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Cent. Eur. J. Phys. ´ 11(6) ´ 2013 ´ 779-791
DOI: 10.2478/s11534-013-0219-z
Central European Journal of Physics
Existence, uniqueness and limit property of solutions
to quadratic Erdélyi-Kober type integral equations of
fractional order
Research Article
1,2∗ 2† ˘ 3,4‡
JinRong Wang , Chun Zhu , Michal Feckan
1 Department of Mathematics, Guizhou Normal College,
Guiyang, Guizhou 550018, P.R. China
2 Department of Mathematics, Guizhou University,
Guiyang, Guizhou 550025, P.R. China.
3 Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics,
ComeniusUniversity,
Mlynská dolina, 842 48 Bratislava, Slovakia
4 Mathematical Institute, Slovak Academy of Sciences,
Stefánikova 49, 814 73 Bratislava, Slovakia
Received 14 January 2013; accepted 27 March 2013
Abstract: In this paper, we apply certain measure of noncompactness and fixed point theorem of Darbo type to
derive the existence and limit property of solutions to quadratic Erdélyi-Kober type integral equations of
fractional order with three parameters. Moreover, we also present the uniqueness and another existence
results of the solutions to the above equations. Finally, two examples are given to illustrate the obtained
results.
PACS(2008): 02.30.-f;45.10.Hj
Keywords: quadratic Erdélyi-Kober type integral equations ´ fractional order ´ existence ´ uniquness ´ limit property
© Versita sp. z o.o.
1. Introduction the traditional integer order calculus was mentioned al-
ready in 1695 by Leibnitz and L’Hospital. The subject
of fractional calculus has become a rapidly growing area
Fractional calculus has been introduced since the end of and has found applications in diverse fields ranging from
the nineteenth century by Liouville and Riemann, but the physical sciences, engineering to biological sciences and
concept of non-integer calculus, as a generalization of economics. It draws a great application in nonlinear oscil-
lations of earthquakes, many physical phenomena such as
∗E-mail: sci.jrwang@gzu.edu.cn seepage flow in porous media and in fluid dynamic traffic
†E-mail: czhumath@126.com model.
‡E-mail: Michal.Feckan@fmph.uniba.sk (Corresponding author) Recently, Banaś and Zając [1] studied the solvability of a
779
Existence, uniqueness and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order
functional integral equation of fractional order: where t ∈ R := [0;∞),
Z +
f (t;x(t)) t
x(t) = f (t;x(t)) + 2 (t − s)α−1u(t;s;x(s))ds; 1 > α > 0; β >0; γ > β(1−α)−1; (3)
1 Γ(α) 0 t ∈ [0;1]; α ∈ (0;1);(1)
f , f and u are three functions will be defined later.
1 2
where Γ(·) is the gamma function. Here, the term (t−s)α−1 Obviously, the equation (1) is a particular case of the
can be named as Riemann-Liouville singular kernel since equation (2) when β = 1 and γ = 0.
it appears in the standard Riemann-Liouville fractional We will use certain measure of noncompactness from [1]
integral of order α of a continuous function y defined by and some of the methods in [46] to derive new existence
and limit property of solutions to the equation (2) with
Z t restriction. Moreover, we also give the uniqueness and
Iαy(t) = 1 (t − s)α−1y(s)ds; t > 0; α > 0: other existence results of the solutions to the equation
Γ(α) 0 (2).
The rest of this paper is organized as follows. In Section
Moreover, Erdélyi-Kober fractional integrals, Hadamard 2, some notations and preparation results are given. In
fractional integrals and Riesz fractional integrals are also Section 3, the existence and limit property of solutions
widely used to describe the medium with non-integer mass to the equation (2) are obtained. In Section 4, we give
dimension. One can also find more details of such frac- sufficient conditions to derive the uniqueness and other
tional integrals in physics, viscoelasticity, electrochem- existence results of the solutions to the equation (2). In
istry and porous media [2–8]. In the past ten years, the final Section 5, two examples are given to demonstrate
there has been a significant development in the field of the applicability of our results.
fractional differential (integral, evolution) equations and
related controls problems, one can see the monographs 2. Preliminaries
[9, 12–16] and the papers [10, 11, 17–38].
The Erdélyi-Kober fractional integral [39] of a continuous In this section we collect some definitions and results
function y is defined by which will be needed in our investigations.
Weshall work in the Banach space BC(R ) consisting of
γ;α t−β(γ+α) Z t β β α−1 βγ β +
Iβ y(t) = Γ(α) 0 (t −s ) s y(s)d(s ); all real functions defined, continuous and bounded on R+
and furnished with the standard norm ||x|| := sup{|x(t)| :
β β α−1 t ∈ R+}. Next, for any given continuous function f :
R ×R → R, it is defined a Nemytskij operator F :
where α;β and γ > 0. The term (t −s ) can be named +
as Erdélyi-Kober singular kernel. Obviously, Riemann- BC(R+) → BC(R+) by (Fx)(t) := f(t;x(t));t ∈ R for
Liouville singular kernel is a special case of Erdélyi- x ∈ BC(R+). It is well-known that F is continuous [40].
Kober singular kernel which implies that Erdélyi-Kober Now we recall some results from the theory of measures
fractional integral can better describe the memory prop- of noncompactness [41, 42]. Let E be a real Banach space
erty than Riemann-Liouville fractional integral. Thus, with a norm || · ||. Denote Br := {x ∈ X : ||x|| ≤ r}.
quadratic integral equations involving Erdélyi-Kober sin- For any X ⊂ E, X, ConvX denote the closure and convex
gular kernels maybe better applicable in the theory of closure of X, respectively. Moreover, ME denotes the
kinetic theory of gases [44] and in the theory of neutron family of all nonempty and bounded subsets of E, and NE
transport [45]. Let us pay attention to the fact that only a all relatively compact subsets.
few papers investigated the existence and local stability
of solutions of Erdélyi-Kober type integral equations of Definition 1.
fractional order on an unbounded interval [46]. Amappingµ : ME →R+ =[0;∞)issaidtobeameasure
Motivated by the above fact, we are going to study the of noncompactness in E with a kernel kerµ ⊂ NE, if the
following quadratic Erdélyi-Kober type integral equations following properties hold:
of fractional order: (i) kerµ = {X ∈ ME : µ(X) = 0} 6= ∅.
Z t x(t) = f1(t;x(t))+ (ii) X ⊂ Y =⇒ µ(X) ≤ µ(Y), µ(X) = µ(X),
f (t;x(t)) β β α−1 γ µ(ConvX) = µ(X) and µ(λX +(1−λ)Y) ≤ λµ(X)+
2Γ(α) 0 (t −s ) s u(t;s;x(s))ds; (2) (1 −λ)µ(Y) for λ ∈ [0;1].
780
˘
JinRong Wang, Chun Zhu, Michal Feckan
(iii) If {X }∞ ⊂ M is such that X ⊂ X , X = 3. Existence and limit property of
n n=1 E n+1 n n
X , (n = 1;2;···) and lim µ(X ) = 0, then
n n→∞ n
T∞ X 6=∅. solutions
n=1 n In this section, we will investigate existence and limit
We shall use a measure of noncompactness in the space property of solutions to the equation (2) by using the fixed
BC(R ) [41, 42]. In order to define this measure let us
+ point theorem of Darbo type via measure of noncompact-
fix a nonempty bounded subset X of the space BC(R+) ness in the above section. We introduce the following
and a positive number T. For x ∈ X and ε > 0 denote assumptions (see [1]):
ωT(x;ε) := sup{|x(t) − x(s)| : t;s ∈ [0;T];|t − s| ≤ ε}:
Further, let us put ωT(X;ε) := sup{ωT(x;ε) : x ∈ X}, (H ) f ∈ C(R ×R;R), i = 1;2 and there are nonde-
ωT(X) := lim ωT(X;ε); ω (X) := lim ωT(X): Next, 1 i +
0 ε→0 0 T→∞ 0 creasing functions k : R → R such that
for a fixed T > 0 and for a function x ∈ X, we define i + +
β (X) := sup{β (x) : x ∈ X}, where β (x) := sup{|x(s)−
T T T
x(t)| : s ≥ T;t ≥ T}. Let β(X) := lim β (X). Finally, |f (t; x) − f (t;y)| ≤ k (r)|x − y|; i = 1;2
T→∞ T i i i
we define the measure of noncompactness in the space
BC(R ) by
+ for any t ∈ R+, r ≥ 0 and x;y ∈ [−r;r]. Moreover,
f (t; 0) ∈ BC(R ); i = 1;2.
µ(X) := ω0(X)+β(X): (4) i +
The kernel kerµ of this measure consists of all sets X ∈ (H2) u ∈ C(△ × R;R) and there exists n ∈ C(△;R+)
MBC(R+) such that functions belonging to X are locally and a nondecreasing function φ ∈ C(R+;R+) with
equicontinuous on R+ and tend to their limits at infinity φ(0) = 0 such that
uniformly with respect to the set X.
The following basic equality [43] will be used in the se- |u(t;s;x) − u(t;s;y)| ≤ n(t;s)φ(|x − y|)
quel. for all (t;s) ∈ △ and x;y ∈ R. Here △ := {(t;s) ∈
Lemma2. R ×R :t≥s}.
Let α;β;γ and p be constants such that α > 0, p(γ−1)+ + +
1 > 0 and p(β −1)+1 > 0. Then It is clear that (2) is equivalent to a fixed point problem
Z t α α p(β−1) p(γ−1)
0 (t −s ) s ds = x = V(x); (5)
tθ Bp(γ −1)+1;p(β −1)+1;
α α t ∈ R := [0;+∞); when an operator V is defined by
+
where (Vx)(t) := (F x)(t) + (F x)(t)(Ux)(t); t ∈ R
Z 1 ξ−1 η−1 1 2 +
B(ξ;η) = 0 s (1 −s) ds; (Re(ξ) > 0;Re(η) > 0) with (F x)(t) := f (t;x(t)), (F x)(t) := f (t;x(t)) and
is the well-known Beta function and θ = p[α(β−1)+γ− 1 1 2 2
1] + 1: (Ux)(t) := 1 Z t(tβ −sβ)α−1sγu(t;s;x(s))ds:
To end this section, we state a fixed point theorem of Γ(α) 0
Darbo type [41] which will be used in the sequel. Our aim is to solve (5) on BC(R+) by following a method
Theorem3. of [1]. For the sake of convenience, we shall split our main
Let Q ⊆ E be a nonempty, bounded, closed and convex result into several lemmas. Obviously by (H1), we get the
set and T : Q → Q be a continuous mapping. If there first result.
exists a constant k ∈ [0;1) such that µ(T X) ≤ kµ(X) for
any nonempty subset X ⊆ Q, then T has a fixed point in
the set Q. Moreover, the set fix T of all fixed points of T Lemma4.
belonging to Q is a member of the family kerµ. Assume (H ) then F ;F ∈ C(BC(R );BC(R )).
1 1 2 + +
781
Existence, uniqueness and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order
To we show U ∈ C(BC(R+);BC(R+)), and then V ∈ Lemma5.
C(BC(R );BC(R )) as well, we introduce functions n(t) Assume (H ) and (3), then n(t) and u(t) are continuous
+ + 2
and u(t) on R+ as follows functions on R+.
n(t) = Z0t n(t;s)(tβ − sβ)α−1sγds; Proof. It is enough to check the continuity of n(t),
u(t) = Z t |u(t;s;0)|(tβ − sβ)α−1sγds: since the same method can be applied for u(t). First, by
0 (6), we have a continuity at t = 0. Next, fix arbitrarily
Since n := max{n(t;s) : s ∈ [0;t]} < ∞ for any t > 0, T > 0;ε > 0 and t1;t2 ∈ [0;T] such that |t2 − t1| ≤ ε
t with t < t . We denote
by Lemma 2 and (3), we have 1 2
Z t Z t ωT(n;ε) := sup{|n(s ;s )−n(s ;s )| : s ;s ;s ∈ [0;T];
β β α−1 γ β β α−1 γ 1 2 3 1 3 1 2 3
0 n(t;s)(t −s ) s ds ≤ nt 0 (t −s ) s ds s ≤s ;s ≤s ;|s −s |≤ε}:
nttβ(α−1)+γ+1Bγ +1;α < ∞: (6) 3 1 3 2 1 2
β β Since γ > β(1 − α) − 1 and α > 0, we can take ζ > 1
Consequently, n(·) and u(·) are well defined. Moreover, such that ζγ > ζβ(1−α)−1 and ζ(α −1)+1 > 0. Set
we show: ζ∗ := ζ . By Lemma 2 and Hölder inequality, we have
ζ−1
Z t Z t Z t
2 β β α−1 γ 1 β β α−1 γ 1 β β α−1 γ
|n(t ) − n(t )| ≤ n(t ;s)(t −s ) s ds− n(t ;s)(t −s ) s ds + n(t ;s)(t −s ) s ds
2 1 0 2 2 0 1 2 0 1 2
Z t Z t Z t
1 β β α−1 γ 2 β β α−1 γ 1 β β α−1 γ
− n(t ;s)(t −s ) s ds ≤ n(t ;s)(t −s ) s ds+ n(t ;s)(t −s ) s ds
0 1 1 t1 2 2 0 2 2
Z t Z t Z t
1 β β α−1 γ 1 β β α−1 γ 1 β β α−1 γ
− n(t ;s)(t −s ) s ds + n(t ;s)(t −s ) s ds− n(t ;s)(t −s ) s ds
0 1 2 0 1 2 0 1 1
Z t Z t Z t
2 β β α−1 γ 1 β β α−1 γ 1 β β α−1
≤n (t −s ) s ds+ |n(t ;s) − n(t ;s)|(t − s ) s ds+ n(t ;s) (t −s )
t2 t1 2 0 2s 1 2 0 1 2
Z t Z t
√ 2 1
β β α−1 γ ζ∗ ζ β β ζ(α−1) ζγ T β β α−1 γ
−(t1 −s ) s ds ≤ nt t2 − t1 (t2 − s ) s ds+ ω1(n;ε)(t2 −s ) s ds
2 t1 0
Z t Z t Z t
+nt 1(tβ − sβ)1−αsγds − 2(tβ − sβ)1−αsγds +nt 2(tβ − sβ)α−1sγds
1 0 1 0 2 1 t1 2
v
u ζγ+1
uTζβ(α−1)+ζγ+1B ; ζ(α − 1) + 1 β(α−1)+γ+1
√ ζ
≤(nt +nt ) ζ∗ εt β +ωT(n;ε)T B γ+1;α
1 2 β v 1 β β
u ζγ+1
β(α−1)+γ+1 β(α−1)+γ+1 uTζβ(α−1)+ζγ+1B ; ζ(α − 1) + 1
√ ζ
+nt t2 −t1 B γ+1;α ≤(nt +nt )ζ∗ εt β
1 β β 1 2 β
Bγ+1;α
+ β ωT(n;ε)Tβ(α−1)+γ+1 +ωT(h;ε)n ; (7)
β 1 t1
where h(t) := tβ(α−1)+γ+1 is continuous on [0;T]. By (H2), pleted.
limε→0 ωT(n;ε) = 0, so using (7) we derive that n is con-
1
tinuous on [0;T]. Since T is arbitrary, the proof is com- Now we are ready to prove the following result.
782
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