integro-differential equation with not instantaneous impulse

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International Journal of Advances in Applied Mathematics and Mechanics

Existence result of fractional functional

integro-differential equation with not instantaneous impulse

Research Article

Ganga Ram Gautam¹∗, Jaydev Dabas¹

1Department of Applied Science and Engineering, IIT Roorkee, Saharanpur Campus, Saharanpur-247001, India.

Received 03 October 2013; accepted (in revised version) 26 February 2014

Abstract:

In this investigation, we have established the existence and uniqueness results of solutions for class of an abstract fractional functional integro-differential equations with state dependent delay subject to not instantaneous impulse by using fixed point theorems. One example is presented to illustrate the main results of the paper.

MSC:

26A33• 34K05 • 34A12 • 26A33

Keywords:

Fractional order differential equations• Functional differential equations • Impulsive conditions • Fixed point the- orems

2014 IJAAMM all rights reserved.c

1. Introduction

Due to the memory and hereditary property, fractional derivatives are more applicable than the ordinary derivative and with the different conditions such as initial, impulsive and nonlocal so these derivatives became more realistic. The frac- tional differential equations with impulsive effects have been appeared as in natural description evolution processes for many real problems. The impulsive effects can be shown in many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control model in economics, pharmacokinetics and frequency modu- lated system etc. See[1,2,4,5,13,18] for current updates of the theory.

Fractional order functional differential equations originate in several field of applied mathematics and science. Re- cently, fractional functional differential equations with state dependent delay seems frequently in modeling of equa- tions, panorama of natural phenomena and porous media for details developments of the theory one can see the papers [6–12,14,19].

∗ Corresponding author. E-mail:[email protected]

Byszewski[16] have established the existence, uniqueness of mild and classical solutions for a nonlocal Cauchy problem.

As remarked by the author the nonlocal condition can be more realistic than the standard initial conditions to describe some physical phenomena. See[16,17] for more detail and reference therein.

Feckan et al. [2] have presented a counterexample to show an essence error in the the formula of the solutions to the impulsive Cauchy problems used by several authors in literature[5,13]. Further, Wang et al. [18] established sufficient conditions for existence of the solutions for the following fractional differential equations

cD_t^qu(t ) = f (t , u(t )), t ∈ J⁰, q∈ (1, 2),

∆u(tk) = yk,∆u⁰(tk) = ¯yk k= 1,2,··· ,m, u(0) = u0, ¯u(0) = ¯u0,

and by applying fixed point theorems. In their subsequent study authors[18] consider the following problem with non- linear impulsive condition of the form:

cD_t^qu(t ) = f (t , u(t )), t ∈ J⁰, q∈ (1, 2),

∆u(tk) = Ik(u(t_k⁻)),∆u⁰(tk) = Jk(u(t_k⁻)) k = 1,2,··· ,m, u(0) = u0, ¯u(0) = ¯u0.

Nonlinear impulsive conditions arises where abrupt change occur in any state at certain time of moments for example in phenomena involving thresholds, bursting rhythm models in medicine and biology etc. Recently, not instantaneous impulsive condition first time used by author’s Hernandez and O’Regan[4] for the following problem of the form:

u⁰(t ) = Au(t ) + f (t , u(t )), t ∈ (si, t_i+1], i = 0,1,··· ,N , u(t ) = gi(t , u(t )), t ∈ (ti, si], i = 1,2,··· ,N , u(0) = x0,

The dynamical system where impulses start abruptly at the certain time and their action continues on the finite time interval, modeled as not instantaneous differential equations.

Our present work motivated by the papers[2,4,18], we study not instantaneous impulsive fractional functional integro- differential equation of the form:

D_t^αy(t ) = J_t^2−αf(t , y_{ρ(t ,yt)}, B(y_{ρ(t ,yt)}), t ∈ (si, t_i+1], i = 0,1,··· ,N , (1) y(t ) = gi(t , y (t )), y⁰(t ) = qi(t , y (t )), t ∈ (ti, s_i], i = 1,2,··· ,N , (2) y(t ) + u(y ) = φ(t ), y⁰(t ) + v (y ) = ϕ(t ), t ∈ (−∞,0], (3)

where D_t^αis Caputo’s derivative of orderα ∈ (1,2] and J^2−αis Riemann-Liouville fractional integral. y⁰denotes the deriva- tive of y with respect to t and operational interval J= [0,T ],0 < T < ∞. f : J ×Bh×Bh→ X , u, v : X → X are given func- tions.Bhis a abstract phase space and ytthe element ofBhdefined by yt(θ ) = y (t +θ ), θ ∈ (−∞,0]. The term B(y_{ρ(t ,yt)}) is

given by B(yρ(t ,yt)) =Rt

0K(t , s )(yρ(s,ys))d s, where K ∈ C (D ,R⁺), is the set of all positive functions which are continuous on D= {(t , s ) ∈ R²: 0≤ s ≤ t < T } and B^∗= sup_{t∈[0,t ]}Rt

0 K(t , s )d s < ∞. Here 0 = t0= s0< t1≤ s1≤ t2< ··· < tN≤ sN≤ tN+1= T, are pre-fixed numbers, g_i, q_i∈ C ((ti, s_i] × X ; X ) for all i = 1,2,··· ,N .

To the best of our knowledge, our results are new for the fractional functional integro-differential equation with state dependent delay subject to not instantaneous impulsive condition. This work is divided in to four sections. In section two we include the basic definitions and assumptions, next in third section contains the main results and in last section an example is presented.

2. Preliminary

Let(X ,k · kX) be a complex Banach space of functions with the norm ky kX= supt∈J{|y (t )| : y ∈ X }. For infinite delay we use abstract phase spaceBhas defined in[15] details are as follow:

Assume that h :(−∞,0] → (0,∞) is a continuous functions with l =R0

−∞h(s )d s < ∞, t ∈ (−∞,0]. For any a > 0, we define

B= {ψ : [−a,0] → X such that ψ(t ) is bounded and measurable},

and equipped the spaceBwith the normkψk[−a,0]= sup_{s∈[−a ,0]}kψ(s )kX,∀ ψ ∈B. Let us define

B_h= {ψ : (−∞,0] → X , s.t. for any c > 0,ψ |_{[−c ,0]}∈B&

Z0

−∞

h(s )kψk_[s,0]d s< ∞}.

IfB_h is endowed with the normkψkB_h =R0

−∞h(s )kψk_[s,0]d s ,∀ ψ ∈B_h, then it is clear that(B_h,k.kB_h) is a complete Banach space.

To treat the impulsive conditions, we consider the following setting

B⁰_h:= P C ((−∞,T ]; X ), T < ∞,

be a Banach space of all such functions y :(−∞,T ] → X , which are continuous every where except for a finite number of points ti∈ (0, T ), i = 1, 2, . . . , N , at which y (ti⁺) and y (t_i⁻) exists and endowed with the norm

ky kB⁰_h= sup{ky (s )kX: s∈ J } + kφkBh, y∈B⁰_h,

wherek · kB⁰_h to be a semi-norm inB⁰_h.

For a function y∈B⁰_hand i∈ {0, 1, ..., N }, we introduce the function ¯yi∈ C ([ti, ti+1]; X ) given by

¯ y_i(t ) =







y(t ), for t ∈ (ti, ti+1], y(t_i⁺), for t = ti,

and setting

B⁰⁰_h:= P C¹((−∞,T ]; X ), T < ∞,

be a Banach space of all such functions y :(−∞,T ] → X , which are continuously differentiable every where except for a finite number of points t_i∈ (0, T ), i = 1, 2, . . . , N , at which y⁰(t_i⁺) and y⁰(t_i⁻) exists and endowed with the semi-norm

ky k_B⁰⁰

h= sup

t∈[0,T ]

{ky (s )kX,ky⁰(s )kX} + kφkB_h, y∈B⁰⁰_h.

For a function y∈B⁰⁰_hand i∈ {0, 1, ..., N }, we introduce the function ¯yi∈ C¹([ti, t_i+1]; X ) given by

¯ y_i(t ) =







y⁰(t ), for t ∈ (ti, ti+1], y⁰(t_i⁺), for t = ti.

If function y :(−∞,T ] → X such that y ∈B⁰⁰_hthen for all t ∈ [0, T ], the following conditions hold:

(C1) yt∈Bh.

(C2) ky (t )kX≤ H kytkBh.

(C3) kytkB_h ≤ K (t ) sup{ky (s )kX : 0≤ s ≤ t } + M (t )kφkB_h, where H > 0 is constant; K ,M : [0,∞) → [0,∞), K (·) is continuous, M(·) is locally bounded and K ,M are independent of y (t ).

(C4_φ) The function t→ φtis well defined and continuous from the set

ℜ(ρ⁻) = {ρ(s,ψ) : (s,ψ) ∈ [0,T ] ×Bh},

intoB_hand there exists a continuous and bounded function J^φ:ℜ(ρ⁻) → (0,∞) such that kφtkB_h≤ J^φ(t )kφkB_h

for every t∈ ℜ(ρ⁻).

Lemma 2.1 ([5], lemma 3.6).

Let y :(−∞,T ] → X be a function such that y ∈B⁰⁰_hwith y0= φ, y |J_k∈ C¹(Jk, X) and if (C4_φ) hold, then

kyskB_h≤ (Mb+ J^φ)kφkB_h+ Kbsup{ky (θ )kX;θ ∈ [0,max{0, s }]}, s ∈ ℜ(ρ⁻) ∪ J ,

where

J^φ= sup

t∈ℜ(ρ⁻)J^φ(t ), Mb= sup

s∈[0,T ]

M(s ) and Kb= sup

s∈[0,T ]

K(s ).

Definition 2.1.

Caputo’s derivative of orderα > 0 for a function f : [a,∞) → R is defined as

aD_t^αf(t ) = 1 Γ (n − α)

Zt a

(t − s )^n−α−1f⁽ⁿ⁾(s )d s =aJ_t^n−αf⁽ⁿ⁾(t ),

where a≥ 0, n ∈ N . It is clear that derivative of constant function is zero.

Definition 2.2.

The Riemann-Liouville fractional integral operator of orderα > 0, for a function f ∈ L¹(R⁺, X) is defined by

aJ_t⁰f(t ) = f (t ),aJ_t^αf(t ) = 1 Γ (α)

Zt a

(t − s )^α−1f(s )d s, α > 0, t > 0,

where a≥ 0, n ∈ N and Γ (·) is the Euler gamma function.

Lemma 2.2 ([5], lemma 3.3 ).

Forα > 0, solution of fractional differential equations with lower limit not zeroaJ_t^αD_t^αy(t ) = y (t ) + c0+ c1(t − a) + c2(t − a)²+ c3(t −a)³+···+ c_n−1(t −a)ⁿ⁻¹where c_i∈ R, i = 0, 1, · · · , n − 1, n = [α] + 1 and [α] represent the integral part of the real numberα.

Our following result is based definition 2.1 in[4].

Lemma 2.3.

A function y :(−∞,T ] → X s.t. y ∈B⁰⁰_h is called a solution of the problem (1)-(3) if y(0) = φ(0), y⁰(0) = ϕ(0), y (t ) = gj(t , y (t )), y⁰(t ) = qj(t , y (t )) for t ∈ (tj, sj], j = 1,2,··· ,N , and satisfying the following integral equation

y(t ) =¨ φ(0) − u(y ) + (ϕ(0) − v (y ))t + R₀^t(t − s )f (s, y_ρ(s,ys), B y_ρ(s,ys))d s, t ∈ [0, t1], gi(si, y(si)) + qi(si, y(si))t +Rt

s_i(t − s )f (s, y_ρ(s,ys), B y_ρ(s,ys))d s, t∈ [si, ti+1], for every i= 1,2,··· ,N .

3. Main results

To prove our results we shall assume the functionρ : [0,T ] ×Bh→ (−∞, T ] is continuous and φ, ϕ ∈Bh. If y ∈Bhwe defined ¯y :(−∞,T ) → X as the extension of y to (−∞,T ] such that ¯y(t ) = φ. We definedey :(−∞,T ) → X such that

ye= y + x where x : (−∞,T ) → X is the extension of φ ∈Bhsuch that x(t ) = φ(0) for t ∈ [0,T ]. In additional if y⁰∈Bhwe defined ¯y⁰:(−∞,T ) → X as the extension of y⁰to(−∞,T ] such that ¯y⁰(t ) = ϕ. We definedye⁰:(−∞,T ) → X such that ye⁰= y⁰+ x⁰where x⁰:(−∞,T ) → X is the extension of ϕ ∈Bhsuch that x⁰(t ) = ϕ(0) for t ∈ [0,T ]. Now we introduce the following assumption.

(H1) f : J ×Bh×Bh→ X is jointly continuous function and there exist positive constants Lf 1, Lf 1such that

kf (t , ψ, ϕ) − f (t , ξ, χ)kX≤ Lf 1kψ − ξkB_h+ Lf 2kϕ − χkB_h,∀ ψ, ϕ, χ, ξ ∈B_h.

(H2) f is continuous and there exist positive constants M1, M2such that

kf (t , ψ, ϕ)kX ≤ M1kψkBh+ M2kϕkBh,∀ ψ, ϕ ∈Bh.

(H3) The functions u, v are continuous and there are positive constants Lu, L_vsuch that

ku(x ) − u(y )kX≤ Lukx − y kX;kv (x ) − v (y )kX≤ Lvkx − y kX,∀ x , y ∈ X .

(H4) The functions u, v are continuous and there are positive constants M3, M4such that

ku(y )kX≤ M3ky kX;kv (y )kX≤ M4ky kX,∀ x , y ∈ X .

(H5) The functions gi, q_iare continuous and there are positive constants L_gi, L_qi such that

kgi(t , x ) − gi(t , y )kX≤ Lgikx − y kX;kqi(t , x ) − qi(t , y )kX≤ Lqikx − y kX,

for all x , y∈ X , t ∈ (ti, si] and each i = 1,2,··· ,N .

(H6) The functions gi, qiare continuous and there are positive constants M5, M6such that

kgi(t , y )kX≤ M5ky kX;kqi(t , y )kX≤ M6ky kX,

for all x , y∈ X , t ∈ (ti, s_i] and each i = 1,2,··· ,N .

Theorem 3.1.

Assume the condition(H1),(H3) and (H5) are satisfied and constant

∆ = max{(Lu+ T Lv+ Kb

T²

2 (Lf1+ Lf2B^∗)), Lgi+ T Lqi+ Kb

T²

2(Lf 1+ Lf 2B^∗)} < 1,

for i= 1,··· ,N . Then there exists a unique solution y (t ) of the problem (1)-(3) on J .

Proof. Let ¯φ, ¯ϕ : (−∞,T ) → X be the extension of φ,ϕ to (−∞,T ] such that ¯φ(t ) = φ(0), ¯ϕ(0) = ϕ(0) on J . Consider the spaceB⁰⁰⁰_h = {y ∈B⁰⁰_h: y(0) = φ(0), y⁰(0) = ϕ(0)} and y (t ) = φ(t ), y⁰(t ) = ϕ(t ) for t ∈ (−∞,0] endowed with the uniform convergence topology. Let us consider an operator P :B⁰⁰⁰_{h →}B⁰⁰⁰_h defined as P y(t ) = gi(t , ¯y (t )) for t ∈ (ti, s_i] and

P y(t ) =







φ(0) − u( ¯y ) + (ϕ(0) − v ( ¯y ))t +Rt

0(t − s )f (s, ¯y_{ρ(s, ¯ys)}, B ¯y_{ρ(s, ¯ys)})d s, t ∈ [0, t1], g_i(si, ¯y(si)) + qi(si, ¯y(si))t +Rt

s_i(t − s )f (s, ¯yρ(s, ¯ys), B ¯y_{ρ(s, ¯ys)})d s, t∈ [si, t_i+1],

(4)

where ¯y :(−∞,T ] → X is such that ¯y(0) = φ, ¯y⁰(0) = ϕ and ¯y = y on J . It is obvious that P is well defined. Now, we show that the operator P has a fixed point. Let y(t ), y^∗(t ) ∈B⁰⁰⁰_h and t ∈ [0, t1], we have

kP y − P y^∗kX ≤ ku( ¯y ) − u( ¯y^∗)kX+ T kv ( ¯y ) − v ( ¯y^∗)kX

+ Zt

0

(t − s )kf (s, ¯y_{ρ(s, ¯ys)}, B ¯y_{ρ(s, ¯ys)}) − f (s, ¯y_{ρ(s, ¯y}^∗ s∗), B ¯y_{ρ(s, ¯y}^∗ s∗))kXd s

≤ (Lu+ T Lv+ Kb

T²

2 (Lf 1+ Lf 2B^∗))ky − y^∗kB⁰⁰⁰_h.

For t∈ [si, t_i+1], we have

kP y − P y^∗kX ≤ kgi(si, ¯y(si)) − gi(si, ¯y^∗(si))kX+ kqi(si, ¯y(si)) − qi(si, ¯y^∗(si))kXT +

Zt s_i

(t − s )kf (s, ¯yρ(s, ¯ys), B ¯y_{ρ(s, ¯ys)}) − f (s, ¯y_{ρ(s, ¯y}^∗ ∗ s), B ¯y_{ρ(s, ¯y}^∗ ∗

s))kXd s

≤ (Lgi+ T Lqi+ Kb

T²

2 (Lf 1+ Lf 2B^∗))ky − y^∗kB⁰⁰⁰_h.

For t∈ (tj, s_j], we get

kP y − P y^∗kX ≤ Lg_jky − y^∗kB⁰⁰⁰_h, j= 1,2,··· ,N .

Gathering above results, we obtain

kP y − P y^∗kX ≤ ∆ky − y^∗kB⁰⁰⁰_h.

Since∆ < 1, which implies that P is a contraction map and there exists a unique fixed point which is the solution of system (1)-(3) on J . This completes the proof of the theorem.

Theorem 3.2.

Let the assumptions(H2),(H4) and (H6) are satisfied. Then the problem (1)-(3) has at least one solution y(t ) on J .

Proof. Consider the operator P :B⁰⁰⁰_h →B⁰⁰⁰_h, defined by (4) in theorem3.1. We shall show P has a fixed point inB⁰⁰⁰_h. First, we shall show that P is continuous, so we consider a sequence yⁿ→ y inB⁰⁰⁰_h, then for[0, t1]

kP (yⁿ) − P (y )kX ≤ ku( ¯yⁿ) − u( ¯y )kX+ T kv ( ¯yⁿ) − v ( ¯y )kX

+ Zt

0

(t − s )kf (s, ¯yⁿ_{ρ(s, ¯y}ns), B ¯yⁿ_{ρ(s, ¯y}ns)) − f (s, ¯y_{ρ(s, ¯ys)}, B ¯y_{ρ(s, ¯ys)})kXd s .

For t∈ [si, t_i+1], we have

kP (yⁿ) − P (y )kX ≤ kgi(si, ¯yⁿ(si)) − gi(si, ¯y(si))kX

+T kqi(si, ¯yⁿ(si)) − qi(si, ¯y(si))kX

+ Zt

s_i

(t − s )kf (s, ¯yⁿ_{ρ(s, ¯y}ns), B ¯yⁿ_{ρ(s, ¯y}ns)) − f (s, ¯y_{ρ(s, ¯ys)}, B ¯y_{ρ(s, ¯ys)})kXd s .

Since f , u , v, g_iand q_iare continuous functions, then we have

kP (yⁿ) − P (y )kX→ 0 as n → ∞,

which show that P is continuous.

LetBr= {y ∈B⁰⁰⁰_h :ky kX≤ r } be a closed bounded and convex subset ofB⁰⁰⁰_h. Now, it is easy to prove that P maps bounded set into bounded set inBr. To do this we have for[0, t1]

kP (y )(t )kX ≤ kφ(0)k + ku( ¯y )k + T (kϕ(0)k + kv ( ¯y )k) + Zt

0

(t − s )f (s, ¯yρ(s, ¯ys), B ¯y_{ρ(s, ¯ys)})d s

≤ kφ(0)k + Lur+ T (kϕ(0)k + Lvr) +T²

2(M1+ M2B^∗)r^∗.

For t∈ [si, t_i+1], we have

kP (y )(t )kX ≤ kgi(si, ¯y(si))kX+ T kqi(si, ¯y(si))kX+ Zt

s_i

(t − s )kf (s, ¯y_{ρ(s, ¯ys)}, B ¯y_{ρ(s, ¯ys)})kXd s

≤ M5r+ T M6r+T²

2 (M1+ M2B^∗)r^∗,

where r^∗= (Mb+ J^φ)kφkBh+ Kbr. Which implies that P maps bounded set into bounded set in Br.

Next, we shall show that P maps bounded sets into equi-continuous sets inBr. Let l1, l2∈ [0, t1] with l1< l2, we have

k(P y )(l2) − (P y )(l1)kX ≤ (l2− l1)(kϕ(0)k + kv ( ¯y )k) +

Zl1 0

(l2− l1)kf (s, ¯y_{ρ(s, ¯ys)}, B ¯y_{ρ(s, ¯ys)})kXd s

+ Zl₂

l1

(l2− s )kf (s , ¯yρ(s, ¯ys), B ¯y_{ρ(s, ¯ys)})kXd s

≤ (l2− l1)(kϕ(0)k + M4r) + (l2− l1)T (M1+ M2B^∗)r^∗ +(l2− l1)²

2 (M1+ M2B^∗)r^∗.

Let l₁, l₂∈ (si, t_k+1] with l1< l2, k= 1,2,··· ,m, then we have

k(P y )(l2) − (P y )(l1)kX ≤ (l2− l1)kqi(si, ¯y(si))kX

Zl₁ s_i

(l2− l1)kf (s, ¯yρ(s, ¯ys), B ¯y_{ρ(s, ¯ys)})kXd s

+ Zl₂

l₁

(l2− s )kf (s , ¯yρ(s, ¯ys), B ¯y_{ρ(s, ¯ys)})kXd s

≤ (l2− l1)M6r+ (l2− l1)T (M1+ M2B^∗)r^∗ +(l2− l1)²

2 (M1+ M2B^∗)r^∗,

as l₂→ l1, thenkP (y )(l2) − P (y )(l1)kX → 0. This implies that P is equi-continuous on all t ∈ J in Br. Thus, by Arzela- Ascoli Theorem, it follows that P is completely continuous. Therefore, by Schauder fixed point theorem, the operator P has a fixed point, which in turn implies that problem (1)-(3) has at least one solution on J . This is complete the proof of theorem.

4. Example

Consider the following nonlinear impulsive fractional functional initial value problem

D_t³²y(t ) = 1 Γ (2 − α)

Z t 0

(t − s )^1−α• Z^s

−∞

e^{2(ν−s )}y(ν − σ(ky k))

24 dν

+ Zξ

0

cos(γ − ξ)y(γ − σ(ky k)) 25 dγ˜

d s ,(t , y ) ∈ ∪^N_i=1[si, t_i+1] × [0,π], (5)

y(t ) +

r

X

k=1

cky(sk) = φ(t ), t ∈ (−∞,0], y ∈ [0,π], (6)

y⁰(t ) +

r

X

k=1

d_ky(sk) = ψ(t ), t ∈ (−∞,0], y ∈ [0,π], (7)

y(t ) = Gi(t , y ); y⁰(t ) = Hi(t , y ), t ∈ (ti, si]. (8)

For the phase spaceBh, let h(s ) = e^2s, s< 0 then l =R0

−∞h(s )d s =¹₂< ∞,for t ∈ (−∞,0] and define

kφkB_h= Z0

−∞

h(s ) sup

θ ∈[s,0]kφ(θ )kL²d s . Hence for(t ,φ) ∈ [0,1] ×Bh,

Let y :(−∞,T ] → L²[0,π] such that y ∈Bh. Setting

ρ(t ,φ) = t − σ(kφ(0)k), (t ,φ) ∈ J ×Bh,

we have

f(t ,φ, Bφ) = Z0

−∞

e^2(ν)[φ 24+

Zξ 0

cos(ξ − γ)φ 25dγ]d ν,

u(y ) =

r

X

k=1

c_ky(sk); v (y ) =

r

X

k=1

d_ky(sk),

g_i(t , y ) = Gi(t , y ); qi(t , y ) = Hi(t , y ),

then the above equations (5)-(8) can be written in the abstract form as (1)-(3). Further more, we can see that for (t ,φ,ξ),(t ,ψ,ν) ∈ J ×Bh×Bh, we get

kf (t , φ, B φ) − f (t , ψ, B ψ)kL2

=



 Zπ

0

¨Z0

−∞

e^{2(s )}kφ 24−ψ

24kd s + Z0

−∞

e^{2(s )} Zξ

0

k cos(γ − ξ)kφ 25−ψ

25kd γd s

«² d y





1/2

≤



 Zπ

0

¨Z0

−∞

e^{2(s )}kφ 24−ψ

24kd s + Z0

−∞

e^{2(s )}kφ 25−ψ

25kd s

«2

d y





1/2

≤



 Zπ

0

¨1 24

Z0

−∞

e^{2(s )}supkφ − ψkd s + 1 25

Z0

−∞

e^{2(s )}supkφ − ψkd s

«2

d y





1/2

≤pπ

24kφ − ψk +pπ

25kφ − ψk.

Hence function f satisfies(H1). Similarly we can show that the functions gi, qi, u , v satisfy(H3),(H5). All the condition of theorem3.1have fulfilled so we deduced that the system (5)-(6) has a unique solution on[0,1].

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