Research Article A Generalized Nonlinear Sum-Difference Inequality of Product Form

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1 Joural of Applied Mahemaics Volume 03, Aricle ID 47585, 7 pages hp://dx.doi.org/0.55/03/47585 Research Aricle A Geeralized Noliear Sum-Differece Iequaliy of Produc Form YogZhou Qi ad Wu-Sheg Wag School of Mahemaics ad Saisics, Hechi Uiversiy, Guagxi, Yizhou , Chia Correspodece should be addressed o Wu-Sheg Wag; wag4896@6.com Received 9 Augus 03; Acceped November 03 Academic Edior: Mehme Sezer Copyrigh 03 Y. Qi ad W.-S. Wag. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. We esablish a geeralized oliear discree iequaliy of produc form, which icludes boh ocosa erms ouside he sums ad composie fucios of oliear fucio ad ukow fucio wihou assumpio of moooiciy. Upper boud esimaios of ukow fucios are give by echique of chage of variable, amplificaio mehod, differece ad summaio, iverse fucio, ad he dialecical relaioship bewee cosas ad variables. Usig our resul we ca solve boh he discree iequaliy i Pachpae (995). Our resul ca be used as ools i he sudy of differece equaios of produc form.. Iroducio Beig a impora ool i he sudy of exisece, uiqueess, boudedess, sabiliy, ad oher qualiaive properies of soluios of differeial equaios ad iegral equaios, various geeralizaios of Growall iequaliies [, ] ad heir applicaios have araced grea ieress of may mahemaicias (such as [3 6]). Some rece works ca be foud, for example, i [7 0] ad some refereces herei. Alog wih he developme of he heory of iegral iequaliies ad he heory of differece equaios, more aeio is paid o some discree versios of Growall-Bellma ype iequaliies (such as [3, 4, 3]). Some rece works ca be foud, for example, i [4 4] ad some refereces herei. Pachpae [4] obaied he explici boud o he ukow fucio of he followig sum-differece iequaliy: u () (c + f (s) u (s))(c + g (s) u (s)). () Pachpae [3] obaied he esimaio of he ukow fucio of he followig iequaliy: u () (c + f (s) u (s))(c + g (s) u (s)). () The, he esimaio ca be used o sudy he boudedess, asympoic behavior, ad slow growh of he soluios of he sum-differece equaio: x () =k(p () + (p () + f ( s) x (s)) f ( s) x (s)). However, he boud give o such iequaliies i [3, 4] is odireclyapplicableihesudyofceraisum-differece equaios. I is desirable o esablish ew iequaliies of he above ype, which ca be used more effecively i he sudy of cerai classes of sum-differece equaios of produc form. I his paper, we esablish a ew iegral iequaliy of produc form u () (p () + (p () + f (, s) φ (u (s))) f (, s) φ (u (s))), N 0, where p i, f i, φ i (i =, ) may o be moooe. For φ, φ, we employ a echique of moooizaio o cosruc (3) (4)

2 Joural of Applied Mahemaics wo fucios; he secod possesses sroger moooiciy ha he firs. We ca demosrae ha iequaliies () ad (), cosidered i [3, 4], respecively, ca also be solved wih our resul. Fially, we expoud ha we ca give esimaio of soluios of a class of sum-differece equaios of produc form.. Mai Resul ad Proof I his secio, we proceed o solve he discree iequaliy (4) ad prese explici bouds o he embedded ukow fucio. Le N 0 := {0,,,...}, N := {,,...},adn b a := {a,a +,a +,...,a + = b} (a N 0,,b N). For fucio z(), is differece is defied by Δz = z( + ) z(). Obviously,helieardiffereceequaioΔz() = b() wih he iiial codiio z( 0 ) = 0 has he soluio z() = s= 0 b(s). For coveiece, i he sequel we complemearily defie ha 0 s= 0 b(s) = 0. Firs of all, we moooize some give fucios p i, f i, φ i i he sum; le q i () = max p i (), N 0 g i (, s) = max f i (, s), i =,, N 0 (5) where q i ad g i are all odecreasig i (i =, ) ad saisfy Le q i () p i () 0, g i (, s) f i (, s) 0. (6) w (u) = max s [0,u] φ i (s), w (u) := max { φ (s) s [0,u] w (s) }w (u), (7) where w i (u) is odecreasig i u (i =, ) ad w (u)/w (u) is also odecreasig i u ad saisfies where W respecively. w i (u) φ i (u), i =,, (8) x ds W (x) = x 0 w (s), x 0 >0, (9) x W (x) := x 0 w (W (s))ds w (W (s)), x 0 >0, (0) x ds W 3 (x) := x 0 w (W (W (s))), x 0 >0, (), W deoe he iverse fucio of W, W, Theorem. Le f, g be oegaive ad give fucios o N 0.Supposehau is a oegaive ad ukow fucio. The, he discree iequaliy (4) gives u () W 3 (A () +B()))), N b 0, () where W, W, W 3 are defied by (9), (0),ad(),respecively, W, W, W 3 deoe he iverse fucios of W, W, W 3, respecively, A () =W 3 (W (W (q () q ())+ B () = + (g (, ) q () g (, )), g (, s) +g (, ) q () g (, )) ad b is he larges aural umber such ha b=max { N 0 :A() +B() Dom 3 ), W 3 (A () +B()) Dom ), g (, s)), (3) W 3 (A () +B())) Dom )}. (4) Proof. Usig (5), (6), (7), ad (8), we observe ha u () (p () + (p () + (q () + f (, s) φ (u (s))) (q () + f (, s) φ (u (s))) g (, s) w (u (s))) (q (T) + (q (T) + g (, s) w (u (s))) g (T, s) w (u (s))) g (T, s) w (u (s))), N T 0, (5) where T N J 0 is chose arbirarily. Le V() deoe he fucio o he righ-had side of (5), amely, V () =(q (T) + (q (T) + g (T, s) w (u (s))) g (T, s) w (u (s))), N T 0, (6) which is a oegaive ad odecreasig fucio o N T 0 wih V(0) = q (T)q (T).The(4)isequivaleo u () V (), N T 0. (7)

3 Joural of Applied Mahemaics 3 Usig he differece formula Δx () y () =x(+) Δy () +y() Δx () (8) ad he moooiciy of w i ad V, from(6) ad(7), we observe ha ΔV () =(q (T) + Δ(q (T) + g (T, s) w (u (s))) +Δ(q (T) + (q (T) + g (T, s) w (u (s))) g (T, s) w (u (s))) g (T, s) w (u (s))) =(q (T) + g (T, s) w (u (s))) g (T, ) w (u ()) +g (T, ) w (u ()) (q (T) + g (T, s) w (u (s))) (q (T) + g (T, s) w (V (s))) g (T, ) w (V ()) +g (T, ) w (V ()) (q (T) + g (T, s) w (V (s))) =w (V ()) (q (T) g (T, ) +q (T) g (T, ) w (V ()) w (V ()) +g (T, ) w (V ()) w (V ()) g (T, s) w (V (s)) +g (T, ) g (T, s) w (V (s))) w (V ()) (q (T) g (T, ) +q (T) g (T, ) w (V ()) w (V ()) +(g (T, ) g (T, s) +g (T, ) g (T, s)) w (V ())), for all N T 0.From(9), we have N T 0, (9) ΔV () w (V ()) q (T) g (T, ) +q (T) g (T, ) w (V ()) w (V ()) +(g (T, ) g (T, s) +g (T, ) g (T, s))w (V ()), N T 0. (0) O he oher had, by he mea value heorem for iegrals, for arbirarily give iegers, + N T 0, here exiss ξ i he ope ierval (V(), V( + )) such ha W (V (+)) W (V ()) V(+) = V() ds ΔV () = w (s) w (ξ) ΔV () w (V ()), NT 0, () where W is defied by (9). From (0)ad(), we have W (V (+)) W (V ()) q (T) g (T, ) +q (T) g (T, ) w (V ()) w (V ()) +(g (T, ) g (T, s) +g (T, ) g (T, s))w (V ()), ()

4 4 Joural of Applied Mahemaics for all N T 0.Byseig=i () ad subsiuig = 0,,,..., successively, we obai W (V ()) W (V (0)) + + q (T) g (T, ) q (T) g (T, ) w (V ()) w (V ()) From (4), we obai Δx () =q (T) g (T, ) w (V ()) w (V ()) +(g (T, ) g (T, s) +g (T, ) g (T, s))w (V ()) + (g (T, ) g (T, s) W (V (0)) g (T, ) g (T, s))w (V ()) q (T) g (T, ) q (T) g (T, ) w (V ()) w (V ()) (g (T, ) g (T, s) +g (T, ) g (T, s))w (V ()), N T 0. (3) q (T) g (T, ) w (x ())) w (x ())) From (6), we have +(g (T, ) g (T, s) w (x ()))Δx() w (x ())) q (T) g (T, ) +g (T, ) g (T, s))w (x ())), N T 0. (6) Le x() deoe he fucio o he righ-had side of (3); amely, x () =W (V (0)) + q (T) g (T, ) +(g (T, ) g (T, s) +g (T, ) g (T, s))w (x ())), (7) + + q (T) g (T, ) w (V ()) w (V ()) (g (T, ) g (T, s) +g (T, ) g (T, s))w (V ()), N T 0. (4) N T 0. Oce agai, performig he same procedure as i (), (), ad (3), (7)gives W (x ()) W (x (0)) + + q (T) g (T, ) (g (T, ) g (T, s) The x(0) = W (V(0))+ q (T)g (T, ), x is a oegaive ad odecreasig fucio o N T 0,ad(3)isequivaleo V () W (x ()), N T 0. (5) +g (T, ) g (T, s)) w (x ()))

5 Joural of Applied Mahemaics 5 W (x (0)) + + q (T) g (T, ) (g (T, ) g (T, s) +g (T, ) g (T, s)) w (x ())), N T 0, (8) where W is defied by (0). Le z() deoe he fucio o he righ-had side of (8); amely, z () =W (x (0)) + + q (T) g (T, ) (g (T, ) g (T, s) +g (T, ) g (T, s))w (x ())), N T 0. (9) The z(0) = W (x(0))+ q (T)g (T, ), z is a oegaive ad odecreasig fucio o N T 0,ad(8)isequivaleo x () W (z ()), N T 0. (30) From (9)ad(30), we obai Δz () =(g (T, ) g (T, s) +g (T, ) g (T, s))w (x ())) (g (T, ) g (T, s) +g (T, ) g (T, s))w for all N T 0.From(3), we have Δz () w (W (W (z ()))) g (T, ) g (T, s) +g (T, ) g (T, s), (z ()))), N T 0. (3) (3) Oce agai, performig he same procedure as i (), (), ad (3), (3)gives W 3 (z ()) W 3 (z (0)) + (g (T, ) g (T, s) +g (T, ) g (T, s)), N T 0. Usig (7), (5), ad (30), from (33)wehave u () V () W (x ()) W (z ())) W 3 (W 3 (z (0)) + (g (T, ) g (T, s) W 3 (W 3 (W (x (0)) + (33) +g (T, ) g (T, s))))) + q (T) g (T, )) (g (T, ) g (T, s) +g (T, ) g (T, s))))) W 3 (W 3 (W (W (q (T) q (T)) q (T) g (T, )) q (T) g (T, )) (g (T, ) g (T, s) +g (T, ) g (T, s))))), N T 0. (34)

6 6 Joural of Applied Mahemaics As =T,(34)yields u (T) W 3 (W 3 (W (W (q (T) q (T)) (g (T, ) g (T, s) q (T) g (T, )) q (T) g (T, )) +g (T, ) g (T, s))))). (35) Sice T N, adt b is chose arbirarily i (35), he esimaio () is derived. This complees he proof of Theorem. 3. Applicaio We cosider a sum-differece equaio of produc form From (36), we have x () =(a() + (b() + x () (a() + f (, s) φ (x (s))) (b() + g (, s) φ (x (s))), N 0. f (, s) φ ( x (s) )) g (, s) φ ( x (s) )), N 0. (36) (37) Le u() = x(), p () = a(), p () = b(), f (, s) = f(, s), adf (, s) = g(, s) i (37); he (37) ishe iequaliy of he form (4). Applyig our resul we ge he esimaio of soluio of he sum-differece equaios of produc form (36). Ackowledgmes This research was suppored by Naioal Naural Sciece Foudaio of Chia (Projec o. 608) ad Guagxi Naural Sciece Foudaio (0GXNSFAA053009). Refereces [] T. H. Growall, Noe o he derivaives wih respec o a parameer of he soluios of a sysem of differeial equaios, Aals of Mahemaics, vol. 0, o. 4, pp. 9 96, 99. [] R. Bellma, The sabiliy of soluios of liear differeial equaios, Duke Mahemaical Joural, vol.0,pp , 943. [3] B. G. Pachpae, O a ew iequaliy suggesed by he sudy of cerai epidemic models, Joural of Mahemaical Aalysis ad Applicaios,vol.95,o.3,pp ,995. [4] B. G. Pachpae, O some ew iequaliies relaed o cerai iequaliies i he heory of differeial equaios, Joural of Mahemaical Aalysis ad Applicaios,vol.89,o.,pp.8 44, 995. [5]R.P.Agarwal,S.Deg,adW.Zhag, Geeralizaioofa rearded Growall-like iequaliy ad is applicaios, Applied Mahemaics ad Compuaio, vol.65,o.3,pp.599 6, 005. [6] W.-S. Cheug, Some ew oliear iequaliies ad applicaios o boudary value problems, Noliear Aalysis: Theory, Mehods & Applicaios,vol.64,o.9,pp. 8,006. [7] W.-S. Wag, A geeralized rearded Growall-like iequaliy i wo variables ad applicaios o BVP, Applied Mahemaics ad Compuaio,vol.9,o.,pp.44 54,007. [8] R. P. Agarwal, C. S. Ryoo, ad Y.-H. Kim, New iegral iequaliies for ieraed iegrals wih applicaios, Joural of Iequaliies ad Applicaios, vol.007,aricleid4385,8 pages, 007. [9] A. Abdeldaim ad M. Yakou, O some ew iegral iequaliies of Growall-Bellma-Pachpae ype, Applied Mahemaics ad Compuaio,vol.7,o.0,pp ,0. [0] W. S. Wag, D. Huag, ad X. Li, Geeralized rearded oliear iegral iequaliies ivolvig ieraed iegrals ad a applicaio, Joural of Iequaliies ad Applicaios, vol. 03, aricle 376, 03. [] T. E. Hull ad W. A. J. Luxemburg, Numerical mehods ad exisece heorems for ordiary differeial equaios, Numerische Mahemaik,vol.,o.,pp.30 4,960. [] D. Wille ad J. S. W. Wog, O he discree aalogues of some geeralizaios of Growall s iequaliy, vol. 69, pp , 965. [3] B.G.PachpaeadS.G.Deo, Sabiliyofdiscree-imesysems wih rearded argume, Uilias Mahemaica,vol.4,pp.5 33, 973. [4] E. Yag, A ew oliear discree iequaliy ad is applicaio, Aals of Differeial Equaios,vol.7,o.3,pp.6 67, 00. [5] B. G. Pachpae, O some fudameal iegral iequaliies ad heir discree aalogues, Joural of Iequaliies i Pure ad Applied Mahemaics,vol.,o.,aricle5,3pages,00. [6] F. W. Meg ad W. N. Li, O some ew oliear discree iequaliies ad heir applicaios, Joural of Compuaioal ad Applied Mahemaics,vol.58,o.,pp ,003. [7] W.-S. Cheug ad J. Re, Discree o-liear iequaliies ad applicaios o boudary value problems, Joural of Mahemaical Aalysis ad Applicaios,vol.39,o.,pp ,006. [8] B. G. Pachpae, Iegral ad Fiie Differece Iequaliies ad Applicaios, vol. 05, Elsevier Sciece, Amserdam, The Neherlads, 006.

7 Joural of Applied Mahemaics 7 [9] W. Sheg ad W. N. Li, Bouds o cerai oliear discree iequaliies, Joural of Mahemaical Iequaliies, vol.,o., pp.79 86,008. [0] Q.-H. Ma ad W.-S. Cheug, Some ew oliear differece iequaliies ad heir applicaios, Joural of Compuaioal ad Applied Mahemaics,vol.0,o.,pp ,007. [] W. S. Wag, A geeralized sum-differece iequaliy ad applicaios o parial Differece equaios, Advaces i Differece Equaios,vol.008,aricle,008. [] W. S. Wag, Esimaio o cerai oliear discree iequaliy ad applicaios o boudary value problem, Advaces i Differece Equaios, vol. 009, Aricle ID , 009. [3] W. S. Wag, Z. Li, ad W. S. Cheug, Some ew oliear rearded sum-differece iequaliies wih applicaios, Advaces i Differece Equaios, vol. 0, aricle 4, 0. [4] H. Zhou, D. Huag, ad W. S. Wag, Some ew differece iequaliies ad a applicaio o discree-ime corol sysems, Joural of Applied Mahemaics, vol. 0, AricleID 4609,4pages,0.

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