A Nonlinear Sturm Picone Comparison Theorem for Dynamic Equations on Time Scales

Transcription

1 International Journal of Difference Equations IJDE). ISSN Volume 2 Number ), pp Research India Publications A Nonlinear Sturm Picone Comparison Theorem for Dnamic Equations on Time Scales Boris Belinski Department of Mathematics, Universit of Tennessee at Chattanooga, Chattanooga, TN 37403, USA Boris-Belinski@utc.edu John R. Graef Department of Mathematics, Universit of Tennessee at Chattanooga, Chattanooga, TN 37403, USA John-Graef@utc.edu Sonja Petrović Department of Mathematics, Universit of Kentuck, Lexington, KY 40506, USA petrovic@ms.uk.edu Abstract The authors derive an analog of the well-known Picone identit but for nonlinear dnamic equations on time scales. As a consequence, the obtain a nonlinear comparison theorem in the spirit of the classical Sturm Picone comparison theorem. Comparison results ielding the nonoscillation of all solutions of nonlinear equations are also obtained. AMS subject classification: 34C10, 34C15, 39A11. Kewords: Comparison theorem, dnamic equations, nonoscillation, oscillation, Sturm Picone theorem, time scales. Research supported in part b the Office of Academic Research Computing Services of the Universit of Tennessee at Chattanooga. Received Januar 31, 2007; Accepted June 20, 2007

2 26 Boris Belinski, John R. Graef, Sonja Petrović 1. Introduction In this paper, we consider dnamic equations on time scales of the form pt)x t) ) + q t,x σ t), x σ t)) ) = r t,x σ t), x σ t)) ) 1.1) give a nonlinear comparison theorem in the spirit of the classical Sturm Picone comparison theorem. The literature on oscillation nonoscillation of solutions of differential, difference, now dnamic equations is a long rich one stretching back to the work of Sturm in the 1830 s. While we are especiall interested here in results ielding the nonoscillation of all solutions, special cases of the theorems below include the well-known Sturm Picone comparison theorem. We first derive a nonlinear Picone tpe identit, as a consequence, we will be able to obtain some new oscillation nonoscillation results for equations of the tpe 1.1). We will assume that the functions p : T R q, r : T R 2 R are rd-continuous, where T is a time scale, i.e., a closed subset of the real numbers R. The forward backward jump operators σ, ρ : T T are defined in the usual wa b σt) = inf{s T : s>t} ρt) = sup{s T : s<t}, respectivel, supplemented with inf :=sup T sup :=inf T. The function µt) = σt) t is called the graininess function. We also use the usual notation that x σ t) = xσt)). The point t T is left-dense, left-scattered, right-dense, or right-scattered if ρt) = t, ρt)<t, σt) = t, orσt)>t, respectivel. The smbols [a,b], [a,b), etc. denote time scale intervals, for example, [t 1,t 2 ]={t T : t 1 t t 2 }, where t 1, t 2 T with t 1 <ρt 2 ). We next define what is meant b a generalized zero of a solution. Definition 1.1. A solution x of equation 1.1) has a generalized zero at t if either or t is left-scattered xt) = 0, pρt))xρt))xt) < 0. It will be convenient to classif solutions according to their oscillator behavior as follows. Definition 1.2. A solution x of equation 1.1) will be called nonoscillator if there exists T T such that x has no generalized zeros for t T.

3 Nonlinear Sturm Picone Theorem 27 Definition 1.3. A solution x of equation 1.1) will be called oscillator if for ever t 1 T there exists t 2 T with t 2 >t 1 pρt 2 ))xρt 2 ))xt 2 )<0. Definition 1.4. A solution x of equation 1.1) will be said to be a Z-tpe solution if it has arbitraril large generalized zeros but is ultimatel nonnegative or nonpositive. Remark 1.5. Notice that if p>0 or p<0) x is a Z-tpe solution, then eventuall xt) = 0 at each of its generalized zeros. For additional basic facts about dnamic equations on time scales, we refer the reader to the monograph of Bohner Peterson [2]. 2. Comparison Results We consider the pair of equations pt)x t) ) + q t,x σ t), x σ t)) ) = r t,x σ t), x σ t)) ) 2.1) Pt) t) ) + Q t, σ t), σ t)) ) = 0, 2.2) where are rd-continuous. We assume that p, P : T R q, Q, r : T R 2 R pt) Pt) > 0 2.3) qt,u 1,u 2 ) u 1 Qt, v 1,v 2 ) v 1 for all t, u 1, u 2, v 1, v 2 with u 1 = 0 = v ) Define G : T R b { xt)[pt)x t)t) Pt)xt) } t)] Gt) =. t)

4 28 Boris Belinski, John R. Graef, Sonja Petrović Suppressing the argument t,wehave G = Now ) x px Px ) + xσ px σ Px ) = x x σ px Px ) + xσ σ [ px ) σ + px P ) x σ Px ) ] = p x ) 2 2 σ P xx σ p xx σ + P x2 ) 2 σ + xσ σ [ qt,x σ,x σ t))) σ + rt,x σ,x σ t))) σ + px + Qt, σ, σ t)))x σ Px ] = p x ) 2 2 σ P xx σ p xx σ + P x2 ) 2 σ + xσ [ px σ Px ] + xσ σ [ qt,x σ,x σ t))) σ + Qt, σ, σ t)))x σ ] + rt,x σ,x σ t)))x σ. x σ [ qt,x σ σ,x σ t))) σ + Qt, σ, σ t)))x σ ] Using the fact that we obtain p x ) 2 2 σ Since P xx σ = 1 σ [σ x σ ) 2 ] [ Qt, σ, σ t))) σ qt,xσ,x ] σ t))) x σ. u σ t) = ut) + µt)u t), 2.5) p xx σ + P x2 ) 2 σ + xσ [ px σ Px ] = 1 [ px σ ) 2 2 pxx + px x + µx ) ] + 1 [ Pxx σ + Px 2 ) 2 Px x + µx ) ] = 1 [ px σ ) px ) 2 µ 2Pxx + Px 2 ) 2 Px ) 2 µ ]. x x ) 2 = x ) 2 2 2xx + x 2 ) 2,

5 Nonlinear Sturm Picone Theorem 29 we can write 2Pxx + Px 2 ) 2 Px ) 2 µ = Px x ) 2 Px ) 2 2 Px 2 ) 2 + Px 2 ) 2 Px ) 2 µ. Hence, 1 [ px σ ) px ) 2 µ 2Pxx + Px 2 ) 2 Px ) 2 µ ] = Px x ) 2 σ + 1 σ [ p P )x ) p P )x ) 2 µ ] = Px x ) 2 σ + 1 σ [p P )x ) 2 + µ)] = Px x ) 2 σ + 1 σ p P )x ) 2 σ = Px x ) 2 σ + p P )x ) 2. Therefore, [ Qt, σ, σ t))) Gt) = σ qt,xσ,x ] σ t))) x σ x σ ) 2 + Px x ) 2 σ + p P )x ) 2 + rt,x σ,x σ t)))x σ. 2.6) We will make use of the nonlinear Picone tpe identit 2.6) in proving our comparison theorems. Remark 2.1. The linear version of 2.6) was derived in [9]. In what follows, we will assume that not all inequalities in our hpotheses simultaneousl become equalities on an open set. Theorem 2.2. In addition to conditions 2.3) 2.4), assume that rt,u,v) ) If equation 2.1) has a solution x with two generalized zeros at t 1 t 2 in the interval [a,σ 2 b)] with xt) > 0 for t t 1,t 2 ), then ever solution of 2.2) has a generalized zero in [a,σ 2 b)]. Proof. Let x be a solution of 2.1) with two generalized zeros in [a,σ 2 b)) let be a solution of 2.2) that has no generalized zero in [a,σ 2 b)). Suppose that the two generalized zeros of xt) occur at the points σc) σd), with a c σc)<d σd) σ 2 b)

6 30 Boris Belinski, John R. Graef, Sonja Petrović xt) > 0 for t c, d]. For convenience, we define the auxiliar function u : T R b 0, a t c, ut) = xt), c < t d, 0, d < t σ 2 b). Integrating the nonlinear Picone identit 2.6) with the function x replaced b u, we obtain σ 2 b) [ ] u I = pu Pu ) t) t. Observing that using the fact that we have σ 2 b) a = a c a + σt) I = µc)gc) + t σc) c d + + σc) σd) d σ 2 b) + σd) f s) s = µt)ft), 2.8) d σc) Gt) t + µd)gd). 2.9) We will consider three cases depending on the nature of the points c d. Case I: If both c d are right-dense, then u σ c) = uc) = xc) = 0 = xd) = ud) µc) = µd) = 0, so I = ud)[pd)u d)d) Pd)u d)]/d) uc)[pc)u c)c) Pc)u c)]/c) = ) Case II: If c is right-dense d is right-scattered, then xc) = x σ c) = µc) = 0, xd) > 0, x σ d) < 0. Since ud) = xd), we see that u d) = uσ d) ud) µd) = xd) µd), a simple calculation shows that ) u d) = xd) µd)d).

7 Nonlinear Sturm Picone Theorem 31 From 2.9), we have I = ud)[pd)u d)d) P d)ud) d)]/d) { u ) + µd) d) [ } pu Pu ] uσ d)) [ d) + pu Pu ] d) σd)) = ud) [ pd)u d)d) P d)ud) d) ) /d) xd) ] d) pd)u d)d) P d)ud) d)) = 0. Case III: If both c d are right-scattered, then uc) = 0, u σ c) = x σ c), ud) = xd), u σ d) = 0, µc) > 0, µd) > 0. Hence, u c) = uσ c) uc) µc) = xσ c) µc) Direct calculations give u d) = xd) µd). [ ] u c) = xσ c) µc) σ c) [ ] u d) = xd) µd)d). In this case, 2.9) becomes ) ) u u pu Pu ) d) pu Pu ) σ c)) [ u ) ) ] u + µc) c)pu Pu ) σ c) + c)pu Pu ) c) [ u ) ) u σ + µd) d)pu Pu )d) + d)pu Pu ) d)]

8 32 Boris Belinski, John R. Graef, Sonja Petrović = ) u d)pu Pu )d) ) u σ c)pu Pu ) σ c) x σ c) + µc) µc) σ c) pu Pu ) σ c) + µd) xd) µd)d) pu Pu )d) = xd) d) pu Pu )d) xσ c) σ c) pu Pu ) σ c) + xσ c) σ c) pu Pu ) σ c) xd) d) pu Pu )d) = 0. In each of the above three cases, we see that an integration of the left-h side of the nonlinear Picone identit 2.6) from a to σ 2 b) is 0, but in view of conditions 2.3) 2.4), the integral of the right-h side is positive. Therefore, must have a generalized zero in the interval [a,σ 2 b)). The proof of the following result should now be clear. Theorem 2.3. In addition to conditions 2.3) 2.4), assume that rt,u,v) ) If equation 2.1) has a solution x with two generalized zeros at t 1 t 2 in the interval [a,σ 2 b)] with xt) < 0 for t t 1,t 2 ), then ever solution of 2.4) has a generalized zero in [a,σ 2 b)]. The following corollaries are immediate consequences of the above theorems. Corollar 2.4. If conditions 2.3) 2.4) 2.7), 2.11)) hold equation 2.1) has an oscillator or nonnegative nonpositive) Z-tpe solution, then ever solution of 2.2) is oscillator or Z-tpe. Corollar 2.5. Assume that conditions 2.3) 2.4) 2.7), 2.11)) hold. If there is a solution of 2.2) with no generalized zeros in [a,σ 2 b)], then no solution of 2.1) that is nonnegative nonpositive) can vanish more than once there. Of special interest here are nonoscillation results, so we have the following theorem. Theorem 2.6. Suppose conditions 2.3) 2.4) 2.7) hold qt,u,v) 0 for u ) If equation 2.2) has a nonoscillator solution, then ever solution of 2.1) is nonoscillator. Proof. It follows from Corollar 2.4 that no solution of 2.1) is oscillator or nonnegative Z-tpe. Suppose x is a nonpositive Z-tpe solution of 2.1), sa xt 1 ) = 0 xt) 0

9 Nonlinear Sturm Picone Theorem 33 for t t 1. From equation 2.1), we have pt)x t)) = rt,x σ t), x σ t))) qt,x σ t), x σ t))) 0 for t t 1. This implies that pt)x t) is eventuall of one sign which is impossible for a Z-tpe solution. Remark 2.7. A result analogous to Theorem 2.6 can be obtained b replacing 2.7) with 2.11) 2.12) with qt,u,v) 0 for u ) We can obtain additional results in this same spirit b replacing conditions 2.7) 2.11) with the single condition This ields the following theorem. urt,u,v) 0 for u = ) Theorem 2.8. Assume that conditions 2.3), 2.4), 2.14) hold. i) If equation 2.1) has an oscillator or Z-tpe solution, then ever solution of 2.2) is oscillator. ii) If uqt,u,v) 0 for u = 0 equation 2.2) has a nonoscillator solution, then ever solution of 2.1) is nonoscillator. 3. Consequences of the Main Results Further Discussion If, in equations 2.1) 2.2), we have rt,u,v) r 1 t), qt,u,v) = q 1 t)u, Qt,u,v)= Q 1 t)u, then our pair of equations become pt)x t)) + q 1 t)x σ t) = r 1 t) 3.1) P t) t)) + Q 1 t) σ t) = ) As a consequence of the results here, we then have the following corollar. Corollar 3.1. Suppose that pt) Pt) 0 0 q 1 t) Q 1 t), 3.3)

10 34 Boris Belinski, John R. Graef, Sonja Petrović that equation 3.2) is nonoscillator. If either r 1 t) 0orr 1 t) 0, then ever solution of equation 3.1) is nonoscillator. This corollar was first proved for differential equations b Švec [11] in 1963, rediscovered b Hammett [6] Keener [7] extended b Graef Spikes [4]. If the time scale T is the set of real numbers R, then Theorem 2.2 or Theorem 2.3) becomes a nonlinear generalization of the well-known interlacing theorem. All the results here agree with the usual Sturm Picone comparison results for unforced linear differential equations see, for example, Swanson [12]). If the time scale T is the integers Z, then we obtain the corresponding results for difference equations see, for example, Agarwal [1] or Kelle Peterson [8] in the linear case). Generalizations of the Sturm Picone comparison theorem to higher order nonlinear difference equations can be found in Graef, Miciano Cariño, Qian [3], of course it would be of interest to extend the results here in that direction as well. We also wish to point out that for the nonlinear equations pt)x t)) + q 1 t)f x σ t)) = 0 3.4) P t) t)) + Q 1 t)f σ t)) = 0 3.5) again with 3.3) holding 0 < f u) u Fv) v for all u, v with u = 0 = v, 3.6) we have that if equation 3.5) has a nonoscillator solution, then ever solution of equation 3.4) is nonoscillator. For example, if 3.3) holds the linear equation P t) t)) + Q 1 t) σ t) = 0 3.7) is nonoscillator, then ever solution of the nonlinear equations pt)x t)) x σ t)) 3 + q 1 t) 1 + x σ = 0, 3.8) t)) 2 pt)x t)) x σ t)) 3 + q 1 t) 1 + x σ t)) 2 =±1 + sin2 t)cos 2 x σ t), 3.9) pt)x t)) x σ t)) 3 + q 1 t) 1 + x σ t)) 2 = 1 cos t)xσ t)) ) is nonoscillator. We note that if T = R, then the results here include those of Graef Spikes [5] for ordinar differential equations as a special case, are new in the case T = Z of difference equations.

11 Nonlinear Sturm Picone Theorem 35 References [1] R. P. Agarwal, Difference Equations Inequalities, Second Edition, Dekker, New York, [2] M. Bohner A. Peterson, Dnamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, [3] J.R. Graef, A. Miciano Cariño C. Qian, A general comparison result for higher order nonlinear difference equations with deviating arguments, J. Difference Equ. Appl., 8: , [4] J.R. Graef P.W. Spikes, Nonoscillation theorems for forced second order nonlinear differential equations, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 598): , [5] J.R. Graef P.W. Spikes, Comparison nonoscillation results for perturbed nonlinear differential equations, Ann. Mat. Pura Appl., 1164): , [6] M.E. Hammett, Oscillation Nonoscillation Theorems for Nonhomogeneous Linear Differential Equations of Second Order, Ph.D. Dissertation, Auburn Universit, [7] M.S. Keener, On the solutions of certain linear nonhomogeneous second-order differential equations, Applicable Anal., 1:57 63, [8] W.G. Kelle A. Peterson, Difference Equations: An Introduction with Applications, Second Edition, Academic Press, New York, [9] S. Petrović, Oscillation of Solutions of Dnamic Equations on Time Scales: Sturm Picone Comparison Theorem, An Undergraduate Honors Project, Universit of Tennessee at Chattanooga, Chattanooga, Tennessee, [10] C. Sturm, Sur les équations différentielles linéaires du second ordré, J. Pures Appl. Math., 1: , [11] M. Švec, On various properties of the solutions of third- fourth-order linear differential equations, in Differential Equations Their Applications, Proceedings of a Conference Held in Prague, September 1962, Academic Press, New York, 1963, pp [12] C.A. Swanson, Comparison Oscillation Theor of Linear Differential Equations, Academic Press, New York, 1968.