y| + v(x)|z| + w(x)

o

.Q

cu "D

cu cS co

o

o co co

cu o c cu

-I—' Ü2 x LU

"03

-I—'

cu

for all x e [0, 1] and (y, -) e

333

co

We have

that leads to the relation

u(x) + v(x) = xa

1

aa + 1

<

1

2n-1 ( 1 \b

E2n-k=1 \ kb+l

Setting a = b = 2, inequality (24) becomes:

1

2a + 1

<

1

V2

v-^2n-1 2^k=1

2k+1

Z ( 2, 2 ) - Z ( 2, 2n + 1 )

where

Z(s,q) := £

k=0

(k + q)s

(24)

(25)

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 (a,n), setting a = n, for instance one obtains that inequality (25) is satisfied whenever

1 < n < 7. (26)

Letting a = n2 we obtain that inequality (25) is satisfied whenever

n > 1. (27)

For the above choice of parameters, the existence of a nontrivial solution £(x) e C2n[0,1] is guaranteed for problem (23).

1

1

References

Ahmad B. & Ntouyas, S.K. 2012. A study of higher-order nonlinear ordinary differential equations with four-point nonlocal integral boundary conditions. Journal of Applied Mathematics and Computing, 39, pp.97-108. Available at: https://doi.or g/10.1007/s12190-011-0513-0.

Bekri Z. & Benaicha, S. 2018. Nontrivial solution of a nonlinear sixth-order boundary value problem. Waves, Wavelets and Fractals, 4(1), pp.10-18. Available at: https://doi.org/10.1515/wwfaa-2018-0002.

Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. New

cn

York, NY: Dover. Online ISBN: 9780486319209. £

<M m !± CL

Deimling, K. 1985. Nonlinear Functional Analysis. Berlin, Heidelberg: Springer. Available at: https://doi.org/10.1007/978-3-662-00547-7. Online ISBN: 978-3-66200547-7. E

Fabiano, N., Nikolic, N., Shanmugam, T., Radenovic, S. & Citakovic, N. 2020. Tenth order boundary value problem solution existence by fixed point theorem. Journal of Inequalities and Applications, art.number:166. Available at: https://doi. org/10.1186/s13660-020-02429-2. ^

<1J

Isac, G. 2006. Leray-Schauder Type Alternatives, Complementarity Problems * and Variational Inequalities. Boston, MA: Springer. Available at: https://doi.org/10 .1007/0-387-32900-5. Online ISBN: 978-0-387-32900-0.

£ о <u

о

L

<u

"D

Ma, R. 2000. Existence and uniqueness theorems for some fourth-order non- ¿> linear boundary value problems. International Journal of Applied Mathematics and J^ Computer Science, 23, art.ID:739631. Available at: https://doi.org/10.1155/S016 50 1171200003057.

Shanmugam, T., Muthiah, M. & Radenovic, S. 2019. Existence of Positive Solution for the Eighth-Order Boundary Value Problem Using Classical Version of Leray-Schauder Alternative Fixed Point Theorem. Axioms, 8(4), art.number:129. Available at: https://doi.org/10.3390/axioms8040129.

Zvyagin, V.G. & Baranovskii, E.S. 2010. Topological degree of condensing multi-valued perturbations of the (S)+-class maps and its applications. Journal of ® Mathematical Sciences, 170, pp.405-422. Available at: https://doi.org/10.1007/s1 0958-010-0094-8.

о

.Q

СУЩЕСТВОВАНИЕ РЕШЕНИЯ КРАЕВОЙ ЗАДАЧИ ОБЩЕГО ПОРЯДКА С ИСПОЛЬЗОВАНИЕМ ТЕОРЕМЫ ЛЕРЕ-ШАУДЕРА О НЕПОДВИЖНОЙ ТОЧКЕ &

го

Никола Фабиано3, Вахид Парванех6 р

а независимый исследователь, г Рим, Италия, корреспондент, б Исламский университет Азад, факультет математики, филиал Гилан-э-Герб, г Гилан-э-Герб, 8

Исламская Республика Иран

<и о

с ф

-ь-'

41 х ш

"го

-ь-' ф

335

J2

го

РУБРИКА ГРНТИ: 27.00.00 МАТЕМАТИКА:

27.29.19 Краевые задачи и задачи на собственные значения для обыкновенных дифференциальных уравнений и систем уравнений, 27.29.21 Аналитическая теория

обыкновенных дифференциальных уравнений и систем уравнений, 27.39.00 Функциональный анализ ВИД СТАТЬИ: оригинальная научная статья

Резюме:

Введение/цель: В данной статье приведено существование производящей функции Грина для решения краевой задачи произвольного порядка, которая встречается во многих явлениях тепловой конвекции как в атмосфере, в океанах, так и на поверхности Солнца.

Методы: В статье применена теорема неподвижной точки Лере - Шаудера, с целью подтверждения существования неподвижной точки в данной задаче.

Результаты: Доказано существование решения по произвольному порядку и предлагаются некоторые практические примеры.

Выводы: Краевая задача имеет решение по произвольному п-му порядку.

Ключевые слова: неподвижная точка, краевая задача, теорема Лере - Шаудера о неподвижной точке.

РЕШЕТЕ ПРОБЛЕМА ГРАНИЧНЕ ВРЕДНОСТИ ОПШТЕГ РЕДА ^Е КОРИСТИ ТЕОРЕМУ НЕПОКРЕТНЕ ТАЧКЕ ТИПА LERAY-SCHAUDER

Никола Фабиано3, Вахид Парванех6

а независни истраживач, Рим, Итали]а, ауторза преписку,

б Исламски универзитет Азад, Оде^е^е за математику, Огранак Гилан-Е-Гхарб, Гилан-Е-Гхарб, Исламска Република Иран

ОБЛАСТ: математика

ВРСТА ЧЛАНКА: оригинални научни рад

337

со со I

со <м

со <±

CL

e" ф

О eu

Сажетак:

Увод/цил>: У раду се приказу/е посто]ак>е генеричке Гэино-ве функци]е за проблем граничне вредности произвол>ног реда ко\и се }авъа код многих по}ава конвекци]е топлоте у, на пример, атмосфери, океанима и на површини Сунца.

Методе: Користи се теорема непокретне тачке типа Leray-Schauder како би се утврдило посто}аше непокретне тачке у наведеном проблему.

и

си

Резултати: Приказано }е решете за произвоъан ред про- * блема. Предложени су неки практични примери.

Закъучак: Гэанични проблем има решете за произвоъни n-ти ред. от

Къучне речи: непокретна тачка, проблем граничне вредности, теорема непокретне тачке типа Leray-Schauder.

о CL

eu

"О

о

> го

Paper received on / Дата получения работы / Датум приема чланка: 03.12.2020.

Manuscript corrections submitted on / Дата получения исправленной версии работы / ф

Датум достав^а^а исправки рукописа: 01.03.2021. о

Paper accepted for publishing on / Дата окончательного согласования работы / Датум Œ

коначног прихвата^а чланка за об]ав^ива^е: 03.03.2021. Ф

© 2021 The Authors. Published by Vojnotehnicki glasnik / Military Technical Courier (http://vtg.mod.gov.rs, http://втг.мо.упр.срб). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/rs/).

ro

>

о

J2

© 2021 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military Technical Courier" (http://vtg.mod.gov.rs, httpV/втг.мо.упр.срб). Данная статья в открытом доступе и распространяется в соответствии с лицензией "Creative Commons" (http://creativecommons.org/licenses/by/3.0/rs/).

го

© 2021 Аутори. Обjавио Воjнотехнички гласник / Vojnotehnicki glasnik / Military Technical Courier о

(http://vtg.mod.gov.rs, httpV/втг.мо.упр.срб). Ово jе чланак отвореног приступа и дистрибуира се у складу са Creative Commons лиценцом (http://creativecommons.org/licenses/by/3.0/rs/).

о сл го

ч—

О Ф О

С ф

-ь-'

сл х Ш

"03

-ь-' ф

J2

го