Appl. Math. Inf. Sci. 7, No. 6, 289-293 (203) 289 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/0.2785/amis/070608 Oscillation heorems for Second-Order Nonlinear Dynamic Equation on ime Scales M.amer Şenel Department of Mathematics, Faculty of Sciences, Erciyes University, 38039, Kayseri, urkey Received: 27 Feb. 203, Revised: 25 Jun. 203, Accepted: 26 Jun. 203 Published online: Nov. 203 Abstract: his paper concerns the oscillation of solutions to the second order non-linear dynamic equation (r(t)x (t)) p(t) f(x σ (t))g(x (t))=0 on a time scale which is unbounded above. By using a generalized Riccati transformation integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates. Keywords: Oscillation, Dynamic equations, ime scales, Second-order. Introduction In this paper, we consider the second order nonlinear dynamic equation (r(t)x (t)) p(t) f(x σ (t))g(x (t))=0, () p, r real-valued, non-negative, right-dense continuous function on a time scale. It is the purpose of this paper to give oscillation criteria for equation (). hroughout this paper, we will assume the following hypotheses: (A) p,r C rd (t 0, ),R ) such that t 0 r(s) s=, (A2) g : R rd-continuous g(u) c>0, (A3) f : R is continuously differentiable satisfies u f(u)>0, k > 0, u 0. f(u) u Much recent attention has been given to dynamic equations on time scales, or measure chains, we refer the reader to the lmark paper of Hilger 2 for a comprehensive treatment of the subject. Since then, several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. the references cited therein. A book on the subject of time scales by Bohner Peterson 3 summarizes organizes much of the time scale calculus. In recent years there has been much research activity concerning the oscillation nonoscillation of solutions of some different equations on time scales. We refer the reader to the papers 2, 4 6, 8,. In 6, the authors consider the second order dynamic equation (p(t)x (t)) q(t)x σ (t)=0 f or t a,b give necessary sufficient conditions for oscillation of all solutions on unbounded time scales. Unfortunately, the oscillation criteria are restricted in usage since additional assumptions have to be imposed on the unknown solutions. In 0, the authors consider the same equation suppose that there exists some t 0 such that p is bounded above on t 0, ), inf{µ(t) : t t 0, )}>0 use the Riccati equation to prove that if t 0 q(t) t =, then every solution is oscillatory ont 0, ). In 5, the authors consider (p(t)x (t)) q(t)( f ox σ )=0 f or t a,b, p q are positive, real-valued rd-continuous functions. Corresponding author e-mail: senel@erciyes.edu.tr, tsenel@gmail.com c 203 NSP
290 M.. Şenel: Oscillation heorems for Second-Order Nonlinear... 2 Preliminary results Lemma 2.. Let x(t) be a nonoscillatory solution of () (A)-(A3) hold. hen there exist 0 such that x(t)>0, x (t)>0, (r(t)x (t)) < 0 f or t 0. Proof. Suppose that x(t) is nonoscillatory solution of () without loss of generality, assume x(t)>0 for t 0. Assume that x (t)<0 for all large t. hen without loss of generality x (t) < 0 for all t 0. If x(t) > 0, then f(x σ (t))>0. From () (r(t)x (t)) =p(t) f(x σ (t))g(x (t))<0, (r(t)x (t)) < 0. (2) Define y(t)=r(t)x (t). So y(t) is decreasing. Assume that there exist 0 with y( )=κ < 0. hen r(t)x (t)=y(t) y( )=κ f or all t therefore x (t) κ r(t) f or all t hen an integration for t > 2 gives x(t) x( 2 )κ s as t 2 r(s) which is a contradiction. Hence x (t) is not negative for all large t thus x (t)>0 for all t 0. 3 Main results heorem 3. Assume that (A)-(A3) hold. Furthermore, assume that there exist a positive real rd-functions differentiable functions such that k cz(s)p(s) 4 (z (t)) 2 z(s) then every solution of () is oscillatory. s=, (3) nonoscillatory solution of (). Without loss of generality, we may assume that x(t) > 0 for t > 0. We shall eventually negative is similar. From Lemma 2., x (t)>0 for t > 0. Define the function w(t) by the Riccati substitution w(t) := r(t)x (t), t. (4) x(t) hen w(t)>0 satisfies w (t)= (r(t)x (t)) σ x(t) x(t) (r(t)x (t)) w (t)= z (t)x(t)x (t) x(t)x σ (r(t)x (t)) σ (t) x(t) (p(t) f(xσ (t))g(x (t))). From Lemma 2., x(t)>0, x (t)>0, (r(t)x (t)) < 0 so x σ (t)>x(t), (r(t)x (t)) σ < r(t)x (t). We get w (t) z (t) wσ (t) (t)(r(t)x (t)) σ x (x σ (t)) 2 w (t) k cp(t)z (t) wσ (t) x σ (t) p(t) f(xσ (t))c (5) x (t)(w σ (t)) 2 () 2 (r(t)(x (t))) σ ) k cp(t)z (t) wσ (t) (w σ (t)) 2 () 2 r σ (t) w (t) k cp(t) r σ (t)(z (t)) 2 4 then r σ (t) (6) (7) 2 w σ (t) r σ (t) 2 z (t), w (t) k cp(t) r σ (t)(z (t)) 2. (8) 4 Integrating (8) from to t w(t)w() k cp(t) 4 k cp(t) 4 r σ (t)(z (t)) 2 r σ (t)(z (t)) 2 s, s w()w(t)<w(). aking the limsup of both sides of above inequality as t, we obtain a contradiction to condition (3). he proof is complete. Corollary 3.2. Assume that (A)-(A3) hold. If k cp(s) s=, (9) then every solution () is oscillatory. Example 3.3. Consider the dynamic equation (tx (t)) µ(t) xσ (t)((x σ (t)) 2 )((x (t)) 2 )=0, t. r(t)= t, p(t)= µ(t), f(xσ (t))=x σ (t)((x σ (t)) 2 ), g(x (t))=(x (t)) 2. If we take : i)=z σ(t)= t µ(t)= p(t)=, ii) = {2 N : n Z} {0} σ(t) = 2t µ(t) = t c 203 NSP
Appl. Math. Inf. Sci. 7, No. 6, 289-293 (203) / www.naturalspublishing.com/journals.asp 29 p(t)= t, all conditions of Corollary 3.2 are satisfied. Hence it is oscillatory. Corollary 3.4. Assume that (A)-(A3) hold. If there is λ such that k cs λ p(s) r σ (t)((s λ ) ) 2 4 s λ s=, (0) then every solution () is oscillatory. Now, let us introduce the class of functions R which will be extensively used in the sequel. Let D 0 {(t,s) 2 : t > s t 0 } D {(t,s) 2 : t s t 0 }. he function H C rd (D,R) is said belong to the classrif (i)h(t,t)=0, t t 0, H(t,s)>0, on D 0, (ii) H has a continuous -partial derivative H s (t,s) ond 0 with respect to the second variable. (H is rd-continuous function if H is rd-continuous function in t s.) heorem 3.6. Assume that (A)-(A3) hold. Let be a positive differentiable function let H : D R be an rd-continuous function such that H belongs to the class R satisfies H(t,) C(t,s)=r σ (s)(z σ (s)b(t,s)) 2, B(t,s)=H s (t,s)h(t,s) z (s) z σ (s). k ch(t,s)z(s)p(s) 4 C(t,s) s=, () hen every solution of () is oscillatory. nonoscillatory solution of (). Without loss of generality, we may assume that x(t) > 0 for t > 0. We shall eventually negative is similar. From Lemma 2. (7), x (t)>0 for t > 0, it follows that w (t) k cp(t) z (t) wσ (t) We multiply both sides by H(t,s) to get (w σ (t)) 2 () 2 r σ (t). (2) H(t,s)w (t) k ch(t,s)p(t)h(t,s)z (t) wσ (t) (w σ (t)) 2 H(t, s) () 2 r σ (t). (3) Using the integration by parts formula, we have k ch(t,s)z(s)p(s) s H(t,t)w(t) from H(t,t)=0, we obtain k ch(t,s)z(s)p(s) s k ch(t,s)z(s)p(s) s H(t,)w() H(t,)w() Hs (t,s)w σ (s) s H(t,s)z (s) wσ (s) z σ (s) s ((w σ (s)) 2 (z σ (s)) 2 r σ (s) s H(t,)w() Hs (t,s)h(t,s) z (s) z σ w σ (s) s (s) ((w σ (s)) 2 (z σ (s)) 2 r σ (s) s, B(t,s)w σ (s) s ((w σ (s)) 2 (z σ (s)) 2 r σ (s) s. herefore, by completing the square as in heorem 3., we obtain k ch(t,s)z(s)p(s) s H(t,)w() 4 r σ (s)(z σ (s)) 2 B 2 (t,s) s Hence, we obtain w σ (s) r σ (s) z σ (s) 2 k ch(t,s)z(s)p(s) s H(t,)w() C(t,s)=r σ (s)(z σ (s)b(t,s)) 2. r σ (s) H(t,s)z(s) B(t,s)zσ (s) = H(t,)w() 2 s. 4 rσ (s)(z σ (s)b(t,s)) 2 s 4 C(t,s) s hen for all t, we have k ch(t,s)z(s)p(s) 4 C(t,s) s H(t,)w() this implies that H(t,) k ch(t,s)z(s)p(s) 4 C(t,s) s w() which contradicts (). he proof is complete. As consequences of heorem 3.6 we get the following. Corollary 3.7. Suppose that the assumptions of heorem 3.6 hold. If t limsup k ch(t,s)p(s) 4 H(t,) rσ (s)(h s (t,s)) 2 s=, then every solution of () is oscillatory. c 203 NSP
292 M.. Şenel: Oscillation heorems for Second-Order Nonlinear... Corollary 3.8. Let the assumption () in heorem 3.6 be replaced by H(t,) r σ (s)(z σ (s)) 2 H(t,) 4 k ch(t,s)z(s)p(s)=, hen every solution of () is oscillatory. ( ) H(t,s)z (s) 2 H z σ (s) s (t,s) s<. Lemma 3.9. Let H(t,s)=(t s) n, (t,s) D with n>, we see that H belongs to the class R. Hence ((t s) n ) n(t σ(s)) n. Proof. We consider the following two cases: Case. If µ(t)=0 then ((t s) m ) =m(t s) m. Case 2. If µ(t) 0, then we have ((t s) m ) = µ(s) (t σ(s))m (t s) m = σ(s)s (t s)m (t σ(s)) m. (4) Using the Hardy, Littlewood Polya inequality x m y m γy m (xy) for all x y>0 m, we have (t s) m (t σ(s)) m m(t σ(s)) m (σ(s)s). (5) hen, from (4) (5), we have ((t s) m ) m(t σ(s)) m. Corollary 3.0. Assume that (A)-(A3) hold. Let =, H :D R be rd-continuous function such that H belongs to the class R. If t n k c(t s) n p(s) rσ (s) 4 (n(t σ(s))n ) 2 s=, f or n>, then equation () is oscillatory ont 0, ). 4 Assume that f is differentiable. In this section, we assume that f (u) k 2 for u 0 some k 2 > 0. heorem 4.. Assume that (A)-(A3) hold. Furthermore, assume that there exists a positive real rd-continuous function such that cz(s)p(s) 4 then every solution of () is oscillatory. r(s)(z (t)) 2 s=, (6) k 2 z(s) nonoscillatory solution of (). Without loss of generality, we may assume that x(t) > 0 for t > 0. We shall eventually negative is similar. From Lemma 2., x (t)>0 for t > 0. Define the function w(t) by w(t) := r(t)x (t) f(x(t)), t. (7) hen w(t) satisfies w (t)=(r(t)(x (t))) σ f(x(t)) f(x(t)) (r(t)x (t)). (8) In view of (), f(x σ ) f(x) Lemma 2., we have w (t) z (t) (r(t)x (t)) σ f(x σ (t)) f (x(t))(r(t)x (t)) σ f 2 (x σ (t)) cp(t) f(xσ (t)) f(x σ (t)) w (t) cp(t)z (t) wσ (t) () 2 f (x(t))(w σ (t)) 2 (r(t)x (t)) σ. Using chain rule 4 f (x(t))= f (x(τ))x (t) k 2 x (t), τ t,σ(t). We have w (t) cp(t)z (t) wσ (t) () 2 k 2 x (t)(w σ (t)) 2 r(t)x (t) cp(t)z (t) wσ (t) k 2 (w σ (t)) 2 () 2 r(t) cp(t) r(t)(z (t)) 2 4 k 2 then k2 w σ (t) r(t) 2 w (t) cp(t) r(t)(z (t)) 2. 4 k 2 (9) r(t) k 2 z (t) 2, We proceed as in the proof of heorem 3. we obtain a contradiction. Corollary 4.2. Assume that (A)-(A3) hold. If limsup cs λ p(s) r(t)((sλ ) ) 2 s=, λ, 4k 2 r(s) then every solution of () is oscillatory. Different choices of lead to different corollaries of the above theorem. heorem 4.3. Assume that (A)-(A3) hold. Let be a positive differentiable function let H : D R be an rd-continuous function such that H belongs to the class R H(t,) r(s) cp(s) 4k E2 (t,s) s=, 2z(s)H(t,s) E(t,s)=H(t,s)z (t)h s (t,s). c 203 NSP
Appl. Math. Inf. Sci. 7, No. 6, 289-293 (203) / www.naturalspublishing.com/journals.asp 293 hen every solution of () is oscillatory. Proof. he proof is similar to that of heorem 3.6 hence is omitted. As an immediate consequence of heorem 4.3 using =, H(t,s) = (t s) m m = n, we get the following two results. Corollary 4.4. Assume that (A)-(A3) hold. he condition in heorem 4.3 is replaced by H(t,) H(t,) ch(t,s)z(s)p(s) s= r(s)(hs (t,s)z σ (s)h(t,s)z (s)) 2 k 2 H(t,s)z(s) s<, then every solution of () is oscillatory on t 0, ). Corollary 4.5. Assume that (A)-(A3) hold. If for n>2 limsup t n t n c(t s) n p(s) s= r(s)((n)(tσ(s)) n2 ) 2 s<, t s, 4k 2 (ts) n then every solution of () is oscillatory on, ). Acknowledgement his work was supported by Research Fund of the Erciyes University. Project Number: FBA--339. he author is grateful to the anonymous referee for a careful checking of the details for helpful comments that improved this paper. 5 M. Bohner, S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math., 34, 239-254 (2004). 6 O. Došlý S. Hilger, A necessary sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, J. Comput. Appl. Math., 4, 47-58 (2002). 7 Yuri V. Rogovchenko, Oscillation Criteria for Certain Nonlinear Differential Equations, J. Math. Anal. Appl., 229, 399-46 (999). 8 O. Došlý R. Hilscher, Disconjugacy, transformations quadratic functionals for symplectic dynamic systems on time scales, J. Differ. Equations Appl., 7, 265-295 (200). 9 L. Erbe A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, in Boundary value problems related topics, Math. Comput. Modelling, 32, 57-585 (2000). 0 L. Erbe A. Peterson, Riccati equations on a measure chain, in Proceedings of dynamic systems applications (G. S. Ladde, N. G. Medhin M. Sambham, eds.), Dynamic Publishers, Atlanta, GA, 3, 93-99 (200). G. Sh. Guseinov B. Kaymakçalan, On a disconjugacy criterion for second order dynamic equations on time scales, J. Comput. Appl. Math., 4, 87-96 (2002). 2 S. Hilger, Analysis on measure chains A unified approach to continuous discrete calculus, Results Math., 8, 8-56 (990). M. amer Şenel is Associate Professor Department of Mathematics of Faculty of Science at Erciyes University. His research interests are in the areas of applied mathematics oscillation theory of differential dynamic equations. He is referee editor of mathematical journals. References R. P. Agarwal, M. Bohner, D. O Regan A. Peterson, Dynamic equations on time scales: A survey, J. Comput. Appl. Math., 4, -26 (2002). 2 E. Akın, L. Erbe, B. Kaymakçalan A. Peterson, Oscillation results for a dynamic equation on a time scale, J. Differ. Equations Appl., 7, 793-80 (200). 3 M. Bohner, A. Peterson, Dynamic Equations on ime Scales: An Introduction with Applications, Birkhäuser, Boston, (200). 4 M. Bohner, L. Erbe, A. Peterson, Oscillation for second order dynamic equations on time scale, J. Math. Anal. Appl., 30, 49-507 (2005). c 203 NSP