Nonlinear fractional integro-differential equations ... - Semantic Scholar
Journal of Computational and Applied Mathematics 249 (2013) 51–56
Contents lists available at SciVerse ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Nonlinear fractional integro-differential equations on unbounded domains in a Banach space Lihong Zhang a , Bashir Ahmad b , Guotao Wang a,∗ , Ravi P. Agarwal c,b a
School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, People’s Republic of China
b
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
c
Department of Mathematics, Texas A and M University, Kingsville, TX 78363-8202, USA
article
info
Article history: Received 9 November 2012 Received in revised form 16 January 2013 MSC: 34A08 34B15 34B40 45J05
abstract In this paper, by employing the fixed point theory and the monotone iterative technique, we investigate the existence of a unique solution for a class of nonlinear fractional integrodifferential equations on semi-infinite domains in a Banach space. An explicit iterative sequence for approximating the solution of the boundary value problem is derived. An error estimate is also given. © 2013 Elsevier B.V. All rights reserved.
Keywords: Nonlinear fractional integro-differential equations Banach space Explicit iterative sequence Error estimate Infinite interval
1. Introduction The extensive application of fractional calculus in the mathematical modelling of physical, engineering and biological phenomena has motivated several researchers to explore theoretical as well as practical aspects of the subject. It is learnt through experimentation that the integral and derivative operators of fractional order do share some of the characteristics exhibited by the processes associated with complex systems having long-memory in time. Thus fractional models are the natural substitutes of the classical integer-order model for such systems. Fractional calculus also provides an excellent tool to describe the hereditary properties of various materials and processes. Concerning the development of theory, methods and applications of fractional calculus, we refer the books [1–5]. Some recent results on fractional differential equations with finite domain, for instance, can be found in papers [6–14] and the references cited therein. Though much of the work on fractional calculus deals with finite domain yet there is a considerable development on the topic involving unbounded domain [15–23]. In this paper, we investigate the existence of solutions for a fractional nonlinear integro-differential equation of mixed type on a semi-infinite interval in a Banach space E:
Dα u(t ) + f (t , u(t ), Tu(t ), Su(t )) = θ , u(0) = u′ (0) = u′′ (0) = · · · = u(n−2) (0) = θ ,
n − 1 < α ≤ n, n ∈ N, n ≥ 2, Dα−1 u(∞) = u∞ ,
∗
(1.1)
Corresponding author. Tel.: +86 18935042188. E-mail addresses: [email protected] (L. Zhang), [email protected] (B. Ahmad), [email protected] (G. Wang), [email protected] (R.P. Agarwal). 0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.02.010
52
L. Zhang et al. / Journal of Computational and Applied Mathematics 249 (2013) 51–56
where t ∈ J = [0, +∞), f ∈ C [J × E × E × E , E ], u∞ ∈ E, Dα is the Riemann–Liouville fractional derivatives, θ refers to the zero in the space E and
(Tu)(t ) =
t
k(t , s)u(s)ds,
(Su)(t ) =
∞
h(t , s)u(s)ds, 0
0
with k(t , s) ∈ C [D, R], h(t , s) ∈ C [D0 , R], D = {(t , s) ∈ R2 | 0 ≤ s ≤ t }, D0 = {(t , s) ∈ J × J }. 2. Preliminaries and some lemmas In this section, we will introduce notations, definitions and some useful lemmas, which play an important role in obtaining the main results of this paper. We begin with some basic definitions [1,2]. Definition 2.1. The Riemann–Liouville fractional derivative of order δ for a continuous function f : [0, ∞) → R is defined by
1
δ
D f (t ) =
Γ (n − δ)
d
n
dt
t
(t − s)n−δ−1 f (s)ds,
n = [δ] + 1,
0
provided the right hand side is defined pointwise on (0, ∞). Definition 2.2. The Riemann–Liouville fractional integral of order δ for a continuous function f : [0, ∞) → R is defined by I f (t ) =
t
1
δ
Γ (δ)
(t − s)δ−1 f (s)ds,
δ > 0,
0
provided that the integral exists. Throughout this paper, we assume that the following condition holds: (H ) there exists constants k∗ and h∗ such that t
∗
|k(t , s)|ds < ∞,
k = sup t ∈J
h = sup t ∈J
0
1
∗
(1 + t α−1 )
∞
|h(t , s)|(1 + sα−1 )ds < ∞
0
and ∞
|h(t ′ , s) − h(t , s)|(1 + sα−1 )ds = 0,
lim
t ′ →t
t, t′ ∈ J.
0
Define
∥u(t )∥ FC (J , E ) = u ∈ C (J , E ) : sup
Contents lists available at SciVerse ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Nonlinear fractional integro-differential equations on unbounded domains in a Banach space Lihong Zhang a , Bashir Ahmad b , Guotao Wang a,∗ , Ravi P. Agarwal c,b a
School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, People’s Republic of China
b
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
c
Department of Mathematics, Texas A and M University, Kingsville, TX 78363-8202, USA
article
info
Article history: Received 9 November 2012 Received in revised form 16 January 2013 MSC: 34A08 34B15 34B40 45J05
abstract In this paper, by employing the fixed point theory and the monotone iterative technique, we investigate the existence of a unique solution for a class of nonlinear fractional integrodifferential equations on semi-infinite domains in a Banach space. An explicit iterative sequence for approximating the solution of the boundary value problem is derived. An error estimate is also given. © 2013 Elsevier B.V. All rights reserved.
Keywords: Nonlinear fractional integro-differential equations Banach space Explicit iterative sequence Error estimate Infinite interval
1. Introduction The extensive application of fractional calculus in the mathematical modelling of physical, engineering and biological phenomena has motivated several researchers to explore theoretical as well as practical aspects of the subject. It is learnt through experimentation that the integral and derivative operators of fractional order do share some of the characteristics exhibited by the processes associated with complex systems having long-memory in time. Thus fractional models are the natural substitutes of the classical integer-order model for such systems. Fractional calculus also provides an excellent tool to describe the hereditary properties of various materials and processes. Concerning the development of theory, methods and applications of fractional calculus, we refer the books [1–5]. Some recent results on fractional differential equations with finite domain, for instance, can be found in papers [6–14] and the references cited therein. Though much of the work on fractional calculus deals with finite domain yet there is a considerable development on the topic involving unbounded domain [15–23]. In this paper, we investigate the existence of solutions for a fractional nonlinear integro-differential equation of mixed type on a semi-infinite interval in a Banach space E:
Dα u(t ) + f (t , u(t ), Tu(t ), Su(t )) = θ , u(0) = u′ (0) = u′′ (0) = · · · = u(n−2) (0) = θ ,
n − 1 < α ≤ n, n ∈ N, n ≥ 2, Dα−1 u(∞) = u∞ ,
∗
(1.1)
Corresponding author. Tel.: +86 18935042188. E-mail addresses: [email protected] (L. Zhang), [email protected] (B. Ahmad), [email protected] (G. Wang), [email protected] (R.P. Agarwal). 0377-0427/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cam.2013.02.010
52
L. Zhang et al. / Journal of Computational and Applied Mathematics 249 (2013) 51–56
where t ∈ J = [0, +∞), f ∈ C [J × E × E × E , E ], u∞ ∈ E, Dα is the Riemann–Liouville fractional derivatives, θ refers to the zero in the space E and
(Tu)(t ) =
t
k(t , s)u(s)ds,
(Su)(t ) =
∞
h(t , s)u(s)ds, 0
0
with k(t , s) ∈ C [D, R], h(t , s) ∈ C [D0 , R], D = {(t , s) ∈ R2 | 0 ≤ s ≤ t }, D0 = {(t , s) ∈ J × J }. 2. Preliminaries and some lemmas In this section, we will introduce notations, definitions and some useful lemmas, which play an important role in obtaining the main results of this paper. We begin with some basic definitions [1,2]. Definition 2.1. The Riemann–Liouville fractional derivative of order δ for a continuous function f : [0, ∞) → R is defined by
1
δ
D f (t ) =
Γ (n − δ)
d
n
dt
t
(t − s)n−δ−1 f (s)ds,
n = [δ] + 1,
0
provided the right hand side is defined pointwise on (0, ∞). Definition 2.2. The Riemann–Liouville fractional integral of order δ for a continuous function f : [0, ∞) → R is defined by I f (t ) =
t
1
δ
Γ (δ)
(t − s)δ−1 f (s)ds,
δ > 0,
0
provided that the integral exists. Throughout this paper, we assume that the following condition holds: (H ) there exists constants k∗ and h∗ such that t
∗
|k(t , s)|ds < ∞,
k = sup t ∈J
h = sup t ∈J
0
1
∗
(1 + t α−1 )
∞
|h(t , s)|(1 + sα−1 )ds < ∞
0
and ∞
|h(t ′ , s) − h(t , s)|(1 + sα−1 )ds = 0,
lim
t ′ →t
t, t′ ∈ J.
0
Define
∥u(t )∥ FC (J , E ) = u ∈ C (J , E ) : sup
