Research Article A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations

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1 Hndaw Publshng Corporaton Advances n Dfference Equatons Volume 008, Artcle ID , pages do:0.55/008/ Research Artcle A Generalzed Sum-Dfference Inequalty and Applcatons to Partal Dfference Equatons Wu-Sheng Wang Department of Mathematcs, Hech College, Guangx, Yzhou , Chna Correspondence should be addressed to Wu-Sheng Wang, wang4896@6.com Receved October 007; Accepted 9 January 008 Recommended by Rgoberto Medna We establsh a general form of sum-dfference nequalty n two varables, whch ncludes both two dstnct nonlnear sums wthout an assumpton of monotoncty and a nonconstant term outsde the sums. We employ a technque of monotonzaton and use a property of stronger monotoncty to gve an estmate for the unknown functon. Our result enables us to solve those dscrete nequaltes consdered by Cheung and Ren 006. Furthermore, we apply our result to a boundary value problem of a partal dfference equaton for boundedness, unqueness, and contnuous dependence. Copyrght q 008 Wu-Sheng Wang. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.. Introducton Beng an mportant tool n the study of dfferental equatons and ntegral equaton, varous generalzatons of Gronwall nequalty, and ther applcatons have attracted great nterests of many mathematcans see 3 5. Some recent works can be found, for example, n 6 9 and some references theren. Along wth the development of the theory of ntegral nequaltes and the theory of dfference equatons, more attentons are pad to some dscrete versons of Gronwall-type nequaltes see, e.g., 0 for some early works.foundn 3, the unknown functon u n the fundamental form of sum-dfference nequalty u n a n f s u s s 0. can be estmated by u n a n n s 0 f s. Pang and Agarwal 4 consdered the nequalty u n P u 0 αu s Qg s u s ],. s 0

2 Advances n Dfference Equatons where α, P,andQare nonnegatve constants and u and g are nonnegatve functons defned on {,,...,T} and {,,...,T }, and they estmated that u n α n Pu 0 n s 0 Qg s, for all 0 n T. Another form of sum-dfference nequalty, u n c f s u s w u s ) f s u s ],.3 s 0 where c s a constant, f and f are both real-valued nonnegatve functons defned on N 0 {0,,,...},andw s a contnuous nondecreasng functon defned on u 0, such that w u > 0on u 0, and w u 0 0, for a real constant u 0, was estmated by Pachpatte 5 as u n Ω Ω c n s 0 f s n s 0 f s, whereω u : u u 0 ds/w s. Recently, dscretzaton see 6, 7 was also made for Ou-Yang s nequalty 8.In 6, the nequalty of two varables, u m, n c a s, t u s, t b s, t u s, t w u s, t ), was dscussed. Later, ths result was generalzed n 7 to the nequalty u p m, n c a s, t u q s, t b s, t u q s, t w u s, t ),.5 where c 0andp>q>0areall constant, a and b are both nonnegatve real-valued functons defned on a lattce n Z,andw s a contnuous nondecreasng functon satsfyng w u > 0, for all u>0. In ths paper, we establsh a more general form of sum-dfference nequalty ψ u m, n ) a m, n f s, t ϕ u s, t ) for nonnegatve ntegers m, n.in.6, we replace the constant c n.5 wth a functon a m, n and replace the functons u p, u q, u q w u n.5 wth the more general form of functons ψ u, ϕ u, ϕ u, respectvely. Moreover, we do not requre the monotoncty of ϕ and ϕ. We employ a technque of monotonzaton and use a property of stronger monotoncty to overcome the dffculty from nonmonotoncty so as to gve an estmate for the unknown functon u.our result enables us to solve the dscrete nequalty.5 and other nequaltes consdered n 7. Furthermore, we apply our result to a boundary value problem of a partal dfference equaton for boundedness, unqueness, and contnuous dependence.. Man result Throughout ths paper, let R denote the set of all real numbers, R 0,, and N 0 {0,,,...}. Gvenm 0,n 0 N 0,M,N N 0 { }, consder two lattces I m 0,M N 0 and J n 0,N N 0 of nteger ponts n R. LetΛ I J N 0. For any s, t Λ, let Λ s,t denote the sublattce m 0,s n 0,t Λ of Λ. For functons w m, z m, n, m,n N 0, ther frst-order dfferences are defned by Δw m w m w m, Δ w m, n w m,n w m, n, and Δ z m, n z m, n z m, n. Obvously, the lnear dfference equaton Δx m b m wth the ntal condton x m 0 0 has the soluton m b s. For convenence, n the sequel we complementarly defne that m 0 b s

3 Wu-Sheng Wang 3 Our basc assumptons for nequalty.6 are gven n the followng. H ψ s a strctly ncreasng contnuous functon on R satsfyng that ψ u > 0, for all u>0. H All ϕ, are contnuous functons on R and postve on 0,. H 3 a m, n 0onΛ. H 4 All f, are nonnegatve functons on Λ. Wth gven functons ϕ, ϕ,andψ, we defne { w u : max ϕ τ }, τ 0,u. { } ϕ τ w u : max w u, τ 0,u w τ. ) u W u, u : u w ψ x ), ) u W u, u : u w ψ x ),.3.4 where u > 0, are gven constants. Sometmes we smply let W u denote W u, u when there s no confuson. Obvously, W and W are both strctly ncreasng n u>0and therefore the nverses W, are well defned, contnuous, and ncreasng. Theorem.. Suppose that H ) H 4 )holdandu m, n s a nonnegatve functon on Λ satsfyng.6. Then, u m, n ψ { W W Υ m, n ) f s, t ]}.5 for m, n Λ m,n, a sublattce n Λ,where Υ m, n : W W Υ m, n ) Υ m, n : a m ) m 0,n 0 ] f s, t, a ) ) s,n 0 a s, n0 a m, t a m, t.6 and m,n Λ s arbtrarly gven on the boundary of the lattce { U : m, n Λ : W Υ m, n ) f s, t u } ),,..7 w ψ x Remark.. Dfferent choces of u n W, do not affect our results. For postve constants v / u,,, let W u u v /w ψ x. Obvously, W u W u W u

4 4 Advances n Dfference Equatons and W v W v W u. It follows that W W Υ m, n n f s, t W W Υ m, n n f s, t, that s, we obtan the same expresson n.5 f we replace W wth W. Moreover, by replacng W wth W, the condton n the defnton of U n our theorem reads m )) W Υ m,n n f s, t v w ψ x ), the left-hand sde of whch s equal to W u W Υ m,n m n f s, t and the rght-hand sde of whch equals u v w ψ x ) u w ψ x ) W u u w ψ x ). Comparson between both sdes mples that.8 s equvalent to the condton gven n the defnton of U n our theorem wth m, n m,n. Remark.3. If we choose ψ u u p, ϕ u u q,ϕ u u q w u, f s, t a s, t, and f s, t b s, t wth p>q>0n.6 and restrct a m, n to be a constant c, thenwecan apply Theorem. to nequalty.5 as dscussed n Proof of theorem Frst of all, we monotonze some gven functons ϕ n the sums. Obvously, w s and w s, defned by ϕ and ϕ n. and., are nondecreasng and nonnegatve functons and satsfy w s ϕ s,,. Moreover, we can check that the rato w s /w s s also nondecreasng. Therefore, from.6 we get ψ u m, n ) ) a m, n f s, t w u s, t, m, n Λ. 3. We frst dscuss n the case that a m, n > 0, for all m, n Λ. It means that Υ m, n > 0, for all m, n Λ. In such a crcumstance, Υ s postve and nondecreasng on Λ and satsfes Υ m, n a m ) m 0,n 0 ) ) a s,n0 a s, n0 a m, t a m, t a m, n. 3. Because ψ s strctly ncreasng, from 3. we have u m, n ψ Υ m, n ] ) f s, t w u s, t 3.3 ψ Υ m, n z m, n ), m, n Λ, where z m, n f s, t w u s, t ). 3.4

5 Wu-Sheng Wang 5 From the propertes of f and w, we see that z s nonnegatve and nondecreasng n each varable on Λ. Snce Υ s nondecreasng, for arbtrarly fxed par of ntegers K, L Λ m,n, we observe from 3.3 that u m, n ψ Υ K, L z m, n ), m, n Λ K,L. 3.5 Moreover, we note that w s nondecreasng and satsfes w u > 0, for u>0,, and that Υ K, L z m, n > 0. It mples by 3.5 that Δ Υ K, L z m, n ) n ) w t n )) 0 f m, t w u m, t n ) ψ Υ K, L z m, n w t n )) 0 f m, t w u m, t ψ Υ K, L z m, n w ψ Υ K, L z m, n )) f m, t f m, t θ ψ Υ K, L z m, t )), 3.6 where θ u : w u w u. 3.7 On the other hand, by the mean-value theorem for ntegrals, for arbtrarly gven m, n, m,n Λ K,L there exsts ξ n the open nterval Υ K, L z m, n, Υ K, L z m,n such that W Υ K, L z m,n ) W Υ K, L z m, n ) z m,n Υ K,L z m,n Υ K,L du w ψ u ) Δ Υ K, L z m, n ) w ψ ξ ) Δ Υ K, L z m, n ) w ψ Υ K, L z m, n )) by the monotoncty of w and ψ. It follows from 3.6 and 3.8 that W Υ K, L z m,n ) W Υ K, L z m, n ) 3.8 f m, t f m, t θ ψ Υ K, L z m, t )). 3.9 Keep n fxed and substtute m wth s n 3.9. Then, takng the sum on both sdes of 3.9 over s m 0,m 0,m 0,...,m, we get W Υ K, L z m, n ) W Υ K, L ) f s, t f s, t θ ψ Υ K, L z s, t )), 3.0

6 6 Advances n Dfference Equatons for all m, n Λ K,L, where we note from the defnton of z m, n n 3.3 and the remark about m 0 n the second paragraph of Secton that z m 0,n 0. For convenence, let Ξ m, n : W Υ K, L z m, n ), σ m, n : W Υ K, L ) f s, t Then, 3.0 can be rewrtten as Ξ m, n σ K, L f s, t θ ψ W Ξ s, t ))), 3.3 for all m, n Λ K,L, where we note that σ K, L σ m, n, for all m, n Λ K,L.Letg m, n denote the functon on the rght-hand sde of 3.3, whch s obvously a postve functon and nondecreasng n each varable. Snce the composton θ ψ W u s also nondecreasng n u, by 3.3, that s the fact that Ξ m, n g m, n,wehave ) n Δ g m, n θ t n ψ W ))) 0 f m, t θ ψ W ))) Ξ m, t g m, n θ ψ W ))) g m, n n f m, t θ ψ W ))) g m, t f m, t. θ ψ W g m, n ))) In order to estmate the left-hand sde of 3.4 further, we consder the followng ntegral: g m,n g m,n θ ψ W x )) g m,n g m,n w ψ W x )) w ψ W x )) W g m,n W g m,n W W w ψ x ) )) )) g m,n W W g m, n, where we note the defntons of W,W, and θ n.3,.4, and 3.7. Applyng the meanvalue theorem to 3.5, we see that for arbtrarly gven m, n, m,n Λ K,L, there exsts η n the open nterval g m, n,g m,n such that g m,n ) ) g m,n θ ψ W x )) Δ g m, n θ ψ W η )) Δ g m, n θ ψ W ))). 3.6 g m, n Thus, t follows from 3.4, 3.5, and 3.6 that W W )) )) g m,n W W g m, n f m, t, 3.7

7 Wu-Sheng Wang 7 for all m, n Λ K,L. Furthermore, usng the same procedure as done for 3.9, we keep n fxed and settng m s n 3.7. Then, summng up both sdes of 3.7 over s m 0,m 0,m 0,...,m, we get W W m )) )) g m, n W W σ K, L f s, t, 3.8 for all m, n Λ K,L, where we note the fact that g m 0,n σ K, L and the defnton of σ n 3.. By the monotoncty of W and ψ, the fact that Ξ m, n g m, n, gven n 3.3, and nequalty 3.8, weobtanfrom 3.5 that u m, n ψ Υ K, L z m, n ) ψ W { ψ W ) Ξ m, n ψ W )) g m, n ]} m )) W W σ K, L f s, t, 3.9 for all m, n Λ K,L, where we note the defntons of Ξ n 3. and g just after 3.3. Ths result also mples the partcular case that u K, L ψ { W W W W Υ K, L ) K )) L f s, t K ]} L f s, t. 3.0 For the arbtrary choce of K, L Λ m,n, t also mples that.5 holds for all m, n Λ m,n. The remander case s that a m, n 0, for some m, n Λ.Let Υ,ε m, n Υ m, n ε, 3. where ε>0 s an arbtrary small number. Obvously, Υ,ε m, n > 0, for all m, n Λ. Usng the same arguments as above, where Υ m, n s replaced wth Υ,ε m, n, weget u m, n ψ { W W W W Υ,ɛ m, n ) )) f s, t ]} f s, t, 3. for all m, n Λ m,n. Lettng ε 0,weobtan.5 because of contnuty of Υ,ε n ε and contnuty of W and W,for,. Ths completes the proof. Remark that m and n le on the boundary of the lattce U. In partcular,.5 s true for all m, n Λ when all w s, satsfy u /w ψ x, so we may take m M, n N.

8 8 Advances n Dfference Equatons 4. Applcatons to a dfference equaton In ths secton, we apply our result to the followng boundary value problem smply called BVP for the partal dfference equaton: Δ Δ ψ z m, n ) F m, n, z m, n ), m, n Λ, z m, n 0 ) f m, z m0,n ) g n, m, n Λ, where Λ : I J s defned as n the begnnng of Secton, ψ C 0 R, R s a strctly ncreasng odd functon satsfyng ψ u > 0, for u>0, F : Λ R R satsfes F m, n, u ) ) h m, n ϕ u h m, n ϕ u 4. for gven functons h,h : Λ R and ϕ C 0 R, R, satsfyng ϕ u > 0, for u>0, and functons f : I R and g : J R satsfy f m 0 g n 0 0. Obvously, 4. s a generalzaton of the BVP problem consdered n 7, Secton 3. So the results of 7 cannot be appled mmedately. In what follows we frst apply our man result to dscuss boundedness of solutons of 4.. Corollary 4.. All solutons z m, n of BVP 4. have the estmate { z m, n ψ W W Υ m, n ) ]} h s, t, 4.3 for all m, n Λ m,n,wherem,n are gven as n Theorem. and u { { ϕ ψ τ ) } W u / { ϕ ψ τ )} { max ϕ ψ τ )}}, max τ 0,x max τ 0,τ τ 0,x 4. W u u Υ m, n W { / max ϕ ψ τ )}, τ 0,x W Υ m, n ) ] h t, s, 4.4 Υ m, n ψ f s ) ψ f s ) ψ g t ) ψ g t ). Proof. Clearly, the dfference equaton of BVP 4. s equvalent to ψ z m, n ) ψ f m ) ψ g n ) It follows that ψ z m, n ) ψ f m ) ψ g n ) F s, t, z s, t ). h s, t ϕ z s, t ) 4.5 h s, t ϕ z s, t ) by 4..Leta m, n ψ f m ψ g n. Snce ψ z m, n ψ z m, n, 4.6 s of the form.6. Applyng Theorem. to nequalty 4.6, we obtan the estmate of z m, n as gven n ths corollary. 4.6

9 Wu-Sheng Wang 9 Corollary 4. gves a condton of boundedness for solutons. Concretely, f Υ m, n <, h s, t <, h s, t <, for all m, n Λ m,n, then every soluton z m, n of BVP 4. s bounded on Λ m,n. Next, we dscuss the unqueness of solutons for BVP Corollary 4.. Suppose addtonally that F m, n, u ) F m, n, u ) h m, n ϕ ψ u ) ψ u ) ) h m, n ϕ ψ u ) ψ u ) ), 4.8 for u,u R and m, n Λ : I J, wherei m 0,M N 0,J n 0,N N 0 as assumed n the begnnng of Secton wth natural numbers M and N, h,h are both nonnegatve functons defned on the lattce Λ, ϕ,ϕ C 0 R, R are both nondecreasng wth the nondecreasng rato ϕ /ϕ such that ϕ 0 0, ϕ u > 0, for all u>0and 0 ds/ϕ s, for,, and ψ C 0 R, R s a strctly ncreasng odd functon satsfyng ψ u > 0, foru>0. Then, BVP 4. has at most one soluton on Λ. Proof. Assume that both z m, n and z m, n are solutons of BVP 4.. From the equvalent form 4.5 of 4.,wehave ψ z m, n ) ψ z m, n ) h s, t ϕ ψ z s, t ) ψ z s, t ) ) h s, t ϕ ψ z s, t ) ψ z s, t ) ), for all m, n Λ, whch s an nequalty of the form.6, wherea m, n 0. Applyng Theorem. wth the choce that u u, we obtan an estmate of the dfference ψ z m, n ψ z m, n n the form.5, whereυ m, n 0, because a m, n 0. Furthermore, by the defnton of W we see that It follows that lmw u, u lm W u u 0,,. 4.0 W Υ m, n ) h s, t snce m<m,n<n. Thus, by 4.0 Υ m, n W W Υ m, n ) ] h s, t Smlarly, we get W Υ m, n n h s, t and therefore ] W W Υ m, n ) h s, t

10 0 Advances n Dfference Equatons Thus, we conclude from.5 that ψ z m, n ψ z m, n 0, mplyng that z m, n z m, n, for all m, n Λ snce ψ s strctly ncreasng. It proves the unqueness. Remark 4.3. If h 0orh 0n 4.8, the concluson of Corollary 4. also can be obtaned. Fnally, we dscuss the contnuous dependence of solutons of BVP 4. on the gven functons F, f, andg. Consder a varaton of BVP 4. Δ Δ ψ z m, n ) F m, n, z m, n ), m, n Λ, z m, n 0 ) f m, z m0,n ) g n, m, n Λ, 4.4 where ψ C 0 R, R s a strctly ncreasng odd functon satsfyng ψ u > 0foru>0, F C 0 Λ R, R, and f : I R, g : J R are functons satsfyng f m 0 g n 0 0. Corollary 4.4. Let F be a functon as assumed n the begnnng of Secton 4 and satsfy 4. and 4.8 on the same lattce Λ as assumed n Corollary 4.. Suppose that the three dfferences max f f, max g g, max F s, t, u F s, t, u 4.5 m I n J s,t,u Λ R are all suffcently small. Then, soluton z m, n of BVP 4.4 s suffcently close to the soluton z m, n of BVP 4.. Proof. By Corollary 4., thesolutonz m, n s unque. By the contnuty and the strct monotoncty of ψ, we suppose that maxψ ) ) f m ψ f m <ɛ, maxψ g n ) ψ g n ) <ɛ, m I n J max F s, t, u F s, t, u 4.6 <ɛ, s,t,u I J R where ɛ>0sasmall number. By the equvalent dfference equaton 4.5 and nequalty 4.8, we get ψ z m, n ) ψ z m, n ) ψ f m ) ψ f m ) ψ g n ) ψ g n ) ɛ F s, t, z s, t ) F s, t, z s, t ) { m ) )} m m 0 n n 0 ɛ F s, t, z s, t ) F s, t, z s, t ) F s, t, z s, t ) F s, t, z s, t ) h s, t ϕ ψ z s, t ) ψ z s, t ) h s, t ϕ ψ z s, t ) ψ z s, t ), 4.7

11 Wu-Sheng Wang that s, an nequalty of the form.6. Applyng Theorem. to 4.7,weobtan ψ z m, n ) ψ z m, n ) W W Υ m, n ) ] h s, t, 4.8 for all m, n Λ m,n,wherem,n are gven as n Theorem., Υ m, n W W Υ m, n ) ] h t, s, 4.9 Υ m, n { m m 0 ) n n 0 )} ɛ. By 4.0 we see that Υ m, n 0, as ɛ 0. It follows from 4.8 that lm ɛ 0 ψ z m, n ψ z m, n 0, and hence z m, n depends contnuously on F, f, and g snce ψ s strctly ncreasng. Our requrement on the small dfference, F F n Corollary 4.4, s stronger than the condton n 7, Theorem 3.3, but ours may be easer to check because one has to verfy the nequalty n hs condton for each soluton z m, n of BVP 4.4. Acknowledgments The author thanks Professor Wenan Zhang Schuan Unversty for hs valuable dscusson. The author also thanks the referees for ther helpful comments and suggestons. Ths project s supported by Foundaton of Guangx Natural Scence of Chna and by Foundaton of Natural Scence and Key Dscplne of Appled Mathematcs of Hech College of Chna. References R. Bellman, The stablty of solutons of lnear dfferental equatons, Duke Mathematcal Journal, vol. 0, no. 4, pp , 943. T. H. Gronwall, Note on the dervatves wth respect to a parameter of the solutons of a system of dfferental equatons, Annals of Mathematcs, vol. 0, no. 4, pp. 9 96, D. Baĭnov and P. Smeonov, Integral Inequaltes and Applcatons, vol.57ofmathematcs and Its Applcatons, Kluwer Academc Publshers, Dordrecht, The Netherlands, D. S. Mtrnovć, J. E. Pečarć, and A. M. Fnk, Inequaltes Involvng Functons and Ther Integrals and Dervatves, vol. 53 of Mathematcs and Its Applcatons, Kluwer Academc Publshers, Dordrecht, The Netherlands, B. G. Pachpatte, Inequaltes for Dfferental and Integral Equatons, vol. 97 of Mathematcs n Scence and Engneerng, Academc Press, San Dego, Calf, USA, R. P. Agarwal, S. Deng, and W. Zhang, Generalzaton of a retarded Gronwall-lke nequalty and ts applcatons, Appled Mathematcs and Computaton, vol. 65, no. 3, pp , O. Lpovan, Integral nequaltes for retarded Volterra equatons, Journal of Mathematcal Analyss and Applcatons, vol. 3, no., pp , Q.-H. Ma and E.-H. Yang, On some new nonlnear delay ntegral nequaltes, Journal of Mathematcal Analyss and Applcatons, vol. 5, no., pp , W. Zhang and S. Deng, Projected Gronwall-Bellman s nequalty for ntegrable functons, Mathematcal and Computer Modellng, vol. 34, no. 3-4, pp , T. E. Hull and W. A. J. Luxemburg, Numercal methods and exstence theorems for ordnary dfferental equatons, Numersche Mathematk, vol., no., pp. 30 4, 960.

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