Conditions for the existence of at least three positive solutions to the
nonlinear first-order problem with a nonlinear nonlocal boundary condition
given by
  && y'(t) - p(t)y(t) = \sum_{i=1}^m f_i\big(t,y(t)\big), \quad t\in[0,1],
  && \lambda y(0) = y(1) + \sum_{j=1}^n \Phi_j(\tau_j,y(\tau_j)), \quad
\tau_j\in[0,1], are discussed, for sufficiently large $\lambda>1$. The
Leggett-Williams fixed point theorem is utilized.Comment: outline, 6 page

Anderson, Douglas R.

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arXiv

Douglas R. Anderson

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Existence of three solutions for a first-order problem with nonlinear nonlocal boundary conditions

Conditions for the existence of at least three positive solutions to the nonlinear first-order problem with a nonlinear nonlocal boundary condition given by are discussed, for sufficiently large Î» &gt; 1 and r â\u89¥ 0. The Leggett-Williams fixed point theorem is utilized

Douglas R Anderson

Accepted Manuscript
Existence of three solutions for a first-order problem with nonlinear
nonlocal boundary conditions
Douglas R. Anderson
PII: S0022-247X(13)00570-2
DOI: http://dx.doi.org/10.1016/j.jmaa.2013.06.025
Reference: YJMAA 17688
To appear in: Journal of Mathematical Analysis and
Applications
Received date: 25 October 2012
Please cite this article as: D.R. Anderson, Existence of three solutions for a first-order problem
with nonlinear nonlocal boundary conditions, J. Math. Anal. Appl. (2013),
http://dx.doi.org/10.1016/j.jmaa.2013.06.025
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EXISTENCE OF THREE SOLUTIONS FOR A FIRST-ORDER
PROBLEM WITH NONLINEAR NONLOCAL BOUNDARY
CONDITIONS
DOUGLAS R. ANDERSON
Abstract. Conditions for the existence of at least three positive solutions to the nonlinear
first-order problem with a nonlinear nonlocal boundary condition given by
y′(t)− r(t)y(t) =
m∑
i=1
fi
(
t, y(t)
)
, t ∈ [0, 1],
λy(0) = y(1) +
n∑
j=1
Λj(τj , y(τj)), τj ∈ [0, 1],
are discussed, for sufficiently large λ > 1 and r ≥ 0. The Leggett-Williams fixed point
theorem is utilized.
1. Introduction
We are interested in the first-order boundary value problem with nonlinear nonlocal
boundary condition given by
y′(t)− r(t)y(t) =
m∑
i=1
fi
(
t, y(t)
)
, t ∈ [0, 1], (1.1)
λy(0) = y(1) +
n∑
j=1
Λj(τj, y(τj)), τj ∈ [0, 1], (1.2)
where r : [0, 1] → [0,∞) is continuous; the nonlocal points satisfy 0 ≤ τ1 < τ2 < · · · < τn ≤
1; the nonlinear functions Λj : [0, 1] × [0,∞) → [0,∞) satisfy
0 ≤ yψj(t, y) ≤ Λj(t, y) ≤ yΨj(t, y), t ∈ [0, 1], y ∈ [0,∞), (1.3)
for some positive continuous (possibly nonlinear) functions ψj ,Ψj : [0, 1]× [0,∞) → [0,∞);
the scalar λ satisfies
λ >

1 +
n∑
j=1
βj

 exp
(∫ 1
0
r(η)dη
)
> 1, βj := max
[0,1]×[0,C]
Ψj(t, y) (1.4)
for some real constant C > 0; and the nonlinear functions fi : [0, 1] × [0,∞) → [0,∞) are
all continuous. We also set
αj := min
[0,1]×[B,λB]
ψj(t, y) (1.5)
2010 Mathematics Subject Classification. 34B05.
Key words and phrases. positive solutions, cone, fixed point theorem, nonlinear boundary condition,
Leggett-Williams theorem.
*Manuscript
2 D. R. ANDERSON
for some constant real B > 0 for later reference. Note that by continuity and compactness
αj and βj exist and satisfy βj > αj > 0.
Some of the motivation for this paper and the study of problem (1.1), (1.2) is as follows.
First-order equations with various boundary conditions, including multi-point and nonlocal
conditions, are of recent interest. For example, we cite the following papers. Zhao and Sun
[14] were concerned with the first-order PBVP (if T = R)
y′(t) + r(t)y(t) = λf
(
t, y(t)
)
, t ∈ [0, 1], (1.6)
y(0) = y(1). (1.7)
Tian and Ge [12] investigated the first-order three-point problem (if T = R)
y′(t) + r(t)y(t) = λf
(
t, y(t)
)
, t ∈ [0, 1], (1.8)
y(0)− αy(η) = γy(1), (1.9)
while Gao and Luo [2] were interested in the problem (if T = R)
y′(t) + r(t)y(t) = λf
(
t, y(t)
)
, t ∈ [0, 1], (1.10)
y(0) =
n∑
j=1
γjy(tj); (1.11)
similarly Anderson [1] studied the first-order problem (if T = R)
y′(t) + r(t)y(t) = λf
(
t, y(t)
)
, t ∈ [0, 1], (1.12)
y(0) = y(1) +
n∑
j=1
γjy(tj). (1.13)
In a related paper, Nieto and R. Rodrguez-López [8] considered
y′(t) + r(t)y(t) = λf
(
t
)
, t ∈ [0, 1], (1.14)
λy(t0) =
n∑
j=1
γjy(tj). (1.15)
Gilbert [3] looked at (if T = R)
y′(t) = λf
(
t, y(t)
)
, a.e. t ∈ [0, 1], (1.16)
y(0) = y(1), or y(0) = y0 (1.17)
using measure theory and ∆-Carathéodory functions. Goodrich [4] analyzed the p-Laplacian
problem (if T = R)
φp
(
y′(t)
)
= h(t)f (y(t)) , t ∈ [0, 1], (1.18)
y(0) = Ψ(y) or y(0) = B0
(
y′(1)
)
or y(0) =
(
y′(1)
)m
, (1.19)
while Graef and Kong [6] explored the related p-Laplacian problem (if T = R)
φp
(
y′(t)
)
= f (t, y(t)) , t ∈ [0, 1], (1.20)
y(0) = B
(
y′(1)
)
. (1.21)
FIRST-ORDER NONLINEAR PROBLEM 3
Otero-Espinar and Vivero [9] gave a general view of different kinds of weak first-order dis-
continuous boundary value problems on an arbitrary time scale with nonlinear functional
boundary value conditions. Shu and Deng [11] proved the existence of three positive solu-
tions to (if T = R)
y′(t) = λf
(
y(t)
)
, t ∈ [0, 1], (1.22)
y(0) = γy(1). (1.23)
Zhao [13] applied a monotone iteration method to the problem (if T = R)
y′(t) + r(t)y(t) = λf
(
t, y(t)
)
, t ∈ [0, 1], (1.24)
y(0) = g(y(1)), (1.25)
where g denotes a nonlinear boundary condition. Precup and Trif [10] dealt with the
existence, localization and multiplicity of positive solutions to non-local problems for first
order differential systems of the form
y′(t) = f
(
t, y(t)
)
, t ∈ [0, 1], (1.26)
y(0) = α[y], (1.27)
where α is linear and continuous. In a second-order problem, Goodrich [5] was concerned
with (if T = R)
y′′ = −λf (t, y(t)) , t ∈ [0, 1], (1.28)
y(0) = ψ(y), y(1) = 0. (1.29)
where ψ is a nonlocal boundary condition. One can see from these representative works
that problem (1.1) with the nonlinear nonlocal boundary condition (1.2) is new.
2. at least three positive solutions
In this section we establish the existence of at least three positive solutions for the
boundary value problem (1.1), (1.2), by finding fixed points for operators on cones in a
Banach space; the theorem we will use below is the Leggett-Williams fixed point theorem
[7].
A nonempty closed convex set P contained in a real Banach space S is called a cone if
it satisfies the following two conditions:
(i) if y ∈ P and ζ ≥ 0 then ζy ∈ P ;
(ii) if y ∈ P and −y ∈ P then y = 0.
The cone P induces an ordering ≤ on S by x ≤ y if and only if y − x ∈ P . An operator
K is said to be completely continuous if it is continuous and compact (maps bounded sets
into relatively compact sets). A map θ is a nonnegative continuous concave functional on
P if it satisfies the following conditions:
(i) θ : P → [0,∞) is continuous;
4 D. R. ANDERSON
(ii) θ(tx+ (1− t)y) ≥ tθ(x) + (1− t)θ(y) for all x, y ∈ P and 0 ≤ t ≤ 1.
Let
PC := {y ∈ P : ‖y‖ < C} (2.1)
and
P (θ,A,B) := {y ∈ P : A ≤ θ(y), ‖y‖ ≤ B}. (2.2)
Theorem 2.1 (Leggett-Williams). Let P be a cone in the real Banach space S, K : PC →
PC be completely continuous, and θ be a nonnegative continuous concave functional on P
with θ(y) ≤ ‖y‖ for all y ∈ PC . Suppose there exist constants 0 < A < B < B† ≤ C such
that the following conditions hold:
(i) {y ∈ P (θ,B,B†) : θ(y) > B} 6= ∅ and θ(Ky) > B for all y ∈ P (θ,B,B†);
(ii) ‖Ky‖ < A for ‖y‖ ≤ A;
(iii) θ(Ky) > B for y ∈ P (θ,B,C) with ‖Ky‖ > B†.
Then K has at least three fixed points y1, y2, and y3 in PC satisfying:
‖y1‖ < A, θ(y2) > B, A < ‖y3‖ with θ(y3) < B.
In the next theorem we use the following notation. Let
G(t, s) =
exp
(∫ t
s r(η)dη
)
λ− exp
(∫ 1
0 r(η)dη
) ×



λ : 0 ≤ s < t ≤ 1,
exp
(∫ 1
0 r(η)dη
)
: 0 ≤ t ≤ s ≤ 1,
(2.3)
denote a function that will be shown (in the proof of Theorem 2.2 below) to be the corre-
sponding Green function for our boundary value problem,
M :=
1∫ 1
0 G(1, s)ds

1−
exp
(∫ 1
0 r(η)dη
) ∑n
j=1 βj
λ− exp
(∫ 1
0 r(η)dη
)

 > 0 (2.4)
for βj from (1.4), and
N :=
1∫ 1
0 G(0, s)ds

1−
∑n
j=1 αj
λ− exp
(∫ 1
0 r(η)dη
)

 > 0 (2.5)
for αj from (1.5). Both M and N are positive by (1.4).
Theorem 2.2. Suppose that (1.3) and (1.4) hold, and suppose that there exist constants
0 < A < B < λB ≤ C such that
(F1) fi(t, y) < MA/m for t ∈ [0, 1], y ∈ [0, A],
(F2) fi(t, y) > NB/m for t ∈ [0, 1], y ∈ [B,λB],
(F3) fi(t, y) ≤MC/m for t ∈ [0, 1], y ∈ [0, C],
FIRST-ORDER NONLINEAR PROBLEM 5
for i = 1, 2, · · · ,m, where M and N are given above by (2.4) and (2.5), respectively. Then
the boundary value problem (1.1), (1.2) has at least three positive solutions y1, y2, y3 satis-
fying
‖y1‖ = y1(1) < A, B < θ(y2) = y2(0), ‖y3‖ = y3(1) > A with θ(y3) = y3(0) < B,
where θ is given below in (2.6).
Proof. Let the Banach space S = C[0, 1] be endowed with the sup norm, and define the
cone P ⊂ S by
P = {y ∈ S : y ≥ 0, y increasing}.
Let the nonnegative continuous concave functional θ : P → [0,∞) by defined by
θ(y) := min
t∈[0,1]
y(t) = y(0), y ∈ P. (2.6)
Using (2.3), define the operator K : P → S by
Ky(t) =
m∑
i=1
∫ 1
0
G(t, s)fi(s, y(s))ds +
exp
(∫ t
0 r(η)dη
) ∑n
j=1 Λj(τj, y(τj))
λ− exp
(∫ 1
0 r(η)dη
)
=
m∑
i=1
∫ t
0
λ exp
(∫ t
s r(η)dη
)
λ− exp
(∫ 1
0 r(η)dη
)fi(s, y(s))ds
+
m∑
i=1
∫ 1
t
exp
(∫ 1
0 r(η)dη
)
exp
(∫ t
s r(η)dη
)
λ− exp
(∫ 1
0 r(η)dη
) fi(s, y(s))ds
+
exp
(∫ t
0 r(η)dη
) ∑n
j=1 Λj(τj, y(τj))
λ− exp
(∫ 1
0 r(η)dη
) .
Note that if y ∈ P , the fact that the fi are positive, (1.3), and (1.4) imply that Ky(t) ≥ 0
for t ∈ [0, 1]. We claim that fixed points of K are solutions of (1.1), (1.2). If y = Ky, then
λy(0)− y(1) =
m∑
i=1
∫ 1
0
λ exp
(∫ 1
s r(η)dη
)
λ− exp
(∫ 1
0 r(η)dη
)fi(s, y(s))ds +
λ
∑n
j=1 Λj(τj, y(τj))
λ− exp
(∫ 1
0 r(η)dη
)
−
m∑
i=1
∫ 1
0
λ exp
(∫ 1
s r(η)dη
)
λ− exp
(∫ 1
0 r(η)dη
)fi(s, y(s))ds
−
exp
(∫ 1
0 r(η)dη
) ∑n
j=1 Λj(τj , y(τj))
λ− exp
(∫ 1
0 r(η)dη
)
=
n∑
j=1
Λj(τj , y(τj)),
6 D. R. ANDERSON
so the boundary condition (1.2) is satisfied. Moreover,
y′(t) = (Ky)′(t) = r(t)y(t) +
m∑
i=1
fi(t, y(t)) > 0, t ∈ [0, 1], (2.7)
implies that y is increasing and (1.1) is met. Thus we see that Ky ∈ P , that is to say,
K : P → P . In addition, K is completely continuous using a standard argument.
We now show that all of the conditions of Theorem 2.1 are satisfied, whereB† is replaced
by λB. For all y ∈ P we have y(0) = θ(y) ≤ ‖y‖ = y(1). Using (2.1), if y ∈ PC , then
‖y‖ ≤ C and assumption (F3) implies fi(t, y(t)) ≤MC/m for t ∈ [0, 1] and i = 1, 2, · · · ,m.
By (2.7) we know that Ky is increasing; as a result,
‖Ky‖ = Ky(1)
=
m∑
i=1
∫ 1
0
G(1, s)fi(s, y(s))ds +
exp
(∫ 1
0 r(η)dη
) ∑n
j=1 Λj(τj , y(τj))
λ− exp
(∫ 1
0 r(η)dη
)
≤
m∑
i=1
∫ 1
0
G(1, s)
MC
m
ds+
exp
(∫ 1
0 r(η)dη
) ∑n
j=1 y(τj)Ψj(τj, y(τj))
λ− exp
(∫ 1
0 r(η)dη
)
≤ MC
∫ 1
0
G(1, s)ds +
exp
(∫ 1
0 r(η)dη
)
C
∑n
j=1 Ψj(τj , y(τj))
λ− exp
(∫ 1
0 r(η)dη
)
≤ C

M
∫ 1
0
G(1, s)ds +
exp
(∫ 1
0 r(η)dη
) ∑n
j=1 βj
λ− exp
(∫ 1
0 r(η)dη
)

 = C,
where we used (1.3), (1.4), and (2.4). Therefore K : PC → PC . In the same way, if y ∈ PA,
then assumption (F1) yields fi(t, y(t)) < MA/m for t ∈ [0, 1] and i = 1, 2, · · · ,m. If we
mimic the argument above, then it follows that K : PA → PA, as the maximum for Ψj
on [0, 1] × [0, A] is less than or equal to βj given in (1.4) since A < C, for j = 1, 2, · · · , n.
Hence, condition (ii) of Theorem 2.1 is satisfied.
To check condition (i) of Theorem 2.1, choose yP (t) ≡ λB for t ∈ [0, 1]. Using (2.2),
then yP ∈ P (θ,B, λB) and θ(yP ) = yP = λB > B, so that {y ∈ P (θ,B, λB) : θ(y) > B} 6=
∅. Consequently, if y ∈ P (θ,B, λB), then B ≤ y(t) ≤ λB for t ∈ [0, 1]. From assumption
(F2) we have that fi(t, y(t)) > NB/m for t ∈ [0, 1] and i = 1, 2, · · · ,m; by the definitions
of θ in (2.6) and the cone P , we have
θ(Ky) = Ky(0)
=
m∑
i=1
∫ 1
0
G(0, s)fi(s, y(s))ds +
∑n
j=1 Λj(τj , y(τj))
λ− exp
(∫ 1
0 r(η)dη
)
> NB
∫ 1
0
G(0, s)ds +
B
∑n
j=1 αj
λ− exp
(∫ 1
0 r(η)dη
) = B,
FIRST-ORDER NONLINEAR PROBLEM 7
for N in (2.5) and αj in (1.5). As a result we have
θ(Ky) > B, y ∈ P (θ,B, λB),
so that condition (i) of Theorem 2.1 holds.
Lastly we consider Theorem 2.1 (iii). Suppose y ∈ P (θ,B,C) with ‖Ky‖ > λB. Then,
using (1.4), (2.3), and the definition of θ in (2.6), we see that
θ(Ky) = Ky(0) ≥ 1
λ
Ky(1) =
‖Ky‖
λ
>
λB
λ
= B.
Consequently all the conditions hold in Theorem 2.1, and its conclusion follows. 
3. closing comments
Although we deal with problem (1.1), (1.2) on the real unit interval, the boundary value
problem and accompanying techniques introduced in this work can be readily extended to
related difference equations and dynamic equations on time scales.
The main conditions here are (1.3) on the nonlinear functions, and on λ in (1.4). Note
that these assumptions are fairly mild. We do not assume that the Λj are completely
separable, nor that they are asymptotically linear. As for λ, we merely have it bounded
below, leaving an unbounded range of possible values that it may assume.
Future authors may wish to explore (1.1), (1.2) using monotone iteration methods, up-
per and lower solution techniques, existence of solutions under singularity, or semi-positone
conditions, and so on. We leave the details to interested readers.
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Department of Mathematics, Concordia College, Moorhead, MN 56562 USA
E-mail address: andersod@cord.edu


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Existence of three solutions for a first-order problem with nonlinear non-local boundary conditions

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