Approximate Controllability for Impulsive Riemann-Liouville Fractional ...

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 639492, 17 pages http://dx.doi.org/10.1155/2013/639492

Research Article Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions Zhenhai Liu and Maojun Bin Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis, College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China Correspondence should be addressed to Zhenhai Liu; [email protected] Received 21 July 2013; Revised 25 September 2013; Accepted 2 November 2013 Academic Editor: Naseer Shahzad Copyright © 2013 Z. Liu and M. Bin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the control systems governed by impulsive Riemann-Liouville fractional differential inclusions and their approximate controllability in Banach space. Firstly, we introduce the 𝑃𝐶1−𝛼 -mild solutions for the impulsive Riemann-Liouville fractional differential inclusions in Banach spaces. Secondly, by using the fractional power of operators and a fixed point theorem for multivalued maps, we establish sufficient conditions for the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions, which is a generalization and continuation of the recent results on this issue. At the end, we give an example to illustrate the application of the abstract results.

1. Introduction The concept of controllability plays an important part in the analysis and design of control systems. Since Kalman [1] first introduced its definition in 1963, controllability of the deterministic and stochastic dynamical control systems in finite-dimensional and infinite-dimensional spaces is well developed in different classes of approaches, and more details can be found in papers [2–4]. Some authors [5–7] have studied the exact controllability for nonlinear evolution systems by using the fixed point theorems. In [5–7], to prove the controllability results for fractional-order semilinear systems, the authors made an assumption that the semigroup associated with the linear part is compact. But if 𝐶0 -semigroup 𝑇(𝑡) is compact or the operator 𝐵 is compact, then the controllability operator is also compact and hence the inverse of it does not exist if the state space 𝑉 is infinite dimensional [8]. Thus, it is shown that the concept of exact controllability is difficult to be satisfied in infinite-dimensional space. Therefore, it is important to study the weaker concept of controllability, namely, approximate controllability for differential equations. In these years, several researchers [9–17] have studied it for control systems.

In [13], Sakthivel et al. studied on the approximate controllability of semilinear fractional differential systems: 𝑐

𝐷𝛼 𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) + 𝑓 (𝑡, 𝑥 (𝑡)) ,

𝑡 ∈ 𝐽 = [0, 𝑇] ,

𝑥 (0) = 𝑥0 , (1) where 𝑐 𝐷𝑡𝛼 is Caputo’s fractional derivative of 0 < 𝛼 < 1 and 𝐴 is the infinitesimal generator of a 𝐶0 -semigroup 𝑇(𝑡) of bounded operators on the Hilbert space 𝑋; the control function 𝑢(⋅) is given in 𝐿2 (𝐽, 𝑈); 𝑈 is a Hilbert space; 𝐵 is a bounded linear operator from 𝑈 to 𝑋; 𝑓 : 𝐽 × 𝑋 → 𝑋 is a given function satisfying some assumptions and 𝑥0 is an element of the Hilbert space 𝑋. In [16], Sukavanam and Kumar researched approximate controllability of fractional-order semilinear delay systems: 𝑑𝛼 𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) + 𝑓 (𝑡, 𝑥𝑡 , 𝑢 (𝑡)) , 𝑡𝛼 𝑥0 (𝜃) = 𝜙 (𝜃) ,

𝑡 ∈ [0, 𝜏] ,

(2)

𝜃 ∈ [−ℎ, 0] ,

where 1/2 < 𝛼 < 1; 𝐴 : 𝐷(𝐴) ⊆ 𝑉 → 𝑉 is a closed linear operator with dense domain 𝐷(𝐴) generating

2

Abstract and Applied Analysis

a 𝐶0 -semigroup 𝑆(𝑡); the state 𝑥(⋅) takes values in the Banach ̂ 𝐵 is a space 𝑉; the control function 𝑢(⋅) takes values in 𝑉; 2 2 ̂ bounded linear operator from 𝐿 ([0, 𝜏]; 𝑉) to 𝐿 ([0, 𝜏]; 𝑉); ̂ → 𝑉 is nonlinear. the operator 𝑓 : [0, 𝜏] × 𝐶([−ℎ, 0]; 𝑉) × 𝑉 If 𝑥 : [−ℎ, 𝜏] → 𝑉 is a continuous function, then 𝑥𝑡 : [−ℎ, 0] → 𝑉 is defined as 𝑥𝑡 (𝜃) = 𝑥(𝑡 + 𝜃) for 𝜃 ∈ [−ℎ, 0] and 𝜙 ∈ 𝐶([−ℎ, 0]; 𝑉). In [18], Rykaczewski studied the approximate controllability of an inclusion of the form 𝑥̇ (𝑡) ∈ 𝐴𝑥 (𝑡) + 𝐹 (𝑡, 𝑥 (𝑡)) + 𝐵𝑢 (𝑡) ,

𝑡 ∈ 𝐽 = [0, 𝑏] ,

𝑥 (0) = 𝑥0 ,

(3)

where 𝐴 is a linear operator which generates a compact semigroup, 𝐹 is u.h.c. multivalued perturbation with weakly compact values, and the state 𝑥(⋅) takes values in the Hilbert space 𝐻. 𝑈 is a Hilbert space of all admissible controls. 𝐵 : 𝑈 → 𝐻 is a continuous linear operator. Fractional differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, and economics. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth; see [19–27] for example. As a consequence there was an intensive development of the theory of differential equations of fractional order. One can see the monographs of Kilbas et al. [28] and Podlubny [29] and the references therein. The definitions of Riemann-Liouville fractional derivatives or integrals with initial conditions are strong tools to resolve some fractional differential problems in the real world. Heymans and Podlubny [30] have verified that it was possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals, and such initial conditions are more appropriate than physically interpretable initial conditions. Furthermore, they have investigated that the impulse response with RiemannLiouville fractional derivatives was seldom used in the fields of physics, such as viscoelasticity. In recent years, many authors [18, 27, 31] were devoted to mild solutions to fractional evolution equations with Caputo fractional derivative, and there have been a lot of interesting works. As for the study of the fractional differential systems with Caputo fractional derivative, we can refer to [27, 31, 32] for the existence results. Its approximate controllability was considered in [9, 13–16]. The approximate controllability of Caputo fractional inclusion systems has been investigated by [10]. We know that differential inclusions are strong tools to solve some problems in various fields of engineering, physics, and optimal control; see [10, 32–35]. However, the approximate controllability for the impulsive fractional differential evolution inclusion with Riemann-Liouville fractional derivatives is still an untreated topic in the literature. Motivated by the above work, in this paper, we consider the following system: 𝐷𝑡𝛼 𝑥 (𝑡) ∈ 𝐴𝑥 (𝑡) + 𝐹 (𝑡, 𝑥 (𝑡)) + 𝐵𝑢 (𝑡) , 𝑡 ∈ 𝐽󸀠 = (0, 𝑏] \ {𝑡1 , 𝑡2 , . . . , 𝑡𝑚 } ,

󵄨󵄨 − Δ𝐼01−𝛼 + 𝑥 (𝑡)󵄨 󵄨󵄨𝑡=𝑡 = 𝐼𝑘 (𝑥 (𝑡𝑘 )) , 𝑘

𝑘 = 1, 2, . . . , 𝑚,

󵄨 𝐼𝑡1−𝛼 𝑥 (𝑡)󵄨󵄨󵄨󵄨𝑡=0 = 𝑥0 ∈ 𝑋, (4) where 1/2 < 𝛼 ≤ 1 and 𝐷𝑡𝛼 denotes the Riemann-Liouville fractional derivative of order 𝛼 with the lower limit zero. 𝐹 : 𝐽 × 𝑋 → P(𝑋) := 2𝑋 \ {0} is a nonempty, bounded, closed, and convex multivalued map. 𝐴 : 𝐷(𝐴) ⊆ 𝑋 → 𝑋 is the infinitesimal generator of a 𝐶0 -semigroup 𝑇(𝑡) (𝑡 ≥ 0) on a Banach space 𝑋. 0 = 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑚 < 𝑡𝑚+1 = 𝑏, 𝐼𝑘 : 1−𝛼 + 1−𝛼 − 1−𝛼 + 𝑋 󳨃→ 𝑋, Δ𝐼01−𝛼 + 𝑥(𝑡𝑘 ) = 𝐼0+ 𝑥(𝑡𝑘 ) − 𝐼0+ 𝑥(𝑡𝑘 ), 𝐼0+ 𝑥(𝑡𝑘 ) and 1−𝛼 − 1−𝛼 𝐼0+ 𝑥(𝑡𝑘 ) denote the right and the left limits of 𝐼0+ 𝑥(𝑡) at 𝑡 = 𝑡𝑘 , 𝑘 = 1, 2, . . . , 𝑚. The control function 𝑢(𝑡) takes value in 𝑉 = 𝐿𝑝 ([0, 𝑏]; 𝑈), 𝑝 > 1/𝛼, and 𝑈 is a Banach space; 𝐵 is a linear operator from 𝑉 to 𝐿𝑝 ([0, 𝑏]; 𝑋). The purpose of this paper is to provide some suitable sufficient conditions for the existence of mild solutions and approximate controllability results for the impulsive fractional abstract Cauchy problems with Riemann-Liouville fractional derivatives. The main tools used in our study are fixed point theorem, semigroup theory for multivalued maps, and the theory from fractional differential equations. The rest of this paper is organized as follows. In Section 2, we present some preliminaries to prove our main results. In Section 3, by applying some standard fixed point principles, we prove the existence of the mild solutions for semilinear fractional differential equations, and the approximate controllability of the system (4) is proved. In Section 4, we give an example to illustrate our main results.

2. Preliminaries In this section, we introduce some basic definitions and preliminaries which are used throughout this paper. The norm of a Banach space 𝑋 will be denoted by ‖ ⋅ ‖𝑋 . 𝐿 𝑏 (𝑋, 𝑌) denotes the space of bounded linear operators from 𝑋 to 𝑌. For the uniformly bounded 𝐶0 -semigroup 𝑇(𝑡) (𝑡 ≥ 0), we set 𝑀 := sup𝑡∈[0,∞) ‖𝑇(𝑡)‖𝐿 𝑏 (𝑋) < ∞. Let 𝐶(𝐽, 𝑋) denote the Banach space of all 𝑋 value continuous functions from 𝐽 = [0, 𝑏] to 𝑋 with the norm ‖𝑥‖𝐶 = sup𝑡∈𝐽 ‖𝑥(𝑡)‖𝑋 . Let 𝐶1−𝛼 (𝐽, 𝑋) = {𝑥 : 𝑡1−𝛼 𝑥(𝑡) ∈ 𝐶(𝐽, 𝑋)} with the norm ‖𝑥‖𝐶1−𝛼 = sup {𝑡1−𝛼 ‖𝑥 (𝑡)‖𝑋 : 𝑡 ∈ 𝐽} .

(5)

Obviously, the space 𝐶1−𝛼 (𝐽, 𝑋) is a Banach space. To define the mild solutions of (4), we also consider the Banach space 𝑃𝐶1−𝛼 (𝐽, 𝑋) = {𝑥 : (𝑡 − 𝑡𝑘 )1−𝛼 𝑥(𝑡) ∈ 𝐶((𝑡𝑘 , 𝑡𝑘+1 ], 𝑋), lim𝑡 → 𝑡𝑘+ (𝑡 − 𝑡𝑘 )1−𝛼 𝑥(𝑡) exist, and 𝑥(𝑡𝑘− ) with 𝑥(𝑡𝑘− ) = 𝑥(𝑡𝑘 ), 𝑘 = 0, 1, 2, . . . , 𝑚} with the norm ‖𝑥‖𝑃𝐶1−𝛼 = max {sup (𝑡 − 𝑡𝑘 )

1−𝛼

‖𝑥 (𝑡)‖𝑋 : 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ] ,

𝑘 = 0, 1, 2, . . . , 𝑚} .

(6)

It is easily known that the space 𝑃𝐶1−𝛼 (𝐽, 𝑋) is a Banach space.


3

Let (𝑋, 𝑑) be a metric space. We use the notations

Proof. If 𝑡 ∈ [0, 𝑡1 ], then, by Lemma 3, we easily get 𝛼−1 󵄨 𝑡 . 𝐼0𝛼+ 𝐷0𝛼+ 𝑥 (𝑡) = 𝑥 (𝑡) − 𝑥1−𝛼 (𝑡)󵄨󵄨󵄨𝑡=0 Γ (𝛼)

𝑃𝑐𝑙 (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 is closed} , 𝑃𝑏 (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 is bounded} , 𝑃𝑐V (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 is convex} ,

(7)

Firstly, let us recall the following basic definitions from fractional calculus. For more details, one can see [28, 29]. Definition 1. The integral 𝐼𝑡𝛼 𝑓 (𝑡) =

𝑡 1 ∫ (𝑡 − 𝑠)𝛼−1 𝑓 (𝑠) 𝑑𝑠, Γ (𝛼) 0

If 𝑡 ∈ (𝑡1 , 𝑡2 ], since 𝐼𝑡𝛼 𝐷𝑡𝛼 𝑥 (𝑡) =

𝑃𝑐𝑝 (𝑋) = {𝑌 ∈ P (𝑋) : 𝑌 is compact} .

𝑡 𝑑 1 = { ∫ (𝑡 − 𝑠)𝛼 𝐿 𝐷𝑡𝛼 𝑥 (𝑠) 𝑑𝑠} , 𝑑𝑡 Γ (𝛼 + 1) 0

then, by (13), we have

(8)

is called Riemann-Liouville fractional integral of order 𝛼, where Γ is the gamma function.

=

𝑡 𝑑 1 ∫ (𝑡 − 𝑠)𝛼 {𝐼01−𝛼 + 𝑥 (𝑠)} 𝑑𝑠 Γ (𝛼 + 1) 0 𝑑𝑠

=

1 Γ (𝛼 + 1) Γ (1 − 𝛼)

Definition 2. For a function 𝑓(𝑡) given in the interval [0, ∞), the expression 𝑑 𝑛 𝑡 1 ( ) ∫ (𝑡 − 𝑠)𝑛−𝛼−1 𝑓 (𝑠) 𝑑𝑡, 𝐷𝑡𝛼 𝑓 (𝑡) = Γ (𝑛 − 𝛼) 𝑑𝑡 0

𝑡

× ∫ (𝑡 − 𝑠)𝛼 0

(9)

=

× ∫ (𝑡 − 𝑠)𝛼 0

+

Lemma 3 (see [28]). Let 𝛼 > 0, 𝑚 = [𝛼] + 1, and let 𝑥𝑚−𝛼 (𝑡) = 𝐼𝑡𝑚−𝛼 𝑥(𝑡) be the fractional integral of order 𝑚 − 𝛼. If 𝑥(𝑡) ∈ 𝐿1 (𝐽, 𝑋) and 𝑥𝑚−𝛼 (𝑡) ∈ 𝐴𝐶𝑚 (𝐽, 𝑋), then one has the following equality: 𝑥(𝑚−𝑘) (0) 𝛼−𝑘 𝐼𝑡𝛼 𝐷𝑡𝛼 𝑥 (𝑡) = 𝑥 (𝑡) − ∑ 𝑚−𝛼 𝑡 . Γ (𝛼 − 𝑘 + 1) 𝑘=1

𝑡

𝑡1

+ (10)

Lemma 4. Let 0 < 𝛼 ≤ 1, and let 𝑥1−𝛼 (𝑡) = 𝐼𝑡1−𝛼 𝑥(𝑡) be the fractional integral of order 1 − 𝛼. If 𝑥(𝑡) ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋) and 𝑥1−𝛼 (𝑡) ∈ 𝑃𝐶(𝐽, 𝑋), then one has the following equality: 𝐼𝑡𝛼 𝐷𝑡𝛼 𝑥 (𝑡)

(11)

where Δ𝑥1−𝛼 (𝑡𝑘 ) = 𝑥1−𝛼 (𝑡𝑘+ ) − 𝑥1−𝛼 (𝑡𝑘− ), 𝑘 = 1, 2, . . . , 𝑚.

𝑡

𝑡1

𝑑 𝑠 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏 𝑑𝑠 𝑑𝑠 𝑡1

1 Γ (𝛼) Γ (1 − 𝛼) 𝑡1

𝑠

0

0

× ∫ (𝑡 − 𝑠)𝛼−1 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏 𝑑𝑠 +

1 Γ (𝛼 + 1) Γ (1 − 𝛼)

󵄨󵄨𝑠=𝑡1 𝑠 󵄨 × [(𝑡 − 𝑠)𝛼 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏]󵄨󵄨󵄨 󵄨󵄨𝑠=0 0

𝑡 ∈ [0, 𝑡1 ] ,

𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ] ,

𝑑 𝑡1 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏 𝑑𝑠 𝑑𝑠 0

1 Γ (𝛼 + 1) Γ (1 − 𝛼)

× ∫ (𝑡 − 𝑠)𝛼 =

𝑑 𝑠 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏 𝑑𝑠 𝑑𝑠 0

1 Γ (𝛼 + 1) Γ (1 − 𝛼)

× ∫ (𝑡 − 𝑠)𝛼

𝑚

In order to study the 𝑃𝐶-mild solutions of (4) in Banach space 𝑃𝐶1−𝛼 (𝐽, 𝑋), we give the following results which will be used throughout this paper.

𝑑 𝑠 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏 𝑑𝑠 𝑑𝑠 0

1 Γ (𝛼 + 1) Γ (1 − 𝛼) 𝑡1

where 𝑛 = [𝛼] + 1 and [𝛼] denotes the integer part of number 𝛼, is called the Riemann-Liouville fractional derivative of order 𝛼.

𝛼−1 󵄨󵄨 𝑡 { 𝑥 − 𝑥 (𝑡) (𝑡) 󵄨 { 1−𝛼 󵄨𝑡=0 Γ (𝛼) , { { { { { { { { 𝑚 Δ𝑥1−𝛼 (𝑡𝑖 ) ={ 𝛼−1 𝑥 − (𝑡 − 𝑡𝑖 ) ∑ (𝑡) { { { Γ (𝛼) { 𝑘=1 { { 𝛼−1 { { 󵄨 𝑡 { − 𝑥1−𝛼 (𝑡)󵄨󵄨󵄨𝑡=0 , Γ (𝛼) {

𝑡 1 ∫ (𝑡 − 𝑠)𝛼−1 𝐿 𝐷𝑡𝛼 𝑥 (𝑠) 𝑑𝑠 Γ (𝛼) 0

𝑡 1 ∫ (𝑡 − 𝑠)𝛼 𝐿 𝐷𝑡𝛼 𝑥 (𝑠) 𝑑𝑠 Γ (𝛼 + 1) 0

𝛼 > 0,

(12)

+

1 Γ (𝛼) Γ (1 − 𝛼) 𝑡

𝑡1

𝑡1

0

× ∫ (𝑡 − 𝑠)𝛼−1 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏 𝑑𝑠 +

1 Γ (𝛼 + 1) Γ (1 − 𝛼)

(13)

4

Abstract and Applied Analysis 󵄨󵄨𝑠=𝑡 𝑡1 󵄨 × [(𝑡 − 𝑠)𝛼 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏]󵄨󵄨󵄨 󵄨󵄨𝑠=𝑡1 0

̂ is the Laplace of 𝑥 defined by where 𝑥(𝜆) ∞

𝑥̂ (𝜆) = ∫ 𝑒−𝜆𝑡 𝑥 (𝑡) 𝑑𝑡,

1 + Γ (𝛼) Γ (1 − 𝛼)

0

𝑡

𝑠

𝑡1

𝑡1

𝑠

1−𝛼−1

× [(𝑡 − 𝑠) ∫ (𝑠 − 𝜏) 𝑡1

=

󵄨󵄨𝑠=𝑡 󵄨 𝑥 (𝜏) 𝑑𝜏]󵄨󵄨󵄨󵄨 󵄨󵄨𝑠=𝑡1

𝑡 ∈ (0, 𝑏] ,

0

𝜏

𝛼−1

1−𝛼−1

𝑡1

𝑡

(𝑠 − 𝜏)

𝑑𝑠

0

𝑡1

𝑥 (𝑡) 𝑡

{ 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 + ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) ℎ (𝑠) 𝑑𝑠, { { { 0 { { { 𝑡 ∈ [0, 𝑡1 ] , { { { 𝑚 { { 𝛼−1 1−𝛼 𝛼−1 = {𝑡 𝑇𝛼 (𝑡) 𝑥0 + ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) Δ𝐼0+ 𝑥 (𝑡𝑘 ) { { 𝑘=1 { { 𝑡 { { + ∫ − 𝑠)𝛼−1 𝑇 − 𝑠) ℎ 𝑑𝑠, { (𝑡 (𝑠) { 𝛼 (𝑡 { { 0 { 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ] , {

1 Γ (𝛼) Γ (1 − 𝛼) 𝑡

𝑡

𝑡1

𝜏

× ∫ 𝑥 (𝜏) 𝑑𝜏 ∫ (𝑡 − 𝑠)𝛼−1 (𝑠 − 𝜏)1−𝛼−1 𝑑𝑠 𝑥1−𝛼 (0) 𝛼 Δ𝑥1−𝛼 (𝑡1 ) 𝛼 𝑡 − (𝑡 − 𝑡1 ) Γ (𝛼 + 1) Γ (𝛼 + 1) 𝑡

0

∞

Thus, by (13) and (14), we get 𝑥1−𝛼 (0) 𝛼−1 Δ𝑥1−𝛼 (𝑡1 ) 𝛼−1 𝑡 − (𝑡 − 𝑡1 ) . Γ (𝛼) Γ (𝛼) (15)

𝑇𝛼 (𝑡) = 𝛼 ∫ 𝜃𝜉𝛼 (𝜃) 𝑇 (𝑡𝛼 𝜃) 𝑑𝜃, 0

1 𝜉𝛼 (𝜃) = 𝜃−1−(1/𝛼) 𝜛𝛼 (𝜃−1/𝛼 ) , 𝛼 Γ (𝑛𝛼 + 1) 1∞ 𝜛𝛼 (𝜃) = ∑ (−1)𝑛−1 𝜃−𝑛𝛼−1 sin (𝑛𝜋𝛼) , 𝜋 𝑛=1 𝑛!

(21)

𝜃 ∈ (0, ∞) ,

Similarly, if 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ], 𝑘 = 2, . . . , 𝑚, we can get 𝐼𝑡𝛼 𝐷𝑡𝛼 𝑥 (𝑡) 𝑚

𝛼−1 (16) Δ𝑥1−𝛼 (𝑡𝑘 ) 𝛼−1 󵄨 𝑡 (𝑡 − 𝑡𝑘 ) − 𝑥1−𝛼 (𝑡)󵄨󵄨󵄨𝑡=0 . Γ (𝛼) Γ (𝛼) 𝑘=1

= 𝑥 (𝑡) − ∑

The proof is completed. The Laplace transform formula for the Riemann-Liouville fractional integral is defined by 1 𝐿 {𝐼𝑡𝛼 𝑥 (𝑡) ; 𝜆} = 𝛼 𝑥̂ (𝜆) , 𝜆

(20)

where 𝑘 = 1, 2, . . . , 𝑚,

𝑥1−𝛼 (0) 𝛼 Δ𝑥1−𝛼 (𝑡1 ) 𝛼 𝑡 − (𝑡 − 𝑡1 ) . Γ (𝛼 + 1) Γ (𝛼 + 1) (14)

𝐼𝑡𝛼 𝐷𝑡𝛼 𝑥 (𝑡) = 𝑥 (𝑡) −

(19)

then, 𝑥 satisfies the following equation:

× ∫ 𝑥 (𝜏) 𝑑𝜏 ∫ (𝑡 − 𝑠)𝛼−1 (𝑠 − 𝜏)1−𝛼−1 𝑑𝑠

= ∫ 𝑥 (𝜏) 𝑑𝜏 −

𝑘 = 1, 2, . . . , 𝑚,

󵄨 𝐼𝑡1−𝛼 𝑥 (𝑡)󵄨󵄨󵄨󵄨𝑡=0 = 𝑥0 ∈ 𝑋,

1 + Γ (𝛼) Γ (1 − 𝛼)

−

𝑘 = 1, 2, . . . , 𝑚,

𝑘

× ∫ 𝑥 (𝜏) 𝑑𝜏 ∫ (𝑡 − 𝑠)

+

𝑡 ≠ 𝑡𝑘 ,

󵄨 Δ𝐼𝑡1−𝛼 𝑥 (𝑡)󵄨󵄨󵄨󵄨𝑡=𝑡 = 𝐼𝑘 (𝑥 (𝑡𝑘− )) , 𝑡1

𝑐 is a constant.

𝐷𝑡𝛼 𝑥 (𝑡) = 𝐴𝑥 (𝑡) + ℎ (𝑡) ,

1 Γ (𝛼) Γ (1 − 𝛼) 𝑡1

(18)

Lemma 5. Let 𝛼 ∈ (0, 1] and ℎ ∈ 𝐿𝑝 (𝐽, 𝑋), 𝑝 > 1/𝛼; if 𝑥(𝑡) ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋), 𝑥1−𝛼 (𝑡) ∈ 𝑃𝐶(𝐽, 𝑋), and 𝑥 is a solution of the following problem:

1 Γ (𝛼 + 1) Γ (1 − 𝛼) 𝛼

|𝑥 (𝑡)| ≤ 𝑐𝑒 ,

𝑅𝑒𝜆 > 𝜔,

× ∫ (𝑡 − 𝑠)𝛼−1 ∫ (𝑠 − 𝜏)1−𝛼−1 𝑥 (𝜏) 𝑑𝜏 𝑑𝑠 +

𝜔𝑡

(17)

where 𝜉𝛼 is a probability density function defined on (0, ∞); that is, 𝜉𝛼 (𝜃) ≥ 0,

𝜃 ∈ (0, ∞) ,

∞

∫ 𝜉𝛼 (𝜃) 𝑑𝜃 = 1. 0

(22)

Proof. We observe that 𝑥(⋅) can be decomposed to 𝑝(⋅) + 𝑞(⋅), where 𝑝 is the continuous mild solution for 𝐷𝑡𝛼 𝑝 (𝑡) = 𝐴𝑝 (𝑡) + ℎ (𝑡) ,

𝑡 ∈ (0, 𝑏] ,

󵄨 𝐼𝑡1−𝛼 𝑝 (𝑡)󵄨󵄨󵄨󵄨𝑡=0 = 𝑥0 ∈ 𝑋,

(23)


5

and 𝑞 is the 𝑃𝐶1−𝛼 -mild solution for

Hence, it follows from (28) and (30) that

𝐷𝑡𝛼 𝑞 (𝑡) = 𝐴𝑞 (𝑡) , 𝑡 ∈ (0, 𝑏] ,

𝑡 ≠ 𝑡𝑘 ,

∞

𝑘 = 1, 2, . . . , 𝑚,

󵄨 Δ𝐼𝑡1−𝛼 𝑞 (𝑡)󵄨󵄨󵄨󵄨𝑡=𝑡 = 𝑦𝑘 ,

𝑘 = 1, 2, . . . , 𝑚,

𝑘

󵄨 𝐼𝑡1−𝛼 𝑞 (𝑡)󵄨󵄨󵄨󵄨𝑡=0

(24)

𝛼

∫ 𝑒−𝜆 𝑡 𝑇 (𝑡) 𝑥0 𝑑𝑡 0

∞

𝛼

= ∫ 𝑒−(𝜆𝑠) 𝑇 (𝑠𝛼 ) 𝑥0 𝑑𝑠𝛼

= 0 ∈ 𝑋.

∞

Indeed, by adding together (23) and (24), it follows by (19). Since 𝑝 is continuous, then 𝑝(𝑡𝑘+ ) = 𝑝(𝑡𝑘− ), 𝑘 = 1, 2, . . . , 𝑚. On the other hand, any solution of (19) can be decomposed to (23) and (24). So we show the results by the following. At first, we calculate the mild solution of (23). Apply Riemann-Liouville fractional integral operator on both sides of (23); then, by Lemma 3, we get 𝑝 (𝑡) = =

󵄨 𝐼𝑡1−𝛼 𝑝 (𝑡)󵄨󵄨󵄨󵄨𝑡=0 Γ (𝛼)

(𝑡 = 𝑠𝛼 )

0

𝛼

= 𝛼 ∫ 𝑒−(𝜆𝑠) 𝑇 (𝑠𝛼 ) 𝑠𝛼−1 𝑥0 𝑑𝑠 0

∞

= 𝛼∬ 𝜛𝛼 (𝜃) 𝑒−𝜆𝑠𝜃 𝑇 (𝑠𝛼 ) 𝑠𝛼−1 𝑥0 𝑑𝜃 𝑑𝑠 0

∞

= 𝛼∬ 𝜛𝛼 (𝜃) 𝑒−𝜆𝑢 𝑇 ( 0

𝑡𝛼−1 + 𝐼𝑡𝛼 𝐴𝑝 (𝑡) + 𝐼𝑡𝛼 ℎ (𝑡)

∞

0

0

= ∫ 𝑒−𝜆𝑢 [𝛼 ∫ 𝜛𝛼 (𝜃) 𝑇 ( (25)

𝛼−1

∞

𝑢𝛼 𝑠𝛼−1 ) 𝑥 𝑑𝑢 𝑑𝜃 𝜃𝛼 𝜃𝛼 0

𝑡 𝑥 + 𝐼𝛼 𝐴𝑝 (𝑡) + 𝐼𝑡𝛼 ℎ (𝑡) . Γ (𝛼) 0 𝑡

∞

(𝑢 = 𝜃𝑠)

𝑢𝛼 𝑠𝛼−1 ) 𝑥 𝑑𝜃] 𝑑𝑢, 𝜃𝛼 𝜃𝛼 0

𝛼

∫ 𝑒−𝜆 𝑡 𝑇 (𝑡) ̂ℎ (𝜆) 𝑑𝑡 0

∞

That is,

∞

𝛼

= ∫ 𝑒−𝜆 𝑡 𝑇 (𝑡) [∫ 𝑒−𝜆𝑠 ℎ (𝑠) 𝑑𝑠] 𝑑𝑡

𝑡 𝑡𝛼−1 1 𝑥0 + 𝑝 (𝑡) = ∫ (𝑡 − 𝑠)𝛼−1 [𝐴𝑝 (𝑠) + ℎ (𝑠)] 𝑑𝑠. Γ (𝛼) Γ (𝛼) 0 (26)

𝑝̂ (𝜆) = ∫ 𝑒−𝜆𝑡 𝑝 (𝑡) 𝑑𝑡, 0

0

0

0

∞

𝛼

0

= ∭ 𝛼𝜛𝛼 (𝜃) 𝑒−𝜆(]+𝑠) 𝑇 (

0

0

Consider the one-sided stable probability density

𝜛𝛼 (𝜃) 𝑑𝜃 = 𝑒

,

𝛼 ∈ (0, 1) .

∞

0

𝑠

×

(29)

(𝜏 − 𝑠)𝛼 ) 𝜃𝛼

(𝜏 − 𝑠)𝛼−1 ℎ (𝑠) 𝑑𝜏 𝑑𝜃 𝑑𝑠 𝜃𝛼

∞

𝜏

∞

0

0

0

= ∫ 𝑒−𝜆𝜏 [𝛼 ∫ ∫ 𝜛𝛼 (𝜃) 𝑇 (

whose Laplace transformation is given by −𝜆𝛼

∞

(30)

(] = 𝜇𝜃)

]𝛼 ]𝛼−1 ) ℎ (𝑠) 𝑑] 𝑑𝜃 𝑑𝑠 𝜃𝛼 𝜃𝛼

= ∬ ∫ 𝛼𝜛𝛼 (𝜃) 𝑒−𝜆𝜏 𝑇 (

𝜃 ∈ (0, ∞) ,

−𝜆𝜃

]𝛼 ) 𝜃𝛼

]𝛼−1 −𝜆𝑠 𝑒 ℎ (𝑠) 𝑑] 𝑑𝜃 𝑑𝑠 𝜃𝛼

∞

𝛼

Γ (𝑛𝛼 + 1) 1∞ 𝜛𝛼 (𝜃) = ∑ (−1)𝑛−1 𝜃−𝑛𝛼−1 sin (𝑛𝜋𝛼) , 𝜋 𝑛=1 𝑛!

×

(28)

= ∫ 𝑒−𝜆 𝑡 𝑇 (𝑡) 𝑥0 𝑑𝑡 + ∫ 𝑒−𝜆 𝑡 𝑇 (𝑡) ̂ℎ (𝜆) 𝑑𝑡.

0

∞

∞

−1 ̂

(𝑡 = 𝜇𝛼 )

= ∭ 𝛼𝜛𝛼 (𝜃) 𝑒−𝜆𝜇𝜃 𝑇 (𝜇𝛼 ) 𝜇𝛼−1 𝑒−𝜆𝑠 ℎ (𝑠) 𝑑𝜃 𝑑𝑠 𝑑𝜇 = ∭ 𝛼𝜛𝛼 (𝜃) 𝑒−𝜆] 𝑇 (

= (𝜆𝛼 𝐼 − 𝐴) 𝑥0 + (𝜆𝛼 𝐼 − 𝐴) ℎ (𝜆)

∫ 𝑒

(31) 𝛼

0

̂ℎ (𝜆) = ∫ 𝑒−𝜆𝑡 ℎ (𝑡) 𝑑𝑡 (27)

−1

∞

𝛼

0

∞

1 1 1 𝑥 + 𝐴𝑝̂ (𝜆) + 𝛼 ̂ℎ (𝜆) 𝜆𝛼 0 𝜆𝛼 𝜆 ∞

∞

= ∬ 𝑒−𝜆 𝑡 𝑇 (𝑡) 𝑒−𝜆𝑠 ℎ (𝑠) 𝑑𝑠 𝑑𝑡 = ∬ 𝑒−(𝜆𝜇) 𝑇 (𝜇𝛼 ) 𝑞𝜇𝛼−1 𝑒−𝜆𝑠 ℎ (𝑠) 𝑑𝑠 𝑑𝜇

to (26), we obtain 𝑝̂ (𝜆) =

0

∞

Let 𝜆 > 0; taking the Laplace transformations ∞

0

×

(𝜏 = ] + 𝑠)

(𝜏 − 𝑠)𝛼 ) 𝜃𝛼

(𝜏 − 𝑠)𝛼−1 ℎ (𝑠) 𝑑𝜃 𝑑𝑠] 𝑑𝜏. 𝜃𝛼

6

Abstract and Applied Analysis According to the above work, we get ∞

∞

0

0

𝑝̂ (𝜆) = ∫ 𝑒−𝜆𝑡 [𝛼 ∫ 𝜛𝛼 (𝜃) 𝑇 ( 𝑡

∞

0

0

where

𝑡𝛼 𝑡𝛼−1 ) 𝑥 𝑑𝜃 𝜃𝛼 𝜃𝛼 0

+ 𝛼 ∫ ∫ 𝜛𝛼 (𝜃) 𝑇 ( ×

𝜒𝑘 (𝑡) = {

(𝑡 − 𝑠)𝛼 ) 𝜃𝛼

(𝑡 − 𝑠)𝛼−1 ℎ (𝑠) 𝑑𝜃 𝑑𝑠] 𝑑𝑡. 𝜃𝛼 (32)

𝑡𝛼 𝑡𝛼−1 𝑝 (𝑡) = 𝛼 ∫ 𝜛𝛼 (𝜃) 𝑇 ( 𝛼 ) 𝛼 𝑥0 𝑑𝜃 𝜃 𝜃 0

𝐼𝑘 (𝑥 (𝑡𝑘− )) 𝑒−𝜆𝑡𝑘 1 + 𝛼 𝐴̂ 𝑞 (𝜆) . 𝛼 𝜆 𝜆 𝑘=1 𝑚

𝑞̂ (𝜆) = ∑

𝑚

0

−1

𝑞̂ (𝜆) = ∑ (𝜆𝛼 𝐼 − 𝐴) 𝐼𝑘 (𝑥 (𝑡𝑘− )) 𝑒−𝜆𝑡𝑘 .

∞

1 −1−(1/𝛼) 𝜛𝛼 (𝜃−1/𝛼 ) 𝜃𝑇 (𝑡𝛼 𝜃) 𝑡𝛼−1 𝑥0 𝑑𝜃 𝜃 𝛼

Notice that the Laplace transform of 𝑡𝛼−1 𝑇𝛼 (𝑡)𝑦𝑘 is (𝜆𝛼 𝐼 − 𝐴)−1 𝑦𝑘 . Thus one can calculate the mild solution of (24) as 𝑚

𝛼−1

𝑞 (𝑡) = ∑ 𝜒𝑘 (𝑡) (𝑡 − 𝑡𝑖 ) 𝑘=1

𝑡 ∞ 1 + 𝛼 ∫ ∫ (𝑡 − 𝑠)𝛼−1 𝜃−1−(1/𝛼) 𝜛𝛼 (𝜃−1/𝛼 ) 𝜃𝑇 𝛼 0 0

(33)

𝑚

𝛼−1

+ ∑ 𝜒𝑘 (𝑡) (𝑡 − 𝑡𝑘 )

∞

(41)

𝑥 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0

Let 1 𝜉𝛼 (𝜃) = 𝜃−1−(1/𝛼) 𝜛𝛼 (𝜃−1/𝛼 ) , 𝛼

𝑇𝛼 (𝑡 − 𝑡𝑖 ) 𝐼𝑘 (𝑥 (𝑡𝑘− )) .

By the above work, the 𝑃𝐶1−𝛼 -mild solution of (19) is given by

× ((𝑡 − 𝑠)𝛼 𝜃) ℎ (𝑠) 𝑑𝜃 𝑑𝑡.

𝑘=1

𝑇𝛼 (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘− ))

(42)

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) ℎ (𝑠) 𝑑𝑠.

(34)

𝑇𝛼 (𝑡) = 𝛼 ∫ 𝜃𝜉𝛼 (𝜃) 𝑇 (𝑡𝛼 𝜃) 𝑑𝜃. 0

0

That is,

Then, we get

𝑥 (𝑡)

𝑡

𝑝 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 + ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) ℎ (𝑠) 𝑑𝑠. (35) 0

Now we calculate the 𝑃𝐶1−𝛼 -mild solution of (24). Applying Riemann-Liouville fractional integral operator on both sides of (24), then by Lemma 4, we get 𝑞 (𝑡) 𝑡 1 { ∫ (𝑡 − 𝑠)𝛼−1 𝐴𝑞 (𝑠) 𝑑𝑠, { { Γ (𝛼) { 0 { { { { 𝑚 𝐼 (𝑥 (𝑡− )) 𝛼−1 𝑘 = {∑ 𝑘 (𝑡 − 𝑡𝑖 ) { Γ (𝛼) { { 𝑘=1 { 𝑡 { { { + 1 ∫ (𝑡 − 𝑠)𝛼−1 𝐴𝑞 (𝑠) 𝑑𝑠, { Γ (𝛼) 0

(40)

𝑘=1

(𝑡 − 𝑠)𝛼 (𝑡 − 𝑠)𝛼−1 + 𝛼 ∫ ∫ 𝜛𝛼 (𝜃) 𝑇 ( ) ℎ (𝑠) 𝑑𝜃 𝑑𝑡 𝜃𝛼 𝜃𝛼 0 0 = 𝛼∫

(39)

That is,

∞

∞

(38)

Let 𝜆 > 0; taking the Laplace transformation to (37), we obtain

Now, we can invert the Laplace transform to (20) and obtain

𝑡

0, 𝑡 ≤ 𝑡𝑘 , 1, 𝑡 > 𝑡𝑘 .

𝑡 ∈ [0, 𝑡1 ] ,

𝑡

{ 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 + ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) ℎ (𝑠) 𝑑𝑠, { { { 0 { { { 𝑡 ∈ [0, 𝑡1 ] , { { { 𝑚 { { 𝛼−1 − 𝛼−1 = {𝑡 𝑇𝛼 (𝑡) 𝑥0 + ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘 )) { { 𝑘=1 { { 𝑡 { { + ∫ − 𝑠)𝛼−1 𝑇 − 𝑠) ℎ 𝑑𝑠, { (𝑡 (𝑠) { 𝛼 (𝑡 { { 0 { 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ] , {

(36)

(43)

where 𝑘 = 1, 2, . . . , 𝑚, ∞

𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ] .


The above equation (36) can be rewritten as 𝐼𝑘 (𝑥 (𝑡𝑘− )) 𝛼−1 (𝑡 − 𝑡𝑘 ) 𝜒𝑘 (𝑡) Γ (𝛼) 𝑘=1

𝜉𝛼 (𝜃) =

𝑚

𝑞 (𝑡) = ∑

𝑡 1 + ∫ (𝑡 − 𝑠)𝛼−1 𝐴𝑞 (𝑠) 𝑑𝑠, Γ (𝛼) 0

(37)

1 −1−(1/𝛼) 𝜛𝛼 (𝜃−1/𝛼 ) , 𝜃 𝛼

Γ (𝑛𝛼 + 1) 1∞ 𝜛𝛼 (𝜃) = ∑ (−1)𝑛−1 𝜃−𝑛𝛼−1 sin (𝑛𝜋𝛼) , 𝜋 𝑛=1 𝑛! 𝜃 ∈ (0, ∞) ,

(44)


7

where 𝜉𝛼 is a probability density function defined on (0, ∞); that is, 𝜉𝛼 (𝜃) ≥ 0,

𝜃 ∈ (0, ∞) ,

∞

∫ 𝜉𝛼 (𝜃) 𝑑𝜃 = 1. 0

(45)

This completes the proof of the lemma. According to Lemma 5, we give the following definition. Definition 6. A function 𝑥 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋) is called a mild solution of (4) if 𝐼𝑡1−𝛼 𝑥(𝑡)|𝑡=0 = 𝑥0 and there exits 𝑓 ∈ 𝐿1 (𝐽 < 𝑋) such that 𝑓(𝑡) ∈ 𝐹(𝑡, 𝑥(𝑡)) a.e. on 𝑡 ∈ 𝐽 and 𝑥 (𝑡) 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 { { 𝑡 { { { + ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) [𝐵𝑢 (𝑠) + 𝑓 (𝑠, 𝑥 (𝑠))] 𝑑𝑠, { { { 0 { { { { 𝑡 ∈ [0, 𝑡1 ] , { { 𝑚 = { 𝛼−1 𝛼−1 − {𝑡 𝑇𝛼 (𝑡) 𝑥0 + ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘 )) { { { 𝑘=1 { { 𝑡 { { 𝛼−1 { { { + ∫ (𝑡 − 𝑠) 𝑇𝛼 (𝑡 − 𝑠) [𝐵𝑢 (𝑠) + 𝑓 (𝑠, 𝑥 (𝑠))] 𝑑𝑠, { 0 { 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ] , { (46) where 𝑘 = 1, 2, . . . , 𝑚, ∞


1 𝜉𝛼 (𝜃) = 𝜃−1−(1/𝛼) 𝜛𝛼 (𝜃−1/𝛼 ) , 𝛼 1∞ Γ (𝑛𝛼 + 1) 𝜛𝛼 (𝜃) = ∑ (−1)𝑛−1 𝜃−𝑛𝛼−1 sin (𝑛𝜋𝛼) , 𝜋 𝑛=1 𝑛!

(47)

A multivalued map 𝐺 : 𝑋 → P(𝑋) is convex (closed) valued if 𝐺(𝑥) is convex (closed) for all 𝑥 ∈ 𝑋. 𝐺 is bounded on bounded sets if 𝐺(𝐵) = ⋃𝑥∈𝐵 . 𝐺(𝑥) is bounded on 𝑋 for any bounded set 𝐵 of 𝑋; that is, sup𝑥∈𝐵 {sup{‖𝑦‖ : 𝑦 ∈ 𝐺(𝑥)}} < ∞. 𝐺 is called upper semicontinuous (u.s.c.) on 𝑋 if, for each 𝑥0 ∈ 𝑋, the set 𝐺(𝑥0 ) is a nonempty closed subset of 𝑋, and if, for each open set 𝑈 of 𝑋 containing 𝐺(𝑥0 ), there exists an open neighborhood 𝑉 of 𝑥0 such that 𝐺(𝑉) ⊆ 𝑈. 𝐺 is said to be completely continuous if 𝐺(𝐵) is relatively compact for every 𝐵 ∈ 𝑃𝑏 (𝑋). If the multivalued map 𝐺 is completely continuous with nonempty compact values, then 𝐺 is u.s.c. if and only if 𝐺 has a closed graph (i.e., 𝑥𝑛 → 𝑥∗ , 𝑦𝑛 → 𝑦∗ , 𝑦𝑛 ∈ 𝐺(𝑥𝑛 ) imply 𝑦∗ ∈ 𝐺(𝑥∗ )). We say that 𝐺 has a fixed point if there is a 𝑥 ∈ 𝑋 such that 𝑥 ∈ 𝐺(𝑥). A multivalued map 𝐺 : 𝐽 → 𝑃𝑐𝑙 (𝑋) is said to be measurable if for each 𝑥 ∈ 𝑋 the function 𝑌 : 𝐽 → 𝑅+ defined by 𝑌(𝑡) = 𝑑(𝑥, 𝐺(𝑡)) = inf{‖𝑥 − 𝑧‖ : 𝑧 ∈ 𝐺(𝑡)} is measurable. Definition 8. The system (4) is said to be exactly controllable on 𝐽, if, for all 𝑥0 , 𝑥1 ∈ 𝑋, there exists a control 𝑢 ∈ 𝐿𝑝 (𝐽, 𝑈) (𝑝 > 1/𝛼) such that the mild solution of (4) satisfies 𝑥(0; 𝑢) = 𝑥0 and 𝑥(𝑏; 𝑢) = 𝑥1 . Definition 9. The system (4) is said to be approximately controllable on the interval 𝐽, if, for all 𝑥0 ∈ 𝑋, one has R(𝑏, 𝑥0 ) = 𝑋, where R(𝑏, 𝑥0 ) = {𝑥(𝑏; 𝑢) : 𝑢 ∈ 𝐿𝑝 (𝐽, 𝑈) (𝑝 > 1/𝛼), 𝑥(0; 𝑢) = 𝑥0 } is the reachable set of system (4) with the initial values 𝑥0 , 𝑥1 at the terminal time 𝑏. It is convenient at this point to introduce two relevant operators: 𝑏

Γ0𝑏 = ∫ (𝑏 − 𝑠)2𝛼−2 𝑇𝛼 (𝑏 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑠) 𝑑𝑠, 0

𝜃 ∈ (0, ∞) ,

1 < 𝛼 ≤ 1, 2

where 𝜉𝛼 is a probability density function defined on (0, ∞); that is, 𝜉𝛼 (𝜃) ≥ 0, 𝜃 ∈ (0, ∞) ,

∞

∫ 𝜉𝛼 (𝜃) 𝑑𝜃 = 1. 0

(48)

Due to the work of the paper [31], we have the following result. Lemma 7. The operator 𝑇𝛼 (𝑡) has the following properties. (i) For any fixed 𝑡 ≥ 0, 𝑇𝛼 (𝑡) is linear and bounded operator; that is, for any 𝑥 ∈ 𝑋, 𝑀 󵄩 󵄩󵄩 ‖𝑥‖ . 󵄩󵄩𝑇𝛼 (𝑡) 𝑥󵄩󵄩󵄩 ≤ Γ (𝛼)

−1

𝑅 (𝑎, Γ0𝑏 ) = (𝑎𝐼 + Γ0𝑏 ) ,

(50)

𝑎 > 0,

where 𝐵∗ denotes the adjoint of 𝐵 and 𝑇𝛼∗ (𝑡) is the adjoint of 𝑇𝛼 (𝑡). It is straightforward that the operator Γ0𝑏 is a linear bounded operator. We consider the following linear fractional differential system: 𝑐

𝐷𝛼 𝑥 (𝑡) = 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) ,

1 < 𝛼 ≤ 1, 𝑡 ∈ 𝐽 = (0, 𝑏] , 2

󵄨 𝐼𝑡1−𝛼 𝑥 (𝑡)󵄨󵄨󵄨󵄨𝑡=0 = 𝑥0 ∈ 𝑋.

(51)

(49)

Lemma 10. The linear fractional differential system (51) is approximately controllable on 𝐽 if and only if 𝑎𝑅(𝑎, Γ0𝑏 ) → 0 as 𝑎 → 0+ in the strong operator topology.

Now, we also introduce some basic definitions on multivalued maps. For more details, see [36–38].

The proof of this lemma is a straightforward adaptation of the proof of [3].

(ii) 𝑇𝛼 (𝑡) (𝑡 ≥ 0) is strongly continuous.

8


Lemma 11. Let 𝐸 be a Banach space and let W ⊂ 𝐿1 (𝐽, 𝐸) be integrably bounded. If, for all 𝑡 ∈ 𝐽, there is a relatively weakly compact set 𝐶(𝑡) ⊂ 𝐸 such that 𝑤(𝑡) ∈ 𝐶(𝑡) for every 𝑤 ∈ W, then W is relatively weakly compact in 𝐿1 (𝐽, 𝐸). Lemma 12 (Lasota and Opial [39]). Let 𝐽 be a compact real interval and let 𝑋 be a Banach space. The multivalued map 𝐹 : 𝐽 × 𝑋 → P𝑏,𝑐𝑙,𝑐V (𝑋) is measurable to 𝑡 for each fixed 𝑥 ∈ 𝑋, u.s.c. to 𝑥 for each 𝑡 ∈ 𝐽, and for each 𝑥 ∈ 𝐶(𝐽, 𝑋) the set 𝑆𝐹,𝑥 = {𝑓 ∈ 𝐿1 (𝐽, 𝑋) : 𝑓(𝑡) ∈ 𝐹(𝑡, 𝑥(𝑡)), 𝑡 ∈ 𝐽} is nonempty. Let Γ be a linear continuous mapping from 𝐿1 (𝐽, 𝑋) to 𝐶(𝐽, 𝑋); then, the operator Γ ∘ 𝑆𝐹 : 𝐶 (𝐽, 𝑋) 󳨀→ P𝑏,𝑐𝑙,𝑐V (𝐶 (𝐽, 𝑋)) , 𝑥 󳨃󳨀→ (Γ ∘ 𝑆𝐹 ) (𝑥) = Γ (𝑆𝐹,𝑥 )

(52)

is a closed graph operator in 𝐶(𝐽, 𝑋) × 𝐶(𝐽, 𝑋).

3. Main Results In this section, we present our main result on approximate controllability of system (4). To do this, we first prove the existence of solutions for fractional control system. Secondly, we show that, under certain assumptions, the approximate controllability of (4) is implied by the approximate controllability of the corresponding linear system. For convenience, let us introduce some notations: 𝛽=(

1 − 𝛾 (𝛼−𝛾)/(1−𝛾) 1−𝛾 ) , 𝑏 𝛼−𝛾

𝑀3 𝑀𝐵2 𝑏2𝛼−1 𝛽 𝑀𝛽 . Λ= + Γ (𝛼) 𝑎 (2𝛼 − 1) (Γ (𝛼))3

(56)

𝐻(4): There exist constants 𝑑𝑘 > 0, 𝑘 = 1, 2, . . . , 𝑚, with (𝑀/Γ(𝛼)) ∑𝑚 𝑘=1 𝑑𝑘 < 1 such that 1−𝛼 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩𝐼𝑘 (𝑥) − 𝐼𝑘 (𝑦)󵄩󵄩󵄩 ≤ 𝑑𝑘 (𝑡 − 𝑡𝑘 ) 󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩𝑋 ,

∀𝑥, 𝑦 ∈ 𝑋. (57)

Theorem 14. If the conditions 𝐻(1)–𝐻(4) are held, then the system (4) has a mild solution.

𝐵𝑟 = {𝑥 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋) : ‖𝑥‖ ≤ 𝑟, 𝑟 > 0}

𝐻(2): 𝐹 is a multivalued map satisfying 𝐹 : 𝐽 × 𝑋 → P𝑐𝑝,𝑐V (𝑋) which is measurable to 𝑡 for each fixed 𝑥 ∈ 𝐷, u.s.c. to 𝑥 for each 𝑡 ∈ 𝐽, and for each 𝑥 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋) the set (54)

is nonempty. 𝐻(3): There exist a function 𝑃(𝑡) ∈ 𝐿1/𝛾 (𝐽, 𝑅+ ), 𝛾 ∈ (0, 𝛼), and a nondecreasing continuous function 𝜓 : → 𝑅+ , such that 󵄩 󵄩 ‖𝐹 (𝑡, 𝑥 (𝑡))‖𝑋 = sup {󵄩󵄩󵄩𝑓 (𝑡)󵄩󵄩󵄩𝑋 : 𝑓 (𝑡) ∈ 𝐹 (𝑡, 𝑥 (𝑡))} (55) ≤ 𝑃 (𝑡) 𝜓 (‖𝑥‖𝐷) ,

(58)

on the space 𝑃𝐶1−𝛼 (𝐽, 𝑋). We easily know that 𝐵𝑟 is a bounded, closed, and convex set in 𝑃𝐶1−𝛼 (𝐽, 𝑋). For 𝑎 > 0, for all 𝑥(⋅) ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋), 𝑥1 ∈ 𝑋, we take the control function as 𝑢 (𝑡) = (𝑏 − 𝑡)𝛼−1 𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 ) 𝑃 (𝑥 (⋅)) ,

(59)

where 𝑃 (𝑥 (⋅)) = 𝑥1 − 𝑡𝛼−1 𝑇𝛼 (𝑏) 𝑥0 𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 )

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑏

(53)

𝐻(1): 𝑇𝛼 (𝑡) is compact, ‖𝑎𝑅(𝑎, Γ0𝑏 )‖ ≤ 1.

𝑆𝐹,𝑥 = {𝑓 ∈ 𝐿 (𝐽, 𝑋) : 𝑓 (𝑡) ∈ 𝐹 (𝑡, 𝑥 (𝑡))}

𝜓 (𝑟) ‖𝑃‖𝐿2 = 𝜌 < 1. 𝑟

𝑘=1

Before stating and proving our main results, we introduce the following assumptions.

1

lim inf

𝑟→∞

Proof. We consider a set

Lemma 13 (see [37]). Let 𝐷 be a bounded, convex, and closed subset in the Banach space 𝐸 and let 𝐺 : 𝐷 → 2𝑋 \ {0} be a u.s.c. condensing multivalued map. If, for every 𝑥 ∈ 𝐷, 𝐺(𝑥) is a closed and convex set in 𝐷, then 𝐺 has a fixed point.

𝑀𝑏 = ‖𝐵‖ ,

for a.e. 𝑡 ∈ 𝐽, for all 𝑥 ∈ 𝐷, and for each 𝑟 > 0, there exists 0 < 𝜌 < 1, such that

− ∫ (𝑏 − 𝑠)𝛼−1 𝑇𝛼 (𝑏 − 𝑠) 𝑓 (𝑠) 𝑑𝑠, 0

𝑓 ∈ 𝑆𝐹,𝑥 . (60)

By this control, we define the operator Φ𝑎 : 𝑃𝐶1−𝛼 (𝐽, 𝑋) → P(𝑃𝐶1−𝛼 (𝐽, 𝑋)) as follows: Φ𝑎 (𝑥) = {𝜏 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋) : 𝜏 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠) 𝑑𝑠 0

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝑢 (𝑠) 𝑑𝑠, 0

𝑓 ∈ 𝑆𝐹,𝑥 , 𝑡 ∈ (0, 𝑡1 ] , 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑚

𝛼−1

+ ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝑘=1

𝐼𝑘 (𝑥 (𝑡𝑘− ))


9

𝑡

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠) 𝑑𝑠

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 )

0

0

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1

× [𝑥1 − 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0

0

× 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝑢 (𝑠) 𝑑𝑠,

𝑚

− ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝑘=1

𝑓 ∈ 𝑆𝐹,𝑥 , 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ] , 𝑘 = 1, 2, . . . , 𝑚.} .

𝑏

− ∫ (𝑏 − 𝜇)

(61)

0

𝜏𝑖 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑚

+ ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 )

𝛼−1

𝑘=1

𝑇𝛼 (𝑏 − 𝜇)

(63) Since 𝑆𝐹,𝑥 is convex (because 𝐹 has convex values), 𝜆𝑓1 + (1 − 𝜆)𝑓2 ∈ 𝑆𝐹,𝑥 ; thus, 𝜆𝜏1 (𝑡) + (1 − 𝜆)𝜏2 ∈ Φ𝑎 (𝑥). Step 2. For each 𝑎 > 0, there is a positive constant 𝑟0 = 𝑟(𝑎), such that Φ𝑎 (𝐵𝑟0 ) ⊂ 𝐵𝑟0 . If this is not true, then there exists 𝑎 > 0 such that, for every 𝑟 > 0, there exists a 𝑥 ∈ 𝐵𝑟 such that Φ𝑎 (𝑥) ⊈ 𝐵𝑟 ; that is,

𝐼𝑘 (𝑥 (𝑡𝑘− ))

󵄩 󵄩󵄩 󵄩󵄩Φ𝑎 (𝑥)󵄩󵄩󵄩 ≡ sup {‖𝜏‖𝑃𝐶1−𝛼 (𝐽,𝑋) : 𝜏 ∈ Φ𝑎 (𝑥)} > 𝑟,

𝑡

𝜏 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓𝑖 (𝑠) 𝑑𝑠 0

𝑡

𝛼−1

𝑚

+ ∫ (𝑡 − 𝑠)

𝛼−1

0

𝑇𝛼 (𝑡 −

𝑠) 𝐵𝐵∗ 𝑇𝛼∗

𝐼𝑘 (𝑥 (𝑡𝑘− ))

× [𝜆𝑓1 (𝜇) + (1 − 𝜆) 𝑓2 (𝜇)] 𝑑𝜇] 𝑑𝑠.

We will show that, for all 𝑎 > 0, the operator Φ𝑎 : 𝑃𝐶1−𝛼 (𝐽, 𝑋) → P(𝑃𝐶1−𝛼 (𝐽, 𝑋)) has a fixed point. For the sake of convenience, we subdivide the proof into several steps. Step 1. For each 𝑎 > 0, the operator Φ𝑎 (𝑥) is convex for each 𝑥 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋). In fact, if 𝜏1 , 𝜏2 ∈ Φ𝑎 (𝑥), then, for each 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ], 𝑘 = 1, 2, . . . , 𝑚, there exist 𝑓1 , 𝑓2 ∈ 𝑆𝐹,𝑥 such that

𝛼−1

(𝑏 −

𝛼−1

+ ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 )

𝑡) 𝑅 (𝑎, Γ0𝑏 )

𝑘=1

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑡

𝛼−1

× [𝑥1 − 𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠) 𝑑𝑠

𝑇𝛼 (𝑡) 𝑥0

0

𝑡

𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝑘=1 𝑏

− ∫ (𝑏 − 𝜇) 0

𝛼−1


𝐼𝑘 (𝑥 (𝑡𝑘− ))

0

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

𝑇𝛼 (𝑏 − 𝜇) 𝑓𝑖 (𝜇) 𝑑𝜇] 𝑑𝑠,

𝑚

𝑖 = 1, 2. (62)

𝑘=1 𝑏

− ∫ (𝑏 − 𝜇) 0

Let 0 ≤ 𝜆 ≤ 1; then, for each 𝑡 ∈ (𝑡𝑘 , 𝑡𝑘+1 ], 𝑘 = 1, 2, . . . , 𝑚, we have

𝛼−1

= 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 + ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝑘=1

𝑡

𝐼𝑘 (𝑥 (𝑡𝑘− ))

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) [𝜆𝑓1 (𝑠) + (1 − 𝜆) 𝑓2 (𝑠)] 𝑑𝑠 0

𝛼−1

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑇𝛼 (𝑏 − 𝜇) 𝑓 (𝜇) 𝑑𝜇] 𝑑𝑠, (64)

for some 𝑆𝐹,𝑥 . By using Holder’s inequality and 𝐻(3), we have

𝜆𝜏1 (𝑡) + (1 − 𝜆) 𝜏2 (𝑡) 𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 )

𝑡

∫ (𝑡 − 𝑠)𝛼−1 ‖𝑢 (𝑠)‖ 𝑑𝑠 0

≤

𝑀𝑀𝐵 𝑏2𝛼−1 𝑎Γ (𝛼) (2𝛼 − 1)

10

Abstract and Applied Analysis 󵄩 󵄩 󵄩 󵄩 𝑀 󵄩󵄩𝑥 󵄩󵄩 × [󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩 + 󵄩 0 󵄩 Γ (𝛼)

Thus, 1−𝛼

𝑟≤

𝑚

𝑀 𝛼−1 󵄩 󵄩 +∑ (𝑑𝑘 ‖𝑥‖ + 𝑚(𝑏 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩) Γ (𝛼) 𝑘=1

𝑀𝑏𝛼−1 (𝑏 − 𝑡𝑘 ) Γ (𝛼) 𝑚

𝑀 (𝑑 ‖𝑥‖ Γ (𝛼) 𝑘 𝑘=1

×∑

𝑀𝜓 (𝑟) 𝛽 + ‖𝑃‖𝐿1/𝛾 ] . Γ (𝛼)

𝛼−1

+ 𝑚(𝑏 − 𝑡𝑘 )

(65)

1−𝛼

+

‖𝜏 (𝑡)‖

𝑏2𝛼−1 𝑀2 𝑀𝐵2 (𝑏 − 𝑡𝑘 )

(67)

𝑎 (2𝛼 − 1) (Γ (𝛼))2

󵄩 󵄩 󵄩 󵄩 𝑀 󵄩󵄩𝑥 󵄩󵄩 × [ 󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩 + 󵄩 0 󵄩 Γ (𝛼)

󵄩󵄩 𝛼−1 󵄩 󵄩󵄩𝑡 𝑇𝛼 (𝑡) 𝑥0 󵄩󵄩󵄩 + (𝑡 − 𝑡𝑘 )1−𝛼 󵄩 󵄩

1−𝛼

≤ (𝑡 − 𝑡𝑘 ) 𝑚

𝑚

𝑀 𝛼−1 󵄩 󵄩 (𝑑𝑘 ‖𝑥‖ + 𝑚(𝑏 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩)] Γ (𝛼) 𝑘=1

𝛼−1 󵄩 󵄩 󵄩 󵄩 × ∑ 󵄩󵄩󵄩𝑇𝛼 (𝑡 − 𝑡𝑘 )󵄩󵄩󵄩 (𝑡 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (𝑥 (𝑡𝑘− ))󵄩󵄩󵄩

+ ∑

𝑘=1

1−𝛼

+ 𝜓 (𝑟) Λ‖𝑃‖𝐿1/𝛾 .

+ (𝑡 − 𝑡𝑘 ) 𝑡

Dividing both sides by 𝑟 and taking the low limit as 𝑟 → ∞, we get

󵄩 󵄩󵄩 󵄩 × ∫ (𝑡 − 𝑠)𝛼−1 󵄩󵄩󵄩𝑇𝛼 (𝑡 − 𝑠)󵄩󵄩󵄩 󵄩󵄩󵄩𝑓 (𝑠)󵄩󵄩󵄩 𝑑𝑠 0

1−𝛼

+ (𝑡 − 𝑡𝑘 )

1 ≤ lim inf 𝑟→∞

𝑡

󵄩 󵄩 × ∫ (𝑡 − 𝑠)𝛼−1 󵄩󵄩󵄩𝑇𝛼 (𝑡 − 𝑠)󵄩󵄩󵄩 ‖𝐵‖ ‖𝑢 (𝑠)‖ 𝑑𝑠 1−𝛼

𝑀𝑏𝛼−1 (𝑏 − 𝑡𝑘 ) Γ (𝛼)

1−𝛼 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑥0 󵄩󵄩 + (𝑏 − 𝑡𝑘 )

𝑚

𝑀 𝛼−1 󵄩 󵄩 (𝑑𝑘 ‖𝑥‖ + 𝑚(𝑏 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩) Γ (𝛼) 𝑘=1 1−𝛼

𝑡

∫ (𝑡 − 𝑠) 0

1−𝛼

𝑀𝑀𝐵 (𝑏 − 𝑡𝑘 ) + Γ (𝛼) 𝑀𝑏𝛼−1 (𝑏 − 𝑡𝑘 ) Γ (𝛼)

(66)

0

𝜏𝑛 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0

𝑃 (𝑠) 𝜓 (𝑟) 𝑑𝑠

∫ (𝑡 − 𝑠)

1−𝛼

≤

𝑡

𝛼−1

𝛼−1

𝑚

𝑘=1

𝑢 (𝑠) 𝑑𝑠

𝑡

+ ∫ (𝑡 − 𝑠)

1−𝛼 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑥0 󵄩󵄩 + (𝑏 − 𝑡𝑘 )

0

+

𝑏

𝑀

𝑀𝐵2 (𝑏

1−𝛼

− 𝑡𝑘 )

𝑎 (2𝛼 − 1) (Γ (𝛼))

2

󵄩 󵄩 󵄩 󵄩 𝑀 󵄩󵄩𝑥 󵄩󵄩 × [ 󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩 + 󵄩 0 󵄩 Γ (𝛼) 𝑚

𝑀 𝛼−1 󵄩 󵄩 (𝑑𝑘 ‖𝑥‖ + 𝑚(𝑏 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩)] Γ (𝛼) 𝑘=1

+ ∑

+ 𝜓 (𝑟) Λ‖𝑃‖𝐿1/𝛾 .

(69) 𝑇𝛼 (𝑡 − 𝑠) 𝑓𝑛 (𝑠) 𝑑𝑠

𝑡

𝑚

2

𝛼−1

𝐼𝑘 (𝑥 (𝑡𝑘 ))

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝑢𝑛 (𝑠) 𝑑𝑠, 0

×∑

2𝛼−1

𝛼−1

+ ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 )

𝑀 𝛼−1 󵄩 󵄩 (𝑑𝑘 ‖𝑥‖ + 𝑚(𝑏 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩) Γ (𝛼) 𝑘=1

(68)

Step 3. Φ𝛼 (𝑥) is closed for each 𝑥 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋). Indeed, for each given 𝑥 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋), let {𝜏𝑛 }𝑛≥0 ⊂ Φ𝑎 (𝑥) such that 𝜏𝑛 → 𝜏 in 𝑃𝐶1−𝛼 (𝐽, 𝑋). Then, there exists 𝑓𝑛 ∈ 𝑆𝐹,𝑥 such that, for each 𝑡 ∈ 𝐽,

×∑

𝑀(𝑏 − 𝑡𝑘 ) + Γ (𝛼)

𝜓 (𝑟) ‖𝑃‖𝐿1/𝛾 , 𝑟

which is a contradiction to 𝐻(3). Thus, for each 𝑎 > 0, there exists 𝑟0 such that Φ𝑎 (𝑥) maps 𝐵𝑟0 into itself.

0

≤

󵄩 󵄩󵄩 󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩)

1−𝛼

Then, we obtain (𝑡 − 𝑡𝑘 )

1−𝛼 󵄩󵄩 󵄩󵄩 󵄩󵄩𝑥0 󵄩󵄩 + (𝑏 − 𝑡𝑘 )

where 𝑢𝑛 (𝑡) = 𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 ) × [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0 𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝑏

𝐼𝑘 (𝑥 (𝑡𝑘− ))

− ∫ (𝑏 − 𝑠)𝛼−1 𝑇𝛼 (𝑏 − 𝑠) 𝑓𝑛 (𝑠) 𝑑𝑠] . 0

(70)


11 𝑡

Because of [40, Proposition 3.1], 𝑆𝐹,𝑥 is weakly compact in 𝐿1 (𝐽, 𝑋) which implies that 𝑓𝑛 converges weakly to some 𝑓 ∈ 𝑆𝐹,𝑥 in 𝐿1 (𝐽, 𝑋). Thus, 𝑢𝑛 ⇀ 𝑢, and

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 0

× 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 )

𝑢 (𝑡) = 𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 )

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

𝑚

𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 )

(71)

𝑘=1

𝑏

− ∫ (𝑏 − 𝜇)

𝛼−1

0

𝑏

− ∫ (𝑏 − 𝑠)𝛼−1 𝑇𝛼 (𝑏 − 𝑠) 𝑓 (𝑠) 𝑑𝑠] . 0

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑇𝛼 (𝑏 − 𝜇) 𝑓 (𝜇) 𝑑𝜇] 𝑑𝑠,

𝑓 ∈ 𝑆𝐹,𝑥 , 𝑡 ∈ 𝐽} .

Then, for each 𝑡 ∈ 𝐽,

(73)

𝜏𝑛 (𝑡) 󳨀→ 𝜏 (𝑡) 𝛼−1

=𝑡

According to [41, Corollary 2.2.1], we will prove that Φ1𝑎 is a contraction operator, while Φ2𝑎 is a completely continuous operator. Let us begin proving that Φ1𝑎 is a contraction operator. For any 𝑥, 𝑦 ∈ 𝑋, we obtain

𝑇𝛼 (𝑡) 𝑥0

𝑚

+ ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 )

𝛼−1

𝑘=1

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠) 𝑑𝑠 0

𝑡


(72)

0

󵄩 󵄩󵄩 1 󵄩󵄩(Φ𝑎 𝑥) (𝑡) − (Φ1𝑎 𝑦) (𝑡)󵄩󵄩󵄩 󵄩 󵄩 󵄩󵄩 𝑚 󵄩󵄩 𝛼−1 ≤ 󵄩󵄩󵄩 ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝐼𝑘 (𝑥 (𝑡𝑘− )) 󵄩󵄩𝑘=1 󵄩 𝑚

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

− ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝑘=1

𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝑏

− ∫ (𝑏 − 𝜇)

𝛼−1

0

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑇𝛼 (𝑏 − 𝜇) 𝑓 (𝜇) 𝑑𝜇] 𝑑𝑠.

Thus, we show that 𝜏 ∈ Φ𝑎 (𝑥). Step 4. Φ𝑎 is u.s.c and condensing. We decompose Φ𝑎 as Φ𝑎 = Φ1𝑎 + Φ2𝑎 , where the operators 1 Φ𝑎 and Φ2𝑎 are defined by 𝑚

𝛼−1

(Φ1𝑎 𝑥) (𝑡) = ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 ) 𝑘=1

𝛼−1

𝐼𝑘 (𝑥 (𝑡𝑘− )) ,

𝑡 ∈ 𝐽, 𝑘 = 1, 2, . . . , 𝑚, Φ2𝑎 (𝑥) = {𝜏 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋) : 𝜏 (𝑡)

≤

󵄩󵄩

󵄩󵄩 𝐼𝑘 (𝑦 (𝑡𝑘− ))󵄩󵄩󵄩 󵄩

(74)

󵄩󵄩

𝑀 𝑚 󵄩󵄩 󵄩 ∑ 𝑑 󵄩𝑥 − 𝑦󵄩󵄩󵄩 . Γ (𝛼) 𝑘=1 𝑘 󵄩

Then, Φ1𝑎 is a contraction operator, since (𝑀/Γ(𝛼)) ∑𝑚 𝑘=1 𝑑𝑘 < 1. Next, we prove that Φ2𝑎 is u.s.c and completely continuous. We subdivide the proof into several claims. Claim 1. There exists a positive constant 𝑟 such that Φ2𝑎 (𝐵𝑟 ) ⊆ 𝐵𝑟 . By employing the technique used in Step 2, one can easily show that there exists 𝑟 > 0 such that Φ2𝑎 (𝐵𝑟 ) ⊆ 𝐵𝑟 . Claim 2. Φ2𝑎 (𝐵𝑟 ) is a family of equicontinuous functions. Let 0 ≤ 𝑠 ≤ 𝑡1 ≤ 𝑡2 ≤ 𝑏. For each 𝑥 ∈ 𝐵𝑟 , 𝜑 ∈ Φ2𝑎 (𝑥), there exists 𝑓 ∈ 𝑆𝐹,𝑥 such that 𝜏 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑡

𝛼−1

=𝑡

𝑇𝛼 (𝑡) 𝑥0 𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠) 𝑑𝑠 0

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠) 𝑑𝑠

(75)

0

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝑢 (𝑠) 𝑑𝑠, 0

𝑡 ∈ 𝐽.

12


Then, we have 󵄩󵄩 󵄩 󵄩󵄩𝜏 (𝑡2 ) − 𝜏 (𝑡1 )󵄩󵄩󵄩 󵄩󵄩 𝑡1 󵄩󵄩 𝛼−1 𝛼−1 󵄩 󵄩 ≤ 󵄩󵄩󵄩∫ [(𝑡2 − 𝑠) − (𝑡1 − 𝑠) ] 𝑇𝛼 (𝑡2 − 𝑠) 𝑓 (𝑠) 𝑑𝑠󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 𝑡1 󵄩󵄩 𝛼−1 󵄩 󵄩 × 󵄩󵄩󵄩∫ (𝑡1 − 𝑠) [𝑇𝛼 (𝑡2 − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)] 𝑓 (𝑠) 𝑑𝑠󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 𝑡2 󵄩󵄩 󵄩 󵄩 𝛼−1 + 󵄩󵄩󵄩󵄩∫ (𝑡2 − 𝑠) 𝑇𝛼 (𝑡2 − 𝑠) 𝑓 (𝑠) 𝑑𝑠󵄩󵄩󵄩󵄩 󵄩󵄩 𝑡1 󵄩󵄩 󵄩󵄩󵄩 𝑡1 󵄩󵄩󵄩 𝛼−1 𝛼−1 × 󵄩󵄩󵄩∫ [(𝑡2 − 𝑠) − (𝑡1 − 𝑠) ] 𝑇𝛼 (𝑡2 − 𝑠) 𝑢 (𝑠) 𝑑𝑠󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩󵄩󵄩 𝑡1 󵄩󵄩󵄩 𝛼−1 × 󵄩󵄩󵄩∫ (𝑡1 − 𝑠) [𝑇𝛼 (𝑡2 − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)] 𝑢 (𝑠) 𝑑𝑠󵄩󵄩󵄩 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 𝑡2 󵄩󵄩 󵄩 󵄩 𝛼−1 + 󵄩󵄩󵄩󵄩∫ (𝑡2 − 𝑠) 𝑇𝛼 (𝑡2 − 𝑠) 𝑢 (𝑠) 𝑑𝑠󵄩󵄩󵄩󵄩 󵄩󵄩 𝑡1 󵄩󵄩

in the uniform operator topology for 𝑡 > 0; we can directly obtain 𝐼1 and 𝐼3 tending to zero independently of 𝑥 ∈ 𝐵𝑟 as 𝑡2 → 𝑡1 . Applying the absolute continuity of the Lebesgue integral, we have 𝐼4 , 𝐼5 , and 𝐼6 tending to zero independently of 𝑥 ∈ 𝐵𝑟 as 𝑡2 → 𝑡1 . Therefore, Φ2𝑎 (𝐵𝑟 ) ⊂ 𝑃𝐶1−𝛼 (𝐽, 𝑋) is equicontinuous. Claim 3. The set Π(𝑡) = {𝜏(𝑡) : 𝜏 ∈ Φ2𝑎 (𝐵𝑟 )} ⊂ 𝑋 is relatively compact for each 𝑡 ∈ 𝐽. Let 0 < 𝑡 ≤ 𝑏 be fixed. For 𝑥 ∈ 𝐵𝑟 and 𝜏 ∈ Φ2𝑎 (𝑥), there exists 𝑓 ∈ 𝑆𝐹,𝑥 such that, for each 𝑡 ∈ 𝐽, 𝑡

𝜏 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 + ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓 (𝑠) 𝑑𝑠 0

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 ) 0

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

≤ 𝐼1 + 𝐼2 + 𝐼3 + 𝐼4 + 𝐼5 + 𝐼6 .

𝑚

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 )

(76)

𝑘=1

By using Hölder’s inequality and assumption 𝐻(3), we get

𝑏

0

𝜏𝜀,𝛿 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0

1 − 𝛾 (𝛼−𝛾)/(1−𝛾) 1−𝛾 ) ], 𝑡 𝛼−𝛾 2

+ 𝛼∫

𝑡−𝜀

0

𝐼2 = 𝜓 (𝑟) ‖𝑃‖𝐿1/𝛾 (

+ 𝛼∫ (77)

∞

∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜉𝛼 (𝜃) 𝑇 ((𝑡 − 𝑠)𝛼 𝜃) 𝑓 (𝑠) 𝑑𝜃 𝑑𝑠 𝛿

𝑡−𝜀

0

∞

∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜉𝛼 (𝜃) 𝑇 ((𝑡 − 𝑠)𝛼 𝜃) 𝛿

× 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑠) 𝑅 (𝑎, Γ0𝑏 )

1−𝛾 𝑀𝜓 (𝑟) ‖𝑃‖𝐿1/𝛾 1 − 𝛾 (𝛼−𝛾)/(1−𝛾) [ ] , (𝑡2 − 𝑡1 ) Γ (𝛼) 𝛼−𝛾

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

𝑀𝑀𝐵 𝑡2 𝛼−1 𝛼−1 ∫ [(𝑡2 − 𝑠) − (𝑡1 − 𝑠) ] ‖𝑢 (𝑠)‖ 𝑑𝑠, Γ (𝛼) 𝑡1 󵄩 󵄩 𝐼5 = 𝑀𝐵 sup 󵄩󵄩󵄩𝑇𝛼 (𝑡2 − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)󵄩󵄩󵄩 𝑠∈[0,𝑡1 ]

𝐼4 =

𝑡1

𝛼−1

× ∫ (𝑡1 − 𝑠) 0

𝐼6 =

𝑇𝛼 (𝑏 − 𝜇) 𝑓 (𝜇) 𝑑𝜇] 𝑑𝑠,

For all 𝜀 ∈ (0, 𝑡) and for all 𝛿 > 0, define

1−𝛾 1−𝛾 (𝛼−𝛾)/(1−𝛾) +( ) (𝑡2 − 𝑡1 ) 𝛼−𝛾

1 − 𝛾 (𝛼−𝛾)/(1−𝛾) 1−𝛾 ) 𝑡 𝛼−𝛾 1 󵄩 󵄩 × sup 󵄩󵄩󵄩𝑇𝛼 (𝑡2 − 𝑠) − 𝑇𝛼 (𝑡1 − 𝑠)󵄩󵄩󵄩 , 𝑠∈[0,𝑡1 ]

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑡 ∈ 𝐽. (78)

1 − 𝛾 (𝛼−𝛾)/(1−𝛾) 1−𝛾 ) × [( 𝑡 𝛼−𝛾 1

𝐼3 =

𝛼−1

− ∫ (𝑏 − 𝜇)

𝑀𝜓 (𝑟) ‖𝑃‖𝐿1/𝛾 𝐼1 = Γ (𝛼)

−(

𝛼−1

‖𝑢 (𝑠)‖ 𝑑𝑠,

𝑀𝑀𝐵 𝑡2 𝛼−1 ∫ (𝑡 − 𝑠) ‖𝑢 (𝑠)‖ 𝑑𝑠. Γ (𝛼) 𝑡1 2

It is easy to see that 𝐼2 tends to zero independently of 𝑥 ∈ 𝐵𝑟 as 𝑡2 → 𝑡1 . Note that, from Lemma 10, 𝑇𝛼 (𝑡) is continuous

𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝑏

𝛼−1

− ∫ (𝑏 − 𝜇) 0

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑇𝛼 (𝑏 − 𝜇) 𝑓 (𝜇) 𝑑𝜇] 𝑑𝜃 𝑑𝑠

= 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 + 𝑇 (𝜀𝛼 𝛿) × {𝛼 ∫

𝑡−𝜀

0

∞

∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜉𝛼 (𝜃) 𝑇 𝛿

× ((𝑡 − 𝑠)𝛼 𝜃 − 𝜀𝛼 𝛿) 𝑓 (𝑠) 𝑑𝜃 𝑑𝑠

Abstract and Applied Analysis 𝑡−𝜀

+ 𝛼∫

0

13

∞

Claim 4. Φ2𝛼 has a closed graph. Let 𝑥𝑛 → 𝑥∗ (𝑛 → ∞), 𝜏𝑛 ∈ Φ2𝑎 (𝑥𝑛 ), 𝜏𝑛 → 𝜏∗ (𝑛 → ∞). We will prove that 𝜏∗ ∈ Φ2𝑎 (𝑥∗ ). Since 𝜏𝑛 ∈ Φ2𝑎 (𝑥𝑛 ), there exists 𝑓𝑛 ∈ 𝑆𝐹,𝑥𝑛 , such that, for each 𝑡 ∈ 𝐽,

∫ 𝜃(𝑡 − 𝑠)𝛼−1 𝜉𝛼 (𝜃) 𝑇 ((𝑡 − 𝑠)𝛼 𝜃 − 𝜀𝛼 𝛿) 𝛿

× 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑠) 𝑅 (𝑎, Γ0𝑏 ) × [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

𝜏𝑛 (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑡

𝑚

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝛼−1

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓𝑛 (𝑠) 𝑑𝑠

𝐼𝑘 (𝑥 (𝑡𝑘− ))

0

𝑡

𝑏

𝛼−1

− ∫ (𝑏 − 𝜇) 0


𝑇𝛼 (𝑏 − 𝜇) 𝑓 (𝜇) 𝑑𝜇]𝑑𝜃 𝑑𝑠} .

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

(79) By the compactness of 𝑇(𝜀𝛼 𝛿) (𝜀𝛼 𝛿 > 0), we obtain the set Π𝜀,𝛿 (𝑡) = {𝜑𝜀,𝛿 (𝑡) : 𝜏 ∈ Φ2𝑎 (𝐵𝑟 )}

𝑚

𝑘=1

(80)

𝑏

− ∫ (𝑏 − 𝜇)

which is relatively compact in 𝑋 for all 𝜀 ∈ (0, 𝑡) and 𝛿 > 0. Moreover, we have 󵄩 󵄩󵄩 󵄩󵄩𝜏 (𝑡) − 𝜏𝜀,𝛿 (𝑡)󵄩󵄩󵄩 󵄩 󵄩 ≤{

𝜏∗ (𝑡) = 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓∗ (𝑠) 𝑑𝑠 0

𝑀 𝛼−1 󵄩 󵄩 +∑ (𝑑𝑘 ‖𝑥‖ + 𝑚(𝑏 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩) Γ (𝛼) 𝑘=1

+

𝑡


𝛿 𝑀𝜓 (𝑟) 𝛽 ‖𝑃‖𝐿1/𝛾 ] } ∫ 𝜃𝜉𝛼 (𝜃) 𝑑𝜃 Γ (𝛼) 0

𝑀𝜓 (𝑟) 1 − 𝛾 (𝛼−𝛾)/(1−𝛾) 1−𝛾 ) 𝜀 ‖𝑃‖𝐿1/𝛾 ( Γ (𝛼) 𝛼−𝛾 𝑀2 𝑀𝐵2 𝜀2𝛼−1

𝑎 (2𝛼 − 1) (Γ (𝛼)) 󵄩 󵄩 󵄩 󵄩 𝑀 󵄩󵄩𝑥 󵄩󵄩 × [󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩 + 󵄩 0 󵄩 Γ (𝛼)

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0 (81)

𝑚

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝑏

𝑚

𝑀 𝛼−1 󵄩 󵄩 (𝑑𝑘 ‖𝑥‖ + 𝑚(𝑏 − 𝑡𝑘 ) 󵄩󵄩󵄩𝐼𝑘 (0)󵄩󵄩󵄩) Γ (𝛼) 𝑘=1 𝑀𝜓 (𝑟) 𝛽 ‖𝑃‖𝐿1/𝛾 ] . Γ (𝛼)

The right-hand side of the above inequality tends to zero as 𝜀 → 0. Therefore, there are relatively compact sets arbitrarily close to the set Π(𝑡), 𝑡 > 0. Hence the set Π(𝑡), 𝑡 > 0 is also relatively compact in 𝑋. As a consequence of Claims 1– 3 together with the Arzola-Ascoli theorem, we can conclude that Φ2𝑎 is completely continuous.

𝛼−1

− ∫ (𝑏 − 𝜇)

2

+∑ +

𝑇𝛼 (𝑏 − 𝜇) 𝑓𝑛 (𝜇) 𝑑𝜇] 𝑑𝑠. (82)

𝑚

+

𝛼−1

0

𝐼𝑘 (𝑥 (𝑡𝑘− ))

We must prove that there exists 𝑓∗ ∈ 𝑆𝐹,𝑥∗ , such that, for each 𝑡 ∈ 𝐽,

𝛼𝑀2 𝑀𝐵2 𝑏2𝛼−1 𝛼𝑀𝜓 (𝑟) 𝛽 ‖𝑃‖𝐿1/𝛾 + Γ (𝛼) 𝑎 (2𝛼 − 1) Γ (𝛼) 󵄩󵄩 󵄩󵄩 󵄩 󵄩 𝑀 󵄩𝑥 󵄩 × [󵄩󵄩󵄩𝑥1 󵄩󵄩󵄩 + 󵄩 0 󵄩 Γ (𝛼)

+

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 )

0

𝛼−1

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑇𝛼 (𝑏 − 𝜇) 𝑓∗ (𝜇) 𝑑𝜇] 𝑑𝑠. (83)

Since 𝜏𝑛 → 𝜏∗ (𝑛 → ∞), we can obtain 󵄩󵄩 󵄩󵄩 󵄩󵄩 (𝜏𝑛 (𝑡) − 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 󵄩󵄩󵄩 󵄩 𝑡

− ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 ) 0

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0 𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝐼𝑘 (𝑥 (𝑡𝑘− ))] 𝑑𝑠)

14

Abstract and Applied Analysis Therefore, Φ2𝑎 has a closed graph. Since Φ2𝑎 is a completely continuous multivalued map with compact value, we have that Φ2𝑎 is u.s.c. Thus Φ𝑎 = Φ1𝑎 + Φ2𝑎 is u.s.c and condensing. Therefore, applying Lemma 13, we conclude that Φ𝑎 has a fixed point 𝑥(⋅) on 𝐵𝑟0 . Thus, the fractional control system (4) has a mild solution on 𝐽. The proof is complete.

− (𝜏∗ (𝑡) − 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑡

− ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑡) 𝑅 (𝑎, Γ0𝑏 ) 0

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0 𝑚

𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 )

The following result concerns the approximate controllability of that problem (4). We assume that the following assumption be held.

𝑘=1

󵄩󵄩 󵄩󵄩 × 𝐼𝑘 (𝑥 (𝑡𝑘− )) ] 𝑑𝑠) 󵄩󵄩󵄩 󳨀→ 0, 󵄩󵄩 󵄩

as 𝑛 󳨀→ ∞.

𝐻󸀠 (3): There exists a positive constant 𝐿 such that ‖𝐹(𝑡, 𝑥(𝑡))‖ ≤ 𝐿 for all (𝑡, 𝑥) ∈ 𝐽 × 𝑋.

(84)

Theorem 15. Assume that assumptions 𝐻(1), 𝐻(2), 𝐻󸀠 (3), and 𝐻(4) are satisfied and the linear system (24) is approximately controllable on 𝐽. Then system (4) is approximately controllable on 𝐽.

Consider the linear continuous operator Γ : 𝐿1/𝛾 (𝐽, 𝑋) 󳨀→ 𝐶1−𝛼 (𝐽, 𝑋) , 𝑡

(Γ𝑓) (𝑡) = ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 0

× [𝑓 (𝑠) − 𝐵(𝑏 − 𝑠)𝛼−1 𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑠) 𝑅 (𝑎, Γ0𝑏 ) 𝑏

× ∫ 𝑇𝛼 (𝑏 − 𝜂) 𝑓 (𝜂) 𝑑𝜂] 𝑑𝑠. 0

(85) Clearly it follows from Lemma 13 that Γ ∘ 𝑆𝐹 is a closed graph operator. Moreover, we have

= 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0 𝑚

𝛼−1

𝑘=1

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝑓𝑎 (𝑠) 𝑑𝑠

0

0

𝑡

+ ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑠) 𝑅 (𝑎, Γ0𝑏 )

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

0

𝛼−1

𝑘=1

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

𝐼𝑘 (𝑥 (𝑡𝑘− ))] 𝑑𝑠 ∈ Γ (𝑆𝐹,𝑥𝑛 ) .

𝑚

(86)

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 )

Since 𝑥𝑛 → 𝑥∗ , it follows from Lemma 13 that

𝑏

𝛼−1

− ∫ (𝑏 − 𝜇) 0

𝑡

− ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑠) 𝑅 (𝑎, Γ0𝑏 ) 0

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑇𝛼 (𝑏 − 𝜇) 𝑓𝑎 (𝜇) 𝑑𝜇] 𝑑𝑠.

Define

× [𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0

𝑃 (𝑓𝑎 ) 𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝛼−1

𝑘=1

𝜏∗ (𝑡) − 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0

𝑚

𝐼𝑘 (𝑥 (𝑡𝑘− ))

𝑡

− ∫ (𝑡 − 𝑠)𝛼−1 𝑇𝛼 (𝑡 − 𝑠) 𝐵𝐵∗ 𝑇𝛼∗ (𝑏 − 𝑠) 𝑅 (𝑎, Γ0𝑏 )

𝑚

𝑥𝑎 (𝑡)

+ ∑ 𝑇𝛼 (𝑡 − 𝑡𝑘 ) (𝑡 − 𝑡𝑘 )

𝜏𝑛 (𝑡) − 𝑡𝛼−1 𝑇𝛼 (𝑡) 𝑥0

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 )

Proof. By employing the technique used in Theorem 14, we can easily show that, for all 0 < 𝑎 < 1, the operator Φ𝑎 has a fixed point in 𝐵𝑟0 , where 𝑟0 = 𝑟(𝑎). Let 𝑥𝑎 (⋅) be a fixed point of Φ𝑎 in 𝐵𝑟0 . Any fixed point of Φ𝑎 is a mild solution of (4); this means that there exists 𝑓𝑎 ∈ 𝑆𝐹,𝑥𝑎 such that, for each 𝑡 ∈ 𝐽󸀠 ,

𝐼𝑘 (𝑥 (𝑡𝑘− ))] 𝑑𝑠 ∈ Γ (𝑆𝐹,𝑥∗ ) . (87)

= 𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0 𝑚

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝛼−1

𝐼𝑘 (𝑥 (𝑡𝑘− ))

(88)


15

𝑏

where 𝐷𝑡𝛼 denotes the Riemann-Liouville fractional deriva-

0

tive, the operator 𝐻 is defined by (𝐻𝑥)(𝑡) = ∫0 ℎ(𝑡, 𝑠, 𝑥(𝑠))𝑑𝑠. By a similar way, we can get the approximate controllability of the system (95).

− ∫ (𝑏 − 𝑠)𝛼−1 𝑇𝛼 (𝑏 − 𝑠) 𝑓𝑎 (𝑠) 𝑑𝑠,

𝑏

for some 𝑓𝑎 ∈ 𝑆𝐹,𝑥𝑎 . (89) Noting that 𝐼 − Γ0𝑏 𝑅(𝑎, Γ0𝑏 ) = 𝑎𝑅(𝑎, Γ0𝑏 ), we get 𝑥𝑎 (𝑏) = 𝑥1 − 𝑎𝑅 (𝑎, Γ0𝑏 ) 𝑃 (𝑓𝑎 ) .

(90)

By assumption 𝐻󸀠 (3), 𝑏

󵄩2 󵄩 ∫ 󵄩󵄩󵄩𝑓𝑎 (𝑠)󵄩󵄩󵄩 𝑑𝑠 ≤ 𝐿2 𝑏. 0

(91)

Consequently the sequence {𝑓𝑎 } is uniformly bounded in 𝐿1/𝛾 (𝐽, 𝑋). Thus, there is a subsequence, still denoted by {𝑓𝑎 }, that converges weakly to, say, 𝑓 in 𝐿1/𝛾 (𝐽, 𝑋). Denoting ℎ = 𝑥1 − 𝑏𝛼−1 𝑇𝛼 (𝑏) 𝑥0 𝑚

𝐷𝑡𝛼 𝑥 (𝑡) ∈ 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) + 𝐹 (𝑡, 𝑥𝜌(𝑡,𝑥𝑡 ) ) , 𝑡 ∈ (0, 𝑏] ,

1 < 𝛼 ≤ 1, 2

󵄨󵄨 − Δ𝐼01−𝛼 + 𝑥 (𝑡)󵄨 󵄨󵄨𝑡=𝑡 = 𝐼𝑘 (𝑥 (𝑡𝑘 )) , 𝑘

(96)

𝑘 = 1, 2, . . . , 𝑚,

󵄨󵄨 𝐼01−𝛼 + 𝑥 (𝑡)󵄨 󵄨󵄨𝑡=0 = 𝜙 ∈ 𝑋, 𝛼−1

− ∑ 𝑇𝛼 (𝑏 − 𝑡𝑘 ) (𝑏 − 𝑡𝑘 ) 𝑘=1

𝐼𝑘 (𝑥 (𝑡𝑘− ))

(92)

𝑏

− ∫ (𝑏 − 𝑠)𝛼−1 𝑇𝛼 (𝑏 − 𝑠) 𝑓 (𝑠) 𝑑𝑠, 0

we see that 󵄩󵄩 󵄩 𝛼 󵄩󵄩𝑃 (𝑓 ) − ℎ󵄩󵄩󵄩

󵄩󵄩 𝑏 󵄩󵄩󵄩 󵄩 = 󵄩󵄩󵄩󵄩∫ (𝑏 − 𝑠)𝛼−1 𝑇𝛼 (𝑏 − 𝑠) [𝑓𝑎 (𝑠) − 𝑓 (𝑠)] 𝑑𝑠󵄩󵄩󵄩󵄩 (93) 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 𝑏 󵄩󵄩 󵄩 󵄩 ≤ sup 󵄩󵄩󵄩󵄩∫ (𝑏 − 𝑠)𝛼−1 𝑇𝛼 (𝑏 − 𝑠) [𝑓𝛼 (𝑠) − 𝑓 (𝑠)] 𝑑𝑠󵄩󵄩󵄩󵄩 . 󵄩󵄩 0 0≤𝑡≤𝑏 󵄩 󵄩 By the Ascoli-Arzela theorem, we can show that the linear ⋅ operator 𝑔 → ∫0 (⋅ − 𝑠)𝛼−1 𝑇𝛼 (⋅ − 𝑠)𝑔(𝑠)𝑑𝑠 : 𝐿1/𝛾 (𝐽, 𝑋) → 𝑃𝐶1−𝛼 (𝐽, 𝑋) is compact; consequently the right-hand side of (93) tends to zero as 𝑎 → 0+ . This implies that 󵄩󵄩 𝑎 󵄩󵄩 󵄩󵄩𝑥 (𝑏) − 𝑥1 󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩󵄩𝑎𝑅 (𝑎, Γ0𝑏 ) 𝑃 (𝑓𝑎 )󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩󵄩𝑎𝑅 (𝑎, Γ0𝑏 ) (ℎ)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑎𝑅 (𝛼, Γ0𝑏 ) (𝑃 (𝑓𝑎 ) − ℎ)󵄩󵄩󵄩󵄩 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩󵄩𝑎𝑅 (𝑎, Γ0𝑏 ) (ℎ)󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑃 (𝑓𝑎 ) − ℎ󵄩󵄩󵄩 󳨀→ 0, as 𝑎 󳨀→ 0+ . (94)

Remark 16. In [9], Ganesh et al. have studied the approximate controllability of fractional integrodifferential evolution equations. If our problem (4) can be changed as 𝐷𝑡𝛼 𝑥 (𝑡) ∈ 𝐴𝑥 (𝑡) + 𝐵𝑢 (𝑡) + 𝐹 (𝑡, 𝑥 (𝑡) , (𝐻𝑥) (𝑡)) , 1 < 𝛼 ≤ 1, 2

󵄨󵄨 𝐼01−𝛼 + 𝑥 (𝑡)󵄨 󵄨󵄨𝑡=0 = 𝑥0 + 𝑔 (𝑥) ∈ 𝑋,

where 𝐷𝑡𝛼 is the Riemann-Liouville fractional derivative, 𝐴 is the infinitesimal generator of a 𝐶0 -semigroup {𝑇(𝑡), 𝑡 ≥ 0} on a Banach space 𝑋, the state 𝑥(⋅) takes values in 𝑋, the control function 𝑢(⋅) is given in 𝐿𝑝 (𝐽, 𝑈) (𝑝 > 1/𝛼), 𝑈 is a Banach space of admissible control, and 𝐵 : 𝑈 → 𝑋 is a bounded linear operator. 𝐹 : 𝐽 × 𝐷 → P(𝑋) := 2𝑋 \ {0} with 𝐷 := 𝐶((−∞, 0], 𝑋) is a multivalued map. For a continuous function 𝑥 : 𝐽∗ := (−∞, 𝑏] → 𝑋, 𝑥𝑡 is the element of 𝐷 defined by 𝑥𝑡 (𝑠) = 𝑥(𝑡 + 𝑠), −∞ ≤ 𝑠 ≤ 0, and 𝜙 ∈ 𝐷. Further, 𝜌 : [0, 𝑏] × P(𝑋) → (−∞, 𝑏] are appropriate nonlinear functions.

4. An Example In this final section, we give an example to illustrate our abstract results. Example 1. Consider the following fractional partial differential inclusion with control 𝐷𝑡𝛼 𝑥 (𝑡, 𝑦) ∈

𝜕2 𝑥 (𝑡, 𝑦) + 𝐹 (𝑡, 𝑥 (𝑡, 𝑦)) + 𝐵𝑢 (𝑡, 𝑦) , 𝜕𝑦2

1 𝑡 ∈ 𝐽󸀠 = [0, 1] \ { } , 𝑦 ∈ [0, 𝜋] , 2 󵄨 󵄨󵄨 󵄨𝑥 (𝑦)󵄨󵄨󵄨 1 󵄨 Δ𝐼01−𝛼 𝑥 ( , 𝑦) = + 󵄨 , 𝑦 ∈ [0, 𝜋] , 󵄨 2 2𝜋 + 󵄨󵄨󵄨𝑥 (𝑦)󵄨󵄨󵄨 𝑥 (𝑡, 0) = 𝑥 (𝑡, 𝜋) = 0,

This proves the approximate controllability of system (4).

𝑡 ∈ (0, 𝑏] ,

Remark 17. In [15], Rathinasamy and Yong have researched the approximate controllability of fractional differential equations with state-dependent delay. Our problem (4) can be adopted to the impulsive fractional differential inclusion system with state-dependent delay:

(95)

󵄨󵄨 𝐼01−𝛼 + 𝑥 (𝑡, 𝑦)󵄨 󵄨󵄨𝑡=0 = 𝑥0 (𝑦) ,

(97)

𝑡 ∈ 𝐽 = [0, 1] , 𝑡 ∈ [0, 1] , 𝑦 ∈ [0, 𝜋] ,

where 𝐷𝛼 is the Riemann-Liouville fractional partial derivative of order 3/5, 𝐽 = [0, 1]. Let 𝑋 = 𝑈 = 𝐿2 [0, 𝜋], define; the operator 𝐴 by 𝐴𝑥 = 𝑥𝑦𝑦 with the domain 𝐷 (𝐴) = {𝑥 ∈ 𝑋 : 𝑥, 𝑥󸀠 are absolutely continuous, 𝑥𝑦𝑦 ∈ 𝑋, 𝑥 (𝑡, 0) = 𝑥 (𝑡, 𝜋) = 0} .

(98)

16


Then, ∞

𝐴𝑥 = − ∑ 𝑛2 ⟨𝑥, 𝑒𝑛 ⟩ 𝑒𝑛 ,

𝑥 ∈ 𝐷 (𝐴) ,

(99)

𝑛=1

where 𝑒𝑛 (𝑦) = √2/𝜋 sin 𝑛𝑦, 0 ≤ 𝑦 ≤ 𝜋, 𝑛 = 1, 2, . . ., is the orthogonal set of eigenvectors of 𝐴. It is easily shown that operator 𝐴 generates a strongly continuous semigroup {𝑇(𝑡) : 𝑡 ≥ 0} on 𝑋, which are compact, analytic, and selfadjoint in 𝑋. It can be easily seen that the linear system corresponding to (97) is approximately controllable on [0, 1] [42]. Define 𝑥(𝑡)(𝑦) = 𝑥(𝑡, 𝑦); let 𝐵 : 𝑈 → 𝑋 be defined by 𝐵𝑢(𝑡)(𝑦) = 𝐵𝑢(𝑡, 𝑦). Then system (97) can be written in the abstract form given by (4). We assume that 𝐹 : 𝐽×𝐷 → P(𝑋) satisfy the following conditions. (𝐹1 ): 𝐹 : 𝐽 × 𝐷 → P𝑐𝑝,𝑐V (𝑋) is measurable to 𝑡 for each fixed 𝑥 ∈ 𝐷, u.s.c. to 𝑥 for a.e. 𝑡 ∈ 𝐽, and for each 𝑥 ∈ 𝑃𝐶1−𝛼 (𝐽, 𝑋) the set 𝑆𝐹,𝑥 = {𝑓 ∈ 𝐿1 (𝐽, 𝑋) : 𝑓 (𝑡) ∈ 𝐹 (𝑡, 𝑥)}

(100)

is nonempty. (𝐹2 ): There exists a positive constant 𝐿 such that ‖𝐹(𝑡, 𝑥)‖ ≤ 𝐿 for all (𝑡, 𝑥) ∈ 𝐽 × 𝐷, since, for any 𝑥, 𝑦 ∈ 𝐴𝐶(𝐽, 𝑋), we have 󵄩 󵄩 󵄩 󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 󵄩󵄩 1−𝛼 󵄩󵄩Δ𝐼𝑡 𝑥 (𝑡𝑘 , 𝑧) − Δ𝐼𝑡1−𝛼 𝑦 (𝑡𝑘 , 𝑧)󵄩󵄩󵄩 ≤ 󵄩 . 󵄩 󵄩 2

(101)

Obviously, the hypotheses of Theorem 15 are fulfilled. Thus, the system (97) is approximately controllable on [0, 1].

5. Conclusions In this paper, we study the approximate controllability for impulsive fractional differential inclusions with RiemannLiouville fractional derivatives in Banach spaces. We mainly use fixed point theorem and semigroup theory; we obtain some new mild solutions for this kind of problems (4). An illustrative example is also discussed to show the effectiveness of the results in this paper. In the near future, we will consider Riemann-Liouville fractional partial differential problems, which will be more complicated.

Acknowledgments The project was supported by NNSF of China Grant nos. 11271087 and 61263006, the Scientific Research Foundation of Guangxi University for Nationalities no. 2012QD014, and the Innovation Project of Guangxi University for Nationalities no. gxun-chx2013083.

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The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com


Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014


Volume 2014

Volume 2014


Volume 2014

Discrete Mathematics

Journal of

Volume 2014


Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com


Volume 2014


Volume 2014


Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization



Volume 2014

Volume 2014