Some nonlinear dynamic inequalities on time scales

Transcription

1 Proc. Indian Acad. Sci. Math. Sci.) Vol. 117, No. 4, November 2007, Printed in India Some nonlinear dynamic inequalities on time scales WEI NIAN LI 1,2 and WEIHONG SHENG 1 1 Deartment of Mathematics, Binzhou University, Shandong , Peole s Reublic of China 2 Deartment of Mathematics, Qufu Normal University, Shandong , Peole s Reublic of China wnli@263.net; wh-sheng@163.com MS received 5 November 2005 Abstract. The aim of this aer is to investigate some nonlinear dynamic inequalities on time scales, which rovide exlicit bounds on unknown functions. The inequalities given here unify and extend some inequalities in B G Pachatte, On some new inequalities related to a certain inequality arising in the theory of differential equation, J. Math. Anal. Al ) ). Keywords. Time scales; nonlinear; dynamic inequality; dynamic equation. 1. Introduction In 1988, Stefan Hilger 10] introduced the calculus on time scales which unifies continuous and discrete analysis. Since then many authors have exounded on various asects of the theory of dynamic equations on time scales. Recently, there has been much research activity concerning the new theory. For examle, we refer the reader to the aers 1, 6 9, 14, 15], the monograhes 4, 5] and the references cited therein. At the same time, a few aers 2, 3, 11] have studied the theory of dynamic inequalities on time scales. In this aer, we investigate some nonlinear dynamic inequalities on time scales, which unify and extend some inequalities by Pachatte in 13]. This aer is organized as follows: In 2 we give some reliminary results with resect to the calculus on time scales, which can also be found in 4, 5]. In 3 we deal with our nonlinear dynamic inequalities on time scales. In 4 we give an examle to illustrate our main results. 2. Some reliminaries on time scales In what follows, R denotes the set of real numbers, Z denotes the set of integers and C denotes the set of comlex numbers. A time scale T is an arbitrary nonemty closed subset of R. The forward jum oerator σ on T is defined by σt) := inf{s T: s > t} T for all t T. In this definition we ut inf =su T, where is the emty set. If σt)>t, then we say that t is right-scattered. Ifσt) = t and t<su T, then we say that t is right-dense. 545

2 546 Wei Nian Li and Weihong Sheng The backward jum oerator, left-scattered and left-dense oints are defined in a similar way. The graininess μ: T 0, ) is defined by μt) := σt) t. The set T κ is derived from T as follows: If T has a left-scattered maximum m, then T κ = T {m}; otherwise, T κ = T.Forf : T R and t T κ, we define f t) to be the number rovided it exists) such that given any ε>0, there is a neighborhood U of t with fσt)) fs)] f t)σt) s] ε σt) s for all s U. We call f t) the delta derivative of f at t. Remark 2.1. f is the usual derivative f if T = R and the usual forward difference f defined by f t) = ft + 1) ft))ift = Z. Theorem 2.1. Assume f, g: T R and let t conclusions: T κ. Then we have the following i) If f is differentiable at t, then f is continuous at t. ii) If f is continuous at t and t is right-scattered, then f is differentiable at t with f t) = fσt)) ft). μt) iii) If f is differentiable at t and t is right-dense, then f ft) fs) t) = lim. s t t s iv) If f is differentiable at t, then fσt))= ft)+ μt)f t). v) If f and g are differentiable at t, then the roduct fg: T R is differentiable at t with fg) t) = f t)gt) + f σ t))g t). We say that f : T R is rd-continuous rovided f is continuous at each right-dense oint of T and has a finite left-sided limit at each left-dense oint of T. As usual, the set of rd-continuous functions is denoted by C rd. A function F : T R is called an antiderivative of f : T R rovided F t) = ft) holds for all t T κ. In this case we define the Cauchy integral of f by b a f t) t = Fb) Fa) for a,b T. Theorem 2.2. If a,b,c T,α R, and f, g C rd, then i) b a ft)+ gt)] t = b a f t) t + b a gt) t; ii) b a αf )t) t = α b a f t) t; iii) b a f t) t = a b f t) t;

3 Some nonlinear dynamic inequalities on time scales 547 iv) b a f t) t = c a f t) t + b c f t) t; v) a a f t) t = 0; vi) if ft) 0 for all a t b, then b a f t) t 0. We say that : T R is regressive rovided 1 + μt)t) 0 for all t T. We denote by R the set of all regressive and rd-continuous functions. We define the set of all ositively regressive functions by R + ={ R: 1+ μt)t) > 0 for all t T}. For h>0, we define the cylinder transformation ξ h : C h Z h by ξ h z) = 1 h log1 + zh), where log is the rincial logarithm function, and { C h = z C: z 1 }, Z h = h { z C: π h < Imz) π h For h = 0, we define ξ 0 z) = z for all z C. If R, then we define the exonential function by ) e t, s) = ex ξ μτ) τ)) τ for s, t T. s Theorem 2.3. If R and t 1 T fixed, then the exonential function e,t 1 ) is the unique solution of the initial value roblem x = t)x, xt 1 ) = 1 on T. Theorem 2.4. If R, then i) e t, t) 1 and e 0 t, s) 1; ii) e σ t), s) = 1 + μt)t))e t, s); iii) if R +, then e t, )>0 for all t T. }. Remark 2.2. It is easy to see that, if T = R, the exonential function is given by e t, s) = e s τ)dτ, e α t, s) = e αt s), e α t, 0) = e αt for s, t R, where α R is a constant and : R R is a continuous function; if T = Z, the exonential function is given by t 1 e t, s) = 1 + τ)], e α t, s) = 1 + α) t s, e α t, 0) = 1 + α) t τ=s for s, t Z with s<t, where α 1 is a constant and : Z R is a sequence satisfying t) 1 for all t Z. Theorem 2.5. If R and a,b,c T, then b a t)e c, σ t)) t = e c, a) e c, b).

4 548 Wei Nian Li and Weihong Sheng Theorem 2.6. Let t 1 T κ and k: T T κ R be continuous at t, t), where t T κ with t>t 1. Assume that k t, ) is rd-continuous on t 1, σ t)]. If for any ε>0, there exists a neighborhood U of t, indeendent of τ t 1, σ t)], such that kσt), τ) ks, τ) k t, τ)σ t) s) ε σt) s for all s U, where k denotes the derivative of k with resect to the first variable. Then imlies gt) := g t) = t 1 kt, τ) τ t 1 k t, τ) τ + kσt), t). The following theorem is a foundational result in the theory of dynamic inequalities. For convenience of notation, throughout this aer, we always assume that T, T 0 =, ) T. Theorem 2.7. Suose u, b C rd,a R +. Then imlies u t) at)ut) + bt) for all t T 0 ut) u )e a t, ) + e a t, σ τ))bτ) τ for all t T Main results In this section, we deal with dynamic inequalities on time scales. Throughout this section, let >1 be a real constant. The following lemma, which is roved in 12], is useful in our main results. Lemma 3.1. Assume that 1 + q 1 = 1. Then x 1 y 1 q x + y q for x,y R + = 0, ). Theorem 3.1. Assume that u, a, b, g, h C rd, u, a, b, g, h are nonnegative. Then ut)) at) + bt) gτ)uτ)) + hτ)uτ)] τ for all t T 0 3.1) which imlies { )] } 1 + aτ) 1 ut) at) + bt) aτ)gτ)+ hτ) e bm t, σ τ)) τ where for all t T 0, 3.2) mt) = gt) + ht). 3.3)

5 Some nonlinear dynamic inequalities on time scales 549 Proof. Define a function zt) by zt) = gτ)uτ)) + hτ)uτ)] τ for all t T ) Then z ) = 0 and 3.1) can be restated as ut)) at) + bt)zt) for all t T ) Using Lemma 3.1, from 3.5), we easily obtain ut) at) + bt)zt)) 1 1) 1 at) Combining 3.4) 3.6), we get + bt) zt) + 1 for all t T ) z t) = gt)ut)) + ht)ut) 1 + at) gt)at) + bt)zt)] + ht) + bt) ) zt) = at)gt) + at) + 1 ] ht) + bt)mt)zt) for all t T 0, 3.7) where mt) is defined as in 3.3). Using Theorem 2.7 and noting z ) = 0, from 3.7) we obtain )] 1 + aτ) zt) aτ)gτ)+ hτ) e bm t, σ τ)) τ for all t T ) Clearly, the desired inequality 3.2) follows from 3.5) and 3.8). The roof is comlete. Remark 3.1. If T = R, then the inequality established in Theorem 3.1 reduces to the inequality established by Pachatte in Theorem 1a 1 ) of 13]. Letting T = Z, from Theorem 3.1, we easily obtain Theorem 3c 1 ) in 13]. COROLLARY 3.1 Assume that u, h C rd,u,h 0.Ifβ 0 is a real constant, then imlies ut)) β + hτ)uτ) τ for all t T 0 3.9) ut) { 1 + β)e h t, ) 1)} 1/ for all t T 0, 3.10) where ht) = ht). 3.11)

6 550 Wei Nian Li and Weihong Sheng Proof. Using Theorem 3.1, it follows from 3.9) that ut) = { β + hτ) 1 + β { β β) } 1/ e h/ t, σ τ)) τ } hτ) 1/ e h/t, σ τ)) τ ={β β)e h t, ) e h t, t)]}1/ ={β β)e h t, ) + 1 β} 1/ ={ 1 + β)e h t, ) 1)} 1/ for all t T 0, where the second equation holds because of Theorem 2.5, and the third equation holds because of Theorem 2.4i). This comletes the roof. Using Corollary 3.1, it is not difficult to obtain the following result. COROLLARY 3.2 Let k>0and T = kz 0, ). Ifu is a nonnegative function defined on T, β 0 and γ>0are real constants. Then t k 1 ut)) β + kγ ukτ) for all t T 3.12) τ=0 imlies { ut) 1 + β) 1 + kγ ) } t/k 1/ 1) for all t T. 3.13) Theorem 3.2. Assume that u, a, b, g, h C rd,u,a,b,g,hare nonnegative. If kt, s) is defined as in Theorem 2.6 such that kσt), t) 0 and k t, s) 0 for t,s T with s t, then ut)) at) + bt) kt, τ)gτ)uτ)) + hτ)uτ)] τ for all t T ) imlies where ut) { } 1/ at) + bt) e A t, σ τ))bτ) τ for all t T 0, 3.15) At) = kσ t), t)bt) gt) + ht) ) + k t, τ)bτ) gτ) + hτ) ) τ 3.16)

7 Some nonlinear dynamic inequalities on time scales 551 and 1 + at) Bt) = kσt), t) at)gt) + ht) + k t, τ) aτ)gτ)+ hτ) )] 1 + aτ) )] τ 3.17) for all t T 0. Proof. Define a function zt) by zt) = kt, τ)gτ)uτ)) + hτ)uτ)] τ for all t T ) Then z ) = 0. As in the roof of Theorem 3.1, we easily obtain 3.5) and 3.6). Using Theorem 2.5 and combining 3.18), 3.5) and 3.6), we have z t) = kσt), t)gt)ut)) + ht)ut)] + 1 kσt), t) at)gt) + ht) 1 + k t, τ) aτ)gτ)+ hτ) +bτ) gτ) + hτ) ) ] zτ) τ kσ t), t)bt) gt) + ht) ) kσt), t) at)gt) + ht) + k t, τ) aτ)gτ)+ hτ) = At)zt) + Bt) for all t T 0. k t, τ)gτ)uτ)) + hτ)uτ)] τ + at) ) ) ] + bt) zt) + aτ) ) k t, τ)bτ) 1 + at) )] + aτ) Therefore, using Theorem 2.7 and noting z ) = 0, we get zt) gt) + ht) gτ) + hτ) ) ] τ zt) )] τ e A t, σ τ))bτ) τ for all t T ) It is easy to see that the desired inequality 3.15) follows from 3.5) and 3.19). The roof of Theorem 3.2 is comlete. Remark 3.2. Clearly, if T = R, then the inequality established in Theorem 3.2 reduces to the inequality established by Pachatte in Theorem 1a 3 ) of 13]. Letting T = Z in Theorem 3.2, we easily obtain Theorem 3c 3 ) in 13].

8 552 Wei Nian Li and Weihong Sheng COROLLARY 3.3 Suose that ut), at) and kt, s) are defined as in Theorem 3.2.Ifat) is nondecreasing for all t T 0, then imlies ut)) at) + kt, τ)uτ) τ for all t T ) where ut) { 1 + at)]eāt, ) 1)} 1/ for all t T 0, 3.21) Āt) = 1 ) kσt), t) + k t, τ) τ for all t T ) Proof. Letting bt) = 1, gt) = 0 and ht) = 1 in Theorem 3.2, we obtain At) = 1 ) kσt), t) + k t, τ) τ = Āt) for all t T ) and Bt) = 1 { } kσt), t) 1 + at)] + k t, τ) 1 + aτ)] τ 1 + at) { } kσt), t) + k t, τ) τ = 1 + at)]āt) for all t T 0, 3.24) where the inequality holds because at) is nondecreasing for all t T 0. Therefore, by Theorem 3.2, using 3.23) and 3.24), we easily get { } 1/ ut) at) + e A t, σ τ))bτ) τ { } 1/ at) + eāt, σ τ)) 1 + aτ)]āτ) τ { } 1/ at) at)] eāt, σ τ))āτ) τ ={at) at)]eāt, ) eāt, t)]} 1/ ={ 1 + at)]eāt, ) 1)} 1/ for all t T 0. The roof of Corollary 3.3 is comlete. 4. An alication In this section, we resent an examle to illustrate our main results.

9 Some nonlinear dynamic inequalities on time scales 553 Examle 4.1. Consider the dynamic equation u t)) = Mt, ut)), t T 0, 4.1) where >1 is a constant, M: T 0 R R is a continuous function. Assume that Mt, ut)) ht) ut), 4.2) where ht) is as defined in Corollary 3.1. If ut) is a solution of eq. 4.1), then where ut) { 1 + C )e h t, ) 1)} 1/ for all t T 0, 4.3) C = u ), ht) = ht). 4.4) In fact, the solution ut) of eq. 4.1) satisfies the following equivalent equation: u t) = C + Mτ, uτ)) τ, t T ) Using assumtion 4.2), we have u t) C + hτ) uτ) τ, t T ) Now a suitable alication of Corollary 3.1 to 4.6) yields 4.3). Acknowledgements This work is suorted by the Natural Science Foundation of Shandong Province Y2007A05), the National Natural Science Foundation of China , ), the Project of Science and Technology of the Education Deartment of Shandong Province J06P51), and the Science Foundation of Binzhou University 2006Y01, BZXYLG200708). The authors are very grateful to the referee for his valuable comments and suggestions. References 1] Agarwal R P, Bohner M, O Regan D and Peterson A, Dynamic equations on time scales: A survey, J. Comut. Al. Math ) ] Agarwal R P, Bohner M and Peterson A, Inequalities on time scales: A survey, Math. Inequal. Al ) ] Akin Bohner E, Bohner M and Akin F, Pachatte inequalities on time scales, J. Inequal. Pure Al. Math ), Art. 6 Online: htt://jiam.vu.edu.au/) 4] Bohner M and Peterson A, Dynamic equations on time scales: An introduction with alications Boston: Birkhäuser) 2001)

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