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Journal of Applied Mathematics
Volumeย 2012ย (2012), Article IDย 193061, 21 pages
http://dx.doi.org/10.1155/2012/193061
Research Article

Analysis of a System for Linear Fractional Differential Equations

Fang Wang,1,2ย Zhen-hai Liu,3ย and Ping Wang4

1School of Mathematical Science and Computing Technology, Central South University, Hunan, Changsha 410075, China
2School of Mathematical Science and Computing Technology, Changsha University of Science and Technology, Hunan, Changsha 410076, China
3School of Mathematical Science and Computing Technology, Guangxi University for Nationalities, Guangxi, Nanning 530006, China
4Xiamen Institute of Technology, Huaqiao University, Fujian, Xiamen 361021, China

Received 3 March 2012; Revised 21 July 2012; Accepted 23 July 2012

Academic Editor: Chein-Shanย Liu

Copyright ยฉ 2012 Fang Wang et al. 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.

Abstract

The main purpose of this paper is to obtain the unique solution of the constant coefficient homogeneous linear fractional differential equations ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต and the constant coefficient nonhomogeneous linear fractional differential equations ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก)+๐ท,๐‘‹(๐‘Ž)=๐ต if ๐‘ƒ is a diagonal matrix and ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] and prove the existence and uniqueness of these two kinds of equations for any ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š) and ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Then we give two examples to demonstrate the main results.

1. Introduction

System of fractional differential equations has gained a lot of interest because of the challenges it offers compared to the study of system of ordinary differential equations. Numerous applications of this system in different areas of physics, engineering, and biological sciences have been presented in [1โ€“3]. The differential equations involving the Riemman-Liouville differential operators of fractional order 0<๐‘ž<1 appear to be more important in modeling several physical phenomena and therefore seem to deserve an independent study of their theory parallel to the well-known theory of ordinary differential equations. The existence and uniqueness of solution for fractional differential equations with any ๐‘‹(๐‘ก)โˆˆ๐ถ[๐‘ก0,๐‘‡]ร—๐ถ[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ[๐‘ก0,๐‘‡] have been studied in many papers, see [4โ€“28]. In [4] Daftradar-Gejji and Babakhani have studied the existence and uniqueness of ๐ท๐‘ž0๎€ท๐‘‹(๐‘ก)โˆ’๐‘‹0๎€ธ=๐‘ƒ๐‘‹(๐‘ก),(1.1) where ๐ท๐‘ž0 denotes the standard Riemman-Liouville fractional derivative, 0<๐‘ž<1,๐‘‹(๐‘ก)=(๐‘ฅ1(๐‘ก),๐‘ฅ2(๐‘ก),โ€ฆ,๐‘ฅ๐‘š(๐‘ก))๐‘‡, ๐‘‹(0)=๐‘‹0=(๐‘ฅ10,๐‘ฅ20,โ€ฆ,๐‘ฅ๐‘š0)๐‘‡, ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š) which is an m dimensional linear space. They have obtained that the system (1.1) has a unique solution defined on [0,๐‘‡] if ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š) and ๐‘‹(๐‘ก)โˆˆ๐ถ[๐‘ก0,๐‘‡]ร—๐ถ[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ[๐‘ก0,๐‘‡]. In [17] Belmekki et al. have studied the existence of periodic solution for some linear fractional differential equation in ๐ถ1โˆ’๐‘ž[0,1]. In [21] Ahmad and Nieto have studied the Riemann-Liouwille fractional differential equations with fractional boundary conditions. In comparison with the earlier results of this type we get more general assumptions. We assume ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] instead of ๐‘‹(๐‘ก)โˆˆ๐ถ[๐‘ก0,๐‘‡]ร—๐ถ[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ[๐‘ก0,๐‘‡] and consider the following system of fractional differential equations: ๐ท๐‘ž๐‘ก0๐ท๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต,๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก)+๐ท,๐‘‹(๐‘Ž)=๐ต,(1.2) where ๐ท๐‘ž0 denotes the standard Riemman-Liouville fractional derivative, 0<๐‘ž<1, ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š), ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—โ‹ฏร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,,๐‘‡(1.3)(๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…๐‘š and ๐ท is a constant vector. We completely generalize the results in [4] and obtain the new results if ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Furthermore, we also obtain some results of the unique solution of the homogeneous and nonhomogeneous initial value problems with the classical Mittag-Leffler special function [5] which is similar to the ordinary differential equations. Now we introduce the first Mittag-Leffler function ๐‘’๐‘ž(๐‘กโˆ’๐‘ก0) defined by ๐‘’๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ=+โˆž๎“๐‘˜=1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘˜๐‘žโˆ’1.ฮ“(๐‘˜๐‘ž)(1.4) The function ๐‘’๐‘ž(๐‘กโˆ’๐‘ก0) belongs to ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Indeed, taking the norm in ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡], we have โ€–โ€–๐‘’๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธโ€–โ€–1โˆ’๐‘žโ‰ค+โˆž๎“๐‘˜=1๎€ท๐‘‡โˆ’๐‘ก0๎€ธ(๐‘˜โˆ’1)๐‘žฮ“(๐‘˜๐‘ž)<+โˆž.(1.5) The formula remains valid for ๐‘žโ†’1โˆ’. In this case, ๐‘’1(๐‘กโˆ’๐‘ก0)=exp(๐‘กโˆ’๐‘ก0). Then we introduce the second Mittag-Leffler function ๐ธ๐‘ž(๐‘กโˆ’๐‘ก0) defined by ๐ธ๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ=+โˆž๎“๐‘˜=1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘˜๐‘ž.ฮ“(๐‘˜๐‘ž+1)(1.6) The formula remains also valid for ๐‘žโ†’1โˆ’. In this case, ๐ธ1(๐‘กโˆ’๐‘ก0)=exp(๐‘กโˆ’๐‘ก0)โˆ’1.

The paper is organized as follows. In Section 2 we recall the definitions of fractional integral and derivative and related basic properties and preliminary results used in the text. In Section 3 we obtain the unique solution of the constant coefficient homogeneous and nonhomogeneous linear fractional differential equations for ๐‘ƒ being the diagonal matrix. In Section 4 we prove the existence and uniqueness of these two kinds of equations for any ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š). In Section 5 we give some specific examples to illustrate the results.

2. Definitions and Preliminary Results

Let us denote by ๐ถ[๐‘ก0,๐‘‡] the space of all continuous real functions defined on [๐‘ก0,๐‘‡], which turns out to be a Banach space with the norm โ€–๐‘ฅโ€–=max๐‘กโˆˆ[๐‘ก0,๐‘‡]||||.๐‘ฅ(๐‘ก)(2.1)

We define similarly another Banach space ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡], in which function ๐‘ฅ(๐‘ก) is continuous on (๐‘ก0,๐‘‡] and (๐‘กโˆ’๐‘ก0)1โˆ’๐‘ž๐‘ฅ(๐‘ก) is continuous on [๐‘ก0,๐‘‡] with the norm: โ€–๐‘ฅโ€–1โˆ’๐‘ž=max๐‘กโˆˆ[๐‘ก0,๐‘‡]๎€ท๐‘กโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž||||.๐‘ฅ(๐‘ก)(2.2)

๐ฟ[๐‘ก0,๐‘‡] is the space of real functions defined on [๐‘ก0,๐‘‡] which are Lebesgue integrable on [๐‘ก0,๐‘‡].

Obviously ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]โŠ‚๐ฟ(๐‘ก0,๐‘‡).

The definitions and results of the fractional calculus reported below are not exhaustive but rather oriented to the subject of this paper. For the proofs, which are omitted, we refer the reader to [6] or other texts on basic fractional calculus.

Definition 2.1 (see [6]). The fractional primitive of order ๐‘ž>0 of function ๐‘ฅ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] is given by ๐ผ๐‘ž๐‘ก01๐‘ฅ(๐‘ก)=๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘žโˆ’1๐‘ฅ(๐‘ )๐‘‘๐‘ .(2.3)
From [17] we know ๐ผ๐‘ž๐‘ก0๐‘ฅ(๐‘ก) exists for all ๐‘ž>0, when ๐‘ฅโˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]; consider also that when ๐‘ฅโˆˆ๐ถ[๐‘ก0,๐‘‡] then ๐ผ๐‘ž๐‘ก0๐‘ฅ(๐‘ก)โˆˆ๐ถ[๐‘ก0,๐‘‡] and moreover ๐ผ๐‘ž๐‘ก0๐‘ฅ๎€ท๐‘ก0๎€ธ=0.(2.4)

Definition 2.2 (see [6]). The fractional derivative of order 0<๐‘ž<1 of a function ๐‘ฅ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] is given by ๐ท๐‘ž๐‘ก01๐‘ฅ(๐‘ก)=๐‘‘ฮ“(1โˆ’๐‘ž)๎€œ๐‘‘๐‘ฅ๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )โˆ’๐‘ž๐‘ฅ(๐‘ )๐‘‘๐‘ .(2.5)
We have ๐ท๐‘ž๐‘ก0๐ผ๐‘ž๐‘ก0๐‘ฅ(๐‘ก)=๐‘ฅ(๐‘ก) for all ๐‘ฅ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡].

Lemma 2.3 (see [6]). Let 0<๐‘ž<1. If one assumes ๐‘ฅ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡], then the fractional differential equation ๐ท๐‘ž๐‘ก0๐‘ฅ(๐‘ก)=0(2.6) has ๐‘ฅ(๐‘ก)=๐‘(๐‘กโˆ’๐‘ก0)๐‘žโˆ’1,๐‘โˆˆ๐‘…, as solutions.

From this lemma we can obtain the following law of composition.

Lemma 2.4 (see [6]). Assume that ๐‘ฅ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] with a fractional derivative of order 0<๐‘ž<1 that belongs to ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Then ๐ผ๐‘ž๐‘ก0๐ท๐‘ž๐‘ก0๎€ท๐‘ฅ(๐‘ก)=๐‘ฅ(๐‘ก)+๐‘๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1,(2.7) for some ๐‘โˆˆ๐‘…. When the function ๐‘ฅ is in ๐ถ[๐‘ก0,๐‘‡], then ๐‘=0.

Lemma 2.5 (see [6]). Let ๐‘ˆ be a nonempty closed subset of a Banach space ๐ธ, and let ๐›ผ๐‘›โ‰ฅ0 for every ๐‘› and such โˆ‘โˆž๐‘›=0๐›ผ๐‘› converges. Moreover, let the mapping ๐ดโˆถ๐‘ˆโ†’๐‘ˆ satisfy the inequality โ€–๐ด๐‘›๐‘ขโˆ’๐ด๐‘›๐‘ฃโ€–โ‰ค๐›ผ๐‘›โ€–๐‘ขโˆ’๐‘ฃโ€–,(2.8) for every ๐‘›โˆˆ๐‘ and any ๐‘ข,๐‘ฃโˆˆ๐‘ˆ. Then, ๐ด has a uniquely defined fixed point ๐‘ขโˆ—. Furthermore, for any ๐‘ข0โˆˆ๐‘ˆ, the sequence (๐ด๐‘›๐‘ข0)โˆž๐‘›=1 converges to this fixed point ๐‘ขโˆ—.

Lemma 2.6 (see [12]). Let ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š) and have real eigenvalues ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘Ÿ. Then there exists a basis of ๐‘…๐‘š in which the matrix representation of ๐‘ƒ assumes Jordan form, that is, the matrix of ๐‘ƒ is made of diagonal blocks of the form diag(๐ฝ1,๐ฝ2,โ€ฆ,๐ฝ๐‘Ÿ), where each ๐ฝ๐‘– consists of diagonal blocks of the form โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œ†๐‘–0โ‹ฏ001๐œ†๐‘–โ‹ฏ0001โ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ1๐œ†๐‘–โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(2.9)

Lemma 2.7 (see [12]). Let ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š) and have complex eigenvalues ๐œ‡๐‘—=๐›ผ๐‘—+๐‘–๐›ฝ๐‘—,๐‘—=1,2,โ€ฆ,๐‘Ÿ, with multiplicity. Then there exists a basis of ๐‘…๐‘š, where ๐‘ƒ has matrix form ๎๐ฝdiag(1,๎๐ฝ2๎๐ฝ,โ€ฆ,๐‘Ÿ), where each ๎๐ฝ๐‘– consists of diagonal blocks of the type โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐ผ๐ท0โ‹ฏ002๐ทโ‹ฏ000๐ผ2โ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ๐ผ2๐ทโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐›ผ,๐ท=๐‘–โˆ’๐›ฝ๐‘–๐›ฝ๐‘–๐›ผ๐‘–โŽžโŽŸโŽŸโŽ ,๐ผ2=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 1001.(2.10)

Lemma 2.8 (see [12]). Let ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š). Then ๐‘…๐‘š has a basis giving ๐‘ƒ a matrix representation composed of diagonal blocks of type ๐ฝ๐‘– and/or matrices ๎๐ฝ๐‘–, where ๐ฝ๐‘– and ๎๐ฝ๐‘– are as defined in the preceding lemmas.

Now, we will introduce Lemma 2.9 to prove the following Theorem 4.4 in Section 4.

Lemma 2.9. Let 0<๐‘ž<1. Assume that ๐‘ฅ(๐‘ก) and ๐‘“(๐‘ก) belong to ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Then For the initial value problem ๐ท๐‘ž๐‘ก0๐‘ฅ(๐‘ก)=๐œ†๐‘ฅ(๐‘ก)+๐‘“(๐‘ก),๐‘ฅ(๐‘Ž)=๐‘(2.11) has a unique solution ๐‘ฅ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] provided ๐‘ก0<๐‘Ž<๐‘Ž0, where ๐‘Ž0 is a suitable constant depending on ๐‘ก0, ๐‘ž, and ๐œ†.

Proof. The initial value problem (2.11) will be solved in two steps.โ€‰ (1) Local existence.โ€‰Our problem is equivalent to the problem of determination of fixed points of the following operator: ๎€ท๐ด๐‘ฅ(๐‘ก)=๐‘๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+1๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘žโˆ’1(๐œ†๐‘ฅ(๐‘ )+๐‘“(๐‘ ))๐‘‘๐‘ ,(2.12)โ€‰with ๎‚ต1๐‘=๐‘โˆ’๎€œฮ“(๐‘ž)๐‘Ž๐‘ก0(๐‘Žโˆ’๐‘ )๐‘žโˆ’1๎‚ถ๎€ท(๐œ†๐‘ฅ(๐‘ )+๐‘“(๐‘ ))๐‘‘๐‘ ๐‘Žโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž.(2.13)โ€‰It is immediate to verify that ๐ดโˆถ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]โ†’๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] is also well defined. Indeed, |||๎€ท๐‘กโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž|||โ‰ค||๐œ†||||||1๐ด๐‘ฅ(๐‘ก)๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘žโˆ’1๎€ท๐‘ โˆ’๐‘ก0๎€ธ๐‘žโˆ’1๎€ท๐‘ โˆ’๐‘ก0๎€ธ1โˆ’๐‘ž||||+||||1๐‘ฅ(๐‘ )๐‘‘๐‘ ๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘žโˆ’1๎€ท๐‘ โˆ’๐‘ก0๎€ธ๐‘žโˆ’1๎€ท๐‘ โˆ’๐‘ก0๎€ธ1โˆ’๐‘ž||||||๐œ†||๐‘“(๐‘ )๐‘‘๐‘ โ‰ค|๐‘|+โ€–๐‘ฅโ€–1โˆ’๐‘ž|||๐ผ๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1|||+โ€–๐‘“โ€–1โˆ’๐‘ž|||๐ผ๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1|||||๐œ†||โ‰ค|๐‘|+โ€–๐‘ฅโ€–1โˆ’๐‘žฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘กโˆ’๐‘ก0๎€ธ๐‘ž+โ€–๐‘“โ€–1โˆ’๐‘žฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘กโˆ’๐‘ก0๎€ธ๐‘ž,(2.14)โ€‰for ๐‘ฅ(๐‘ก) and ๐‘“(๐‘ก) belong to โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡].โ€‰Then we can also prove ๐ด is a contraction operator. Indeed, ๎€ท๐‘กโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž||||โ‰ค||๐œ†||๐ด๐‘ฅ(๐‘ก)โˆ’๐ด๐‘ฆ(๐‘ก)ฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘Žโˆ’๐‘ก0๎€ธ๐‘žโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐‘ž+||๐œ†||ฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘กโˆ’๐‘ก0๎€ธ๐‘žโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐‘žโ‰ค||๐œ†||ฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘Žโˆ’๐‘ก0๎€ธ๐‘žโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐‘ž+||๐œ†||ฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘‡โˆ’๐‘ก0๎€ธ๐‘žโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐‘ž,(2.15)โ€‰for all ๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Let us assume ||๐œ†||ฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘Žโˆ’๐‘ก0๎€ธ๐‘ž<12,(2.16)โ€‰that is, ๐‘Ž<๐‘Ž0=๎‚ตฮ“(2๐‘ž)2||๐œ†||๎‚ถฮ“(๐‘ž)1/๐‘ž+๐‘ก0.(2.17)โ€‰Taking ๐‘‡โˆ’๐‘Ž>0 sufficiently small, we also have ||๐œ†||ฮ“(๐‘ž)๎€ทฮ“(2๐‘ž)๐‘‡โˆ’๐‘ก0๎€ธ๐‘ž<12,(2.18) and then โ€–๐ด๐‘ฅ(๐‘ก)โˆ’๐ด๐‘ฆ(๐‘ก)โ€–1โˆ’๐‘žโ‰ค๐ฟโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๐‘ž,(2.19)โ€‰with ๐ฟ<1. Therefore ๐ด is a contraction operator. This shows that initial problem (2.11) has a unique solution.โ€‰ (2) Continuation of solution.โ€‰Since we know the value of ๐‘ฅ(๐‘ก) on (๐‘ก0,๐‘Ž], then we can compute ๐‘โˆ—=๎‚ต1๐‘โˆ’๎€œฮ“(๐‘ž)๐‘Ž๐‘ก0(๐‘Žโˆ’๐‘ )๐‘žโˆ’1๎‚ถ๎€ท(๐œ†๐‘ฅ(๐‘ )+๐‘“(๐‘ ))๐‘‘๐‘ ๐‘Žโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž.(2.20)โ€‰We can solve the integral problem ๐‘ฆ(๐‘ก)=๐‘โˆ—๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+1๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘žโˆ’1(๐œ†๐‘ฆ(๐‘ )+๐‘“(๐‘ ))๐‘‘๐‘ ,(2.21)โ€‰obtaining a unique solution ๐‘ฆ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] for all ๐‘‡>๐‘ก0. Now ๐‘ฅ(๐‘ก) and ๐‘ฆ(๐‘ก) agree on (๐‘ก0,๐‘Ž]. Thus the solution admits ๐‘ฆ(๐‘ก) as its continuation. Hence the proof of Lemma 2.9 is complete.

3. Initial Value Problem: Continuous Solutions on (๐‘ก0,๐‘‡]

We open this section with some basic examples, concerning the case when the solutions in ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] are submitted to an initial condition.

Theorem 3.1. Let 0<๐‘ž<1. For all (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…๐‘š the initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=0,๐‘‹(๐‘Ž)=๐ต(3.1) admits ๎€ท๐‘‹(๐‘ก)=๐ต๐‘Žโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1,(3.2) as unique solution in ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡].

Proof. According to Lemma 2.4, the initial value problem (3.1) is equivalent to the following equations: ๎€ท๐‘‹(๐‘ก)=๐ถ๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1๎€ท,๐ถ=๐ต๐‘Žโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž.(3.3) Hence the proof of Theorem 3.1 is complete.

Theorem 3.2. Let 0<๐‘ž<1. Assume ๎€ท๐‘“๐น(๐‘ก)=1(๐‘ก),๐‘“2(๐‘ก),โ€ฆ,๐‘“๐‘š๎€ธ(๐‘ก)๐‘‡โˆˆ๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—โ‹ฏร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป.,๐‘‡(3.4) Then for all (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…๐‘š the initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐น(๐‘ก),๐‘‹(๐‘Ž)=๐ต(3.5) has a unique solution in ๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—โ‹ฏร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,,๐‘‡(3.6) given by ๎€ท๐‘ฅ๐‘‹(๐‘ก)=1(๐‘ก),๐‘ฅ2(๐‘ก),โ€ฆ,๐‘ฅ๐‘š๎€ธ(๐‘ก)๐‘‡,(3.7) with ๐‘ฅ๐‘–(๎‚ต๐‘๐‘ก)=๐‘–โˆ’1๎€œฮ“(๐‘ž)๐‘Ž๐‘ก0(๐‘Žโˆ’๐‘ )๐‘žโˆ’1๐‘“๐‘–(๎‚ถ๎€ท๐‘ )๐‘‘๐‘ ๐‘Žโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+1๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘žโˆ’1๐‘“๐‘–(๐‘ )๐‘‘๐‘ ,(๐‘–=1,2,โ€ฆ,๐‘š).(3.8)

Proof. According to Lemma 2.4, the initial value problem (3.5) is equivalent to the following equations: ๎€ท๐‘‹(๐‘ก)=๐ถ๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+1๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘žโˆ’1๎‚ต1๐น(๐‘ )๐‘‘๐‘ ,๐ถ=๐ตโˆ’ฮ“๎€œ(๐‘ž)๐‘Ž๐‘ก0(๐‘Žโˆ’๐‘ )๐‘žโˆ’1๎‚ถ๎€ท๐น(๐‘ )๐‘‘๐‘ ๐‘Žโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž.(3.9) Hence the proof of Theorem 3.2 is complete.

The result remains true even if ๐‘žโ†’1โˆ’. In this case, (3.5) is reduced to the ordinary differential equations ๐‘‹๎…ž(๐‘ก)=๐น(๐‘ก),๐‘‹(๐‘Ž)=๐ต,(3.10) which have a unique solution in ๐ถ๎€บ๐‘ก0๎€ป๎€บ๐‘ก,๐‘‡ร—๐ถ0๎€ป๎€บ๐‘ก,๐‘‡ร—โ‹ฏร—๐ถ0๎€ป,๐‘‡,(3.11) given by ๎€ท๐‘ฅ๐‘‹(๐‘ก)=1(๐‘ก),๐‘ฅ2(๐‘ก),โ€ฆ,๐‘ฅ๐‘š๎€ธ(๐‘ก)๐‘‡,(3.12) with ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–โˆ’๎€œ๐‘Ž๐‘ก0๐‘“๐‘–(๎€œ๐‘ )๐‘‘๐‘ +๐‘ก๐‘ก0๐‘“๐‘–(๐‘ )๐‘‘๐‘ ,(๐‘–=1,2,โ€ฆ,๐‘š).(3.13)

Theorem 3.3. Let 0<๐‘ž<1. For all (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…๐‘š the initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต,(3.14) where โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œ†๐‘ƒ=10โ‹ฏ000๐œ†2โ‹ฏ0000โ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ0๐œ†๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ (3.15) has a unique solution in ๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—โ‹ฏร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,,๐‘‡(3.16) given by ๎€ท๐‘ฅ๐‘‹(๐‘ก)=1(๐‘ก),๐‘ฅ2(๐‘ก),โ€ฆ,๐‘ฅ๐‘š๎€ธ(๐‘ก)๐‘‡,(3.17) with ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๐‘’๐‘žโˆ’1๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๎€ธ๎‚๐‘’๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚,(๐‘–=1,2,โ€ฆ,๐‘š).(3.18)

Proof. We can write (3.27) in the following form: ๐ท๐‘ž๐‘ก0๐‘ฅ1(๐‘ก)=๐œ†1๐‘ฅ1๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฅ2(๐‘ก)=๐œ†2๐‘ฅ2โ‹ฎ๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฅ๐‘š(๐‘ก)=๐œ†๐‘š๐‘ฅ๐‘š๐‘ฅ(๐‘ก),1(๐‘Ž)=๐‘1,โ€ฆ,๐‘ฅ๐‘š(๐‘Ž)=๐‘๐‘š.(3.19) According to Lemma 2.4,๐ท๐‘ž๐‘ก0๐‘ฅ๐‘–(๐‘ก)=๐œ†๐‘–๐‘ฅ๐‘–(๐‘ก)(3.20) is equivalent to the following equations: ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+๐ผ๐‘ž๐‘ก0๎€ท๐œ†๐‘–๐‘ฅ๐‘–๎€ธ,(๐‘ก)(3.21) for some ๐‘๐‘–โˆˆ๐‘…. From (3.21) we obtain, by iteration, ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๎ƒฉ๎€ทฮ“(๐‘ž)๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+๐œ†ฮ“(๐‘ž)๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ2๐‘žโˆ’1๐œ†ฮ“(2๐‘ž)+โ‹ฏ+๐‘–๐‘›โˆ’1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘›๐‘žโˆ’1๎ƒชฮ“(๐‘›๐‘ž)+๐œ†๐‘›๐‘–๐ผ๐‘ก๐‘›๐‘ž0๐‘ฅ๐‘–(๐‘ก).(3.22) Letting ๐‘›โ†’+โˆž, โ€–๐œ†๐‘›๐‘–๐ผ๐‘ก๐‘›๐‘ž0๐‘ฅ๐‘–(๐‘ก)โ€–1โˆ’๐‘žโ†’0 if ๐‘ฅ๐‘–(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Indeed, โ€–โ€–๐œ†๐‘›๐‘–๐ผ๐‘ก๐‘›๐‘ž0๐‘ฅ๐‘–(โ€–โ€–๐‘ก)1โˆ’๐‘žโ‰ค||๐œ†๐‘›๐‘–||โ€–โ€–๐‘ฅ๐‘–โ€–โ€–(๐‘ก)1โˆ’๐‘žฮ“(๐‘ž)๎€ทฮ“((๐‘›+1)๐‘ž)๐‘กโˆ’๐‘ก0๎€ธ๐‘›๐‘ž.(3.23)
On the other hand, ๐‘’๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ=+โˆž๎“๐‘˜=1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘˜๐‘žโˆ’1,ฮ“(๐‘˜๐‘ž)(3.24) then we can obtain ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–ฮ“(๐‘ž)๐œ†๐‘–(1/๐‘žโˆ’1)๐‘’๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚.(3.25) Since ๐‘ฅ๐‘–(๐‘Ž)=๐‘๐‘–, ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๐‘’๐‘žโˆ’1๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๎€ธ๎‚๐‘’๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚,(๐‘–=1,2,โ€ฆ,๐‘š).(3.26) Hence the proof of Theorem 3.3 is complete.
The result remains valid even if ๐‘žโ†’1โˆ’. In this case, ๐‘‹๎…ž(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต(3.27) has a unique solution in ๐ถ๎€บ๐‘ก0๎€ป๎€บ๐‘ก,๐‘‡ร—๐ถ0๎€ป๎€บ๐‘ก,๐‘‡ร—โ‹ฏร—๐ถ0๎€ป,๐‘‡,(3.28) given by ๎€ท๐‘ฅ๐‘‹(๐‘ก)=1(๐‘ก),๐‘ฅ2(๐‘ก),โ€ฆ,๐‘ฅ๐‘š๎€ธ(๐‘ก)๐‘‡,(3.29) with ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๎€ทexpโˆ’๐œ†๐‘–๎€ท๐‘Žโˆ’๐‘ก0๎€ท๐œ†๎€ธ๎€ธexp๐‘–๎€ท๐‘กโˆ’๐‘ก0,๎€ธ๎€ธ(๐‘–=1,2,โ€ฆ,๐‘š).(3.30)

Theorem 3.4. Let 0<๐‘ž<1. For all (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…๐‘š the initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก)+๐ท,๐‘‹(๐‘Ž)=๐ต,(3.31) where โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œ†๐‘ƒ=10โ‹ฏ000๐œ†2โ‹ฏ0000โ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ0๐œ†๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(3.32) and ๐ท=(๐‘‘1,๐‘‘2,โ€ฆ,๐‘‘๐‘š) has a unique solution in ๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,๐‘‡ร—โ‹ฏร—๐ถ1โˆ’๐‘ž๎€บ๐‘ก0๎€ป,,๐‘‡(3.33) given by ๎€ท๐‘ฅ๐‘‹(๐‘ก)=1(๐‘ก),๐‘ฅ2(๐‘ก),โ€ฆ,๐‘ฅ๐‘š๎€ธ(๐‘ก)๐‘‡,(3.34) with ๐‘ฅ๐‘–๎‚€๐‘(๐‘ก)=๐‘–โˆ’๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๎€ธ๐‘’๎‚๎‚๐‘žโˆ’1๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๎€ธ๎‚๐‘’๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚+๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚(๐‘–=1,2,โ€ฆ,๐‘š).(3.35)

Proof. We can write (3.44) in the following form:๐ท๐‘ž๐‘ก0๐‘ฅ1(๐‘ก)=๐œ†1๐‘ฅ1(๐‘ก)+๐‘‘1,๐ท๐‘ž๐‘ก0๐‘ฅ2(๐‘ก)=๐œ†2๐‘ฅ2(๐‘ก)+๐‘‘2,โ‹ฎ๐ท๐‘ž๐‘ก0๐‘ฅ๐‘š(๐‘ก)=๐œ†๐‘š๐‘ฅ๐‘š(๐‘ก)+๐‘‘๐‘š๐‘ฅ1(๐‘Ž)=๐‘1,โ€ฆ,๐‘ฅ๐‘š(๐‘Ž)=๐‘๐‘š.(3.36)
According to Lemma 2.4, the equation ๐ท๐‘ž๐‘ก0๐‘ฅ๐‘–(๐‘ก)=๐œ†๐‘–๐‘ฅ๐‘–(๐‘ก)+๐‘‘๐‘–(3.37) is equivalent to the following equations: ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+๐ผ๐‘ž๐‘ก0๎€ท๐œ†๐‘–๐‘ฅ๐‘–๎€ธ(๐‘ก)+๐ผ๐‘ž๐‘ก0๎€ท๐‘‘๐‘–๎€ธ,(3.38) for some ๐‘๐‘–โˆˆ๐‘…. From (3.38) we obtain, by iteration, ๐ผ๐‘ž๐‘ก0๐œ†๐‘–๐‘ฅ๐‘–(๐‘ก)=๐ผ๐‘ž๐‘ก0๎‚€๐‘๐‘–๐œ†๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1๎‚+๐ผ๐‘ก2๐‘ž0๎€ท๐œ†2๐‘–๐‘ฅ๐‘–๎€ธ(๐‘ก)+๐ผ๐‘ก2๐‘ž0๎€ท๐œ†๐‘–๐‘‘๐‘–๎€ธ,๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+๐ผ๐‘ž๐‘ก0๎‚€๐œ†๐‘–๐‘๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1๎‚+๐ผ๐‘ก2๐‘ž0๎€ท๐œ†2๐‘–๐‘ฅ๐‘–๎€ธ(๐‘ก)+๐ผ๐‘ž๐‘ก0๎€ท๐‘‘๐‘–๎€ธ+๐ผ๐‘ก2๐‘ž0๎€ท๐œ†๐‘–๐‘‘๐‘–๎€ธ,๐ผ๐‘ž๐‘ก0๐‘ฅ๐‘–(๐‘ก)=๐ผ๐‘ž๐‘ก0๎‚€๐‘๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1๎‚+๐ผ๐‘ก2๐‘ž0๎€ท๐œ†๐‘–๐‘ฅ๐‘–๎€ธ(๐‘ก)+๐ผ๐‘ก2๐‘ž0๎€ท๐‘‘๐‘–๎€ธ,๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–๎ƒฉ๎€ทฮ“(๐‘ž)๐‘กโˆ’๐‘ก0๎€ธ๐‘žโˆ’1+๐œ†ฮ“(๐‘ž)๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ2๐‘žโˆ’1๐œ†ฮ“(2๐‘ž)+โ‹ฏ+๐‘–๐‘›โˆ’1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘›๐‘žโˆ’1๎ƒชฮ“(๐‘›๐‘ž)+๐‘‘๐‘–๎ƒฉ๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘ž+๐œ†ฮ“(๐‘ž+1)๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ2๐‘ž๐œ†ฮ“(2๐‘ž+1)+โ‹ฏ+๐‘–๐‘›โˆ’1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘›๐‘ž๎ƒชฮ“(๐‘›๐‘ž+1)+๐œ†๐‘›๐‘–๐ผ๐‘ก๐‘›๐‘ž0๐‘ฅ๐‘–(๐‘ก).(3.39)
Letting ๐‘›โ†’+โˆž, โ€–๐œ†๐‘›๐‘–๐ผ๐‘ก๐‘›๐‘ž0๐‘ฅ๐‘–(๐‘ก)โ€–1โˆ’๐‘žโ†’0 if ๐‘ฅ๐‘–(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Indeed, โ€–โ€–๐œ†๐‘›๐‘–๐ผ๐‘ก๐‘›๐‘ž0๐‘ฅ๐‘–โ€–โ€–(๐‘ก)1โˆ’๐‘ž=max๎€บ๐‘ก๐‘กโˆˆ0๎€ป,๐‘‡๎€ท๐‘กโˆ’๐‘ก0๎€ธ1โˆ’๐‘ž||||๐œ†๐‘›๐‘–1๎€œฮ“(๐‘ž)๐‘ก๐‘ก0(๐‘กโˆ’๐‘ )๐‘›๐‘žโˆ’1๐‘ฅ๐‘–||||โ‰ค||๐œ†(๐‘ )๐‘‘๐‘ ๐‘›๐‘–||โ€–โ€–๐‘ฅ๐‘–โ€–โ€–(๐‘ก)1โˆ’๐‘žฮ“(๐‘ž)ฮ“๎€ท((๐‘›+1)๐‘ž)๐‘กโˆ’๐‘ก0๎€ธ๐‘›๐‘ž.(3.40)
On the other hand, ๐‘’๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ=+โˆž๎“๐‘˜=1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘˜๐‘žโˆ’1,๐ธฮ“(๐‘˜๐‘ž)๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ=+โˆž๎“๐‘˜=1๎€ท๐‘กโˆ’๐‘ก0๎€ธ๐‘˜๐‘žฮ“.(๐‘˜๐‘ž+1)(3.41)
Then we can obtain ๐‘ฅ๐‘–(๐‘ก)=๐‘๐‘–ฮ“(๐‘ž)๐œ†๐‘–(1/๐‘žโˆ’1)๐‘’๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚+๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚.(3.42)
We know that ๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ๐‘ž(๐œ†๐‘–1/๐‘ž(๐‘กโˆ’๐‘ก0)) is satisfied for the fractional nonhomogeneous linear differential equation ๐ท๐‘ž๐‘ก0๐‘ฅ๐‘–(๐‘ก)=๐œ†1๐‘ฅ๐‘–(๐‘ก)+๐‘‘๐‘–. So we can also deduce that the general solution of the fractional nonhomogeneous linear differential equation is equal to the general solution of the corresponding homogeneous linear differential equation plus the special solution of the nonhomogeneous linear differential equation. If ๐‘‹(๐‘Ž)=๐ต, ๐‘ฅ๐‘–(๐‘Ž)=๐‘๐‘–, then ๐‘ฅ๐‘–๎‚€๐‘(๐‘ก)=๐‘–โˆ’๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๎€ธ๐‘’๎‚๎‚๐‘žโˆ’1๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๎€ธ๎‚๐‘’๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚+๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ๐‘ž๎‚€๐œ†๐‘–1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚(๐‘–=1,2,โ€ฆ,๐‘š).(3.43) Hence the proof of Theorem 3.4 is complete.

The result remains valid even if ๐‘žโ†’1โˆ’. In this case, ๐‘‹๎…ž(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก)+๐ท,๐‘‹(๐‘Ž)=๐ต,(3.44) where โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œ†๐‘ƒ=10โ‹ฏ000๐œ†2โ‹ฏ0000โ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ0๐œ†๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(3.45) and ๐ท=(๐‘‘1,๐‘‘2,โ€ฆ,๐‘‘๐‘š) has a unique solution in ๐ถ๎€บ๐‘ก0๎€ป๎€บ๐‘ก,๐‘‡ร—๐ถ0๎€ป๎€บ๐‘ก,๐‘‡ร—โ‹ฏร—๐ถ0๎€ป,๐‘‡,(3.46) given by ๎€ท๐‘ฅ๐‘‹(๐‘ก)=1(๐‘ก),๐‘ฅ2(๐‘ก),โ€ฆ,๐‘ฅ๐‘š๎€ธ(๐‘ก)๐‘‡,(3.47) with ๐‘ฅ๐‘–๎€ท๐‘(๐‘ก)=๐‘–โˆ’๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ1๎€ท๐œ†๐‘–๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ๎€ธ1โˆ’1๎€ท๐œ†๐‘–๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ1๎€ท๐œ†๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎€ธ+๐‘‘๐‘–๐œ†๐‘–โˆ’1๐ธ1๎€ท๐œ†๐‘–๎€ท๐‘กโˆ’๐‘ก0=๎€ท๐‘๎€ธ๎€ธ๐‘–+๐‘‘๐‘–๐œ†๐‘–โˆ’1๎€ท๎€ทexpโˆ’๐œ†๐‘Žโˆ’๐‘ก0๎€ท๐œ†๎€ธ๎€ธ๎€ธexp๐‘–๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎€ธโˆ’๐‘‘๐œ†โˆ’1,(๐‘–=1,2,โ€ฆ,๐‘š).(3.48)

4. Existence and Uniqueness of the Solution

In Section 3 we have obtained the unique solution of the constant coefficient homogeneous and nonhomogeneous linear fractional differential equations for ๐‘ƒ being the diagonal matrix. In the present section we will prove the existence and uniqueness of these two kinds of equations for any ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š).

Theorem 4.1. Let 0<๐‘ž<1 and ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š). If the matrix ๐‘ƒ has distinct real eigenvalues, then for all (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…๐‘š the initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต(4.1) has the unique solution ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡].

Proof. Since the matrix ๐‘ƒ has distinct real eigenvalues, there exists an invertible matrix ๐‘„ such that ๐‘„โˆ’1โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐œ†๐‘ƒ๐‘„=10โ‹ฏ000๐œ†2โ‹ฏ0000โ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ0๐œ†๐‘šโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(4.2) where ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘š are the eigenvalues of the matrix ๐‘ƒ. If we define ๐‘Œ(๐‘ก)=๐‘„โˆ’1๐‘‹(๐‘ก), ๐ท๐‘ž๐‘ก0๐‘Œ(๐‘ก)=๐ท๐‘ž๐‘ก0๐‘„โˆ’1๐‘‹(๐‘ก)=๐‘„โˆ’1๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘„โˆ’1๐‘ƒ๐‘‹(๐‘ก)=๐‘…๐‘Œ(๐‘ก),(4.3) with ๐‘…=diag(๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘š). From the above Theorem 3.3 we know the initial value problem ๐ท๐‘ž๐‘ก0๐‘Œ(๐‘ก)=๐‘…๐‘Œ(๐‘ก),๐‘Œ(๐‘Ž)=๐‘„โˆ’1๐ต(4.4) has a unique solution ๐‘Œ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] defined on [๐‘ก0,๐‘‡]. Then ๐‘‹(๐‘ก)=๐‘„๐‘Œ(๐‘ก) uniquely solves the equations (4.1), where ๐‘กโˆˆ[๐‘ก0,๐‘‡]. Hence the proof of Theorem 4.1 is complete.

Theorem 4.2. Let 0<๐‘ž<1. For all (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…2 the initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต,(4.5) where โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๐‘ƒ=๐›ผโˆ’๐›ฝ๐›ฝ๐›ผ(4.6) has the unique solution ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] defined on [๐‘ก0,๐‘‡].

Proof. Let us define ๐‘(๐‘ก)=๐‘ฅ1(๐‘ก)+๐‘–๐‘ฅ2(๐‘ก),๐œ‡=๐›ผ+๐‘–๐›ฝ.(4.7) We can find that (4.5) is equivalent to the following equation ๐ท๐‘ž๐‘ก0๐‘(๐‘ก)=๐œ‡๐‘(๐‘ก),๐‘(๐‘Ž)=๐‘ฅ1(๐‘Ž)+๐‘ฅ2(๐‘Ž)=๐‘1+๐‘–๐‘2.(4.8) Obviously, ๐‘(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] if ๐‘ฅ1(๐‘ก) and ๐‘ฅ2(๐‘ก) belong to ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. From the above Theorem 3.3 in Section 3, we know the complex Equation (4.7) has a unique solution defined on [๐‘ก0,๐‘‡]. Hence the proof of Theorem 4.2 is complete.

Theorem 4.3. Let 0<๐‘ž<1 and ๐‘ƒโˆˆ๐‘…2. If ๐‘ƒ has eigenvalues ๐›ผยฑ๐‘–๐›ฝ, for all (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…2 the initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต(4.9) has a unique solution ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡].

Proof. Since ๐‘ƒ has eigenvalues ๐›ผยฑ๐‘–๐›ฝ, there exists an invertible matrix ๐‘„ such that ๐‘ƒ=๐‘„๐‘†๐‘„โˆ’1 where โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๐‘†=๐›ผโˆ’๐›ฝ๐›ฝ๐›ผ.(4.10) Define ๐‘Œ(๐‘ก)=๐‘„โˆ’1๐‘‹(๐‘ก),(4.11) then ๐ท๐‘ž๐‘ก0๐‘Œ(๐‘ก)=๐ท๐‘ž๐‘ก0๐‘„โˆ’1๐‘‹(๐‘ก)=๐‘„โˆ’1๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘„โˆ’1๐‘ƒ๐‘‹(๐‘ก)=๐‘†๐‘Œ(๐‘ก).(4.12) From the above Theorem 4.2, we know the initial value problem ๐ท๐‘ž๐‘ก0๐‘Œ(๐‘ก)=๐‘†๐‘Œ(๐‘ก),๐‘Œ(๐‘Ž)=๐‘„โˆ’1๐ต(4.13) has a unique solution defined on [๐‘ก0,๐‘‡]. Hence the proof of result is complete.

Theorem 4.4. Let 0<๐‘ž<1 and ๐‘ƒโˆˆ๐‘…๐‘š be an elementary Jordan matrix: โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๐œ†0โ‹ฏ001๐œ†โ‹ฏ0001โ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ1๐œ†.(4.14) The initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต(4.15) has a unique solution ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] provided (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘Ž0]ร—๐‘…๐‘š, where ๐‘Ž0 is a suitable constant depending on ๐‘ก0, ๐‘ž, and ๐œ†.

Proof. From the (4.15), we can write the equations in the following form: ๐ท๐‘ž๐‘ก0๐‘ฅ1(๐‘ก)=๐œ†๐‘ฅ1๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฅ2(๐‘ก)=๐‘ฅ1(๐‘ก)+๐œ†๐‘ฅ2โ‹ฎ๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฅ๐‘š(๐‘ก)=๐‘ฅ๐‘šโˆ’1(๐‘ก)+๐œ†๐‘ฅ๐‘š๐‘ฅ(๐‘ก),1(๐‘Ž)=๐‘1,โ€ฆ,๐‘ฅ๐‘š(๐‘Ž)=๐‘๐‘š.(4.16)
Consider the first equation ๐ท๐‘ž๐‘ก0๐‘ฅ1(๐‘ก)=๐œ†๐‘ฅ1(๐‘ก),๐‘ฅ1(๐‘Ž)=๐‘1.(4.17)
We can obtain the solution of this equation ๐‘ฅ1(๐‘ก)=๐‘1๐‘’๐‘žโˆ’1๎€ท๐œ†1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ๐‘ž๎€ท๐œ†1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0.๎€ธ๎€ธ(4.18)
Consider the second equation ๐ท๐‘ž๐‘ก0๐‘ฅ2(๐‘ก)=๐‘ฅ1(๐‘ก)+๐œ†๐‘ฅ2(๐‘ก),๐‘ฅ2(๐‘Ž)=๐‘2,(4.19) where now ๐‘ฅ1(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] is a known function. Since ๐‘ฅ1(๐‘ก),๐‘ฅ2(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡], according to Lemma 2.9, (4.19) has a unique solution in ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Now ๐‘ฅ1(๐‘ก) and ๐‘ฅ2(๐‘ก) are known functions which will be substituted in ๐ท๐‘ž๐‘ก0๐‘ฅ3(๐‘ก)=๐‘ฅ2(๐‘ก)+๐œ†๐‘ฅ3(๐‘ก),๐‘ฅ3(๐‘Ž)=๐‘3(4.20) and so on. Thus the system of equations given in (4.15) has unique solution in ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡].

Theorem 4.5. Let 0<๐‘ž<1 and ๐‘ƒโˆˆ๐ฟ(๐‘…๐‘š). The initial value problem ๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘ƒ๐‘‹(๐‘ก),๐‘‹(๐‘Ž)=๐ต(4.21) has the unique solution ๐‘‹(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡] provided (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘Ž0]ร—๐‘…๐‘š, where ๐‘Ž0 is a suitable constant depending on ๐‘ก0, ๐‘ž, and ๐œ†.

Proof. In view of Lemma 2.8, there exists an invertible matrix ๐‘„ such that ๐‘„โˆ’1๐‘ƒ๐‘„ is composed of diagonal blocks of the type ๐ฝ๐‘– and ๎๐ฝ๐‘–, as defined in the preceding Lemmas 2.7 and 2.8. Let ๐ต=๐‘„โˆ’1๐‘ƒ๐‘„ and ๐‘Œ(๐‘ก)=๐‘„โˆ’1๐‘‹(๐‘ก). Consider the initial value problem: ๐ท๐‘ž๐‘ก0๐‘Œ(๐‘ก)=๐ท๐‘ž๐‘ก0๐‘„โˆ’1๐‘‹(๐‘ก)=๐‘„โˆ’1๐ท๐‘ž๐‘ก0๐‘‹(๐‘ก)=๐‘„โˆ’1๐‘ƒ๐‘‹(๐‘ก)=๐ต๐‘Œ(๐‘ก),๐‘Œ(๐‘Ž)=๐‘„โˆ’1๐‘‹(๐‘Ž)=๐‘„โˆ’1๐ต.(4.22) Then in view of Theorems 4.1โ€“4.5, (4.22) has a unique solution: ๐‘Œ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]. Therefore (4.21) has a unique solution ๐‘„โˆ’1๐‘Œ(๐‘ก)โˆˆ๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡]ร—โ‹ฏร—๐ถ1โˆ’๐‘ž[๐‘ก0,๐‘‡].

Remark 4.6. All the above results are valid for ๐‘žโ†’1โˆ’. Moreover, we can also discuss the case if ๐‘Ž=๐‘ก0, in this case, we cannot consider the usual initial condition ๐‘ฅ(๐‘ก0)=๐‘, but lim๐‘กโ†’๐‘ก0(๐‘กโˆ’๐‘ก0)1โˆ’๐‘ž๐‘ฅ(๐‘ก)=๐‘. We can also obtain some similar results by the same method, So we did not give the detailed process and conclusion in this paper.

5. Illustrative Examples

In this section, we give some specific examples to illustrate the above results.

Example 5.1. Consider the following system, where 0<๐‘ž<1, ๐‘กโˆˆ[๐‘ก0,๐‘‡], (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…3,๐ต=(๐‘1,๐‘2,๐‘3),๐ท๐‘ž๐‘ก0๐‘ฅ1(๐‘ก)=3๐‘ฅ1(๐‘ก)โˆ’๐‘ฅ2(๐‘ก)+๐‘ฅ3๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฅ2(๐‘ก)=โˆ’๐‘ฅ1(๐‘ก)+5๐‘ฅ2(๐‘ก)โˆ’๐‘ฅ3๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฅ3(๐‘ก)=๐‘ฅ1(๐‘ก)โˆ’๐‘ฅ2(๐‘ก)+3๐‘ฅ3๐‘ฅ(๐‘ก),1(๐‘Ž)=๐‘1,๐‘ฅ2(๐‘Ž)=๐‘2,๐‘ฅ3(๐‘Ž)=๐‘3.(5.1) Here โŽ›โŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽ ๐‘ƒ=3โˆ’11โˆ’15โˆ’11โˆ’13,(5.2) having the eigenvalues 2, 3, and 6. Choose the eigenvectorsโ€‰โ€‰๐‘”1=(1,0,โˆ’1)๐‘‡,๐‘”2=(1,1,1)๐‘‡, and ๐‘”3=(1,โˆ’2,1)๐‘‡. Then โŽ›โŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽ 200030006=๐‘„โˆ’1โŽ›โŽœโŽœโŽœโŽœโŽโŽžโŽŸโŽŸโŽŸโŽŸโŽ 3โˆ’11โˆ’15โˆ’11โˆ’13๐‘„,(5.3) where ๐‘„โˆ’1=โŽ›โŽœโŽœโŽœโŽœโŽœโŽ1210โˆ’213131316โˆ’1316โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(5.4)
Define the ๐‘Œ=๐‘„โˆ’1๐‘‹. Then the system of equation in ๐‘Œ is decoupled, namely, ๐ท๐‘ž๐‘ก0๐‘ฆ1(๐‘ก)=2๐‘ฆ1๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฆ2(๐‘ก)=3๐‘ฆ2๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฆ3(๐‘ก)=6๐‘ฆ3๐‘ฆ(๐‘ก),11(๐‘Ž)=2๐‘1โˆ’12๐‘3,๐‘ฆ21(๐‘Ž)=3๐‘1+13๐‘2+13๐‘3,๐‘ฆ31(๐‘Ž)=6๐‘1โˆ’13๐‘2+16๐‘3.(5.5)
In view of (3.30), we can obtain ๐‘ฆ1๎‚€1(๐‘ก)=2๐‘1โˆ’12๐‘3๎‚๐‘’๐‘žโˆ’1๎€ท21/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ๐‘ž๎€ท21/๐‘ž๎€ท๐‘กโˆ’๐‘ก0,๐‘ฆ๎€ธ๎€ธ2๎‚€1(๐‘ก)=3๐‘1+13๐‘2+13๐‘3๎‚๐‘’๐‘žโˆ’1๎€ท31/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ๐‘ž๎€ท31/๐‘ž๎€ท๐‘กโˆ’๐‘ก0,๐‘ฆ๎€ธ๎€ธ3๎‚€1(๐‘ก)=6๐‘1โˆ’13๐‘2+16๐‘3๎‚๐‘’๐‘žโˆ’1๎€ท61/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ๐‘ž๎€ท61/๐‘ž๎€ท๐‘กโˆ’๐‘ก0.๎€ธ๎€ธ(5.6)
Hence ๐‘ฅ1(๐‘ก)=๐‘ฆ1(๐‘ก)+๐‘ฆ2(๐‘ก)+๐‘ฆ3๐‘ฅ(๐‘ก),2(๐‘ก)=๐‘ฆ2(๐‘ก)โˆ’2๐‘ฆ3๐‘ฅ(๐‘ก),3(๐‘ก)=โˆ’๐‘ฆ1(๐‘ก)+๐‘ฆ2(๐‘ก)+๐‘ฆ3(๐‘ก).(5.7)

Example 5.2. Consider the following system, where 0<๐‘ž<1, ๐‘กโˆˆ[๐‘ก0,๐‘‡], (๐‘Ž,๐ต)โˆˆ(๐‘ก0,๐‘‡]ร—๐‘…2,๐ต=(๐‘1,๐‘2), ๐ท๐‘ž๐‘ก0๐‘ฅ1(๐‘ก)=โˆ’2๐‘ฅ1(๐‘ก)โˆ’๐‘ฅ2๐ท(๐‘ก),๐‘ž๐‘ก0๐‘ฅ2(๐‘ก)=13๐‘ฅ1(๐‘ก)+4๐‘ฅ2๐‘ฅ(๐‘ก),1(๐‘Ž)=๐‘1,๐‘ฅ2(๐‘Ž)=๐‘2.(5.8) Here โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ๐‘ƒ=โˆ’2โˆ’1134(5.9) having the eigenvalues 1ยฑ2๐‘–. Choose the eigenvectorsโ€‰โ€‰ ๐‘”1=(1,โˆ’3โˆ’2๐‘–)๐‘‡, and ๐‘”2=(1,โˆ’3+2๐‘–)๐‘‡, Then โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ 1+2๐‘–001โˆ’2๐‘–=๐‘„โˆ’1โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ โˆ’2โˆ’1134๐‘„,(5.10) where โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ ,๐‘„๐‘„=11โˆ’3โˆ’2๐‘–โˆ’3+2๐‘–โˆ’1=โŽ›โŽœโŽœโŽ2+3๐‘–4๐‘–42โˆ’3๐‘–4โˆ’๐‘–4โŽžโŽŸโŽŸโŽ .(5.11) Define the ๐‘Œ=๐‘„โˆ’1๐‘‹. Then the system of equation in ๐‘Œ is decoupled, namely, ๐ท๐‘ž๐‘ก0๐‘ฆ1(๐‘ก)=(1+2๐‘–)๐‘ฆ1(๐‘ก),๐‘ฆ11(๐‘Ž)=2๐‘1+๎‚€34๐‘1+14๐‘2๎‚๐‘–๐ท๐‘ž๐‘ก0๐‘ฆ2(๐‘ก)=(1โˆ’2๐‘–)๐‘ฆ2(๐‘ก),๐‘ฆ21(๐‘Ž)=2๐‘1โˆ’๎‚€34๐‘1+14๐‘2๎‚๐‘–.(5.12) In view of (3.30), we can obtain ๐‘ฆ1๎‚€1(๐‘ก)=2๐‘1+๎‚€34๐‘1+14๐‘2๎‚๐‘–๎‚๐‘’๐‘žโˆ’1๎€ท(1+2๐‘–)1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ๐‘ž๎€ท(1+2๐‘–)1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0๐‘ฆ๎€ธ๎€ธ2๎‚€1(๐‘ก)=2๐‘1โˆ’๎‚€34๐‘1+14๐‘2๎‚๐‘–๎‚๐‘’๐‘žโˆ’1๎€ท(1โˆ’2๐‘–)1/๐‘ž๎€ท๐‘Žโˆ’๐‘ก0๐‘’๎€ธ๎€ธ๐‘ž๎€ท(1โˆ’2๐‘–)1/๐‘ž๎€ท๐‘กโˆ’๐‘ก0.๎€ธ๎€ธ(5.13) Hence ๐‘ฅ1(๐‘ก)=๐‘ฆ1(๐‘ก)+๐‘ฆ2๐‘ฅ(๐‘ก)2๎€ท๐‘ฆ(๐‘ก)=โˆ’31(๐‘ก)+๐‘ฆ2๎€ธ๎€ท๐‘ฆ(๐‘ก)โˆ’2๐‘–1(๐‘ก)โˆ’๐‘ฆ2๎€ธ.(๐‘ก)(5.14)

Acknowledgments

The authors are highly grateful for the referee's careful reading and comments on this paper. The present paper was supported by the NNSF of China Grants no. 11271087 and no. 61263006.

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