Semi-linear impulsive higher order boundary value problems

Semi-linear impulsive higher order boundary value problems

Dedicated to Professor Jeffrey R. L. Webb on the occasion of his 75th birthday

Feliz Minhós

and Rui Carapinha

1Departamento de Matemática, Escola de Ciências e Tecnologia, Universidade de Évora

2Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação Avançada, Universidade de Évora,

Rua Romão Ramalho, 59, 7000-671 Évora, Portugal Received 18 May 2020, appeared 21 December 2020

Communicated by Alberto Cabada

Abstract. This paper considers two-point higher order impulsive boundary value prob- lems, with a strongly nonlinear fully differential equation with an increasing homeo- morphism. It is stressed that the impulsive effects are defined by very general functions, that can depend on the unknown function and its derivatives, till ordern−1.

The arguments are based on the lower and upper solutions method, together with Leray–Schauder fixed point theorem. An application, to estimate the bending of a one- sided clamped beam under some impulsive forces, is given in the last section.

Keywords: higher order boundary value problems, generalized impulsive conditions, upper and lower solutions, fixed point theory.

2020 Mathematics Subject Classification: 34B37, 34B10, 34B15.

1 Introduction

In this article we study the two point boundary value problem composed by the one-dimensional φ-Laplacian equation

(φ(u⁽ⁿ⁻¹⁾(t)))⁰+q(t)f(t,u(t), . . . ,u⁽ⁿ⁻¹⁾(t)) =0, t ∈[a,b]\{t₁, . . . ,t_m}, (1.1) where φis an increasing homeomorphism such that φ(0) = 0 and φ(_R) = _R, q∈ L^∞[a,b]is a positive function and f : [a,b]×_Rⁿ → _Ris a L¹-Carathéodory function, together with the boundary conditions

u^(j)(a) =A_j, u⁽ⁿ⁻¹⁾(b) =B, j=0, 1, . . . ,n−2, A_j,B∈_R, (1.2)

BCorresponding author. Email:fminhos@uevora.pt

and the impulsive conditions

∆u⁽ⁱ⁾(t_k) =I_i,k(u(t_k),u⁰(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)), i=0, 1, . . . ,n−2,

∆φ(u⁽ⁿ⁻¹⁾(t_k)) =I_n−1,k(u(t_k),u⁰(t_k), , . . . ,u⁽ⁿ⁻¹⁾(t_k)), (1.3) being∆u⁽ⁱ⁾(t_k) = u⁽ⁱ⁾(t⁺_k )−u⁽ⁱ⁾(t⁻_k ),i= 0, 1, . . . ,n−1, k = 1, 2, . . . ,m, I_i,k ∈ C(_Rⁿ,R), andt_k fixed points such thata=t₀<t₁< t₂ <· · ·< t_m < t_m+1= b.

Impulsive boundary value problems have been studied by many authors where it is high- lighted the huge possibilities of applications to phenomena where a sudden variation hap- pens. Indeed, these types of jumps occur in different areas such as population dynamics, engineering, control, and optimization theory, medicine, ecology, biology and biotechnology, economics, pharmacokinetics, and many other fields.

From a large number of items existent in the literature on classical impulsive differential problems, we mention, for instance, [1,2,17,19–21] and the references therein. The most applied arguments are based on critical point theory and variational methods [18,22,26], fixed point theory on cones [6,29], bifurcation results [13,15], and upper and lower solutions techniques suggested in [4,5,12,14].

In the last years, p-Laplacian andφ-Laplacian operators have been applied to semi-linear, quasi-linear, and strongly nonlinear differential equations, in singular and regular cases, in- creasing the range of theoretical and practical applications, as it can be seen, for example, in [3,11,25,27,28,30] and in their references. However, impulsive problems with this type of nonlinear differential equations are scarce.

In [16], the third order differential equation

(φ(u⁰⁰(t)))⁰+q(t)f(t,u(t),u⁰(t),u⁰⁰(t)) =0, t∈[a,b]\{t₁, . . . ,tn},

is studied, whereφis an increasing homeomorphism,q∈ C([a,b])withq>0 , f ∈ C([a,b]× R³,R), the two-point boundary conditions

u(a) =A, u⁰(a) =B, u⁰⁰(b) =C, A,B,C∈_R, and the impulsive effects are given by

∆u(t_k) =I_1k(t_k,u(t_k),u⁰(t_k)),

∆u⁰(t_k) =I_2k(t_k,u(t_k),u⁰(t_k),u⁰⁰(t_k)),

∆φ(u⁰⁰(t_k)) =I_3k(t_k,u(t_k),u⁰(t_k),u⁰⁰(t_k)),

wherek=1, 2, . . . ,n, I_1k ∈ C([a,b]×_R²,R), andI_ik∈ C([a,b]×_R³,R),i=2, 3.

In this work, we found a method that allows generalizing the above results to higher-order boundary value problems with impulsive functions, depending not only on the unknown function but also on its derivatives till ordern−1. To best of our knowledge, it is the first time, where such nonlinear higher-order problems are considered with this type of generalized impulsive functions.

This paper is organized in the following way: Section 2 contains the functional framework, some definitions and an explicit form for the solution of the associated homogeneous problem.

Section 3 presents the main existence and localization theorem obtained via lower and upper solutions technique and a fixed point theorem. The last section gives a technique to estimate the bending of a one-sided clamped beam under some impulsive forces and how it can be obtained some qualitative data about its variation.

2 Definitions and preliminary results

Let

PCⁿ⁻¹[a,b] =

(u:u∈ Cⁿ⁻¹([a,b];R)fort 6=t_k,u⁽ⁱ⁾(t_k) =u⁽ⁱ⁾(t⁻_k ),u⁽ⁱ⁾(t⁺_k ) exists fork=1, 2, . . . ,m, andi=0, 1, . . . ,n−1

) . DenoteX:=PCⁿ⁻¹[a,b]. ThenX is a Banach space with norm

kuk_X =max{ku⁽ⁱ⁾k_∞, i=0, 1, . . . ,n−1}, where

kwk_∞ = _sup

a≤t≤b

|w(t)|_.

Defining J := [a,b]_and J⁰ = J\{t₁, . . . ,t_m}, for a solutionuof problem (1.1)–(1.3) one should consideru(t)∈E, where

E:= PCⁿ⁻¹(J)∩Cⁿ(J⁰).

Next lemma provides a uniqueness result for a linear problem related to (1.1)–(1.3).

Lemma 2.1. For v∈ PC[a,b],the problem composed by the differential equation

(φ(u⁽ⁿ⁻¹⁾(t)))⁰+v(t) =0 (2.1) together with conditions(1.2),(1.3), has a unique solution given by

u(t) =

n−2 i

∑

"

A_i+

∑

k:tk<t

I_i,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

#(t−a)ⁿ⁻²⁻ⁱ (n−2−i)!

!

+

Z _t

a

(t−s)ⁿ⁻²

(n−2)! φ⁻¹ φ(B) +

Z _b

s v(r)dr−

∑

k:t_k>s

I_n−1,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

! ds.

Proof. Integrating the differential equation (2.1) fort∈(t_m,b]we get, by (1.2), φ

u⁽ⁿ⁻¹⁾(t)=φ(B) +

Z _b

tm

v(s)ds. (2.2)

By integration of (2.1) fort∈(t_m−1,tm]one has by (2.2) φ

u⁽ⁿ⁻¹⁾(t)=

Z _tm

t v(s)ds−I_n−1,m

t_m,u(t_m)_{, . . . ,}u⁽ⁿ⁻¹⁾(t_m)+φ

u⁽ⁿ⁻¹⁾ t⁺_m

=φ(B)−I_n−1,m

t_m,u(t_m), . . . ,u⁽ⁿ⁻¹⁾(t_m)+

Z _b

t v(s)ds and so,

u⁽ⁿ⁻¹⁾(t) =φ⁻¹

φ(B)−I_n−1,m

tm,u(tm), . . . ,u⁽ⁿ⁻¹⁾(tm)+

Z _b

t v(s)ds

. Therefore, fort ∈[a,b], we have

u⁽ⁿ⁻¹⁾(t) =φ⁻¹ φ(B) +

Z _b

t v(s)ds−

∑

k:tk>t

I_n−1,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

!

. (2.3)

Integrating (2.3), fort∈ [a,t₁], u⁽ⁿ⁻²⁾(t) =A_n−2+

Z _t

a φ⁻¹ φ(B) +

Z _b

s v(r)dr−

∑

k:tk>s

I_n−1,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

!!

ds.

By integration of (2.3) on(t₁,t₂]and (1.3) u⁽ⁿ⁻²⁾(t) = I_n−2,1

t₁,u(t₁), . . . ,u⁽ⁿ⁻¹⁾(t₁) +

Z _t

t1

φ⁻¹ φ(B) +

Z _b

s v(r)dr−

∑

k:tk>s

I_n−1,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

!!

ds.

Therefore, fort ∈[a,b], u⁽ⁿ⁻²⁾(t) =

∑

k:t_k<t

I_n−2,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)+An−2

+

Z _t

a φ⁻¹ φ(B) +

Z _b

s v(r)dr−

∑

k:tk>s

I_n−1,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

!!

ds.

Following the same method, by iterate integrations and (1.3), we obtain fort∈ [a,b] u(t) =

n−2 i

∑

"

A_i+

∑

k:tk<_t

I_i,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

#(t−a)ⁿ⁻²⁻ⁱ (n−2−i)!

!

+

Z _t

a

(t−s)ⁿ⁻²

(n−2)! φ⁻¹ φ(B) +

Z _b

s v(s)ds−

∑

k:tk>s

I_n−1,k

t_k,u(t_k), . . . ,u⁽ⁿ⁻¹⁾(t_k)

! ds.

Lower and upper solutions will play a key role in our method, and they are defined as it follows:

Definition 2.2. A functionα(t)∈Ewithφ(α⁽ⁿ⁻¹⁾(t))∈PC¹[a,b]is a lower solution of problem (1.1), (1.2), (1.3) if

























 (φ

α⁽ⁿ⁻¹⁾(t))⁰+q(t)f

t,α(t),α⁰(t), . . . ,α⁽ⁿ⁻¹⁾(t)≥0 α^(j)(a)≤ A_j, j=0, 1, . . . ,n−2,

α⁽ⁿ⁻¹⁾(b)≤ B

∆α⁽ⁱ⁾(t_k)≤ I_i,k(α(t_k), . . . ,α⁽ⁿ⁻¹⁾(t_k)), i=0, 1, . . . ,n−3,

∆α⁽ⁿ⁻²⁾(t_k)> I_n−2,k(α(t_k), . . . ,α⁽ⁿ⁻¹⁾(t_k))

∆φ(α⁽ⁿ⁻¹⁾(t_k))> I_n−1,k(α(t_k), . . . ,α⁽ⁿ⁻¹⁾(t_k)),

(2.4)

fork =1, 2, . . . ,m.

A function β(t)∈ Esuch thatφ(β⁽ⁿ⁻¹⁾(t))∈ PC¹[a,b]is an upper solution of (1.1)–(1.3) if it satisfies the opposite inequalities.

To control the derivativeu⁽ⁿ⁻¹⁾(t)we will apply the Nagumo condition:

Definition 2.3. AnL¹-Carathéodory function f :[a,b]×_Rⁿ→_Rsatisfies a Nagumo condition related to a pair of functionsγ,Γ∈E, withγ⁽ⁱ⁾(t)≤ _Γ⁽ⁱ⁾(t), fori=0, 1, . . . ,n−2, andt∈ [a,b], if there exists a function ψ:C([0,+_∞),]0,+_∞))such that

|f(t,x₀,x₁, . . . ,x_n−1)| ≤ψ(|x_n−1|), for all(t,x₀,x₁, . . . ,x_n−1)∈S (2.5) with

S:={(t,x₀,x₁, . . . ,x_n−1)∈ [a,b]×_Rⁿ:γ⁽ⁱ⁾(t)≤x_i ≤_Γ⁽ⁱ⁾(t),i=0, 1, . . . ,n−2}, and

Z +_∞ φ(µ)

ds ψ(φ⁻¹(s)) >

Z _b

a q(s)ds, (2.6)

where

µ:= max

k=0,1,2,...,m

(

Γ⁽ⁿ⁻²⁾(t_k+1)−γ⁽ⁿ⁻²⁾(t_k) t_k+1−t_k

,

γ⁽ⁿ⁻²⁾(t_k+1)−_Γ⁽ⁿ⁻²⁾(t_k) t_k+1−t_k

) . From the Nagumo condition we deduce ana prioriestimation foru⁽ⁿ⁻¹⁾(t):

Lemma 2.4. If the L¹-Carathéodory function f : [a,b]×_Rⁿ → _R satisfies a Nagumo condition in the set S,referred to the functionsγandΓ,then there is N ≥ µ> 0such that every solution u of the differential equation(1.1)verifiesku⁽ⁿ⁻¹⁾k_∞ ≤ N.

Proof. Letu(t)be a solution of (1.1) such that

γ⁽ⁱ⁾(t)≤u⁽ⁱ⁾(t)≤_Γ⁽ⁱ⁾(t), fori=0, 1, . . . ,n−2 andt∈ [a,b]. By the Mean Value Theorem, there existsη₀ ∈(t_k,t_k+1)with

u⁽ⁿ⁻¹⁾(η₀) = ^u

(n−2)(t_k+1)−u⁽ⁿ⁻²⁾(t_k)

t_k+1−t_k , withk=0, 1, 2, . . . ,m.

Moreover,

−N≤ −µ≤ ^γ

(n−2)(t_k+1)−_Γ⁽ⁿ⁻²⁾(t_k)

t_k+1−t_k ≤u⁽ⁿ⁻¹⁾(η₀) _(2.7)

≤ ^Γ

(n−2)(t_k+1)−γ⁽ⁿ⁻²⁾(t_k)

t_k+1−t_k ≤ µ≤ N.

If

u⁽ⁿ⁻¹⁾(t)≤ Nfor every t∈[a,b], the proof is complete.

On the contrary, assume that there isτ ∈ [a,b] such that

u⁽ⁿ⁻¹⁾(τ) > N. Consider the case where u⁽ⁿ⁻¹⁾(τ)> N. Therefore there is η₁ such that u⁽ⁿ⁻¹⁾(η₁) = N. Suppose, without loss of generality, thatη₀<η₁. So,

u⁽ⁿ⁻¹⁾(t)>_{0 and} u⁽ⁿ⁻¹⁾(η₀)≤ u⁽ⁿ⁻¹⁾(t)≤ N, fort∈[η₀,η₁]. So

|φ(u⁽ⁿ⁻¹⁾(t))|=|q(t)f(t,u(t)_{, . . . ,}u⁽ⁿ⁻¹⁾(t))| ≤q(t)|ψ(u⁽ⁿ⁻¹⁾(t))|_{, for}t ∈[η₀,η₁],

and Z _φ(N)

φ(u⁽ⁿ⁻¹⁾(η₀))

ds ψ(φ⁻¹(s)) ≤

Z _η1

η₀

|(φ(u⁽ⁿ⁻¹⁾(t))⁰| ψ(u⁽ⁿ⁻¹⁾(t)) ^dt

=

Z _η1

η0

|q(t)f(t,u(t), . . . ,u⁽ⁿ⁻¹⁾(t))|

ψ(u⁽ⁿ⁻¹⁾(t)) ^dt≤

Z _η1

η0

q(t)dt<

Z _b

a q(t)dt.

Asu⁽ⁿ⁻¹⁾(η₀)≤µ< N, by the monotony ofφ,

φ(u⁽ⁿ⁻¹⁾(η₀))≤φ(µ) and, by (2.6),

Z _φ(N) φ(u⁽ⁿ⁻¹⁾(η0))

ds ψ(φ⁻¹(s)) ≥

Z _φ(N) φ(µ)

ds ψ(φ⁻¹(s)) >

Z _b

a q(t)dt which leads to a contradiction.

The other cases, that is, u⁽ⁿ⁻¹⁾(τ) > Nwithη₁ < η₀, and u⁽ⁿ⁻¹⁾(τ) < −Nwith η₀ <

η₁ orη₁<η₀, follow the same arguments to obtain a contradiction.

Therefore

u⁽ⁿ⁻¹⁾(t) ≤ N, fort ∈[a,b].

Forward, in our method, we will use the following lemma, given in [23]:

Lemma 2.5. For v,w∈C(I)such that v(x)≤w(x), for every x ∈ I, define q(x,u) =max{v, min{u,w}}.

Then, for each u∈C¹(I)the next two properties hold:

(a) _dx^dq(x,u(x))exists for a.e. x∈ I.

(b) If u,u_m ∈C¹(I)and u_m →u in C¹(I)then d

dxq(x,u_m(x))→ ^d

dxq(x,u(x)) for a.e. x∈ I.

We recall the classical Schauder’s fixed point theorem:

Theorem 2.6. Let M be a nonempty, closed, bounded and convex subset of a Banach space X, and suppose that T:M → M is a compact operator. Then T as at least one fixed point in M.

3 Existence and localization result

The main result is an existence and localization theorem, as it provides not only the existence of solutions but also some of its qualitative properties.

Theorem 3.1. Suppose that there are α and β lower and upper solutions, respectively, of problem (1.1)–(1.3)such that

α⁽ⁿ⁻²⁾(t)≤ β⁽ⁿ⁻²⁾(t), for t∈[a,b].

Assume that the L¹-Carathéodory function f :[a,b]×_Rⁿ →_Rsatisfies a Nagumo condition, related toαandβ, and verifies

f(t,α(t)_{, . . . ,}α⁽ⁿ⁻³⁾(t))_,y,z)≤ f(t,x₀, . . . ,x_n−1)≤ f(t,β(t)_{, . . . ,}β⁽ⁿ⁻³⁾(t)_,y,z)_{, (3.1)}

forα⁽ⁱ⁾(t)≤ x_i ≤ β⁽ⁱ⁾(t),for i =0, . . . ,n−3,and fixed (y,z)∈_R². Moreover, if the impulsive functions satisfy

I_j,k(α(t_k), . . . ,α⁽ⁿ⁻¹⁾(t_k))≤ I_j,k(x₀, . . . ,x_n−1)≤ I_j,k(β(t_k), . . . ,β⁽ⁿ⁻¹⁾(t_k)), (3.2) for j=0, . . . ,n−3,α⁽ⁱ⁾(t_k)≤x_i ≤ β⁽ⁱ⁾(t_k), for i=0, 1, . . . ,n−2,k=1, 2, . . . ,m,

and

I_n−2,k(α(t_k), . . . ,α⁽ⁿ⁻³⁾(t_k),y,z)≥ I_n−2,k(x₀, . . . ,x_n−1) (3.3)

≥ I_n−2,k(β(t_k)_{, . . . ,}β⁽ⁿ⁻³⁾(t_k)_,y,z)

forα⁽ⁱ⁾(t)≤ x_i ≤β⁽ⁱ⁾(t),for i=0, . . . ,n−3,and fixed(y,z)∈_R², then problem (1.1)–(1.3)has at least one solution u ∈E, such that

α⁽ⁱ⁾(_t)≤u⁽ⁱ⁾(_t)≤ β⁽ⁱ⁾(_t) _{for i} =0, 1, . . . ,n−2and −N≤ u⁽ⁿ⁻¹⁾(_t)≤ N, for t ∈[a,b]and N given by(2.7).

Proof. Define the continuous functionsδ_i, fori=0, 1, . . . ,n−2,

δ_i(t,u⁽ⁱ⁾(t)) =











β⁽ⁱ⁾(t), u⁽ⁱ⁾(t)≥ β⁽ⁱ⁾(t)

u⁽ⁱ⁾(t), α⁽ⁱ⁾(t)≤u⁽ⁱ⁾(t)≤ β⁽ⁱ⁾(t) α⁽ⁱ⁾(t), u⁽ⁱ⁾(t)≤α⁽ⁱ⁾(t)

and consider the following modified and perturbed equation (φ(u⁽ⁿ⁻¹⁾(t)))⁰+q(t)f

t,δ₀(t,u(t)), . . . ,δn−2(t,u⁽ⁿ⁻²⁾(t)), d dt

δn−2(t,u⁽ⁿ⁻²⁾(t))

+ ^δn−2(t,u⁽ⁿ⁻²⁾(t))−u⁽ⁿ⁻²⁾(t)

1+|u⁽ⁿ⁻²⁾(t)−δ_n−2(t,u⁽ⁿ⁻²⁾(t))| =0, (3.4) coupled with boundary conditions (1.2) and the truncated impulsive conditions, for i = 0, 1, . . . ,n−2,

∆u⁽ⁱ⁾(t_k) = I_i,k

δ₀(t_k,u(t_k))_{, . . . ,}δ_n−2(t_k,u⁽ⁿ⁻²⁾(t_k)_,

d

dt(δ_n−2(t_k,u⁽ⁿ⁻²⁾(t_k)))

:= I_i,k^∗ (t_k),

∆φ(u⁽ⁿ⁻¹⁾(t)) = I_n−1,k

δ₀(t_k,u(t_k)), . . . ,δ_n−2(t_k,u⁽ⁿ⁻²⁾(t_k)),

d

dt(δ_n−2(t_k,u⁽ⁿ⁻²⁾(t_k)))

:= I_n^∗_−1,k(t_k).

(3.5)

Define the operatorT:E→ Eby T(u)(t):=

n−2 i

∑

"

A_i+

∑

k:t_k<t

I_i,k^∗

#(t−a)ⁿ⁻²⁻ⁱ (n−₂−i)!

!

+

Z _t

a

(_t−s)ⁿ⁻²

(n−2)! φ⁻¹ φ(B) +

Z _b

s v(s)ds−

∑

k:t_k>s

I_n^∗_−1,k

! ds.

By Lemma2.1, it is clear that the fixed points ofT,u∗, are solutions of the initial problem (1.1)-(3.5), if they verify

α⁽ⁱ⁾(t)≤u⁽_∗ⁱ⁾(t)≤ β⁽ⁱ⁾(t)_{, for}t ∈[a,b]_andi=0, 1, . . . ,n−_2.

AsTis compact, by Schauder’s fixed point theorem, Thas a fixed pointu∈ E, which is a solution of (3.4), (1.2), (3.5). To prove that this solution verifies

α⁽ⁱ⁾(t)≤u⁽ⁱ⁾(t)≤ β⁽ⁱ⁾(t)_{, for}t ∈[a,b]_{, and}i=0, 1, . . . ,n−_2, suppose, by contradiction, that, fori=n−2, there is t∈[a,b]such that

u⁽ⁿ⁻²⁾(t)> β⁽ⁿ⁻²⁾(t). Defineζ ∈ [a,b]as

sup

t∈[a,b]

(u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t)):=u⁽ⁿ⁻²⁾(ζ)−β⁽ⁿ⁻²⁾(ζ))>0. (3.6)

By (1.2) and Definition 2.2, u⁽ⁿ⁻²⁾(a)−β⁽ⁿ⁻²⁾(a) ≤ 0, then ζ 6= a. On the other hand u⁽ⁿ⁻¹⁾(b)−β⁽ⁿ⁻¹⁾(b)<0 and thenζ 6= b, by (3.6).

Thereforeζ ∈ ]a,b[_.

Case 1: Assume that there is p∈ {1, 2, . . . ,m}such thatζ ∈(t_p,t_p+1)_. Considere>0 small enough such that

u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t)>0 and u⁽ⁿ⁻¹⁾(t)−β⁽ⁿ⁻¹⁾(t)≤0, fort∈ (ζ,ζ+e). (3.7) Therefore, by (3.1) and (3.7), for allt ∈(ζ,ζ+e), we have the following contradiction

0≥ φ

u⁽ⁿ⁻¹⁾(t)⁰−φ

β⁽ⁿ⁻¹⁾(t)⁰

≥ −q(t)f

t,δ₀(t,u(t)), . . . ,δ_n−2(t,u⁽ⁿ⁻²⁾(t)), d dt

δ_n−2(t,u⁽ⁿ⁻²⁾(t))

− ^δn−2

t,u⁽ⁿ⁻²⁾(t)−u⁽ⁿ⁻²⁾(t) 1+u⁽ⁿ⁻²⁾(t)−δ_n−2 t,u⁽ⁿ⁻²⁾(t)

+q(t)f

t,β(t), . . . ,β⁽ⁿ⁻¹⁾(t)

= −q(t)f

t,δ0(t,u(t)), . . . ,δn−3(t,u(t)),β⁽ⁿ⁻²⁾(t),β⁽ⁿ⁻¹⁾(t)

− ^β

(n−2)(t)−u⁽ⁿ⁻²⁾(t) 1+u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t)

+q(t)f

t,β(t), . . . ,β⁽ⁿ⁻¹⁾(t)

≥ −q(t)f

t,β(t), . . . ,β⁽ⁿ⁻¹⁾(t)− ^β

(n−2)(t)−u⁽ⁿ⁻²⁾(t) 1+u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t) +q(t)f

t,β(t), . . . ,β⁽ⁿ⁻¹⁾(t)= ^u

(n−2)(t)−β⁽ⁿ⁻²⁾(t) 1+u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t)

>0.

Case 2: Consider that there existsk∈ {1, 2, . . . ,m}such that, or

tmax∈[a,b]

u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t):=u⁽ⁿ⁻²⁾(t⁻_k )−β⁽ⁿ⁻²⁾(t⁻_k )>0 (3.8) or

sup

t∈[a,b]

u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t):=u⁽ⁿ⁻²⁾(t⁺_k )−β⁽ⁿ⁻²⁾(t⁺_k )>0. (3.9) If (3.8) holds, then

∆

u⁽ⁿ⁻²⁾(t)−β⁽ⁿ⁻²⁾(t)≤0

and, by (3.3) and Definition2.2, we have the contradiction 0≥_∆u⁽ⁿ⁻²⁾(t_k)−_∆β⁽ⁿ⁻²⁾(t_k) =I_n^∗_−2,k−_∆β⁽ⁿ⁻²⁾(t_k)

= I_n−2,k

δ₀(t,u(t)), . . . ,δ_n−3(t,u(t)),β⁽ⁿ⁻²⁾(t),β⁽ⁿ⁻¹⁾(t)−_∆β⁽ⁿ⁻²⁾(t_k)

≥ I_n−2,k

t_k,β(t_k), . . . ,β⁽ⁿ⁻¹⁾(t_k)−_∆β⁽ⁿ⁻²⁾(t_k)>0.

Consider now (3.9). So, there ise>0 such that, fort∈ (t_k,t_k+e), u⁽ⁿ⁻¹⁾(t)−β⁽ⁿ⁻¹⁾(t)≤0,

and the arguments follow by the same technique as in Case 1, to have u⁽ⁿ⁻²⁾(t)≤ β⁽ⁿ⁻²⁾(t), ∀t ∈[a,b].

To prove that u⁽ⁿ⁻²⁾(t)≥α⁽ⁿ⁻²⁾(t), ∀t∈ [a,b], the method is similar. Therefore α⁽ⁿ⁻²⁾(t)≤u⁽ⁿ⁻²⁾(t)≤ β⁽ⁿ⁻²⁾, fort∈ [a,b].

Integrating the first inequality in[a,t₁], we have

α⁽ⁿ⁻³⁾(t)≤u⁽ⁿ⁻³⁾(t)−u⁽ⁿ⁻³⁾(a) +α⁽ⁿ⁻³⁾(a) (3.10)

=u⁽ⁿ⁻³⁾(t)−A_n−3+α⁽ⁿ⁻³⁾(a)≤ u⁽ⁿ⁻³⁾(t). For t∈(t₁,t₂], by (3.2) and (3.10),

α⁽ⁿ⁻³⁾(t)≤u⁽ⁿ⁻³⁾(t)−u⁽ⁿ⁻³⁾(t₁⁺) +α⁽ⁿ⁻³⁾(t⁺₁)

≤u⁽ⁿ⁻³⁾(t)−I_n^∗_−3,1(t₁)−u⁽ⁿ⁻³⁾(t₁) +I_n−3,1

t₁,α(t₁), . . . ,αⁿ⁻¹(t₁)+α⁽ⁿ⁻³⁾(t₁)

≤u⁽ⁿ⁻³⁾(t)−I_n^∗_−3,1(t₁) +I_n−3,1

t₁,α(t₁), . . . ,α⁽ⁿ⁻¹⁾(t₁)

≤u⁽ⁿ⁻³⁾(t).

Applying this method for each interval(t_k,t_k+1],k =2, . . . ,m, we obtain α⁽ⁿ⁻³⁾(t)≤u⁽ⁿ⁻³⁾(t), ∀t∈ [a,b],

and, by the same technique,

β⁽ⁿ⁻³⁾(t)≥ u⁽ⁿ⁻³⁾(t), ∀t ∈[a,b]. By iteration of these arguments, we conclude

α⁽ⁱ⁾(t)≤u⁽ⁱ⁾(t)≤ β⁽ⁱ⁾(t), fori=0, 1, . . . ,n−2, andt ∈[a,b]. The estimation

u⁽ⁿ⁻¹⁾(t)≤ Nis a trivial consequence of Lemma2.4.

4 Estimation for the bending of one-sided clamped beam under im- pulsive effects

Problems related to beam structures and especially beams that support some forces as im- pulses, are part of a vast field of investigation in boundary value problems theory, see, for example, [7–10,24].

In this application we consider a model to describe the bending of a beam with length L>1, given by the fourth-order equation

EI

Au⁽⁴⁾(x) + ³ 2

3

q

u⁰(x)|u⁰⁰(x)| −ku(x)−γu⁰⁰⁰(x) =0, forx∈ ]0,L[, (4.1) whereE>0 is the Young modulus,I >0 the mass moment of inertia, A>0 the cross section area,k > 0 the tension of a spring force vertically applied on the beam, and γ >0 the shear force coefficient.

At the end points the behavior of the beam is given by the following boundary conditions u(0) =0, u⁰(0) =1, u⁰⁰(0) =0, u⁰⁰⁰(L) =0, (4.2) meaning that the beam is clamped on the left end side.

For clearance, we consider only one moment of impulse which occurs at t₁ = 1. The im- pulsive effects are given by generalized functions with dependence on the unknown function itself, and on several derivatives till order three,

∆u(1) =u(1) +u⁰(1)−2u⁰⁰(1)−u⁰⁰⁰(1)

∆u⁰(1) =u(1) +u⁰(1)−2u⁰⁰(1)−u⁰⁰⁰(1) (4.3)

∆u⁰⁰(₁) =−u(₁)−u⁰(₁) +_u⁰⁰(₁) +_5u⁰⁰⁰(₁)−1

∆u⁰⁰⁰(1) =u(1)−u⁰(1) +u⁰⁰(1) +u⁰⁰⁰(1)−1.

This problem (4.1)-(4.3) is a particular case of (1.1)-(1.3) with[a,b] = [0,L],n=4, f(x,y0,y₁,y2,y3) = ^A

EI 3

2

√3

y₁|y2| −ky0−γy3

, (4.4)

φ(w) =w,q(t)≡1,m=1,t₁ =1, and the impulsive functions given by I_0,1(w₀,w₁,w₂,w₃) =w₀+w₁−2w₂−w₃

I_1,1(w0,w₁,w2,w3) =w0+w₁−2w2−w3

I_2,1(w₀,w₁,w₂,w₃) =−w₀−w₁+w₂+5w₃−1 I_3,1(w₀,w₁,w₂,w₃) =w₀−w₁+w₂+w₃−1.

As a numeric example we can consider A= 1, EI = 1, k = 1, γ = 6, L = 2. In this case, the continuous functions

α(x) =0, β(x) = ^x

3

6 +x²+x, for x∈[0, 2],

are, respectively, lower and upper solutions of problem (4.1)–(4.3), according to Definition3.6.

In fact, forα(x)≡0 the inequalities are trivially satisfied and forβ, we have, β(₀) =_0, β⁰(₀) =_1, β⁰⁰(₀) =₂>_0, β⁰⁰⁰(₂) =₁>_0,

∆β(1) =0≥ β(0) +β⁰(0)−2β⁰⁰(0)−β⁰⁰⁰(0) =−⁴ 3,

∆β⁰(1) =0≥ β(0) +β⁰(0)−2β⁰⁰(0)−β⁰⁰⁰(0) =−⁴ 3,

∆β⁰⁰(1) =0<−β(0)−β⁰(0) +β⁰⁰(0) +5β⁰⁰⁰(0)−1= ⁴ 3

∆β⁰⁰⁰(1) =0< β(0)−β⁰(0) +β⁰⁰(0) +β⁰⁰⁰(0)−1= ⁵ 3.

The nonlinear part f(x,y0,y₁,y2,y3), given by (4.4), verifies a Nagumo condition on the set S∗=

((t,y0,y₁,y2,y3)∈ [0, 2]×_Rⁿ: 0≤y0≤ ^x₆³ +x²+x,

0≤y₁≤ ^x₂² +2x+1, 0≤y₂≤ x+2 )

with

µ=max

β⁰⁰(2),

β⁰⁰(0),

β⁰⁰(1) =4, ψ(|y₃|):=|y₃|+²²

3 , and

Z +_∞ µ

ds

s+²²₃ = +_∞>

Z _L

0 1ds= L.

Moreover, f is nondecreasing on y₀ and, by Theorem3.1, there exists a solution u(x) of problem (4.1)–(4.3) such that

α⁽ⁱ⁾(x)6^u(i)(x)6^β(i)(x), i=0, 1, 2, forx∈ [0, 2], that is

0≤ u(x)≤ ^x

3

6 +x²+x, 0≤ u⁰(x)≤ ^x

2

2 +2x+1,

0≤ u⁰⁰(x)≤x+2, forx ∈[0, 2].

Figure 4.1: Strip ofulocalization.