Electronic Journal of Qualitative Theory of Differential Equations Spec. Ed. I, 2009 No. 15, 1–14; http://www.math.u-szeged.hu/ejqtde/

A CANTILEVER EQUATION WITH NONLINEAR BOUNDARY CONDITIONS

Gennaro Infante and Paolamaria Pietramala

Dipartimento di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Cosenza, Italy e-mail: g.infante@unical.it, pietramala@unical.it

Honoring the Career of John Graef on the Occasion of His Sixty-Seventh Birthday

Abstract

We prove new results on the existence of positive solutions for some cantilever equation subject to nonlocal and nonlinear boundary conditions. Our main ingredient is the classical fixed point index.

Key words and phrases: Fixed point index, cone, positive solution, cantilever equation. AMS (MOS) Subject Classifications: Primary 34B18, secondary 34B10, 47H10, 47H30

1 Introduction

In this paper we establish new results on the existence of positive solutions for the fourth order differential equation

u^{(4)}(t) = g(t)f(t, u(t)), t ∈ (0, 1),  (1)

subject to the nonlocal boundary conditions (BCs)

u(0) = u′(0) = u′′(1) = 0, u′′′(1) + k_{0} + B(α[u]) = 0.  (2)

Here k_{0} is a non-negative constant, B is a non-negative continuous function and α[u] is a positive functional given by

α[u] = ∫_{0}^{1} u(s) dA(s),

involving a Stieltjes integral.

Equation (1) models the stationary states of the deflection of an elastic beam. Beam equations have been studied recently by several authors under different boundary conditions. For example, the BCs

u(0) = 0, u(1) = 0, u′(0) = 0, u′(1) = 0,

correspond to both ends of the beam being clamped, these have been studied recently in [32, 37, 42, 43, 48]; the BCs

u(0) = 0, u(1) = 0, u′′(0) = 0, u′′(1) = 0,

model a bar with hinged ends and have been studied for example in [5, 7, 9, 12, 14, 19, 35, 36, 37, 42, 43, 46]. Also the BCs

u(0) = 0, u′(0) = 0, u(1) = 0, u′′(1) = 0,

have been studied in [1, 42, 49] and correspond to the left end being clamped and the right end being hinged. The conditions

u(0) = 0, u′(0) = 0, u′′(1) = 0, u′′′(1) = 0

model the so-called cantilever bar, that is a bar clamped on the left end and where the right end is free to move with vanishing bending moment and shearing force. These types of BCs have been investigated in [4, 47] and, in particular, [4] provides a detailed insight on the physical motivation for this problem.

The BCs (2) describe a cantilever beam with forces acting on its right end, for example,

• u′′′(1) + k_{0} = 0 models a force acting in 1,

• u′′′(1) + k_{1}u(1) = 0 describes a spring in 1,

• u′′′(1)+B(u(1)) = 0 models a spring with a strongly nonlinear rigidity (this could happen, for example, due to the type of material),

• u′′′(1) + B(u(η)) = 0 describes a feedback mechanism, where the spring reacts to the displacement registered in a point η of the beam.

Thus the condition

u′′′(1) + k_{0} + B(α[u]) = 0

covers a variety of cases and includes, as special cases when k_{0} = 0 and B(w) = w, multi-point and integral boundary conditions, that are widely studied objects also in the case of fourth order BVPs, see for example [8, 11, 13, 15, 16, 18, 29, 30, 39]. BCs of nonlinear type are widely studied objects in the case of second and fourth order equations, see for example [3, 6, 10, 20, 21, 23, 25, 31, 45] and references therein. The study of positive solutions of BVPs that involve Stieltjes integrals has been done, in the case of positive measures, in [27, 28]. Signed measures were used in [41] and in the subsequent papers [40, 42]; here, as in [23, 25], we are forced, due to some inequalities involved in our theory, to restrict our attention to positive measures only.

Our approach is to rewrite the BVP (1)-(2) as a perturbed Hammerstein integral equation of the form

u(t) = γ(t)(k_{0} + B(α[u])) + ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds.

This type of integral equation has been investigated, when k_{0} = 0 and B(w) ≡ w, in [38, 41, 43, 44], when B(w) ≡ w in [22, 24, 27], and when k_{0} = 0 in [21, 23, 25, 45]. In order to utilize classical fixed point index theory, we provide here a modification of the results of [27, 41] in order to prove the existence of multiple positive solutions of the BVP (1)-(2).

In a last remark we briefly illustrate how our approach may also be applied to the nonlinear BCs

u(0) = k_{0} + B(α[u]), u′(0) = 0, u′′(1) = 0, u′′′(1) = 0,  (3)
u(0) = 0, u′(0) = k_{0} + B(α[u]), u′′(1) = 0, u′′′(1) = 0,  (4)
u(0) = 0, u′(0) = 0, u′′(1) = k_{0} + B(α[u]), u′′′(1) = 0.  (5)

As in [42], where a different set of BCs were investigated, we point out that these nonlocal boundary conditions can be interpreted as feedback controls; in particular, the BCs (3) can be seen as a control on the displacement in the left end, the BCs (4) would be a device handling the angular attitude at the left end, whereas the BCs (5) describe a control on the bending moment at the right end.

2 Positive Solutions of the Fourth-Order BVPs

We firstly describe in details our approach for the BVP

u^{(4)}(t) = g(t)f(t, u(t)), t ∈ (0, 1),  (6)
u(0) = u′(0) = u′′(1) = 0, u′′′(1) + k_{0} + B(α[u]) = 0.  (7)

Throughout the paper we assume that:

(C_{1}) f : [0, 1] × [0, ∞) → [0, ∞) satisfies Carathéodory conditions, that is, for each u, t ↦ f(t, u) is measurable and for almost every t, u ↦ f(t, u) is continuous, and for every r > 0 there exists a L^{∞}-function ϕ_{r} : [0, 1] → [0, ∞) such that

f(t, u) ≤ ϕ_{r}(t) for almost all t ∈ [0, 1] and all u ∈ [0, r],

(C_{2}) g ∈ L^{1}[0, 1], g ≥ 0 for almost every t ∈ [0, 1],

(C_{3}) A is a function of bounded variation and dA is a positive measure,

(C_{4}) B : [0, ∞) → [0, ∞) is continuous and there exist δ_{1}, δ_{2} ≥ 0 such that

δ_{1}w ≤ B(w) ≤ δ_{2}w for every w ∈ [0, ∞).

The homogeneous BVP that corresponds to the BCs

u(0) = u′(0) = u′′(1) = u′′′(1) = 0,

has been studied in [4, 47], by means of an Hammerstein integral equation of the type

u(t) = ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds,  (8)

where G_{0} is the Green’s function associated with these BCs, that is

G_{0}(t, s) = \begin{cases}\frac{1}{6}(3t^{2}s − t^{3}), & s ≥ t \\ \frac{1}{6}(3s^{2}t − s^{3}), & s ≤ t. \end{cases}

In order to rewrite the BVP (6)-(7) as a perturbation of the integral equation (8), we observe that the function

γ(t) = \frac{1}{6}(3t^{2} − t^{3}) for all t ∈ [0, 1],

is the unique solution of the BVP

γ^{(4)}(t) = 0, γ(0) = γ′(0) = γ′′(1) = 0, γ′′′(1) + 1 = 0.

This allows us to associate with the BVP (6)-(7) the perturbed Hammerstein integral equation

u(t) = γ(t)(k_{0} + B(α[u])) + ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds,  (9)

where γ and G_{0} are as above, and utilize the classical fixed point index for compact maps (see for example [2] or [17]).

Since the derivative of the function G_{0} with respect to t is non-negative for all t ∈ [0, 1], G_{0} is a non-decreasing function of t that attains its maximum when t = 1. Then we set

Φ(s) := max_{0≤t≤1} G_{0}(t, s) = G_{0}(1, s) = \frac{1}{2}s^{2} − \frac{1}{2}s^{3}.

We now look for a suitable interval [a, b] ⊂ [0, 1] and for the constants c_{0}, c_{γ} > 0 that satisfy the inequalities

G_{0}(t, s) ≥ c_{0}Φ(s), for every (t, s) ∈ [a, b] × [0, 1],
γ(t) ≥ c_{γ}∥γ∥, for every t ∈ [a, b].

Since the derivative of the function G_{0}(t, s)/Φ(s) with respect to s is non-negative for all s ∈ [0, 1], the function G_{0}(t, s)/Φ(s) is a non-decreasing function of s.

If we set

c_{0}(t) = min\{\frac{2t}{3 − t}, \frac{1}{2}t^{2}(3 − t)\} = \frac{1}{2}t^{2}(3 − t),

we may take, for [a, b] ⊂ (0, 1],

c_{0} := min_{a≤t≤b} c_{0}(t) = \frac{1}{2}a^{2}(3 − a).

We observe that ∥γ∥ = \frac{1}{3}, min_{t∈[a,b]} γ(t) = γ(a) and therefore we have

γ(t) ≥ 3γ(a)∥γ∥, for every t ∈ [a, b],

so that we can take

c_{γ} := 3γ(a).

Thus, for an arbitrary [a, b] ⊂ (0, 1], we may set

c := min\{c_{0}, c_{γ}\} = \frac{1}{2}a^{2}(3 − a).  (10)

By a solution of the BVP (6)-(7) we mean a solution of the corresponding integral equation (9) and we work in the Banach space C[0, 1] endowed with the usual supremum norm ∥u∥ := sup{|u(t)| : t ∈ [0, 1]} and the above hypotheses enable us to utilize the cone

K = {u ∈ C[0, 1] : min_{t∈[a,b]} u(t) ≥ c∥u∥},

where [a, b] ⊂ (0, 1] and c is as in (10).

If Ω is a bounded open subset of K (in the relative topology) we denote by \overline{Ω} and ∂Ω the closure and the boundary relative to K. We write

K_{r} = \{u ∈ K : ∥u∥ < r\} and \overline{K}_{r} = \{u ∈ K : ∥u∥ ≤ r\}.

We consider now the map T : C[0, 1] → C[0, 1] defined by

T u(t) := γ(t)(k_{0} + B(α[u])) + F u(t),

where

F u(t) := ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds.

Theorem 2.1 If the hypotheses (C_{1})-(C_{4}) hold for some r > 0, then T maps \overline{K}_{r} into K. When these hypotheses hold for each r > 0, T maps K into K. Moreover, T is a compact map.

Proof. Take u ∈ \overline{K}_{r}. Then we have, for t ∈ [0, 1],

T u(t) = γ(t)(k_{0} + B(α[u])) + ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds

≤ γ(t)(k_{0} + B(α[u])) + ∫_{0}^{1} Φ(s)g(s)f(s, u(s)) ds,

therefore

∥T u∥ ≤ ∥γ∥(k_{0} + B(α[u])) + ∫_{0}^{1} Φ(s)g(s)f(s, u(s)) ds.

Then we have

min_{t∈[a,b]} T u(t) ≥ c[∥γ∥(k_{0} + B(α[u])) + ∫_{0}^{1} Φ(s)g(s)f(s, u(s)) ds] ≥ c∥T u∥.

Hence we have T u ∈ K for every u ∈ \overline{K}_{r}. Moreover, the map T is compact, since it is a sum of two compact maps. In fact, the compactness of F is well-known and the perturbation γ(t)(k_{0} + B(α[u])) is compact since γ and B are continuous and it maps a bounded set into a bounded subset of a 1-dimensional space. □

We make use of the following numbers

f^{0,ρ} := sup_{0≤u≤ρ, 0≤t≤1} \frac{f(t, u)}{ρ}, f_{ρ,ρ/c} := inf_{ρ≤u≤ρ/c, a≤t≤b} \frac{f(t, u)}{ρ},

\frac{1}{m} := sup_{t∈[0,1]} ∫_{0}^{1} G_{0}(t, s)g(s) ds, \frac{1}{M(a, b)} := inf_{t∈[a,b]} ∫_{a}^{b} G_{0}(t, s)g(s) ds,

use the notation

G(s) := ∫_{0}^{1} G_{0}(t, s) dA(t),

and assume

(C_{5}) 1 − δ_{2}α[γ] > 0.

First, we give a condition that implies the index is 1 on the set K_{ρ}.

Lemma 2.1 Assume that there exists ρ > 0 such that

(I_{ρ}^{1}) the following inequality holds:

\frac{k_{0}}{3ρ(1 − δ_{2}α[γ])} + f^{0,ρ}(\frac{δ_{2}}{3(1 − δ_{2}α[γ])} ∫_{0}^{1} G(s)g(s) ds + \frac{1}{m}) < 1.  (11)

EJQTDE Spec. Ed. I, 2009 No. 15

Then the fixed point index, i_{K}(T, K_{ρ}), is 1.

Proof. We show that

λu ≠ T u for every u ∈ ∂K_{ρ} and for every λ ≥ 1,

which implies that the index is 1 on the set K_{ρ}. In fact, if there exists λ ≥ 1 and u ∈ ∂K_{ρ} such that

λu(t) = T u(t) = γ(t)(k_{0} + B(α[u])) + ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds

then we have

λu(t) ≤ γ(t)(k_{0} + δ_{2}α[u]) + ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds  (12)

and

λα[u] ≤ α[γ](k_{0} + δ_{2}α[u]) + ∫_{0}^{1} G(s)g(s)f(s, u(s)) ds.

Hence we obtain

(λ − δ_{2}α[γ])α[u] ≤ k_{0}α[γ] + ∫_{0}^{1} G(s)g(s)f(s, u(s)) ds.

Substituting into (12) gives

λu(t) ≤ \frac{λk_{0}γ(t)}{λ − δ_{2}α[γ]}+\frac{γ(t)δ_{2}}{λ − δ_{2}α[γ]} ∫_{0}^{1} G(s)g(s)f(s, u(s)) ds+∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds.

Taking the supremum for t ∈ [0, 1] gives

λρ ≤ \frac{λk_{0}∥γ∥}{λ − δ_{2}α[γ]} + \frac{∥γ∥δ_{2}}{λ − δ_{2}α[γ]} ∫_{0}^{1} G(s)g(s)ρf^{0,ρ} ds + sup_{t∈[0,1]} ∫_{0}^{1} G_{0}(t, s)g(s)ρf^{0,ρ} ds

≤ \frac{k_{0}∥γ∥}{1 − δ_{2}α[γ]} + \frac{∥γ∥δ_{2}}{1 − δ_{2}α[γ]} ∫_{0}^{1} G(s)g(s)ρf^{0,ρ} ds + sup_{t∈[0,1]} ∫_{0}^{1} G_{0}(t, s)g(s)ρf^{0,ρ} ds.

Thus we have,

λ ≤ \frac{k_{0}}{3ρ(1 − δ_{2}α[γ])} + f^{0,ρ}(\frac{δ_{2}}{3(1 − δ_{2}α[γ])} ∫_{0}^{1} G(s)g(s) ds + \frac{1}{m}) < 1.

This contradicts the fact that λ ≥ 1 and proves the result. □

In order to give a condition that implies the fixed point index is equal to 0, we make use of the open set

V_{ρ} = {u ∈ K : min_{t∈[a,b]} u(t) < ρ}.

V_{ρ} is equal to the set called Ω_{ρ/c} in [33]. Note that K_{ρ} ⊂ V_{ρ} ⊂ K_{ρ/c}.

8 G. Infante & P. Pietramala Lemma 2.2 Assume that there exists ρ > 0 such that (I_{ρ}^{0}) the following inequality holds:

\frac{k_{0}c}{3ρ(1 − δ_{1}α[γ])} + (\frac{cδ_{1}}{3(1 − δ_{1}α[γ])} ∫_{a}^{b} G(s)g(s) ds + \frac{1}{M(a, b)})f_{ρ,ρ/c} > 1. (13)

Then we have i_{K}(T, V_{ρ}) = 0.

Proof. Let e(t) ≡ 1 for t ∈ [0, 1]. Then e ∈ K. We prove that

u ≠ T (u) + λe for all u ∈ ∂V_{ρ} and λ ≥ 0,

which ensures that the index is 0 on the set V_{ρ}. In fact, if this does not happen, there exist u ∈ ∂V_{ρ} and λ ≥ 0 such that u = T u + λe, that is

u(t) = γ(t)(k_{0} + B(α[u])) + ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds + λ.

Then we have

u(t) ≥ γ(t)(k_{0} + δ_{1}α[u]) + ∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds + λ,  (14)

and

α[u] ≥ α[γ](k_{0} + δ_{1}α[u]) + ∫_{0}^{1} G(s)g(s)f(s, u(s)) ds + α[λ]
≥ α[γ](k_{0} + δ_{1}α[u]) + ∫_{0}^{1} G(s)g(s)f(s, u(s)) ds.

Hence we obtain

(1 − δ_{1}α[γ])α[u] ≥ k_{0}α[γ] + ∫_{0}^{1} G(s)g(s)f(s, u(s)) ds.

Substituting into (14) gives

u(t) ≥ \frac{k_{0}γ(t)}{1 − δ_{1}α[γ]}+\frac{γ(t)δ_{1}}{1 − δ_{1}α[γ]} ∫_{0}^{1} G(s)g(s)f(s, u(s)) ds+∫_{0}^{1} G_{0}(t, s)g(s)f(s, u(s)) ds+λ.

Then we have, for t ∈ [a, b],

u(t) ≥ \frac{k_{0}c_{γ}∥γ∥}{1 − δ_{1}α[γ]} + \frac{c_{γ}∥γ∥δ_{1}}{1 − δ_{1}α[γ]} ∫_{a}^{b} G(s)g(s)ρf_{ρ,ρ/c} ds + ∫_{a}^{b} G_{0}(t, s)g(s)ρf_{ρ,ρ/c} ds + λ,

EJQTDE Spec. Ed. I, 2009 No. 15

and therefore

min_{t∈[a,b]} u(t) ≥ \frac{k_{0}c}{3ρ(1 − δ_{1}α[γ])} + f_{ρ,ρ/c}(\frac{cδ_{1}}{3(1 − δ_{1}α[γ])} ∫_{a}^{b} G(s)g(s) ds + \frac{1}{M(a, b)}) + λ.

By (I_{ρ}^{0}) we have that min_{t∈[a,b]} u(t) > ρ + λ ≥ ρ. This contradict the fact that u ∈ ∂V_{ρ} and proves the result. □

We can now state a result for the existence of one or two positive solutions for the integral equation (9).

Theorem 2.2 Equation (9) has a positive solution in K if either of the following conditions hold.

(S_{1}) There exist ρ_{1}, ρ_{2} ∈ (0, ∞), with ρ_{1} < ρ_{2}, such that (I_{ρ_{1}}^{1}) and (I_{ρ_{2}}^{0}) hold.

(S_{2}) There exist ρ_{1}, ρ_{2} ∈ (0, ∞), with ρ_{1} < cρ_{2}, such that (I_{ρ_{1}}^{0}) and (I_{ρ_{2}}^{1}) hold.

Equation (9) has two positive solutions in K if either of the following conditions hold.

(D_{1}) There exist ρ_{1}, ρ_{2}, ρ_{3} ∈ (0, ∞), with ρ_{1} < ρ_{2} < cρ_{3}, such that (I_{ρ_{1}}^{1}), (I_{ρ_{2}}^{0}) and (I_{ρ_{3}}^{1}) hold.

(D_{2}) There exist ρ_{1}, ρ_{2}, ρ_{3} ∈ (0, ∞), with ρ_{1} < cρ_{2} and ρ_{2} < ρ_{3}, such that (I_{ρ_{1}}^{0}), (I_{ρ_{2}}^{1}) and (I_{ρ_{3}}^{0}) hold.

We omit the proof which follows simply from properties of fixed point index, for details of similar proofs see [26, 34]. Note that, if the nonlinearity f has a suitable oscillatory behavior, one may establish, by the same method, the existence of more than two positive solutions (we refer the reader to [27, 33] to see the type of results that may be stated).

Example 2.1 We now assume that g ≡ 1, α[u] = u(ξ), ξ ∈ (0, 1) and we make, as in [25], the choice

B(w) = \{ \begin{matrix}\frac{1}{4}w, 0 ≤ w ≤ 1, \\ \frac{1}{8}w + \frac{1}{8}, w ≥ 1. \end{matrix}

In this case we have δ_{1} = \frac{1}{8} and δ_{2} = \frac{1}{4},

α[γ] = ∫_{0}^{1} γ(t) dA(t) = \frac{1}{6}(3ξ^{2} − ξ^{3})

and

∫_{0}^{1} G(s) ds = ξ^{2}(−\frac{1}{24}ξ^{2} − \frac{1}{6}ξ + \frac{1}{4}).

By direct calculation one gets m = 8. The ‘optimal’ [a, b], the interval for which

M := M(a, b) = \frac{12}{3a^{2}b^{2} − 2a^{3}b − a^{4}}

EJQTDE Spec. Ed. I, 2009 No. 15

is a minimum, is given by the interval

[\frac{−3 + \sqrt{33}}{4}, 1],  (15)

this gives M = 22.032 and c = 0.545.

Remark 2.1 Until now we have discussed in details the case of the BCs (2). We point out that the same approach may be applied to the BCs (3), (4), (5).

In all these cases the growth assumptions on the nonlinearities, that are the conditions equivalent to (11) and (13), can be written in the form

\frac{k_{0}∥γ∥}{(1 − δ_{2}α[γ])ρ} + f^{0,ρ}(\frac{δ_{2}∥γ∥}{(1 − δ_{2}α[γ])} ∫_{0}^{1} G(s)g(s) ds + \frac{1}{m}) < 1,

and

\frac{k_{0}c_{γ}∥γ∥}{(1 − δ_{1}α[γ])ρ} + (\frac{c_{γ}δ_{1}∥γ∥}{(1 − δ_{1}α[γ])} ∫_{a}^{b} G(s)g(s) ds + \frac{1}{M(a, b)})f_{ρ,ρ/c} > 1,

where

BCs (2)  BCs (3)  BCs (4) BCs (5)
γ(t) (3t^{2} − t^{3})/6  1  t  t^{2}/2
c_{γ} a^{2}(3 − a)/2  1  a  a^{2}/2
c a^{2}(3 − a)/2 a^{2}(3 − a)/2 a^{2}(3 − a)/2 a^{2}/2

This also illustrates that the cone K, given by the constant c, varies according to the non-homogeneous BCs considered.

Acknowledgments

The authors would like to thank Professor Salvatore Lopez of the Faculty of Engineering, Università della Calabria, for shedding some light on the physical interpretation of this problem, and Dr. Mirosława Zima for constructive comments.

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