Oscillatory Behavior of Solutions for Forced Second Order Nonlinear Functional Integro-Dynamic Equations on Time Scales

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1 J. An. Num. Theor. 4, No. 2, 5- (26) 5 Journl of Anlysis & Number Theory An Interntionl Journl Oscilltory Behvior of Solutions for Forced Second Order Nonliner Functionl Integro-Dynmic Equtions on Time Scles H. A. Agw, Ahmed M. M. Khodier Heb A. Hssn Deprtment of Mthemtics, Fculty of Eduction, Ain Shms University, Roxy, Ciro, Egypt. Received: 8 Feb. 26, Revised: 4 Jun. 26, Accepted: 2 Jun. 26 Published online: Jul. 26 Abstrct: In this pper, we del with the oscilltory behvior of forced second order nonliner functionl integro-dynmic equtions of the form (r(t)x (t)) = e(t)± p(t)x γ (τ(t)) k(t,s) f(s,x(τ(s))) s, (r(t)x (t)) = e(t) p(t)x(τ(t)) k(t,s) f(s,x(τ(s))) s, on time scles T, where r(t), p(t) e(t) re right dense continuous (rd-continuous) functions on T. Oscilltion behvior of these equtions dose not studied before. Our results improve extend some results estblished by Grce et l. 3]. We lso give some exmples to illustrte our min results. Keywords: Oscilltion, integro-dynmic equtions, time scles. Mthemtics Subject Clssifiction (2): 34N5, 34C, 45D5. Introduction In recent yers, there hs been n incresing interest in studying the oscilltion nonoscilltion of dynmic equtions on time scles Hilger introduced the theory of time scle which ws expected to unify continuous discrete clculus. We refer the reder to the books 8,9], ppers -3], 5-7], the references cited therein. Reserch on oscilltion theory for integro-dynmic equtions is limited due to lck of techniques vilble on time scles (see 4] -3]). The min gol of this pper is to estblish some new criteri for the oscilltory behvior of forced second order nonliner functionl integro-dynmic equtions on time sclestof the form (r(t)x (t)) = e(t)± p(t)x γ (τ(t)) k(t,s) f(s,x(τ(s))) s, (.) (r(t)x (t)) = e(t) p(t)x(τ(t)) k(t,s) f(s,x(τ(s))) s. (.2) Corresponding uthor e-mil: heb li7@yhoo.com c 26 NSP

2 6 H. A. Agw et l.: Oscilltory behvior of solutions for forced... 2 Some Preliminries on time scles A time scle T is n rbitrry nonempty closed subset of the rel numbers R. On ny time scle T, the forwrd bckwrd jump opertors re defined by σ(t)=inf{s T,s> t} ρ(t)=sup{s T,s< t}. A point t T is sid to be left-dense if ρ(t) = t, right-dense if σ(t) = t, left-scttered if ρ(t) < t, right-scttered if σ(t) > t. The grininess function µ for time scle T is defined by µ(t) = σ(t)t. The set T k is defined by T k =Tm if T hs left-scttered mximum m, Otherwise,T k =T. A function f : T R is clled rd-continuous provided tht it is continuous t right-dense points of T its left-sided limits exist t left-dense points oft. The set of ll rd-continuous functions is denoted by C rd (T,R). By Crd (T,R), we men the set of functions whose delt derivtive belong to C rd (T,R). A function f :T R is regressive provided tht µ(t) f(t) for ll t T k, holds. The set of ll regressive rd-continuous functions f :T R is denoted by R = R(T)=R(T,R). If q R, then we define the exponentil function e q (t,s) by e q (t,s)=exp( t s ξ µ(τ) (q(τ)) τ) for s,t T, where the cylinder function ξ h (z) is defined by ξ h (z)= h Log(zh). For function f :T R (the rngerof f my be ctully replced by ny Bnch spce), the delt derivtive f is defined by f (t)= f(σ(t)) f(t) σ(t)t, provided f is continuous t t t is right-scttered. If t is not right-scttered, then the delt derivtive f (t) is defined by f f(σ(t)) f(t) f(t) f(s) (t)= lim = lim s t t s s t t s provided tht this limit exists. A function f :, b] R is sid to be differentible if its derivtive exists. The derivtive f the shift f σ of function f re relted by f σ = f(σ(t))= f(t) µ(t) f (t). The delt derivtive rules of the product the quotient of two differentible functions f g re given by ( f.g) (t)= f (t)g(t) f σ (t)g (t)= f(t)g (t) f (t)g σ (t) ( f g ) (t)= f (t)g(t) f(t)g (t) g(t)g σ (t), gg σ. The integrtion by prts formul reds b f(t)g (t) t = f(t)g(t)] b b f (t)g σ (t) t or, b f σ (t)g (t) t = f(t)g(t)] b b f (t)g(t) t the infinite integrl is defined by 3 Bsic Lemms f(s) s= lim f(s) s. b b Lemm 3.(4]) If X Y re nonnegtive, then X λ (λ)y λ λxy λ, λ <, X λ (λ )Y λ λxy λ, λ >, with equlity holding iff X = Y. Lemm 3.2(]) Let y, f C rd (T,R), z R (T,R), z α R. If then y(t) α t f(s) s t z(ξ)y(ξ) ξ] s for ll t T, where p(t)= f(t) t z(s) s. 4 Min results y(t) αe p (t,t ) for ll t T, In this section, we give some new oscilltion criteri for equtions (.) (.2). We begin by introducing the clss of functions I which will be used in the proof of the first prt of this section. Let D = {(t,s) T T : t > s t }, D = {(t,s) T T : t s t }. A function H C rd (D,R) belongs to the clss I, if it stiesfies the following conditions: (C )H(t,t)=, t t, H(t,s)> on D, (C 2 )H hs non positive continuous -prtil derivtive H s(t,s) non negtive continuous second-order -prtil derivtive H 2 s 2 (t,s). (C 3 )H s(t,t)=, lim H s (t,t ) H(t,t o ) = O(). c 26 NSP

3 J. An. Num. Theor. 4, No. 2, 5- (26) / Oscilltion criteri for Eq. (.): In the following, we estblish oscilltion criteri for Eq. (.) subject to the following conditions: (H )r,e, p :T R k :T T R re rd-continuous, k(t,s) for t > s there exist rd-continuous functions,m :T (, ) such tht k(t,s) (t)m(s) for ll t s. (H 2 )γ is quotient of odd positive integers such tht <γ <. (H 3 ) f : T R R is continuous there exist rd-continuous function q : T (, ) rel number β with < β < such tht <x f(t,x) q(t) x β for x, t. (4.) Proof. Let x(t) be nonoscilltory solution of (.). We my ssume tht x(t) >, x(τ(t)) > for ll t >. Using H H 3 in (.), we hve e(t)=(r(t)x (t)) p(t)x γ (τ(t)) k(t, s) f(s, x(τ(s))) s where p (t)=mx{±p(t),}. Hence Setting, (r(t)x (t)) p (t)x γ (τ(t)) k(t, s) f(s, x(τ(s))) s k(t, s) f(s, x(τ(s))) s, (4.7) e(t) (r(t)x (t)) p (t)x γ (τ(t)) (t) m(s) f(s, x(τ(s))) s(t) m(s) f(s, x(τ(s))) s. (4.8) (H 4 )τ :T T lim τ(t)=. In the following, we denote A(t)=e(t)(β)β β/(β) (t) g β/(β) (s)m /(β) (s)q /(β) (s) s, (4.2) where g :, ) T (, ) is given rd- continuous function. then, k = mx{ f(t,x(τ(t))), t, ] T }< k 2 =k t m(s) s, e(t) (r(t)x (t)) p (t)x γ (τ(t))k 2 (t) (t) m(s)q(s)x β (τ(s)) s A solution x(t) of (.) or (.2) is sid to be oscilltory if, for every t >, we hve inf x(t)<<sup x(t). t t t t Otherwise, it is sid to be nonoscilltory. Theorem 4.Assume tht τ(t) t, H -H 4 hold there exists kernel function H(t,s) such tht H(t, ) limsup H(t, ) liminf H(t, ) where, (H s (t,s)) s, (4.3) H(t,σ(s))(s) s H(t,σ(s))A(s)k 2 (s)] s τ(t) H(t,σ(s))A(s)k 2 (s)] s τ(t) τ(u)g(u) u] s<, (4.4) G(t,s) s]=, (4.5) G(t,s) s]=, (4.6) G(t,s)=(γ )γ γ/(γ) (H s (t,σ (s))r(σ (s))) s (σ (s)) ] γ/(γ) (H(t,σ(τ (s)))p (τ (s))(τ (s)) ) /(γ )], σ (s) τ (s) re the inverse functions of σ(s) τ(s) respectively, p (t) = mx{±p(t),}. Then every solution x(t)=o(t) of Eq. (.) is oscilltory. Using Lemm 3., we get (r(t)x (t)) p (t)x γ (τ(t))k 2 (t) (t) g(s)x(τ(s)) m(s)q(s)x β (τ(s))] s(t) g(s)x(τ(s)) s. (4.9) g(s)x(τ(s))m(s)q(s)x β (τ(s)) (β )β β/(β) g β/(β) (s)m /(β) (s)q /(β) (s). (4.) Now, From (4.) in (4.9), we obtin A(t) (r(t)x (t)) p (t)x γ (τ(t))k 2 (t)(t) g(s)x(τ(s)) s. (4.) Multiplying (4.) by H(t, σ(s)) integrting from t to, we hve H(t, σ(s))a(s) s H(t, σ(s))(x (s)) s H(t, σ(s))p (s)x γ (τ(s)) s s k 2 H(t, σ(s))(s) s H(t, σ(s))(s) g(u)x(τ(u)) u] s. (4.2) Using integrtion by prts two times, we hve H(t,σ(s))(x (s)) s=h(t, )r( )x ( ) H s (t,s)x (s) s =H(t, )r( )x ( )H s (t, )r( )x( ) (H s (t,s)) s x(σ(s)) s = A(t, ) (H s (t,s)) s x(σ(s)) s, (4.3) where A(t, )=H(t, )r( )x ( )H s (t, )r( )x( ). c 26 NSP

4 8 H. A. Agw et l.: Oscilltory behvior of solutions for forced... From (4.3) in (4.2), we get H(t, σ(s))a(s) s A(t,t ) (H s (t, s)) s x(σ(s)) s Since τ(t) t, then H(t, σ(s))p (s)x γ (τ(s)) sk 2 H(t, σ(s))(s) s s H(t, σ(s))(s) g(u)x(τ(u)) u] s σ(t) = A(t, ) σ( ) (H s (t, σ (s))r(σ (s))) s x(s)(σ (s)) s τ(t) τ( ) H(t, σ(τ (s)))p (τ (s))x γ (s)(τ (s)) s s k 2 H(t, σ(s))(s) s H(t, σ(s))(s) g(u)x(τ(u)) u] s. (4.4) τ(t) H(t, σ(s))a(s) s A(t,t ) σ( ) (H s (t, σ (s))r(σ (s))) s x(s)(σ (s)) s τ(t) τ( ) H(t, σ(τ (s)))p (τ (s))x γ (s)(τ (s)) s s k 2 H(t, σ(s))(s) s H(t, σ(s))(s) g(u)x(τ(u)) u] s B(t, )k 2 H(t, σ(s))(s) s s H(t, σ(s))(s) g(u)x(τ(u)) u] s τ(t) (H s (t, σ (s))r(σ (s))) s (σ (s)) x(s) H(t, σ(τ (s)))p (τ (s))(τ (s)) x γ (s)] s, (4.5) where B(t, ) = A(t, ) σ( ) (H s (t,σ (s))r(σ (s))) s x(s)(σ (s)) s τ( ) H(t,σ(τ (s)))p (τ (s))x γ (s)(τ (s)) s. Applying Lemm 3., we get (H s (t, σ (s))r(σ (s))) s (σ (s)) x(s)h(t, σ(τ (s)))p (τ (s))(τ (s)) x γ (s) (γ )γ γ/(γ) (H s (t, σ (s))r(σ (s))) s (σ (s)) ] γ/(γ) (H(t, σ(τ (s)))p (τ (s))(τ (s)) ) /(γ ) = G(t, s) (4.5) becomes τ(t) H(t, σ(s))a(s) s B(t,t ) G(t, s) sk t 2 H(t, σ(s))(s) s H(t, σ(s))(s) s τ(u)g(u) x(τ(u)) τ(u) u] s. τ(t) H(t,σ(s))A(s)k 2 (s)] s G(t,s) s (4.6) s B(t, ) H(t,σ(s))(s) τ(u)g(u) x(τ(u)) τ(u) u] s. (4.7) Multiplying (4.6) by H (t, ), using (4.4), tking the lower limit of (4.6), we get contrdiction with(4.6). This completes the proof. Theorem 4.2Assume tht τ(t) t, H -H 4 hold there exists kernel function H(t,s) such tht limsup H(t, ) (H s (t,s)) s, (4.8) H(t,σ(s))(s) s τ(u)g(u) u] <, (4.9) τ(t) H(t,σ(τ (s)))s γ p (τ (s))(τ (s)) s< (4.2) H(t, ) t limsup liminf where, t H(t, ) t H(t, ) H(t,σ(s))A(s)k 2 (s)] s H(t,σ(s))A(s)k 2 (s)] s G(t, s)=(γ )γ γ/(γ) (H s (t, σ (s))r(σ (s))) s (σ (s)) ] γ/(γ) G(t,s) s] =, (4.2) τ( ) G(t,s) s] =, (4.22) τ( ) (H(t, σ(τ (s)))p (τ (s))(τ (s)) ) /(γ )], σ (s) τ (s) re the inverse functions of σ(s) τ(s) respectively, p (t) = mx{±p(t),}. Then every solution x(t) = O(t) of Eq. (.) is oscilltory. Proof. Let x(t) be nonoscilltory solution of (.). We my ssume tht x(t)>, x(τ(t))> for ll t. Proceeding s in the proof of Theorem 4. to get (4.4) σ(t) H(t,σ(s))A(s) s A(t, ) (H s (t,σ (s))r(σ (s))) s x(s)(σ (s)) s σ( ) τ(t) H(t,σ(τ (s)))p (τ (s))x γ (s)(τ (s)) s τ( ) since τ(t) t, then k 2 H(t,σ(s))(s) s s H(t,σ(s))(s) g(u)x(τ(u)) u] s. H(t, σ(s))a(s) s A(t,t ) σ( ) (H s (t, σ (s))r(σ (s))) s x(s)(σ (s)) s τ( ) H(t, σ(τ (s)))p (τ (s))x γ (s)(τ (s)) s τ(t) H(t, σ(τ (s)))p (τ (s))x γ (s)(τ (s)) s t s k 2 H(t, σ(s))(s) s H(t, σ(s))(s) g(u)x(τ(u)) u] s. τ(t) H(t,σ(s))A(s) s B(t, ) H(t,σ(τ (s)))p (τ (s))x γ (s)(τ (s)) s t (H s (t,σ (s))r(σ (s))) s x(s)(σ (s)) τ( ) H(t,σ(τ (s)))p (τ (s))x γ (s)(τ (s)) ] s s k 2 H(t,σ(s))(s) s H(t,σ(s))(s) g(u)x(τ(u)) u] s, where, B(t, )= A(t, ) τ( ) σ( ) (H s (t,σ (s))r(σ (s))) s x(s)(σ (s)) s. Therefore, H(t,σ(s))A(s)k 2 (s)] s G(t,s) s B(t, ) H(t,σ(s))(s) τ(t) t s τ( ) τ(u)g(u) x(τ(u)) τ(u) u] s H(t,σ(τ (s)))s γ p (τ (s))( x(s) s )γ (τ (s)) s, (4.23) Multiplying (4.23) by H (t, ), using (4.9), (4.2) tking the lower limit of (4.23), we get contrdiction with(4.22). This completes the proof. c 26 NSP

5 J. An. Num. Theor. 4, No. 2, 5- (26) / 9 Exmple 4Consider (T=R) the integro-differentil eqution x (t)=t 3 ±t 2 sintx γ x β (s) (t) (t 4 )(s 8 s, t, ) (4.24) where, <γ,β <. Here, r(t) =, e(t) = t 3, p(t) = t 2 sint, k(t,s) = (t 4 )(s 8 ), (t) =, m(s) =, f(x) = x β, τ(t) = t. To pply t 4 s 8 Theorem 4., let g(t) = m(t) H(t, s) = t s. Therefore (H (t,s)) =, s s H(t,s)(s) τ(u)g(u)du]ds= H(t,s)(s) u 7 du]ds = H(t,s)(s) 6 s 5 t 5 ]ds = (t s) 6 s 4 s 5 t 5 ]ds. H(t, ) H(t,s)(s) s τ(t) G(t,s)ds=, τ(u)g(u)du]ds<, H(t,s)A(s)k 2 (s)]ds = (t s)s 3 7 (β)β/(β) s t 7s4]k 2 s 4]ds s t. Therefore, Eq. (4.24) is oscilltory. 4.2 Oscilltion criteri for Eq. (.2): Here, we estblish oscilltion criteri for Eq. (.2) subject to the following conditions: (M )r,e, p :T R k :T T R re rd-continuous, r(t) >, k(t,s) for t > s there exist rd-continuous functions b,n :T (, ) such tht k(t,s) b(t)n(s) for ll t s. (M 2 ) f : T R R is continuous there exist rd-continuous function u : T (, ) rel number λ with λ > such tht x f(t,x) u(t) x λ for x, t. (4.25) (M 3 )τ :T Twith τ(t) t lim τ(t)=. In the following, we denote h ± (t)= e(t)±(λ )λ λ/(λ) b(t) v λ/(λ) (s)n /(λ) (s)u /(λ) (s) s, (4.26) where v C rd (T,(o, )). Now, we give sufficient conditions under which nonoscilltory solutions x(t) of (.2) stisfying x(t)=o() t. Theorem 4.3Let λ >, M M 3 hold for ll t > such tht s<. (4.27) If, t τ(s) t τ( ) lim inf t t s t b(ξ) ξ] s<, (4.28) p (τ (ξ))(τ (ξ)) ξ] s<. (4.29) s t h (ξ) ξ] s< t s t h (ξ) ξ] s>, (4.3) then every nonoscilltory solution x(t) of (.2) stisfies x(t)<. (4.3) Proof. Let x(t) be nonoscilltory solution of (.2). We my ssume tht x(t) >, x(τ(t)) > for ll t, for some >. Using M M 2 in (.2), we hve (r(t)x (t)) = e(t) p(t)x(τ(t)) where k(t,s) f(s,x(τ(s))) s k(t,s) f(s,x(τ(s))) s e(t) p(t)x(τ(t))b(t)c n(s) s b(t) n(s)u(s)x λ (τ(s)) s e(t) p (t)x(τ(t))c 2 b(t) b(t) n(s)u(s)x λ (τ(s)) s, (4.32) c := min{ f(t,x(τ(t))) : t, ] T }, c 2 =c n(s) s p (t)=mx{, p(t)}. c 26 NSP

6 H. A. Agw et l.: Oscilltory behvior of solutions for forced... (r(t)x (t)) e(t) p (t)x(τ(t))c 2 b(t) b(t) n(s)u(s)x λ (τ(s)) s e(t) p ( t)x(τ(t))c 2 b(t) b(t) v(s)x(τ(s))n(s)u(s)x λ (τ(s))] s, (4.33) Applying Lemm 3. to v(s)x(τ(s)) n(s)u(s)x λ (τ(s)) with X =(nu) λ x, Y =( λ v(nu) λ) λ, we obtin v(s)x(τ(s))n(s)u(s)x λ (τ(s)) (λ )λ λ/(λ) v λ/(λ) (s)n /(λ) (s)u /(λ) (s). Therefore, (r(t)x (t)) h (t) p (t)x(τ(t))c 2 b(t). (4.34) Integrting (4.34) from to t, we hve r(t)x (t) r( )x ( ) h (s) s p (s)x(τ(s)) sc 2 b(s) s. (4.35) Therefore, x (t) r()x ( ) h (s) s r(t) r(t) p (s)x(τ(s)) s c 2 b(s) s r(t) r(t) = r()x ( ) h (s) s r(t) r(t) τ(t) p (τ (s))x(s)(τ (s)) s c 2 b(s) s. r(t) τ( ) r(t) (4.36) Integrting from to t, we get x(t) x( )r( )x ( ) x(t) K s s h (ξ) ξ] sc 2 s τ(s) p (τ (ξ))x(ξ)(τ (ξ)) ξ] s. τ( ) b(ξ) ξ] s (4.37) τ(s) p (τ (ξ))x(ξ)(τ (ξ)) ξ] s, τ( ) where K is n upper bound for the expression (4.38) x( )r( )x ( ) c 2 s s b(ξ) ξ] s, s h (ξ) ξ s for t. Applying Lemm 3.2 to inequlity (4.38) then using condition (4.29), we get x(t)<. (4.39) If x(t) is eventully negtive, we cn set y = x. y stisfies Eq. (.2) with e(t) replced by e(t) f(t,x) by f(t,y). In similr wy, we get (x(t))<. (4.4) From (4.39) (4.4), we conclude tht (4.3) holds. Theorem 4.4Let λ >, M M 3, (4.27), (4.28), (4.29) (4.3) hold. If lim inf t s t h (ξ) ξ] s=, t s t h (ξ) ξ] s=, for ll t, then every solution of (.2) is oscilltory. Proof. Assume tht (.2) is nonoscilltory on t, ) T. Then there is solution x of (.2) point t, ) T such tht x(t) x(τ(t)) re of the sme sign on, ) T. Consider the cse x(t) x(τ(t)) re positive on, ) T. The proof when x is eventully negtive is similr. Proceeding s in the proof of Theorem 4.3, we get x(t) x( )r( )x ( ) s s h (ξ) ξ sc 2 s b(ξ) ξ] s τ(s) p (τ (ξ))x(ξ)(τ (ξ)) ξ] s. τ( ) Clerly, the conclusion of Theorem 4.3 holds. the second the lst two integrls in the bove inequlity re bounded. Finlly, tking liminf s t using (??), we get contrdiction with the fct tht x(t) is eventully positive. This contrdiction completes the proof. Theorem 4.5Let λ =, M M 3, (4.27), (4.28), (4.29) hold. If, t s lim inf τ(u) b(u) t τ(t ) n(τ (ξ))u(τ (ξ))(τ (ξ)) ξ] ξ] s<, (4.4) t s t e(ξ) ξ] s=, t s t e(ξ) ξ] s=, for ll t, then every solution of (.2) is oscilltory. c 26 NSP

7 J. An. Num. Theor. 4, No. 2, 5- (26) / References ] R. P. Agrwl, M. Bohner, D. O Regn A. Peterson, Dynmic equtions on time scles: survey, J. Comput. Appl. Mth. 4 (22), ] R. P. Agrwl, M. Bohner S. H. Sker, Oscilltion of second order dely dynmic equtions, Cn. Appl. Mth. Q. 3 (25) -8. 3] R. P. Agrwl, D. O Regn S. H. Sker, Philos-type oscilltion criteri for second order for hlf-liner dynmic equtions on time scles, Rocky Mountin J. Mth. 37 (27) ] R. P. Agrwl, S. R. Grce, D. O Regn A. Zfer, Oscilltory symptotic behvior of solutions for second order nonliner integro-dynmic equtions on time scles, Elec. J. Diff. Equ. 24 (5) (24) -2. 5] H. A. Agw, A. M. M. Khodier Heb A. Hssn, Oscilltion of second-order nonliner dely dynmic equtions on time scles, Interntionl Journl of Differentil Equtions, vol. 2, Article ID 8638, 5 pges, (2). 6] H. A. Agw, Ahmed M. M. Khodier Heb A. Hssn, Oscilltion of second-order nonliner dely dynmic equtions with dmping on time scles, Interntionl Journl of Differentil Equtions, vol. 24, Article ID (24), pges. 7] H. A. Agw, Ahmed M. M. Khodier Heb A. Hssn, Oscilltion theorems of second order nonliner dely dynmic equtions on time scles, EJMAA. 3 (2) (25) -. 8] M. Bohner A. Peterson, Dynmic Equtions on Time Scles: An introduction with pplictions, Birkhäuser, Boston, 2. 9] M. Bohner A. Peterson, Advnces in Dynmic Equtions on Time Scles, Birkhäuser, Boston, 23. ] S. K. Choi N. Koo, On Gronwll-type inequlity on time scles, J. Chung. Mth. Soci. 23 () (2) ] S. R. Grce, J. R. Gref A. Zfer, Oscilltion of integrodynmic equtions on time scles, J. Appl. Mth Lett. 26 (23) ] S. R. Grce, M. A. El-Beltgy S. A. Deif, Asymptotic behvior of non-oscilltory solutions of sedcond order integro-dynmic equtions on time scles, J. Appl. Comput. Mth. 2 (23) -4. 3] S. R. Grce M. A. El-Beltgy, Oscilltory behvior of solutions of certin integrodynmic equtions of second oredr on time scles, J. Abstr. Appl. Anls. vol. 24, Article ID (24), 5 pges. 4] G. H. Hrdy, J. E. Littlewood G. Poly, Inequlities, Cmbridge Univ. Press, Cmbridge, 952. H. A. Agw hs received Ph.D degree in 993 from deprtment of mthemtics, fculty of science t Ain Shms university. He is professor of pure mthemtics in deprtment of mthemtics vice den of eduction students ffirs, fculty of eduction, Ain Shms university. He hs published more thn 35 reserch ppers in Interntionl journls. He hs lso ttended delivered tlks in mny ntionl interntionl conferences. His min reserch interests re: functionl differentil equtions, dynmic equtions, time scles, inequlities. Ahmed M. M. Khodier He is lecturer of pure mthemtics in deprtment of mthemtics director of the open eduction center, fculty of eduction, Ain Shms university. He hs published more thn 25 reserch ppers in Interntionl journls. His min reserch interests re: Numericl nlysis, functionl differentil equtions, dynmic equtions, time scles. Heb A. Hssn received MSc degree in 22 from deprtment of mthemtics, fculty of eduction, Ain Shms university. She is ssistnt techer of pure mthemtics in deprtment of mthemtics, fculty of eduction, Ain Shms university. She hs published 4 reserch ppers 3 reserch ppers re ccepted in Interntionl journls. Her min reserch interests re: functionl differentil equtions, dynmic equtions, time scles. c 26 NSP