Vol 5, No 3 (2014)

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Available online throug

ISSN 2229 – 5046

INTEGRAL INEQUALITIES

IN TWO INDEPENDENT VARIABLES FOR RETARDED EQUATION

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

T

he object of this paper is to establish some new nonlinear retarded integral inequalities in two independent variables which provide explicit bounds on unknown functions. The inequalities given here can be used as handy tools in the theory of partial differential equations with time delays.

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

1.INTRODUCTION

The integral inequalities involving functions of one and more than one independent variables which provide explicit bounds on unknown functions play a fundamental role in the development of the theory of differential equations.

The Gronwall-Bellman 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)ds (1.1) for some constant 𝑐𝑐 ≥0, then

𝑢𝑢(𝑡𝑡)≤ 𝑐𝑐exp�∫ 𝑓𝑓_𝑎𝑎𝑡𝑡 (𝑠𝑠)𝑑𝑑𝑠𝑠�, 𝑡𝑡 ∈[𝑎𝑎,𝑏𝑏] (1.2)

Inequality (1.2), 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 equation.

Due to its importance over the years many generalizations and analogous results of (1.1), have been established

(𝑠𝑠𝑒𝑒𝑒𝑒,𝑒𝑒𝑒𝑒. , [1−14]).

In this paper, we establish some new integral inequalities involving functions of two independent variables. Our results extend some results in [7] and [9].

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₊), a and c be nondecreasing in each variables, 𝑓𝑓_𝑖𝑖,ℎ_𝑖𝑖,𝑒𝑒_𝑖𝑖 ∈ 𝐶𝐶(Δ,𝑅𝑅₊), i=1,2,...,n and let

𝛼𝛼𝑖𝑖 ∈ 𝐶𝐶1(𝐽𝐽1,𝐽𝐽1) be nondecreasing with 𝛼𝛼𝑖𝑖(𝑡𝑡)≤ 𝑡𝑡, i=1,2,...,n. and 𝛽𝛽𝑖𝑖 ≤ 𝐶𝐶1(𝐽𝐽2,𝐽𝐽2) be nondecreasing with 𝛽𝛽𝑖𝑖(𝑡𝑡)≤ 𝑡𝑡, i=1,2,...,n. Suppose that 𝑞𝑞> 0 is a constant and 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

𝜑𝜑(𝑢𝑢(𝑥𝑥,𝑦𝑦)) ≤ 𝑎𝑎(𝑥𝑥,𝑦𝑦) +𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫ni=1 _𝛼𝛼𝛼𝛼_𝑖𝑖₍𝑖𝑖(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎[𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡)

�𝜓𝜓�𝑢𝑢(𝑠𝑠,𝑡𝑡)�+∫_𝛼𝛼s_𝑖𝑖₍_𝑥𝑥₀₎∫_𝛽𝛽t_𝑖𝑖₍_𝑦𝑦₀₎ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓�𝑢𝑢(𝜎𝜎,𝜂𝜂)�𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎�+𝑒𝑒𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 (2.1)

for all (𝑥𝑥,𝑦𝑦)∈ Δ, then

Corresponding author:

Jayashree Patil*

1

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𝑢𝑢(𝑥𝑥,𝑦𝑦) ≤ 𝜑𝜑−1_{_𝐺𝐺−1_[_Φ−1₍_Φ₍_𝑘𝑘₍_𝑥𝑥 0,𝑦𝑦))

+𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫i=1n _𝛼𝛼𝛼𝛼_𝑖𝑖𝑖𝑖₍(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(1 +∫_𝛼𝛼s_𝑖𝑖₍_𝑥𝑥₀₎∫_𝛽𝛽t_𝑖𝑖₍_𝑦𝑦₀₎ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠)]} (2.2)

for all (𝑥𝑥,𝑦𝑦)∈[𝑥𝑥₀,𝑥𝑥₁] × [𝑦𝑦₀,𝑦𝑦₁],

where,

𝑘𝑘(𝑥𝑥0,𝑦𝑦) =𝐺𝐺(𝑎𝑎(𝑥𝑥,𝑦𝑦)) +𝑐𝑐(𝑥𝑥,𝑦𝑦)� � � 𝑒𝑒𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 𝛽𝛽𝑖𝑖(𝑦𝑦)

𝛽𝛽𝑖𝑖(𝑦𝑦0)

𝛼𝛼𝑖𝑖(𝑥𝑥)

𝛼𝛼𝑖𝑖(𝑥𝑥0)

n

i=1 𝐺𝐺(𝑟𝑟) =∫ _[_𝜑𝜑−1𝑑𝑑𝑠𝑠₍_𝑠𝑠_)]𝑞𝑞, 𝑟𝑟 ≥ 𝑟𝑟0> 0

r

𝑟𝑟0 (2.3)

Φ(𝑟𝑟) =∫ _𝜓𝜓₍_𝜑𝜑−1₍𝑑𝑑𝑠𝑠_𝐺𝐺−1₍_𝑠𝑠₎₎₎, 𝑟𝑟 ≥ 𝑟𝑟0> 0

r

𝑟𝑟0 (2.4)

when 𝐺𝐺−1 and Φ−1 denote the inverse functions of 𝐺𝐺, Φ and (𝑥𝑥₁,𝑦𝑦₁)∈ Δ is so chosen that

Φ(𝑘𝑘(𝑥𝑥0,𝑦𝑦)) +𝑐𝑐(𝑥𝑥,𝑦𝑦)� � � 𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(1 + 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 ∈ 𝐷𝐷𝐷𝐷𝐷𝐷(Φ−1)

t

s

Proof: Suppose that 𝑎𝑎(𝑥𝑥,𝑦𝑦) > 0. Fixing numbers 𝑥𝑥̅ and 𝑦𝑦� with 𝑥𝑥₀≤ 𝑥𝑥 ≤ 𝑥𝑥̅ and 𝑦𝑦₀≤ 𝑦𝑦 ≤ 𝑦𝑦�, define a positive function 𝑧𝑧(𝑥𝑥,𝑦𝑦) as,

𝑧𝑧(𝑥𝑥,𝑦𝑦) =𝑎𝑎(𝑥𝑥̅,𝑦𝑦�) +𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)∑ ∫ni=1 _𝛼𝛼𝛼𝛼_𝑖𝑖𝑖𝑖₍(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎[𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡) × (𝜓𝜓(𝑢𝑢(𝑠𝑠,𝑡𝑡))

+∫_𝛼𝛼s_𝑖𝑖₍_𝑥𝑥₀₎∫_𝛽𝛽t_𝑖𝑖₍_𝑦𝑦₀₎ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)+𝑒𝑒𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑢𝑢𝑞𝑞(𝑠𝑠,𝑡𝑡)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 (2.5)

Then 𝑧𝑧(𝑥𝑥,𝑦𝑦) > 0, 𝑧𝑧(𝑥𝑥₀,𝑦𝑦) =𝑧𝑧(𝑥𝑥,𝑦𝑦₀) =𝑎𝑎(𝑥𝑥̅,𝑦𝑦�) and (2.1) can be restated

𝑢𝑢(𝑥𝑥,𝑦𝑦)≤ 𝜑𝜑−1₍_𝑧𝑧₍_𝑥𝑥_,_𝑦𝑦₎₎ _(2.6)

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

+∫ ∫_𝛽𝛽t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) +𝑒𝑒_𝑖𝑖(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡)𝑢𝑢𝑞𝑞(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡)]𝛼𝛼_𝑖𝑖′(𝑥𝑥)𝑑𝑑𝑡𝑡

𝑖𝑖(𝑦𝑦0)

Using (2.6) in above inequality,we deduce

𝐷𝐷1𝑧𝑧(𝑥𝑥,𝑦𝑦)≤ 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)[𝜑𝜑−1(𝑧𝑧(𝑥𝑥,𝑦𝑦))]𝑞𝑞� � [𝑓𝑓𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡) 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

× (𝜓𝜓(𝑢𝑢(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡)) +∫ ∫_𝛽𝛽t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) +𝑒𝑒_𝑖𝑖(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡)]𝛼𝛼_𝑖𝑖′(𝑥𝑥)𝑑𝑑𝑡𝑡

𝑖𝑖(𝑦𝑦0)

𝛼𝛼𝑖𝑖(𝑥𝑥0) (2.7)

Using the monotonicity of 𝜑𝜑−1 and 𝑧𝑧, we deduce

[𝜑𝜑−1₍_𝑧𝑧₍_𝑥𝑥_,_𝑦𝑦_))]𝑞𝑞_≥_[_𝜑𝜑−1₍_𝑧𝑧₍_𝑥𝑥

0,𝑦𝑦0))]𝑞𝑞 = [𝜑𝜑−1(𝑎𝑎(𝑥𝑥,𝑦𝑦))]𝑞𝑞> 0

From the definition of 𝐺𝐺 and (2.7), we have

𝐷𝐷1𝐺𝐺�𝑧𝑧(𝑥𝑥,𝑦𝑦)�=_[_𝜑𝜑₋𝐷𝐷₁1₍𝑧𝑧_𝑧𝑧(₍𝑥𝑥_𝑥𝑥,_,𝑦𝑦_𝑦𝑦)_))]_𝑞𝑞

≤ 𝑐𝑐(𝑥𝑥̅,𝑦𝑦�)� � [𝑓𝑓𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡) 𝛽𝛽𝑖𝑖(𝑦𝑦)

n

i=1

× (𝜓𝜓(𝑢𝑢(𝛼𝛼(𝑥𝑥),𝑡𝑡)) +∫ ∫_𝛽𝛽t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) +𝑒𝑒_𝑖𝑖(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡)]𝛼𝛼_𝑖𝑖′(𝑥𝑥)𝑑𝑑𝑡𝑡

𝑖𝑖(𝑦𝑦0)

Keeping 𝑦𝑦 fixed in (2.8) and integrating from 𝑥𝑥₀ to 𝑥𝑥, 𝑥𝑥 ∈ 𝐽𝐽₁ and making change of variable on right-hand side, we have

𝐺𝐺(𝑧𝑧(𝑥𝑥,𝑦𝑦)≤ 𝐺𝐺(𝑎𝑎(𝑥𝑥,𝑦𝑦) +𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫ni=1 _𝛼𝛼𝛼𝛼_𝑖𝑖𝑖𝑖₍(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎[𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡) × (𝜓𝜓(𝑢𝑢(𝑠𝑠,𝑡𝑡)) +∫ ∫_𝛽𝛽t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎) +𝑒𝑒_𝑖𝑖(𝑠𝑠,𝑡𝑡)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

𝑖𝑖(𝑦𝑦0)

s

(3)

𝐺𝐺(𝑧𝑧(𝑥𝑥,𝑦𝑦)≤ 𝐺𝐺(𝑎𝑎(𝑥𝑥,𝑦𝑦) +𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫ ∫𝛽𝛽𝛽𝛽𝑖𝑖(𝑖𝑖(𝑦𝑦𝑦𝑦0))𝑒𝑒𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 ∑𝑖𝑖𝑛𝑛=1∫_𝛼𝛼𝛼𝛼_𝑖𝑖𝑖𝑖₍(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎𝑒𝑒𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 𝛼𝛼𝑖𝑖(𝑥𝑥)

n

i=1

+𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫ ∫_𝛽𝛽𝛽𝛽𝑖𝑖(𝑦𝑦)[𝑓𝑓_𝑖𝑖(𝑠𝑠,𝑡𝑡) × (𝜓𝜓(𝑢𝑢(𝑠𝑠,𝑡𝑡))

𝑖𝑖(𝑦𝑦0)

n i=1

+∫ ∫t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 𝛽𝛽𝑖𝑖(𝑦𝑦0)

s

Now, define a function 𝑘𝑘(𝑥𝑥,𝑦𝑦) by

𝑘𝑘(𝑥𝑥,𝑦𝑦) =𝐺𝐺(𝑎𝑎(𝑥𝑥,𝑦𝑦) +𝑐𝑐(𝑥𝑥,𝑦𝑦)� � � 𝑒𝑒𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 𝛽𝛽𝑖𝑖(𝑦𝑦�)

𝛼𝛼𝑖𝑖(𝑥𝑥̅)

n

i=1

+𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫i=1n _𝛼𝛼𝛼𝛼_𝑖𝑖𝑖𝑖₍(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎[𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(𝜓𝜓(𝑢𝑢(𝑠𝑠,𝑡𝑡)) +∫_𝛼𝛼s_𝑖𝑖₍_𝑥𝑥₀₎∫_𝛽𝛽t_𝑖𝑖₍_𝑦𝑦₀₎ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

Then

𝑘𝑘(𝑥𝑥0,𝑦𝑦) =𝐺𝐺(𝑎𝑎(𝑥𝑥,𝑦𝑦) +𝑐𝑐(𝑥𝑥,𝑦𝑦)� � � 𝑒𝑒𝑖𝑖(𝑠𝑠,𝑡𝑡)𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠 𝛽𝛽𝑖𝑖(𝑦𝑦�)

𝛼𝛼𝑖𝑖(𝑥𝑥̅)

n

i=1

and (2.9) can be restated as

𝑧𝑧(𝑥𝑥,𝑦𝑦)≤ 𝐺𝐺−1[𝑘𝑘(𝑥𝑥,𝑦𝑦)] (2.10)

We know 𝑢𝑢(𝑥𝑥,𝑦𝑦)≤ 𝜑𝜑−1(𝑧𝑧(𝑥𝑥,𝑦𝑦))≤ 𝜑𝜑−1(𝐺𝐺−1(𝑘𝑘(𝑥𝑥,𝑦𝑦))), also 𝜓𝜓 is nondecreasing, we obtained

𝜓𝜓[𝑢𝑢(𝜎𝜎,𝜂𝜂)]≤ 𝜓𝜓[𝜑𝜑−1₍_𝑧𝑧₍_𝜎𝜎_,_𝜂𝜂_))]_{≤ 𝜓𝜓}_[_𝜑𝜑−1₍_𝑧𝑧₍_𝛼𝛼

𝑖𝑖(𝑡𝑡),𝑡𝑡))]≤ 𝜓𝜓[𝜑𝜑−1(𝐺𝐺−1(𝑘𝑘(𝑠𝑠,𝑡𝑡)))]

for 𝜎𝜎 ∈[𝛼𝛼_𝑖𝑖(𝑡𝑡₀),𝛼𝛼_𝑖𝑖(𝑡𝑡)]and 𝜂𝜂 ∈[𝛽𝛽_𝑖𝑖(𝑡𝑡₀),𝛽𝛽_𝑖𝑖(𝑡𝑡)].

It is easy to observe that 𝑘𝑘(𝑥𝑥,𝑦𝑦) is continuous nondecreasing function for all 𝑥𝑥 ∈ 𝐽𝐽₁,𝑦𝑦 ∈ 𝐽𝐽₂.

𝐷𝐷1(𝑘𝑘(𝑥𝑥,𝑦𝑦)) =𝑐𝑐(𝑥𝑥,𝑦𝑦) ∑ ∫ni=1 _𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎[𝑓𝑓𝑖𝑖(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡) × (𝜓𝜓(𝜑𝜑−1�𝑧𝑧(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡)� +∫ ∫t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝜓𝜓(𝑢𝑢(𝜎𝜎,𝜂𝜂))𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)𝑑𝑑𝑡𝑡]𝛼𝛼′_𝑖𝑖(𝑥𝑥)𝑑𝑑𝑡𝑡

𝐷𝐷1(𝑘𝑘(𝑥𝑥,𝑦𝑦))≤ 𝑐𝑐(𝑥𝑥,𝑦𝑦) 𝜓𝜓(𝜑𝜑−1(𝐺𝐺−1(𝑘𝑘(𝛼𝛼𝑖𝑖(𝑥𝑥),𝑡𝑡))))

+∑ ∫ [𝑓𝑓_𝑖𝑖(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡) × (1 +∫ ∫_𝛽𝛽t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝛼𝛼′_𝑖𝑖(𝑥𝑥)𝑑𝑑𝑡𝑡

𝑖𝑖(𝑦𝑦0)

𝛽𝛽𝑖𝑖(𝑦𝑦)

n

𝑖𝑖=1

From definition of Φ and using the monotonicity of 𝜑𝜑−1,𝜓𝜓,𝐺𝐺−1 and 𝑘𝑘, we deduce

𝑑𝑑

𝑑𝑑𝑥𝑥 �Φ�𝑘𝑘(𝑥𝑥,𝑦𝑦)��=

𝐷𝐷1�𝑘𝑘(𝑥𝑥,𝑦𝑦)� 𝜓𝜓 �𝜑𝜑−1_�𝐺𝐺−1_�𝑘𝑘₍_𝑥𝑥_,_𝑦𝑦₎_��

≤ 𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫ [𝑓𝑓_𝑖𝑖(𝛼𝛼_𝑖𝑖(𝑥𝑥),𝑡𝑡) × (1 +∫ ∫_𝛽𝛽t ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝛼𝛼′_𝑖𝑖(𝑥𝑥)𝑑𝑑𝑡𝑡

𝑖𝑖(𝑦𝑦0)

n

i=1 (2.11)

Integrating the inequality in (2.11) from 𝑥𝑥₀ to 𝑥𝑥 and making change of variable on right-hand side, we deduce

Φ(𝑘𝑘(𝑥𝑥,𝑦𝑦))≤ Φ(𝑘𝑘(𝑥𝑥0,𝑦𝑦)) +𝑐𝑐(𝑥𝑥,𝑦𝑦)� � � [𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡) × (1� � ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠

t

s

n

i=1

Using (2.6), (2.10) and above inequality, we get

𝑢𝑢(𝑥𝑥,𝑦𝑦)≤ 𝜑𝜑−1_{_𝐺𝐺−1_[_Φ−1₍_Φ₍_𝑘𝑘₍_𝑥𝑥

0,𝑦𝑦)) +𝑐𝑐(𝑥𝑥,𝑦𝑦)∑ ∫ni=1 _𝛼𝛼𝛼𝛼_𝑖𝑖𝑖𝑖₍(_𝑥𝑥𝑥𝑥₀)₎∫_𝛽𝛽𝛽𝛽_𝑖𝑖𝑖𝑖₍(_𝑦𝑦𝑦𝑦₀)₎[𝑓𝑓𝑖𝑖(𝑠𝑠,𝑡𝑡)(1 +∫_𝛼𝛼𝑠𝑠_𝑖𝑖₍_𝑥𝑥₀₎‍∫_𝛽𝛽𝑡𝑡_𝑖𝑖₍_𝑦𝑦₀₎‍ℎ𝑖𝑖(𝜎𝜎,𝜂𝜂)𝑑𝑑𝜂𝜂𝑑𝑑𝜎𝜎)]𝑑𝑑𝑡𝑡𝑑𝑑𝑠𝑠)]} (2.12)

Taking 𝑥𝑥=𝑥𝑥̅,𝑦𝑦=𝑦𝑦� in the inequality (2.12), since 𝑥𝑥̅ and 𝑦𝑦� are arbitrary, we get required inequality. If 𝑎𝑎(𝑥𝑥,𝑦𝑦) = 0, we carry out the above procedure with 𝜀𝜀> 0 instead of 𝑎𝑎(𝑥𝑥,𝑦𝑦) and subsequently letting 𝜀𝜀 →0. Hence proof is completed.

Remark:(1) If we put ℎ_𝑖𝑖(𝜎𝜎,𝜂𝜂) = 0 in above theorem, we get Theorem 2.1 in [9]

3. APPLICATION

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(𝜑𝜑(𝑧𝑧(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

(4)

with the initial boundary conditions

𝜑𝜑(𝑧𝑧(𝑥𝑥,𝑦𝑦0)) =𝑒𝑒1(𝑥𝑥), 𝜑𝜑(𝑧𝑧(𝑥𝑥0,𝑦𝑦)) =𝑒𝑒2(𝑦𝑦), 𝑒𝑒1(𝑥𝑥0) =𝑒𝑒2(𝑦𝑦0) = 0

where 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 0 < (𝑙𝑙_𝑖𝑖)′(𝑥𝑥)≤1,0 < (𝐷𝐷_𝑖𝑖)′(𝑦𝑦)≤1,𝑙𝑙_𝑖𝑖(𝑥𝑥₀) =𝑥𝑥₀,𝐷𝐷_𝑖𝑖(𝑦𝑦₀) =𝑦𝑦₀,𝑖𝑖= 1,⋅⋅⋅,𝑛𝑛.

The following theorem deals with a boundedness on the solution of the problem (3.1)

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),𝑒𝑒_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|)) +𝑒𝑒𝑖𝑖(𝑥𝑥,𝑦𝑦)|𝑢𝑢𝑖𝑖|𝑞𝑞} (3.2)

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|) (3.3) and

|𝑒𝑒1(𝑥𝑥) +𝑒𝑒2(𝑦𝑦)|≤ 𝑎𝑎(𝑥𝑥,𝑦𝑦) (3.4)

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−𝑙𝑙′𝑖𝑖(𝑥𝑥), 𝑁𝑁𝑖𝑖= max𝑦𝑦∈𝐽𝐽2

1

1−𝐷𝐷′𝑖𝑖(𝑦𝑦), 𝑖𝑖= 1,2,⋅⋅⋅,𝑛𝑛 (3.5)

if 𝑧𝑧(𝑥𝑥,𝑦𝑦) is any solution of problem (3.1) with the initial boundary condition, then

|𝑧𝑧(𝑥𝑥,𝑦𝑦)|≤ 𝜑𝜑−1_{_𝐺𝐺−1_[_Φ−1₍₍_𝑘𝑘₍_𝑥𝑥 0,𝑦𝑦))

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

t

𝑦𝑦0−𝐷𝐷𝑖𝑖(𝑦𝑦0)

s

𝑥𝑥0−𝑙𝑙𝑖𝑖(𝑥𝑥0)

�

𝑦𝑦−𝐷𝐷𝑖𝑖(𝑦𝑦)

𝑥𝑥−𝑙𝑙𝑖𝑖(𝑥𝑥)

n=1

i=1

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

𝑘𝑘(𝑥𝑥0,𝑦𝑦) = [𝐺𝐺|𝑎𝑎(𝑥𝑥,𝑦𝑦)|] +� � � 𝑒𝑒𝑖𝑖(𝜎𝜎,𝜏𝜏)𝑑𝑑𝜏𝜏𝑑𝑑𝜎𝜎. 𝑦𝑦−𝐷𝐷𝑖𝑖(𝑦𝑦)

𝑥𝑥−𝑙𝑙𝑖𝑖(𝑥𝑥)

𝑥𝑥0−𝑙𝑙𝑖𝑖(𝑥𝑥0)

n

i=1 𝑓𝑓𝑖𝑖 =𝑓𝑓𝑖𝑖(𝜎𝜎+𝑙𝑙𝑖𝑖(𝑠𝑠),𝜏𝜏+𝐷𝐷𝑖𝑖(𝑡𝑡))𝑀𝑀𝑖𝑖𝑁𝑁𝑖𝑖

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

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

𝜎𝜎,𝑠𝑠,𝜙𝜙,𝜉𝜉 ∈ 𝐽𝐽1,𝜏𝜏,𝑡𝑡,𝛾𝛾,𝜂𝜂 ∈ 𝐽𝐽2.

Proof: It is easy to see that the solution 𝑧𝑧(𝑥𝑥,𝑦𝑦) of problem (3.1) with the initial boundary condition satisfies the equivalent integral equation

𝜑𝜑(𝑧𝑧(𝑥𝑥,𝑦𝑦)) =𝑒𝑒1(𝑥𝑥) +𝑒𝑒2(𝑦𝑦) +� � 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 (3.6)

Using (3.2), (3.3), (3.4), (3.5) and making change of variable on right hand side of the above inequality, we have

�𝜑𝜑�𝑧𝑧(𝑥𝑥,𝑦𝑦)�� ≤ 𝑎𝑎(𝑥𝑥,𝑦𝑦) +� � � [|𝑧𝑧(𝜎𝜎,𝜏𝜏)|𝑞𝑞_𝑓𝑓 𝑖𝑖(𝜎𝜎,𝜏𝜏) 𝑦𝑦−𝐷𝐷𝑖𝑖(𝑦𝑦)

𝑥𝑥0−𝐷𝐷𝑖𝑖(𝑦𝑦0)

x−li(x)

x0−li(x0)

n

i=1

0−𝐷𝐷𝑖𝑖(𝑦𝑦0)

𝜎𝜎

𝑥𝑥0−𝑙𝑙𝑖𝑖(𝑥𝑥0) (3.7)

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

(5)

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

𝜎𝜎,𝑠𝑠,𝜙𝜙,𝜉𝜉 ∈ 𝐽𝐽1,𝜏𝜏,𝑡𝑡,𝜈𝜈,𝜂𝜂 ∈ 𝐽𝐽2.

Now, immediate application of the inequality established in Theorem 2.1 to the inequality (3.7) yields the result.

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