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International Journal of Engineering Research and Applications (IJERA) um www.ijera.com
20. Feb 2013
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### Gz3113501354

1. Samuel A. Iyase / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.1350-1354 On The Existence Of Periodic Solutions Of Certain Fourth Order Differential Equations With Dealy Samuel A. Iyase Department of Mathematics, Computer Science and Information Technology, Igbinedion University, Okada, P.M.B. 0006, Benin City, Edo State, Nigeria. Abstract We derive existence results for the and Lk([0, 2 ]) of continuous, k times periodic boundary value problem continuously differentiable or measurable real x (iv )  a  f ( x)   cx  g (t , x( x   )  p(t ) x  x  functions whose kth power of the absolute value is Labesgue integrable. x(0) = x(2π), x (0) = x (2  ),  0) =  ( 2  ),   x x  (0) =  (2  ) x x We shall also make use of the sobolev space using degree theoretic methods. The uniqueness of defined by periodic solutions is also examined. H 2k = { {x : [0,2 ]  R x, x are  Keywords and Phrases: “Periodic Solutions, absolutely continuous on [ 0,2 ] and, Caratheodony Conditions, Fourth Order  L2[ 0,2 ] 2 Differential Equations with delay. 2000 x with norm x H2 = 2 Mathematics Subject Classification: 34B15; 2 1 2 1 2   2 34C15, 34C25. ( x 2 (t )dt ) 2  x i (t ) dt . 2 o 2 i 1 o 1. Introduction i d x In this paper we study the periodic boundary value xi  problem dt i + c x + g(t, x(t - )] = p(t) ( iv) x + ax  f ( x )    x  2. The Linear cases (1.1) In this section we shall first consider the x(0) = x(2π), x (0) = x (2    ),  0) =  ( 2  ), x x equation:  (0) =  (2  ) x x x(iv) (t) + a  (t) + b  (t) + c x (t) + dx(t- ) = 0 x x  with fixed delay   [0,2 ) Where c ≠ 0 is a (2.1) constant, p: [0, 2 ]→R and g: [0, 2 ]x  → x(0) = x(2  ), x (0) = x (2  ),  (0) =  (2  ),   x x  are 2π periodic in t and g satisfies certain  (0) =  (2  ) x x Caratheodory conditions. The unknown function x: [0, 2 ]→R is defined Where a, b, c, d, are constants Lemma 2.1 Let c ≠ 0 and Let a/c < 0 for 0 < t <  by x(t -  ) = x( 2 -(t- ) Suppose that: The differential equation x  a .+ b  + ( iv ) x x 0 < d/c < n, n ≥ 1 (2.2) h(x) x + g(t, x(t -  )) = p(t)  Then (2.1) has no non-trivial 2  periodic solution (1.2) for any fixed   [0,2 ). In which b < 0 is a constant was the object of a recent study [6]. Proof Results on the existence and uniqueness of t By substituting x(t) = e with  = in, i2 = -1. 2 periodic solutions were established subject to We can see that the conclusion of the Lemma is certain resonant conditions on g. Fourth order true if Ф(n,  ) = an 3 – cn + d Sin n  ≠ 0 for all n differential equations with delay occur in a variety of physical problems in fields such as Biology, ≥ 1 and   [0, 2 ) (2.3) Physics, Engineering and Medicine. In recent year, By (2.2) we have there have been many publications involving a d differential equation with delay; see for example c-1 ф(n, ) = n3-n + Sin n  ≤ c c [1,2,4,5,6,8,9]. However, as far as we know, there a 3 d a are few results on the existence and uniqueness of n  n   n3  0 periodic solution to [1.1]. c c c In what follows we shall use the spaces Therefore (n, ) ≠0 and the result follows C([0, 2 ]), Ck([0, 2 ]) ‘ 1350 | P a g e
2. Samuel A. Iyase / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp. L1 [0, 2 ] we shall write If x  = 2 2 2 1 2 1 1 x 2 0 x(t )dt, ~(t )  x(t )  x  ~ (t )dt   (t )~x(t   )dt  2  (t ) x dt  x2  2 x x 2 o 0 0 So that 2 1  (t ) x~(t   )dt 2  0 ~ (t ) dt  0 x 2 o x We shall consider next the delay equation Using the identity  a  b  cx  d (t ) x(t   ) = 0 ( iv ) x x x  ( a  b) 2 a2 b2 ab    (2.4) 2 2 2 We get x(0) = x( 2 ), x (0) =,  x ( 2 ),  (0) =  ( 2 ),  x x =  (0) =  ( 2 ) x x 1 2 1 2 1 2  ~ (t )dt  2  x2  (t ) x dt  2  (t ) x~(t   )dt 2 Where a , b, c are constants and d  1 x L 2 2 0 0 0 Here the coefficient d in (2.4) is not necessarily constant. We have he following results which apart from being of independent interest are also useful in the non-linear case involving (1.1) 1 + Lemma 2.2 Let c ≠0 and let a/c < 0 Set Γ(t) = 2 2   c-1d(t)  2 [ x(t   )  ~(t )]2 ~ 2 ~ 2 (t   ) x x x L  Suppose that 2   (t )   0 < Γ(t)<1 (2.5) 0 2 2 2 Then for arbitrary constant b the equation (2.4) x2   x~(t   )  dt 1 admits in H 2 only the trivial solution for every x   [0, 2 ). 2 We note that a and c are not arbitrary. 1 = 2 Proof   2 2 If x  H 2 is a possible solution of (2.4) then on ~ 2 (t )dt  1 (t ) ~ 2  ~ 2 (t   ) dt +    2 x x  1 x  2 0 multiplying (2.4) by x + ~ (t) and integrating over x 0  x(t   )  ~(t ) 2 [0, 2 ] noting that (t )  x 2 dt 1   2 x 1 2 2 2  (x  x(t))  ~ 0 2 0 Using (2.5) we get  c 1 x (iv )  a  b   21 x x  a  2 2 (t )dt x 0   c o 2 2 (t ) ~ 2 ~ 2  x (t )dt  21     1 ~2 x  x (t   ) dt 2 We have that 0 0 2 0= 2  From the periodicity of ~ we have that x (x  ~(t) c 1[ x (iv )  a  b   x  (t ) x(t   ) dt 1 2   x x x  2 2  x (t )dt   x (t   )dt 0  ~2  ~2 = - 0 0 2 2 Hence  ( x  x (t ))x  (t ) x(t   )dt  1 a ~2 1   (t )dt  2  x ~  2 2 c 0  [ ~  (t   ) (t ) ~ 2 (t   )]dt 1 1 2 0 0 2 2 x x + 0 2 2  (x   ~ 2 (t )  (t ) ~ 2 (t   )]dt   ( x  ~(t )){x(t )  (t ) x(t   )}dt  1 1 2 x  2 [ 21 x 0 o (2.6) Using (2.5) we can see that the last expression is non-negative hence 1351 | P a g e
3. Samuel A. Iyase / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp. 2 x (iv )  a  f ( x)   cx  g (t , x(t   ))  p(t ) x  x  0  1 [ 21 2   ( ~ (t   )  (t ) ~ (t   )]dt 0  x2 x2 (3.1)   ~ H1 x(0)  x(2 ), x(0)  x(2 ), (0)  (2 ),(0)  (2 ) 2 x   x x x x 2 By Lemma 1 of [8] where  > 0 is a constant. where f :    is a continuous function and This implies ~ = 0 a .e and that x = x . x But a constant map cannot be a solution of (2.4) g : [0,2 ]x   is such since (t )  0 that g(. x) is a measurable on 0,2  for each Thus x = 0 x   and g (t ,.) is continuous on  for almost each t [0,2  ] Theorem 2.1 Let all the conditions of Lemma 2.2 hold and let We assume moreover that for each r > 0  be related to  by Lemma 2.2.Suppose there exists Yr 1 L 2 such that V  L 2 2 further that satisfies g (t, x)  r (t ) (3.2) 0  V (t )  (t )   a.e t  0,2 where    > for a.e t  [0,2 ] and all x [r , r ] such a g is 0 then said to satisfy the 2  ( x  ~(t ))c Caratheodory’s condition. [ x (iv )  a  f ( x) ] x  V (t ) x(t   ) dt 1  1 x x  x  2 0 Theorem 3.1 2 Let c ≠ 0 and let a/c < 0  (   ) ~ x 1 (2.7) Suppose that g is a caratheodory function H 2 satisfying the inequalities Proof c 1 xg (t , x)  0 ( x  r) (3.3) We have from the proof of Lemma 2.2 that g (t , x ) 2  (t ) ( x  ~(t ))c 1 x (iv )  a  f ( x)   x  V (t ) x(t   ) dt Lim sup (3.4) 1 cx  x 2   x x  x  0 2 2  ~ 2 (t   )  V (t ) ~ 2 (t   ) dt  1 ( 1  (~Uniformly(ta.e. (t ))[0,2 ] 1 1     ( x x x (t )  V ) ~ t dt 2 x 2 where r > 0 is 2 2 0 2 2 0 constant and   L 2 is such that 2 0< 2 2 1  ( 21  ( ~ 2 (t   )  (t ) ~ 2 (t   ))dt   ( 21  x x 2  ~ (t   ))dt x 2 Suppose p  L2  is 2 such that 2 2 0 0 1 2 p 2  0 p(t )dt = 0 then for arbitrary 1 1  (x   + ( ~ 2 (t )  (t ) ~ 2 (t ))dt   ( x 1 continuous function f the boundary value 2 2 2 2 problem (3.1) has at least one solution for every   [0,2 ) 0 2  x (t )dt  ~2 Proof 0 From condition (2.5) , Lemman 2.2 and Let be related to as in Lemma Wirtinger’s inequality we have 2.2 so that by (3.3) and (3.4) there exists a constant R1 such that for 1 x    ~ H1   ~ x   ~ H 1   ~ H 1  (   ) ~ H 1 0 x x x (3.5) 2 L2  2 2 2 2 Define by 3. The Non Linear Case We shall consider the non-linear delay equation 1352 | P a g e
4. Samuel A. Iyase / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp. (cx) 1 g (t , x)   ~ 2  (   p )( x  ~ ) x H1 x  1 2 2 2 2 2 ~ (t , x)  (cR1 ) g (t , R1 )  ~ 2  ( x  ~ x H1 x H1 ) y  1 2 2 2  (cR1 ) g (t , R1 ) Thus (t )  ~ 2  2 ( x  ~ 2 ) x H1 x H1 (3.11) x  R1 2  2 With  0 independent of x and  0  x  R1 .Integrating (3.9) over [0,2π}We obtain  R1  x  0 2 2 (1   )  (t ) x(t   )dt  c   g (t , x(t   )) 1 x0 0 0 (3.6) (3.12) Then Since Γ(t) > 0 we derive that 0 < ~(t , x) ≤ Γ(t) + y 2 (3.7) 1 2  (t )dt    0 0 (3.13) for a .e t  [0,2 ] and all x   . Moreover the Hence if x(t) > r for all t  [0,2π], (3.3) and (3.12) function ~(t , x) satisfy Caratheodory’s conditions y implies that (1   )   0 contradicting   0. and : [0,2 π] x  →  defined by Similarly if x(t) < - r for all t  [0,2π] we reach a contradiction. g (t , x(t   ))  g (t , x(t   ))  cx(t   ) ~(t , x(t   )) ~ y Thus there exists a, t1,  [0,2π] such that (3.8) x(t1 )  r. Le t2 be such that is such that for a.e t  [0,2π] and all x   . ~(t, x(t -  ))   (t ) for some  (t )  g t2 x = x(t2 )  x(t1 )   x( s)ds.  This Let   [0,1] be such that t1 + a + λf( x )  ] + x +(1 - λ) Γ(t)x(t-τ) + λ ~ x  r  2 ~ H 1 (iv ) -1 c [x x  x  y implies that x 2 (t, x(t-τ)) x(t   ) Substituting this in (3.11) we get x ~  c 1 (1   )b  c 1 g (t , x(t   ))  c 1p(t )  0 ~2 c ~ x H1 1 x H1 2 2 (3.9) For  = 0 we obtain (2.1) which by Lemma 2.2 or ~ H 1  c1 , c1  0 x (3.14) 2 admits only the trivial solution Now For λ = 1 we get the original equation (1.1). To prove that equation (3.1) has at least one solution, we show according to the Leray-Shauder Method x H 1  x  ~ H 1  r  (2  1)c1  c 2 x 2 2 that the possible solution of the family of (3.15) equations (3.9) are apriori bounded in C3[0,2π] Thus independently of   [0,1]. x 2  c3 , c3  0  (3.16) Notice that by (3.5) one has From (3.16)we have 0 < (1- λ) Γ(t) + λ ~ (t,x(t-τ)) < Γ(t) + y (3.10) x   c4 , c4  0 (3.17) Then using Theorem 2.1 with V(t) = (1- λ)Γ(t) + λ ~ (t, x(t-τ)) and Cauchy Schwarz inequality we y  Multiplying (3.9) by - x (t ) and integrating over get [0,2π] we have 1  2  a ( x 2  1   x 2 x    2 x 2  p 2 x 2 ) 2 2 0 = x     2 2 1  ( x  ~(t )){c  [ x (iv )  a  f ( x) ]  x  1 x x  x  2 0 Hence ~ ~ (1-λ)Γ(t) x(t-τ) +  (t,x(t-τ)+ g (t , x(t   )  2  c5 , c5  0 x (3.18) +(1-λ) b  - λp(t)}dt. And thus x x   c6 , c6  0  (3.19) 1353 | P a g e
5. Samuel A. Iyase / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp. Multiplying (3.9) by  (t ) and integrating over x Where  (t )  L2  is defined by 2 [0,2π]  (t ) = We get g (t , x1 (t   ))  g (t , x 2 (t   )) 1 x 2  f ( )   2  1   2  2 x   c  2  2  p 2  2  b  2 c( x1  x 2 ) 2 2  x x x x x x If u = x1 - x2 ≠ 0 and since 0 < β(t)  Γ(t) for Thus a.e t  [0,2π] then using the arguments of theorem 2.1 we have that u = 0 and thus x1  2  c7 , c7  0 x (3.20) = x2 a. e. And hence    c8 , c8  0 x (3.21) REFERENCES [1] F. Ahmad. Linear delay differential Also equations with a positive and negative x (iv )  c9 , c9  0 (3.22) term. Electronic Journal of Differential  Equations. Vol. 2003 (2003) No. 9, 1-6. Since (0)  (2 ) there exists to x x  [0,2π] [2] J.G. Dix. Asymptotic behaviour of Such that (t o )  0 Hence x solutions to a first order differential equations with variable delays.    c10 ,c10 > 0 x (3.23) Computer and Mathematics with applications Vol 50 (2005) 1791 - 1800 From (3.17), (3.19), (3.22) and (3.23) our result [3] R. Gaines and J. Mawhin, Coincidence follows. degree and Non-linear differential equations, Lecture Notes in Math, 4. Uniqueness Result No.568 Springer Berlin, (1977). [4] S.A. Iyase: On the existence of periodic If in (1.1), f ( x)  b a constant. The following  solutions of certain third order Non- uniqueness results holds. linear differential equation with delay. Journal of the Nigerian Mathematical Theorem 4.1 Society Vol. 11, No. 1 (1992) 27 - 35 Let a, b, c, be constants with c ≠ 0 a/c < 0. [5] S.A. Iyase. Non-resonant oscillations for Suppose g is a caratheodony function satisfying some fourth-order differential equations with delay. Mathematical Proceedings of theRoyal Irish Academy, Vol.99A, No.1, g (t , x1 (t   ))  g (t , x 2 (t   )) 1999, 113-121 0  (t ) [6] S.A. Iyase and P.O.K. Aiyelo, Resonant c( x1  x 2 ) oscillation of certain fourth order For a.e., t  [0,2π] and all x1, x2 R x1 ≠ x2 Nonlinear differential equations with where Γ  L delay, International Journal of 2 2 Mathematics and Computation Vol.3 No. Then the boundary value problem J09, June 2009 p67-75 [7] Oguztoreli and Stein, An analysis of x iv  a  b  cx  g (t , x(t   ))  p(t ) x x  oscillation in neuromuscular systems. Journal of Mathematical Biology 2 1975 x(0)  x(2 ), x(0)  x(2 ), (0)  (2 ),(0)  (2 )   x x x x 87-105. (4.1) [8] E.De. Pascal and R. Iannaci: Periodic has art most one solution. solutions of generalized Lienard equations with delay, Proceedings equadiff 82, Proof Wurzburg (1982) 148 - 156 Let u = x1 –x2 for any two solutions x1, x2 of (4.1). [9] H.O. Tejumola, Existence of periodic Then u satisfies the boundary value problem solutions of certain third order non-linear c 1[u (iv )  a  bu]  u   (t )u(t   )  0 u   differential equations with delay. Journal of Nigerian Mathematical Society Vol. 7 u(0)  u(2 ), u(0)  u(2 ), u(0)  u(2 ),(0)  (2 )     u u (1988) 59-66 1354 | P a g e
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