APPROXIMATE FUNCTIONAL INEQUALITIES BY ADDITIVE MAPPINGS

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1 Joural of Mathematical Iequalities Volume 6, Number 3 0, doi:0.753/jmi APPROXIMATE FUNCTIONAL INEQUALITIES BY ADDITIVE MAPPINGS HARK-MAHN KIM, JURI LEE AND EUNYOUNG SON Commuicated by J. Pečarić Abstract. Let be a give positive iteger, G a -divisible abelia group, X a ormed space ad f : G X. We prove a geeralized Hyers-Ulam stabitity of the followig fuctioal iequality f x+ f y+fz f + z + ϕx,y,z, x,y,z G, which has bee itroduced i [3], uder some coditios o ϕ : G G G [0,.. Itroductio The stability problem of equatios origiated from a questio of Ulam [9] cocerig the stability of group homomorphisms. We are give a group G ad a metric group G with metric ρ,. Give ε > 0, does there exist a δ > 0 such that if f : G G satisfies ρ f xy, f x f y < δ for all x,y G, the a homomorphism h : G G exists with ρ f x,hx < ε for all x G? I 94, D. H. Hyers [4] cosidered the case of approximately additive mappigs betwee Baach spaces ad proved the followig result. Suppose that E ad E are Baach spaces ad f : E E satisfies the followig coditio: there if a ε 0 such that f x + y f x f y ε for all x,y E. The the limit hx =lim f x exists for all x E ad there exists a uique additive mappig h : E E such that f x hx ε. Moreover, if f tx is cotiuous i t R for each x E, the the mappig h is R-liear. Mathematics subject classificatio 00: 39B7, 39B8. Keywords ad phrases: Cauchy Jese iequality, geeralized Hyers Ulam stability. This study was supported by the Basic Research Program through the Natioal Research Foudatio of Korea fuded by the Miistry of Educatio, Sciece ad Techology No c D l,zagreb Paper JMI

2 46 H. KIM,J.LEE AND E. SON The method which was provided by Hyers, ad which produces the additive mappig h, was called a direct method. This method is the most importat ad most powerful tool for studyig the stability of various fuctioal equatios. I 978, Th. M. Rassias [5] provided a geeralizatio of Hyers Theorem which allows the Cauchy differece to be ubouded. Let E ad E be two Baach space ad f : E E be a mappig such that f tx is cotiuous i t R for each fixed x. Assume that there exists ε > 0ad0 p < such that f x + y f x f y ε x p + y p The there exists a uique liear mappig T : E E such that f x Tx p ε x p for all x E. I 990, Th. M. Rassias [6] durig the 7th Iteratioal Symposium o Fuctioal Equatios asked the questio whether such a theorem ca also be proved for p. This result is also true for p < 0. I 99, Z. Gajda [] followig the same approach as i [5], gave a affirmative solutio to this questio for p >. It was show by Z. Gajda [], as well as by Th. M. Rassias ad P. Semrl [6], that oe caot prove a Th. M. Rassias type theorem whe p =. The couterexamples of Z. Gajda [], as well as of Th. M. Rassias ad P. Semrl [6], have stimulated several mathematicias to ivet ew defiitios of approximately additive or approximately liear mappigs, cf. P. Gǎvruta [] ad S. Jug [8], who amog others studied the stability of fuctioal equatios. I 994, a geeralized result of Rassias theorem was obtaied by P. Gǎvruta i []. Let G be a -divisible abelia group N i.e., a a : G G is a surjectio ad X be a ormed space with orm. Now, for a mappig f : G X, we cosider the followig geeralized Cauchy-Jese equatio x + y f x+ f y+fz=f + z, x,y,z G, which has bee itroduced i the referece [3]. First of all, we recall some result i the paper [3]. PROPOSITION.. For a mappig f : G X, the followig statemets are equivalet. a f is additive. x + y b f x+ f y+fz=f c f x+ f y+fz x + y + z, x,y,z G. + z, x,y,z G.

3 STABILITY OF FUNCTIONAL INEQUALITY 463 The geeralized Hyers-Ulam stability of fuctioal equatio b ad fuctioal iequality c has bee preseted i the paper [3] for a special case =. I this paper, we are goig to improve the theorems of [3] without usig the oddess of approximate additive fuctios cocerig the fuctioal iequality c for the geeral case.. geeralized Hyers-Ulam stability of c From ow o, let G be a -divisible abelia group for some positive iteger ad f : G Y ad let Y be a Baach space. THEOREM.. Let ϕ : G 3 R + satisfy lim k k ϕ k x, k y, k z=0 for all x,y,z G ad ˇϕx,z := i+ ϕ i+ x,0, i z+ϕ i+ x,0, i z <, for all x,z G. Suppose that a mappig f : G Y satisfies the fuctioal iequality f x+ f y+fz + z + ϕx,y,z for all x,y,z G. The there exists a uique additive mappig h : G Y such that f x hx ˇϕx,x+ ϕx, x,0 + f 0 3 Proof. For all x G, lettig y = x, z = 0i ad dividig both sides by, we have f x+ f x ϕx, x,0 + f 0 4 Replacig x by x ad lettig y = 0adz = x i, we get f x+f x+ f 0 ϕx,0, x+ f 0 5 Replacig x by x i 5, oe has f x+fx+ f 0 ϕ x,0,x+ f 0 6 Putgx= f x f x. Associatig 5 with 6 yields gx gx ϕx,0, x+ϕ x,0,x + f 0 that is, gx gx ϕx,0, x+ϕ x,0,x + f 0 7

4 464 H. KIM,J.LEE AND E. SON It follows from 7that gl x l gm x m m k=l k gk x k+ gk+ x m g = k k=l k x gk+ x m [ k=l k+ ϕ k+ x,0, k x+ϕ k+ x,0, k x + ] k f 0 8 for all oegative itegers m ad l with m > l 0adx G. Sice the right had side of 8 teds to zero as l, we obtai the sequece { gk x } is Cauchy for all k x G. Because of the fact that Y is a Baach space it follows that the sequece { gk x } coverges i Y. Therefore we ca defie a fuctio h : G Y by k g k x hx= lim k k, x G. Moreover, lettig l = 0adm i 8 yields gx hx ˇϕx,x+ f 0 Hece we have f x f x hx ˇϕx,x+ f 0 9 It follows from 4ad9 that f x hx ˇϕx,x+ ϕx, x,0 + f 0 It follows from that hx+hy hx + y = hx+hy+h x y = lim k k gk x+g k y+g k x + y = lim k k f k x+ f k y+ f k x + y f k x f k y f k x + y lim k k f0 +ϕ k x, k y, k x+y+ f0 +ϕ k x, k y, k x+y + f k x + y + f k x + y + f k x + y + f k x + y lim k k ϕ k x, k y, k x + y + ϕ k x, k y, k x + y +ϕ k x + y,0, k x + y + ϕ k x + y,0, k x + y = 0

5 STABILITY OF FUNCTIONAL INEQUALITY 465 for all x,y G.Thisimpliesthat hx+hy= hx + y for all x,y G. Hece the mappig h is additive. Next, let h : G Y be aother additive mappig satisfyig f x h x ˇϕx,x+ ϕx, x,0 The we have hx h x = k hk x k h k x + f 0 k hk x f k x + f k x h k x k ˇϕ k x, k x+ ϕk x, k x,0 + k f 0 [ ] = i=k i+ ϕ i+ x,0, i x+ϕ i+ x,0, i x + ϕk x, k x,0 k + k f 0 for all k N ad all x G. Takig the limit as k, we coclude that hx=h x This completes the proof. Supposethat X is a ormed space i the followig corollaries. If we put ϕx, y, z := θ x p y q z t ad ϕx,y,z := θ x p + y q + z t i Theorem., respectively, the we get the followig Corollaries. ad.3. COROLLARY.. Let p+q+t <, p,q,t > 0 ad θ > 0. If a mappig f : X Y satisfies the followig fuctioaliequality f x+ f y+fz + z + θ x p y q z t for all x,y,z X, the f is additive. COROLLARY.3. Let 0 < p,q,t <, θ > 0. If a mappig f : X Y satisfies the followig fuctioal iequality f x+ f y+fz + z + θ x p + y q + z t for all x,y,z X, the there exists a uique additive mappig h : X Y such that [ p f x hx θ p + x p + x q + t x t] + f 0 for all x X.

6 466 H. KIM,J.LEE AND E. SON The followig corollary is a immediate cosequece of Theorem.. COROLLARY.4. For ay fixed positive iteger, suppose that a mappig f : G Y satisfies the iequality f x+ f y+fz f + z ε for all x,y,z G, where ε 0. The there exists a uique additive mappig h : G Y satisfyig the iequality f x hx ε + ε + f 0 We ca similarly prove aother stability theorem uder a somewhat differet coditio as follows: lim k REMARK.5. Let ϕ : G 3 R + satisfy ˇϕx,y,z := i ϕi+ x,0, i z+ϕ i x, i y,0 <, ϕ k x, k y, k z=0forallx,y,z G. If f : G Y is a mappig such that k f x+ f y+fz + z + ϕx,y,z for all x,y,z G, the there exists a uique additive mappig h : G Y such that f x hx ˇϕx,x,x+ + f 0 Proof. Replacig x by x ad lettig y = 0adz = x i, we get f x+f x+ f 0 ϕx,0, x+ f 0 0 Forallx G, lettig y = x, z = 0i, we have f x+ f x+f0 ϕx, x,0+ f 0 The we obtai the followig iequality f x fx ϕx,0, x+ϕx, x,0+ + f 0 The rest of the proof is similar to proof of Theorem.. We may obtai more simple ad sharp approximatio tha that of Theorem. for the stability result uder the oddess coditio.

7 STABILITY OF FUNCTIONAL INEQUALITY 467 COROLLARY.6. Let ϕ : G 3 R + satisfy lim k k ϕ k x, k y, k z=0 for all x,y,z G ad ˇϕx,z := i+ ϕi+ x,0, i z <, for all x,z G. Suppose that f : G Y is a mappig such that f x= fx for all x G ad f x+ f y+fz + z + ϕx,y,z for all x,y,z G, the there exists a uique additive mappig h : G Y such that f x hx ˇϕx,x We may obtai a stability result of fuctioal equatio b by the similar way as i the proof of Theorem.. COROLLARY.7. Let ϕ : G 3 R + satisfy ˇϕx,z := i+ ϕ i+ x,0, i z+ϕ i+ x,0, i z <, for all x,z G ad lim k ϕ k x, k y, k z=0 for all x,y,z G. If f : G Yisa mappig such that f 0=0 ad k f x+ f y+fz f + z ϕx,y,z for all x,y,z G, the there exists a uique additive mappig h : G Y such that f x hx ˇϕx,x+ ϕx, x,0 Now, we cosider aother stability result of fuctioal iequality c i the followigs. THEOREM.8. Let ϕ : G 3 R + satisfy ϕx,z := i x ϕ i,0, z i+ + ϕ x i,0, z i+ <, for all x,z G ad lim k k ϕ x, y, z =0 for all x,y,z G. If f : G Y is a k k mappig such that k f x+ f y+fz + z + ϕx,y,z 3

8 468 H. KIM,J.LEE AND E. SON for all x,y,z G. The there exists a uique additive mappig h : G Y such that f x hx ϕx,x+ ϕx, x,0 4 Proof. Note that f 0 =0siceϕ0,0,0=0. Let y = x, z = 0i3 ad dividig both sides by, we have f x+ f x ϕx, x,0 Replacig x by x ad lettig y = 0adz = x i 3, we get Replacig x by x i 6, we get 5 f x+f x ϕx,0, x 6 f x+fx ϕ x,0,x 7 f x f x.usig6ad7 yields the fuctioal iequal- Putgx= ity gx gx ϕx,0, x+ϕ x,0,x Replacig x by x,weget x gx g ϕ x,0, x + ϕ x,0, x 8 The remaiig roof is similar to the correspodig proof of Theorem.. This completes the proof. Supposethat X is a ormed space i the followig corollaries. If we put ϕx, y, z := θ x p y q z t ad ϕx,y,z := θ x p + y q + z t i Theorem.8, respectively, the we get the followig Corollaries.9 ad.0. COROLLARY.9. Let p + q + t >, p,q,t > 0, θ > 0. If a mappig f : X Y satisfies the followig fuctioal iequality f x+ f y+fz x + y for all x,y,z X, the f is additive. + z + θ x p y q z t

9 STABILITY OF FUNCTIONAL INEQUALITY 469 COROLLARY.0. Let p,q,t >, θ > 0. If a mappig f : X Y satisfies the followig fuctioal iequality f x+ f y+fz + z + θ x p + y q + z t for all x,y,z X, the there exists a uique additive mappig h : X Y such that [ p f x hx θ p + x p + x q + x t] t for all x X. We ca similarly prove aother stability theorem uder somewhat differet coditios as follows: REMARK.. Let ϕ : G 3 R + satisfy ϕx,y,z := i x ϕ i,0, z x i+ + ϕ, y i+ i+,0 <, ad lim k k ϕ x, y, z = 0forallx,y,z G. If f : G Y is a mappig such that k k k f x+ f y+fz + z + ϕx,y,z for all x,y,z G, the there exists a uique additive mappig h : G Y such that f x hx ϕx,x,x We may obtai more simple ad sharp approximatio tha that of Theorem.8 for the stability result uder the oddess coditio. REMARK.. Let ϕ : G 3 R + satisfy ϕx,z := x i ϕ i,0, z i+ <, for all x,z G ad lim k k ϕ x, y, z = 0forallx,y,z G. If a mappig f : k k G Y is odd ad f x+ f y+fz + z + ϕx,y,z for all x,y,z G, the there exists a uique additive mappig h : G Y such that k f x hx ϕx,x

10 470 H. KIM,J.LEE AND E. SON We may alteratively obtai a stability result of fuctioal equatio b by the similar way as i the proof of Theorem.8. COROLLARY.3. Let ϕ : G 3 R + satisfy ϕx,z := i x ϕ i,0, z i+ + ϕ x i,0, z i+ <, for all x,z G ad lim k k ϕ x k, y k, z k =0 for all x,y,z G. If f : G Y is a mappig such that x + y f x+ f y+fz f + z ϕx,y,z for all x,y,z G, the there exists a uique additive mappig h : G Y such that f x hx ϕx,x+ ϕx, x,0 Ackowledgemet. The authors are very grateful for the referees valuable commet. Correspodece should be addressed to Euyoug So. REFERENCES [] Z. GAJDA, O the stability of additive mappigs, Iter. J. Math. Math. Sci. 4 99, [] P. GǍVRUTA, A geeralizatio of the Hyers Ulam Rassias stability of approximately additive mappigs, J. Math. Aal. Appl , [3] Z.-X. GAO, H.-X. CAO, W.-T. ZHENG AND LU XU, Geeralized Hyers Ulam Rassias stability of fuctioal iequalities ad fuctioal equatios, J. Math. Iequal. 3009, [4] D. H. HYERS, O the stability of the liear fuctioal equatio, Proc. Nat. Acad. Sci. U.S.A. 7 94, 4. [5] TH. M. RASSIAS, O the stability of the liear mappig i Baach spaces, Proc. Amer. Math. Soc , [6] TH. M. RASSIAS, The stability of mappigs ad related topics, I Report o the 7th ISFE, Aequ. Math , [7] TH. M. RASSIAS AND P. ŠEMRL, O the behaviour of mappigs which do ot satisfy Hyers Ulam Rassias stability, Proc. Amer. Math. Soc. 4 99, [8] S. JUNG, O Hyers Ulam Rassias stability of approximately additive mappigs, J. Math. Aal. Appl , 6. [9] S.M.ULAM, A Collectio of the Mathematical Problems, Itersciece Publ. New York, 960. Joural of Mathematical Iequalities jmi@ele-math.com

11 STABILITY OF FUNCTIONAL INEQUALITY 47 Received October 6, 0 Hark-Mah Kim Departmet of Mathematics, Chugam Natioal Uiversity 79 Daehago, Yuseog-gu, Daejeo , Korea hmkim@cu.ac.kr Juri Lee Departmet of Mathematics, Chugam Natioal Uiversity 79 Daehago, Yuseog-gu, Daejeo , Korea aas@hamail.et Euyoug So Departmet of Mathematics, Chugam Natioal Uiversity 79 Daehago, Yuseog-gu, Daejeo , Korea sey8405@hamail.et Joural of Mathematical Iequalities jmi@ele-math.com