Confining a non-negative solution between a lower and upper solution for a sixth-degree boundary value problem Текст научной статьи по специальности «Математика»

Nour Bachir University Center, Institute of Sciences,

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Confining a non-negative solution between a lower and upper solution for a f

sixth-degree boundary value problem

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Zouaoui Bekria, Nicola Fabianob >

Mohammad Esmael Sameic, Stojan Radenovicd -§

a University of Oran 1, Laboratory of Fundamental and Applied Mathematics, Oran

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Department of Sciences and Technology, El Bayadh; a

People's Democratic Republic of Algeria e-mail: zouaouibekri@yahoo.fr, E

ORCID iD: ©https://orcid.org/0000-0002-2430-6499 £

b University of Belgrade, "Vinca" Institute of Nuclear Sciences - National Institute of the Republic of Serbia, Belgrade, Republic of Serbia, e-mail: nicola.fabiano@gmail.com, corresponding author, ORCID iD: ©https://orcid.org/0000-0003-1645-2071 c Bu-Ali Sina University, Faculty of Basic Science, Department of Mathematic, Hamedan, Islamic Republic of Iran, e-mail: mesamel@basu.ac.ir, ORCID iD: ©https://orcid.org/0000-0002-5450-3127 ®

University of Belgrade, Faculty of Mechanical Engineering, Belgrade, Republic of Serbia, e-mail: radens@beotel.rs,

ORCID iD: ©https://orcid.org/0000-0001-8254-6688

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DOI: https://doi.org/10.5937/vojtehg72-48996

FIELD: mathematics

ARTICLE TYPE: original scientific paper o

Abstract: >

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Introduction/purpose: The aim of the paper is to prove the existence of JJ solutions for a special case of the sixth-order boundary value problem. Methods: The Leray-Schauder fixed point theorem is used in order to determine lower and upper bound solutions. Results: Lower and upper bound solutions have been found. Conclusions: The sixth-order boundary value problem admits solutions. §

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> Introduction

Key words: Leray-Schauder nonlinear alternative, Green's function, fixed point theorem, lower and upper solutions, boundary value problem.

Many modern mathematical studies are based on modeling phenom-o£ ena in various applied branches, and simulating the results. Often, it is

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E important to evaluate the obtained results at least with approximate so-o lutions. One of these cases concerns BVPs related to sixth-degree differential equations derived mostly from physical phenomena and applied

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1 Some authors have addressed the problem of obtaining positive solu-lju tions to a sixth-degree BVP by employing multiple methods, such as the > spectral operator theory, the general bifurcation theorem, the minimization ¡< theorem, and the fixed point theory in the cone, see references (Möller &

Zinsou, 2013; Tersian & Chaparova, 2002; Li, 2012; Agarwal et al., 2013; Ji etal., 2012; Kovacs & Guedda, 2014; Zhang & An, 2010; Mirzaei, 2016; Yang, 2019). Not a long time ago, the authors of this paper analyzed sixth-<5 order BVPs with different boundary conditions at two points (Bekri & Be-^ naicha, 2018, 2020). In 2018, see reference (Bekri & Benaicha, 2018), the o same authors formed the conditions for finding the existence of solutions

for a sixth-degree BVP:

-{(6)(e) + h (e,{(e),{"(e)) =0, 0 < e < 1, {(0) = w (0)= {'' (0) = 0,

I r(i) = e(4)(i) = e(5)(i) = 0,

where h e C([0,1] x R2). Suitable operators are employed in the proof of the nonlinear Leray-Schauder alternative. Also, in 2020, (Bekri & Benaicha, 2020), created all the necessary factors to prove the existence of non-negative solutions for a BVP of the sixth-order

-{(6) (e) + z(e)h e {(e), {'(e), {''(e), {''' (e), {(4) (e), {(5) (e)) = 0, 0 < e < 1, {(0)= {' (1)= {' '(0) = 0, {'' '(1)= {(4)(0) = {(5)(1) = 0,

where h e C([0,1] x [0, +rc>) x [0, +rc>) x (-<x>, 0] x (-<x>, 0] x [0, +rc>) x [0, +rc>) —> [0, +rc>)). The authors focused on the fixed point theorem

of Leray-Schauder and on a nonlinear version of the theorems of Leray- S

Schauder type. For more examples, see (Chabanea et al., 2022; Fabiano ^

& Parvaneh, 2021; Samei et al., 2021b; Fabiano et al., 2020; Samei etal., ^

2021a; Boutiara et al., 2021; Santra et al., 2023; Houas & Samei, 2023; ^

Hammad et al., 2022; Amiri & Samei, 2022; Boutiara et al., 2023). I

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This work demonstrates the existence of a solution for sixth-degree six- <u point BVPs S

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-£(6)(e) + h (e,£(eU(5)(e)) =o, 0 <e< 1,

e(o) = c (o) = c' (0) = o, (1)

r (1) = C(4)(1) = C(5)(o) = o, I

where h e C([o, 1] x R2), and

-£(6)(e) + h (e,£(e),£'(e),C(5)(e)) = o, o < e < 1,

C(o) = C (o) = C' (o) = o, (2)

C" (1) = c(4)(1)= C(5)(o) = o,

where h e C([o, 1] x R3).

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This article is structured as follows. The section on the basic concepts ^ reviews the definition and three lemmas to realize the analysis. Then, the part on the main results presents some desired results because there are ^ solutions for two nonlinear sixth-order BVPs. The methodology used here is based on the Schauder fixed point theorem and the lower and upper solution method by creating the aforementioned estimates. The fourth, final part, Application, offers a few similar examples to demonstrate the findings.

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Basic concepts

Within this section, a BVP (1) with the condition h e C([o, 1] x R2,R), and A = C6[o, 1] is put forward. Some lemmas that define the role of the nonlinear term are summarized and lower or upper solutions are generated.

Definition 1. Assume Z,n e C2[o, 1] n A lower and upper solution of the BVP (1), respectively, if for o < e < 1, 6

- Z(6)(e) + h(e,((e),Z(5)(e)) > o, *

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^ c(0) = C'(0) = C"(0) = <r(1) = Z(4)(1) = 0, Z(5)(0) > 0,

- n(6)(e) + h(e,n(e),n(5\e)) < 0, ^ n(0) = n (0) = n'(0) = n'''(i) = n(4)(i) = 0, n(5)(0) < 0.

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£ J e(5)(e) = k(e), 0 <e< 1,

§ I e(0) = e ' (0) = e ''(0) = 0, e ' ''(1) = e(4)(1) = 0, ()

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Lemma 1. Let it be k e C([0,1]). So the five-point BVP

e(e)= [ x(£,Q)k(Q)dg,

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where x : [0,1] x [0,1] ^ [0, +rc>) is the Green's function defined by

, , 1 ( e3(4Q - e), 0 < e < q < 1,

x(e, Q) = — \ (4)

24 I Q2(Q2 + 6e2 - 4eQ), 0 < q < e < 1.

<5 Define the integral operator P : A ^ A by

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* Pe(e) = 24 I'S Q2 (q2 + 6e2 - 4eQ) kQ dQ + ^ f e3(

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o According to Lemma 1, if the operator P admits a fixed point in A, it implies that the BVP (1) has a solution. Thus, one found a fixed point of P in A. Using the theorem of Ascoli-Arzela, one checks that P is an operator which is completely continuous. Therefore, a nonlinear alternative of Leray-Schauder is given.

Lemma 2. (Deimling, 1985) Let A be a Banach space of and W be a subset that is open and bounded of A, 0 e W. Let P : W ^ A be a completely continuous operator. Therefore, either

(i) 3 e e dW and v > 1 in which P (e) = ve, or

(ii) 3 e* a fixed point such that e* e W.

Lemma 3. By x(e, q) given by (4), the following holds

(J 11 iH <1 for e'Qe (0'1),

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(J3) q = x(q,q)q^Q, such that q > 0. Jo

Main results

This section introduces some concepts in the form of lemmas indicating some bounds on the nonlinear term and the construction of lower or upper solutions. Consider that h e C([0,1] x R2), and h e C([0,1] x R3).

E(0) = 0, (5)

5(e) - h (e, (0H)(e),0(e)) — 0, 0 <e< 1, (6)

5(0) - 0. (6)

From this, it holds that n(e) = (#5)(e) is an upper solution of the BVP (1).

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Lemma 4. Let there be a real w — 0 in which $

h(e, v) < w, for 0 < e < 1, qwV < i < qw, —w < v < 0

so the BVP (1) admits an upper solution. <o

Proof. We put E(e) = £(5)(e), so problem (1) is parallel to the equivalent problem

' E'(e) = h (e, ($E)(e), E(e)), 0 <e< 1,

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where (#E)(e) = /Q x(e, g)E(g) dQ and x(e, q) is given by (4). It is obvious § that the bound on h ensures that 5(e) = we realizes the inequality

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Lemma 5. Let there be a real number k < 0 such that

h(e, v) — k, for 0 < e < 1, qK < i < qnV., 0 < v < —k, so the BVP (1) has a lower solution. co

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Proof. Suppose that r(e) = ne. Then, it is obvious that the bound on h ensures that r(e) = ne provides

r'(e) - h (e, ($r)(e), r(e)) < 0, 0 < e < 1,

£ \ r(0) < 0, (7)

this inequality proves that Z(e) = (#r)(e) is a lower solution of the BVP (1).

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o Corollary 1. The upper solution n(e) and the lower solution Z(e) of the

° BVP (1) can be written in a simple manner:

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n(e) = _ e6 __e4 + e3

n(e) = 720e - 48e + 18e '

>_ C(e) = ^e6 - -e4 + -e3.

DC z ( ) 720 48 + 18

Theorem 1. Let there be two real numbers w > 0 > k, w > |k| such that

h(e, ¡,v) < w, 0 < e < 1, qwV < i < qw, —w < v < 0, (8) h(e, i,v) > k, 0 < e < 1, qK < i < qnV, 0 < v < —k. (9)

o If h(e, i, v) is nondecreasing in so the BVP (1) admits a solution {(e) for

which

° Z(e) < {(e) < n(e), Z(5)(e) < {(5)(e) < n(5)(e), e € [0,1],

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where

Z(e) = k[ x(e, q)q dp, n(e) = w[ x(e,Q)Q dp, q = 48•

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Proof. According to formulas (8) and (9), Lemmas 4 and 5 together produce the result that the BVP (1) accepts an upper solution n(e) and a lower solution Z(e). Define r = Z(5), E = n(5), therefore r'(e) = k < w = E'. In another way, w > 0 > k implies Z(5)(e) < n(5)(e). This implies that x(e, q) > 0 and, depending on the boundary condition, Z(e) < n(e), for 0 < e <1. Now, introduce the following operators A, ©, where A : dom A = {X € C*(0,1) n C[0,1] ^ C[0,1] : X(0) = 0r is a derivative operator such that (AX)(e) = X'(e), e € (0,1), and 8 : C[0,1] ^ C[0,1] is

a continuous operator for which

AE = 0E, S e dom(A), (10) S

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where 0 is given by e

(0E)(e) = h (minjs, ma^E, r}}) , minjs, ma^E, r}}) . (11) ^

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This shows that if E* is a solution of (10), then r(e) < E*(e) < S(e), e e [o, 1]. Therefore, E*(e) is a solution of (7). In addition C*(e) = (tfE*)(e) is a solution of the BVP (1) realizing Z(e) < C*(e) < n(e). If r(e) > E*(e), there exists ei e [o, 1] such that r(ei) > E*(ei). Also, E*(o) = o = Z(5)(o) = r(o) = o, as E* and r are continuous, so we have an interval (e2,e3) in & which r(e2) = E*(e2) and r(e) > E*(e), for e e (e2,e3). Then, for e e

[e2,e3],

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(BE*) (e) = h min {h, max {E*, r} }) , min {h, max {E*, r} })

= h (e, fl (min |h, max {E*, r} }) , r) .

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There exists Y(e) = E*(e) — r(e), h and fl are monotonic, and one has |

r (e) = C(6)(e) < h (^e,Z(e),Z(5)(e))=h(e,flr(e),r(e)), e e [0,1],

E* ' = h (minjH, max {E*, r}}) ,

so Y'(e) — 0, e e [e2,e3]. Consequently, T(e2) = 0 implies Y(e) — 0, e e [e2,e3]. From the relation r(e) < E*(e), e e [e2,e3]. This contradicts the given hypothesis. So, r(e) < E*(e), e e [0,1]. The proof of H(e) — E*(e), e e [0,1], is analogous to r(e) < E*(e).

One will now introduce Q : C[0,1] ^ [0,1], given by

(QE)(e)= f(BE)(Q)dQ, e e [0,1].

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It is obvious that the operator is continuous and a fixed point of Q is a solution of the BVP (10). Let there be a set = {p e C[0,1] : ||p|| < w}, and is easy to observe that H, r e . Now, one demonstrates that <3 q : ^ is a completely continuous operator. By formula (9), one io"

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That is, )} is bounded uniformly and equi-continuous. According

to the theorem of Arzela-Ascoli, it is affirmed that Q : ^ is a completely continuous operator, and the Schauder fixed point theorem ensures

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C(e)= I x(z,Q)Z*(Q)dQ, 0

is a solution of problem (1) such that Z(e) < {(e) < n(e), Z(5)(e) < {(5)(e) < n(5)(e), e € [0,1]. □

£ Now, let us consider problem (2). Suppose that h € C([0,1] x R3) is a

^ continuous function. It is possible to obtain lemmas analogue to Lemmas 4 and 5. This is evident by observing that the equation of problem (2) is parallel to

| i E'(e) = h(e, (tfE)(e), (nE)(e), E(e)), 0 <e< 1,

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where

(#E)(e) = f1 x(£, q)E(q) dQ, (nE)£ = E(q) dQ. 00

Lemma 6. Let there be a real number w > 0 such that

h(e,ß,v,r) < w, for 0 < £ < 1, ^, 2v e [0,w], t e [—w, 0]. So the BVP (2) accepts an upper solution.

Lemma 7. Let there be a real number k < 0 such that

h(£,ß,v,T) > k, for 0 < £ < 1, n e [sk, 0], 2v,t e [0, —k]. So the BVP (2) admits a lower solution.

In an analogous fashion, we can write the same proof for Lemmas 6 and 7 in the same way, as formulated in Lemmas 4 and 5. The similarity in Theorem 1 allows obtaining the following results of problem (2).

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Theorem 2. Let there be two real numbers w — 0 — k, w — |k| such that

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solution C(e) such that

Z(e) < C(e) < n(e), Z(5)(e) < C(5)(e) < n(5)(e), e e [o, 1],

where

Z(e) = k[ x(e, q)q dp, n(e) = w[ x(e,Q)Q dp, Ç = 48• Jo Jo

Application

Some examples are given in order to illustrate the results obtained above.

Example 1. Extrapolate the following application of the BVP

—C(6) + 48 [e + cose(e) + C(5)(e)] =0, 0 <e< 1,

e(0) = e ' (0) = e ' » = 0, (14)

r '(1) = e(4)(1) = e(5)(0) = 0.

There is h(e, i,v) = 48 [e + cos i + v]. It is not difficult to verify that the values of w = 24, k = 0 meet conditions (8) and (9), respectively; hence,

Z(e) = 0, n(e) =24 f x(e,Q)QdQ,

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rem 1, problem (14) admits a positive solution e* such that

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o < c*(e) < 24/ x(e, q)qdQ.

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represented by the lower and upper solutions of (14). According to Theo- S

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merical solution to (14) and its estimate given by the Green's function. w"

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Figure 1 - The lower plot is the numerical solution of (14), multiplied by a large coefficient 4 x 103 for comparison purposes, while the upper plot is the estimate of the above, given by the Green's function. Observe how the two curves present the same behavior as functions of e.

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Example 2. Extrapolate the following application of the BVP

-e(6) + 50 [e + sin e (e) + cos (e ' (e)) + e(5)(e)J =0, 0 <e< 1,

e (0) = e' (0) = e' '(0) = 0, e'' '(1) = e(4)(i) = e(5)(0) = 0.

(15)

There is h(e,^,v,r) = 50 [e + sin n+cos(v)4 + t]. It is not difficult to verify that the values w = 25, k = 0 meet conditions (12) and (13), respectively; hence,

Z (e) = 0, n(e)=25/ x(e, q)q dQ,

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are the lower and upper solutions of (15). According to Theorem 2, problem (15) admits a positive solution e* such that

0 < e*(e) < 25 / x(e, Q)Qdg. Jo

Figure (2) shows the desired result of the comparison of the numerical solution to (15) and its estimate given by the Green's function.

Figure 2 - The lower plot is the numerical solution of (15), multiplied by a large coefficient 4 x 103 for comparison purposes, while the upper plot is the estimate of the above, given by the Green's function. Observe how the two curves present the same behavior as functions of e.

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Value Problems, 2010, art.number:878131. Available at: ^

https://doi.org/10.1155/2010/878131. ^

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Confinar una solución no negativa entre una solución superior e inferior para un problema de valor límite de sexto grado _

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Zouaoui Bekria, Nicola Fabianob,

Mohammad Esmael Sameic, Stojan Radenovicd .g

aa a Universidad de Orán 1, Laboratorio de Matemáticas Fundamentales ^

y Aplicadas, Oran; Centro Universitario Nour Bachir, Instituto de

Ciencias, Departamento de Ciencias y Tecnología, El Bayadh;

República Argelina Democrática y Popular ro

b Universidad de Belgrado, Instituto de Ciencias Nucleares "Vinca" -Instituto Nacional de la República de Serbia,

Belgrado, República de Serbia, autor de correspondencia Q

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c Universidad Bu-Ali Sina, Facultad de Ciencias Básicas, Departamento de Matemáticas, Hamedan, República Islámica del Irán

d Universidad de Belgrado, Facultad de Ingeniería Mecánica,

o Belgrado, República de Serbia

§ CAMPO: matemáticas o

™ TIPO DE ARTICULO: artículo científico original

Sí Resumen:

3 Introducción/objetivo: El objetivo del artículo es demostrar la

§ existencia de soluciones para un caso especial del problema del valor límite de sexto orden.

Métodos: El teorema del punto fijo de Leray-Schauder se utiliza para determinar las soluciones de límite superior.

Resultados: Se han encontrado soluciones de límite inferior y

superior.

Conclusión: El problema del valor límite de sexto orden admite soluciones.

Palabras claves: alternativa no lineal de Leray-Schauder, función de Green, teorema del punto fijo, soluciones inferior y superior, problema de valor límite.

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_ Ограничение неотрицательного решения между нижним и

^ верхним решениями в краевых задачах шестого порядка

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Зуауи Бекриа, Никола Фабиано,

О Мохаммад Исмаэль Самейв, Стоян Раденовичг

Университет Орана 1, Лаборатория фундаментальной и прикладной математики, г. Оран; Университетский центр Нур Башир Эль-Баяд, Институт естественных наук, факультет естественных наук и технологий, г Эль-Баяд; Алжирская Народно-Демократическая Республика,

Белградский университет, Институт ядерных исследований «Винча» - Институт государственного значения для Республики Сербия, г Белград, Республика Сербия, корреспондент Университет Бу-Али Сина, факультет фундаментальных наук, математический факультет, г Хамадан, Исламская Республика Иран

Белградский университет, факультет машиностроения, г. Белград, Республика Сербия

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РУБРИКА ГРНТИ: 27.25.17 Метрическая теория функций,

27.29.15 Общая теория обыкновенных |

дифференциальных уравнений и систем уравнений, о.

27.39.15 Линейные пространства,

снабженные топологией, порядком и другими структурами

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Введение/цель: Доказательство существования решения частного случая краевой задачи шестого порядка.

Методы: Теорема Лере-Шаудера о неподвижной точке ис- ° пользуется в определении нижней и верхней границы реше- $ ния задачи. ст

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Результаты: В ходе исследования были найдены решения нижней и верхней границы.

Выводы: Краевая задача шестого порядка допускает решения.

Ключевые слова: нелинейная альтернатива Лере-Шаудера, функция Грина, теорема о неподвижной точке, нижние и верхние решения, краевая задача.

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Зуауи Бекриа, Никола Фабиано1 Мохамед Есмаел Самеив, Сто]ан Раденови^ г

а Универзитету Орану 1, Лаборатори]а за основну и приме^ену математику, Оран; Универзитетски центар„Нур Башир", Институт за науке, Оде^е^е за науку и технологи|у, Ел Ба]ад; Народна "5

Демократска Република Алжир,

Универзитет у Београду, Институт за нуклеарне науке „Винча" -Национални институт Републике Срби]е, Београд, Република Срби]а, ауторза преписку в Универзитет BASU, Факултет основних наука, Департман за математику, Хамедан, Исламска Република Иран

г Универзитету Београду, Машински факултет, Београд, Република Срби]а

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КАТЕГОРША (ТИП) ЧЛАНКА: оригинални научни рад

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Увод/цил>: Доказивак>е посто]ак>а решена за посебан слу-g чаj граничног проблема шестог реда.

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^ Методе: Лера}-Шаудерова теорема о фиксно] тачки кори-

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Резултати: Прона^ена су решена док>е и гор^е границе. Закъучак: Гэанични проблем шестог реда дозвол>ава ре-

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ш Гринова функци]а, теорема фиксне тачке, док>а и гор^а

решена, проблем граничних вредности.

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Paper received on: 30.01.2024. Manuscript corrections submitted on: 07.06.2024. Paper accepted for publishing on: 08.06.2024.

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