On the lower and upper solutions method for the problem of elastic beam with hinged ends

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On the lower and upper solutions method for the problem of elastic beam with hinged ends R. Vrabel Slovak University of Technology in Bratislava, Faculty of Materials Science and Technology, Hajdoczyho 1, 917 01 Trnava, Slovakia

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 17 January 2014 Available online xxxx Submitted by P.J. McKenna

We establish the existence theorem for solutions of fourth-order nonlinear equations with the Lidstone boundary conditions which model deflections of an elastic beam (curve) with the hinged ends. The new lower and upper solutions method is formulated and then applied to a class of singularly perturbed problems and here is shown that the solutions uniformly converge to the solution of a reduced problem in the interior points of elastic curve and formula for an approximate solution without the use of numerical or iterative methods is derived. © 2014 Elsevier Inc. All rights reserved.

Keywords: Fourth-order ordinary differential equation Boundary value problem Schauder fixed point theorem Lower and upper solution Singular perturbation

1. Motivation and introduction In this article we are interested in establishing the existence of solution of the fourth-order differential equation which models the stationary states of the deflection of an elastic beam, namely, the ordinary differential equation   y (4) (x) + λy  (x) = h x, y(x) ,

λ<0

(1)

subject to the Lidstone boundary conditions y(0) = y(1) = y  (0) = y  (1) = 0

(2)

which correspond to hinged ends, when there is no bending moment at the ends. In Eq. (1), λ = k1 + k2 , where k1 < k2 < 0 are the real constants and h is a continuous, monotone decreasing in the second variable function in the area specified below in Theorem 7. Without loss of generality we can write Eq. (1) in the form E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2014.08.004 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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  y (4) (x) + (k1 + k2 )y  (x) + k1 k2 y(x) = f x, y(x)

(3)

with f (x, y) = h(x, y) + k1 k2 y. Due to the importance of fourth-order boundary value problems (BVPs) in describing a large class of elastic deflection that topic is covered extensively in the literature that deals with the existence and behavior of solutions for such problem subject to the two- or multi-point or integral boundary conditions and was investigated by several techniques; see e.g. [3] and [18]. In [3], G. Chai studied the nonlocal integral BVP for fourth-order equation applying the fixed point theorem on the cone. D. O’Regan in [12] discussed the problem of existence of solution to two-point BVP for fourth-order equation with the possible singularities in the nonlinear term. J. Webb, G. Infante and D. Franco in [16] studied the existence of multiple positive solutions of the fourth-order differential equation subject to various boundary conditions and under suitable conditions on the nonlinear term. Recently, K. Ozen and K. Orucoglu in [13] investigated the approximations converging to the unique solution of a multi-point BVP given by a linear fourth-order ordinary differential equation with variable coefficients involving nonlocal integral conditions by using reproducing kernel method. In [4], to investigate influences of the internal resonance of a hinged-clamped beam with a random excitation the authors reduced a partial differential equation of nonstationary motion and boundary conditions to a system of coupled non-linear ordinary differential equations using Galerkin procedure. We also remark that, in many papers, the nonlinear term f must have a suitable behavior at infinity and at zero [10] and/or must fulfil the requirements on the monotonicity (mixed monotonicity, for example see e.g. [6]). In the papers [14,18] the authors developed the monotone method in the presence of lower and upper solutions for the fourth-order BVP and for f satisfying some monotonicity properties. The method of lower and upper solutions allows us to ensure the existence of a solution for the considered problem lying between the lower and the upper solution, that is, we get information about the existence and location of the solutions. So the original problem of finding a solution is replaced by that of finding two well-ordered barrier solutions that satisfy some suitable inequalities. To our best knowledge, there is no appropriate lower and upper solutions method for the BVP (3), (2). The fundamental problem lies in that either the signs of inequality in the differential inequalities are opposite [7] as would be appropriate for construction of the barrier functions or having opposite monotonicity requirement with respect to the variables in nonlinear term f , e.g. [6,14,18]. As mentioned before, we consider the mathematical model of the vertical deflection of the elastic curve y(x) which is subjected to a nonuniform load   h(x) = w(x) − f x, y(x), y  (x) .

(4)

In (4), the function f (x, y(x), y  (x)) is the nonlinear spring force upward, which depends not only on the beam deflection y but also on the position x and the curvature of the beam at this point, and w(x) denotes the applied loading downward, if w ≥ 0. For simplicity, the weight of the beam is neglected. Recall that the beam is modeled as a one-dimensional object, the elastic curve y(x). Denoting by EI the flexural rigidity of the beam – E and I are Young modulus and the mass moment of inertia calculated with respect to the axis which passes through the centroid of the cross-section and which is perpendicular to the applied loading, respectively, the vertical deflection y(x), according to the Euler–Bernoulli beam theory, is governed by a fourth-order ordinary differential equation (assuming E and I to be constants) EIy (4) (x) = −h(x), which becomes the following nonlinear differential equation for the deflection y:

(5)

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  EIy (4) (x) = f x, y(x), y  (x) − w(x).

3

(6)

Thus, the problem under consideration is a special case of Eq. (6). The rest of the article is organized as follows: In Sections 2 and 3 we present some technical results needed for the proofs of non-negativity of the Green’s function and the existence of solution. The new theorem regarding the method of lower and upper solution is formulated and proved by the Schauder fixed point theorem in Section 4 (Theorem 7). In Section 5 we apply this theorem to the singularly perturbed problem for fourth-order differential equation subject to the Lidstone boundary conditions. The achieved results show the effectiveness of the method for investigation of asymptotic behavior of solutions and their approximation. 2. Technical results For later reference, in this brief section we summarize some technical results. a) The function g(ζ, x, s) =

sinh[ζ(s − 1)] sinh[ζx] ζ sinh[ζ]

(7)

is negative, monotone increasing function as a function of the variable ζ for ζ > 0 and every pair of parameters (x, s), 0 < x ≤ s < 1. Further, g(0) = (s − 1)x and g(∞) = 0. b) The function gv (ζ, x) =

sinh[ζx] sinh[ζ]

(8)

is positive, monotone decreasing as a function of the variable ζ for ζ > 0 and x ∈ (0, 1]. c) The function sinh[ζx] gv (ζ, x) = 2 2 ζ ζ sinh[ζ]

(9)

is positive, monotone decreasing as a function of the variable ζ for ζ > 0 and x ∈ (0, 1]. 3. Green’s function for L(y) ≡ y (4) + (k1 + k2 )y  + k1 k2 y Lemma 1. The BVP (2) for L(y) ≡ y (4) (x) + (k1 + k2 )y  (x) + k1 k2 y(x) = 0

(10)

with k1 < k2 < 0 has only trivial solution. √ √ Proof. The roots of the characteristic equation for (10) are the real numbers ± −k1 , ± −k2 . Now the claim of the lemma immediately follows from the fact that the determinant     1 1 1 1   √ √ √   e−√−k1 −k2 − −k2 −k1 e e e   D=    −k√ −k −k −k 1 2 √ √2 √1    −k1 e− −k1 −k2 e− −k2 −k2 e −k2 −k1 e −k1        = 2(k2 − k1 )2 cosh[ −k1 + −k2 ] − cosh[ −k1 − −k2 ] is positive (i.e. nonzero). 2

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4

Lemma 2. Let k1 < k2 < 0. Then the Green’s function G(x, s) for the linear problem L(y) = 0 with the boundary conditions (2) satisfies G(x, s) ≥ 0

for

0 ≤ x, s ≤ 1.

Proof. The Green’s function associated to this problem with the Lidstone boundary conditions is √ √ ⎧ sinh[ −k1 (s−1)] sinh[ −k1 x] 1 √ √ ⎪ [ − ⎪ (k2 −k1 ) −k1 sinh[ −k1 ] ⎪ ⎪ ⎨ for 0 ≤ x < s ≤ 1 G(x, s) = √ √ sinh[ −k1 (x−1)] sinh[ −k1 s] ⎪ 1 √ √ ⎪ (k −k [ − ⎪ −k1 sinh[ −k1 ] ⎪ ⎩ 2 1) for 0 ≤ s < x ≤ 1

√ √ sinh[ −k2 (s−1)] sinh[ −k2 x] √ √ ] −k2 sinh[ −k2 ] √ √ sinh[ −k2 (x−1)] sinh[ −k2 s] √ √ ] −k2 sinh[ −k2 ]

(11)

or

G(x, s) =

√

1 (k2 −k1 ) [g( −k1 , x, s) √ 1 (k2 −k1 ) [g( −k1 , s, x)

√ − g( −k2 , x, s)] for 0 ≤ x < s ≤ 1 √ − g( −k2 , s, x)] for 0 ≤ s < x ≤ 1

(12)

where g(ζ, x, s) =

sinh[ζ(s − 1)] sinh[ζx] . ζ sinh[ζ]

(13)

The function g(ζ, x, s) is monotone increasing (Section 2a) for every 0 < x ≤ s < 1 fixed and ζ > 0, therefore, taking into consideration the symmetry of function G, the Green’s function for the linear problem L(y) = 0, (2) is non-negative on the square [0, 1]2 . 2 Lemma 3 (Maximum principle). Let L(˜ y (x)) ≥ 0 for    y˜ ∈ Ψ = y ∈ C 4 [0, 1] ; y satisfies (2) . Then y˜(x) ≥ 0 on [0, 1]. ˜ y (x)) ≥ 0 on [0, 1]. The y˜ is a solution of BVP L(˜ y (x)) = h(x), (2) for an Proof. Let y˜ ∈ Ψ satisfies L(˜ ˜ appropriate continuous function h(x) ≥ 0. Thus the function y˜ is a solution of an integral equation

1 ˜ G(x, s)h(s)ds.

y=

(14)

0

From Lemma 2, the function y˜ is nonnegative on [0, 1]. 2 4. Method of lower and upper solutions We recall the linear differential operator L mentioned in the following definition is introduced in (10). Definition 4. The function α ∈ C 4 ([0, 1]) is said to be a lower solution for the BVP (3), (2) if     L α(x) ≤ f x, α(x) and

for x ∈ (0, 1)

(15)

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α(0) ≤ 0,

α(1) ≤ 0,

α (0) ≥ 0,

5

α (1) ≥ 0.

(16)

An upper solution β ∈ C 4 ([0, 1]) is defined analogously by reversing the inequalities in (15) and (16). Remark 1. If we denote     gα (x) = L α(x) − f x, α(x) ,

    gβ (x) = L β(x) − f x, β(x) ,

x ∈ [0, 1]

(17)

then gα (x) ≤ 0 and gβ (x) ≥ 0 for x ∈ [0, 1].

(18)

Now let vi (x), i = α, β be the solutions of (10) satisfying the same boundary conditions as α and β, respectively, that is, vi (0) = i(0),

vi (1) = i(1),

vi (0) = i (0),

vi (1) = i (1) i = α, β.

(19)

With respect to Lemma 1 the functions vi (x), i = α, β are uniquely determined as vi (x) =

1 √ √ 2(k1 − k2 ) sinh[ −k1 ] sinh[ −k2 ]          · i(0) k1 sinh −k2 (1 − x) sinh[ −k1 ] − k2 sinh −k1 (1 − x) sinh[ −k2 ]       + i(1) k1 sinh[ −k2 x] sinh[ −k1 ] − k2 sinh[ −k1 x] sinh[ −k2 ]         + i (0) sinh −k2 (1 − x) sinh[ −k1 ] − sinh −k1 (1 − x) sinh[ −k2 ]       + i (1) sinh[ −k2 x] sinh[ −k1 ] − sinh[ −k1 x] sinh[ −k2 ] , i = α, β.

Taking into account Section 2c for first two large brackets and Section 2b for second two brackets we have that vα (x) ≤ 0

and vβ (x) ≥ 0 on the interval [0, 1].

(20)

Hence if G(x, s) is the Green’s function for the BVP (10), (2) then for a lower solution α of (3) we have, in view of Lemma 2, (18), (20), the following implications

1 L(α) = f (x, α) + gα (x)

=⇒

α(x) = vα (x) +

  G(x, s)f s, α(s) ds +

0

=⇒

1 G(x, s)gα (s)ds 0

α(x) ≤ T α(x) on [0, 1],

and by a similar way we obtain β(x) ≥ T β(x) on [0, 1] where T : C([0, 1]) → C 4 ([0, 1]) is the operator defined by

1 T φ(x) =

  G(x, s)f s, φ(s) ds,

0 ≤ x ≤ 1.

(21)

0

The meaning of the operator T is based on the equivalence of the problem (3), (2) to the integral equation

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1 y(x) =

  G(x, s)f s, y(s) ds,

0 ≤ x ≤ 1.

(22)

0

where G is the Green’s function for (10), (2). The existence of solution of the BVP (3), (2) will be proved by using the Schauder fixed point theorem [1,15]. For completeness and later references, let us recall Schauder theorem: Theorem 5. Let X be a Banach space, B be a closed and convex subset of X, and T : B → B be a continuous map. Then T has at least one fixed point in B if the image T (B) := {T x; x ∈ B} is precompact. First we prove the following lemma. Lemma 6. Let there exists a constant L such that   f (x, y) ≤ L for (x, y) ∈ [0, 1] × R. Then the BVP (3), (2) has a solution. Proof. As usual, let C([0, 1]) denote the Banach space of continuous functions on [0, 1] equipped with the sup-norm y = supx∈[0,1] |f (x)|. Then the operator (21) maps C([0, 1]) into C([0, 1]). If    ∂G(x, s)    M1 = sup G(x, s) and M2 = sup  ∂x  [0,1]×[0,1] [0,1]×[0,1] then we have that T φ ≤ M1 L. Therefore T maps closed, bounded and convex set    B =: φ ∈ C [0, 1] : φ ≤ M1 L into itself. Furthermore T B is compact due to the fact that |(T φ) | ≤ M2 L for all φ ∈ B. Therefore, T B is equicontinuous and so by the Ascoli–Arzela theorem T is a compact operator. Because of the operator T is a Hammerstein operator with continuous kernel G and continuous function f , the operator T is continuous ([2], p. 14). Hence, by the Schauder fixed point theorem T has a fixed point in B. This is a solution of the BVP (3), (2). 2 Theorem 7. Suppose that for the problem (3), (2) exist a lower solution α and an upper solution β such that α(x) ≤ β(x) for x ∈ [0, 1]. If f : [0, 1] × [α(x), β(x)] → R is continuous and satisfies f (x, u1 ) ≤ f (x, u2 )

for α(x) ≤ u1 ≤ u2 ≤ β(x) and x ∈ [0, 1]

(23)

then there exists a solution y(x) for BVP (3), (2) satisfying α(x) ≤ y(x) ≤ β(x)

for 0 ≤ x ≤ 1.

(24)

Proof. Define the function F on [0, 1] × R by setting ⎧ ⎨ f (x, β(x)) for y > β(x) F (x, y) = f (x, y) for α ≤ y ≤ β(x) ⎩ f (x, α(x)) for y < α(x).

(25)

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Since F is continuous and bounded on [0, 1] ×R, by Lemma 6, there exists a solution of BVP L(y) = F (x, y), (2). We now show that the inequality (24) is true. We have           L y(x) − β(x) = L y(x) − L β(x) ≤ F x, y(x) − f x, β(x) ≤ 0. Thus L(y(x) − β(x)) = h1 (x) ≤ 0 for x ∈ [0, 1], that is, from Lemma 2 and (20)

1 y(x) − β(x) = −vβ (x) +

G(x, s)h1 (s)ds ≤ 0 0

for 0 ≤ x ≤ 1. This implies that y(x) ≤ β(x) on [0, 1]. By a similar way,           L y(x) − α(x) = L y(x) − L α(x) ≥ F x, y(x) − f x, α(x) ≥ 0. Thus L(y(x) − α(x)) = h2 (x) ≥ 0 for x ∈ [0, 1], and hence

1 y(x) − α(x) = −vα (x) +

G(x, s)h2 (s)ds ≥ 0. 0

The proof is complete. 2 Remark 2. Under the condition (23) the operator T is a monotone increasing in the Banach space C([0, 1]) which is partially ordered by the cone P of nonnegative functions P = {y ∈ C([0, 1]); y(x) ≥ 0 for x ∈ [0, 1]}, in the sense that φ1 ≤ φ2 implies T φ1 ≤ T φ2 . Regarding the monotone (or mixed monotone) mappings, many authors have investigated the existence and uniqueness of the fixed point for (mixed) monotone operators and obtained a lot of interesting results. In [9], the authors formulated an intermediate value theorem for the monotone increasing operators in a Banach space ordered by a positive cone. The existence and uniqueness of the fixed point for mixed monotone operators in an ordered Banach space was studied in [20,17,19,11], for example. For nonmonotone mappings Kellog proves in [8] the theorem which ensures the uniqueness of the fixed point in the Schauder theorem. 5. Application of Theorem 7 to a class of singularly perturbed problem In this section we apply Theorem 7 to the class of the singularly perturbed equations L (y) ≡ 2 y (4) (x) + (k1 + k2 )y  (x) + k1 k2 y(x) = f (x, y)

(26)

subject to the Lidstone boundary conditions (2), where k1 , k2 are the real numbers (k1 < k2 < 0), the function f is continuous, and the parameter is assumed to be positive and small, 0 <  1. We will investigate the existence of solutions and their asymptotic behavior as → 0+ . We show, that under some hypotheses the solution y of the singularly perturbed BVP (26), (2) exist for every sufficiently small values of singular perturbation parameter , say ∈ (0, 0 ], and y (x) converges uniformly on every compact subinterval of (0, 1) to the solution of a reduced problem k1 k2 y = f (x, y), obtained by setting = 0 in (26), as → 0+ . As far as we know, when the literature concerning the singularly perturbed differential equations of the second-order is quite huge, very few studies concern the fourth-order BVPs. Recently, in [5], the authors

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considered a class of singularly perturbed BVPs for fourth-order differential equations. The BVP was reduced into a weakly coupled system of two differential equations, in order to solve them numerically. Before proceeding further, we summarize for our convenience, the notations used in this section. Let Dδ (u) denotes the set   (x, y); 0 ≤ x ≤ 1, y − u(x) ≤ d(x) ,



where d(x) is the positive, continuous and convex function on the interval [0, 1] such that ⎧ ⎨ |u(0)| + δ d(x) = δ ⎩ |u(1)| + δ

for 0 ≤ x ≤ 2δ , for δ ≤ x ≤ 1 − δ, for 1 − 2δ ≤ x ≤ 1,

where δ is a small positive constant. Throughout this section, we will assume that (H1) There is a constant η such that 0≤

∂f (x, y) ≤ η < k1 k2 ∂y

for (x, y) ∈ Dδ (u).

(27)

(H2) The reduced problem k1 k2 y = f (x, y) has C 4 solution u defined on [0, 1]. Remark 3. The assumption (27) implies that the operator T :

1 Tφ =

  G(x, s)f s, φ(s) ds,

0

where G is the Green’s function for (10), (2) is monotone increasing operator in the partially ordered Banach space C([0, 1]), ordered by the cone of the nonnegative functions. Indeed, If φ1 ≤ φ2 then

1 T φ1 − T φ2 =

     G(x, s) f s, φ1 (s) − f s, φ2 (s)

0

1 G(x, s)

=

  ∂f  s, θ(s) φ1 (s) − φ2 (s) ds ≤ 0. ∂y

0

Remark 4. The variable y can locally be solved for x by the Implicit Function Theorem. We are thus just assuming that such a solution can be made globally over [0, 1] and is continuous up to the fourth derivative on [0, 1]. Let λ1 < λ2 < 0 < −λ2 < −λ1 are the real roots of the characteristic equation associated with the linear singularly perturbed equation L (y) − ηy = 0, that is,

(28)

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 λ1,2,3,4 = ±

9

    1 1 ˜ √ , −k1 − k2 ± (k1 − k2 )2 + 4η = O 2 ±

or in other words, λ1,2,3,4 is “of the same order” as lim

→0+

1 √ 

for → 0+ , in the sense that

λ1,2,3,4 √ = μ > 0, ±1/

μ = ∞.

Now we can formulate the main result concerning the existence, uniqueness and asymptotic behavior of the solutions for (26), (2). Theorem 8. Suppose that the hypotheses (H1) and (H2) hold and the function f is continuous in Dδ (u). Then the singularly perturbed BVP (26), (2) has in Dδ (u) for every ∈ (0, 0 ] a unique solution satisfying 4    y (x) − u(x) ≤ Γ ( ) + Ai (x, )

for 0 ≤ x ≤ 1,

(29)

i=1

where Ai , i = 1, 2, 3, 4 are the unique solutions of the linear equation L (y) − ηy = 0 with the boundary conditions for for for for

A1 (x, ): A2 (x, ): A3 (x, ): A4 (x, ):

y(0) = |u(0)|, y(1) = 0, y  (0) = 0, y  (1) = 0, y(0) = 0, y(1) = |u(1)|, y  (0) = 0, y  (1) = 0, y(0) = 0, y(1) = 0, y  (0) = |u (0)|, y  (1) = 0, y(0) = 0, y(1) = 0, y  (0) = 0, y  (1) = |u (1)|,

and Γ ( ) ≥

1 k1 k2 − η

  max  2 u(4) (x) + (k1 + k2 )u (x) ≥ 0

x∈[0,1]

(30)

is a constant. Remark 5. The functions Ai , i = 1, 2, 3, 4 can be explicitly calculated and it is easy to verify that A1 (x, ) > 0 A2 (x, ) > 0 A3 (x, ) < 0 A4 (x, ) < 0

and and and and

˜ 2 (λ2 − λ2 )−1 eλ2 x ) A1 (x, ) = |u(0)|O(λ 1 1 2 ˜ 2 (λ2 − λ2 )−1 eλ2 (1−x) ) A2 (x, ) = |u(1)|O(λ 1 1 2 ˜ 2 − λ2 )−1 eλ2 x ) A3 (x, ) = |u (0)|O((λ 2 1 ˜ 2 − λ2 )−1 eλ2 (1−x) ) A4 (x, ) = |u (1)|O((λ 2 1

4 for x ∈ [0, 1]. As a result, the function Γ ( ) + i=1 Ai (x, ) is positive and is less than the function d(x) on [0, 1] for sufficiently small , say, ∈ (0, 0 ]. With aim to get better picture and understanding of investigated phenomenon (an asymptotic behavior of solutions), the functions Ai , i = 1, 2, 3, 4 are shown on Fig. 1. Now we return to the proof of Theorem 8. Proof. The existence of solution: Let us define the lower and upper solutions for the problem BVP (26), (2) as follows: α (x) = u(x) −

4  i=1

Ai (x, ) − Γ ( )

(31)

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Fig. 1. The functions Ai , i = 1, 2, 3, 4 and the sum of all functions Ai for λ1,2,3,4 = ±15, ±9 and |u(0)| = |u(1)| = |u (0)| = |u (1)| = 1 (from top-left to bottom).

and β (x) = u(x) +

4 

Ai (x, ) + Γ ( ).

(32)

i=1

Obviously, the functions α and β satisfy the boundary conditions of Definition 4 required for the lower and upper solutions to the BVP (26), (2) and α (x) ≤ β (x) for x ∈ [0, 1]. It remains to show that     L α (x) ≤ f x, α (x)

    and L β (x) ≥ f x, β (x) .

ˆ y) =: k1 k2 y − f (x, y). According to the Lagrange theorem, Let h(x,

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ˆ     ˆ y) = h ˆ x, u(x) + ∂ h(x, θ(x)) y − u(x) h(x, ∂y where (x, θ(x)) is a point between (x, y) and (x, u(x)). Thus     L α (x) − f x, α (x)  (4) 4  Ai (x, ) − Γ ( ) = 2 u(x) − i=1



+ (k1 + k2 ) u(x) −

= u

(x) −

2

 Ai (x, ) − Γ ( )

i=1

 2 (4)

4 

4 

Ai (x, )

  ˆ x, α (x) +h

(4)

 4 



+ (k1 + k2 )u (x) − (k1 + k2 )

i=1



≤ u

−

2

4 

(4)

+ (k1 + k2 )u − (k1 + k2 )

Ai

− (k1 k2 − η)

 4 

 4 



i=1

Ai (x, )

i=1

 4  ˆ θ1 (x))  ∂ h(x, − Ai (x, ) + Γ ( ) ∂y i=1 2 (4)



 Ai

i=1

 − (k1 k2 − η)Γ ( )

Ai

i=1

= 2 u(4) + (k1 + k2 )u − (k1 k2 − η)Γ ( )   ≤  2 u(4) (x) + (k1 + k2 )u (x) − (k1 k2 − η)Γ ( ). Analogously for β we have     L β (x) − f x, β (x)  4 (4)  4    2 (4) 2  Ai (x, ) + (k1 + k2 )u (x) + (k1 + k2 ) Ai (x, ) = u (x) + i=1

i=1

 4  ˆ θ2 (x))  ∂ h(x, + Ai (x, ) + Γ ( ) ∂y i=1 

≥ u

2 (4)

+

2

4 

(4)



Ai

+ (k1 + k2 )u + (k1 + k2 )

i=1

+ (k1 k2 − η)

 4 

 4 

Ai

i=1

 Ai



+ (k1 k2 − η)Γ ( )

i=1

= 2 u(4) + (k1 + k2 )u + (k1 k2 − η)Γ ( )   ≥ − 2 u(4) (x) + (k1 + k2 )u (x) + (k1 k2 − η)Γ ( ). Thus, for Γ ( ) ≥ k1 k12 −η maxx∈[0,1] [| 2 u(4) (x) + (k1 + k2 )u (x)|] we obtain the required differential inequalities for a lower solution α and an upper solution β . On the basis of Theorem 7, the BVP (26), (2) has a solution lying between α and β on [0, 1] and for every ∈ (0, 0 ].

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The uniqueness of solution: The (local) uniqueness of the solutions in Dδ (u) for (26), (2) follows from the fact that the function f on the basis of assumption (H1) is Lipschitz continuous with Lipschitz constant lf = η < k1 k2 and

1 lim+ 1/ 2

G(x, s,

→0



−k1 / ,



 −k2 / )ds =

0

1 k1 k2

monotonically from below

0

for x ∈ (0, 1) for x = 0 and x = 1

which can be verified bydirect computation.  The function G(x, s, −k1 / , −k2 / ) is defined by the right side of (11) replacing k1 and k2 by k1 / and k2 / , respectively. Now let y¯ , yˆ ∈ C 4 ([0, 1]) be two solutions for (26), (2) lying in Dδ (u). Then for every fixed x ∈ [0, 1] we have   y¯ (x) − yˆ (x) ≤ 1/ 2

1 G(x, s,



−k1 / ,



     −k2 / )f s, y¯ (s) − f s, yˆ (s) ds

0

1 = 1/

2

G(x, s,



−k1 / ,



−k2 / )

  ∂f  s, θ(s) y¯ (s) − yˆ (s)ds ∂y

0

    η ≤ max y¯ (s) − yˆ (s) < max y¯ (s) − yˆ (s) k1 k2 s∈[0,1] s∈[0,1] for every sufficiently small , ∈ (0, 0 ]. It leads to the contradiction because of the function |¯ y (x) − yˆ (x)| is continuous and thus it attains its maximum value on [0, 1] at some point x0 ∈ [0, 1]. The proof of theorem is complete. 2 Remark 6. We can obtain a more precise estimate for solutions of the above considered problem if we use the following construction of the lower and upper bounds of solution in the dependence on the sign of solution u of a reduced problem at x = 0 and x = 1. Concretely, i) for u(0) ≥ 0, u(1) ≥ 0 we define the lower and upper solution as follows: α ˜  (x) = u(x) − A1 (x, ) − A2 (x, ) − A3 (x, ) − A4 (x, ) − Γ ( ) β˜ (x) = u(x) + A3 (x, ) + A4 (x, ) + Γ ( ) ii) for u(0) ≤ 0, u(1) ≤ 0: α ˜  (x) = u(x) − A3 (x, ) − A4 (x, ) − Γ ( ) β˜ (x) = u(x) + A1 (x, ) + A2 (x, ) + A3 (x, ) + A4 (x, ) + Γ ( ) iii) for u(0) ≤ 0, u(1) ≥ 0: α ˜  (x) = u(x) − A2 (x, ) − A3 (x, ) − A4 (x, ) − Γ ( ) β˜ (x) = u(x) + A1 (x, ) + A3 (x, ) + A4 (x, ) + Γ ( ) iv) for u(0) ≥ 0, u(1) ≤ 0:

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π Fig. 2. Solution of BVP 2 y (4) + (k1 + k2 )y  + arctan y = ( π 2 x − 4 ), (2) with k1 = −2, k2 = −1 and  = 0.0001 (the solid line). The dashed and dotted lines represent the approximate solution y˜ (x) (with η = 5/3) and solution of reduced problem, the function π u(x) = tan( π 2 x − 4 ), respectively.

α ˜  (x) = u(x) − A1 (x, ) − A3 (x, ) − A4 (x, ) − Γ ( ) β˜ (x) = u(x) + A2 (x, ) + A3 (x, ) + A4 (x, ) + Γ ( ). It is readily verified by calculus that these functions satisfy the conditions required for the lower and upper solutions for the BVP (26), (2). Thus we can approximate the solutions of the BVP (26), (2) without using any numerical scheme which are confronted with the problem – to capture the solution within the boundary layers. We define a boundary layer as a region near endpoints of the interval of the independent variable in which a solution is changing rapidly while away from the layer the solution behaves regularly and varies gradually. The function y˜ (x) = u(x) −A1 (x, ) −A2 (x, ) (˜ y (x) = u(x) +A1 (x, ) +A2 (x, ); y˜ (x) = u(x) +A1 (x, ) − A2 (x, ); y˜ (x) = u(x) − A1 (x, ) + A2 (x, )) in the case i) (ii; iii; iv) can serve as a good approximation for the solution of the BVP (26), (2) if is small enough. Example 1. Consider the BVP 2 y (4) + (k1 +k2 )y  +arctan y = ( π2 x− π4 ), (2), where k1 < k2 < 0 are such that k1 k2 > 1. Rewrite above differential equation in the form L (y(x)) = k1 k2 y−arctan y+( π2 x − π4 ). It is readily ˜ 2 )−1 verified by calculus that the right side satisfies the assumption (H1) and (H2) with η = k1 k2 −(1 +(1 + δ) and a small constant δ˜ > 0. The constant δ in the definition of Dδ (u) and, consequently, 0 depend on the ˜ The solution of the reduced problem is u(x) = tan( π x − π ). Thus on the basis of Theorem 8, value of δ. 2 4 there exists 0 such that for every ∈ (0, 0 ] the BVP under consideration has in Dδ (u) a unique solution and these solutions uniformly converge to the solution u of reduced problem on every compact subinterval of (0, 1) for → 0+ . The region in the neighborhood of the points x = 0 and x = 1, where the solution is changing rapidly, taking into account that u(0) = 0 = u(1), is called a boundary layer (Fig. 2). Acknowledgment I would like to express my gratitude to the referee for all the valuable and constructive comments. References [1] J. Andres, L. Gorniewicz, From the Schauder fixed-point theorem to the applied multivalued Nielsen theory, in: Topological Methods in Nonlinear Analysis, in: Journal of the Juliusz Schauder Center, vol. 14, 1999, pp. 229–238. [2] N.A. Bobylev, Y.M. Burman, S.K. Korovin, Approximation Procedures in Nonlinear Oscillation Theory, Walter de Gruyter, Berlin, 1994.

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