Vol 5, No 3 (2014)

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Available online throug

ISSN 2229 – 5046

ON A NEW NONLINEAR RETARDED INEQUALITIES APPLICABLE

TO CERTAIN RETARDED PARTIAL INTEGRODIFFERENTIAL EQUATIONS

Jayashree Patil*

1

and D. B. Dhaigude

2

1

Vasantrao Naik Mahavidyalaya, Aurangabad 431003, (M.S.), India

2

Department of mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad (M.S.) India

(Received on: 02-03-14; Revised & Accepted on: 15-03-14)

ABSTRACT

I

n this article a two dimensional nonlinear retarded integral inequalities which can be used as ready and powerful tool in the analysis of class of integrodiffrerntial equations is presented. Application of nonlinear retarded inequality is also presented.

Keywords and Phrases: Retarded, integral inequality, Boundedness, explicit bound, integrodifferential.

1. INTRODUCTION

The celebrated Gronwall-Bellman [4,10] inequality states that if 𝑢𝑢 and 𝑓𝑓 are nonnegative continuous functions on an interval [𝑎𝑎,𝑏𝑏] satisfying

𝑢𝑢(𝑡𝑡)≤ 𝑐𝑐+∫ 𝑓𝑓_𝑎𝑎𝑡𝑡 (𝑠𝑠)𝑢𝑢(𝑠𝑠)𝑑𝑑𝑠𝑠 (1) for some constant 𝑐𝑐 ≥0, then

𝑢𝑢(𝑡𝑡)≤ 𝑐𝑐 exp�∫ 𝑓𝑓_𝑎𝑎𝑡𝑡 (𝑠𝑠)𝑑𝑑𝑠𝑠�, 𝑡𝑡 ∈[𝑎𝑎,𝑏𝑏] (2)

Inequality (1), provides an explicit bound on the unknown function and hence furnishes a handy tool in the study of qualitative and quantitative properties of solution of differential and integral equations, it has become one of the very few classic and most influential results in the theory and applications of inequalities. Due to its fundamental importance many generalizations and analogous results of(1), have been established.Such inequalities are in general known as Gronwall-Bellman type inequalities in the literature[see 1-3,5-8,11-15,].

The aim of the paper is to extend results which proved by Kim [9] to obtain a new generalizations some formal famous inequalities, which can be used as handy tools to study qualitative as well as quantitative properties of solutions of some nonlinear partial integrodifferential equations.

2 MAIN RESULTS

Let R denotes the set of real numbers,𝑅𝑅₊= [0,∞). Also, 𝐽𝐽₁= [𝑥𝑥₀,𝑋𝑋) and 𝐽𝐽₂= [𝑦𝑦₀,𝑌𝑌) be the given subset of R,Δ=𝐽𝐽₁×𝐽𝐽₂.𝐷𝐷₁𝑧𝑧(𝑥𝑥,𝑦𝑦),𝐷𝐷₂𝑧𝑧(𝑥𝑥,𝑦𝑦) be partial derivative of z with respective x and y respectively.

Theorem: 2.1 Let 𝑢𝑢,𝑎𝑎,𝑐𝑐 ∈ 𝐶𝐶(Δ,𝑅𝑅₊),𝑎𝑎 and 𝑐𝑐 be nondecreasing in each variables 𝑓𝑓_𝑖𝑖,𝑔𝑔_𝑖𝑖,ℎ_𝑖𝑖 ∈ 𝐶𝐶(Δ,𝑅𝑅₊),𝑖𝑖= 1,⋅⋅⋅,𝑛𝑛 and let 𝛼𝛼_𝑖𝑖 ∈ 𝐶𝐶1(𝐽𝐽₁,𝐽𝐽₁) be nondecreasing with 𝛼𝛼_𝑖𝑖(𝑡𝑡)≤ 𝑡𝑡,𝑖𝑖= 1,⋅⋅⋅,𝑛𝑛 and 𝛽𝛽_𝑖𝑖 ∈ 𝐶𝐶1(𝐽𝐽₂,𝐽𝐽₂) be nondecreasing with

𝛽𝛽𝑖𝑖(𝑡𝑡)≤ 𝑡𝑡,𝑖𝑖= 1,⋅⋅⋅,𝑛𝑛. Suppose 𝑝𝑝>𝑞𝑞> 0 are constants 𝜑𝜑 ∈ 𝐶𝐶(𝑅𝑅+,𝑅𝑅+) is an increasing function with 𝜑𝜑(∞) =∞ and 𝜓𝜓(𝑢𝑢) is a nondecreasing continuous function for 𝑢𝑢 ∈ 𝑅𝑅 with 𝜓𝜓(𝑢𝑢) > 0 for 𝑢𝑢> 0. If

𝑢𝑢𝑝𝑝₍_𝑥𝑥_,_𝑦𝑦₎_{≤ 𝑎𝑎}₍_𝑥𝑥_,_𝑦𝑦_{) +}_𝑐𝑐₍_𝑥𝑥_,_𝑦𝑦₎_{� �} _[_𝑓𝑓

𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡) (𝜓𝜓(𝑢𝑢(𝑠𝑠,𝑡𝑡)) 𝛼𝛼𝑖𝑖(𝑥𝑥)

𝛼𝛼𝑖𝑖(𝑥𝑥0) 𝑛𝑛

𝑖𝑖=1

+∫ ∫𝑡𝑡 ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) +𝑔𝑔_𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

𝛽𝛽𝑖𝑖(𝑦𝑦0)

s

𝛼𝛼𝑖𝑖(𝑥𝑥0) (3)

Corresponding author: Jayashree Patil*

1

(2)

Integrodifferential Equations/ IJMA- 5(3), March-2014.

for all (𝑥𝑥,𝑦𝑦)∈ Δ, then

𝑢𝑢(𝑥𝑥,𝑦𝑦)≤[𝐺𝐺1−1(𝐺𝐺1(𝑘𝑘1(𝑥𝑥0,𝑦𝑦)) +𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫ ∫ 𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(1 +∫ ∫ ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠)] 1

𝑝𝑝−𝑞𝑞

t 𝛽𝛽𝑖𝑖(𝑦𝑦0)

s 𝛼𝛼𝑖𝑖(𝑥𝑥0) 𝛽𝛽𝑖𝑖(𝑦𝑦)

𝛽𝛽𝑖𝑖(𝑦𝑦0) 𝛼𝛼𝑖𝑖(𝑥𝑥)

𝛼𝛼𝑖𝑖(𝑥𝑥0)

n

i=1 (4)

for all 𝑥𝑥 ∈[𝑥𝑥₀,𝑥𝑥₁] × [𝑦𝑦₀,𝑦𝑦₁], where

𝑘𝑘1(𝑥𝑥0,𝑦𝑦) = [𝑎𝑎(𝑥𝑥,𝑦𝑦)]

𝑝𝑝−𝑞𝑞

𝑝𝑝 ₊𝑝𝑝 − 𝑞𝑞

𝑝𝑝 𝑐𝑐(𝑥𝑥,𝑦𝑦)� � � 𝑔𝑔𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1 𝐺𝐺1(𝑟𝑟) =� 𝑑𝑑𝑠𝑠

𝜓𝜓(𝑠𝑠𝑝𝑝−𝑞𝑞1 ₎

, 𝑟𝑟 ≥ 𝑟𝑟0> 0. r

𝑟𝑟0

𝐺𝐺1−1 denotes the inverse function of 𝐺𝐺1 and (𝑥𝑥1,𝑦𝑦1)∈ Δ is chosen so that

𝐺𝐺1(𝑘𝑘1(𝑥𝑥0,𝑦𝑦)) +𝑐𝑐(𝑥𝑥,𝑦𝑦)� � � [𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(1 + 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 ∈ 𝐷𝐷𝐷𝐷𝐷𝐷(𝐺𝐺1−1) t

s

Proof: Let us first assume that 𝑎𝑎(𝑥𝑥,𝑦𝑦) > 0. Fixing any number 𝑥𝑥̅ and 𝑦𝑦� with 𝑥𝑥₀≤ 𝑥𝑥 ≤ 𝑥𝑥̅ and 𝑦𝑦₀≤ 𝑦𝑦 ≤ 𝑦𝑦�, define a positive function 𝑧𝑧(𝑥𝑥,𝑦𝑦) as right hand side of (3). Then

𝑧𝑧(𝑥𝑥,𝑦𝑦) =𝑎𝑎(𝑥𝑥̅,𝑦𝑦�) +𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)� � � [𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡) (𝜓𝜓(𝑢𝑢(𝑠𝑠,𝑡𝑡)) 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

+∫_𝛼𝛼s_𝑖𝑖₍_𝑥𝑥₀₎∫_𝛽𝛽t_𝑖𝑖₍_𝑦𝑦₀₎ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) +𝑔𝑔𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

Then 𝑧𝑧(𝑥𝑥,𝑦𝑦) > 0, 𝑧𝑧(𝑥𝑥₀,𝑦𝑦) =𝑧𝑧(𝑥𝑥,𝑦𝑦₀) =𝑎𝑎(𝑥𝑥̅,𝑦𝑦�) and (3) can be restated

𝑢𝑢(𝑥𝑥,𝑦𝑦)≤[𝑧𝑧(𝑥𝑥,𝑦𝑦)]𝑝𝑝1 ₍₅₎

clearly, 𝑧𝑧(𝑥𝑥,𝑦𝑦) is continuous nondecreasing function for all 𝑥𝑥 ∈ 𝐽𝐽₁, 𝑦𝑦 ∈ 𝐽𝐽₂ and

𝐷𝐷1𝑧𝑧(𝑥𝑥,𝑦𝑦) =𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)� � [𝑓𝑓𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)𝑢𝑢𝑞𝑞(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)(𝜓𝜓(𝑢𝑢(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)) 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

+� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) +𝑔𝑔𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)𝑢𝑢𝑞𝑞(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)]𝛼𝛼𝑖𝑖′(𝑥𝑥)𝑑𝑑𝑡𝑡 t

Using (5), we deduce

𝐷𝐷1𝑧𝑧(𝑥𝑥,𝑦𝑦) ≤ 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)[𝑧𝑧(𝑥𝑥,𝑦𝑦)]

𝑞𝑞

𝑝𝑝_{� �} _[_𝑓𝑓

𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)(𝜓𝜓(𝑢𝑢(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)) + 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) t

𝛼𝛼𝑖𝑖(𝑥𝑥0) +𝑔𝑔𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)]𝛼𝛼𝑖𝑖′(𝑥𝑥)𝑑𝑑𝑡𝑡

Using the monotonicity of 𝑧𝑧(𝑡𝑡), we deduce

𝐷𝐷1𝑧𝑧(𝑥𝑥,𝑦𝑦)

[𝑧𝑧(𝑥𝑥,𝑦𝑦)]𝑝𝑝𝑞𝑞≤ 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)� � [𝑓𝑓𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)(𝜓𝜓(𝑢𝑢(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)) +

n

i=1

� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) t

+𝑔𝑔_𝑖𝑖(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡)]𝛼𝛼_𝑖𝑖′(𝑥𝑥)𝑑𝑑𝑡𝑡 (6)

Keeping 𝑦𝑦 fixed and integrating inequality (6) from 𝑥𝑥₀ to 𝑥𝑥 and making change of variables, we get

𝑧𝑧𝑝𝑝−𝑞𝑞𝑝𝑝 ₍_𝑥𝑥_,_𝑦𝑦₎_≤₍_𝑎𝑎₍_𝑥𝑥̅_,_𝑦𝑦�₎₎𝑝𝑝−𝑞𝑞𝑝𝑝 ₊𝑝𝑝−𝑞𝑞

𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)∑ ∫ ∫ [𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡) (𝜓𝜓(𝑧𝑧 1

𝑝𝑝₍_𝑠𝑠_,_𝑡𝑡₎₎

n

i=1

+∫ ∫ ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑧𝑧 1

𝑝𝑝₍_𝜎𝜎_,_𝜂𝜂₎₎_{𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎}_{) +}_𝑔𝑔

𝑖𝑖(𝑠𝑠,𝑡𝑡)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 t

s

𝛼𝛼𝑖𝑖(𝑥𝑥0) (7)

Now, define a function 𝑘𝑘₁(𝑥𝑥,𝑦𝑦) by

𝑘𝑘1(𝑥𝑥,𝑦𝑦) = (𝑎𝑎(𝑥𝑥̅,𝑦𝑦�))

𝑝𝑝−𝑞𝑞

𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)� � � 𝑔𝑔𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

𝛽𝛽𝑖𝑖(𝑦𝑦�)

𝛽𝛽𝑖𝑖(𝑦𝑦0) 𝛼𝛼𝑖𝑖(𝑥𝑥̅)

n

i=1

+𝑝𝑝−𝑞𝑞_𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)∑ ∫ ∫ 𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(𝜓𝜓(𝑧𝑧 1

𝑝𝑝₍_𝑠𝑠_,_𝑡𝑡₎₎

𝛼𝛼𝑖𝑖(𝑥𝑥0) 𝑛𝑛

i=1 +∫ ∫ ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑧𝑧

1

𝑝𝑝₍_𝜎𝜎_,_𝜂𝜂₎₎_{𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎}₎_{𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠}

(3)

Then, (7) can be restated as

[𝑧𝑧(𝑥𝑥,𝑦𝑦)]𝑝𝑝1_≤_[_𝑘𝑘₁₍_𝑥𝑥_,_𝑦𝑦_)]𝑝𝑝−𝑞𝑞1 ₍₈₎

We know, 𝑢𝑢(𝑥𝑥,𝑦𝑦)≤[𝑧𝑧(𝑥𝑥,𝑦𝑦)]

1

𝑝𝑝 _≤_[_𝑘𝑘₁₍_𝑥𝑥_,_𝑦𝑦_)]𝑝𝑝−𝑞𝑞1 _{and since}_𝜓𝜓_{is nondecreasing,}

𝜓𝜓[𝑢𝑢(𝜎𝜎,𝜂𝜂)]≤ 𝜓𝜓[𝑧𝑧1𝑝𝑝₍_𝜎𝜎_,_𝜂𝜂_)]_{≤ 𝜓𝜓}_[_𝑘𝑘

1

𝑝𝑝−𝑞𝑞₍_𝑠𝑠_,_𝑡𝑡_)]_for_{𝜎𝜎 ∈}_[_𝛼𝛼

𝑖𝑖(𝑡𝑡0),𝛼𝛼𝑖𝑖(𝑡𝑡)],and 𝜂𝜂 ∈[𝛽𝛽𝑖𝑖(𝑡𝑡0),𝛽𝛽𝑖𝑖(𝑡𝑡)]

𝑘𝑘1(𝑥𝑥,𝑦𝑦)≤(𝑎𝑎(𝑥𝑥̅,𝑦𝑦�))

𝑝𝑝−𝑞𝑞

𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)� � � 𝑔𝑔𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

𝛽𝛽𝑖𝑖(𝑦𝑦�)

n

i=1

+𝑝𝑝−𝑞𝑞_𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)∑ ∫ ∫ 𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(𝜓𝜓(𝑘𝑘1

1

𝑝𝑝−𝑞𝑞₍_𝑠𝑠_,_𝑡𝑡₎₎

n

i=1

+∫ ∫ ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑘𝑘₁

1

𝑝𝑝 −𝑞𝑞₍_𝜎𝜎_,_𝜂𝜂₎₎_{𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎}₎_{𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠}

s 𝛼𝛼𝑖𝑖(𝑥𝑥0)

Here, we observe that 𝑘𝑘(𝑥𝑥,𝑦𝑦) is a continuous nondecreasing function for all 𝑥𝑥 ∈ 𝐽𝐽₁,𝑦𝑦 ∈ 𝐽𝐽₂ and

𝐷𝐷1𝑘𝑘1(𝑥𝑥,𝑦𝑦)≤𝑝𝑝 − 𝑞𝑞_{𝑝𝑝 𝑐𝑐}(𝑥𝑥̅,𝑦𝑦�)� � [𝑓𝑓𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)(𝜓𝜓 �𝑘𝑘1

1

𝑝𝑝−𝑞𝑞₍_𝑠𝑠_,_𝑡𝑡₎_�

n

i=1

+� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑘𝑘1

1

𝑝𝑝−𝑞𝑞₍_𝜎𝜎_,_𝜂𝜂₎₎_{𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎}_)]_𝛼𝛼′

𝑖𝑖(𝑥𝑥)𝑑𝑑𝑡𝑡 t

𝐷𝐷1𝑘𝑘1(𝑥𝑥,𝑦𝑦)≤𝑝𝑝−𝑞𝑞_𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)𝜓𝜓(𝑘𝑘1

1

𝑝𝑝−𝑞𝑞₍_𝑥𝑥_,_𝑦𝑦₎₎_{∑ ∫} _[_𝑓𝑓

𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)(1 +∫_𝛼𝛼𝛼𝛼_𝑖𝑖𝑖𝑖₍(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽t_𝑖𝑖₍_𝑦𝑦₀₎ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝛼𝛼′𝑖𝑖(𝑥𝑥)𝑑𝑑𝑡𝑡 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1 (9)

Using monotonicity of 𝑘𝑘₁ and 𝜓𝜓₁, we have

𝐷𝐷1𝑘𝑘1(𝑥𝑥,𝑦𝑦)

𝜓𝜓(𝑘𝑘₁

1

𝑝𝑝 −𝑞𝑞₍_𝑥𝑥_,_𝑦𝑦₎₎≤

𝑝𝑝 − 𝑞𝑞

𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)� � [𝑓𝑓𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)(1 +

n

i=1

� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝛼𝛼′𝑖𝑖(𝑥𝑥)𝑑𝑑𝑡𝑡 t

From the definition of 𝐺𝐺₁ and Keeping 𝑦𝑦 fixed in (9) and integrating from 𝑥𝑥₀ to 𝑥𝑥 and making change of variables, we have

𝐺𝐺1(𝑘𝑘1(𝑥𝑥,𝑦𝑦))≤ 𝐺𝐺1(𝑘𝑘1(𝑥𝑥0,𝑦𝑦)) +𝑝𝑝−𝑞𝑞_𝑝𝑝 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)∑ ∫i=1n _𝛼𝛼𝛼𝛼_𝑖𝑖₍𝑖𝑖(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎[𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡) (1 +∫_𝛼𝛼s_𝑖𝑖₍_𝑥𝑥₀₎∫_𝛽𝛽t_𝑖𝑖₍_𝑦𝑦₀₎ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 (10)

Using (8),(10) in (5), we have

𝑢𝑢(𝑥𝑥,𝑦𝑦)≤{𝐺𝐺1−1[𝐺𝐺1�𝑘𝑘1(𝑥𝑥0,𝑦𝑦)�+𝑝𝑝 − 𝑞𝑞_{𝑝𝑝 𝑐𝑐}(𝑥𝑥̅,𝑦𝑦�)� � � [𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(1 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

+� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠]} 1

𝑝𝑝−𝑞𝑞

t

s

Taking 𝑥𝑥=𝑥𝑥̅,𝑦𝑦=𝑦𝑦� in the above inequality since 𝑥𝑥̅ and 𝑦𝑦� are arbitrary, we get required inequality.

Remark: (1) If we put ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂) = 0 in above Theorem , we get corollary 2.2 in [9]

3 APPLICATION TO PARTIALINTEGRODIFFERENTIAL EQUATIONS

In this section we show that our one of the result is useful in proving global existence of the solution of certain non-linear partial integrodifferential equation with time delay. Consider the nonlinear integrodifferential equation involving several retarded arguments,

𝐷𝐷2(𝑧𝑧𝑝𝑝−1(𝑥𝑥,𝑦𝑦)𝐷𝐷1𝑧𝑧(𝑥𝑥,𝑦𝑦)) =𝐹𝐹(𝑥𝑥,𝑦𝑦,𝑧𝑧(𝑥𝑥 − 𝑙𝑙1(𝑥𝑥),𝑦𝑦 − 𝐷𝐷1(𝑦𝑦)),⋅⋅⋅,𝑧𝑧(𝑥𝑥 − 𝑙𝑙𝑛𝑛(𝑥𝑥),𝑦𝑦 − 𝐷𝐷𝑛𝑛(𝑦𝑦)),

∫ ∫ 𝐻𝐻_𝑦𝑦𝑦𝑦 _𝑖𝑖(𝑥𝑥,𝑦𝑦,𝑠𝑠,𝑡𝑡,𝑧𝑧(𝑠𝑠 − 𝑙𝑙₁(𝑠𝑠),𝑡𝑡 − 𝐷𝐷₁(𝑡𝑡)),⋅⋅⋅,⋅⋅⋅ 𝑧𝑧(𝑠𝑠 − 𝑙𝑙_𝑛𝑛(𝑠𝑠),𝑡𝑡 − 𝐷𝐷_𝑛𝑛(𝑡𝑡)))𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠)

0 𝑥𝑥

𝑥𝑥0 (11)

with the initial boundary conditions

𝑧𝑧𝑝𝑝₍_𝑥𝑥_,_𝑦𝑦

0) =𝑒𝑒1(𝑥𝑥), 𝑧𝑧𝑝𝑝(𝑥𝑥0,𝑦𝑦) =𝑒𝑒2(𝑦𝑦), 𝑒𝑒1(𝑥𝑥0) =𝑒𝑒2(𝑦𝑦0) = 0

where 𝑝𝑝> 1 is constant, 𝐹𝐹 ∈ 𝐶𝐶(Δ×𝑅𝑅𝑛𝑛×𝑅𝑅,𝑅𝑅),𝐻𝐻_𝑖𝑖 ∈ 𝐶𝐶(Δ×Δ×𝑅𝑅𝑛𝑛,𝑅𝑅),𝑒𝑒₁∈ 𝐶𝐶1(𝐽𝐽₁,𝑅𝑅)𝑒𝑒₂∈ 𝐶𝐶1(𝐽𝐽₂,𝑅𝑅 and 𝑙𝑙_𝑖𝑖 ∈ 𝐶𝐶1(𝐽𝐽₁,𝐽𝐽₁),𝐷𝐷_𝑖𝑖 ∈ 𝐶𝐶1(𝐽𝐽₂,𝐽𝐽₂) such that

(4)

Integrodifferential Equations/ IJMA- 5(3), March-2014.

The following theorem deals with a boundedness on the solution of the problem (11)

Theorem: 3.1 Assume that 𝐹𝐹:Δ×𝑅𝑅𝑛𝑛×𝑅𝑅 → 𝑅𝑅 is a continuous function for which there exists continuous nonnegative functions 𝑓𝑓_𝑖𝑖(𝑥𝑥,𝑦𝑦),𝑔𝑔_𝑖𝑖(𝑥𝑥,𝑦𝑦);𝑖𝑖= 1,⋅⋅⋅ 𝑛𝑛 such that

|𝐹𝐹(𝑥𝑥,𝑦𝑦,𝑢𝑢1,𝑢𝑢2,⋅⋅⋅ 𝑢𝑢𝑛𝑛,𝑣𝑣)|≤ ∑ni=1|𝑢𝑢𝑖𝑖|𝑞𝑞{𝑓𝑓𝑖𝑖(𝑥𝑥,𝑦𝑦)(𝜓𝜓(|𝑢𝑢𝑖𝑖| + |𝑣𝑣|)) +𝑔𝑔𝑖𝑖(𝑥𝑥,𝑦𝑦)|𝑢𝑢𝑖𝑖|𝑞𝑞} (12)

and the function 𝐻𝐻_𝑖𝑖:Δ×Δ×𝑅𝑅𝑛𝑛 → 𝑅𝑅 is a continuous function for which there exist continuous nonnegative function

ℎ𝑖𝑖(𝑥𝑥,𝑦𝑦),𝑖𝑖= 1,⋅⋅⋅,𝑛𝑛 such that

|𝐻𝐻𝑖𝑖(𝑥𝑥,𝑦𝑦,𝑠𝑠,𝑡𝑡,𝑢𝑢1,𝑢𝑢2⋅⋅⋅ 𝑢𝑢𝑛𝑛)|≤ ∑ ℎni=1 𝑖𝑖(𝑥𝑥,𝑦𝑦)𝜓𝜓(|𝑢𝑢𝑖𝑖|) (13)

and

|𝑒𝑒1(𝑥𝑥) +𝑒𝑒2(𝑦𝑦)|≤ 𝑎𝑎(𝑥𝑥,𝑦𝑦) (14)

where 𝑎𝑎(𝑥𝑥,𝑦𝑦)∈ 𝐶𝐶(Δ,𝑅𝑅₊) is nondecreasing in each variables, 𝑝𝑝>𝑞𝑞> 0 are constants and 𝜓𝜓(𝑢𝑢) is a nondecreasing continuous function for 𝑢𝑢 ∈ 𝑅𝑅 with 𝜓𝜓(𝑢𝑢) > 0 for 𝑢𝑢> 0. Let

𝑀𝑀𝑖𝑖 = max_{𝑥𝑥∈𝐽𝐽} 1

1

1−𝑙𝑙′𝑖𝑖(𝑥𝑥), 𝑁𝑁𝑖𝑖 = max𝑦𝑦∈𝐽𝐽2

1

1−𝐷𝐷′𝑖𝑖(𝑦𝑦), 𝑖𝑖= 1,2,⋅⋅⋅,𝑛𝑛 (15)

if 𝑧𝑧(𝑥𝑥,𝑦𝑦)is any solution of problem (11) with the initial boundary condition, then

|𝑧𝑧(𝑥𝑥,𝑦𝑦)| = {𝐺𝐺1−1[𝐺𝐺1(𝑘𝑘1(𝑥𝑥0,𝑦𝑦)) + (𝑝𝑝 − 𝑞𝑞)� � � 𝑓𝑓𝑖𝑖(𝜎𝜎,𝜏𝜏) 𝑦𝑦−𝐷𝐷𝑖𝑖(𝑦𝑦)

𝑦𝑦0−𝐷𝐷𝑖𝑖(𝑦𝑦0) 𝑥𝑥−𝑙𝑙𝑖𝑖(𝑥𝑥)

𝑥𝑥0−𝑙𝑙𝑖𝑖(𝑥𝑥0)

n

i=1

(1 +∫ ∫ ℎ_𝑖𝑖(𝜙𝜙,𝛾𝛾)𝑑𝑑𝛾𝛾𝑑𝑑𝜙𝜙)𝑑𝑑𝜏𝜏𝑑𝑑𝜎𝜎]}

1

𝑝𝑝−𝑞𝑞

t 𝑦𝑦0−𝐷𝐷𝑖𝑖(𝑦𝑦0)

s 𝑥𝑥0−𝑙𝑙𝑖𝑖(𝑥𝑥0)

for all (𝑥𝑥,𝑦𝑦)∈[𝑥𝑥₀,𝑥𝑥₁] × [𝑦𝑦₀,𝑦𝑦₁], where 𝐺𝐺 is in Theorem 2.1 and

𝑘𝑘(𝑥𝑥0,𝑦𝑦) = [𝑎𝑎(𝑥𝑥,𝑦𝑦)]

𝑝𝑝−𝑞𝑞

𝑝𝑝 _{+ (}_{𝑝𝑝 − 𝑞𝑞}₎_{� �} _� _𝑔𝑔_𝑖𝑖₍_𝜎𝜎_,_𝜏𝜏₎_{𝑑𝑑𝜏𝜏𝑑𝑑𝜎𝜎}_.

𝑦𝑦−𝐷𝐷𝑖𝑖(𝑦𝑦)

n

i=1

𝑓𝑓𝑖𝑖 =𝑓𝑓𝑖𝑖(𝜎𝜎+𝑙𝑙𝑖𝑖(𝑠𝑠),𝜏𝜏+𝐷𝐷𝑖𝑖(𝑡𝑡))𝑀𝑀𝑖𝑖𝑁𝑁𝑖𝑖

ℎ𝑖𝑖 =𝑔𝑔𝑖𝑖(𝜙𝜙+𝑙𝑙𝑖𝑖(𝜉𝜉),𝛾𝛾+𝐷𝐷𝑖𝑖(𝜂𝜂))𝑀𝑀𝑖𝑖𝑁𝑁𝑖𝑖

𝑔𝑔𝑖𝑖 =ℎ𝑖𝑖(𝜎𝜎+𝑙𝑙𝑖𝑖(𝑠𝑠),𝜏𝜏+𝐷𝐷𝑖𝑖(𝑡𝑡))𝑀𝑀𝑖𝑖𝑁𝑁𝑖𝑖

Proof: It is easy to see that the solution 𝑧𝑧(𝑥𝑥,𝑦𝑦) of problem (11) with the initial boundary condition satisfies the equivalent integral equation

𝑧𝑧𝑝𝑝₍_𝑥𝑥_,_𝑦𝑦_{) =}_𝑒𝑒

1(𝑥𝑥) +𝑒𝑒2(𝑦𝑦) +𝑝𝑝 � � 𝐹𝐹(𝑠𝑠,𝑡𝑡,𝑧𝑧(𝑠𝑠 − 𝑙𝑙1(𝑠𝑠),𝑡𝑡 − 𝐷𝐷1(𝑡𝑡)),⋅⋅⋅,𝑧𝑧(𝑠𝑠 − 𝑙𝑙𝑛𝑛(𝑠𝑠),𝑡𝑡 − 𝐷𝐷𝑛𝑛(𝑡𝑡)), 𝑦𝑦

𝑦𝑦0 𝑥𝑥

𝑥𝑥0

∫ ∫ 𝐻𝐻_𝑥𝑥𝑠𝑠₀ _𝑦𝑦𝑡𝑡₀ 𝑖𝑖(𝑠𝑠,𝑡𝑡,𝜉𝜉,𝜂𝜂,𝑧𝑧(𝜉𝜉 − 𝑙𝑙1(𝜉𝜉),𝜂𝜂 − 𝐷𝐷1(𝜂𝜂)),⋅⋅⋅⋅⋅⋅,𝑧𝑧(𝜉𝜉 − 𝑙𝑙𝑛𝑛(𝜉𝜉),𝜂𝜂 − 𝐷𝐷𝑛𝑛(𝜂𝜂)))𝑑𝑑𝜂𝜂𝑑𝑑𝜉𝜉)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 (16)

using (12),(13),(14),(15) and making change of variable we have

|𝑧𝑧𝑝𝑝₍_𝑥𝑥_,_𝑦𝑦_)|_{≤ 𝑎𝑎}₍_𝑥𝑥_,_𝑦𝑦_{) +}_{𝑝𝑝 � �} _� _[|_𝑧𝑧₍_𝜎𝜎_,_𝜏𝜏_)|𝑞𝑞_𝑓𝑓 𝑖𝑖(𝜎𝜎,𝜏𝜏) 𝑦𝑦−𝐷𝐷𝑖𝑖(𝑦𝑦)

𝑥𝑥0−𝐷𝐷𝑖𝑖(𝑦𝑦0) 𝑥𝑥−𝑙𝑙𝑖𝑖(𝑥𝑥)

𝑥𝑥0−𝑙𝑙𝑖𝑖(𝑥𝑥0) 𝑛𝑛

𝑖𝑖=1

0−𝐷𝐷𝑖𝑖(𝑦𝑦0) 𝜎𝜎

𝑥𝑥0−𝑙𝑙𝑖𝑖(𝑥𝑥0) (17)

𝑓𝑓𝑖𝑖 =𝑓𝑓𝑖𝑖(𝜎𝜎+𝑙𝑙𝑖𝑖(𝑠𝑠),𝜏𝜏+𝐷𝐷𝑖𝑖(𝑡𝑡))𝑀𝑀𝑖𝑖𝑁𝑁𝑖𝑖,

ℎ𝑖𝑖 =ℎ𝑖𝑖(𝜙𝜙+𝑙𝑙𝑖𝑖(𝜉𝜉),𝜈𝜈+𝐷𝐷𝑖𝑖(𝜂𝜂))𝑀𝑀𝑖𝑖𝑁𝑁𝑖𝑖,

𝑔𝑔𝑖𝑖 =𝑔𝑔𝑖𝑖(𝜎𝜎+𝑙𝑙𝑖𝑖(𝑠𝑠),𝜏𝜏+𝐷𝐷𝑖𝑖(𝑡𝑡))𝑀𝑀𝑖𝑖𝑁𝑁𝑖𝑖,

𝜎𝜎,𝑠𝑠,𝜙𝜙,𝜉𝜉 ∈ 𝐽𝐽1,𝜏𝜏,𝑡𝑡,𝜈𝜈,𝜂𝜂 ∈ 𝐽𝐽2.

Now, applying Theorem 2.1 to above inequality, we get

|𝑧𝑧(𝑥𝑥,𝑦𝑦)| = {𝐺𝐺1−1[𝐺𝐺1(𝑘𝑘1(𝑥𝑥0,𝑦𝑦))

+(𝑝𝑝 − 𝑞𝑞)� � � 𝑓𝑓𝑖𝑖(𝜎𝜎,𝜏𝜏)

𝑦𝑦−𝐷𝐷𝑖𝑖(𝑦𝑦)

n

i=1

(1 +� � ℎ𝑖𝑖(𝜙𝜙,𝛾𝛾)𝑑𝑑𝛾𝛾𝑑𝑑𝜙𝜙)𝑑𝑑𝜏𝜏𝑑𝑑𝜎𝜎]} 1

𝑝𝑝−𝑞𝑞

t

𝑦𝑦0−𝐷𝐷𝑖𝑖(𝑦𝑦0)

s

(5)

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