Refined Hyers-Ulam approximation for Jensen and Euler-Lagrange Mappings

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1 Reed Hyes-Ul oo o Jese d Eule-Lgge Mgs Joh Mchel Rsss d M (Sk Joh Rsss Pedgogcl Dee E E Seco o Mhecs d Iocs Nol d Cods Uvesy o Ahes 0 Hocous S. Ahes GREECE e-l: jsss@edu.uo.g URL:h:// d Sscs d Modellg Scece Uvesy o Shclyde Lvgsoe Towe 6 Rchod S. Glsgow Scold UK G XH e-l:@ss.sh.c.uk ΠΕΡΙΛΗΨΗ Το 940 ο διάσηµος Μαθηµατικός S. M. Ul πρότεινε για λύση το φηµισµένο πρόβληµα ευστάθειας που φέρει το όνοµά του. Στη συνέχεια το 94 ο γνωστός Μαθηµατικός D. H. Hyes έλυσε το παραπάνω πρόβληµα του Ul για προσθετικές απεικονίσεις υπό τον όρο ότι θα ισχύει η γνωστή συνθήκη του Hyes για προσεγγιστικά προσθετικές απεικονίσεις. Σ αυτήν την εργασία γενικεύουµε το εν λόγω αποτέλεσµα του Hyes για εναλλακτικές απεικονίσεις τύπου Jese υπό τον όρο ότι θα ισχύει µία ασθενέστερη συνθήκη από την γνωστή συνθήκη του Hyes ως προς γινόµενα δυνάµεων ορισµένων o. Αυτή η διαδικασία (ocess οδηγεί στη βελτίωση (eee της γνωστής προσέγγισης του Hyes. Επιπλέον εισάγουµε στα Μαθηµατικά για πρώτη φορά τις εναλλακτικά προσθετικές απεικονίσεις πρώτης και δεύτερης µορφής και ερευνούµε αποτελέσµατα ευστάθειας σχετικά µε το πρόβληµα του Ul. Παρόµοια ερευνούµε Eule - Lgge τετραγωνικές απεικονίσεις και προσεγγιστικά προσθετικές απεικονίσεις που εκφυλίζονται σε γνήσια προσθετικές απεικονίσεις. Τα αποτελέσµατα αυτά µπορούν να εφαρµοστούν στη στοχαστική ανάλυση στα οικονοµικά και ασφαλιστικά µαθηµατικά καθώς επίσης και στη ψυχολογία και κοινωνιολογία.

2 ABSTRACT I 940 S. M. Ul oosed he ous Ul sbly oble. I 94 D. H. Hyes solved hs oble o Cuchy ddve gs subjec o he Hyes codo o oely ddve gs. I hs e we geelze he Hyes esul o he Ul sbly oble o leve Jese ye gs by cosdeg oely leve Jese ye gs ssyg codos weke h he Hyes codo es o oducs o owes o os. Ths ocess leds o eee o he well-kow Hyes oo o he Ul sbly oble. Besdes we oduce leve ddve gs o he s d secod o d vesge ee sbly esuls. Slly we vesge Eule- Lgge qudc gs d oely ddve gs degeeg o geue ddve gs. These sbly esuls c be led sochsc lyss cl d cul hecs s well s sychology d socology. ABSTRAIT E 940 S.M. Ul oosés le oblèe célèbe de sblé d' Ul. E 94 D.H. Hyes ésolu ce oblèe ou les cés dds de Cuchy suje à l codo de Hyes su les cés ovee dds. Ds ce cle ous géélsos le ésul de Hyes ou le oblèe de sblé d' Ul ou le ye le cés de Jese e cosdé le ye ovee le cés de Jese sss des codos lus ble que l codo de Hyes e ees des odus des ussces des oes. Ce ocessus èe à ue éloo de l'oo be coue de Hyes ou le oblèe de sblé d' Ul. Ss coe que ous éseos les cés dds les de l eèe e deuèe oe e éudos des ésuls covebles de sblé ou ces ésuls de sblé. De êe ous éudos des cés qudque d' Eule-Lgge e cés ovee dds se dégéé u cés dds vébles. Ces ésuls de sblé euve êe lqués ds l'lyse sochsque héques cèes e cuelles uss be qu'e l sychologe e l socologe. Key wods d hses: Ul sbly oble Hyes codo Cuchy su Jese g Eule-Lgge g geue ddve g. AMS (MOS Subjec Clssco: 39B.. Ioduco I 940 d 964 S. M. Ul [8] oosed he ous Ul sbly oble: "Whe s ue h by slghly chgg he hyoheses o heoe oe c sll sse h he hess o he heoe es ue o oely ue?"

3 3 I 94 D. H. Hyes [4] solved hs sbly oble o Cuchy ddve gs subjec o he ollowg Hyes codo ( ( ( + δ (HC o oely Cuchy ddve gs : X Y o ed δ 0 d ll X whee X s el oed sce d Y el Bch sce. I 95 D. G. Boug [] ws he secod uho o e he Ul oble o ddve gs. I 978 ccodg o P.M. Gube [3] hs kd o sbly obles s o cul ees obbly heoy d he cse o ucol equos o dee yes. I J. M. Rsss ([7-][4-7] d 003 d 005 M. J. Rsss d he s uho ([3] [6] solved he bove Ul oble o dee gs. I 999 P. Gvu [] sweed queso o ous [9] coceg he sbly o he Cuchy equo. I 998 S.- M. Jug [5] d M. J. Rsss d he s uho ([-3] [6] vesged he Ul sbly o ddve d qudc gs o esced dos. I hs e we geelze he Hyes esul o he Ul sbly oble o leve Jese ye gs by cosdeg oely leve Jese d Jese ye gs ssyg codos weke h he Hyes codo es o oducs o owes o os. Also we oduce leve ddve gs o he s d he secod o d vesge ee sbly esuls. These esuls c be led sochsc lyss cl d cul hecs s well s sychology d socology. I 997 P. Mllv [6] ublshed eesg eeece book o sochsc lyss. Thoughou hs e le X be el oed sce d Y be el Bch sce he cse o ucol equles s well s le X d Y be el le sces o ucol equos. Besdes le us deoe wh { 3...} he se o ul ubes he se o el ubes d o soe ed P ( I X he leve Cuchy su he leve su o s o α β ρ α + β. he we oduce he ollowg leve sus: S0 P ( ( ( ( ( S( P ( + + ( + (

4 4 S( P ( + ( + ( he leve su o secod o + S3( P ( + ( + ( he leve Jese su d he leve Jese ye su. I we cosde he oul ( ( ( ( S4 P + ρ ρ M( δ δ 0 ρ ( < ρ < A ( l{ (. (* ρ > Also we cosde equly ( A( M( (** o ll X. I Q ( he we deoe α Π ( Q δ δ 0 ρ α + β. β Deo.0. A g A : X Y s clled leve Cuchy A sses A( ( + [ A( + A( ] (C o ll X. Deo.. A g A : X Y s clled leve ddve o he s o A sses he ucol equo A( + + A( - -A(- ( o ll X. We oe h ( s equvle o he leve Jese equo + y A A( + A( y (J

5 5 o + y -. A g A : X Y s clled leve Jese g A sses he ucol equo (J. Deo.. A g A : X Y s clled leve ddve o he secod o A sses he ucol equo A( + - A( - -A(- ( o ll X. We oe h ( s equvle o he leve Jese ye equo y A A( A( y (JT o + y -. A g A : X Y s clled leve Jese ye g A sses he ucol equo (JT. Ou sbly esul s he ollowg: Theoe.3. I g : X Y sses he oely leve Jese ye equly S(P Π ( Q (3 4 o ll P X Q he hee ess uque leve Jese ye g A : X Y whch sses he oul (* d he equly (** o ll X. I oeove s esuble o ( s couous o ech ed X o ll X d. A( A(. Jese Mgs d Oule o he Poo ( We oe h he Hyes codo (HC o oely leve Jese ye gs s he coesodg equly (3 whe α β 0 ;hus Π ( Q δ. ( I we elce S4 ( P whs ( P ( 0 3 he bove equly (3he we esly esblsh sbly esuls o he eg ou kds o gs coesodg o he leve sus S ( P ( 03.

6 6 I hese ou cses we ove h M ( o he equly (** hs o be subsued by M( /. (3 To ove Theoe.3we gue s ollows: I c elcg 0 (3oe ges (0 0.Seg 0 (3 we ge ( (. Besdes subsug (3 e ob Theeoe ( + ( δ ρ. o ( ( ( + ( δ ρ ( ( δ ρ ρ. Hece ( ( ( ( ( ( ( ( ρ ρ [ ]( δ ρ o. Assue < ρ <. I oe elces wh he bove geel equly he he ds ( ρ ρ ρ ( ( [ ]( ρ δ o.. Assue ρ > Fo he bove wo geel equles we ove h he sequece { ( } wh ( < ρ < ( { ( ρ > s Cuchy sequece. Fo (*(3 d he coleeess o Y oe oves h well-deed g A: X Y ess such h he g A : X Y sses he leve Jese ye equo (JT. I s esy o ove h

7 7 A( < ρ < A ( { ( A ρ > The es o he oo o he esece d uqueess o A: X Y s oed s sl o ou deled oos eeeces ([8]-[0]. The oo o he ls sseo he bove Theoe.3 s obvous ccodg o ou wok [7] 98. The sgul cse ρ s oe. We ee he ede o P. Gvu [] d ([7]-[0] o logous sgul cses. 3. Eule-Lgge Qudc Mgs The ollowg heoe 3. s well-kow o -desol Eule-Lgge qudc gs. Theoe 3. ([] [7]. Le X be oed le sce d le Y be el colee oed le sce. Assue ddo h : X Y s g o whch hee ess cos c (deede o 0 such h he Eule-Lgge qudc ucol equly holds o ll ( X. The he l [ ] ( + + ( ( + ( c Q ( l ( ess o ll X d ll N d Q : X Y s he uque Eule-Lgge qudc g ssyg he ucol equo o ll ( X such h equly [ ] Q ( + + Q ( Q ( + Q ( ( Q( c holds o ll X. I hs seco we esblsh he Ul sbly o 3-desol qudc gs.

8 8 Deo 3.. Le X be oed le sce d le Y be el colee oed le sce. The g Q : X Y s clled 3-desol Eule-Lgge qudc he ucol equo Q( Q( Q( Q( 3 [ Q( Q( Q( 3 ] (4 holds o ll ( 3 X 3.Noe h g Q s clled qudc becuse he ucol equo Q ( ( Q( (5 holds o ll X d ll N. I c subsuo o 3 0 equo (4 yelds h Q(0 0. Subsug 3 0 oe ges h he ucol equo [ Q( + Q ( 0 4 Q( + Q( 0 ]o Q( ( Q( holds o ll X. The duco o N wh - yelds equo (5. Theoe 3.. Le X be oed le sce d le Y be el colee oed le sce. Assue ddo h : X Y s 3-desol g o whch hee ess cos c (deede o 3 0 such h he Eule-Lgge qudc ucol equly ( ( + + ( + + ( [ ( ( ( 3 ] holds o ll ( 3 X 3. The he l c (6 Q ( l ( (7 ess o ll X d ll N d Q : X Y s he uque 3-desol Eule- Lgge qudc g ssyg he ucol equo (4 such h equly ( Q( 5 c 4 (8 holds o ll X. Moeove ucol dey Q( - Q( holds o ll X d ll N.

9 9 Poo. Subsuo o 3 0 equly (6 yelds h c ( 0. (9 8 Moeove subsug 3 0 equly (6 d eloyg (9 d he gle equly oe cocludes [ ] ( + ( 0 4 ( + ( 0 c o ( - 4 ( - ( 0 c o ( - 4 ( ( - 4 ( - ( 0 + ( 0 o - c c 5 ( 4 ( + c o 8 8 hus oe ges h he bsc ucol equly ( 5 ( c c (. (0 3 holds o ll X whee c (5/4c. Relcg ow wh (0 oe cocludes h ( ( c ( o 4 ( ( c (0 4 ( - holds o ll X.Fucol equles (0 - (0 d he gle equly yeld ( 4 ( ( ( + ( 4 ( o h he ucol equly [ ] c ( + ( 4 4 ( ( c (- holds o ll X. Slly by duco o N wh - he bsc equly (0 cl h he geel ucol equly 4 ( ( c ( - ( holds o ll X d ll N.I c he bsc equly (7 wh - yeld he ucol equly ( ( c ( o he equly ( ( ( ( c ( - ( holds o ll X.Moeove by duco hyohess wh - he geel equly ( oe ges h

10 0 ( - - ( - ( ( c ( - (b holds o ll X.Thus ucol equles ( - (b d he gle equly ly ( ( ( ( ( ( + ( ( o ( ( [ ] ( ( c ( + ( c ( coleg he oo o he equed geel ucol equly (. Cl ow h he sequece { ( } coveges. Noe h o he geel equly ( d he coleeess o Y oe oves h he bove sequece s Cuchy sequece. I c > j > 0 he j j j ( j j ( ( ( ( holds o ll X d ll j N.Seg h j he bove elo d eloyg he geel equly ( oe cocludes h j j j ( j -j ( ( ( h ( h j (- j c ( - o - j j j ( ( c ( - < c -j o j j l - ( ( j coleg he oo h he sequece { } 0 ( coveges. Hece Q Q( s well-deed g v he oul (7.Ths es h he l (7 ess o ll X. I ddo cl h g Q sses he ucol equo (4 o ll ( 3 X 3.I c s cle o he ucol equly (6 d he l (7 h he ollowg equly ( ( + + ( + [ ] ( 4 ( + ( + ( c holds o ll ( 3 X 3 d ll N.Theeoe oe ges [ ( ] [ ( + 3 ] + [ ( + 3 ] l + l l [ 3 ] + l ( 4 l + l + l ( ( ( 3 l ( c 0

11 o g Q sses he equo (4 o ll ( 3 X 3. Thus Q s 3- desol qudc g. I s cle ow o he geel ucol equly ( d he oul (7 h equly (8 holds X coleg he esece oo o hs Theoe 3.. The oo o uqueess s oed s obvous d hus he sbly o hs Theoe 3. s colee. 4. Geue Addve Mgs I hs seco we vesge oe ddve gs degeeg o geue ddve gs. Deo 4.. Le X d Y be el le sces. Le α (α α... α R -{( }. The g A : X Y s clled ddve he ddve ucol equo A holds o evey X (... whee A ( ( s by bu ed d equls o 3... d y ed ( 0 : 0 <. Deo 4.. Le X d Y be el oed le sces. Le α (α α...α R -{( }.The g : X Y s clled oely ddve he oely ddve ucol equly ( ck (... (3 holds o evey (... X whee s by bu ed d equls o 3... wh el cos c 0 (deede o... X y ed ( 0 : 0 < d y ed el ( 0 :

12 (... K K 0 (4 holds o evey (... X. Le 4.. I K s gve v (4 he K 0 o y ed el 0. Poo. I c ke uco F F( ( 0 d R.I s cle h ( ( 0 o R :. Thus F s cove o.theeoe F ( F F o o R : d 0 (... whee s by bu ed d equls o Tkg 0 o X (... d R : we ge o o. Bu s cle h o 0.Theeoe we hve h 0 K o >.Slly o 0 <. Thus F s cocve o ( ( 0 F R : 0<. Theeoe.

13 3 Tkg 0 (... we ge K o 0 < coleg he oo o Le Le us deoe I { ( R : 0 < > d > 0< < } d I { ( R : 0 < 0< < d > > } such h - < o y ( I d - < o y ( I.Noe h oely ddve gs e o ddve cse K d > 0. I hs cse Y s ssued o be colee. Also K 0 0 d he sgul cse K ( 0. Theoe 4.. Le X d Y be oed le sces. Le α (α α... α R -{( }: 0 < whee s by bu ed d equls o 3... Assue ddo h : X Y s oely ddve g ssyg (3 wh 0. Dee ( ( ( Ι ( ( Ι o ll X d N o {0... } whee I { ( R : 0 < > d > 0< < } d I { ( R : 0 < 0< < d > > }. The he oul A ( ( (5 ess o ll X d N o d A : X Y s he uque ddve g ssyg ( A( (5 o ll X.

14 4 Poo. I s useul o he ollowg vesgo o obseve h o (3 wh 0 (... d 0< we ge ( 0 0 (0 0. (6 Now cl o N 0 {0... } h ( ( (6 holds o ll X. Fo 0 s vl. Fo (4 wh ( N {... } we ob 0 K o 0 0 o ( K K (7 o evey X d y ed el R : 0 wh 3...Slly o (4 wh - ( N we ge K o ( K K (8 o evey X d y ed el R : 0 wh 3... Fo (3 d (7 wh ( N we ge ( ( ck (... 0 o ( ( (9 whch s (6 o I holds. Slly o (3 d (8 wh - ( 0 ( N we ob whch s (6 o I holds. ( ( ck (... 0 o Assue (6 s ue d o (9 wh ( ( (0 o lce o we ge: + + ( ( ( (. (

15 5 Slly o (0 wh o lce o we ob: ( ( ( ( ( ( + +. ( These ouls ( d ( by duco ove oul (6. I s obvous o (6 h A dees g A : X Y gve by (5.Flly cl o (3 d (6 we c ge h A : X Y s ddve. I c s cle o he ucol equly (3 he Le 4. d he oul (6 h he ollowg ucol equly ( (... ck holds o ll (... X d ll N 0 wh ( ( : I holds. Theeoe ( ( (... ck o ( ( 0 ck A A becuse - < o y ( I o yeldg h g A : X Y sses he ddve ucol equo (. Slly o (3 he Le 4. d (6 we ge h ( A A ( (... ck holds o ll (... X d ll N 0 wh ( ( : I holds. Theeoe ( ( (... ck o ( ( 0 ck A A

16 6 becuse - < o ( I lyg h A : X Y sses ( coleg he oo h A c be ddve g X. Ths colees he esece oo o he bove Theoe 4.. The Uqueess oo o Theoe 4. s cle becuse A : X Y d A : X Y e wo ddve gs ssyg (5 he A d A ssy A( A ( ( ( 0 o A( A ( o ll X. Reeeces [] D. G. Boug Clsses o soos d bodeg soos Bull. Ae. Mh. Soc. 57 ( [] P. Gvu A swe o queso o Joh M. Rsss coceg he sbly o Cuchy equo. I: Advces Equos d Iequles Hdoc Mh. Sees U.S.A [3] P. M. Gube Sbly o Isoees Ts. Ae. Mh. Soc. U.S.A. 45 ( [4] D. H. Hyes O he sbly o he le ucol equo Poc. N. Acd. Sc. 7 (94-4: The sbly o hooohss d eled ocs "Globl Alyss-Alyss o Molds" Teube - Tee zu Mhek 57 ( [5] S.-M. Jug O he Hyes-Ul sbly o he Fucol Equos h hve he Qudc Poey J. Mh. Al. & Al. ( [6] P. Mllv Sochsc Alyss SgeBel997. [7] J. M. Rsss O Aoo o Aoely Le Mgs by Le Mgs J. Fuc. Al. 46 ( [8] J. M. Rsss O Aoo o Aoely Le Mgs by Le Mgs Bull. Sc. Mh. 08 ( [9] J. M. Rsss Soluo o Poble o Ul J. Ao. Th. 57 ( [0] J. M. Rsss Colee soluo o he ul-desol Poble o Ul Dscuss. Mhe. 4( [] J. M. Rsss Soluo o he Ul Sbly Poble o Eule-Lgge qudc gs J. Mh. Al. Al. 0 ( [] J. M. RsssO he Ul sbly o ed ye gs o esced dos J. Mh. Al. Al. 76( [3] J. M. Rsss d M. J. Rsss O he Ul sbly o Jese d Jese ye gs o esced dos J. Mh. Al. Al. 8( [4] J. M. Rsss Asyoc behvo o ed ye ucol equos Ausl. J.Mh. Al. Alcos (004Issue -. [5] J. M. Rsss The Ul sbly oble oo o oely qudc gs by qudc gs J. Iequ. Pue d Al. Mh.5(004Issue 3-9. [6] J. M. Rsss d M. J. Rsss Asyoc behvou o leve Jese d Jese ye ucol equos Bull. Sc. Mh. 9 (005 Issue [7] J. M. Rsss O he geel qudc ucol equo Bol. Soc. M. Mec (3 ( [8] S. M. Ul "Pobles Mode Mhecs Wley - Iescece New Yok 964 Ch. VI.

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