Existence and stability results for nonlinear fractional order Riemannâ ...
Applied Mathematics and Computation 247 (2014) 319–328
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Existence and stability results for nonlinear fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations q S. Abbas a, M. Benchohra b,c, M. Rivero d, J.J. Trujillo e,⇑ a
Laboratory of Mathematics, University of Saïda, PO Box 138, 20000 Saïda, Algeria Laboratory of Mathematics, University of Sidi Bel-Abbès, PO Box 89, 22000 Sidi Bel-Abbès, Algeria c Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia d Departamento de Matemáticas, Estadística e I.O., University of La Laguna, 38271 La Laguna, Tenerife, Spain e Departamento de Análisis Matemático, University of La Laguna, 38271 La Laguna, Tenerife, Spain b
a r t i c l e
i n f o
a b s t r a c t
Keywords: Volterra–Stieltjes integral equation Fractional integral–differential equations Riemann–Liouville fractional operators Existence and stability of solutions Fixed point
Our aim in this paper is to study the existence and the stability of solutions for Riemann–Liouville Volterra–Stieltjes quadratic integral equations of fractional order. Our results are obtained by using some fixed point theorems. Some examples are provided to illustrate the main results. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Fractional differential and integral equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering and other applied sciences. There has been a significant development in ordinary and partial fractional differential and integral equations in recent years; see the monographs of Abbas et al. [5], Baleanu et al. [7], Diethelm [15], Hilfer [17], Kilbas et al. [18], Lakshmikantham et al. [19], Podlubny [20] and Tarasov [28], and the papers by Abbas et al. [1–3,6], Qian et al. [21–23], Vityuk and Golushkov [29]. Recently interesting results of the stability of the solutions of various classes of integral equations of fractional order have obtained by Abbas et al. [4,5], Banas´ et al. [8–10], Darwish et al. [12], Dhage [13,14] and the references therein. In [8,10], Banas´ et al. proved some existence results for the following nonlinear Volterra–Stieltjes quadratic integral equation
xðtÞ ¼ aðtÞ þ f ðt; xðtÞÞ
Z
t
uðt; s; xðsÞÞds gðt; sÞ;
t P 0;
ð1Þ
0
where a : ½0; 1Þ ! ½0; 1Þ and f : ½0; 1Þ R ! R are given continuous functions. In this paper we establish some sufficient conditions for the existence of solutions of the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations of the form
q
This work has been supported in part by the Government of Spain and FEDER (Grants MTM2010-16499 and MTM2013-41704).
⇑ Corresponding author.
E-mail addresses: [email protected] (S. Abbas), [email protected] (M. Benchohra), [email protected] (M. Rivero), [email protected] (J.J. Trujillo). http://dx.doi.org/10.1016/j.amc.2014.09.023 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
uðt; xÞ ¼ lðt; xÞ þ
f ðt; x; uðt; xÞÞ CðrÞ
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ð2Þ
0
for ðt; xÞ 2 J :¼ ½0; a ½0; b; a; b > 0; r 2 ð0; 1Þ; l : J ! R; g : Rþ Rþ ! R; f : J R ! R; h : J 1 R ! R are given continuous functions, Rþ ¼ ½0; 1Þ; J 1 ¼ fðt; x; sÞ 2 J ½0; a : s 6 tg, and CðÞ is the Euler’s Gamma function. We present two results for the existence of solutions of the Eq. (2). The first one is based on Schauder’s fixed point theorem (Theorem 3.2) and the second one on the nonlinear alternative of Leray–Schauder type (Theorem 3.5). Next, we establish a sufficient condition for the existence and the stability of solutions of the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations of the form
uðt; xÞ ¼ lðt; xÞ þ
f ðt; x; uðt; xÞÞ CðrÞ
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ð3Þ
0
where ðt; xÞ 2 J 0 :¼ Rþ ½0; b; b > 0; r 2 ð0; 1Þ; l : J 0 ! R; g : Rþ Rþ ! R; f : J 0 R ! R; h : J 01 R ! R are given continuous functions and J 01 ¼ fðt; x; sÞ 2 J 0 Rþ : s 6 tg. We use the Schauder fixed point theorem for the existence of solutions of the Eq. (3), and we prove that all solutions are locally asymptotically stable (Theorem 4.3). 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By L1 ðJÞ, we denote the space of Lebesgue-integrable functions u : J ! R with the norm
kuk1 ¼
Z
a
Z
0
b
juðt; xÞjdxdt: 0
Let C :¼ CðJÞ be the Banach space of all continuous functions from J into R endowed with the norm
kukC ¼ sup juðt; xÞj: ðt;xÞ2J
By BC :¼ BCðJ 0 Þ we denote the Banach space of all bounded and continuous functions from J 0 into R equipped with the standard norm
kukBC ¼ sup juðt; xÞj: ðt;xÞ2J 0
For u0 2 C ðor u0 2 BCÞ and g 2 ð0; 1Þ, we denote by Bðu0 ; gÞ, the closed ball in C ðor BCÞ centered at u0 with radius g. Definition 2.1 [27]. Let r 2 ð0; 1Þ and u 2 L1 ðJÞ. The partial Riemann–Liouville integral of order r of uðt; xÞ with respect to t is defined by the expression
Ir0;t uðt; xÞ ¼
1 CðrÞ
Z
t
ðt sÞr1 uðs; xÞds;
for almost all ðt; xÞ 2 J:
0
Analogously, we define the integral
Ir0;x uðt; xÞ ¼
1 CðrÞ
Z
x
ðx sÞr1 uðt; sÞds;
for almost all ðt; xÞ 2 J:
0
Example 2.2. Let k; x 2 ð1; 1Þ and r 2 ð0; 1Þ, then
Ir0;t t k xx ¼
Cð1 þ kÞ kþr x t x ; Cð1 þ k þ rÞ
for almost all ðt; xÞ 2 J:
W If u is a real function defined on the interval ½a; b, then the symbol ba u denotes the variation of u on ½a; b. We say that u is Wb W of bounded variation on the interval ½a; b whenever a u is finite. If w : ½a; b ½c; b ! R, then the symbol qt¼p wðt; sÞ indicates the variation of the function t ! wðt; sÞ on the interval ½p; q ½a; b, where s is arbitrarily fixed in ½c; d. In the same way we W define qs¼p wðt; sÞ. For the properties of functions of bounded variation we refer to [24]. If u and u are two real functions defined on the interval ½a; b, then under some conditions (see [24]) we can define the Stieltjes integral (in the Riemann–Stieltjes sense)
Z
b
uðtÞduðtÞ
a
of the function u with respect to u. In this case we say that u is Stieltjes integrable on ½a; b with respect to u. Several conditions are known guaranteeing Stieltjes integrability [24,26]. One of the most frequently used requires that u is continuous and u is of bounded variation on ½a; b.
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
321
In what follows we will use a few properties of the Stieltjes integral contained in the below given lemma ([25], Section 8.13): Lemma 2.3 [25]. If u is Stieltjes integrable on the interval ½a; b with respect to a function u of bounded variation then
Z Z ! b t b _ uðtÞduðtÞ 6 juðtÞjd u : a a a In the sequel we will also consider Stieltjes integrals of the form
Z
b
uðtÞds gðt; sÞ;
a
and Riemann–Liouville Stieltjes integrals of fractional order of the form
1 CðrÞ
Z
t
ðt sÞr1 uðsÞds gðt; sÞ;
0
where g : Rþ Rþ ! R; r 2 ð0; 1Þ and the symbol ds indicates the integration with respect to s. Let ; – X BC, and let G : X ! X, and consider the solutions of equation
ðGuÞðt; xÞ ¼ uðt; xÞ:
ð4Þ
Inspired by the definition of the attractivity of solutions of integral equations (for instance [9]), we introduce the following concept of attractivity of solutions for Eq. (4). Definition 2.4. Solutions of Eq. (4) are locally attractive if there exists a ball Bðu0 ; gÞ in the space BC such that, for arbitrary solutions v ¼ v ðt; xÞ and w ¼ wðt; xÞ of Eq. (4) belonging to Bðu0 ; gÞ \ X, we have that, for each x 2 ½0; b,
limðv ðt; xÞ wðt; xÞÞ ¼ 0:
ð5Þ
t!1
When the limit (5) is uniform with respect to Bðu0 ; gÞ \ X, solutions of Eq. (4) are said to be uniformly locally attractive (or equivalently that solutions of (4) are locally asymptotically stable). Lemma 2.5 [11]. Let D BC. Then D is relatively compact in BC if the following conditions hold: (a) D is uniformly bounded in BC, (b) The functions belonging to D are almost equicontinuous on Rþ ½0; b, i.e. equicontinuous on every compact of Rþ ½0; b, (c) The functions from D are equiconvergent, that is, given > 0; x 2 ½0; b there corresponds Tð; xÞ > 0 such that juðt; xÞ lim uðt; xÞj < for any t P Tð; xÞ and u 2 D. t!1
3. Local existence results Let us start in this section by defining what we mean by a solution of the Eq. (2). Definition 3.1. We mean by a solution of Eq. (2), every function u 2 C such that u satisfies Eq. (2) on J. The following hypotheses will be used in the sequel: ðH1 Þ The function f is a continuous and there exists a positive function p 2 C such that
jf ðt; x; uÞ f ðt; x; v Þj 6 pðt; xÞju v j; ðH2 Þ ðH3 Þ ðH4 Þ ðH5 Þ
ðt; xÞ 2 J;
u; v 2 R:
For all t1 ; t2 2 ½0; a such that t1 < t2 the function s # gðt2 ; sÞ gðt 1 ; sÞ is nondecreasing on ½0; a. The function s # gð0; sÞ is nondecreasing on ½0; a0. The functions s # gðt; sÞ and t # gðt; sÞ are continuous on ½0; a for each fixed t 2 ½0; a or s 2 ½0; a, respectively. The function h is continuous and there exist continuous functions q ¼ qðt; x; sÞ : J 1 ! Rþ ; U : Rþ ! Rþ such that U is nondecreasing and
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ;
ðt; x; sÞ 2 J 1 ;
u 2 R:
Set
l :¼ sup lðt; xÞ; ðt;xÞ2J
f :¼ sup f ðt; x; 0Þ; ðt;xÞ2J
g ¼ sup
t _
gðt; kÞ;
t2½0;ak¼0
322
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ðt sÞr1 qðt; x; sÞ : CðrÞ ðt;x;sÞ2J 1
p :¼ sup pðt; xÞ
and q :¼ sup
ðt;xÞ2J
Now, we shall prove the following theorem concerning the existence of a solution of the Eq. (2), based on Schauder’s fixed point theorem [16]. Theorem 3.2. Assume that the hypotheses ðH1 Þ ðH5 Þ and the following hypothesis hold ðH6 Þ There exists a constant g > 0, such that
l þ q g UðgÞðf þ p gÞ 6 g.
Then the Eq. (2) has at least one solution in the space C. Proof 3.3. Let us define the operator N : C ! C such that, for any u 2 C,
ðNuÞðt; xÞ ¼ lðt; xÞ þ
f ðt; x; uðt; xÞÞ CðrÞ
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ðt; xÞ 2 J:
ð6Þ
0
It is clear that the operator N maps C into C. The problem of finding the solutions of the Eq. (2) is reduced to finding the solutions of the operator equation NðuÞ ¼ u. Hypothesis ðH6 Þ implies that N transforms the ball Bg :¼ Bð0; gÞ into itself. Indeed, for any u 2 Bg , and for each ðt; xÞ 2 J we have
jf ðt; x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞjjds gðt; sÞj þ ! Z t s _ r1 ðt sÞ jhðt; x; s; uðs; xÞÞjds gðt; kÞ
jðNuÞðt; xÞj 6 jlðt; xÞj þ
0
0
k¼0
6 l þ
Z
f CðrÞ t
Z
t
ðt sÞr1 qðt; x; sÞUðjuðs; xÞjÞds
0
! gðt; kÞ þ
k¼0
r1
ðt sÞ
s _
s _
qðt; x; sÞUðjuðs; xÞjÞds
0
jf ðt; x; uðt; xÞÞ f ðt; x; 0Þj CðrÞ
pðt; xÞjuðt; xÞj CðrÞ
!
gðt; kÞ
k¼0
6 l þ q g UðkukC Þðf þ p kukC Þ 6 l þ q g UðgÞðf þ p gÞ 6 g: Hence, NðuÞ 2 Bg . We shall show that N : Bg ! Bg satisfies the assumptions of Schauder’s fixed point theorem [16]. The proof will be given in several steps. Step 1: N is continuous. Let fun gn2N be a sequence such that un ! u in Bg . Then, for each ðt; xÞ 2 J, we have
jðNun Þðt; xÞ ðNuÞðt; xÞj 6 jf ðt; x; un ðt; xÞÞ f ðt; x; uðt; x:ÞÞj þ jf ðt; x; uðt; xÞÞj
Z 0
Z
t
0 t
ðt sÞr1 jhðt; x; s; un ðs; xÞÞjds gðt; sÞj CðrÞ
r1
ðt sÞ CðrÞ
jhðt; x; s; un ðs; xÞÞ hðt; x; s; uðs; xÞÞjjds gðt; sÞj
6 p q g Uðkun kC Þjun ðt; xÞ uðt; xÞj þ ! s _ hðt; x; s; uðs; xÞÞjds gðt; kÞ
f þ p kukC CðrÞ
k¼0
f þ p g 6 p q g UðgÞjun ðt; xÞ uðt; xÞj þ CðrÞ ! s _ hðt; x; s; uðs; xÞÞjds gðt; kÞ :
Z
t
Z
t
ðt sÞr1 jhðt; x; s; un ðs; xÞÞ
0
ðt sÞr1 jhðt; x; s; un ðs; xÞÞ
0
k¼0
Since un ! u as n ! 1 and h is continuous, (7) gives
kNðun Þ NðuÞkC ! 0 as n ! 1: Step 2: NðBg Þ is bounded. This is clear since NðBg Þ Bg and Bg is bounded. Step 3: NðBg Þ is equicontinuous. Let ðt 1 ; x1 Þ; ðt 2 ; x2 Þ 2 J; t 1 < t 2 ; x1 < x2 and let u 2 Bg . Thus we have
ð7Þ
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S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
jðNuÞðt 2 ; x2 Þ ðNuÞðt 1 ; x1 Þj 6 jlðt 2 ; x2 Þ lðt1 ; x1 Þj þ jf ðt 2 ; x2 ; uðt 2 ; x2 ÞÞ f ðt1 ; x1 ; uðt1 ; x1 ÞÞj ! Z t2 s _ ðt 2 sÞr1 gðt 2 ; kÞ jhðt2 ; x2 ; s; uðs; x2 ÞÞjds CðrÞ 0 k¼0 Z Z t1 t2 ðt sÞr1 ðt 1 sÞr1 2 hðt2 ; x2 ; s; uðs; x2 ÞÞds gðt 2 ; sÞ hðt 1 ; x1 ; s; uðs; x1 ÞÞds gðt 1 ; sÞ: þ jf ðt1 ; x1 ; uðt 1 ; x1 ÞÞj 0 CðrÞ C ðrÞ 0 Thus
jðNuÞðt 2 ; x2 Þ ðNuÞðt 1 ; x1 Þj 6 jlðt 2 ; x2 Þ lðt1 ; x1 Þj þ q g UðgÞjf ðt 2 ; x2 ; uðt2 ; x2 ÞÞ f ðt1 ; x1 ; uðt 1 ; x1 ÞÞj Z Z t1 t2 ðt sÞr1 ðt1 sÞr1 2 þ p g hðt 2 ; x2 ; s; uðs; x2 ÞÞds gðt2 ; sÞ hðt1 ; x1 ; s; uðs; x1 ÞÞds gðt 1 ; sÞ: 0 CðrÞ CðrÞ 0 Hence
jðNuÞðt 2 ; x2 Þ ðNuÞðt 1 ; x1 Þj 6 jlðt 2 ; x2 Þ lðt1 ; x1 Þj þ p q g UðgÞjuðt2 ; x2 Þ uðt1 ; x1 Þj þ
Z
t1
0
Z
t1
p g CðrÞ
! s _ p g r1 r1 gðt 1 ; kÞ þ ðt2 sÞ hðt 2 ; x2 ; s; uðs; x2 ÞÞðt1 sÞ hðt 1 ; x1 ; s; uðs; x1 ÞÞds C ðrÞ k¼0 jðt 2 sÞr1 hðt2 ; x2 ; s; uðs; x2 ÞÞjjds ðgðt 2 ; sÞ gðt1 ; sÞÞj þ p q gUðgÞ
0
t2 _
gðt2 ; kÞ:
k¼t 1
From continuity of l; g; h and as t1 ! t 2 and x1 ! x2 , the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3 together with the Arzelá–Ascoli theorem, we can conclude that N : Bg ! Bg is continuous and compact. From an application of Schauder’s theorem [16], we deduce that N has a fixed point u which is a solution of the Eq. (2). h In the sequel, we need the following theorem. Theorem 3.4 ([16] Nonlinear alternative of Leray–Schauder type). By U and @U we denote the closure of U and the boundary of U respectively. Let X be a Banach space and C a nonempty convex subset of X. Let U be a nonempty open subset of C with 0 2 U and T : U ! C completely continuous operator. Then either (a) T has fixed points. Or (b) There exist u 2 @U and k 2 ð0; 1Þ with u ¼ kTðuÞ. Now, we present another result based on the nonlinear alternative of Leray–Schauder type (Theorem 3.4). Theorem 3.5. Assume that the hypotheses ðH1 Þ ðH4 Þ and the following hypothesis holds ðH7 Þ The function h is continuous and there exist continuous functions k1 ; k2 : J 1 ! Rþ such that
jhðt; x; s; uÞj 6 k1 ðt; x; sÞ þ
k2 ðt; x; sÞ ; 1 þ juj
ðt; x; sÞ 2 J 1 ;
u 2 R;
with
ki :¼ sup
ðt;x;sÞ2J 1
1
CðrÞ
ðt sÞr1 ki ðt; x; sÞ;
i ¼ 1; 2:
If
p g k1 < 1;
ð8Þ
then the Eq. (2) has at least one solution in the space C. Proof 3.6. We shall show that the operator N defined in (6) satisfies all the conditions of Theorem 3.4. As in Theorem 3.2, we can show that N is completely continuous. A priori bounds. We shall show there exists an open set U # C with u – kNðuÞ, for k 2 ð0; 1Þ and u 2 @U. Let u 2 C be such that u ¼ kNðuÞ for some 0 < k < 1. Thus for each ðx; yÞ 2 J, we have
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S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
juðt;xÞj 6 jklðt;xÞj þ
jkf ðt;x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x;s;uðs; xÞÞjjds gðt; sÞj
0
! Z s _ jkjjf ðt;x; uðt; xÞÞ f ðt; x;0Þj t r1 þ ðt sÞ jhðt; x; s;uðs;xÞÞjds gðt; kÞ CðrÞ 0 k¼0 ! Z t s _ f k2 ðt; x;sÞ ds ðt sÞr1 k1 ðt;x; sÞ þ gðt;kÞ 6 l þ 1 þ juðs; xÞj CðrÞ 0 k¼0 ! Z s _ pðt; xÞjuðt; xÞj t k2 ðt; x;sÞ r1 ds ðt sÞ k1 ðt; x;sÞ þ gðt;kÞ 6 l þ f g ðk1 þ k2 Þ þ p g k2 þ p g k1 kukC : þ 1 þ juðs;xÞj CðrÞ 0 k¼0 Then,
kukC 6 l þ f g ðk1 þ k2 Þ þ p g k2 þ p g k1 kukC : Thus, by (8) we get
kukC 6
l þ f g ðk1 þ k2 Þ þ p g k2
1 p g k1
:¼ M:
Set
U ¼ fu 2 C : kukC < M þ 1g: By our choice of U, there is no u 2 @U such that u ¼ kNðuÞ, for k 2 ð0; 1Þ. From Theorem 3.4, we deduce that N has a fixed point u in U which is a solution to the Eq. (2). h
4. Global existence and stability of solutions Definition 4.1. We mean by a solution of Eq. (3), every function u 2 BC such that u satisfies Eq. (3) on J 0 . The following hypotheses will be used in the sequel: ðH01 Þ The function l is in BC. Moreover, assume that lim lðt; xÞ ¼ 0; x 2 ½0; b. t!1 ðH02 Þ The function f is a continuous and there exists a positive function p 2 BC such that
jf ðt; x; uÞ f ðt; x; v Þj 6 pðt; xÞju v j;
ðt; xÞ 2 J;
u; v 2 R:
Moreover, assume that lim f ðt; x; 0Þ ¼ 0; x 2 ½0; b. t!1 ðH03 Þ For all t 1 ; t 2 2 Rþ such that t1 < t 2 the function s # gðt 2 ; sÞ gðt 1 ; sÞ is nondecreasing on Rþ . 0 ðH4 Þ The function s # gð0; sÞ is nondecreasing on Rþ . ðH05 Þ The functions s # gðt; sÞ and t # gðt; sÞ are continuous on Rþ for each fixed t 2 Rþ or s 2 Rþ , respectively. ðH06 Þ The function h is continuous and there exist continuous functions q : J 01 ! Rþ ; U : Rþ ! Rþ such that U is nondecreasing and
ðt; x; sÞ 2 J 01 ;
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ; Moreover, assume that lim
t!1
Rt 0
u 2 R:
ðt sÞr1 qðt; x; sÞds gðt; sÞ ¼ 0.
Remark 4.2. Set
l :¼ sup lðt; xÞ; f :¼ sup f ðt; x; 0Þ; ðt;xÞ2J 0
p :¼ sup pðt; xÞ and ðt;xÞ2J 0
ðt;xÞ2J 0
1 ðt;xÞ2J0 CðrÞ
q :¼ sup
Z
t 0
ðt sÞr1 qðt; x; sÞds
s _
! gðt; kÞ :
k¼0
From hypotheses, we infer that l ; f ; p and q are finite. Now, we shall prove the following theorem concerning the existence and the stability of a solution of the Eq. (3). Theorem 4.3. Assume that the hypotheses ðH01 Þ ðH06 Þ and the following hypothesis hold ðH07 Þ There exists a constant g > 0, such that
l þ q UðgÞðf þ p gÞ 6 g.
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
325
Then the Eq. (3) has at least one solution in the space BC. Moreover, if there exists a constant g > 0, such that
q ðf þ p gÞUðgÞ þ ½q f þ p q ðg þ g ÞUðg þ g Þ 6 g ; then solutions of the Eq. (3) are locally asymptotically stable. Proof 4.4. Let us define the operator N 0 such that, for any u 2 BC,
ðN0 uÞðt; xÞ ¼ lðt; xÞ þ
Z
f ðt; x; uðt; xÞÞ CðrÞ
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ðt; xÞ 2 J 0 :
ð9Þ
0
The operator N 0 maps BC into BC. Indeed the map N 0 ðuÞ is continuous on J 0 for any u 2 BC, and for each ðt; xÞ 2 J 0 we have
jf ðt; x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞjjds gðt; sÞj þ ! Z t s _ r1 ðt sÞ jhðt; x; s; uðs; xÞÞjds gðt; kÞ
jðN0 uÞðt; xÞj 6 jlðt; xÞj þ
0
0
k¼0
6 l þ
Z
f CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞUðjuðs; xÞjÞds
0
t
ðt sÞ
r1
s _
! gðt; kÞ þ
k¼0 s _
qðt; x; sÞUðjuðs; xÞjÞds
0
jf ðt; x; uðt; xÞÞ f ðt; x; 0Þj CðrÞ
pðt; xÞjuðt; xÞj CðrÞ
!
gðt; kÞ
k¼0
! Z s _ f UðkukBC Þ t p kukBC UðkukBC Þ r1 6l þ ðt sÞ qðt; x; sÞds gðt; kÞ þ CðrÞ CðrÞ 0 k¼0 ! Z t s _ ðt sÞr1 qðt; x; sÞds gðt; kÞ
0
k¼0
6 l þ q UðkukBC Þðf þ p kukBC Þ: Thus,
kN0 ðuÞkBC 6 l þ q UðkukBC Þðf þ p kukBC Þ: 0
ð10Þ 0
Hence, N ðuÞ 2 BC. This proves that the operator N maps BC into itself. The problem of finding the solutions of the Eq. (3) is reduced to finding the solutions of the operator equation N 0 ðuÞ ¼ u. Hypothesis ðH07 Þ implies that N 0 transforms the ball Bg :¼ Bð0; gÞ into itself. We shall show that N 0 : Bg ! Bg satisfies the assumptions of Schauder’s fixed point theorem [16]. As in the proof of Theorem 1, we can show that N 0 is continuous, N 0 ðBg Þ is uniformly bounded, and equicontinuous on every compact subset ½0; a ½0; b of J 0 ; a > 0. Step 1: N 0 ðBg Þ is equiconvergent. Let ðt; xÞ 2 Rþ ½0; b and u 2 Bg , then we have
jf ðt; x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞjjds gðt; sÞj þ ! Z t s _ ðt sÞr1 jhðt; x; s; uðs; xÞÞjds gðt; kÞ
jðNuÞðt; xÞj 6 jlðt; xÞj þ
0
0
jf ðt; x; uðt; xÞÞ f ðt; x; 0Þj CðrÞ
k¼0
! Z t s _ f pðt; xÞjuðt; xÞj r1 6 jlðt; xÞj þ ðt sÞ qðt; x; sÞUðjuðs; xÞjÞds gðt; kÞ þ CðrÞ 0 CðrÞ k¼0 ! Z t s _ ðt sÞr1 qðt; x; sÞUðjuðs; xÞjÞds gðt; kÞ 0
k¼0
6 jlðt; xÞj þ
UðgÞðf þ p gÞ CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞds
0
Thus, for each x 2 ½0; b, we get
jðNuÞðt; xÞj ! 0;
as t ! þ1:
Hence,
jðNuÞðt; xÞ ðNuÞðþ1; xÞj ! 0;
as t ! þ1:
s _
! gðt; kÞ :
k¼0
326
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
As a consequence of Step 1 together with Lemma 2.5, we can conclude that N 0 : Bg ! Bg is continuous and compact. From an application of Schauder’s theorem [16], we deduce that N 0 has a fixed point u which is a solution of the Eq. (3). Step 2: The local asymptotic stability of solutions. Now we investigate the local asymptotic stability of solutions of the Eq. (3). Let us assume that u0 is a solution of the Eq. (3) with the conditions of this theorem. Taking u 2 Bðu0 ; g Þ, we have
jðN0 uÞðt; xÞ u0 ðt; xÞj ¼ jðN0 uÞðt; xÞ ðN0 u0 Þðt; xÞj
6 p q Uðg þ g Þjuðt; xÞ u0 ðt; xÞj þ ! s _ hðt; x; s; u0 ðs; xÞÞjds gðt; kÞ
f þ p g CðrÞ
k¼0
6 p q Uðg þ g Þjuðt; xÞ u0 ðt; xÞj þ ! s _ þ Uðju0 ðs; xÞjÞÞds gðt; kÞ
f þ p g CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞ
0
Z
t
ðt sÞr1 qðt; x; sÞðUðjuðs; xÞjÞ
0
k¼0
6 p q g Uðg þ g Þ þ q ðf þ p gÞ½UðgÞ þ Uðg þ g Þ 6 g : Thus we observe that N 0 is a continuous function such that
N0 ðBðu0 ; g ÞÞ Bðu0 ; g Þ: Moreover, if u is a solution of the Eq. (3), then
juðt; xÞ u0 ðt; xÞj ¼ jðN0 uÞðt; xÞ ðN0 u0 Þðt; xÞj 6 p Uðg þ g Þjuðt; xÞ u0 ðt; xÞj
Z
t
ðt sÞ
r1
1 CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞds
0
p g Uðg þ g Þ CðrÞ
s _
jhðt; x; s; uðs; xÞÞ hðt; x; s; u0 ðs; xÞÞjds Z
t
ðt sÞr1 qðt; x; sÞds
0
þ Uðju0 ðs; xÞjÞÞds
s _
!
s _
f þ p g CðrÞ
!
gðt; kÞ
k¼0
!
gðt; kÞ þ
k¼0
f þ p g CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞðUðjuðs; xÞjÞ
0
gðt; kÞ
k¼0
gðt; kÞ þ
k¼0
0
6
!
s _
6 ðp g Uðg þ g Þ þ ðf þ p gÞ½UðgÞ þ Uðg þ g ÞÞ
Z
t
0
! s _ ðt sÞr1 qðt; x; sÞds gðt; kÞ : CðrÞ k¼0
Thus
juðt; xÞ u0 ðt; xÞj 6 ðp g Uðg þ g Þ þ ðf þ p gÞ½UðgÞ þ Uðg þ g ÞÞ
Z 0
t
! s _ ðt sÞr1 qðt; x; sÞds gðt; kÞ : CðrÞ k¼0
ð11Þ
By using (11) we deduce that
lim juðt; xÞ u0 ðt; xÞj ¼ 0:
t!1
Consequently, all solutions of the Eq. (3) are locally asymptotically stable.
h
5. Examples As applications and to illustrate our results, we present two examples. Example 1. Consider the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations
uðt; xÞ ¼ lðt; xÞ þ where r ¼ 14 ;
f ðt; x; uðt; xÞÞ CðrÞ
Z 0
lðt; xÞ ¼ 2þt1 2 ; t; x 2 ½0; 1,
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
for ðt; xÞ 2 J :¼ ½0; 1 ½0; 1;
ð12Þ
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
f ðt; x; uÞ ¼
ext juj ; 1 þ juj
t;
x 2 ½0; 1; u 2 R;
gðt; sÞ ¼ s;
t;
hðt; x; s; uÞ ¼
3 cxjuj ðt sÞ4 sin t sin s; 1 þ juj
c¼
Cð14Þ 8e
327
s 2 ½0; 1; ðt; x; sÞ 2 J 1 ;
J 1 ¼ fðt; x; sÞ 2 ½0; 13 : s 6 tg:
and
The function f is a continuous, f ¼ 0, and
jf ðt; x; uÞ f ðt; x; v Þj 6 ext ju v j;
t;
x 2 ½0; 1; u;
v 2 R:
xt
Then, the assumption ðH1 Þ is satisfies with pðt; xÞ ¼ e ; t; x 2 ½0; 1, and then p ¼ e. Also, we can easily see that the function g satisfies the hypotheses ðH2 Þ ðH4 Þ. The function h satisfies the assumption ðH5 Þ. Indeed, h is continuous and
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ;
ðt; x; sÞ 2 J 1 ;
u 2 R;
where U : Rþ ! Rþ : UðwÞ ¼ w, and 3
qðt; x; sÞ ¼ cxðt sÞ4 sin t sin s;
ðt; x; sÞ 2 J 1 :
Then, 3
ðt sÞ 4 qðt; x; sÞ 1 ¼ : 8e Cð14Þ ðt;x;sÞ2J 1
q ¼ sup
Finally, we can see that the hypothesis ðH6 Þ is satisfies with g ¼ 1. Indeed, we have
l ¼ 12 ; g ¼ 1 and the inequality
l þ q g UðgÞðf þ p gÞ 6 g implies
1 1 2 þ g 6 g; 2 8 which is satisfies for g ¼ 1. Consequently, by Theorem 3.2, the Eq. (12) has a solution defined on ½0; 1 ½0; 1. Example 2. Consider now the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations
uðt; xÞ ¼ lðt; xÞ þ where r ¼ 14 ;
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
for ðt; xÞ 2 J 0 :¼ Rþ ½0; 1;
ð13Þ
0
lðt; xÞ ¼ 2þt1 2 ; ðt; xÞ 2 J0 ,
f ðt; x; uÞ ¼
ext juj ; 1 þ juj
hðt; x; s; uÞ ¼
pffiffiffiffi
p
8eCð14Þ
ðt; xÞ 2 J 0 ;
u 2 R; gðt; sÞ ¼ s; ðt; sÞ 2 R2þ ;
pffiffi 3 cxs 4 juj sin t sin s ; ð1 þ t2 Þð2 þ jujÞ
hðt; x; 0; uÞ ¼ 0; c¼
f ðt; x; uðt; xÞÞ CðrÞ
ðt; xÞ 2 J
ðt; x; sÞ 2 J 01 ;
s – 0 and u 2 R;
and u 2 R;
and J 01 ¼ fðt; x; sÞ 2 J Rþ : s 6 t;
First, we can see that lim t!þ1 f ¼ 0, and
1 2þt 2
¼ 0, then the assumption ðH01 Þ is satisfies and
jf ðt; x; uÞ f ðt; x; v Þj 6 ext ju v j; ðH02 Þ
x 2 ½0; 1g:
ðt; xÞ 2 J 0 ;
l ¼ 12. Next, the function f is a continuous,
u; v 2 R: xt
Then, the assumption is satisfies with pðt; xÞ ¼ e ; ðt; xÞ 2 J, and then p ¼ e. Also, we can easily see that the function g satisfies the hypotheses ðH03 Þ ðH05 Þ. The function h satisfies the assumption ðH06 Þ. Indeed, h is continuous and
328
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
ðt; x; sÞ 2 J 01 ;
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ;
u 2 R;
where U : Rþ ! Rþ : UðwÞ ¼ w, and
qðt; x; sÞ ¼
pffiffi 3 cxs 4 sin t sin s ; 1 þ t2
ðt; x; sÞ 2 J 01 ;
s – 0;
ðt; xÞ 2 J 0 :
qðt; x; 0Þ ¼ 0; Then,
! Z t Z t s _ pffiffi pffiffi Z t 3 3 3 3 ðt sÞr1 qðt; x; sÞds gðt; sÞ 6 4 4 ðt sÞ cxs j sin t sin sjds gðt; kÞ 6 cxj sin t j ðt sÞ 4 s 4 ds 0
0
6
k¼0
pffiffi 2 1 ð4Þ sin t
0
2 1 ð4Þ
cxC pffiffiffiffi
cxC
pffiffi 6 pffiffiffiffiffiffi !0 as t ! 1; p t pt
and
1 q :¼ sup ðt;xÞ2J CðrÞ
Z
t
r1
ðt sÞ
qðt; x; sÞds
0
k¼0
Finally, we can see that hypothesis
s _
ðH07 Þ
! gðt; kÞ
pffiffi cxCð1Þ sin t cCð1Þ 1 6 sup pffiffiffiffi4 pffiffi ¼ pffiffiffi4ffi ¼ : p p 8e t ðt;xÞ2J
is satisfies with g ¼ 1. Indeed, the inequality
l þ q UðgÞðf þ p gÞ 6 g implies 12 þ 18 g2 6 g, which is satisfies for g ¼ 1. Moreover, the inequality
q ðf þ p gÞUðgÞ þ ½q f þ p q ðg þ g ÞUðg þ g Þ 6 g is satisfies with g ¼ g ¼ 1 and gives 58 6 1. Consequently, by Theorem 4.3, Eq. (13) has a solution defined on Rþ ½0; 1 and all solutions are locally asymptotically stable on Rþ ½0; 1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
S. Abbas, M. Benchohra, Darboux problem for implicit impulsive partial hyperbolic differential equations, Electron. J. Differ. Equ. 2011 (2011) 15. S. Abbas, M. Benchohra, On the set of solutions of fractional order Riemann–Liouville integral inclusions, Demonstratio Math. 46 (2013) 271–281. S. Abbas, M. Benchohra, Fractional order Riemann–Liouville integral equations with multiple time delay, Appl. Math. E-Notes 12 (2012) 79–87. S. Abbas, M. Benchohra, J. Henderson, On global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order, Commun. Appl. Nonlinear Anal. 19 (2012) 79–89. S. Abbas, M. Benchohra, G.M. N’Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012. S. Abbas, M. Benchohra, A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Fract. Calc. Appl. Anal. 15 (2) (2012) 168–182. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012. J. Banas´, Existence results for Volterra–Stieltjes quadratic integral equations on an unbounded interval, Math. Scand. 98 (2006) 143–160. J. Banas´, B.C. Dhage, Global asymptotic stability of solutions of a functional integral equation, Nonlinear Anal. Theory Methods Appl. 69 (7) (2008) 1945–1952. J. Banas´, T. Zajaßc, A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl. 375 (2011) 375–387. C. Corduneanu, Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973. M.A. Darwish, J. Henderson, D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional integral equations with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011) 539–553. B.C. Dhage, Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem, Nonlinear Anal. 70 (2009) 2485–2493. B.C. Dhage, Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness, Differ. Equ. Appl. 2 (3) (2010) 299–318. K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, 2010. A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, 2000. A.A. Kilbas, Hari M. Srivastava, Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009. I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999. C. Qian, Global attractivity in nonlinear delay differential equations, J. Math. Anal. Appl. 197 (1996) 529–547. C. Qian, Global attractivity in a delay differential equation with application in a commodity model, Appl. Math. Lett. 24 (2011) 116–121. C. Qian, Y. Sun, Global attractivity of solutions of nonlinear delay differential equations with a forcing term, Nonlinear Anal. 66 (2007) 689–703. I.P. Natanson, Theory of Functions of a Real Variable, Ungar, New York, 1960. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1970. R. Sikorski, Real Functions, PWN, Warsaw, 1958 (in Polish). S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993. V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010. A.N. Vityuk, A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (2004) 318–325.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Existence and stability results for nonlinear fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations q S. Abbas a, M. Benchohra b,c, M. Rivero d, J.J. Trujillo e,⇑ a
Laboratory of Mathematics, University of Saïda, PO Box 138, 20000 Saïda, Algeria Laboratory of Mathematics, University of Sidi Bel-Abbès, PO Box 89, 22000 Sidi Bel-Abbès, Algeria c Department of Mathematics, Faculty of Science, King Abdulaziz University, 21589 Jeddah, Saudi Arabia d Departamento de Matemáticas, Estadística e I.O., University of La Laguna, 38271 La Laguna, Tenerife, Spain e Departamento de Análisis Matemático, University of La Laguna, 38271 La Laguna, Tenerife, Spain b
a r t i c l e
i n f o
a b s t r a c t
Keywords: Volterra–Stieltjes integral equation Fractional integral–differential equations Riemann–Liouville fractional operators Existence and stability of solutions Fixed point
Our aim in this paper is to study the existence and the stability of solutions for Riemann–Liouville Volterra–Stieltjes quadratic integral equations of fractional order. Our results are obtained by using some fixed point theorems. Some examples are provided to illustrate the main results. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Fractional differential and integral equations have recently been applied in various areas of engineering, mathematics, physics and bio-engineering and other applied sciences. There has been a significant development in ordinary and partial fractional differential and integral equations in recent years; see the monographs of Abbas et al. [5], Baleanu et al. [7], Diethelm [15], Hilfer [17], Kilbas et al. [18], Lakshmikantham et al. [19], Podlubny [20] and Tarasov [28], and the papers by Abbas et al. [1–3,6], Qian et al. [21–23], Vityuk and Golushkov [29]. Recently interesting results of the stability of the solutions of various classes of integral equations of fractional order have obtained by Abbas et al. [4,5], Banas´ et al. [8–10], Darwish et al. [12], Dhage [13,14] and the references therein. In [8,10], Banas´ et al. proved some existence results for the following nonlinear Volterra–Stieltjes quadratic integral equation
xðtÞ ¼ aðtÞ þ f ðt; xðtÞÞ
Z
t
uðt; s; xðsÞÞds gðt; sÞ;
t P 0;
ð1Þ
0
where a : ½0; 1Þ ! ½0; 1Þ and f : ½0; 1Þ R ! R are given continuous functions. In this paper we establish some sufficient conditions for the existence of solutions of the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations of the form
q
This work has been supported in part by the Government of Spain and FEDER (Grants MTM2010-16499 and MTM2013-41704).
⇑ Corresponding author.
E-mail addresses: [email protected] (S. Abbas), [email protected] (M. Benchohra), [email protected] (M. Rivero), [email protected] (J.J. Trujillo). http://dx.doi.org/10.1016/j.amc.2014.09.023 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
320
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
uðt; xÞ ¼ lðt; xÞ þ
f ðt; x; uðt; xÞÞ CðrÞ
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ð2Þ
0
for ðt; xÞ 2 J :¼ ½0; a ½0; b; a; b > 0; r 2 ð0; 1Þ; l : J ! R; g : Rþ Rþ ! R; f : J R ! R; h : J 1 R ! R are given continuous functions, Rþ ¼ ½0; 1Þ; J 1 ¼ fðt; x; sÞ 2 J ½0; a : s 6 tg, and CðÞ is the Euler’s Gamma function. We present two results for the existence of solutions of the Eq. (2). The first one is based on Schauder’s fixed point theorem (Theorem 3.2) and the second one on the nonlinear alternative of Leray–Schauder type (Theorem 3.5). Next, we establish a sufficient condition for the existence and the stability of solutions of the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations of the form
uðt; xÞ ¼ lðt; xÞ þ
f ðt; x; uðt; xÞÞ CðrÞ
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ð3Þ
0
where ðt; xÞ 2 J 0 :¼ Rþ ½0; b; b > 0; r 2 ð0; 1Þ; l : J 0 ! R; g : Rþ Rþ ! R; f : J 0 R ! R; h : J 01 R ! R are given continuous functions and J 01 ¼ fðt; x; sÞ 2 J 0 Rþ : s 6 tg. We use the Schauder fixed point theorem for the existence of solutions of the Eq. (3), and we prove that all solutions are locally asymptotically stable (Theorem 4.3). 2. Preliminaries In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. By L1 ðJÞ, we denote the space of Lebesgue-integrable functions u : J ! R with the norm
kuk1 ¼
Z
a
Z
0
b
juðt; xÞjdxdt: 0
Let C :¼ CðJÞ be the Banach space of all continuous functions from J into R endowed with the norm
kukC ¼ sup juðt; xÞj: ðt;xÞ2J
By BC :¼ BCðJ 0 Þ we denote the Banach space of all bounded and continuous functions from J 0 into R equipped with the standard norm
kukBC ¼ sup juðt; xÞj: ðt;xÞ2J 0
For u0 2 C ðor u0 2 BCÞ and g 2 ð0; 1Þ, we denote by Bðu0 ; gÞ, the closed ball in C ðor BCÞ centered at u0 with radius g. Definition 2.1 [27]. Let r 2 ð0; 1Þ and u 2 L1 ðJÞ. The partial Riemann–Liouville integral of order r of uðt; xÞ with respect to t is defined by the expression
Ir0;t uðt; xÞ ¼
1 CðrÞ
Z
t
ðt sÞr1 uðs; xÞds;
for almost all ðt; xÞ 2 J:
0
Analogously, we define the integral
Ir0;x uðt; xÞ ¼
1 CðrÞ
Z
x
ðx sÞr1 uðt; sÞds;
for almost all ðt; xÞ 2 J:
0
Example 2.2. Let k; x 2 ð1; 1Þ and r 2 ð0; 1Þ, then
Ir0;t t k xx ¼
Cð1 þ kÞ kþr x t x ; Cð1 þ k þ rÞ
for almost all ðt; xÞ 2 J:
W If u is a real function defined on the interval ½a; b, then the symbol ba u denotes the variation of u on ½a; b. We say that u is Wb W of bounded variation on the interval ½a; b whenever a u is finite. If w : ½a; b ½c; b ! R, then the symbol qt¼p wðt; sÞ indicates the variation of the function t ! wðt; sÞ on the interval ½p; q ½a; b, where s is arbitrarily fixed in ½c; d. In the same way we W define qs¼p wðt; sÞ. For the properties of functions of bounded variation we refer to [24]. If u and u are two real functions defined on the interval ½a; b, then under some conditions (see [24]) we can define the Stieltjes integral (in the Riemann–Stieltjes sense)
Z
b
uðtÞduðtÞ
a
of the function u with respect to u. In this case we say that u is Stieltjes integrable on ½a; b with respect to u. Several conditions are known guaranteeing Stieltjes integrability [24,26]. One of the most frequently used requires that u is continuous and u is of bounded variation on ½a; b.
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
321
In what follows we will use a few properties of the Stieltjes integral contained in the below given lemma ([25], Section 8.13): Lemma 2.3 [25]. If u is Stieltjes integrable on the interval ½a; b with respect to a function u of bounded variation then
Z Z ! b t b _ uðtÞduðtÞ 6 juðtÞjd u : a a a In the sequel we will also consider Stieltjes integrals of the form
Z
b
uðtÞds gðt; sÞ;
a
and Riemann–Liouville Stieltjes integrals of fractional order of the form
1 CðrÞ
Z
t
ðt sÞr1 uðsÞds gðt; sÞ;
0
where g : Rþ Rþ ! R; r 2 ð0; 1Þ and the symbol ds indicates the integration with respect to s. Let ; – X BC, and let G : X ! X, and consider the solutions of equation
ðGuÞðt; xÞ ¼ uðt; xÞ:
ð4Þ
Inspired by the definition of the attractivity of solutions of integral equations (for instance [9]), we introduce the following concept of attractivity of solutions for Eq. (4). Definition 2.4. Solutions of Eq. (4) are locally attractive if there exists a ball Bðu0 ; gÞ in the space BC such that, for arbitrary solutions v ¼ v ðt; xÞ and w ¼ wðt; xÞ of Eq. (4) belonging to Bðu0 ; gÞ \ X, we have that, for each x 2 ½0; b,
limðv ðt; xÞ wðt; xÞÞ ¼ 0:
ð5Þ
t!1
When the limit (5) is uniform with respect to Bðu0 ; gÞ \ X, solutions of Eq. (4) are said to be uniformly locally attractive (or equivalently that solutions of (4) are locally asymptotically stable). Lemma 2.5 [11]. Let D BC. Then D is relatively compact in BC if the following conditions hold: (a) D is uniformly bounded in BC, (b) The functions belonging to D are almost equicontinuous on Rþ ½0; b, i.e. equicontinuous on every compact of Rþ ½0; b, (c) The functions from D are equiconvergent, that is, given > 0; x 2 ½0; b there corresponds Tð; xÞ > 0 such that juðt; xÞ lim uðt; xÞj < for any t P Tð; xÞ and u 2 D. t!1
3. Local existence results Let us start in this section by defining what we mean by a solution of the Eq. (2). Definition 3.1. We mean by a solution of Eq. (2), every function u 2 C such that u satisfies Eq. (2) on J. The following hypotheses will be used in the sequel: ðH1 Þ The function f is a continuous and there exists a positive function p 2 C such that
jf ðt; x; uÞ f ðt; x; v Þj 6 pðt; xÞju v j; ðH2 Þ ðH3 Þ ðH4 Þ ðH5 Þ
ðt; xÞ 2 J;
u; v 2 R:
For all t1 ; t2 2 ½0; a such that t1 < t2 the function s # gðt2 ; sÞ gðt 1 ; sÞ is nondecreasing on ½0; a. The function s # gð0; sÞ is nondecreasing on ½0; a0. The functions s # gðt; sÞ and t # gðt; sÞ are continuous on ½0; a for each fixed t 2 ½0; a or s 2 ½0; a, respectively. The function h is continuous and there exist continuous functions q ¼ qðt; x; sÞ : J 1 ! Rþ ; U : Rþ ! Rþ such that U is nondecreasing and
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ;
ðt; x; sÞ 2 J 1 ;
u 2 R:
Set
l :¼ sup lðt; xÞ; ðt;xÞ2J
f :¼ sup f ðt; x; 0Þ; ðt;xÞ2J
g ¼ sup
t _
gðt; kÞ;
t2½0;ak¼0
322
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ðt sÞr1 qðt; x; sÞ : CðrÞ ðt;x;sÞ2J 1
p :¼ sup pðt; xÞ
and q :¼ sup
ðt;xÞ2J
Now, we shall prove the following theorem concerning the existence of a solution of the Eq. (2), based on Schauder’s fixed point theorem [16]. Theorem 3.2. Assume that the hypotheses ðH1 Þ ðH5 Þ and the following hypothesis hold ðH6 Þ There exists a constant g > 0, such that
l þ q g UðgÞðf þ p gÞ 6 g.
Then the Eq. (2) has at least one solution in the space C. Proof 3.3. Let us define the operator N : C ! C such that, for any u 2 C,
ðNuÞðt; xÞ ¼ lðt; xÞ þ
f ðt; x; uðt; xÞÞ CðrÞ
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ðt; xÞ 2 J:
ð6Þ
0
It is clear that the operator N maps C into C. The problem of finding the solutions of the Eq. (2) is reduced to finding the solutions of the operator equation NðuÞ ¼ u. Hypothesis ðH6 Þ implies that N transforms the ball Bg :¼ Bð0; gÞ into itself. Indeed, for any u 2 Bg , and for each ðt; xÞ 2 J we have
jf ðt; x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞjjds gðt; sÞj þ ! Z t s _ r1 ðt sÞ jhðt; x; s; uðs; xÞÞjds gðt; kÞ
jðNuÞðt; xÞj 6 jlðt; xÞj þ
0
0
k¼0
6 l þ
Z
f CðrÞ t
Z
t
ðt sÞr1 qðt; x; sÞUðjuðs; xÞjÞds
0
! gðt; kÞ þ
k¼0
r1
ðt sÞ
s _
s _
qðt; x; sÞUðjuðs; xÞjÞds
0
jf ðt; x; uðt; xÞÞ f ðt; x; 0Þj CðrÞ
pðt; xÞjuðt; xÞj CðrÞ
!
gðt; kÞ
k¼0
6 l þ q g UðkukC Þðf þ p kukC Þ 6 l þ q g UðgÞðf þ p gÞ 6 g: Hence, NðuÞ 2 Bg . We shall show that N : Bg ! Bg satisfies the assumptions of Schauder’s fixed point theorem [16]. The proof will be given in several steps. Step 1: N is continuous. Let fun gn2N be a sequence such that un ! u in Bg . Then, for each ðt; xÞ 2 J, we have
jðNun Þðt; xÞ ðNuÞðt; xÞj 6 jf ðt; x; un ðt; xÞÞ f ðt; x; uðt; x:ÞÞj þ jf ðt; x; uðt; xÞÞj
Z 0
Z
t
0 t
ðt sÞr1 jhðt; x; s; un ðs; xÞÞjds gðt; sÞj CðrÞ
r1
ðt sÞ CðrÞ
jhðt; x; s; un ðs; xÞÞ hðt; x; s; uðs; xÞÞjjds gðt; sÞj
6 p q g Uðkun kC Þjun ðt; xÞ uðt; xÞj þ ! s _ hðt; x; s; uðs; xÞÞjds gðt; kÞ
f þ p kukC CðrÞ
k¼0
f þ p g 6 p q g UðgÞjun ðt; xÞ uðt; xÞj þ CðrÞ ! s _ hðt; x; s; uðs; xÞÞjds gðt; kÞ :
Z
t
Z
t
ðt sÞr1 jhðt; x; s; un ðs; xÞÞ
0
ðt sÞr1 jhðt; x; s; un ðs; xÞÞ
0
k¼0
Since un ! u as n ! 1 and h is continuous, (7) gives
kNðun Þ NðuÞkC ! 0 as n ! 1: Step 2: NðBg Þ is bounded. This is clear since NðBg Þ Bg and Bg is bounded. Step 3: NðBg Þ is equicontinuous. Let ðt 1 ; x1 Þ; ðt 2 ; x2 Þ 2 J; t 1 < t 2 ; x1 < x2 and let u 2 Bg . Thus we have
ð7Þ
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jðNuÞðt 2 ; x2 Þ ðNuÞðt 1 ; x1 Þj 6 jlðt 2 ; x2 Þ lðt1 ; x1 Þj þ jf ðt 2 ; x2 ; uðt 2 ; x2 ÞÞ f ðt1 ; x1 ; uðt1 ; x1 ÞÞj ! Z t2 s _ ðt 2 sÞr1 gðt 2 ; kÞ jhðt2 ; x2 ; s; uðs; x2 ÞÞjds CðrÞ 0 k¼0 Z Z t1 t2 ðt sÞr1 ðt 1 sÞr1 2 hðt2 ; x2 ; s; uðs; x2 ÞÞds gðt 2 ; sÞ hðt 1 ; x1 ; s; uðs; x1 ÞÞds gðt 1 ; sÞ: þ jf ðt1 ; x1 ; uðt 1 ; x1 ÞÞj 0 CðrÞ C ðrÞ 0 Thus
jðNuÞðt 2 ; x2 Þ ðNuÞðt 1 ; x1 Þj 6 jlðt 2 ; x2 Þ lðt1 ; x1 Þj þ q g UðgÞjf ðt 2 ; x2 ; uðt2 ; x2 ÞÞ f ðt1 ; x1 ; uðt 1 ; x1 ÞÞj Z Z t1 t2 ðt sÞr1 ðt1 sÞr1 2 þ p g hðt 2 ; x2 ; s; uðs; x2 ÞÞds gðt2 ; sÞ hðt1 ; x1 ; s; uðs; x1 ÞÞds gðt 1 ; sÞ: 0 CðrÞ CðrÞ 0 Hence
jðNuÞðt 2 ; x2 Þ ðNuÞðt 1 ; x1 Þj 6 jlðt 2 ; x2 Þ lðt1 ; x1 Þj þ p q g UðgÞjuðt2 ; x2 Þ uðt1 ; x1 Þj þ
Z
t1
0
Z
t1
p g CðrÞ
! s _ p g r1 r1 gðt 1 ; kÞ þ ðt2 sÞ hðt 2 ; x2 ; s; uðs; x2 ÞÞðt1 sÞ hðt 1 ; x1 ; s; uðs; x1 ÞÞds C ðrÞ k¼0 jðt 2 sÞr1 hðt2 ; x2 ; s; uðs; x2 ÞÞjjds ðgðt 2 ; sÞ gðt1 ; sÞÞj þ p q gUðgÞ
0
t2 _
gðt2 ; kÞ:
k¼t 1
From continuity of l; g; h and as t1 ! t 2 and x1 ! x2 , the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3 together with the Arzelá–Ascoli theorem, we can conclude that N : Bg ! Bg is continuous and compact. From an application of Schauder’s theorem [16], we deduce that N has a fixed point u which is a solution of the Eq. (2). h In the sequel, we need the following theorem. Theorem 3.4 ([16] Nonlinear alternative of Leray–Schauder type). By U and @U we denote the closure of U and the boundary of U respectively. Let X be a Banach space and C a nonempty convex subset of X. Let U be a nonempty open subset of C with 0 2 U and T : U ! C completely continuous operator. Then either (a) T has fixed points. Or (b) There exist u 2 @U and k 2 ð0; 1Þ with u ¼ kTðuÞ. Now, we present another result based on the nonlinear alternative of Leray–Schauder type (Theorem 3.4). Theorem 3.5. Assume that the hypotheses ðH1 Þ ðH4 Þ and the following hypothesis holds ðH7 Þ The function h is continuous and there exist continuous functions k1 ; k2 : J 1 ! Rþ such that
jhðt; x; s; uÞj 6 k1 ðt; x; sÞ þ
k2 ðt; x; sÞ ; 1 þ juj
ðt; x; sÞ 2 J 1 ;
u 2 R;
with
ki :¼ sup
ðt;x;sÞ2J 1
1
CðrÞ
ðt sÞr1 ki ðt; x; sÞ;
i ¼ 1; 2:
If
p g k1 < 1;
ð8Þ
then the Eq. (2) has at least one solution in the space C. Proof 3.6. We shall show that the operator N defined in (6) satisfies all the conditions of Theorem 3.4. As in Theorem 3.2, we can show that N is completely continuous. A priori bounds. We shall show there exists an open set U # C with u – kNðuÞ, for k 2 ð0; 1Þ and u 2 @U. Let u 2 C be such that u ¼ kNðuÞ for some 0 < k < 1. Thus for each ðx; yÞ 2 J, we have
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juðt;xÞj 6 jklðt;xÞj þ
jkf ðt;x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x;s;uðs; xÞÞjjds gðt; sÞj
0
! Z s _ jkjjf ðt;x; uðt; xÞÞ f ðt; x;0Þj t r1 þ ðt sÞ jhðt; x; s;uðs;xÞÞjds gðt; kÞ CðrÞ 0 k¼0 ! Z t s _ f k2 ðt; x;sÞ ds ðt sÞr1 k1 ðt;x; sÞ þ gðt;kÞ 6 l þ 1 þ juðs; xÞj CðrÞ 0 k¼0 ! Z s _ pðt; xÞjuðt; xÞj t k2 ðt; x;sÞ r1 ds ðt sÞ k1 ðt; x;sÞ þ gðt;kÞ 6 l þ f g ðk1 þ k2 Þ þ p g k2 þ p g k1 kukC : þ 1 þ juðs;xÞj CðrÞ 0 k¼0 Then,
kukC 6 l þ f g ðk1 þ k2 Þ þ p g k2 þ p g k1 kukC : Thus, by (8) we get
kukC 6
l þ f g ðk1 þ k2 Þ þ p g k2
1 p g k1
:¼ M:
Set
U ¼ fu 2 C : kukC < M þ 1g: By our choice of U, there is no u 2 @U such that u ¼ kNðuÞ, for k 2 ð0; 1Þ. From Theorem 3.4, we deduce that N has a fixed point u in U which is a solution to the Eq. (2). h
4. Global existence and stability of solutions Definition 4.1. We mean by a solution of Eq. (3), every function u 2 BC such that u satisfies Eq. (3) on J 0 . The following hypotheses will be used in the sequel: ðH01 Þ The function l is in BC. Moreover, assume that lim lðt; xÞ ¼ 0; x 2 ½0; b. t!1 ðH02 Þ The function f is a continuous and there exists a positive function p 2 BC such that
jf ðt; x; uÞ f ðt; x; v Þj 6 pðt; xÞju v j;
ðt; xÞ 2 J;
u; v 2 R:
Moreover, assume that lim f ðt; x; 0Þ ¼ 0; x 2 ½0; b. t!1 ðH03 Þ For all t 1 ; t 2 2 Rþ such that t1 < t 2 the function s # gðt 2 ; sÞ gðt 1 ; sÞ is nondecreasing on Rþ . 0 ðH4 Þ The function s # gð0; sÞ is nondecreasing on Rþ . ðH05 Þ The functions s # gðt; sÞ and t # gðt; sÞ are continuous on Rþ for each fixed t 2 Rþ or s 2 Rþ , respectively. ðH06 Þ The function h is continuous and there exist continuous functions q : J 01 ! Rþ ; U : Rþ ! Rþ such that U is nondecreasing and
ðt; x; sÞ 2 J 01 ;
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ; Moreover, assume that lim
t!1
Rt 0
u 2 R:
ðt sÞr1 qðt; x; sÞds gðt; sÞ ¼ 0.
Remark 4.2. Set
l :¼ sup lðt; xÞ; f :¼ sup f ðt; x; 0Þ; ðt;xÞ2J 0
p :¼ sup pðt; xÞ and ðt;xÞ2J 0
ðt;xÞ2J 0
1 ðt;xÞ2J0 CðrÞ
q :¼ sup
Z
t 0
ðt sÞr1 qðt; x; sÞds
s _
! gðt; kÞ :
k¼0
From hypotheses, we infer that l ; f ; p and q are finite. Now, we shall prove the following theorem concerning the existence and the stability of a solution of the Eq. (3). Theorem 4.3. Assume that the hypotheses ðH01 Þ ðH06 Þ and the following hypothesis hold ðH07 Þ There exists a constant g > 0, such that
l þ q UðgÞðf þ p gÞ 6 g.
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
325
Then the Eq. (3) has at least one solution in the space BC. Moreover, if there exists a constant g > 0, such that
q ðf þ p gÞUðgÞ þ ½q f þ p q ðg þ g ÞUðg þ g Þ 6 g ; then solutions of the Eq. (3) are locally asymptotically stable. Proof 4.4. Let us define the operator N 0 such that, for any u 2 BC,
ðN0 uÞðt; xÞ ¼ lðt; xÞ þ
Z
f ðt; x; uðt; xÞÞ CðrÞ
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
ðt; xÞ 2 J 0 :
ð9Þ
0
The operator N 0 maps BC into BC. Indeed the map N 0 ðuÞ is continuous on J 0 for any u 2 BC, and for each ðt; xÞ 2 J 0 we have
jf ðt; x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞjjds gðt; sÞj þ ! Z t s _ r1 ðt sÞ jhðt; x; s; uðs; xÞÞjds gðt; kÞ
jðN0 uÞðt; xÞj 6 jlðt; xÞj þ
0
0
k¼0
6 l þ
Z
f CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞUðjuðs; xÞjÞds
0
t
ðt sÞ
r1
s _
! gðt; kÞ þ
k¼0 s _
qðt; x; sÞUðjuðs; xÞjÞds
0
jf ðt; x; uðt; xÞÞ f ðt; x; 0Þj CðrÞ
pðt; xÞjuðt; xÞj CðrÞ
!
gðt; kÞ
k¼0
! Z s _ f UðkukBC Þ t p kukBC UðkukBC Þ r1 6l þ ðt sÞ qðt; x; sÞds gðt; kÞ þ CðrÞ CðrÞ 0 k¼0 ! Z t s _ ðt sÞr1 qðt; x; sÞds gðt; kÞ
0
k¼0
6 l þ q UðkukBC Þðf þ p kukBC Þ: Thus,
kN0 ðuÞkBC 6 l þ q UðkukBC Þðf þ p kukBC Þ: 0
ð10Þ 0
Hence, N ðuÞ 2 BC. This proves that the operator N maps BC into itself. The problem of finding the solutions of the Eq. (3) is reduced to finding the solutions of the operator equation N 0 ðuÞ ¼ u. Hypothesis ðH07 Þ implies that N 0 transforms the ball Bg :¼ Bð0; gÞ into itself. We shall show that N 0 : Bg ! Bg satisfies the assumptions of Schauder’s fixed point theorem [16]. As in the proof of Theorem 1, we can show that N 0 is continuous, N 0 ðBg Þ is uniformly bounded, and equicontinuous on every compact subset ½0; a ½0; b of J 0 ; a > 0. Step 1: N 0 ðBg Þ is equiconvergent. Let ðt; xÞ 2 Rþ ½0; b and u 2 Bg , then we have
jf ðt; x; 0Þj CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞjjds gðt; sÞj þ ! Z t s _ ðt sÞr1 jhðt; x; s; uðs; xÞÞjds gðt; kÞ
jðNuÞðt; xÞj 6 jlðt; xÞj þ
0
0
jf ðt; x; uðt; xÞÞ f ðt; x; 0Þj CðrÞ
k¼0
! Z t s _ f pðt; xÞjuðt; xÞj r1 6 jlðt; xÞj þ ðt sÞ qðt; x; sÞUðjuðs; xÞjÞds gðt; kÞ þ CðrÞ 0 CðrÞ k¼0 ! Z t s _ ðt sÞr1 qðt; x; sÞUðjuðs; xÞjÞds gðt; kÞ 0
k¼0
6 jlðt; xÞj þ
UðgÞðf þ p gÞ CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞds
0
Thus, for each x 2 ½0; b, we get
jðNuÞðt; xÞj ! 0;
as t ! þ1:
Hence,
jðNuÞðt; xÞ ðNuÞðþ1; xÞj ! 0;
as t ! þ1:
s _
! gðt; kÞ :
k¼0
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S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
As a consequence of Step 1 together with Lemma 2.5, we can conclude that N 0 : Bg ! Bg is continuous and compact. From an application of Schauder’s theorem [16], we deduce that N 0 has a fixed point u which is a solution of the Eq. (3). Step 2: The local asymptotic stability of solutions. Now we investigate the local asymptotic stability of solutions of the Eq. (3). Let us assume that u0 is a solution of the Eq. (3) with the conditions of this theorem. Taking u 2 Bðu0 ; g Þ, we have
jðN0 uÞðt; xÞ u0 ðt; xÞj ¼ jðN0 uÞðt; xÞ ðN0 u0 Þðt; xÞj
6 p q Uðg þ g Þjuðt; xÞ u0 ðt; xÞj þ ! s _ hðt; x; s; u0 ðs; xÞÞjds gðt; kÞ
f þ p g CðrÞ
k¼0
6 p q Uðg þ g Þjuðt; xÞ u0 ðt; xÞj þ ! s _ þ Uðju0 ðs; xÞjÞÞds gðt; kÞ
f þ p g CðrÞ
Z
t
ðt sÞr1 jhðt; x; s; uðs; xÞÞ
0
Z
t
ðt sÞr1 qðt; x; sÞðUðjuðs; xÞjÞ
0
k¼0
6 p q g Uðg þ g Þ þ q ðf þ p gÞ½UðgÞ þ Uðg þ g Þ 6 g : Thus we observe that N 0 is a continuous function such that
N0 ðBðu0 ; g ÞÞ Bðu0 ; g Þ: Moreover, if u is a solution of the Eq. (3), then
juðt; xÞ u0 ðt; xÞj ¼ jðN0 uÞðt; xÞ ðN0 u0 Þðt; xÞj 6 p Uðg þ g Þjuðt; xÞ u0 ðt; xÞj
Z
t
ðt sÞ
r1
1 CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞds
0
p g Uðg þ g Þ CðrÞ
s _
jhðt; x; s; uðs; xÞÞ hðt; x; s; u0 ðs; xÞÞjds Z
t
ðt sÞr1 qðt; x; sÞds
0
þ Uðju0 ðs; xÞjÞÞds
s _
!
s _
f þ p g CðrÞ
!
gðt; kÞ
k¼0
!
gðt; kÞ þ
k¼0
f þ p g CðrÞ
Z
t
ðt sÞr1 qðt; x; sÞðUðjuðs; xÞjÞ
0
gðt; kÞ
k¼0
gðt; kÞ þ
k¼0
0
6
!
s _
6 ðp g Uðg þ g Þ þ ðf þ p gÞ½UðgÞ þ Uðg þ g ÞÞ
Z
t
0
! s _ ðt sÞr1 qðt; x; sÞds gðt; kÞ : CðrÞ k¼0
Thus
juðt; xÞ u0 ðt; xÞj 6 ðp g Uðg þ g Þ þ ðf þ p gÞ½UðgÞ þ Uðg þ g ÞÞ
Z 0
t
! s _ ðt sÞr1 qðt; x; sÞds gðt; kÞ : CðrÞ k¼0
ð11Þ
By using (11) we deduce that
lim juðt; xÞ u0 ðt; xÞj ¼ 0:
t!1
Consequently, all solutions of the Eq. (3) are locally asymptotically stable.
h
5. Examples As applications and to illustrate our results, we present two examples. Example 1. Consider the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations
uðt; xÞ ¼ lðt; xÞ þ where r ¼ 14 ;
f ðt; x; uðt; xÞÞ CðrÞ
Z 0
lðt; xÞ ¼ 2þt1 2 ; t; x 2 ½0; 1,
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
for ðt; xÞ 2 J :¼ ½0; 1 ½0; 1;
ð12Þ
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
f ðt; x; uÞ ¼
ext juj ; 1 þ juj
t;
x 2 ½0; 1; u 2 R;
gðt; sÞ ¼ s;
t;
hðt; x; s; uÞ ¼
3 cxjuj ðt sÞ4 sin t sin s; 1 þ juj
c¼
Cð14Þ 8e
327
s 2 ½0; 1; ðt; x; sÞ 2 J 1 ;
J 1 ¼ fðt; x; sÞ 2 ½0; 13 : s 6 tg:
and
The function f is a continuous, f ¼ 0, and
jf ðt; x; uÞ f ðt; x; v Þj 6 ext ju v j;
t;
x 2 ½0; 1; u;
v 2 R:
xt
Then, the assumption ðH1 Þ is satisfies with pðt; xÞ ¼ e ; t; x 2 ½0; 1, and then p ¼ e. Also, we can easily see that the function g satisfies the hypotheses ðH2 Þ ðH4 Þ. The function h satisfies the assumption ðH5 Þ. Indeed, h is continuous and
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ;
ðt; x; sÞ 2 J 1 ;
u 2 R;
where U : Rþ ! Rþ : UðwÞ ¼ w, and 3
qðt; x; sÞ ¼ cxðt sÞ4 sin t sin s;
ðt; x; sÞ 2 J 1 :
Then, 3
ðt sÞ 4 qðt; x; sÞ 1 ¼ : 8e Cð14Þ ðt;x;sÞ2J 1
q ¼ sup
Finally, we can see that the hypothesis ðH6 Þ is satisfies with g ¼ 1. Indeed, we have
l ¼ 12 ; g ¼ 1 and the inequality
l þ q g UðgÞðf þ p gÞ 6 g implies
1 1 2 þ g 6 g; 2 8 which is satisfies for g ¼ 1. Consequently, by Theorem 3.2, the Eq. (12) has a solution defined on ½0; 1 ½0; 1. Example 2. Consider now the following fractional order Riemann–Liouville Volterra–Stieltjes quadratic integral equations
uðt; xÞ ¼ lðt; xÞ þ where r ¼ 14 ;
Z
t
ðt sÞr1 hðt; x; s; uðs; xÞÞds gðt; sÞ;
for ðt; xÞ 2 J 0 :¼ Rþ ½0; 1;
ð13Þ
0
lðt; xÞ ¼ 2þt1 2 ; ðt; xÞ 2 J0 ,
f ðt; x; uÞ ¼
ext juj ; 1 þ juj
hðt; x; s; uÞ ¼
pffiffiffiffi
p
8eCð14Þ
ðt; xÞ 2 J 0 ;
u 2 R; gðt; sÞ ¼ s; ðt; sÞ 2 R2þ ;
pffiffi 3 cxs 4 juj sin t sin s ; ð1 þ t2 Þð2 þ jujÞ
hðt; x; 0; uÞ ¼ 0; c¼
f ðt; x; uðt; xÞÞ CðrÞ
ðt; xÞ 2 J
ðt; x; sÞ 2 J 01 ;
s – 0 and u 2 R;
and u 2 R;
and J 01 ¼ fðt; x; sÞ 2 J Rþ : s 6 t;
First, we can see that lim t!þ1 f ¼ 0, and
1 2þt 2
¼ 0, then the assumption ðH01 Þ is satisfies and
jf ðt; x; uÞ f ðt; x; v Þj 6 ext ju v j; ðH02 Þ
x 2 ½0; 1g:
ðt; xÞ 2 J 0 ;
l ¼ 12. Next, the function f is a continuous,
u; v 2 R: xt
Then, the assumption is satisfies with pðt; xÞ ¼ e ; ðt; xÞ 2 J, and then p ¼ e. Also, we can easily see that the function g satisfies the hypotheses ðH03 Þ ðH05 Þ. The function h satisfies the assumption ðH06 Þ. Indeed, h is continuous and
328
S. Abbas et al. / Applied Mathematics and Computation 247 (2014) 319–328
ðt; x; sÞ 2 J 01 ;
jhðt; x; s; uÞj 6 qðt; x; sÞUðjujÞ;
u 2 R;
where U : Rþ ! Rþ : UðwÞ ¼ w, and
qðt; x; sÞ ¼
pffiffi 3 cxs 4 sin t sin s ; 1 þ t2
ðt; x; sÞ 2 J 01 ;
s – 0;
ðt; xÞ 2 J 0 :
qðt; x; 0Þ ¼ 0; Then,
! Z t Z t s _ pffiffi pffiffi Z t 3 3 3 3 ðt sÞr1 qðt; x; sÞds gðt; sÞ 6 4 4 ðt sÞ cxs j sin t sin sjds gðt; kÞ 6 cxj sin t j ðt sÞ 4 s 4 ds 0
0
6
k¼0
pffiffi 2 1 ð4Þ sin t
0
2 1 ð4Þ
cxC pffiffiffiffi
cxC
pffiffi 6 pffiffiffiffiffiffi !0 as t ! 1; p t pt
and
1 q :¼ sup ðt;xÞ2J CðrÞ
Z
t
r1
ðt sÞ
qðt; x; sÞds
0
k¼0
Finally, we can see that hypothesis
s _
ðH07 Þ
! gðt; kÞ
pffiffi cxCð1Þ sin t cCð1Þ 1 6 sup pffiffiffiffi4 pffiffi ¼ pffiffiffi4ffi ¼ : p p 8e t ðt;xÞ2J
is satisfies with g ¼ 1. Indeed, the inequality
l þ q UðgÞðf þ p gÞ 6 g implies 12 þ 18 g2 6 g, which is satisfies for g ¼ 1. Moreover, the inequality
q ðf þ p gÞUðgÞ þ ½q f þ p q ðg þ g ÞUðg þ g Þ 6 g is satisfies with g ¼ g ¼ 1 and gives 58 6 1. Consequently, by Theorem 4.3, Eq. (13) has a solution defined on Rþ ½0; 1 and all solutions are locally asymptotically stable on Rþ ½0; 1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
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