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Communications in Applied Analysis

13

(2009), no. 1, 93–104

ON THE OSCILLATIONS OF FOURTH ORDER FUNCTIONAL

DIFFERENTIAL EQUATIONS

SAID R. GRACE1

, RAVI P. AGARWAL2

, AND SANDRA PINELAS3

1

Department of Engineering Mathematics, Faculty of Engineering Cairo University, Orman, Giza 12221, Egypt

E-mail: srgrace@eng.cu.eg

2

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, U. S. A.

E-mail: agarwal@fit.edu

3

Department of Mathematics, Azores University, R. M˜ae de Deus 9500-321 Ponta Delgada, Portugal

E-mail: sandra.pinelas@clix.pt

ABSTRACT.We establish some sufficient conditions for the oscillations of all solutions of fourth

order functional differential equations

d dt

a(t)

d3

dt3x(t)

α

+q(t)f(x[g(t)]) = 0

and

d dt

a(t)

d3

dt3x(t)

α

=q(t)f(x[g(t)]) +p(t)h(x[σ(t)])

whenR∞

a−1/α(s)ds <. The case whenR∞

a−1/α(s)ds=is also included.

1.

Introduction

This paper deals with the oscillatory behavior of solutions of fourth order

func-tional differential equations

d

dt

a

(t)

d

3

dt

3

x

(t)

α

+

q

(t)

f

(x

[g

(t)]) = 0

(1)

and

d

dt

a

(t)

d

3

dt

3

x

(t)

α

=

q

(t)

f

(x

[g

(t)]) +

p

(t)

h

(x

(t)])

(2)

where the following conditions are assumed to hold:

(i)

α

is the ratio of two positive odd integers;

(ii)

a

(t),

p

(t) and

q

(t)

C

([t

0

,

)

,

(0,

)) ;

(iii)

g

(t) and

σ

(t)

C

1

([t

0

,

)

,

R

),

g

(t)

< t,

σ

(t)

> t,

g

(t)

>

0 and

σ

(t)

>

0 for

t

>

t

0

and lim

t→∞

g

(t) =

;

Referências

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