Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 476392, 9 pages
doi:10.1155/2011/476392
Research Article
Solutions of a Class of Deviated-Advanced
Nonlocal Problems for the Differential Inclusion
x1 t ∈ Ft, xt
A. M. A. El-Sayed,1 E. M. Hamdallah,1 and Kh. W. Elkadeky2
1
2
Faculty of Science, Alexandria University, Alexandria, Egypt
Faculty of Science, Garyounis University, Benghazi, Libya
Correspondence should be addressed to Kh. W. Elkadeky, k welkadeky@yahoo.com
Received 19 January 2011; Accepted 27 April 2011
Academic Editor: Stephen Clark
Copyright q 2011 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 solutions for deviated-advanced nonlocal and integral condition
problems for the differential inclusion x1 t ∈ Ft, xt.
1. Introduction
Problems with nonlocal conditions have been extensively studied by several authors in
the last two decades. The reader is referred to 1–12 and references therein. Consider the
deviated-advanced nonlocal problem
dxt
∈ Ft, xt,
dt
a.e. t ∈ 0, 1,
m
n
ak x φτk α bj x ψ ηj ,
k1
ak , bj > 0,
1.1
1.2
j1
where τk , ηj ∈ 0, 1, α > 0 is a parameter, and ψ and φ are, respectively, deviated and
advanced given functions.
Our aim here is to study the existence of at least one absolutely continuous solution
x ∈ AC0, 1 for the problem 1.1-1.2 when the set-valued function F : R → P R is
L1 -Carathéodory.
2
Abstract and Applied Analysis
As an application, we deduce the existence of a solution for the nonlocal problem of
the differential inclusion 1.1 with the deviated-advanced integral condition
1
x φs ds α
1
0
x ψs ds.
1.3
0
It must be noticed that the following nonlocal and integral conditions are special cases
of our nonlocal and integral conditions
x φτ αx ψ η ,
m
τ, η ∈ 0, 1,
ak x φτk αx ψ η ,
τk , η ∈ 0, 1,
k1
m
ak x φτk 0,
τk ∈ 0, 1,
k1
1
x φs ds αx ψ η ,
η ∈ 0, 1,
0
1
α
0
x ψs ds x φτ ,
1
1.4
τ ∈ 0, 1,
x φs ds 0,
0
1
x ψs ds 0.
0
As an example of the deviated function φ : 0, 1 → 0, 1, we have φt βt, β ∈ 0, 1. As an
example of the advanced function ψ : 0, 1 → 0, 1, we have ψt tβ , β ∈ 0, 1.
2. Preliminaries
The following preliminaries are needed.
Definition 2.1. A set-valued function F : 0, 1 × R → P R is called L1 -Carathéodory if
a t → Ft, x is measurable for each x ∈ R,
b x → Ft, x is upper semicontinuous for almost all t ∈ 0, 1,
c there exists m ∈ L1 0, 1, D, D ⊂ R such that
|Ft, x| sup{|v| : v ∈ Ft, x} ≤ mt,
for almost all t ∈ 0, 1.
Definition 2.2. A single-valued function f : 0, 1 × R → R is called L1 -Carathéodory if
i t → ft, x is measurable for each x ∈ R,
2.1
Abstract and Applied Analysis
3
ii x → ft, x is continuous for almost all t ∈ 0, 1,
iii there exists m ∈ L1 0, 1, D, D ⊂ R such that |f| ≤ m.
Definition 2.3. The set
S1F·,xt f ∈ 0, 1, R : ft, x ∈ Ft, xt for a.e. t ∈ 0, 1
2.2
is called the set of selections of the set-valued function F.
Theorem 2.4. For any L1 -Carathéodory set-valued function F, the set S1F·,xt is nonempty [1, 13].
Theorem 2.5 Carathéodory, 14. Let f : 0, 1 × R → R be L1 -Carathéodory. Then the problem
dxt
ft, xt,
dt
for a.e. t > 0, x0 x0 ,
2.3
has at least one solution x ∈ AC0, T .
3. Existence of Solution
Consider the following assumptions.
i F : 0, 1 × R → P R is L1 -Carathéodory.
ii
n
m
ak .
α bj /
j1
3.1
k1
iii φ : 0, 1 → 0, 1, φt ≤ t is a deviated continuous function.
iv ψ : 0, 1 → 0, 1, ψt ≥ t is an advanced continuous function.
Now we have the following lemma.
Lemma 3.1. Let assumptions (i)-(ii) be satisfied. The solution of the nonlocal problem 1.1-1.2 can
be expressed by the integral equation
⎞
⎛
t
φτk ψηj m
n
fs, xsds − α bj
fs, xsds⎠ fs, xsds,
xt A⎝ ak
k1
0
j1
where ft, x ∈ Ft, x, for all x ∈ R, and A α
0
0
n
j1
bj −
m
k1
ak −1 .
3.2
4
Abstract and Applied Analysis
Proof. From the assumption that the set-valued function F : 0, 1 × R → P R is L1 Carathéodory, then Theorem 2.4 there exists a single-valued selection f : 0, 1 × R → R
such that
d
xt ft, x ∈ Ft, x,
dt
∀x ∈ R.
3.3
This selection ft, x is L1 -Carathéodory.
Integrating 3.3, we get
xt x0 t
3.4
fs, xsds.
0
Let t φτk . Then
m
m
m
ak x φτk ak x0 ak
k1
k1
φτk fs, xsds.
3.5
0
k1
Let t ψηj . Then
ψηj n
n
n
fs, xsds.
α bj x ψ ηj α bj x0 α bj
j1
j1
j1
3.6
0
From 3.5 and 3.6, we obtain
⎛
⎞
φτk ψηj m
n
x0 A⎝ ak
fs, xsds − α bj
fs, xsds⎠,
k1
0
j1
3.7
0
n
−1
where A α nj1 bj − m
m
j1 bj /
k1 ak , α
k1 ak .
Substituting 3.7 into 3.4, we obtain
⎞
t
φτk ψηj m
n
⎠
⎝
ak
fs, xsds − α bj
fs, xsds fs, xsds.
xt A
⎛
k1
0
j1
0
3.8
0
This proves that the solution of the nonlocal problem 1.1-1.2 can be expressed by the
integral equation 3.2.
For the existence of the solution, we have the following theorem.
Theorem 3.2. Assume that (i)–(iv) are satisfied. Then the integral equation 3.2 has at least one
continuous solution x ∈ C0, 1.
Abstract and Applied Analysis
5
Proof. Define a subset Qr ⊂ C0, 1 by
⎞⎫
⎬
Qr x ∈ C0, 1 : |xt| ≤ r, r AM⎝1 ak α
bj ⎠ .
⎭
⎩
j1
k1
⎧
⎨
⎛
m
n
3.9
Clearly, the set Qr is nonempty, closed, and convex.
Let H be an operator defined by
⎞
φτk ψηj t
m
n
fs, xsds − α bj
fs, xsds⎠ fs, xsds.
Hxt A⎝ ak
⎛
0
k1
0
j1
0
3.10
Let x ∈ Qr . Let {xn t} be a sequence in Qr converging to xt, xn t → xt, for all t ∈ I.
Then
⎛
⎞
φτk ψηj m
n
lim Hxn t A⎝ ak lim
fs, xn sds − α bj lim
fs, xn sds⎠
n→∞
n→∞
k1
lim
n→∞
t
0
j1
n→∞
0
3.11
fs, xn sds,
0
By assumptions i-ii and the Lebesgue dominated convergence theorem, we deduce that
lim Hxn t Hxt.
3.12
n→∞
Then H is continuous.
Now, letting x ∈ Qr , then φt ≤ t and ψt ≥ t, we obtain
⎛
⎞
τk
ηj
m
n
fs, xsds − α bj
fs, xsds⎠
Hxt ≤ A⎝ ak
k1
0
j1
0
t
fs, xsds,
0
⎞
⎛
τk
ηj
m
n
fs, xsds α bj
fs, xsds⎠
|Hxt| ≤ A⎝ ak
k1
t
0
j1
0
fs, xsds
0
⎛
⎞
t
τk
ηj
m
n
≤ A⎝ ak
msds α bj
msds⎠ msds
k1
0
j1
0
0
6
Abstract and Applied Analysis
⎛
⎞
m
n
≤ A⎝ ak M α bj M⎠ M
j1
k1
⎛
≤ AM⎝1 m
⎞
n
ak α bj ⎠ ≤ r.
j1
k1
3.13
Then {Hxt} is uniformly bounded in Qr .
Also for t1 , t2 ∈ 0, 1, t1 < t2 such that |t2 − t1 | < δ, we have
Hxt2 − Hxt1 t2
fs, xsds −
0
|Hxt2 − Hxt1 | ≤
t2
t1
fs, xsds,
0
fs, xsds
t1
≤
t2
3.14
msds,
t1
|Hxt2 − Hxt1 | ≤ ε.
Hence the class of functions {Hxt} is equicontinuous. By Arzela-Ascoli’s theorem, {Hxt}
is relatively compact. Since all conditions of Schauder’s theorem hold, then H has a fixed
point in Qr .
Therefore the integral equation 3.2 has at least one continuous solution x ∈ C0, 1.
Now,
⎛
⎞
φτk ψηj m
n
fs, xsds − α bj
fs, xsds⎠
lim xt A lim⎝ ak
t→0
t→0
lim
0
k1
0
j1
t
t→0
3.15
fs, xsds
0
⎛
⎞
φτk ψηj m
n
A⎝ ak
fs, xsds − α bj
fs, xsds⎠ x0.
k1
0
j1
0
Also
⎛
⎞
φτk φηj m
n
x1 limxt A⎝ ak
fs, xsds − α bj
fs, xsds⎠
t→1
k1
0
j1
0
1
fs, xsds.
0
Then the integral equation 3.2 has at least one continuous solution x ∈ C0, 1.
3.16
Abstract and Applied Analysis
7
The following theorem proves the existence of at least one solution for the nonlocal
problem1.1-1.2.
Theorem 3.3. Let (i)–(iv) be satisfied. Then the nonlocal problem 1.1-1.2 has at least one solution
x ∈ AC0, 1.
Proof. From Theorem 3.2 and the integral equation 3.2, we deduce that there exists at least
one solution, x ∈ AC0, 1, of the integral equation 3.2.
To complete the proof, we prove that the integral equation 3.2 satisfies nonlocal
problem 1.1-1.2.
Differentiating 3.2, we get
dx
ft, xt ∈ Ft, xt,
dt
a.e. t ∈ 0, 1.
3.17
Letting t φτk in 3.2, we obtain
m
k1
φτ ψηj m
m
m
n
k
ak x φτk ak A ak 1
fs, xsds − αA ak bj
fs, xsds.
k1
0
k1
k1
j1
0
3.18
Also, letting t ψηj in 3.2, we obtain
φτk n
n
m
α bj x ψ ηj αA bj ak
fs, xsds
j1
j1
k1
0
⎛
⎞
ψηj n
n
α bj ⎝1 − αA bj ⎠
fs, xsds.
j1
j1
3.19
0
And from 3.19 from 3.18, we obtain
m
k1
n
ak x φτk α bj x ψ ηj .
3.20
j1
This complete the proof of the equivalence between the nonlocal problem 1.1-1.2 and the
integral equation 3.2.
This implies that there exists at least one absolutely continuous solution x ∈ AC0, 1
of the nonlocal problem 1.1-1.2.
8
Abstract and Applied Analysis
4. Nonlocal Integral Condition
Let x ∈ 0, 1 be a solution of the nonlocal problem 1.1-1.2. Let ak tk − tk−1 , τk ∈
tk−1 , tk ⊂ 0, 1. Also, let bj tj − tj−1 , ηj ∈ tj−1 , tj ⊂ 0, 1. Then the nonlocal condition
1.2 will be
m
n
tj − tj−1 x ψ ηj .
tk − tk−1 x φτk α
4.1
j1
k1
From the continuity of the solution x of the nonlocal condition 1.2 we obtain
m
n
tj − tj−1 x ψ ηj .
tk − tk−1 x φτk lim α
lim
m→∞
n→∞
k1
4.2
j1
That is, the nonlocal condition 1.2 is transformed to the integral condition
1
x φs ds α
0
1
x ψs ds,
4.3
0
and the solution of the integral equation 3.2 will be
xt A
∗
1 φs
t
0
fθ, xθdθds − α
0
fθ, xθdθds
0
fs, xsds,
1 ψs
0
4.4
−1
∗
A α − 1 .
0
Now, we have the following theorem.
Theorem 4.1. Let assumptions (i)–(iv) of Theorem 3.2 be satisfied. Then the nonlocal problem with
the integral condition
dxt
ft, xt ∈ Ft, xt,
for a.e. t ∈ 0, 1,
dt
1
1
x φs ds α x ψs ds
0
4.5
0
has at least one solution x ∈ AC0, 1 represented by 4.4.
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