Existence of local and global solutions of fractional order differential equations
In this paper we shall study the existence of local and global mild solutions of the fractional order differential equations in an arbitrary Banach space by using the semigroup theory and Schauder’s fixed point theorem. We also give some examples to illustrate the applications of the abstract result...
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irk-123456789-1753162021-02-01T01:28:49Z Existence of local and global solutions of fractional order differential equations Muslim, M. Conca, C. Agarwal, R.P. In this paper we shall study the existence of local and global mild solutions of the fractional order differential equations in an arbitrary Banach space by using the semigroup theory and Schauder’s fixed point theorem. We also give some examples to illustrate the applications of the abstract results. Вивчено питання iснування локальних та глобальних м’яких розв’язкiв диференцiальних рiвнянь дробового порядку в довiльному банаховому просторi з використанням теорiї пiвгруп та теореми Шаудера про нерухому точку. Також наведено кiлька прикладiв, якi iлюструють застосування абстрактного результату. 2011 Article Existence of local and global solutions of fractional order differential equations / M. Muslim, C. Conca, R.P. Agarwal // Нелінійні коливання. — 2011. — Т. 14, № 1. — С. 76-84. — Бібліогр.: 11 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/175316 517.9 en Нелінійні коливання Інститут математики НАН України |
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In this paper we shall study the existence of local and global mild solutions of the fractional order differential equations in an arbitrary Banach space by using the semigroup theory and Schauder’s fixed point theorem. We also give some examples to illustrate the applications of the abstract results. |
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Article |
| author |
Muslim, M. Conca, C. Agarwal, R.P. |
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Muslim, M. Conca, C. Agarwal, R.P. Existence of local and global solutions of fractional order differential equations Нелінійні коливання |
| author_facet |
Muslim, M. Conca, C. Agarwal, R.P. |
| author_sort |
Muslim, M. |
| title |
Existence of local and global solutions of fractional order differential equations |
| title_short |
Existence of local and global solutions of fractional order differential equations |
| title_full |
Existence of local and global solutions of fractional order differential equations |
| title_fullStr |
Existence of local and global solutions of fractional order differential equations |
| title_full_unstemmed |
Existence of local and global solutions of fractional order differential equations |
| title_sort |
existence of local and global solutions of fractional order differential equations |
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Інститут математики НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/175316 |
| citation_txt |
Existence of local and global solutions of fractional order differential equations / M. Muslim, C. Conca, R.P. Agarwal // Нелінійні коливання. — 2011. — Т. 14, № 1. — С. 76-84. — Бібліогр.: 11 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT muslimm existenceoflocalandglobalsolutionsoffractionalorderdifferentialequations AT concac existenceoflocalandglobalsolutionsoffractionalorderdifferentialequations AT agarwalrp existenceoflocalandglobalsolutionsoffractionalorderdifferentialequations |
| first_indexed |
2025-07-15T12:33:57Z |
| last_indexed |
2025-07-15T12:33:57Z |
| _version_ |
1837716301753090048 |
| fulltext |
UDC 517 . 9
EXISTENCE OF LOCAL AND GLOBAL SOLUTIONS
OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS
IСНУВАННЯ ЛОКАЛЬНИХ ТА ГЛОБАЛЬНИХ РОЗВ’ЯЗКIВ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ДРОБОВОГО ПОРЯДКУ
M. Muslim
Birla Inst. Technology and Sci.
Pilani-Goa Campus, Goa India
e-mail: malikiisc@gmail.com
C. Conca
Center Math. Modeling, Univ. Chile, Santiago-Chile
e-mail: cconca@dim.uchile.cl
R. P. Agarwal
Florida Inst. Technology
150 West University Boulevard, Melbourne, FL 32901, USA
e-mail: agarwal@fit.edu
In this paper we shall study the existence of local and global mild solutions of the fractional order di-
fferential equations in an arbitrary Banach space by using the semigroup theory and Schauder’s fixed point
theorem. We also give some examples to illustrate the applications of the abstract results.
Вивчено питання iснування локальних та глобальних м’яких розв’язкiв диференцiальних рiв-
нянь дробового порядку в довiльному банаховому просторi з використанням теорiї пiвгруп та
теореми Шаудера про нерухому точку. Також наведено кiлька прикладiв, якi iлюструють за-
стосування абстрактного результату.
1. Introduction. We consider the following fractional order differential equation in a Banach
space (H, ‖.‖) :
dβu(t)
dtβ
+Au(t) = f(t, u(t)), t ∈ (0, T ],
(1.1)
u(0) = u0,
where A is a closed linear operator defined on a dense set, 0 < β ≤ 1, 0 < T < ∞ and
dβu(t)
dtβ
denotes the derivative of u in the Caputo sense. We assume −A is the infinitesimal generator
of a compact analytic semigroup {S(t) : t ≥ 0} in H and the nonlinear map f is defined from
[0, T ]×H into H satisfying certain conditions to be specified later.
For the initial works on existence and uniqueness of solutions to different type of differential
equations we refer to [1 – 9] and references cited in these papers.
c© M. Muslim, C. Conca, R. P. Agarwal, 2011
76 ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
EXISTENCE OF LOCAL AND GLOBAL SOLUTIONS . . . 77
Jardat et al. [3] have considered the following fractional order differential equation in a
Banach space of the form
dβu(t)
dtβ
= Au(t) + f(t, u(t), Gu(t), Su(t)), t > t0, β ∈ (0, 1],
(1.2)
u(t0) = u0,
where A generates a strongly continuous semigroup. They have used the semigroup and fixed
point method to prove the existence and uniqueness of solutions.
In this paper, we use the Schauder’s fixed point theorem and semigroup theory to prove
the existence of local and global mild solutions to the given problem (1.1). With some extra
assumptions, we can use all the results of this paper to the problem considered by Jardat [3].
The plan of the paper is as follows. Introduction and preliminaries are given respectively, in
the first two sections. In Section 3, we prove the existence of local mild solutions and in Section
4, the existence of global mild solutions for the problem (1.1) is given. In the last section, we
have given some examples
2. Preliminaries. We note that if −A is the infinitesimal generator of an analytic semigroup
then for c > 0 large enough, −(A + cI) is invertible and generates a bounded analytic semi-
group. This allows us to reduce the general case in which −A is the infinitesimal generator
of an analytic semigroup to the case in which the semigroup is bounded and the generator is
invertible. Hence without loss of generality we suppose that
‖S(t)‖ ≤ M for t ≥ 0
and
0 ∈ ρ(−A),
where ρ(−A) is the resolvent set of −A. It follows that for 0 ≤ α ≤ 1, Aα can be defined as
a closed linear invertible operator with domain D(Aα) being dense in H . We have Hκ ↪→ Hα
for 0 < α < κ and the embedding is continuous. For more details on the fractional powers of
closed linear operators we refer to Pazy [10]. It can be proved easily that Hα := D(Aα) is a
Banach space with norm ‖x‖α = ‖Aαx‖ and it is equivalent to the graph norm of Aα.
We notice that CT = C([0, T ], H), the set of all continuous functions from [0, T ] into H, is a
Banach space under the supremum norm given by
‖ψ‖T := sup
0≤η≤T
‖ψ(η)‖, ψ ∈ CT .
We consider the following assumptions:
(H1) −A is the infinitesimal generator of a compact analytic semigroup S(t).
(H2) The nonlinear map f : [0, T ]×H → H is continuous in the first variable and satisfies
the following condition:
‖f(t, x)− f(s, y)‖ ≤ Lf (r)[|t− s|+ ‖x− y‖],
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
78 M. MUSLIM, C. CONCA, R. P. AGARWAL
for all x, y ∈ Br(H,u0), t, s ∈ [0, T ]. Here, Lf : R+ → R+ is a nondecreasing function and for
r > 0
Br(Z, z1) = {z ∈ Z : ‖z − z1‖Z ≤ r},
where (Z, ‖.‖Z) is a Banach space.
We need some basic definitions and properties of the fractional calculus theory.
Definition 2.1. A real function g(x), x > 0, is said to be in the space Cµ, µ ∈ R, if there exists
a real number p(> µ) such that g(x) = xpg1(x), where g1 ∈ C[0,∞), and it is said to be in the
space Cmµ iff g(m) ∈ Cµ, m ∈ N.
Definition 2.2. The Riemann – Liouville fractional integral operator of order β ≥ 0 of a
fucntion g ∈ Cµ, µ ≥ −1 is defined as
Iβg(t) =
1
Γ(β)
t∫
0
(t− θ)β−1g(θ)dθ, t > 0.
Definition 2.3. If the function g ∈ Cm−1 and m is a positive integer then we can define the
fractional derivative of g(t) in the Caputo sense as given below
dβg(t)
dtβ
=
1
Γ(m− β)
t∫
0
(t− θ)m−β−1gm(θ)dθ, m− 1 < β ≤ m, t > 0.
Definition 2.4. By a mild solution of the differential equation (1.1), we mean a continuous
solution u of the following integral equation given below,
u(t) = S(t)u0 +
1
Γ(β)
t∫
0
(t− s)β−1S(t− s)f(s, u(s)) ds.
For more details on mild solution, we refer to [3].
3. Existence of local solutions. To prove the existence of mild solution of the evolution
problem (1.1), we need the following lemma.
Lemma 3.1. The differential equation (1.1) is equivalent to the following integral equation:
u(t) = u0 +
1
Γ(β)
t∫
0
(t− s)β−1(−Au(s))ds+
1
Γ(β)
t∫
0
(t− s)β−1f(s, u(s))ds,
where 0 < t ≤ T.
Proof. For the details, we refer to Lemma 1.1 in [3].
Now, we state the following theorem.
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
EXISTENCE OF LOCAL AND GLOBAL SOLUTIONS . . . 79
Theorem 3.1. Assume the conditions (H1), (H2) are satisfied and u0 ∈ D(A). Then, there
exists a t0, 0 < t0 < T, such that the equation (1.1) has a local mild solution on [0, t0].
Proof. Let R > 0 be such that M‖u0‖ ≤
R
2
and A1 = ‖A−α‖.
Choose t0, 0 < t0 ≤ T such that
t0 <
[
R
2
{
M
βΓ(β)
{Lf (R)[T +R] + ‖f(0, u0)‖}
}−1] 1
β
.
We set
Y = {u ∈ Ct0 : u(0) = u0, ‖u(t)− u0‖ ≤ R, for 0 ≤ t ≤ t0}.
Clearly, Y is a bounded, closed and convex subset of Ct0 .
For any 0 < T̃ ≤ T , we define a mapping F from CT̃ into CT̃ given by
(Fu)(t) = S(t)u0 +
1
Γ(β)
t∫
0
(t− s)β−1S(t− s)f(s, u(s)) ds.
Clearly, F is well defined.
We need to show that F : Y → Y. For any u ∈ Y, we have (Fu)(0) = u0. If t ∈ [0, t0], then
we have,
‖(Fu)(t)− u0‖ ≤ ‖S(t)u0 − u0‖+
1
Γ(β)
t∫
0
(t− s)β−1‖S(t− s)‖‖f(s, u(s))‖ds ≤
≤ R
2
+
M
βΓ(β)
{Lf (R)[T +R] + ‖f(0, u0)‖}tβ0 ≤ R.
Hence, F : Y → Y.
Now we will show that F maps Y into a precompact subset F (Y ) of Y. For this we will
show that for fixed t ∈ [0, t0], Y (t) = {(Fu)(t) : u ∈ Y } is precompact in H and F (Y ) is an
uniformly equicontinuous family of functions. Here, for t = 0, Y (0) = {u0} is precompact inH.
Let t > 0 be fixed. For an arbitrary ε ∈ (0, t), define a mapping Fε on Y by the formula
(Fεu)(t) = S(t)u0 +
1
Γ(β)
t−ε∫
0
(t− s)β−1S(t− s)f(s, u(s))ds =
= S(t)u0 +
S(ε)
Γ(β)
t−ε∫
0
(t− s)β−1S(t− s− ε)f(s, u(s)) ds.
Since S(ε) is compact for every ε > 0, the set Yε(t) = {(Fεu)(t) : u ∈ Y } is precompact in H
for every ε ∈ (0, t), where t ∈ (0, t0].
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
80 M. MUSLIM, C. CONCA, R. P. AGARWAL
Also, we have
‖(Fu)(t)− (Fεu)(t)‖ =
∥∥∥∥∥∥ 1
Γ(β)
t∫
t−ε
(t− s)β−1S(t− s)f(s, u(s)) ds
∥∥∥∥∥∥ ≤ εβR1
for all t ∈ (0, t0], u ∈ Y and R1 =
M
βΓ(β)
{Lf (R)[T + R] + ‖f(0, u0)‖}. Consequently, the set
Y (t), where t ≥ 0, is precompact in H.
For any t1, t2 ∈ (0, t0] with t1 < t2 and u ∈ Y, we have
(Fu)(t2)− (Fu)(t1) = [S(t2)− S(t1)]u0 +
1
Γ(β)
t2∫
t1
(t2 − s)β−1S(t2 − s)f(s, u(s)) ds+
+
−1
Γ(β)
t1∫
0
[(t1 − s)β−1 − (t2 − s)β−1]S(t2 − s)f(s, u(s)) ds+
+
1
Γ(β)
t1∫
0
(t1 − s)β−1[S(t2 − s)− S(t1 − s)] f(s, u(s)) ds =
= I1 + I2 + I3 + I4. (3.1)
Hence,
‖(Fu)(t2)− (Fu)(t1)‖ ≤ ‖I1‖+ ‖I2‖+ ‖I3‖+ I4‖. (3.2)
We have
I1 = [S(t2)− S(t1)]u0.
From Theorem 2.6.13 in Pazy [10], it follows that for every 0 < η < 1 − α, t2 > t1 > 0, we
have
‖I1‖ ≤ A1‖(S(t2)− S(t1))A
αu0‖ ≤ A1CηCα+ηt
−(α+η)
1 (t2 − t1)η‖u0‖ ≤ M1(t2 − t1)η,
whereCη is some positive constant satisfying ‖AηS(t)‖ ≤ Cηt
−η for all t > 0.Also,M1 depends
on t1 and blows up as t1 decreases to zero.
From equation (3.1), we have
‖I2‖ ≤
1
Γ(β)
t2∫
t1
(t2 − s)β−1‖S(t2 − s)‖‖f(s, u(s))‖ds ≤ MA2
βΓ(β)
(t2 − t1)β,
where A2 = {Lf (R)[T +R] + ‖f(0, u0)‖}. We have
I3 =
−1
Γ(β)
t1∫
0
[(t1 − s)β−1 − (t2 − s)β−1]S(t2 − s)f(s, u(s)) ds.
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
EXISTENCE OF LOCAL AND GLOBAL SOLUTIONS . . . 81
Hence,
‖I3‖ ≤
A2A1Cα
Γ(β)
t1∫
0
(t1 − s)λ−1[(t1 − s)−λµ − (t2 − s)−λµ] ds,
where λ = 1− α, µ =
1− β
1− α
and α 6= 1.
Hence, after some calculation, we get
‖I3‖ ≤
A2A1Cα
Γ(β)
µδ1
µ−1(1− c)−λ(1−µ)−1(t2 − t1)λ(1−µ),
where c =
(
1−
(µ
λ
)1
λµ
)
and 0 < δ1 ≤ 1.
Similarly, we get
‖I4‖ ≤
A1A2C1+α
Γ(β)
t1∫
0
(t1 − s)β−1[(t1 − s)−1 − (t2 − s)−1] ds ≤
≤ A1A2C1+α
αΓ(β)
δ2
( 1
β
−1)
(1− c1)−β(t2 − t1)β(1−
1
β
)
,
where C1+α =
(
1− 1
β2
)
, 0 < δ2 ≤ 1 and C1+α is some positive constant satisfying
‖A1+αS(t)‖ ≤ C1+αt
−1−α for all t > 0.
Thus from the above calculations we observe that the right hand side of the inequality (3.2)
tends to zero when t2 − t1 → 0. Hence, F (Y ) is a family of equicontinuous functions. Also,
F (Y ) is bounded. Thus from the Arzela – Ascoli theorem (cf. see Dieudonne [11]), F (Y ) is
precompact. The existence of a fixed point of F in Y is the consequence of Schauder’s fixed
point theorem.
Hence, there exists u ∈ Y, such that for all t ∈ [0, t0], we have
u(t) = S(t)u0 +
1
Γ(β)
t∫
0
(t− s)β−1S(t− s)f(s, u(s))ds, (3.3)
where u(0) = u0.
Applying the similar arguments as above, we see that the function u given by equation (3.3)
is uniformly Hölder continuous on [0, t0]. With the help of the condition (H2), we can show
that the map t 7−→ f1(t, u(t)) is Hölder continuous on [0, t0]. This completesthe proof of the
theorem.
4. Existence of global solutions.
Theorem 4.1. Suppose that 0 ∈ ρ(−A) and −A generates a compact analytic semigroup S(t)
with ‖S(t)‖ ≤ M , for t ≥ 0, u0 ∈ D(A), and the function f1 : [0,∞) × H → H satisfies the
condition (H2). If there is a continuous nondecreasing real valued function k(t) such that
‖f1(t, ψ)‖ ≤ k(t)(1 + ‖ψ‖) for t ≥ 0, ψ ∈ H,
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
82 M. MUSLIM, C. CONCA, R. P. AGARWAL
then the equation (1.1) has a unique mild solution u which exists for all t ≥ 0.
Proof. By Theorem 3.1, we can continue the solution of equation (1.1) as long as ‖u(t)‖
stays bounded. Therefore, we need to show that if u exists on [0, T ), then ‖u(t)‖ is bounded as
t ↑ T.
For t ∈ [0, T ), we have
u(t) = S(t)u0 +
1
Γ(β)
t∫
0
(t− s)β−1S(t− s)f(s, u(s))ds.
From the above equation, we get
‖u(t)‖ ≤ M‖u0‖+
1
Γ(β)
t∫
0
(t− s)β−1‖S(t− s)|‖f(s, u(s))‖ds.
Hence
‖u(t)‖ ≤ C2 + C3
t∫
0
(t− s)(β−1)‖u(s)‖ ds,
where
C2 = M‖u0‖+
1
βΓ(β)
Mk(T )T β
and
C3 =
1
Γ(β)
Mk(T ).
Hence from Lemma 6.7 (Chapter 5 in Pazy [10]), u is a global solution.
To complete the proof of the theorem we only need to show that u is unique on the whole
interval.
Let u1 and u2 be two solutions of the given fractional integral equation (1.1). Then, by a
similar argument as above, we see that
‖u1(t)− u2(t)‖ ≤
≤ 1
Γ(β)
MLf (R)
t∫
0
(t− s)(β−1)‖u1(s)− u2(s)‖ ds.
Hence from Lemma 6.7 (Chapter 5, Pazy [10]), the solution u is unique. This completes the
proof of the theorem.
ISSN 1562-3076. Нелiнiйнi коливання, 2011, т . 14, N◦ 1
EXISTENCE OF LOCAL AND GLOBAL SOLUTIONS . . . 83
5. Examples. Let H = L2((0, 1);R). Consider the following fractional partial differential
equations,
∂β
∂tβ
w(t, x)− ∂2xw(t, x) = F (t, w(t, x)) x ∈ (0, 1), t > 0,
(5.1)
w(0, x) = u0, w(t, 0) = w(t, 1) = 0, t ∈ [0, T ], 0 < T < ∞,
where F is a given functions and 0 < β < 1.
We define an operator A,
Au = −u′′ with u ∈ D(A) = H1
0 (0, 1).
Here clearly the operator A is self-adjoint, with compact resolvent and is the infinitesimal
generator of a compact analytic semigroup S(t). We take α = 1/2, D(A1/2) is a Banach space
with norm
‖x‖1/2 := ‖A1/2x‖, x ∈ D(A1/2),
and we denote this space by H1/2.
The equation (5.1) can be reformulated as the following abstract equation inH = L2((0, 1);R) :
dβu(t)
dtβ
+Au(t) = f(t, u(t)), t > 0,
u(0) = u0,
where u(t) = w(t, .) that is u(t)(x) = w(t, x), t ∈ [0, T ], x ∈ (0, 1), and the function
f : [0, T ]×H → H is given by
f(t, u(t))(x) = F (t, w(t, x)).
We can take f(t, u) = h(t)g(u
′
), where h is Lipschitz continuous and g : H → H is Lipschitz
continuous on H . In particular, we can take g(u) = sinu, g(u) = ξu, g(u) = arctan(u), where
ξ is constant.
Acknowledgement. Second author would like to thanks the CMM Santiago, University of
Chile for the financial support.
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