Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2013, Article ID 823961, 9 pages
http://dx.doi.org/10.1155/2013/823961
Research Article
On a Multipoint Boundary Value Problem for a Fractional Order
Differential Inclusion on an Infinite Interval
Nemat Nyamoradi,1 Dumitru Baleanu,2,3,4 and Ravi P. Agarwal5,6
1
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran
Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, 06530 Ankara, Turkey
3
Institute of Space Sciences, P.O. BOX MG-23, 76900 Magurele, Bucharest, Romania
4
Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204,
Jeddah 21589, Saudi Arabia
5
Department of Mathematics, Texas A & M University-Kingsville, 700 University Boulevard, Kingsville, USA
6
Department of Mathematics, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
2
Correspondence should be addressed to Dumitru Baleanu; dumitru@cankaya.edu.tr
Received 12 January 2013; Accepted 22 January 2013
Academic Editor: José Tenreiro Machado
Copyright © 2013 Nemat Nyamoradi 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 investigate the existence of solutions for the following multipoint boundary value problem of a fractional order differential
𝑚−2
inclusion 𝐷0𝛼+ 𝑢(𝑡) + 𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡)) ∋ 0, 0 < 𝑡 < +∞, 𝑢(0) = 𝑢󸀠 (0) = 0, 𝐷𝛼−1 𝑢(+∞) − ∑𝑖=1 𝛽𝑖 𝑢(𝜉𝑖 ) = 0, where 𝐷0𝛼+ is the standard
𝑚−2
Riemann-Liouville fractional derivative, 2 < 𝛼 < 3, 0 < 𝜉1 < 𝜉2 < ⋅ ⋅ ⋅ < 𝜉𝑚−2 < +∞, satisfies 0 < ∑𝑖=1 𝛽𝑖 𝜉𝑖𝛼−1 < Γ(𝛼), and 𝐹 :
[0, +∞) × R × R → P(R) is a set-valued map. Several results are obtained by using suitable fixed point theorems when the right
hand side has convex or nonconvex values.
1. Introduction
In this paper, we will consider the existence of solutions
for the following multipoint boundary value problem of a
fractional order differential inclusion
𝐷0𝛼+ 𝑢 (𝑡) + 𝐹 (𝑡, 𝑢 (𝑡) , 𝑢󸀠 (𝑡)) ∋ 0,
𝑢 (0) = 𝑢󸀠 (0) = 0,
0 < 𝑡 < +∞,
𝑚−2
𝐷𝛼−1 𝑢 (+∞) − ∑ 𝛽𝑖 𝑢 (𝜉𝑖 ) = 0,
(1)
𝑖=1
𝐷0𝛼+
is the standard Riemann-Liouville fractional
where
derivative, 2 < 𝛼 < 3, 0 < 𝜉1 < 𝜉2 < ⋅ ⋅ ⋅ < 𝜉𝑚−2 < +∞,
𝛼−1
satisfies 0 < ∑𝑚−2
< Γ(𝛼), and 𝐹 : [0, +∞) × R × R →
𝑖=1 𝛽𝑖 𝜉𝑖
P(R) is a set-valued map.
The present paper is motivated by a recent paper of Liang
and Zhang [1], where it is considered problem (1) with 𝐹
single valued, and several existence results are provided.
Fractional differential equations have been of great interest recently. This is because of both the intensive development
of the theory of fractional calculus itself and the applications
of such constructions in various scientific fields such as
physics, mechanics, chemistry, and engineering. For details,
see [2–4] and the references therein.
The existence of solutions of initial value problems for
fractional order differential equations has been studied in
the literature [5–17] and the references therein. The study
of fractional differential inclusions was initiated by El-Sayed
and Ibrahim [18]. Also, recently, several qualitative results for
fractional differential inclusions were obtained in [19–23] and
the references therein.
The aim here is to establish existence results for problem
(1) when the right hand side is convex as well as nonconvex
valued. In the first result (Theorem 21), we consider the case
when the right hand side has convex values and prove an
existence result via nonlinear alternative for Kakutani maps.
In the second result (Theorem 25), we will use the fixed
point theorem for contraction multivalued maps according
to Covitz and Nadler. The paper is organized as follows.
2
Advances in Mathematical Physics
In Section 2 we recall some preliminary facts that we need
in the sequel, and in Section 3 we prove our main results.
Finally, in Section 4, an example is given to demonstrate the
application of one of our main results.
that is, for every sequence {𝑥𝑛 }𝑛∈N ⊂ 𝑋 and {𝑦𝑛 }𝑛∈N ⊂ 𝑌, if
when 𝑛 → ∞, 𝑥𝑛 → 𝑥∗ , 𝑦𝑛 → 𝑦∗ , and 𝑦𝑛 ∈ 𝑇(𝑥𝑛 ), then
𝑦∗ ∈ 𝑇(𝑥∗ ). Conversely, if 𝑇 is completely continuous and has
a closed graph, then it is upper semicontinuous.
2. Preliminaries
For the convenience of the reader, we present here the
following nonlinear alternative of the Leray-Schauder type
and its consequences.
In this section, we present some notations and preliminary
lemmas that will be used in the proof of the main result.
Let (𝑋, 𝑑) be a metric space with the corresponding norm
‖ ⋅ ‖ and let 𝐼 = [0, +∞). We denote by L(𝐼) the 𝜎-algebra
of all Lebesgue measurable subsets of 𝐼, by B(𝑋) the family
of all nonempty subsets of 𝑋, and by P(𝑋) the family of all
Borel subsets of 𝑋. If 𝐴 ⊂ 𝐼 then 𝜒𝐴 : 𝐼 → {0, 1} denotes the
characteristic function of 𝐴. For any subset 𝐴 ⊂ 𝑋, we denote
by 𝐴 the closure of 𝐴.
Recall that the Pompeiu-Hausdorff distance of the closed
subsets 𝐴, 𝐵 ⊂ 𝑋 is defined by the following:
𝑑𝐻 (𝐴, 𝐵) = max {𝑑∗ (𝐴, 𝐵) , 𝑑∗ (𝐵, 𝐴)} ,
𝑑∗ (𝐴, 𝐵) = sup {𝑑 (𝑎, 𝐵) , 𝑎 ∈ 𝐴} ,
(2)
where 𝑑(𝑥, 𝐵) = inf 𝑦∈𝐵 𝑑(𝑥, 𝑦). Define
(i) 𝑇 has a fixed point in 𝑈 or
(ii) there is a 𝑢 ∈ 𝜕𝑈 and 𝜆 ∈ (0, 1) with 𝑢 ∈ 𝜆𝑇(𝑢).
Definition 3. The multifunction 𝑇 : 𝑋 → P(𝑋) is said to
be lower semicontinuous if for any closed subset 𝐶 ⊂ 𝑋, the
subset {𝑠 ∈ 𝑋 : 𝑇(𝑠) ⊂ 𝐶} is closed.
If 𝐹 : [0, +∞) × R × R → P(R) is a set-valued map with
compact values and 𝑥 ∈ 𝐶([0, +∞), R), we define
P (𝑋) = {𝑌 ⊂ 𝑋 : 𝑌 ≠ 0} ,
P𝑏 (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 bounded} ,
P𝑐𝑙 (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 closed} ,
Theorem 2 (nonlinear alternative for Kakutani maps [25]).
Let 𝑋 be a Banach space, 𝐶 a closed convex subset of 𝑋,
𝑈 an open subset of 𝐶, and 0 ∈ 𝑈. Suppose that 𝑇 :
𝑈 → P𝑐𝑙,𝑐V (𝐶) is an upper semicontinuous compact map;
here P𝑐𝑙,𝑐V (𝐶) denotes the family of nonempty, compact convex
subsets of 𝐶. Then, either
(3)
P𝑐𝑝 (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 compact} ,
P𝑐V (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 convex} .
Also, we denote by 𝐶(𝐼, 𝑋) the Banach space of all continuous functions 𝑥 : [0, +∞) → 𝑋 endowed with the norm
|𝑥|𝑐 = sup𝑡∈[0,+∞) |𝑥(𝑡)| and by 𝐿1 ([0, +∞), 𝑋) the Banach
space of all (Bochner) integrable functions 𝑥 : [0, +∞) → 𝑋
endowed with the norm |𝑥|1 = ∫[0,+∞) |𝑥(𝑡)|𝑑𝑡.
Let (𝑋, 𝑑1 ) and (𝑌, 𝑑2 ) be two metric spaces. If 𝑇 : 𝑋 →
P(𝑋) is a set-valued map, then a point 𝑥 ∈ 𝑋 is called a
fixed point for 𝑇 if 𝑥 ∈ 𝑇(𝑥). 𝑇 is said to be bounded on
bounded sets if 𝑇(𝐵) := ∪𝑥∈𝐵 𝑇(𝑥) is a bounded subset of
𝑋 for all bounded sets 𝐵 in 𝑋. 𝑇 is said to be compact if
𝑇(𝐵) is relatively compact for any bounded sets 𝐵 in 𝑋. 𝑇
is said to be totally compact if 𝑇(𝑋) is a compact subset of
𝑋. 𝑇 is said to be upper semicontinuous if for any open set
𝐷 ⊂ 𝑋, the set {𝑥 ∈ 𝑋 : 𝑇(𝑥) ⊂ 𝐷} is open in 𝑋. 𝑇 is called
completely continuous if it is upper semicontinuous and, for
every bounded subset 𝐴 ⊂ 𝑋, 𝑇(𝐴) is relatively compact. It is
well known that a compact set-valued map 𝑇 with nonempty
compact values is upper semicontinuous if and only if 𝑇 has
a closed graph.
We define the graph of 𝑇 to be the set Gr(𝑇) = {(𝑥, 𝑦) ∈
𝑋 × 𝑌, 𝑦 = 𝑇(𝑥)} and recall a useful result regarding
connection between closed graphs and upper semicontinuity.
Lemma 1 (see [24, Proposition 1.2]). If 𝑇 : 𝑋 → P𝑐𝑙 (𝑌) is
upper semicontinuous, then Gr(𝑇) is a closed subset of 𝑋 × 𝑌,
𝑆𝐹 (𝑥) := {𝑓 ∈ 𝐿1 ([0, +∞) , R) : 𝑓 (𝑡)
∈ 𝐹 (𝑡, 𝑥 (𝑡) , 𝑥󸀠 (𝑡)) a.e. [0, +∞)} .
(4)
Then, 𝐹 is of a lower semicontinuous type if 𝑆𝐹 (⋅) is a lower
semicontinuous with closed and decomposable values.
Theorem 4 (see [26]). Let 𝑆 be a separable metric space and
𝐺 : 𝑆 → P(𝐿1 ([0, +∞), R)) be a lower semicontinuous setvalued map with closed decomposable values. Then 𝐺 has a
continuous selection (i.e., there exists a continuous mapping
𝑔 : 𝑆 → 𝐿1 ([0, +∞), R) such that 𝑔(𝑠) ∈ 𝐺(𝑠) for all 𝑠 ∈ 𝑆).
Definition 5. Consider the following.
(i) A set-valued map 𝐺 : [0, +∞) → P(R) with
nonempty compact convex values is said to be measurable if for any 𝑥 ∈ R the function 𝑡 → 𝑑(𝑥, 𝐺(𝑡))
is measurable.
(ii) A set-valued map 𝐹 : [0, +∞) × R × R → P(R)
is said to be Carathéodory if 𝑡 → 𝐹(𝑡, 𝑥, 𝑦) is
measurable for all 𝑥, 𝑦 ∈ R and (𝑥, 𝑦) → 𝐹(𝑡, 𝑥, 𝑦) is
upper semicontinuous for almost all 𝑡 ∈ [0, +∞).
(iii) 𝐹 is said to be 𝐿1 -Carathéodory if for any 𝑙 > 0 there
exists ℎ𝑙 ∈ 𝐿1 ([0, +∞), R) such that sup{|V| : V ∈
𝐹(𝑡, 𝑥, 𝑦)} ≤ ℎ𝑙 (𝑡) a.e. [0, +∞); ∀𝑥, 𝑦 ∈ R.
Finally, the following results are easily deduced from the
theoretical limit set properties.
Advances in Mathematical Physics
3
Lemma 6 (see [27, Lemma 1.1.9]). Let {𝐾𝑛 }𝑛∈N ⊂ 𝐾 ⊂ 𝑋 be a
sequence of subsets where 𝐾 is a compact subset of a separable
Banach space 𝑋. Then,
co (lim sup𝐾𝑛 ) = ⋂ co ( ⋃ 𝐾𝑛 ) ,
𝑛→∞
𝑁>0
(5)
𝑛≥𝑁
Definition 13 (see [31]). The Riemann-Liouville fractional
derivative of order 𝛼 > 0, 𝑛 − 1 < 𝛼 < 𝑛, 𝑛 ∈ N is defined
as
𝐷0𝛼+ 𝑓 (𝑡) =
𝑑 𝑛 𝑡
1
( ) ∫ (𝑡 − 𝑠)𝑛−𝛼−1 𝑓 (𝑠) 𝑑𝑠,
Γ (𝑛 − 𝛼) 𝑑𝑡
0
(9)
where co(𝐴) refers to the closure of the convex hull of 𝐴.
where the function 𝑓(𝑡) has absolutely continuous derivatives
up to order (𝑛 − 1).
Lemma 7 (see [27, Lemma 1.4.13]). Let 𝑋 and 𝑌 be two
metric spaces. If 𝐺 : 𝑋 → P𝑐𝑝 (𝑌) is an upper semicontinuous,
then, for each 𝑥0 ∈ 𝑋,
Lemma 14 (see [31]). The equality 𝐷0+ 𝐼0+ 𝑓(𝑡) = 𝑓(𝑡), 𝛾 > 0
holds for 𝑓 ∈ 𝐿1 (0, 1).
lim sup 𝐺 (𝑥) = 𝐺 (𝑥0 ) .
Lemma 15 (see [31]). Let 𝛼 > 0 and 𝑢 ∈ 𝐶(0, 1) ∩ 𝐿1 (0, 1).
Then, the differential equation
𝑥 → 𝑥0
(6)
Definition 8. Let 𝑋 be a Banach space. A sequence {𝑥𝑛 }𝑛∈N ⊂
𝐿1 ([𝑎, 𝑏], 𝑋) is said to be semicompact if
(a) it is integrably bounded; that is, there exists 𝑞 ∈
𝐿1 ([𝑎, 𝑏], R+ ) such that
󵄨
󵄨󵄨
󵄨󵄨𝑥𝑛 (𝑡)󵄨󵄨󵄨𝐸 ≤ 𝑞 (𝑡) , for a.e. 𝑡 ∈ [𝑎, 𝑏] and every 𝑛 ∈ N, (7)
(b) the image sequence {𝑥𝑛 (𝑡)}𝑛∈N is relatively compact in
𝐸 for a.e. 𝑡 ∈ [𝑎, 𝑏].
The following important result follows from the DunfordPettis theorem (see [28, Proposition 4.2.1]).
Lemma 9. Every semicompact sequence 𝐿1 ([𝑎, 𝑏], 𝑋) is
weakly compact in 𝐿1 ([𝑎, 𝑏], 𝑋).
When the nonlinearity takes convex values, Mazur’s
Lemma, 1933, may be useful.
Lemma 10 (see [29, Theorem 21.4]). Let 𝐸 be a normed space
and {𝑥𝑘 }𝑘∈N ⊂ 𝐸 a sequence weakly converging to a limit 𝑥 ∈
𝐸. Then, there exists a sequence of convex combinations 𝑦𝑚 =
𝑚
∑𝑚
𝑘=1 𝛼𝑚𝑘 𝑥𝑘 with 𝛼𝑚𝑘 > 0 for 𝑘 = 1, 2, . . . , 𝑚 and ∑𝑘=1 𝛼𝑚𝑘 = 1
which converges strongly to 𝑥.
Lemma 11 (see [30]). Let 𝑋 be defined as before and 𝑀 ⊂ 𝑋.
Then 𝑀 is relatively compact in 𝑋 if the following conditions
hold:
(a) 𝑀 is uniformly bounded in 𝑋;
(b) the functions from 𝑀 are equicontinuous on any
compact interval of [0, +∞);
(c) the functions from 𝑀 are equiconvergent; that is, for
any given 𝜖 > 0, there exists a 𝑇 = 𝑇(𝜖) > 0 such that
|𝑓(𝑡) − 𝑓(+∞)| < 𝜖, for any 𝑡 > 𝑇, 𝑓 ∈ 𝑀.
Definition 12 (see [6]). The Riemann-Liouville fractional
integral operator of order 𝛼 > 0, of function 𝑓 ∈ 𝐿1 (R+ ),
is defined as
𝐼0𝛼+ 𝑓 (𝑡) =
𝑡
1
∫ (𝑡 − 𝑠)𝛼−1 𝑓 (𝑠) 𝑑𝑠,
Γ (𝛼) 0
where Γ(⋅) is the Euler gamma function.
(8)
𝛾
𝛾
𝐷0𝛼+ 𝑢 (𝑡) = 0
(10)
has a unique solution 𝑢(𝑡) = 𝑐1 𝑡𝛼−1 +𝑐2 𝑡𝛼−2 +⋅ ⋅ ⋅+𝑐𝑛 𝑡𝛼−𝑛 , 𝑐𝑖 ∈ R,
𝑖 = 1, . . . , 𝑛, where 𝑛 − 1 < 𝛼 < 𝑛.
Lemma 16 (see [31]). Let 𝛼 > 0. Then, the following equality
holds for 𝑢 ∈ 𝐿1 (0, 1), 𝐷0𝛼+ 𝑢 ∈ 𝐿1 (0, 1):
𝐼0𝛼+ 𝐷0𝛼+ 𝑢 (𝑡) = 𝑢 (𝑡) + 𝑐1 𝑡𝛼−1 + 𝑐2 𝑡𝛼−2 + ⋅ ⋅ ⋅ + 𝑐𝑛 𝑡𝛼−𝑛 ,
(11)
𝑐𝑖 ∈ R, 𝑖 = 1, . . . , 𝑛, where 𝑛 − 1 < 𝛼 ≤ 𝑛.
By 𝐴𝐶1 ([0, +∞), R) we denote the space of continuous
real-valued functions whose first derivative exists and it is
absolutely continuous on [0, +∞). In this paper, we will use
the following space 𝐸 to the study (1) which is denoted by
|𝑢 (𝑡)|
,
𝛼−1
1
0≤𝑡<+∞ + 𝑡
𝐸 = {𝑢 ∈ 𝐴𝐶1 ([0, +∞) , R) : sup
(12)
󵄨󵄨 󸀠 󵄨󵄨
󵄨󵄨𝑢 (𝑡)󵄨󵄨
sup 󵄨 𝛼−1󵄨 < +∞} .
0≤𝑡<+∞ 1 + 𝑡
From [32], we know that 𝐸 is a Banach space equipped
with the norm
󵄨󵄨 󸀠 󵄨󵄨
󵄨󵄨𝑢 (𝑡)󵄨󵄨
|𝑢 (𝑡)|
(13)
, sup 󵄨 𝛼−1󵄨 } .
‖𝑢‖ = max { sup
𝛼−1
0≤𝑡<+∞ 1 + 𝑡
0≤𝑡<+∞ 1 + 𝑡
In what follows, 𝐼 = [0, +∞), 𝛼 ∈ (2, 3), and Δ =
𝛼−1
. Next, we need the following technical result
∑𝑚−2
𝑖=1 𝛽𝑖 𝜉𝑖
proved in [1].
Lemma 17 (see [1]). For any ℎ ∈ 𝐿1 ([0, +∞), R), the problem
𝐷0𝛼+ 𝑢 (𝑡) + ℎ (𝑡) = 0,
0 < 𝑡 < ∞, 2 < 𝛼 < 3,
𝑚−2
𝑢 (0) = 𝑢󸀠 (0) = 0, 𝐷𝛼−1 𝑢 (+∞) = ∑ 𝛽𝑖 𝑢 (𝜉𝑖 ) ,
(14)
𝑖=1
has a unique solution 𝑢(𝑡) that
𝑢 (𝑡) = ∫
+∞
0
𝐺 (𝑡, 𝑠) ℎ (𝑠) 𝑑𝑠,
(15)
4
Advances in Mathematical Physics
where
𝐺 (𝑡, 𝑠) = 𝐺1 (𝑡, 𝑠) + 𝐺2 (𝑡, 𝑠) ,
𝐺1 (𝑡, 𝑠) =
𝑡𝛼−1 − (𝑡 − 𝑠)𝛼−1 ,
1
{ 𝛼−1
Γ (𝛼) 𝑡 ,
𝐺2 (𝑡, 𝑠) =
(16)
0 ≤ 𝑠 ≤ 𝑡 < +∞,
(17)
0 ≤ 𝑡 ≤ 𝑠 < +∞,
𝛼−1
∑𝑚−2
𝑖=1 𝛽𝑖 𝑡
𝐺 (𝜉 , 𝑠) .
Γ (𝛼) − Δ 1 𝑖
(18)
Note that 𝐺(𝑡, 𝑠) > 0, ∀𝑡, 𝑠 ∈ [0, +∞), (e.g., Lemma 3.2 in
[1]) and from the definition of 𝐺1 (𝑡, 𝑠), we have the following
(e.g., Remark 3.1 in [1]):
𝐺1 (𝑡, 𝑠)
1
,
≤
1 + 𝑡𝛼−1 Γ (𝛼)
𝐺 (𝑡, 𝑠)
≤ 𝐿1
1 + 𝑡𝛼−1
(19)
for (𝑡, 𝑠) ∈ [0, +∞) × [0, +∞) ,
∑𝑚−2 𝛽 𝜉𝛼−1
1
(1 + 𝑖=1 𝑖 𝑚−2 ) .
Γ (𝛼)
(Γ (𝛼) − Δ)
(20)
for (𝑡, 𝑠) ∈ [0, +∞) × [0, +∞) ,
(21)
where
∑𝑚−2 𝛽 𝜉𝛼−1
2 (𝛼 − 1)
𝐿2 =
(1 + 𝑖=1 𝑖 𝑚−2 ) .
Γ (𝛼)
(Γ (𝛼) − Δ)
𝑀
+∞
max {𝐿 1 , 𝐿 2 } 𝜓 (𝑀) ∫0
𝜑 (𝑠) 𝑑𝑠
> 1.
(25)
Then, problem (1) has at least one solution.
𝑢 (𝑡) ∈ ∫
+∞
𝐺 (𝑡, 𝑠) 𝐹 (𝑠, 𝑢 (𝑠) , 𝑢󸀠 (𝑠)) 𝑑𝑠,
𝑡 ∈ 𝐼,
(26)
where 𝐺(𝑡, 𝑠) is defined by (16) and (17). Consider the setvalued map, 𝑇 : 𝐸 → P(𝑋) is defined by
𝑇 (𝑢) := {V ∈ 𝑋; V (𝑡) = ∫
+∞
0
𝐺 (𝑡, 𝑠) 𝑓 (𝑠) 𝑑𝑠, 𝑓 ∈ 𝑆𝐹 (𝑢)} .
(27)
(22)
We show that 𝑇 satisfies the hypotheses of Theorem 2.
Lemma 18. The function 𝐺(𝑡, 𝑠) defined by (16) satisfies
𝛼−1
∑𝑚−2
𝐺 (𝑡, 𝑠)
𝑖=1 𝛽𝑖 𝜉𝑖
.
lim
=
𝑡 → +∞ 1 + 𝑡𝛼−1
Γ (𝛼) (Γ (𝛼) − Δ)
(H2) There exists a constant 𝑀 > 0 such that
0
Also, one can get
𝜕𝐺 (𝑡, 𝑠) /𝜕𝑡
≤ 𝐿2
1 + 𝑡𝛼−1
(H1) There exists a continuous nondecreasing function 𝜓 :
[0, +∞) → (0, +∞) and 𝜑 ∈ 𝐿1 ([0, +∞), R+ )
such that ‖𝐹(𝑡, 𝑥, 𝑦)‖P := sup{|V(𝑡)/(1 + 𝑡𝛼−1 )| :
V ∈ 𝐹(𝑡, 𝑥, 𝑦)} ≤ 𝜑(𝑡)𝜓(‖𝑥‖), for a.e. 𝑡 ∈
𝐼 and each 𝑥, 𝑦 ∈ R.
Proof. Let 𝑋 = 𝐸 and consider 𝑀 > 0 as in (25). It is obvious
that the existence of solutions to problem (1) is reduced to the
existence of the solutions of the integral inclusion
where
𝐿1 =
Theorem 21. The Carathéodory multivalued map 𝐹 : 𝐼 ×
R × R → P(R) has nonempty, compact, convex values and
satisfies the following.
(23)
By calculation, it is easy to prove that Lemma 18 holds. So,
we omit its proof here.
3. Main Results
Now we are able to present the existence results for problem
(1).
Claim 1. We show that 𝑇(𝑢) ⊂ 𝑋 is convex for any 𝑢 ∈ 𝑋. If
V1 , V2 ∈ 𝑇(𝑢), then, there exist 𝑓1 , 𝑓2 ∈ 𝑆𝐹 (𝑢) such that for any
𝑡 ∈ 𝐼 one has
V𝑖 (𝑡) = ∫
0
Lemma 19 (see [32]). Let 𝑉 = {𝑢 ∈ 𝐸 | ‖𝑢‖ < 𝑙}(𝑙 >
0), 𝑉1 = {𝑢(𝑡)/(1 + 𝑡𝛼−1 ) : 𝑢 ∈ 𝑉}. If 𝑉1 is equicontinuous on
any compact interval of [0, +∞) and equiconvergent at infinity,
then 𝑉 is relatively compact on 𝐸.
Definition 20. 𝑉1 is called equiconvergent at infinity if and
only if for all 𝜖 > 0, there exists V(𝜖) > 0 such that for all
𝑢 ∈ 𝑉1 , 𝑡1 , 𝑡2 ≥ V, it holds
󵄨󵄨 𝑢 (𝑡 )
𝑢 (𝑡2 ) 󵄨󵄨󵄨󵄨
󵄨󵄨
1
󵄨
󵄨󵄨
(24)
−
󵄨󵄨 1 + 𝑡𝛼−1 1 + 𝑡𝛼−1 󵄨󵄨󵄨 < 𝜖.
󵄨
1
2 󵄨
𝐺 (𝑡, 𝑠) 𝑓𝑖 (𝑠) 𝑑𝑠,
𝑖 = 1, 2.
(28)
Let 0 ≤ 𝜆 ≤ 1. Then, for any 𝑡 ∈ 𝐼, we have
(𝜆V1 + (1 − 𝜆) V2 ) (𝑡)
=∫
3.1. The Upper Semicontinuous Case. To obtain the complete
continuity of existence solutions operator, the following
lemma is still needed.
+∞
+∞
0
𝐺 (𝑡, 𝑠) [𝜆𝑓1 (𝑠) + (1 − 𝜆) 𝑓2 (𝑠)] 𝑑𝑠.
(29)
The values of 𝐹 are convex; thus, 𝑆𝐹 (𝑢) is a convex set and
hence 𝜆V1 + (1 − 𝜆)V2 ∈ 𝑇(𝑢).
Claim 2. We show that 𝑇 is bounded on bounded sets of 𝑋.
Let 𝐵 be any bounded subset of 𝑋. Then, there exists 𝑚 > 0
such that ‖𝑢‖ ≤ 𝑚 for all 𝑢 ∈ 𝐵. If V ∈ 𝑇(𝑢), then there exists
+∞
𝑓 ∈ 𝑆𝐹 (𝑢) such that V(𝑡) = ∫0 𝐺(𝑡, 𝑠)𝑓(𝑠)𝑑𝑠. One may write
the following for any 𝑡 ∈ 𝐼:
󵄨󵄨 V (𝑡) 󵄨󵄨
󵄨
+∞ 󵄨󵄨
󵄨
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝐺 (𝑡, 𝑠) 󵄨󵄨󵄨 󵄨󵄨
𝑓 (𝑠)󵄨󵄨󵄨 𝑑𝑠
≤
∫
󵄨󵄨
󵄨
󵄨
󵄨
󵄨
𝛼−1 󵄨󵄨󵄨 󵄨󵄨
󵄨󵄨 1 + 𝑡𝛼−1 󵄨󵄨
󵄨
1
+
𝑡
0
󵄨
󵄨
≤ 𝐿1 ∫
+∞
0
󵄨 󵄨
𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠 ≤ 𝐿 1 󵄨󵄨󵄨𝜑󵄨󵄨󵄨1 𝜓 (𝑚) .
(30)
Advances in Mathematical Physics
5
On the other hand,
󵄨󵄨 󸀠
󵄨
+∞
𝜕𝐺 (𝑡, 𝑠) /𝜕𝑡
󵄨󵄨 V (𝑡) 󵄨󵄨󵄨
󵄨󵄨
󵄨
𝑓 (𝑠) 𝑑𝑠
󵄨󵄨 1 + 𝑡𝛼−1 󵄨󵄨󵄨 ≤ ∫0
1 + 𝑡𝛼−1
󵄨
󵄨
≤ 𝐿2 ∫
+∞
0
𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠
(31)
󵄨 󵄨
≤ 𝐿 2 󵄨󵄨󵄨𝜑󵄨󵄨󵄨1 𝜓 (𝑚) ,
On the other hand, we get
+∞ 󵄨󵄨 𝐺 (𝑡 , 𝑠)
𝐺 (𝑡 , 𝑠) 󵄨󵄨󵄨
󵄨
∫ 󵄨󵄨󵄨󵄨 1 1𝛼−1 − 1 2𝛼−1 󵄨󵄨󵄨󵄨 𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠
1 + 𝑡1 󵄨󵄨
󵄨󵄨 1 + 𝑡1
0
𝑡1
𝑡2
+∞ 󵄨󵄨 𝐺 (𝑡 , 𝑠)
𝐺 (𝑡 , 𝑠) 󵄨󵄨󵄨
󵄨
≤ (∫ + ∫ + ∫ ) 󵄨󵄨󵄨󵄨 1 1𝛼−1 − 1 2𝛼−1 󵄨󵄨󵄨󵄨
1 + 𝑡1 󵄨󵄨
󵄨󵄨 1 + 𝑡1
𝑡2
0
𝑡1
× 𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠
𝑡1
≤ 𝜓 (𝑚) ∫
(𝑡2𝛼−1 − 𝑡1𝛼−1 ) + ((𝑡2 − 𝑠)
1+
0
𝛼−1
− (𝑡1 − 𝑠)
𝛼−1
)
𝑡1𝛼−1
× 𝜑 (𝑠) 𝑑𝑠
and therefore
󵄨󵄨 V (𝑡) 󵄨󵄨 󵄨󵄨󵄨 V󸀠 (𝑡) 󵄨󵄨󵄨
󵄨
󵄨󵄨 󵄨󵄨
󵄨󵄨}
‖V‖ = max {󵄨󵄨󵄨
󵄨,
󵄨󵄨 1 + 𝑡𝛼−1 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 1 + 𝑡𝛼−1 󵄨󵄨󵄨󵄨
𝑡∈𝐼
+ 𝜓 (𝑚) ∫
𝑡2
𝛼−1
(𝑡2𝛼−1 − 𝑡1𝛼−1 ) + (𝑡2 − 𝑠)
1 + 𝑡1𝛼−1
𝑡1
(32)
󵄨 󵄨
≤ max {𝐿 1 , 𝐿 2 } 󵄨󵄨󵄨𝜑󵄨󵄨󵄨1 𝜓 (𝑚) ,
+ 𝜓 (𝑚) ∫
+∞
(𝑡2𝛼−1 − 𝑡1𝛼−1 )
1 + 𝑡1𝛼−1
𝑡2
𝜑 (𝑠) 𝑑𝑠
𝜑 (𝑠) 𝑑𝑠
󳨀→ 0 uniformly as 𝑡1 󳨀→ 𝑡2 .
(34)
for all V ∈ 𝑇(𝑢); that is, 𝑇(𝐵) is bounded.
Claim 3. We show that 𝑇 maps bounded the sets into equicontinuous sets. Let 𝐵 be any bounded subset of 𝑋 as before
and V ∈ 𝑇(𝑢) for some 𝑢 ∈ 𝐵. Then, there exists 𝑓 ∈ 𝑆𝐹 (𝑢)
+∞
such that V(𝑡) = ∫0 𝐺(𝑡, 𝑠)𝑓(𝑠)𝑑𝑠. So, for any 𝑇0 ∈ (0, +∞)
and 𝑡1 , 𝑡2 ∈ [0, 𝑇0 ], without loss of generality, we may assume
that 𝑡2 > 𝑡1 and one can get the following:
󵄨󵄨 V (𝑡 )
V (𝑡2 ) 󵄨󵄨󵄨󵄨
󵄨󵄨
1
󵄨
󵄨󵄨
−
󵄨󵄨 1 + 𝑡𝛼−1 1 + 𝑡𝛼−1 󵄨󵄨󵄨
󵄨
1
2 󵄨
+∞ 󵄨󵄨 𝐺 (𝑡 , 𝑠)
𝐺 (𝑡 , 𝑠) 󵄨󵄨󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨󵄨 1 1𝛼−1 − 1 2𝛼−1 󵄨󵄨󵄨󵄨 𝑓 (𝑠) 𝑑𝑠
1 + 𝑡2 󵄨󵄨
󵄨󵄨 1 + 𝑡1
0
+∞
0
󵄨󵄨 𝐺 (𝑡 , 𝑠) 𝐺 (𝑡 , 𝑠) 󵄨󵄨
󵄨
󵄨
≤ ∫ 󵄨󵄨󵄨󵄨 1 1𝛼−1 − 1 2𝛼−1 󵄨󵄨󵄨󵄨 𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠
1 + 𝑡1 󵄨󵄨
󵄨󵄨 1 + 𝑡1
0
+∞ 󵄨󵄨 𝐺 (𝑡 , 𝑠)
𝐺 (𝑡 , 𝑠) 󵄨󵄨󵄨
󵄨
+ ∫ 󵄨󵄨󵄨󵄨 1 2𝛼−1 − 1 2𝛼−1 󵄨󵄨󵄨󵄨 𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠
1 + 𝑡2 󵄨󵄨
󵄨󵄨 1 + 𝑡1
0
󵄨󵄨 𝑡𝛼−1
𝑡2𝛼−1 󵄨󵄨󵄨󵄨
󵄨󵄨 1
󵄨󵄨
󵄨󵄨
+
−
Γ (Γ (𝛼) − Δ 󵄨󵄨󵄨 1 + 𝑡1𝛼−1 1 + 𝑡2𝛼−1 󵄨󵄨󵄨
∑𝑚−2
𝑖=1
×∫
+∞
0
𝛼−1
𝛽𝑖 𝜉𝑚−2
𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠.
uniformly as 𝑡1 󳨀→ 𝑡2 .
From (34) and (35), we have
󵄨󵄨󵄨 V (𝑡1 )
V (𝑡2 ) 󵄨󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 󳨀→ 0 uniformly as 𝑡1 󳨀→ 𝑡2 . (36)
−
󵄨󵄨
󵄨󵄨 1 + 𝑡1𝛼−1 1 + 𝑡2𝛼−1 󵄨󵄨󵄨
Similar to (36), one can get
󵄨󵄨 V󸀠 (𝑡 )
V󸀠 (𝑡2 ) 󵄨󵄨󵄨󵄨
󵄨󵄨
1
󵄨
󵄨󵄨
−
󵄨󵄨 1 + 𝑡𝛼−1 1 + 𝑡𝛼−1 󵄨󵄨󵄨 󳨀→ 0 uniformly as 𝑡1 󳨀→ 𝑡2 . (37)
󵄨
1
2 󵄨
Therefore, 𝑇(𝐵) is an equicontinuous set in 𝑋.
∫
𝐺1 (𝜉𝑖 , 𝑠) 𝑓 (𝑠) 𝑑𝑠
+∞
(35)
Claim 4. We show that 𝑇 is equiconvergent at ∞. Let V ∈ 𝑇(𝑢)
for some 𝑢 ∈ 𝐵. Then, there exists 𝑓 ∈ 𝑆𝐹 (𝑢) such that V(𝑡) =
+∞
∫0 𝐺(𝑡, 𝑠)𝑓(𝑠)𝑑𝑠. So, we have the following:
∑𝑚−2 𝛽 󵄨󵄨󵄨 𝑡𝛼−1
𝑡𝛼−1 󵄨󵄨󵄨
+ 𝑖=1 𝑖 󵄨󵄨󵄨󵄨 1 𝛼−1 − 2 𝛼−1 󵄨󵄨󵄨󵄨
Γ (𝛼) − Δ 󵄨󵄨 1 + 𝑡1
1 + 𝑡2 󵄨󵄨
×∫
Similar to (34), we have
+∞ 󵄨󵄨 𝐺 (𝑡 , 𝑠)
𝐺 (𝑡 , 𝑠) 󵄨󵄨󵄨
󵄨
∫ 󵄨󵄨󵄨󵄨 1 2𝛼−1 − 1 2𝛼−1 󵄨󵄨󵄨󵄨 𝜑 (𝑠) 𝜓 (‖𝑢‖) 𝑑𝑠 󳨀→ 0
1 + 𝑡2 󵄨󵄨
󵄨󵄨 1 + 𝑡1
0
(33)
+∞
0
+∞
𝑓 (𝑠) 𝑑𝑠 ≤ 𝜓 (𝑚) ∫
0
𝜑 (𝑠) 𝑑𝑠 < +∞,
󵄨󵄨 V (𝑡) 󵄨󵄨
+∞
1
󵄨
󵄨󵄨
=
lim
lim 󵄨󵄨󵄨
𝐺 (𝑡, 𝑠) 𝑓 (𝑠) 𝑑𝑠
∫
󵄨
𝑡 → +∞ 󵄨󵄨 1 + 𝑡𝛼−1 󵄨󵄨󵄨
𝑡 → +∞ 1 + 𝑡𝛼−1 0
=
𝛼−1
+∞
∑𝑚−2
𝑖=1 𝛽𝑖 𝜉𝑖
∫ 𝑓 (𝑠) 𝑑𝑠
Γ (𝛼) (Γ (𝛼) − Δ) 0
−
∑𝑚−2
𝑖=1 𝛽𝑖
Γ (𝛼) (Γ (𝛼) − Δ)
𝜉𝑖
× ∫ (𝜉𝑖 − 𝑠)
0
𝛼−1
𝑓 (𝑠) 𝑑𝑠 < ∞,
(38)
6
Advances in Mathematical Physics
and, similarly, one has
󵄨󵄨 󸀠
󵄨
󵄨 V (𝑡) 󵄨󵄨󵄨
󵄨󵄨 < ∞.
lim 󵄨󵄨󵄨󵄨
𝑡 → +∞ 󵄨 1 + 𝑡𝛼−1 󵄨󵄨
󵄨
󵄨
(39)
Therefore, 𝑇(𝐵) is equiconvergent at infinity.
Therefore, with Lemma 11, Lemma 19 and Claims 2–4, we
conclude that 𝑇 is completely continuous.
Claim 5. 𝑇 is upper semicontinuous. To this end, it is sufficient
to show that 𝑇 has a closed graph. Let V𝑛 ∈ 𝑇(𝑢𝑛 ) such that
V𝑛 → V and 𝑢𝑛 → 𝑢, as 𝑛 → +∞. Then, there exists 𝑚 > 0
such that ‖𝑢𝑛 ‖ ≤ 𝑚. We will prove that V ∈ 𝑇(𝑢) means that
there exists 𝑓𝑛 ∈ 𝑆𝐹 (𝑢𝑛 ) such that, for a.e. 𝑡 ∈ 𝐼, we have
+∞
V𝑛 (𝑡) = ∫0 𝐺(𝑡, 𝑠)𝑓𝑛 (𝑠)𝑑𝑠. Then, we need to show that V ∈
𝑇(𝑢).
Condition (H1) implies that 𝑓𝑛 (𝑡) ∈ 𝜑(𝑡)𝜓(𝑚)𝐵1 (0).
Then, {𝑓𝑛 }𝑛∈N is integrably bounded in 𝐿1 (𝐼, R). Since 𝐹
has compact values, we deduce that {𝑓𝑛 }𝑛 is semicompact.
By Lemma 9, there exists a subsequence, still denoted as
{𝑓𝑛 }𝑛∈N , which converges weakly to some limit 𝑓 ∈ 𝐿1 (𝐼, R).
Moreover, the mapping Γ : 𝐿1 (𝐼, R) → 𝑋 = 𝐸 defined by
Γ (𝑔) (𝑡) = ∫
+∞
0
𝐺 (𝑡, 𝑠) 𝑔 (𝑠) 𝑑𝑠
(40)
is a continuous linear operator. Then, it remains continuous if
these spaces are endowed with their weak topologies [29, 33].
Moreover, for a.e. 𝑡 ∈ 𝐼, 𝑢𝑛 (𝑡) converges to 𝑢(𝑡). Then, we have
+∞
V (𝑡) = ∫
0
𝐺 (𝑡, 𝑠) 𝑓 (𝑠) 𝑑𝑠.
(41)
It remains to prove that 𝑓 ∈ 𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡)), a.e. 𝑡 ∈ 𝐼. Mazur’s
Lemma (see Lemma 10) yields the existence of 𝛼𝑖𝑛 ≥ 0, 𝑖 =
𝑛
𝑛, . . . , 𝑘(𝑛) such that ∑𝑘(𝑛)
𝑖=1 𝛼𝑖 = 1 and the sequence of convex
𝑘(𝑛) 𝑛
combinations 𝑔𝑛 (⋅) = ∑𝑖=1 𝛼𝑖 𝑓𝑖 (⋅) converges strongly to 𝑓 in
𝐿1 . Using Lemma 6, we obtain that
V (𝑡) ∈ ⋂ {𝑔𝑛 (𝑡)},
a.e. 𝑡 ∈ 𝐼
𝑛≥1
𝑛≥1
(42)
⊂ ⋂ co {⋃ 𝐹 (𝑡, 𝑢𝑘 (𝑡) , 𝑢𝑘󸀠 (𝑡))}
×∫
+∞
0
(44)
𝜑 (𝑠) 𝑑𝑠.
Consequently,
‖𝑢‖
+∞
max {𝐿 1 , 𝐿 2 } 𝜓 (‖𝑢‖) ∫0
𝜑 (𝑠) 𝑑𝑠
≤ 1.
(45)
In view of (H2), there exists 𝑀 such that ‖𝑢‖ ≠ 𝑀. Let us
consider the following:
𝑈 := {𝑢 ∈ 𝐴𝐶1 ([0, +∞) , R) : ‖𝑢‖ < 𝑀} .
(46)
Note that the operator 𝑇 : 𝑈 → P(𝐴𝐶1 ([0, +∞)) is
upper semicontinuous and completely continuous. From the
choice of 𝑈, there is no 𝑢 ∈ 𝜕𝑈 such that 𝑢 = 𝜆𝑇(𝑢) for
some 𝜆 ∈ (0, 1). Consequently, by the nonlinear alternative
of the Leray-Schauder type (Theorem 2), we deduce that 𝑇
has a fixed point 𝑢 ∈ 𝑈 which is a solution of the problem (1).
This completes the proof.
3.2. The Lipschitz Case. Now we prove the existence of
solutions for the problem (1) with a nonconvex-valued right
hand side by applying a fixed point theorem for multivalued
maps according to Covitz and Nadler [34].
(a) 𝛾-Lipschitz if and only if there exists 𝛾 > 0 such that
𝑑𝐻(𝑁(𝑥), 𝑁(𝑦)) ≤ 𝛾𝑑(𝑥, 𝑦) for each 𝑥, 𝑦 ∈ 𝑋;
Lemma 23 (Covitz-Nadler, [34]). Let (𝑋, 𝑑) be a complete
metric space. If 𝑁 : 𝑋 → P𝑐𝑙 (𝑋) is a contraction, then
𝐹𝑖𝑥𝑁 ≠ 0.
= co (lim sup𝐹 (𝑡, 𝑢𝑘 (𝑡) , 𝑢𝑘󸀠 (𝑡))) .
𝑘 → +∞
However, the fact that the multivalued 𝑥 → 𝐹(⋅, 𝑥, 𝑥󸀠 ) is
upper semicontinuous and has compact values, together with
Lemma 7, implies that
𝑛 → +∞
≤ max {𝐿 1 , 𝐿 2 } 𝜓 (‖𝑢‖)
(b) a contraction if and only if it is 𝛾-Lipschitz with 𝛾 < 1.
𝑛≥1
lim sup 𝐹 (𝑡, 𝑢𝑛 (𝑡) , 𝑢𝑛󸀠 (𝑡)) = 𝐹 (𝑡, 𝑢 (𝑡) , 𝑢󸀠 (𝑡)) ,
Claim 6. A priori bounds on solutions. Let 𝑢 be a solution
of (1). Then, there exists 𝑓 ∈ 𝐿1 ([0, +∞), R) with 𝑓 ∈ 𝑆𝐹 (𝑢)
+∞
such that 𝑢(𝑡) = ∫0 𝐺(𝑡, 𝑠)𝑓(𝑠)𝑑𝑠. In view of (H1), and using
the computations in Claim 2 above, for each 𝑡 ∈ [0, +∞), we
obtain
󵄨󵄨 𝑢 (𝑡) 󵄨󵄨 󵄨󵄨󵄨 𝑢󸀠 (𝑡) 󵄨󵄨󵄨
+∞
󵄨
󵄨󵄨 󵄨󵄨
󵄨󵄨} ≤ max {𝐿 , 𝐿 } ∫ 𝑓 (𝑠) 𝑑𝑠
,
{󵄨󵄨󵄨
󵄨
󵄨
󵄨
1
2
󵄨󵄨 1 + 𝑡𝛼−1 󵄨󵄨󵄨 󵄨󵄨󵄨 1 + 𝑡𝛼−1 󵄨󵄨󵄨
0
Definition 22. A multivalued operator 𝑁 : 𝑋 → P𝑐𝑙 (𝑋) is
called the following:
⊂ ⋂ co {𝑓𝑘 (𝑡) , 𝑘 ≥ 𝑛}
𝑛≥1
𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡)), a.e. 𝑡 ∈ 𝐼. Thus, V ∈ 𝑇(𝑢), proving that 𝑇 has a
closed graph. Finally, with Lemma 1 and the compactness of
𝑇, we conclude that 𝑇 is upper semicontinuous.
a.e. 𝑡 ∈ 𝐼.
(43)
This along with (42) yields that 𝑓(𝑡) ∈ co𝐹(𝑡, 𝑢(𝑡), 𝑢󸀠 (𝑡)).
Finally, 𝐹(⋅, ⋅, ⋅) has closed, convex values; hence, 𝑓(𝑡) ∈
Definition 24. A measurable multivalued function 𝐹 :
[0, +∞) → P(𝑋) is said to be integrably bounded if there
exists a function 𝑓 ∈ 𝐿1 ([0, +∞), 𝑋) such that for all V ∈ 𝐹(𝑡),
‖V‖ ≤ 𝑓(𝑡) for a.e. 𝑡 ∈ [0, +∞).
Theorem 25. Assume that the following condition holds:
(H4) 𝐹 : 𝐼 × R × R → P𝑐𝑝 (R) such that 𝐹(⋅, 𝑥, 𝑦) :
[0, +∞) → P𝑐𝑝 (R) is measurable for each 𝑥, 𝑦 ∈ R;
Advances in Mathematical Physics
7
(H5) There exist 𝑙1 , 𝑙2 : [0, +∞) → [0, +∞) which are not
identical zero on any closed subinterval of [0, +∞),
and
∫
+∞
0
(1 + 𝑠𝛼−1 ) 𝑙𝑖 (𝑠) 𝑑𝑠 < +∞,
𝑖 = 1, 2,
(48)
for all 𝑥1 , 𝑥2 , 𝑦1 , and 𝑦2 ∈ R with 𝑑(0, 𝐹(𝑡, 0, 0)) ≤ 𝑙1 (𝑡) + 𝑙2 (𝑡)
for almost all 𝑡 ∈ [0, +∞).
Then, the boundary value problem (1) has at least one
solution on 𝐼 = [0, +∞) if
0
(1 + 𝑠𝛼−1 ) (𝑙1 (𝑠) + 𝑙2 (𝑠)) 𝑑𝑠) < 1. (49)
Proof. We transform problem (1) into a fixed point problem. Consider the set-valued map 𝑇 : 𝐴𝐶1 [0, +∞) →
P(𝐴𝐶1 [0, +∞)) defined at the beginning of the proof of
Theorem 21. It is clear that the fixed point of 𝑇 are solutions
of the problem (1).
Note that since the set-valued map 𝐹(⋅, 𝑢(⋅)) is measurable
with the measurable selection theorem (e.g., Theorem III. 6
in [35]) it admits a measurable selection 𝑓 : 𝐼 → R.
Moreover, since 𝐹 is integrably bounded, 𝑓 ∈ 𝐿1 ([0, +∞), R).
Therefore, 𝑆𝐹 (𝑢) ≠ 0.
We will prove that 𝑇 fulfills the assumptions of CovitzNadler contraction principle (Lemma 23).
First, we note that since 𝑆𝐹 (𝑢) ≠ 0, 𝑇(𝑢) ≠ 0 for any 𝑢 ∈
1
𝐴𝐶 ([0, +∞), R).
Second, we prove that 𝑇(𝑢) is closed for any 𝑢 ∈
1
𝐴𝐶 ([0, +∞), R). Let {𝑢𝑛 }𝑛≥0 ∈ 𝑇(𝑢) such that 𝑢𝑛 → 𝑢0
in 𝐴𝐶1 ([0, +∞), R). Then 𝑢0 ∈ 𝐴𝐶1 ([0, +∞), R) and there
exists 𝑓𝑛 ∈ 𝑆𝐹 (𝑢) such that
𝑢𝑛 (𝑡) = ∫
+∞
0
𝐺 (𝑡, 𝑠) 𝑓𝑛 (𝑠) 𝑑𝑠.
(50)
Since 𝐹 has compact values, we may pass onto a subsequence (if necessary) to obtain that 𝑓𝑛 converges to 𝑓 ∈
𝐿1 (([0, +∞), R)) in 𝐿1 (([0, +∞), R)). In particular, 𝑓 ∈
𝑆𝐹 (𝑢), and for any 𝑡 ∈ [0, +∞) we have
𝑢𝑛 (𝑡) 󳨀→ 𝑢0 (𝑡) = ∫
+∞
0
𝐺 (𝑡, 𝑠) 𝑓 (𝑠) 𝑑𝑠,
+∞
0
𝐺 (𝑡, 𝑠) 𝑓1 (𝑠) 𝑑𝑠,
󵄨
󵄨
𝑑𝐻 (𝐹 (𝑡, 𝑥1 , 𝑥1󸀠 ) , 𝐹 (𝑡, 𝑥2 , 𝑥2󸀠 )) ≤ 𝑙1 (𝑡) 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨
󵄨
󵄨
+ 𝑙2 (𝑡) 󵄨󵄨󵄨󵄨𝑥1󸀠 − 𝑥2󸀠 󵄨󵄨󵄨󵄨 ,
(53)
󵄨 󸀠
󵄨
󵄨
󵄨
󵄨󵄨
󸀠󵄨
󵄨󵄨𝑓1 (𝑡) − 𝑓2 (𝑡)󵄨󵄨󵄨 ≤ 𝑙1 (𝑡) 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨 + 𝑙2 (𝑡) 󵄨󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨󵄨 .
+∞
V2 (𝑡) = ∫
0
𝐺 (𝑡, 𝑠) 𝑓2 (𝑠) 𝑑𝑠,
𝑡 ∈ [0, +∞) ,
(56)
and one can get
󵄨󵄨 V (𝑡)
V2 (𝑡) 󵄨󵄨󵄨󵄨
󵄨󵄨 1
−
󵄨
󵄨󵄨
󵄨󵄨 1 + 𝑡𝛼−1 1 + 𝑡𝛼−1 󵄨󵄨󵄨
≤∫
+∞
𝐺 (𝑡, 𝑠) 󵄨󵄨
󵄨
󵄨𝑓 (𝑠) − 𝑓1 (𝑠)󵄨󵄨󵄨 𝑑𝑠
1 + 𝑡𝛼−1 󵄨 1
0
≤ 𝐿1 ∫
+∞
0
≤ 𝐿1 ∫
+∞
0
󵄨
󵄨
󵄨
󵄨
[𝑙1 (𝑠) 󵄨󵄨󵄨𝑥1 (𝑠)−𝑥2 (𝑠)󵄨󵄨󵄨 +𝑙2 (𝑠) 󵄨󵄨󵄨󵄨𝑥1󸀠 (𝑠)−𝑥2󸀠 (𝑠)󵄨󵄨󵄨󵄨] 𝑑𝑠
󵄨󵄨 𝑥 (𝑠) − 𝑥 (𝑠) 󵄨󵄨
󵄨󵄨
󵄨
2
(1 + 𝑠𝛼−1 ) [𝑙1 (𝑠) 󵄨󵄨󵄨 1
󵄨
󵄨󵄨 1 + 𝑠𝛼−1 󵄨󵄨󵄨
󵄨󵄨󵄨 𝑥󸀠 (𝑠) − 𝑥2󸀠 (𝑠) 󵄨󵄨󵄨
󵄨󵄨] 𝑑𝑠
+ 𝑙2 (𝑠) 󵄨󵄨󵄨󵄨 1
󵄨
󵄨󵄨 1 + 𝑠𝛼−1 󵄨󵄨󵄨
󵄩
󵄩
≤ max {𝐿 1 , 𝐿 2 } 󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩 ∫
+∞
0
(1+𝑠𝛼−1 ) (𝑙1 (𝑠)+𝑙2 (𝑠)) 𝑑𝑠.
(57)
Similarly, we have
󵄨󵄨 V󸀠 (𝑡)
V2󸀠 (𝑡) 󵄨󵄨󵄨󵄨
󵄨󵄨 1
󵄩󵄩
󵄩󵄩
󵄨󵄨
󵄨
−
󵄨󵄨 1 + 𝑡𝛼−1 1 + 𝑡𝛼−1 󵄨󵄨󵄨 ≤ max {𝐿 1 , 𝐿 2 } 󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩
󵄨
󵄨
×∫
+∞
(1 + 𝑠𝛼−1 ) (𝑙1 (𝑠) + 𝑙2 (𝑠)) 𝑑𝑠.
(58)
Therefore,
󵄩
󵄩
󵄩󵄩
󵄩
󵄩󵄩V1 − V2 󵄩󵄩󵄩 ≤ max {𝐿 1 , 𝐿 2 } 󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩
+∞
(52)
(55)
Define
0
𝑡 ∈ [0, +∞) .
(54)
hence 𝐻 has nonempty closed values. Moreover, since 𝐻 is
measurable (e.g., Proposition III. 4 in [35]), there exists 𝑓2
which is a measurable selection of 𝐻. It follows that 𝑓2 ∈
𝑆𝐹 (𝑢2 ) and for any 𝑡 ∈ [0, +∞),
(51)
that is, 𝑢0 ∈ 𝑇(𝑢) and 𝑇(𝑢) is closed.
Next we show that 𝑇 is a contraction on 𝐴𝐶1 ([0, +∞), R).
Let 𝑢1 , 𝑢2 ∈ 𝐴𝐶1 ([0, +∞), R) and V1 ∈ 𝑇(𝑢1 ). Then there exist
𝑓1 ∈ 𝑆𝐹 (𝑢1 ) such that
V1 (𝑡) = ∫
󵄨
󵄨
󵄨
󵄨
∩ {𝑢 ∈ R; 󵄨󵄨󵄨𝑓1 (𝑡) − 𝑢󵄨󵄨󵄨 ≤ 𝑙1 (𝑡) 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨
󵄨
󵄨
+𝑙2 (𝑡) 󵄨󵄨󵄨󵄨𝑥1󸀠 − 𝑥2󸀠 󵄨󵄨󵄨󵄨} , 𝑡 ∈ [0, +∞) .
By (H5), we have
󵄨
󵄨
𝑑𝐻 (𝐹 (𝑡, 𝑥1 , 𝑦1 ) , 𝐹 (𝑡, 𝑥2 , 𝑦2 )) ≤ 𝑙1 (𝑡) 󵄨󵄨󵄨𝑥1 − 𝑥2 󵄨󵄨󵄨
󵄨
󵄨
+ 𝑙2 (𝑡) 󵄨󵄨󵄨𝑦1 − 𝑦2 󵄨󵄨󵄨
+∞
𝐻 (𝑡) := 𝐹 (𝑡, 𝑢2 (𝑡) , 𝑢2󸀠 (𝑡))
(47)
such that for almost all 𝑡 ∈ [0, +∞),
max {𝐿 1 , 𝐿 2 } (∫
Consider the set-valued map
×∫
0
(1 + 𝑠𝛼−1 ) (𝑙1 (𝑠) + 𝑙2 (𝑠)) 𝑑𝑠.
(59)
8
Advances in Mathematical Physics
From an analogous reasoning by interchanging the roles of 𝑢1
and 𝑢2 , it follows that
󵄩
󵄩
𝑑𝐻 (𝑇 (𝑢1 ) , 𝑇 (𝑢2 )) ≤ max {𝐿 1 , 𝐿 2 } 󵄩󵄩󵄩𝑥1 − 𝑥2 󵄩󵄩󵄩
×∫
+∞
0
(1 + 𝑠𝛼−1 ) (𝑙1 (𝑠) + 𝑙2 (𝑠)) 𝑑𝑠.
4. Application
Consider the fractional boundary value problem,
󸀠
𝐷05/2
+ 𝑢 (𝑡) + 𝐹 (𝑡, 𝑢 (𝑡) , 𝑢 (𝑡)) ∋ 0,
0 < 𝑡 < +∞,
1
1
1
𝑢 (0) = 𝑢󸀠 (0) = 0, 𝐷3/2 𝑢 (+∞) − 𝑢 ( ) − 𝑢 (1) = 0.
8
8
4
(61)
Here 𝑚 = 4, 𝛼 = 5/2, 𝛽1 = 1/8, 𝛽2 = 1/4, 𝜉1 = 1/8, and
𝜉2 = 1, and 𝐹 : 𝐼 × R × R → P(R) is a multivalued map
given by
3/2
𝐹 (𝑡, 𝑥, 𝑦) = [𝑒 (1 + 𝑡
󵄨5
󵄨󵄨
󵄨󵄨𝑥 + 𝑦󵄨󵄨󵄨
) (󵄨
+ 9) ,
󵄨󵄨𝑥 + 𝑦󵄨󵄨󵄨5 + 5
󵄨
󵄨
󵄨3
󵄨󵄨
󵄨𝑥 + 𝑦󵄨󵄨
2𝑒−𝑡 (1 + 𝑡3/2 ) ( 󵄨 󵄨 󵄨3 󵄨 + 1)] .
󵄨󵄨𝑥 + 𝑦󵄨󵄨 + 3
󵄨
󵄨
(62)
For V ∈ 𝐹, we have
󵄨5
󵄨󵄨
󵄨󵄨 V (𝑡) 󵄨󵄨
󵄨󵄨𝑥 + 𝑦󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨
−𝑡
+ 9) ,
󵄨 ≤ max (𝑒 ( 󵄨
󵄨󵄨
󵄨󵄨 1 + 𝑡3/2 󵄨󵄨󵄨
󵄨󵄨𝑥 + 𝑦󵄨󵄨󵄨5 + 5
󵄨
󵄨
󵄨3
󵄨󵄨
󵄨𝑥 + 𝑦󵄨󵄨
2𝑒−𝑡 ( 󵄨 󵄨 󵄨3 󵄨 + 1))
󵄨󵄨𝑥 + 𝑦󵄨󵄨 + 3
󵄨
󵄨
≤ 10𝑒−𝑡 ,
(63)
𝑥, 𝑦 ∈ R.
Thus,
󵄨󵄨 V (𝑡) 󵄨󵄨
󵄩
󵄨
󵄨󵄨
󵄩󵄩
󵄨 : V ∈ 𝐹 (𝑡, 𝑥, 𝑦)}
󵄩󵄩𝐹(𝑡, 𝑥, 𝑦)󵄩󵄩󵄩P := sup {󵄨󵄨󵄨
󵄨󵄨 1 + 𝑡3/2 󵄨󵄨󵄨
−𝑡
≤ 10𝑒 = 𝜑 (𝑡) 𝜓 (‖𝑥‖) ,
(64)
𝑥, 𝑦 ∈ R,
with 𝜑(𝑡) = 𝑒−𝑡 , 𝜓(‖𝑥‖) = 10.
Also, by direct calculation, we can obtain that 𝐿 1 =
1.01529 and 𝐿 2 = 3.045869. Further, by using the following
condition:
𝑀
The authors thank the referees for their careful reading of this
paper and useful suggestions. This paper was funded by King
Abdulaziz University, under Grant no. (130-1-1433/HiCi).
The authors, therefore, acknowledge technical and financial
support of KAU.
(60)
Since 𝑇 is a contraction, it follows by the Lemma 23 that 𝑇
admits a fixed point which is a solution to problem (1).
−𝑡
Acknowledgments
+∞
max {𝐿 1 , 𝐿 2 } 𝜓 (𝑀) ∫0
𝜑 (𝑠) 𝑑𝑠
> 1,
(65)
we find that 𝑀 > 30.45869. Clearly, all the conditions of
Theorem 21 are satisfied. So, there exists at least one solution
of problem (1) on 𝐼.
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