system of initial value problems involving riemann-liouville sequential ...

Communications in Applied Analysis, 22, No. 3 (2018), 353-368

ISSN: 1083-2564

SYSTEM OF INITIAL VALUE PROBLEMS INVOLVING RIEMANN-LIOUVILLE SEQUENTIAL FRACTIONAL DERIVATIVE

J.A. NANWARE1 AND D.B. DHAIGUDE2 1 Department

of Mathematics Shrikirishna Mahavidyalaya, Gunjoti Dist. Osmanabad (M.S.), 413 606, INDIA 2 Department

of Mathematics Dr. Babasaheb Ambedkar Marathwada University Aurangabad (M.S.), 431 004, INDIA

ABSTRACT: In this paper, system of initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative is studied by using monotone iterative technique coupled with lower-upper solutions. Monotone iterative technique is successfully applied to obtain existence and uniqueness of solutions of system of initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. AMS Subject Classification: 34A12, 34C60, 34A45 Key Words: initial value problems, fractional differential equations, lowerupper solutions, Riemann-Liouville sequential fractional derivative Received: May 2, 2017 ; Accepted: March 22, 2018 ; Published: May 28, 2018. doi: 10.12732/caa.v22i3.2 Dynamic Publishers, Inc., Acad. Publishers, Ltd. http://www.acadsol.eu/caa

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J.A. NANWARE AND D.B. DHAIGUDE

1. INTRODUCTION Fractional differential equations occur more frequently in different research areas and engineering such as physics, chemistry, control of dynamical systems etc [2, 3, 6, 13, 21]. During the last decade many researchers paid attention towards existence and uniqueness results for initial value problems [4, 11, 26], boundary value problems [1, 15, 17], periodic boundary value problem [12, 18, 22] and integral boundary value problem [19, 23]. Some recent results on the theory of fractional differential equations due to Lakshmikantham et. al. can be seen in [7, 8, 9, 10]. Recently, Wei et. al. [24, 25] developed monotone iterative technique for initial value problems and periodic boundary value problems involving Riemann- Liouville sequential fractional derivative and technique is successfully applied to study existence and uniqueness results for initial value problems and periodic boundary value problems. In the year 2012, Nanware and Dhaigude developed monotone method for system of Caputo fractional differential equations with periodic boundary conditions when the function is quasimonotone nondecreasing [5, 14], Riemann-Liouville fractional differential equations with integral boundary conditions when the function on the right is sum of nondecreasing and nonincreasing functions [14, 16] and system of Riemann-Liouville fractional differential equations with integral boundary conditions when the function is quasimonotone nondecreasing [14, 20]. Monotone method is successfully applied to obtain existence and uniqueness of solutions of the problems. In this paper, we shall consider the following system of initial value problems involving Riemann-Liouville sequential fractional derivative when the function on the right hand side is quasimonotone nondecreasing : 2q q q u2 ), t ∈ (0, T ], u1 , D0+ (D0+ ui )(t) = fi (t, u1 , u2 , D0+ (1.1) t1−q ui (t) = ui0 , t1−q (D q ui )(t)|t=0 = ui1 , i = 1, 2. where fi ∈ C([0, T ] × R × R × R × R) is quasimonotone nondecreasing. Monotone method is developed for the initial value problem (1.1). Existence and uniqueness results are obtained for initial value problem (1.1) by using monotone method. The paper is arranged in the following way: In the second section definitions and basic results are considered. Com-

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SYSTEM OF INITIAL VALUE PROBLEMS

parison results and some important Lemmas are also given. In the last section monotone method is developed for system of initial value problems for fractional differential equations involving Riemann-Liouville fractional derivative and technique developed is successfully applied to obtain existence and uniqueness of solution of the problem (1.1).

2. DEFINITIONS AND BASIC RESULTS Let J = [0, T ] be a compact interval on the real axis R, and u(t) be a measurable function, that is, u ∈ L1 (0, T ). Let t ∈ J and q ∈ R (0 < q ≤ 1). The relation between the Riemann-Liouville sequential fractional derivatives and non-sequential Riemann-Liouville fractional derivative [6] is given by        (x − 0)q−1 2q 2q 1−q (t). D u (t) = D u(x) − I u Γ(q)

(2.1)

Definition 2.1. A function ui (t) is called a classical solution of initial value problem (1.1) if: (i) ui (t) is continuous on (0, T ]; t1−q ui (t), t1−q (D q ui )(t) are continuous on [0, T ], and its fractional integrals (I 1−q ui )(t), (I 1−q D q ui )(t) are continuously differentiable for (0, T ], (ii) ui (t) satisfies initial value problem (1.1). Definition 2.2. Define the following classes: C([0, T ]) =

  ui : ui (t) is continuous on [0, T ], ||ui (t)||C = max |ui (t)| ,

C1−q ([0, T ]) =

t∈(0,T ]

 ui ∈ C([0, T ]) : t1−q ui (t) ∈ C([0, T ]), ||ui (t)||C1−q = ||t

q C1−q ([0, T ])

1−q



ui (t)||C ,

  1−q q = ui ∈ C1−q ([0, T ]) : t (D ui )(t) ∈ C([0, T ]) .

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q Definition 2.3. A function v 0 (t) = (v10 , v20 ) ∈ C1−q ([0, T ]) is called a lower solution of initial value problem (1.1) if it satisfies

(D 2q vi0 )(t) ≤ fi (t, v10 , v20 , D q v10 , D q v20 ), t1−q vi0 (t) ≤ vi0 ,

t ∈ (0, T ]

t1−q (D q vi0 )(t)|t=0 ≤ vi1 .

q ([0, T ]) is called a upper Definition 2.4. A function w0 (t) = (w10 , w20 ) ∈ C1−q solution of initial value problem (1.1) if it satisfies

(D 2q wi0 )(t) ≥ fi (t, w10 , w20 , D q w10 , D q w20 ),

t1−q wi0 (t)|t=0 ≥ wi0 ,

t ∈ (0, T ]

t1−q (D q wi0 )(t)|t=0 ≥ wi1 .

Assume that vi0 (t) ≤ wi0 (t), t ∈ (0, T ] : t1−q vi0 (t)|t=0 ≤ t1−q wi0 (t)|t=0 ,

t1−q (D q vi0 )(t)|t=0 ≤ t1−q (D q wi0 )(t)|t=0

q Definition 2.5. Define the sector in space C1−q ([0, T ]):  q [v 0 , w0 ] = ui ∈ C1−q ([0, T ]) | vi0 ≤ ui ≤ wi0 , t ∈ (0, T ] :

t1−q vi0 |t=0 ≤ t1−q ui |t=0 ≤ t1−q wi0 |t=0 , t

1−q

(D q vi0 )|t=0

≤t

1−q

q

(D ui )|t=0 ≤ t

1−q

(D

q

wi0 )|t=0



Following Lemma gives the existence of solution of the linear initial value problem for fractional differential equation Lemma 2.6. value problem

[24] Suppose that u(t) ∈ C1−q ([0, T ]), then the linear initial D q u(t) + M u(t) = σ(t), t1−q u(t)|t=0 = u0 ,

t ∈ (0, T ],

(2.2)

where M ∈ R is a constant and σ(t) ∈ C1−q ([0, T ]), has the following integral representation of solution   u(t) = Γ(q)u0 eq (−M, t) + eq (−M, x) ∗ σ(x) (t), (2.3) where (g ∗ f )(t) =

Z

0

t

g(t − x)f (x)dx,

eq (λ, z) = z q−1 Eq,q (λz q )

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SYSTEM OF INITIAL VALUE PROBLEMS

= z q−1

∞ X k=0

λk

z qk , Γ[(k + 1)q]

where Eq,q (t) =

∞ X k=0

tk Γ[(k + 1)q]

is Mittag-Leffler function of the two parameter. q [24] Suppose that u(t) ∈ C1−q ([0, T ]), then the linear initial

Lemma 2.7. value problem

(D 2q u)(t) + N D q u(t) + M u(t) = σ(t), t1−q u(t)|t=0 = u0 ,

t ∈ (0, T ],

t1−q (D q u)(t)|t=0 = u1

where N, M ∈ R, N 2 ≥ 4M are constants and σ(t) ∈ C1−q ([0, T ]), has the following representation of solution   u(t) = Γ(q)u0 eq (λ2 , t) + Γ(q)(u1 − λ2 u0 ) eq (λ2 , x) ∗ eq (λ1 , x) (t)+   eq (λ2 , x) ∗ eq (λ1 , x) ∗ σ(x) (t), where λ1 =

−N +

√

N 2 − 4M , 2

λ2 =

−N −

√ N 2 − 4M ≤ 0. 2

Lemma 2.8. [24] Prove that:     eq (λ2 , x) ∗ eq (λ1 , x) (t) = eq (λ1 , x) ∗ eq (λ2 , x) (t)   1 = eq (λ1 , x) − eq (λ2 , x) (t), t ∈ R. λ1 − λ2 Lemma 2.9. [24] For 0 ≤ q ≤ 1, there exist positive constants b0n > 0, b1n > P i+1 0, b2n > 0, ..., bnn > 0, such that ωn (kq) = ni=0 bin Ck+i . Pn i+2 i . Hence, we have (k − 1)ωn (kq) = i=0 (i + 2)bn Ck+i n

kq 1X 1 i i kq b C . (1 + kq)(1 + )...(1 + ) = 2 n q i + 1 n k+i i=0

Following comparison results play a vital role in the later section.

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Lemma 2.10. [24] If w(t) ∈ C1−q ([0, T ]) and satisfies the relations D q w(t) + M w(t) ≥ 0, t

1−q

t ∈ (0, T ]

w(t)|t=0 ≥ 0,

where M ∈ R is constant. Then w(t) ≥ 0,

t ∈ (0, T ].

3. MONOTONE METHOD AND APPLICATIONS In this section monotone method is developed for the system of initial value problem for fractional differential equations involving Riemann-Liouville fractional derivative when the function on the right hand side is quasimonotone nondecreasing and monotone method is successfully applied to obtain existence of solution of the initial value problem (1.1). The uniqueness of the solution of the problem is also obtained. Theorem 3.1. Assume that q (i) vi0 , wi0 ∈ C1−q ([0, T ]) are ordered lower and upper solutions of IVP (1.1), f = (f1 , f2 ) ∈ C([0, T ] × R × R × R × R) is quasimonotone nondecreasing

(ii) fi satisfies one-sided Lipschitz condition fi (t, w1 , w2 , D q w1 , D q w2 ) − fi (t, v1 , v2 , D q v1 , D q v2 ) ≥ −Ni (D q wi − D q vi ) − Mi (wi − vi ),

where Ni , Mi ∈ R, Ni2 > 4Mi . (iii) There exist constants Ni , Mi ∈ R, Ni2 > 4Mi such that (ii) holds and for t ∈ (0, T ], vi ≤ yi ≤ yi∗ ≤ wi , D(t) ≤ zi ≤ D ∗ (t), D(t) ≤ zi∗ ≤ D ∗ (t) such that fi (t, y1 , y2 , z1 , z2 ) − fi (t, y1∗ , y2∗ , z1∗ , z2∗ ) ≤ Ni (zi − zi∗ ) + Mi (yi − yi∗ ), q where D(t) = D0+ vi (t) + λi2 (wi (t) − vi (t)),

q D ∗ (t) = D0+ wi (t) − λi2 (wi (t) − vi (t)), q q −Ni + Ni2 − 4Mi −Ni − Ni2 − 4Mi λi1 = ≥ 0 > λi2 = . 2 2

(3.1)

359

SYSTEM OF INITIAL VALUE PROBLEMS q Then there exist sequences {vin (t)}, {win (t)} ⊂ C1−q ([0, T ]) with

vi0 (t) = vi (t), wi0 (t) = wi (t) such that for t ∈ (0, T ], lim v n (t) n→∞ i

= vi (t),

lim win (t) = wi (t)

n→∞

and vi (t), wi (t) are minimal and maximal solutions on the ordered interval [v 0 , w0 ] for initial value problem (1.1) respectively and for any solution ui (t) of initial value problem (1.1) such that ui (t) ∈ Ω, we have vi0 ≤ vi1 ≤ vi2 ≤ ... ≤ vin ≤ vi ≤ ui ≤ wi ≤ win ≤ ... ≤ wi2 ≤ wi1 ≤ wi0 . Proof. Let σ(ηi )(t) = fi (t, η1 , η2 , D q η1 , D q η2 ) + Ni D q ηi (t) + Mi ηi (t),

t ∈ (0, T ].

For any η(t) = (η1 (t), η2 (t)) ∈ Ω, consider the linear initial value problem (D 2q ui )(t) + Ni D q ui (t) + Mi ui (t) = σ(ηi )(t), t

1−q

ui (t)|t=0 =

u0i ,

t

1−q

q

(D ui )(t)|t=0 =

u1i .

t ∈ (0, T ],

(3.2)

By Lemma 2.1 and relation (2.1), linear initial value problem (3.2) has exactly q one solution ui (t) ∈ C1−q ([0, T ]) and is given by ui (t) = (Aηi )(t) =

Γ(q)u0i eq (λi2 , t)

+

Γ(q)(u1i

−



λ2 u0i )

eq (λi2 , x) 

∗

eq (λi2 , x)



eq (λi1 , x)

(t)+

∗ eq (λi1 , x) ∗

 σηi (x) (t),

where λi1 =

−Ni +

q

Ni2 − 4Mi

2

,

λi2 =

−Ni −

q

Ni2 − 4Mi

2

≤ 0.

(D q Aηi )(t) = Γ(q)u0i λi2 eq (λi2 , t)+ Γ(q)(u1i

−

λ2 u0i ) i λ1

  1 i i i i λ eq (λ1 , x) − λ2 eq (λ1 , x) (t)+ − λi2 1

  1 i i i i λ eq (λ1 , x) ∗ σ(ηi )(x) − λ2 eq (λ2 , x) ∗ (σηi )(x) (t). λi1 − λi2 1

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J.A. NANWARE AND D.B. DHAIGUDE

q Then A is an operator from Ω into C1−q ([0, T ]) and ηi (t) is a solution of initial value problem (1.1) if and only if ηi = Aηi . Since λi1 ≥ 0 ≥ λi2 in (3.1), we have   1 i i i i λ eq (λ1 , x) − λ2 eq (λ2 , x) (t) ≥ 0, t ∈ (0, T ]. λi1 − λi2 1

Using Lemma 2.3 and Lemma 2.4, definitions of lower and upper solutions, definition of ui (t) and (D q Aηi )(t), we get vi (t) = (Aηi )(t)   = Γ(q)u0i eq (λi2 , t) + Γ(q)(u1i − λi2 u0i ) eq (λi2 , x) ∗ eq (λi1 , x) (t)+   i i eq (λ2 , x) ∗ eq (λ1 , x) ∗ (σηi )(x) (t) vi (t) ≤ Avi (t) =

Γ(q)u0i eq (λi2 , t)

+

Γ(q)(u1i

−



eq (λi2 , x)

∗ eq (λi1 , x)



eq (λi2 , x)

eq (λi1 , x)

λi2 u0i )

≤ (Aηi )(t) ≤ (Awi )(t) ≤ wi (t),



∗

(t)+

 ∗ σvi (x) (t)

∀ ηi (t) ∈ Ω.

Thus, we have vi ≤ Avi ≤ Aηi ≤ Awi ≤ wi ,

∀

ηi ∈ Ω

(3.3)

and if vi ≤ θi ≤ φi ≤ wi then (σθi ) ≤ (σφi ), Aθi ≤ Aφi , and D q Aθi ≤ D q Aφi .

By Lemma 2.5, we know that, for i = 1, 2 zi1 (t) = D q (Aηi − vi )(t) − λi2 (Aηi − vi )(t) ≥ 0, t ∈ (0, T ], Hence

∀

D q (Aηi )(t) = D q vi (t) + λi2 (Aηi − vi )(t) ≥ D q vi (t) + λi2 (wi − vi )(t)

≥ D(t), t ∈ (0, T ],

∀ ηi ∈ Ω.

ηi ∈ Ω.

(3.4)

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SYSTEM OF INITIAL VALUE PROBLEMS

Similarly, we can show D q (Aηi )(t) ≤ D ∗ (t), t ∈ (0, T ],

∀

Therefore A(Ω) ⊂ Ω.

ηi (t) ∈ Ω.

Now, let vi0 = vi , wi0 = wi , vin = Avin−1 , win = Awin−1 , n = 1, 2, 3, .... From (3.3) and (3.4), we have vi0 ≤ vi1 ≤ vi2 ≤ ... ≤ vin ≤ ... ≤ win ≤ ... ≤ wi2 ≤ wi1 ≤ wi0 , D(t) ≤ D q vi1 ≤ ... ≤ D q vin ≤ ... ≤ D q win ≤ ... ≤ D q wi1 ≤ D ∗ (t). It is clear that the upper sequence {win (t)} is monotone nondecreasing and bounded from below and that lower sequence {vin (t)} is monotone nondecreasing and bounded above. Moreover D q vin (t), D q win (t) ∈ [D(t), D ∗ (t)].   Let Bi =

vin : n = 1, 2, 3, ... . Now we show that the set B is relatively

q compact in C1−q ([0, T ]). For any ηi (t) ∈ Ω, by definition of lower and upper solutions and Lipschitz condition, we have

(D 2q vi )(t) + Ni D q vi (t) + Mi vi (t) ≤ fi (t, v1 , v2 , D q v1 , D q v2 )+

Ni D q vi (t) + Mi vi (t)

≤ fi (t, η1 , η2 , D q η1 , D q η2 ) + Ni D q ηi (t) + Mi ηi (t)

≤ fi (t, w1 , w2 , D q w1 , D q w2 ) + Ni D q wi (t) + Mi wi (t)

q Since B, Ω ⊂ C1−q ([0, T ]) are bounded sets, therefore

σηi (t) = fi (t, η1 , η2 , D q η1 , D q η2 ) + Ni D q ηi (t) + Mi ηi (t)|ηi ∈ Ω is bounded also. Hence there exists a constant L > 0 such that |σ(vin )k = max|t1−q σ(vin (t)| ≤ L, |σ(vin (t))|

This implies

≤ Lt

1−q

,∀

∀,

n = 1, 2, ...

t ∈ (0, T ].

On the other hand, {vin (t)|n ∈ N } satisfies vin (t)

=

Γ(q)u0i eq (λi2 , t) + 

Γ(q)(u1i

−



λi2 u1i )

eq (λi2 , x)

∗



eq (λi1 , x)

 n−1 i i eq (λ2 , x) ∗ eq (λ1 , x) ∗ (σ(vi )(x) (t)

(t)+ (3.5)

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J.A. NANWARE AND D.B. DHAIGUDE

(D q vin )(t) = Γ(q)u0i λ2 eq (λi2 , t)+   1 i i i i λ eq (λ1 , x) − λ2 eq (λ2 , x) (t)+ − − λi2 1   1 i i n−1 i i n−1 λ e (λ , x) ∗ σ(v )(x) − λ e (λ , x) ∗ σ(v )(x) (t) q 1 2 q 2 i i λi1 − λi2 1 (3.6)   Γ(q)(u1i

λi2 u0i ) i λ1

Let G(λij , x) = x1−q eq (λij , x) ∗ σ(vin−1 )(x) ,

x ∈ [0, T ],

j = 1, 2.

Without loss of generality, assume that 0 ≤ x1 < x2 ≤ T , from λi2 < 0 ≤ λi1 , we have G(λi2 , x1 ) − G(λi2 , x2 ) ≤ Li Γ(q) Eq,q (λi2 , xq ) − Eq,q (λi2 , xq ) + 1 2 |λi1 | 2Li Γ(q) (x2 − x1 )q Γ(2q)   Li Γ(q) Li T q q q i i i i + E (λ , x ) − E (λ , x ) and G(λ1 , x1 ) − G(λ1 , x2 ) ≤ q,q q,q 1 1 1 2 + q |λi1 | 2Li Γ(q) Eq,q (λi1 , T q (x2 − x1 )q ) Γ(2q) (3.7) From Eq,q (x) ∈ C([0, T ]), ∀ ǫ > 0, there exists δ = δ(ǫ), when |x1 − x2 | < δ (without loss of generality 0 ≤ x1 < x2 ≤ T ), we have Eq,q (λi1 , xq ) − Eq,q (λi1 , xq ) < ǫ (3.8) 1 2 8L1i Eq,q (λi2 , xq ) − Eq,q (λi2 , xq ) < ǫ (3.9) 1 2 8L2 i ǫ (x2 − x1 )q < (3.10) 8L3i where   |Γ(q)(u1i − λi2 u0i )λi1 | 1 |λi1 |T q = max , i Γ(q) + q |λi1 − λi2 | |λ1 − λi2 | 1 i 0 i |Γ(q)(ui − λ2 ui )λ1 | Li Γ(q) , L2i = max Γ(q)(ui0 λi2 ), |λ1 − λ2 | |λ1 − λ2 |   2Li Γ(q) i i i q 3 |λ2 | + |λ1 |Γ(q)Eq,q (|λ1 |T ) Li = Γ(2q)|λi1 − λi2 | L1i



Using (3.7) to (3.8) in (3.6), we obtain 1−q q n 1−q q q q n 0 i i i x (D vi )(x1 ) − x (D vi )(x2 ) ≤ Γ(q)ui λ2 . Eq,q (λ2 x ) − Eq,q (λ2 x ) + 2 1 2 1

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SYSTEM OF INITIAL VALUE PROBLEMS

 |Γ(q)(u1i − λi2 u0i ) |λi1 ||Eq,q (λi1 xq1 ) − Eq,q (λi1 xq2 )|+ |λi1 − λi2 |  i i q i q , |λ2 ||Eq,q (λ2 x1 ) − Eq,q (λ2 x2 )| +  Li |λ|T q )|Eq,q (λi1 xq1 ) − Eq,q (λi1 xq2 )|+ (Γ(q) + q |λi1 − λi2 |  i q i q Γ(q)|Eq,q (λ2 t1 ) − Eq,q (λ2 t2 )| +   2Li Γ(q) i i i q |λ2 | + |λ1 |Γ(q)Eq,q (|λ1 |T ) (x2 − x1 )q Γ(2q)|λi1 − λi2 | ≤ ǫ.

q Thus B is equicontinuous in C1−q ([0, T ]), by Ascoli-Arzela theorem, we have q that B is relatively compact set of C1−q ([0, T ]),. Similarly we can prove q n that {wi (t)} is relatively compact set of C1−q ([0, T ]). Hence, the sequences n n {vi (t)}, {wi (t)} converges uniformly to vi (t), wi (t) respectively on [0, T ]

lim v n (t) n→∞ i lim D 2q vin (t) n→∞

= vi (t), = D 2q vi (t),

lim win (t) = wi (t),

n→∞

t ∈ [0, T ]

lim D 2q win (t) = D 2q wi (t),

n→∞

t ∈ [0, T ].

Thus by relations (vi0 ≤ vi1 ≤ vi2 ≤ ...), it follows that vi (t) and wi (t) satisfy vi0 ≤ vi1 ≤ vi2 ≤ ... ≤ vin ≤ vi ≤ wi ≤ win ≤ ... ≤ wi2 ≤ wi1 ≤ wi0

D(t) ≤ D q vi1 ≤ D q vi2 ≤ ... ≤ D q vin ≤ D q vi ≤ D q wi ≤ D q win ≤ ... ≤ D q wi2 ≤ D q wi1 ≤ D ∗ (t)

Lastly, we prove that vi (t), wi (t) are extremal solutions of initial value problem (1.1). Since fi , (i = 1, 2) is continuous, the function σ(ηi )(t) is continuous and is monotone nondecreasing in vi (t), the sequence {vin (t)} converges to vi (t) implies that {σ(vin )(t)} converges to σ(vi )(t), t ∈ (0, T ]. Taking limit as n → ∞ of {vin (t)} and using dominated convergence theorem, vi (t) satisfies the integral equation   0 i 1 i 0 i i vi (t) = (Avi )(t) = Γ(q)ui eq (λ2 , t) + Γ(q)(ui − λ2 ui ) eq (λ2 , x) ∗ eq (λ1 , x) (t)+   i i eq (λ2 , x) ∗ eq (λ1 , x) ∗ (σvi )(x) (t).

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J.A. NANWARE AND D.B. DHAIGUDE

Thus vi (t) is an integral representation of the solution of initial value problem. Since fi , (i = 1, 2) is continuous and by Lemma 2.2, it follows that vi (t) is a classical solution of initial value problem (1.1). This proves that the lower sequence {vin (t)} converges to a solution vi (t) of initial value problem (1.1). Similarly we can show that the upper sequence {win (t)} converges to a solution wi (t) of initial value problem (1.1) and satisfies vi (t) ≤ wi (t), D q vi (t) ≤ D q wi (t), t ∈ (0, T ]. Thus by standard arguments it follows that vi0 ≤ vi1 ≤ vi2 ≤ ... ≤ vin ≤ vi ≤ wi ≤ win ≤ ... ≤ wi2 ≤ wi1 ≤ wi0 and hence vi (t) and wi (t) are minimal and maximal solutions of initial value problem (1.1) on the sector [v 0 , w0 ] respectively. Finally, if there exist Ni , Mi such that (3.1) holds and for t ∈ (0, T ], vi ≤ yi ≤ yi∗ ≤ wi , D(t) ≤ zi ≤ D ∗ (t), D(t) ≤ zi∗ ≤ D ∗ (t) such that fi (t, y1 , y2 , z1 , z2 ) − fi (t, y1∗ , y2∗ , z1∗ , z2∗ ) ≤ Ni (zi − zi∗ ) + Mi (yi − yi∗ ) then vi (t) = wi (t) is a unique solution of initial value problem (1.1). It is sufficient to prove vi (t) ≥ wi (t), D q vi (t) ≥ D q wi (t), t ∈ (0, T ]. For this, consider ui (t) = vi (t) − wi (t). Then from IVP (1.1) and above hypothesis, we have (D 2q ui )(t) + Ni D q ui (t) + Mi ui (t) = D 2q (vi − wi )(t) + Ni D q (vi − wi )(t) + Mi (vi − wi )(t)

= D 2q vi (t) − D 2q wi (t) + Ni D q vi − Ni D q wi

+ Mi vi (t) − Mi wi (t) = (φui )(t) ≥ 0,

t ∈ (0, T ];

t1−q ui (t)|t=0 = 0,

t1−q (D q ui )(t)|t=0 = 0.    1 q i i i i (D ui )(t) = i λ1 eq (λ1 , x) − λ2 eq (λ2 , x) ∗ (φui )(x) (t) λ1 − λi2

By Lemma 2.4, it follows that ui (t) ≥ 0, D q vi (t) ≥ D q wi (t), t ∈ (0, T ].

t ∈ (0, T ]. Thus vi (t) ≥ wi (t),

Hence vi (t) = ui (t) = wi (t) is a solution of the IVP (1.1).

Next we prove the uniqueness of solution of the IVP (1.1).

SYSTEM OF INITIAL VALUE PROBLEMS

365

Theorem 3.2. Assume that q (i) vi0 , wi0 ∈ C1−q ([0, T ]) are ordered lower and upper solutions of IVP (1.1), f = (f1 , f2 ) ∈ C([0, T ] × R × R × R × R) is quasimonotone nondecreasing

(ii) fi satisfies Lipschitz condition ∆ ≤ Mi (wi − vi ) + Ni (D q wi − D q vi ),

(3.11)

where ∆ = fi (t, v1 , v2 , D q v1 , D q v2 ) − fi (t, w1 , w2 , D q w1 , D q w2 ); vi , wi ∈ [v 0 , w0 ], D q vi , D q wi ∈ [D(t), D ∗ (t)], and Mi > 0, Ni > 0, Ni2 > 4Mi are Lipschitz constants such that D = D q vi (t) + λi2 (wi (t) − vi (t)), D ∗ = D q wi (t) − λi2 (wi (t) − vi (t)), q q −Ni − Ni2 − 4Mi −Ni + Ni2 − 4Mi ≥ 0 > λ= λi1 = 2 2 2 Then the IVP (1.1) has unique solution in the sector [v 0 , w0 ]. Proof. We know vi0 ≤ wi0 . It is enough to show that vi0 ≥ wi0 . From (3.11), we have −Mi (wi − vi ) − Ni (D q wi − D q vi ) ≤ ∆ ≤ Mi (wi − vi ) + Ni (D q wi − D q vi ) where ∆ = fi (t, w1 , w2 , D q w1 , D q w2 ) − fi (t, v1 , v2 , D q v1 , D q v2 ). This implies vi0 ≥ wi0 . Thus the IVP (1.1) has unique solution in the sector [v 0 , w0 ].

4. ACKNOWLEDGMENTS Authors express sincere thanks to Referee for reviewing the research paper.

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