Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 489353, 12 pages
doi:10.1155/2012/489353
Research Article
Monotonic Positive Solutions of Nonlocal
Boundary Value Problems for a Second-Order
Functional Differential Equation
A. M. A. El-Sayed,1 E. M. Hamdallah,1 and Kh. W. El-kadeky2
1
2
Faculty of Science, Alexandria University, Alexandria, Egypt
Faculty of Science, Garyounis University, Benghazi, Libya
Correspondence should be addressed to Kh. W. El-kadeky, k welkadeky@yahoo.com
Received 13 October 2011; Accepted 5 December 2011
Academic Editor: István Györi
Copyright q 2012 A. M. A. El-Sayed et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We study the existence of at least one monotonic positive solution for the nonlocal boundary value
problem of the second-order
functional differentialequation x t ft, xφt, t ∈ 0, 1, with
n
the nonlocal condition m
j1 bj x ηj x1 , where τk ∈ a, d ⊂ 0, 1,
k1 ak xτk x0 , x 0 ηj ∈ c, e ⊂ 0, 1, and x0 , x1 > 0. As an application the integral and the nonlocal conditions
d
xtdt x0 , x 0 xe − xc x1 will be considered.
a
1. Introduction
The nonlocal boundary value problems of ordinary differential equations arise in a variety of
different areas of applied mathematics and physics.
The study of nonlocal boundary value problems was initiated by Il’in and Moiseev
1, 2. Since then, the non-local boundary value problems have been studied by several
authors. The reader is referred to 3–22 and references therein.
In most of all these papers, the authors assume that the function f : 0, 1 × R → R
is continuous. They all assume that
lim
x→∞
fx
0 or ∞,
x
fx
0 or ∞.
lim
x→0 x
1.1
2
Abstract and Applied Analysis
These assumptions are restrictive, and there are many functions that do not satisfy these
assumptions.
Here we assume that the function f : 0, 1 × R → R is measurable in t ∈ 0, 1 for
all x ∈ R and continuous in x ∈ R for almost all t ∈ 0, 1 is and there exists an integrable
function a ∈ L1 0, 1 and a constant b > 0 such that
ft, x ≤ |at| b|x|,
∀t, x ∈ 0, 1 × D.
1.2
Our aim here is to study the existence of at least one monotonic positive solution for the
nonlocal problem of the second-order functional differential equation
x t f t, x φt ,
t ∈ 0, 1,
1.3
with the nonlocal condition
m
ak xτk x0 ,
x 0 n
bj x ηj x1 ,
1.4
j1
k1
where τk ∈ a, d ⊂ 0, 1, ηj ∈ c, e ⊂ 0, 1, and x0 , x1 > 0.
As an application, the problem with the integral and nonlocal conditions
d
xtdt x0 ,
x 0 xe − xc x1 ,
1.5
a
is studied.
It must be noticed that the nonlocal conditions
m
xτ x0 ,
τ ∈ a, d ,
x 0 x η x1 ,
ak xτk 0,
τk ∈ a, d,
x 0 n
bj x ηj 0,
j1
k1
d
xtdt 0,
η ∈ c, e,
ηj ∈ c, e,
1.6
x 0 xe xc
a
are special cases of our the nonlocal and integral conditions.
2. Integral Equation Representation
Consider the functional differential equation 1.3 with the nonlocal condition 1.4 with the
following assumptions.
Abstract and Applied Analysis
3
i f : 0, 1 × R → R is measurable in t ∈ 0, 1 for all x ∈ R and continuous in
x ∈ R for almost all t ∈ 0, 1 and there exists an integrable function a ∈ L1 0, 1,
and a constant b > 0 such that
ft, x ≤ |at| b|x|,
∀t, x ∈ 0, 1 × D.
2.1
ii φ : 0, 1 → 0, 1 is continuous.
iii b < 1/3 − B, B nj1 bj 1−1 .
iv
m
ak > 0,
n
∀k 1, 2, . . . , m,
bj > 0,
∀j 1, 2, . . . , n.
2.2
j1
k1
Now, we have the following Lemma.
Lemma 2.1. The solution of the nonlocal problem 1.3-1.4 can be expressed by the integral equation
xt A x0 −
m
k1
τk
ak
τk − sf s, x φs ds
0
⎫
⎧
ηj
m
n
⎨
⎬
x − b
f s, x φs ds
B t − A ak τk
⎭
⎩ 1 j1 j 0
k1
t
2.3
t − sf s, x φs ds,
0
where A m
k1
ak −1 , B n
j1
bj 1−1 .
Proof. Integrating 1.3, we get
x t x 0 t
f s, x φs ds.
2.4
0
Integrating 2.4, we obtain
xt x0 x 0t t
t − sf s, x φs ds.
2.5
0
Let t τk , in 2.5, we get
τk
m
n
n
m
ak xτk ak x0 ak τk x 0 ak
τk − sf s, x φs ds,
k1
k1
k1
k1
0
2.6
4
Abstract and Applied Analysis
and we deduce that
x0 A x0 −
m
m
ak τk x 0 −
k1
τk
ak
τk − sf s, x φs ds ,
A
0
k1
m
ak
−1
.
2.7
k1
Substitute from 2.7 into 2.5, we obtain
xt A x0 −
m
τk
ak
k1
t
m
τk − sf s, x φs ds x 0 t − A ak τk
0
k1
2.8
t − sf s, x φs ds.
0
Let t ηj , in 2.4, we obtain
ηj
n
n
n
bj x ηj bj x 0 bj
f s, x φs ds,
j1
j1
x1 − x 0 x 0
0
j1
n
ηj
n
bj bj
f s, x φs ds,
j1
2.9
0
j1
and we deduce that
⎞
ηj
n
x 0 B⎝x1 − bj
f s, x φs ds⎠,
⎛
j1
⎛
⎞−1
n
B ⎝ bj 1⎠ .
0
2.10
j1
Substitute from 2.10 into 2.8, we obtain
τk
m
xt A x0 − ak
τk − sf s, x φs ds
0
k1
B t−A
m
k1
t
ak τk
⎧
⎨
n
⎩
j1
x1 −
ηj
bj
0
⎫
⎬
f s, x φs ds ,
⎭
2.11
t − sf s, x φs ds,
0
which proves that the solution of the nonlocal problem 1.3-1.4 can be expressed by the
integral equation 2.3.
Abstract and Applied Analysis
5
3. Existence of Solution
We study here the existence of at least one monotonic nondecreasing solution x ∈ C0, 1 for
the integral equation 2.3.
Theorem 3.1. Assume that (i)–(iv) are satisfied. Then the nonlocal problem 1.3-1.4 has at least
one solution x ∈ C0, 1.
Proof. Define the subset Qr ⊂ C0, 1 by Qr {x ∈ C : |xt| ≤ r, r Ax0 Bx1 3 −
Ba/1 − 3 − Bb, r > 0}. Clear the set Qr which is nonempty, closed, and convex.
Let H be an operator defined by
τk
m
τk − sf s, x φs ds
Hxt A x0 − ak
0
k1
⎫
⎧
ηj
m
n
⎨
⎬
x − b
f s, x φs ds
B t − A ak τk
⎭
⎩ 1 j1 j 0
k1
t
t − sf s, x φs ds.
0
Let x ∈ Qr , then
|Hxt| ≤ A x0 m
τk
ak
τk − s f s, x φs ds
0
k1
⎫
⎧
ηj
m
n
⎨
⎬
f s, x φs ds
B t − A ak τk
x1 bj
⎭
⎩
0
j1
k1
t
t − sf s, x φs ds
0
≤ A x0 B
⎧
⎨
⎩
1
m
1
ak
x1 |as| bx φs ds
0
k1
n
1
bj
0
j1
⎫
⎬
|as| bx φs ds
⎭
|as| bx φs ds
0
n
≤ Ax0 a b supx φt Bx1 B bj a
t∈I
bB
j1
n
bj supx φt a b supx φt j1
t∈I
t∈I
3.1
6
Abstract and Applied Analysis
≤ Ax0 Bx1 2a 2bx 1 − Ba b1 − Bx
≤ Ax0 Bx1 3 − Ba 3 − Bbr ≤ r,
3.2
then H : Qr → Qr and {Hxt} is uniformly bounded in Qr .
Also for t1 , t2 ∈ 0, 1 such that t1 < t2 , we have
Hxt2 − Hxt1 B t2 − A
m
⎫
ηj
n
⎬
x − b
f s, x φt ds
⎭
⎩ 1 j1 j 0
⎧
⎨
ak τk
k1
t2
t2 − sf s, x φt ds
0
− B t1 − A
m
ak τk
k1
−
t1
⎫
ηj
n
⎬
f s, x φt ds
x − b
⎭
⎩ 1 j1 j 0
⎧
⎨
t1 − sf s, x φt ds
0
⎫
⎧
ηj
n
⎨
⎬
Bt2 − t1 x1 − bj
f s, x φt ds
⎭
⎩
0
j1
t1
t2 − t1 f s, x φt ds
0
t2
t2 − sf s, x φt ds.
t1
Then
⎫
ηj
n
⎬
|Hxt2 − Hxt1 | ≤ B|t2 − t1 | x1 bj
|as| bx φs ds
⎭
⎩
0
j1
⎧
⎨
|t2 − t1 |
t2
t1
t1
|as| bx φs ds
0
t2 − s |as| bx φs ds
3.3
Abstract and Applied Analysis
7
≤ B|t2 − t1 |x1 n
bj a br
j1
|t2 − t1 |a br t2
ads brt2 − t1 .
t1
3.4
The above inequality shows that
|Hxt2 − Hxt1 | −→ 0 as t2 −→ t1 .
3.5
Therefore {Hxt} is equicontinuous. By the Arzelà-Ascoli theorem, {Hxt} is relatively
compact.
Since all conditions of the Schauder theorem hold, then H has a fixed point in Qr
which proves the existence of at least one solution x ∈ C0, 1 of the integral equation 2.3,
where
τk
m
lim xt A x0 − ak
τk − sf s, x φs ds
t→0
0
k1
− BA
m
ak τk
k1
⎧
⎨
n
⎩
j1
x1 −
ηj
bj
0
⎫
⎬
f s, x φs ds x0,
⎭
τk
m
lim− xt A x0 − ak
τk − sf s, x φs ds
t→1
k1
3.6
0
⎫
⎧
ηj
m
n
⎨
⎬
f s, x φs ds
B 1 − A ak τk
x − b
⎭
⎩ 1 j1 j 0
k1
1
1 − sf s, x φs ds x1.
0
To complete the proof, we prove that the integral equation 2.3 satisfies nonlocal problem
1.3-1.4. Differentiating 2.3, we get
⎫
ηj
n
⎬ t f s, x φs ds f s, x φs ds,
x t B x1 − bj
⎭
⎩
0
0
j1
⎧
⎨
x t f t, x φt .
3.7
3.8
8
Abstract and Applied Analysis
Let t τk in 2.3, we obtain
xτk A x0 −
m
τk
ak
τk − sf s, x φs ds 0
k1
τk
τk − sf s, x φs ds,
3.9
0
which proves
m
ak xτk x0 .
3.10
k1
Also let t ηj in 3.7, we obtain
⎫
⎧
ηj
n
⎬ ηj ⎨
x ηj B x1 − bj
f s, x φs ds f s, x φs ds,
⎭
⎩
0
0
j1
3.11
then
⎧
⎫
ηj
ηj
n
n
n
n
⎨
⎬ bj x ηj B bj x1 − bj
f s, x φs ds bj
f s, x φs ds.
⎩
⎭
0
0
j1
j1
j1
j1
3.12
Let t 0 in 3.7, we obtain
⎫
ηj
n
⎬
x 0 B x1 − bj
f s, x φs ds .
⎭
⎩
0
j1
⎧
⎨
3.13
Adding 3.12 and 3.13, we obtain
x 0 n
bj x ηj x1 .
3.14
j1
This implies that there exists at least one solution x ∈ C0, 1 of the nonlocal problem 1.3
and 1.4. This completes the proof.
Abstract and Applied Analysis
9
Corollary 3.2. The solution of the problem 1.3-1.4 is monotonic nondecreasing.
Proof. Let t1 < t2 , we deduce from 2.3 that
τk
m
xt1 A x0 − ak
τk − sf s, x φs ds
0
k1
B t1 − A
m
⎫
ηj
n
⎬
x − b
f s, x φs ds
⎭
⎩ 1 j1 j 0
⎧
⎨
ak τk
k1
t1
t1 − sf s, x φs ds
0
τk
m
< A x0 − ak
τk − sf s, x φs ds
0
k1
B t2 − A
m
⎫
ηj
n
⎬
x − b
f s, x φs ds
⎭
⎩ 1 j1 j 0
⎧
⎨
ak τk
k1
t2
3.15
t2 − sf s, x φs ds xt2 ,
0
which proves that the solution x of the problem 1.3-1.4 is monotonic nondecreasing.
3.1. Positive Solution
Let bj 0, j 1, 2, . . . n and x1 0, then the nonlocal problem condition 1.4 will be
m
x 0 0.
ak xτk x0 ,
3.16
k1
Theorem 3.3. Let the assumptions (i)–(iv) of Theorem 3.1 be satisfied. Then the solution of the
nonlocal problem 1.3–3.16 is positive t ∈ d, 1.
Proof. Let bj 0, j 1, 2, . . . n and x1 0 in the integral equation 2.3 and the nonlocal
condition 1.4, then the solution of the nonlocal problem 1.3–3.16 will be given by the
integral equation
xt A x0 −
m
τk
ak
k1
where A m
k1
ak −1 .
0
τk − sf s, x φs ds t
0
t − s f s, x φs ds,
3.17
10
Abstract and Applied Analysis
Let t ∈ d, 1, then
τk
τk − sf s, x φs ds ≤
t
t − sf s, x φs ds,
τk ≤ t,
0
0
τk
t
m
m
ak
ak t − sf s, x φs ds.
τk − sf s, x φs ds ≤
0
k1
Multiplying by A 0
k1
m
k1
3.18
ak −1 , we obtain
τk
t
m
m
A ak
τk − sf s, x φs ds ≤ A ak t − sf s, x φs ds
k1
0
0
k1
t
t − sf s, x φs ds,
3.19
0
and the solution x of the nonlocal problem 1.3 and 3.16, given by the integral equation
3.17, is positive for t ∈ d, 1. This complete the proof.
Example 3.4. Consider the nonlocal problem of the second-order functional differential
equation 1.3 with two-point boundary condition
x 0 0,
x η x0 ,
η ∈ a, d ⊂ 0, 1.
3.20
Applying our results here, we deduce that the two-point boundary value problem 1.3–
3.20 has at least one monotonic nondecreasing solution x ∈ C0, 1 represented by the
integral equation
xt x0 −
η
η − s f s, x φs ds t
0
t − sf s, x φs ds.
3.21
0
This the solution is positive with t > η.
4. Nonlocal Integral Condition
Let x ∈ C0, 1 be the solution of the nonlocal problem 1.3 and 1.4.
Let ak tk − tk−1 , τk ∈ tk−1 , tk ⊂ a, d ⊂ 0, 1 and let bj ξj − ξj−1 , ηj ∈ ξj−1 , ξj ⊂
c, e ⊂ 0, 1, then
m
k1
tk − tk−1 xτk x0 ,
x 0 n
ξj − ξj−1 x ηj x1 .
j1
4.1
Abstract and Applied Analysis
11
From the continuity of the solution x of the nonlocal problem 1.3 and 1.4, we obtain
lim
m
m→∞
x 0 lim
tk − tk−1 xτk d
xsds,
a
k1
n
n→∞
ξj − ξj−1 x ηj x 0 e
4.2
x sds,
c
j1
and the nonlocal condition 1.4 transformed to the integral condition
d
x 0 xe − xc x1 ,
xsds x0 ,
4.3
a
and the solution of the integral equation 2.3 will be
xt d − a
−1
x0 −
d t
a
t − sf s, x φs ds dt
0
−1
b − c 1 t − 1 x1 −
e t
t
f s, x φs ds dt
c
4.4
0
f s, x φs ds.
0
Now, we have the following theorem.
Theorem 4.1. Let the assumptions (i)–(iv) of Theorem 3.1 be satisfied. Then the nonlocal problem
x t f t, x φt ,
d
xsds x0 ,
t ∈ 0, 1 ,
x 0 xe − xc x1
4.5
a
has at least one monotonic nondecreasing solution x ∈ C0, 1 represented by 4.4.
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