GENERALIZATION OF AN IMPULSIVE NONLINEAR SINGULAR GRONWALL-BIHARI INEQUALITY WITH DELAY

GENERALIZATION OF AN IMPULSIVE NONLINEAR SINGULAR GRONWALL-BIHARI INEQUALITY WITH DELAY

SHENGFU DENG AND CARL PRATHER DEPARTMENT OFMATHEMATICS

VIRGINIAPOLYTECHNICINSTITUTE ANDSTATEUNIVERSITY

BLACKSBURG, VA 24061, USA [email protected] [email protected]

Received 25 August, 2007; accepted 23 May, 2008 Communicated by S.S. Dragomir

ABSTRACT. This paper generalizes a Tatar’s result of an impulsive nonlinear singular Gronwall- Bihari inequality with delay [J. Inequal. Appl., 2006(2006), 1-12] to a new type of inequalities which includesndistinct nonlinear terms.

Key words and phrases: Gronwall-Bihari inequality, Nonlinear, Impulsive.

2000 Mathematics Subject Classification. 26D15, 26D20.

1. INTRODUCTION

In order to investigate problems of the form

x⁰ =f(t, x), t 6=t_k,

∆x=Ik(x), t =tk,

Samoilenko and Perestyuk [6] first used the following impulsive integral inequality u(t)≤a+

Z t c

b(s)u(s)ds+ X

0<t_k<t

ηku(tk), t ≥0.

Then Bainov and Hristova [2] studied a similar inequality with constant delay. In 2004, Hristova [3] considered a more general inequality with nonlinear functions in u. All of these papers treated the functions (kernels) involved in the integrals which are regular. Recently, Tatar [7]

investigated the following singular inequality u(t)≤a(t) +b(t)

Z t 0

k1(t, s)u^m(s)ds+c(t) Z t

0

k2(t, s)uⁿ(s−τ)ds +d(t) X

0<tk<t

η_ku(t_k), t≥0, u(t)≤ϕ(t), t∈[−τ,0], τ >0

(1.1)

279-07

where the kernels k_i(t, s)are defined by k_i(t, s) = (t −s)^βi−1s^γiF_i(s) for β_i > 0 and γ_i >

−1, i = 1,2, the points t_k (called "instants of impulse effect") are in increasing order and limk→∞t_k = +∞. This inequality was called the impulsive nonlinear singular version of the Gronwall inequality with delay by Tatar [7]. In this paper, we will consider an inequality

u(t)≤a(t) +

n

X

i=1

Z bi(t) 0

(t−s)^βi−1s^rif_i(t, s)w_i(u(s))ds

+

m+n

X

j=n+1

Z bj(t) 0

(t−s)^βj−1s^rjf_j(t, s)w_j(u(s−τ))ds +d(t) X

0<tL<t

η_Lu(t_L), t ≥0, (1.2)

u(t)≤ϕ(t), t∈[−τ,0], τ >0,

wheren, mare positive integers, β_l > 0, r_l >−1forl = 1, . . . , n+mandη_L ≥ 0and other assumptions are given in Section 2. This inequality is more general than (1.1) since (1.2) hasn nonlinear terms.

2. MAINRESULTS

Notation: Following [1] and [5], we say w₁ ∝ w₂ for w₁, w₂ : A ⊂ R → R\{0} if ^w_w²

1 is nondecreasing onA. This concept helps us to compare the monotonicity of different functions.

Now we make the following assumptions:

(H1) allw_i (i= 1, . . . , n+m)are continuous and nondecreasing on[0,∞)and positive on (0,∞), andw₁ ∝w₂ ∝ · · · ∝w_n

(H2) a(t)andd(t)are continuous and nonnegative on[0,∞);

(H3) all b_l : [0,∞) → [0,∞) are continuously differentiable and nondecreasing such that 0≤b_l(t)≤ton[0,∞),t_L ≤b_l(t)≤t_L+τfort∈[t_L, t_L+τ]andt_L+τ ≤b_l(t)≤t_L+1 fort ∈[t_L+τ, t_L+1],l= 1, . . . , n+mandL= 0,1,2, . . . wheret₀ = 0. The pointst_L are called instants of impulse effect which are in increasing order, andlimL→∞t_L =∞;

(H4) all f_l(t, s) (l = 1, . . . , n+m) are continuous and nonnegative functions on[0,∞)× [0,∞);

(H5) ϕ(t)is nonnegative and continuous;

(H6) u(t)is a piecewise continuous function fromR→ R⁺ = [0,∞)with points of discon- tinuity of the first kind at the pointst_L ∈ R. It is also left continuous at the pointst_L. This space is denoted byP C(R,R⁺).

Without loss of generality, we will suppose that the t_L satisfy τ < t_L+1 −t_L ≤ 2τ, L = 0,1,2, . . .. As in Remark 3.2 of [7], other cases can be reduced to this one.

Theorem 2.1. Let the above assumptions hold. Suppose that u satisfies (1.2) and is in P C([−τ,∞),[0,∞)). Then ifβ_α >−¹_p + 1andr_α >−¹_p, it holds that

u(t)≤























u_L,0(t), t∈(t_L, t_L+τ], u_L,1(t), t∈(t_L+τ, t_L+1],

u_k,0(t), t∈(t_k, t_k+τ] ift_k+τ ≤T, uk,1(t), t∈(tk+τ, T] iftk+τ < T, u_k,0(t), t∈(t_k, T] ift_k+τ > T,

wheret_k ≤T < t_k+1 and uL,l(t) =

"

W_n⁻¹ Wn(γL,l,n(t)) + Z bn(t)

tL+lτ

(n+m+L+ 1)^q−1c^q_n(t) ˜f_n^q(t, s)ds

!#¹_q ,

γ_L,l,j(t) = W_j−1⁻¹

Wj−1(γL,l,j−1(t)) +

Z bj−1(t) tL+lτ

(n+m+L+ 1)^q−1c^q_j−1(t) ˜f_j−1^q (t, s)ds

, j 6= 1,

γ_L,l,1(t) = (n+m+L+ 1)^q−1

"

˜ a^q(t) +

n

X

i=1

Z tL+lτ 0

c^q_i(t) ˜f_i^q(t, s)w^q_i(φ(s))ds

+

n+m

X

j=n+1

Z bj(t) 0

c^q_j(t) ˜f_j^q(t, s)w_j^q(ψ(s−τ))ds+

L

X

e=1

d˜^q(t)η_e^qu^q_e−1,1(t_e)

# ,

φ(t) =







u_L,0(t), t∈(t_L, t_L+τ], t∈(t_k, t_k+τ] ift_k+τ ≤T, andt∈(tk, T] iftk+τ > T,

u_L,1(t), t∈(t_L+τ, t_L+1]andt ∈(t_k+τ, T] ift_k+τ < T,

ψ(t) =















ϕ(t), t ∈[−τ,0],

u_L,0(t), t∈(t_L, t_L+τ], t∈(t_k, t_k+τ] ift_k+τ ≤T, andt∈(t_k, T] ift_k+τ > T,

uL,1(t), t∈(tL+τ, tL+1]andt∈(tk+τ, T] iftk+τ < T,

˜

a(t) = max

0≤x≤ta(x), f˜_α(t, s) = max

0≤x≤tf_α(x, s), d(t) = max˜

0≤x≤td(x), W_i(u) =

Z u ui

dv w^q_i(v^1q)

, u >0, u_i >0,

cα(t) =t^{p1+βα+rα−1}

Γ(1 +p(β_α−1))Γ(1 +pr_α) Γ(2 +p(β_α+r_α−1))

¹_p ,

forL= 0,1, . . . , k−1,α= 1,2, . . . , n+m,l = 0,1, andi, j = 1, . . . , nwhere ¹_p +¹_q = 1for p >0andq >1, andT is the largest number such that

(2.1) W_j(γ_L,l,j(t)) + Z bj(t)

tL+lτ

(n+m+L+ 1)^q−1c_j(t) ˜f_j^q(t, s)ds≤ Z ∞

uj

dz w_j^q(z^1/q),

for allt ∈ (t_L, t_L+τ], allt ∈ (t_k, t_k+τ]ift_k+τ ≤ T and allt ∈ (t_L, T]ift_k+τ > T as l = 0, or allt∈[tL+τ, tL+1]and allt ∈[tk+τ, T)iftk+τ < T asl= 1wherej = 1, . . . , n, l = 0,1andL= 0,1. . . , k−1.

Before the proof, we introduce a lemma which will play a very important role.

Lemma 2.2 ([1]). Suppose that

(1) all w_i (i = 1, . . . , n) are continuous and nondecreasing on [0,∞) and positive on (0,∞), andw₁ ∝w₂ ∝ · · · ∝w_n.

(2) a(t)is continuously differentiable intand nonnegative on[t₀, t₁),

(3) all b_l are continuously differentiable and nondecreasing such that b_l(t) ≤ t for t ∈ [t₀, t₁)

where t₀, t₁ are constants and t₀ < t₁. If u(t) is a continuous and nonnegative function on [t₀, t₁)satisfying

u(t)≤a(t) +

n

X

i=1

Z bi(t) bi(t0)

f_i(t, s)w_i(u(s))ds, t₀ ≤t < t₁, then

u(t)≤W˜_n⁻¹

"

W˜_n(γ_n(t)) + Z bn(t)

bn(t0)

f˜_n(t, s)ds

#

, t₀ ≤t≤T₁, where

γ_i(t) = ˜W_i−1⁻¹

"

W˜i−1(γi−1(t)) +

Z bi−1(t) bi−1(t0)

f˜i−1(t, s)ds

#

, i= 2,3, . . . , n, γ₁(t) =a(t₀) +

Z t t0

|a⁰(s)|ds, W˜_i(u) = Z u

ui

dz

wi(z), u_i >0, T₁ < t₁ andT₁is the largest number such that

W˜_i(γ_i(T₁)) +

Z bi(T1) bi(t0)

f˜_i(T₁, s)ds ≤ Z ∞

ui

dz

w_i(z), i= 1, . . . , n.

Proof of Theorem 2.1. Sinceβ_α > −¹_p + 1 andr_α > −¹_p forα = 1, . . . , n+m, by Hölder’s inequality we obtain

u(t)≤a(t) +

n

X

i=1

Z t 0

(t−s)^p(βi−1)s^prids

¹_p Z bi(t) 0

f_i^q(t, s)w_i^q(u(s))ds

!¹_q

+

m+n

X

j=n+1

Z t 0

(t−s)^p(βj−1)s^prjds

¹_p Z bj(t) 0

f_j^q(t, s)w^q_j(u(s−τ))ds

!¹_q

+ X

0<t_L<t

d(t)η_Lu(t_L)

≤a(t) +

n

X

i=1

c_i(t)

Z bi(t) 0

f_i^q(t, s)w_i^q(u(s))ds

!¹_q

+

m+n

X

j=n+1

c_j(t)

Z bj(t) 0

f_j^q(t, s)w^q_j(u(s−τ))ds

!¹_q

+ X

0<tL<t

d(t)η_Lu(t_L)

where we useb_α(t)≤tand the definition ofc_α(t). Now we use the following result [4]:

IfA₁, . . . , A_nare nonnegative forn∈N, then forq >1,

(A₁+· · ·+A_n)^q≤n^q−1(A^q₁+· · ·+A^q_n).

Sincet_k≤t ≤T < t_k+1, we have u^q(t)≤(1 +n+m+k)^q−1

"

a^q(t) +

n

X

i=1

c^q_i(t) Z bi(t)

0

f_i^q(t, s)w^q_i(u(s))ds

+

m+n

X

j=n+1

c^q_j(t) Z bj(t)

0

f_j^q(t, s)w^q_j(u(s−τ))ds+

k

X

L=1

d^q(t)η_L^qu^q(t_L)

# .

We note that ˜a(t) ≥ a(t), d(t)˜ ≥ d(t) and f˜_α(t, s) ≥ f_α(t, s) and they are continuous and nondecreasing int. The above inequality becomes

u^q(t)≤(1 +n+m+k)^q−1

"

˜ a^q(t) +

n

X

i=1 k−1

X

L=0

c^q_i(t) Z tL+1

tL

f˜_i^q(t, s)w_i^q(u(s))ds

+ c^q_i(t) Z bi(t)

tk

f˜_i^q(t, s)w_i^q(u(s))ds

!

+

m+n

X

j=n+1 k−1

X

L=0

c^q_j(t) Z tL+1

tL

f˜_j^q(t, s)w_j^q(u(s−τ))ds

+ c^q_j(t) Z bj(t)

t_k

f˜_j^q(t, s)w_j^q(u(s−τ))ds

! +

k

X

L=1

d˜^q(t)η_L^qu^q(t_L)

# . (2.2)

In the following, we apply mathematical induction with respect tok.

(1)k = 0. We note thatt₀ = 0and we have for any fixed˜t∈[0, t₁] (2.3) u^q(t)≤(n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z bi(t)

0

f˜_i^q(˜t, s)w_i^q(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(t)

0

f˜_j^q(˜t, s)w_j^q(u(s−τ))ds

#

fort ∈[0,˜t]sincec_α(t)are nondecreasing.

Now we consider t˜∈ [0, τ] ⊂ [0, t₁] and t ∈ [0,˜t]. Note that 0 ≤ b_j(t) ≤ t so −τ ≤ b_j(t)−τ ≤0fort∈[0,˜t]. Sinceu(t)≤ϕ(t)fort∈[−τ,0], we have

u^q(t)≤z_0,0(t), t∈[0,˜t], where

(2.4) z_0,0(t) = (n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z bi(t)

0

f˜_i^q(˜t, s)w^q_i(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w^q_j(ϕ(s−τ))ds

# . It implies that

(2.5) u(t)≤z_0,0(t)^1/q, t ∈[0,˜t].

Thus, (2.4) becomes

(2.6) z_0,0(t)≤(n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z bi(t)

0

f˜_i^q(˜t, s)w^q_i(z_0,0^1/q(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w^q_j(ϕ(s−τ))ds

# . By Lemma 2.2, (2.6) and (2.1), we have

z_0,0(t)≤W_n⁻¹

"

W_n(˜γ_0,0,n(t)) + Z bn(t)

0

(n+m+ 1)^q−1c_n(˜t) ˜f_n^q(˜t, s)ds

# ,

˜

γ_0,0,j(t) = W_j−1⁻¹

Wj−1(˜γ0,0,j−1(t)) +

Z bj−1(t) 0

(n+m+ 1)^q−1cj−1(˜t) ˜f_j−1^q (˜t, s)ds

#

, j 6= 1,

˜

γ_0,0,1(t) = (n+m+ 1)^q−1

"

˜ a^q(˜t) +

n+m

X

j=n+1

Z bj(˜t) 0

c^q_j(˜t) ˜f_j^q(˜t, s)w^q_j(ψ(s−τ))ds

#

sinceψ(t) =ϕ(t)fort ∈[−τ,0].

Since (2.5) is true for anyt ∈[0,˜t]andγ˜_0,0,j(˜t) =γ_0,0,j(˜t), we have u(˜t)≤z0,0(˜t)^1/q ≤u0,0(˜t).

We know that˜t ∈[0, τ]is arbitrary so we replace˜tbytand get

(2.7) u(t)≤u_0,0(t), fort∈[0, τ].

This implies that the theorem is true fort∈[0, τ]andk = 0.

Fort ∈[τ,˜t]and˜t ∈ [τ, t₁], use the assumption (H3) and then we know thatb_α(τ) = τ and τ ≤b_α(t)≤t₁ fort∈[τ, t₁]andα= 1, . . . , n+m. Thus,

0≤bα(t)−τ ≤t1 −τ ≤τ sinceτ < t₁−t₀ =t₁ ≤2τ. Using this fact, (2.3) and (2.7), we get

u^q(t)≤(n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z τ

0

f˜_i^q(˜t, s)w_i^q(u(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

τ

f˜_i^q(˜t, s)w^q_i(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z τ

0

f˜_j^q(˜t, s)w^q_j(ψ(s−τ))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

τ

f˜_j^q(˜t, s)w^q_j(u(s−τ))ds

#

≤(n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z τ

0

f˜_i^q(˜t, s)w_i^q(u_0,0(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

τ

f˜_i^q(˜t, s)w^q_i(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z τ

0

f˜_j^q(˜t, s)w^q_j(ψ(s−τ))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

τ

f˜_j^q(˜t, s)w^q_j(u_0,0(s−τ))ds

#

≤(n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z τ

0

f˜_i^q(˜t, s)w_i^q(φ(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

τ

f˜_i^q(˜t, s)w^q_i(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w^q_j(ψ(s−τ))ds

#

:=z_0,1(t),

where we use the definitions ofφandψ. Thus,

(2.8) u(t)≤z_0,1^1/q(t), t ∈[τ,˜t].

Therefore,

z_0,1 ≤(n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z τ

0

f˜_i^q(˜t, s)w^q_i(φ(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

τ

f˜_i^q(˜t, s)w^q_i(z_0,1^1/q(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w_j^q(ψ(s−τ))ds

# . Using Lemma 2.2, (2.1) andb_α(τ) =τ, we obtain fort∈[τ,˜t]

z_0,1(t)≤W_n⁻¹

"

W_n(˜γ_0,1,n(t)) + Z bn(t)

τ

(n+m+ 1)^q−1c^q_n(˜t) ˜f_n^q(˜t, s)ds

# ,

˜

γ_0,1,j(t) =W_j−1⁻¹

Wj−1(˜γ0,1,j−1(t))+

Z bj−1(t) τ

(n+m+ 1)^q−1c^q_j−1(˜t) ˜f_j−1^q (˜t, s)ds

#

, j 6= 1,

˜

γ_0,1,1(t) = (n+m+ 1)^q−1

"

˜ a^q(˜t) +

n

X

i=1

Z τ 0

c^q_i(˜t) ˜f_i^q(˜t, s)w_i^q(φ(s))ds

+

n+m

X

j=n+1

Z bj(˜t) 0

c^q_j(˜t) ˜f_j^q(˜t, s)w_j^q(ψ(s−τ))ds

# .

Since (2.8) is true for anyt ∈[τ, t1]andγ˜0,1,1(˜t) = γ0,1,1(˜t), we have u(˜t)≤z_0,1^1/q(˜t)≤u_0,1(˜t).

We know that˜t ∈[τ, t₁]is arbitrary so we replacet˜bytand get

(2.9) u(t)≤u_0,1(t), t∈[τ, t₁].

This implies that the theorem is valid fort ∈[τ, t1]andL= 0.

(2)L= 1. First we considert∈(t₁,t],˜ where˜t∈(t₁, t₁+τ]is arbitrary. Note thatτ < t₂−t₁ ≤ 2τ. (H3) givesb_α(t₁) =t₁andt₁ ≤b_α(t)≤t₁+τ fort∈(t₁, t₁+τ]sot₁−τ ≤b_α(t)−τ ≤t₁ fort ∈(t₁, t₁+τ]. By (2.7) and (2.9), (2.2) can be written as

u^q(t)≤(n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z τ

0

+ Z t1

τ

f˜_i^q(˜t, s)w^q_i(u(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

t1

f˜_i^q(˜t, s)w^q_i(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z τ

0

+ Z t1

τ

f˜_j^q(˜t, s)w_j^q(u(s−τ))ds

+

n

X

j=1

c^q_j(˜t) Z bj(˜t)

t1

f˜_j^q(˜t, s)w^q_j(u(s−τ))ds+ ˜d^q(˜t)η^q₁u^q(t₁)

#

≤(n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z t1

0

f˜_i^q(˜t, s)w^q_i(φ(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

t1

f˜_i^q(˜t, s)w^q_i(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w_j^q(ψ(s−τ))ds+ ˜d^q(˜t)η₁^qu^q_0,1(t₁)

#

:=z_1,0(t),

where we use the definitions ofφandψ so

(2.10) u(t)≤z_1,0^1/q(t), t∈(t₁,t].˜ Thus,

z1,0(t)≤(n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z t1

0

f˜_i^q(˜t, s)w^q_i(φ(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

t1

f˜_i^q(˜t, s)w^q_i(z_1,0^1/q(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w_j^q(ψ(s−τ))ds+ ˜d^q(˜t)η₁^qu^q_0,1(t₁)

# .

By Lemma 2.2, (2.1) andbα(t1) =t1, we obtain fort∈(t1,t]˜ z_1,0(t)≤W_n⁻¹

"

W_n(˜γ_1,0,n(t)) + Z bn(t)

t1

(n+m+ 2)^q−1c^q_n(˜t) ˜f_n^q(˜t, s)ds

# ,

˜

γ_1,0,j(t) =W_j−1⁻¹

Wj−1(˜γ1,0,j−1(t))+

Z bj−1(t) t1

(n+m+ 2)^q−1c^q_j−1(˜t) ˜f_j−1^q (˜t, s)ds

#

, j 6= 1,

˜

γ_1,0,1(t) = (n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

Z t1

0

c^q_i(˜t) ˜f_i^q(˜t, s)w_i^q(φ(s))ds

+

n+m

X

j=n+1

Z bj(˜t) 0

c^q_j(˜t) ˜f_j^q(˜t, s)w_j^q(ψ(s−τ))ds+ ˜d^q(˜t)η₁^qu^q_0,1(t1)

# . Since (2.10) is true for anyt ∈(t₁,˜t]andγ˜_1,0,1(˜t) = γ_1,0,1(˜t), we have

u(˜t)≤z_1,0^1/q(˜t)≤u_1,0(˜t).

We know that˜t ∈(t1, t1+τ]is arbitrary so we replacet˜bytand get (2.11) u(t)≤u_1,0(t), t∈(t₁, t₁+τ].

This implies that the theorem is valid fort ∈(t₁, t₁+τ]andL= 1.

We now considert ∈ [t₁ +τ,t], where˜ t˜∈ [t₁+τ, t₂]is arbitrary. Again, by (H3) we have t₁+τ ≤b_α(t)≤t₂fort∈[t₁+τ, t₂]andb_α(t₁+τ) =t₁+τsot₁ ≤b_α(t)−τ ≤t₂−τ ≤t₁+τ sinceτ < t₂−t₁ ≤2τ. Obviously, by (2.7), (2.9) and (2.11), (2.2) becomes

u^q(t)≤(n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z t1+τ

0

f˜_i^q(˜t, s)w_i^q(u(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

t1+τ

f˜_i^q(˜t, s)w_i^q(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z t1+τ

0

f˜_j^q(˜t, s)w_j^q(u(s−τ))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

t1+τ

f˜_j^q(˜t, s)w_j^q(u(s−τ))ds+ ˜d^q(˜t)η₁^qu^q_1,0(t₁)

#

≤(n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z t1+τ

0

f˜_i^q(˜t, s)w_i^q(φ(s))ds

+

n

X

i=1

c^q_i(˜t) Z bi(t)

t1+τ

f˜_i^q(˜t, s)w_i^q(u(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w^q_j(ψ(s−τ))ds+ ˜d^q(˜t)η₁^qu^q_0,1(t₁)

#

:=z_1,1(t), that is,

(2.12) u(t)≤z_1,1^1/q(t), t∈[t₁+τ,˜t].

Thus,

z_1,1(t)≤(n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

c^q_i(˜t) Z t1+τ

0

f˜_i^q(˜t, s)w^q_i(φ(s))ds

+c^q_i(˜t) Z bi(t)

t1+τ

f˜_i^q(˜t, s)w_i^q(z_1,1^1/q(s))ds

+

m+n

X

j=n+1

c^q_j(˜t) Z bj(˜t)

0

f˜_j^q(˜t, s)w_j^q(ψ(s−τ))ds+ ˜d^q(˜t)η₁^qu^q_0,1(t₁)

# . Using Lemma 2.2, (2.1) andb_α(t₁+τ) = t₁+τ, we obtain fort ∈(t₁,˜t]

z_1,1(t)≤W_n⁻¹

"

W_n(˜γ_1,1,n(t)) + Z bn(t)

t1+τ

(n+m+ 2)^q−1c^q_n(˜t) ˜f_n^q(˜t, s)ds

# ,

˜

γ_1,1,j(t) =W_j−1⁻¹h

Wj−1(˜γ1,1,j−1(t)) +

Z bj−1(t) t1+τ

(n+m+ 2)^q−1c^q_j−1(˜t) ˜f_j−1^q (˜t, s)ds

#

, j 6= 0,

˜

γ_1,1,1(t) = (n+m+ 2)^q−1

"

˜ a^q(˜t) +

n

X

i=1

Z t1+τ 0

c^q_i(˜t) ˜f_i^q(˜t, s)w_i^q(φ(s))ds

+

n+m

X

j=n+1

Z bj(˜t) 0

c^q_j(˜t) ˜f_j^q(˜t, s)w_j^q(ψ(s−τ))ds+ ˜d^q(˜t)η₁^qu^q_0,1(t₁)

# . Since (2.12) is true for anyt ∈(t₁,˜t]andγ˜_1,1,1(˜t) = γ_1,1,1(˜t), we have

u(˜t)≤z_1,1^1/q(˜t)≤u_1,1(˜t).

We know that˜t ∈[t₁+τ, t₂]is arbitrary so we replacet˜bytand get u(t)≤u_1,1(t), t∈[t₁+τ, t₂].

This implies that the theorem is valid fort ∈[t1+τ, t2]andL= 1.

(3) Finally, suppose that the theorem is valid for k, then for k + 1 we redefine φ and ψ by replacingk withk+ 1. In a similar manner as in steps (1) and (2), we can see that the theorem holds fort∈(t_k+1, T]⊂(t_k+1, t_k+2].

The proof is now completed.

REFERENCES

[1] R.P. AGARWAL, S. DENGANDW. ZHANG, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput., 165 (2005), 599–612.

[2] D.D. BAINOVANDS.G. HRISTOVA, Impulsive integral inequalities with a deviation of the argu- ment, Math. Nachr., 171 (1995), 19–27.

[3] S.G. HRISTOVA, Nonlinear delay integral inequalities for piecewise continuous functions and ap- plications, J. Inequal. Pure Appl. Math., 5(4) (2004), Art. 88. [ONLINE:http://jipam.vu.

edu.au/article.php?sid=441].

[4] M. MEDVED, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions, J. Math. Anal. Appl., 214 (1997), 349–366.

[5] M. PINTO, Integral inequalities of Bihari-type and applications, Funkcial. Ekvac., 33 (1990), 387–

403.

[6] A.M. SAMOILENKO AND N.A. PERESTJUK, Stability of the solutions of differential equations with impulsive action, Differential Equations, 13 (1977), 1981–1992 (Russian).

[7] N.E. TATAR, An impulsive nonlinear singular version of the Gronwall-Bihari inequality, J. Inequal.

Appl., 2006 (2006), 1–12.