Fuzzy Dynamic Equations on Time Scales under Generalized Delta Derivative via Contractive-like Mapping Principles

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1 Idia Joural of Sciece ad echology Vol 9(5) DOI: 7485/ijs/6/v9i5/8533 July 6 ISSN (Pri) : ISSN (Olie) : Fuzzy Dyamic Euaios o ime Scales uder Geeralized Dela Derivaive via Coracive-lie Mappig Priciples Ch Vasavi G Suresh Kumar * ad M S N Mury Deparme of Mahemaic K L Uiversiy Gree-Field Vaddeswaram Guur - 55 Adhra Pradesh Idia; vasavilu@gmailcom drgs6@luiversiyi Deparme of Mahemaic Saiiva D No -47 Opposie Sae Ba of Idia Ba Sree Nuzvid Krisha - 5 Adhra Pradesh Idia; drms@gmailcom Absrac Bacgroud/Objecives: he firs order oliear Fuzzy Dyamic Euaios (FDE play a impora role i rece years o model he dyamic sysems wih uceraiies ad vagueess he prese wor deals wih obaiig he exisece ad uiueess crieria for oliear FDEs o ime scales which provides a foudaio for he oliear sudies i he field of FDEs Mehods/Saisical Aalysis: I place of Baach coracio priciple coracive-lie mappig priciples o parial ordered ses is used as a ool o sudy he FDEs o ime scales uder geeralized dela derivaive Fidigs: he oliear FDEs o ime scales usig geeralized dela derivaive are o sudied so far he geeralized dela derivaive is based o four forms which allow us o obai ew soluios for FDEs wih decreasig legh of heir suppor Moreover he differeiabiliy i hird ad fourh forms is lied wih he cocep of swichig pois hese resuls iclude boh coiuous ad discree FDEs uder oe framewor Applicaio/Improvemes: hese resuls are useful o sudy ualiaive ad uaiaive properies for oliear FDEs o ime scales which arise i biological ecoomical ad corol egieerig problems Keywords: Coracive-Lie Mappig Fuzzy Dyamic Euaio Geeralized Dela Derivaive ime Scales Iroducio Fuzzy differeial euaios (Fde are appropriae i he modelig of may real-world pheomea Usig he cocep of H-derivaive defied i Kaleva developed he heory of Fdes For deailed sudy o Fdes ad heir applicaios we refer o 3- Dyamic euaios o ime scales 34 is a emergig area which uifies effecively boh differeial ad differece euaios A example of his ype ca be see i seasoally breedig populaios givig rise o ew o-overlappig geeraios I 5 he auhor iroduced g derivaive ad g iegral sudied he fudameal properies of fuzzy se-valued mappigs o ime scales I 6 he auhor iroduced SH derivaive ad sudied he FDEs o ime scales Preiary properies ad defiiios relaed o fuzzy se-valued mappig fixed poi heorems ad calculus o ime scale are preseed i Secio I Secio 3 he properies of g derivaive are sudied which are ecessary for laer discussio I Secio 4 wih he help of fixed poi heorems i 78 we esablish exisece ad uiueess crieria for -soluio ad g -soluio g of FDEs o ime scales Preiaries Deoe E { u : R [ ]} u is fuzzy covex upper semi coiuou ormal ad suppor [u] is compac For v w E v w) sup dh ([ v] [ w] ) d Where H is he meric defied i For < he * Auhor for correspodece

2 Fuzzy Dyamic Euaios o ime Scales uder Geeralized Dela Derivaive via Coracive-lie Mappig Priciples level se where [ u] { y R / u( y) } P ( R ) P ( R ) is defied as i For ay v w E ad γ R [ v w] [ v] [ w] [ γ v] γ[ v] ] he parial orderig iduced by he se iclusio i E ie v w [ v] [ w] ] v w E he coverse of he parial order is deoed by For ay u v E a w E u v w w is he H-differece of u ad ν which is deoed by uθ v Defiiio A fuzzy-valued fucio G : E is called Huuhara differeiable a a poi if a G ( ) E he is exis i E ) ) ( ) ( ) ( ) Θ G Θ G G he eleme G'( ) is he Huuhara derivaive of G a i he meric space (E D) 8 Defiiio A alerig disace fucio χ : [ ) [ ) is defied as (i)χ is coiuous ad χ( ) χ( ) for (ii) χ( ) Defiiio 3 8 Le Y be a space wih meric d ad g : Y Y be a fucio he g is wealy coracive if for ay alerig disace fucios χ ad φ χ ( d( g( x ) g( x))) χ( d( x x) ϕ( d( x x)) x x Y Lemma 8 If Y is a parial ordered se wih parial orderig ad Y is complee meric space wih meric d Le g : Y Y be a mappig g ( x) g( x ) x x ad saisfyig χ ( d( g( x ) g( x))) χ( d( x x) ϕ( d( x x)) f o r x x () Suppose X saisfies eiher if a o-decreasig seuece { y m} m N is coverge o y Î Y he { y m } y " mî N or ha g is coiuous If y ÎY y g ( y ) he g has a fixed poi y m is coverge if a o-icreasig seuece { } mîn o y Y he y { ym } m N or ha g is coiuous If here exiss y Y such ha y g ( y ) he g has a fixed poi Lemma 8 Uder he assumpio of Lemma if every pair of elemes of X has a upper boud or a lower boud he g has a uiue fixed poi Moreover if x is he fixed poi of g he x X g ( x) x For basic properie fudameal resuls ad oaios o ime scale we follow 4 3 Differeiabiliy ad iegrabiliy: Now we sudy he resuls o g derivaive for FSVF o ime scales which are defied i 5 he g derivaive give i Defiiio 3 i 5 ca be euivalely wrie as Defiiio 3 A FSVF G : E is called g differeiable a if Gg ( ) E if for < < δ such ha ) ( ) ( ))) ( ( ) ) ( ) Θ σ Θ ( ) σ µ µ ( ) provided he H-differeces ) Θ σ ( ) G σ ( ) ) Θ ) exiss (or) ) ( ( ( ) ) ( ) ( ) ( ) σ Θ Θ ( ) σ µ µ ( ) provided he H-differeces (G ( σ ( ) Θ )) ( ) Θ σ ( ) exiss (or) G 3) ( ) ( ))) ( ) ( ) ( ) Θ σ Θ ( ) σ µ µ ( ) provided he H-differeces G ) Θ σ ( )) ) Θ σ ( ))) exiss (or) ( ( ( ) ) ( ( ) F( ) ( ) σ Θ Θ ( ) σ µ µ ( ) Vol 9 (5) July 6 wwwidjsorg Idia Joural of Sciece ad echology

3 Ch Vasavi G Suresh Kumar ad M S N Mury provided he H-differeces G σ ( ) Θ ) σ ( ) Θ ) exiss ( he eleme G g ( ) is called he g derivaive of G a We say ha G is g differeiable if G is differeiable i form (i) ad g 3 g 4 g - differeiable G is differeiable i (ii)(iii)(iv) forms respecively Remar 3 If R he g differeiabiliy coicides wih he H- derivaive defied i Moreover he g differeiabiliy coicides wih he derivaive defied i 4 I is more geeral ha he differeiabiliy iroduced i 3 which coicides wih (i) ad (ii) bu i does cover (iii) ad (iv) Example 3 Le N { : N } Le G : N E defied by G ( ) u u E is a riagular fuzzy umber he G is g differeiable a all N ( ) Θ ) u G g ( ) ( σ ( ) Θ ))) G u µ ( ) Example 3 Le G : E give by ) e p ( ) u where u E is a riagular fuzzy umber he G is g differeiable a all ( e ( σ ( ) ) Θe ( ) ) u p( ) e ( σ ( ) ) Gg ( ) p p p u µ ( ) hu if G is g differeiable or g differeiable a he G is g differeiable as i Defiiio 3 However whe G is g differeiable a as i Defiiio 3 he G may or may o be g or g differeiable which ca be verified from he followig example Example 33 Le G : E give by G ( ) c where c ( 3) E is a riagular fuzzy umber he G is g differeiable for all whe > ad G g ( ) c ( 3) ad G is g differeiable for all whe < ad G g ( ) c ( 3 - -) For eiher (i) or (ii) of Defiiio 3 holds A he H-differeces i (iii) of Defiiio 3 exiss Le [ c deoes [ c Defiiio 3 Le G : [ E ad [ he poi is said o be a swichig poi for he g differeiable fucio G if i saisfies ay oe of he wo codiios below ie for ay eighborhood of < < (I) G is g differeiable o ( ad G is g differeiable o [ σ ( ) ) ie G is o-decreasig o ad o-icreasig o σ ( ) ) or ( ] ] [ (II) G is g differeiable o ( ad G is g differeiable o [ σ ( ) ) ie G is o-icreasig o ad o-decreasig o σ ( ) ) ( ] ] [ heorem 3 Le s G : [ E be a FSVF ad (i)if S is a swichig poi of ype (I) g differeiable of G he G is g differeiable a S 4 (ii)if S is a swichig poi of ype (II) g differeiable of G he G is 3 g differeiable a S Proof If S is righ-scaered ad swichig poi for G of ype I he ( µ ( σ ( s ) Θ s ) ) ( ( s ) Θ s )) ( σ µ (3) (3) he i i (3) exiss if G is g differeiable or 4 g differeiable ad he i i (3) exiss if G is g or 4 g differeiable Hece G is 4 g differeiable I a similar way we ca prove (ii) Defiiio 33 5 he iegral of g G : E o J defied level wise by s G ( s g( s : g SF ( J ) J J J where SF (J ) he se of all g iegrable secors of G o J For properies o g iegral we refer o 5 4 FDEs o ime scales Now we focus our aeio o he followig oliear FDE o ime scale Vol 9 (5) July 6 wwwidjsorg Idia Joural of Sciece ad echology 3

4 Fuzzy Dyamic Euaios o ime Scales uder Geeralized Dela Derivaive via Coracive-lie Mappig Priciples z ( ) )) ) z (4) deoes he g derivaive hroughou G :[ E E is a oliear fucio which is rdcoiuous ad u E E (4) is called FIVP o ime scales Defiiio 4 Le C rd ([ σ(d))] E be he rd-coiuous fuzzy fucios (i)a soluio ) C rd ([ σ(d))] E ) is called a g differeiable soluio for (4) if ) is a ai-derivaive of G ( ) saisfyig (4) A soluio for (4) is called (ii) ()-soluio if i is g differeiable (iii) ()-soluio if i is g differeiable Cosider he parial order o C C rd ([ σ(d))] E ) as g g g( ) g( ) g g C ad Clearly C rd ([ σ(d))] E ) is parial ordered se Lemma 4 8 he followig resuls hold o C rd ([ σ(d))] E ) : (i)if { fm} m N C is odecreasig wih parial orderig such ha f m f i C he f m f m N (ii)if { f } C m m N is oicreasig wih parial orderig such ha f m f i C he m N Lemma 4 Le G be rd-coiuous (i)a fuzzy fucio z C is called a g differeiable soluio o (4) iff i saisfies he iegral euaio ) z (4) (ii)a fuzzy fucio z C is called a g differeiable soluio o (4) iff i saisfies he iegral euaio z ) ( ) [ (43) or ) z Θ ( ) s (44) Defiiio 4 A soluio for (4) is said o be a lower soluio if he fuzzy fucio η C saisfies η ( ) )) σ (d)] ) z If η is g differeiable he η is said o be a lower g differeiable ad if η is g differeiable he i is lower g differeiable A soluio for (4) is said o be a upper soluio if he fuzzy fucio η C saisfies η ( ) F( )) σ (d)] ) z If η is g differeiable (respecively g differeiable) he η is said o be a upper g differeiable soluio (respecively a upper g differeiable soluio) heorem 4 (Local Exisece ad Uiueess heorem) Le G :[ E E be rd- coiuous If here exiss a lower g differeiable soluio η C rd ([ σ(d))] E ) for (4) ad (i)g is o-decreasig wro secod variable ie for v w he G v ) ) ( (ii)for comparable elemes G is wealy coracive ie for alerig disace fucios χ ad φ χ( v) ))) χ( v ) ϕ( v )) if v (45) he a uiue g differeiable soluio z exiss for (4) o [ σ (d)] Proof Defie he operaor A : C C by [ A z]( ) z From Lemma 4 (i) z C is he soluio of (4) if z C is he fixed poi of A Defie D a meric o C by D ( v ) sup { v( ( e ( ) } v C s e ( ) Where > large eough such ha < his meric is euivale o meric D because D ( v ) v ) e ( ) D ( v ) v C However C rd ([ σ(d))] E ) D is a complee meric space From assumpio (i) ad from Lemma we have [ A v]( ) z v ( s z ( s [ A ]( ) wheever v ad [ Hece he 4 Vol 9 (5) July 6 wwwidjsorg Idia Joural of Sciece ad echology

5 Ch Vasavi G Suresh Kumar ad M S N Mury operaor A is o-decreasig Now from (ii) χ( v) ))) χ( v )) v I a corary Assume ha v ) < v ) )) v (46) Sice χ is o-decreasig we have χ( v ) χ ( u) v ))) From (46) χ ( v ) χ( v) ))) v w From (ii) ϕ ( v ) which implies v ) ad hece χ ( v) ))) From Defiiio we ge D ( v) ) which is a coradicio For v cosider D A v A w )( ) sup { [ A v ]( ) [ A w ]( ) e ( ) } ( herefore σ ( d )] sup D v ( ( s e ( ) σ ( d )] sup v ( ) ( ) s e ( ) sup D ( v ) e ( ) s e ( ) [ e ( ) D ( v ) sup w e ( ) σ ( d )] e ( ) ( ) ( ) sup e D v w D ( v ) e ( ) D ( A ) v A w D ( v ) v Crd ([ ) Hece for alerig disace fucio β e ( ) β D ( v ) β( D ( A v A w ) e ( ) β( D ( v ) ( ( ) D v β β D ( v ) holds he if ( ) ( ) ( ) ϕ β β e for v i follows ha ( D A v A w )) β( D ( v w )) φ( D ( v )) β ( By he exisece of lower g differeiable soluio ad Lemma 4 (i) ) ) η ( s z s [ A η]( ) hus η A η Hece A saisfies all hypoheses of Lemma ad Lemma ad herefore A has he uiue fixed poi which iself is he uiue g soluio for (4) z ( ) )) Example 4 Cosider he FIVP where a ( u( ) u() ) 5] ) E u( ) u() We claim his FIVP has uiue soluio for R Clearly v) ) sup v ) r r R Cosider he alerig disace fucio χ ( ) ad r for some r R he all he assumpios ϕ( ) r i heorem 4 are fulfilled ad herefore he FIVP has uiue soluio heorem 4 Le G :[ d ] E E be rd- coiuous If here exiss a lower g differeiable soluio η C rd ([ σ(d))] E ) for (4) Le G be such ha: (i) diam ([ u ] ) diam s [] (ii) G is o-decreasig i he secod variable ie if v he G v ) ) w ( (iii)for comparable eleme G is wealy coracive ie for alerig disace fucios χ ad φ saisfyig (45) he a uiue g differeiable soluio z exiss for (4) o [σ (d)] Vol 9 (5) July 6 wwwidjsorg Idia Joural of Sciece ad echology 5

6 Fuzzy Dyamic Euaios o ime Scales uder Geeralized Dela Derivaive via Coracive-lie Mappig Priciples Proof Defie he operaor A : C C by [ A z]( ) zθ( ) s From Lemma 4 (ii) z is a soluio of (4) Also he operaor A is o-decreasig for u v I a similar way o heorem 4 A fulfills all assumpios of Lemma ad hece from Lemma A has he uiue g soluio for (4) heorem 43 heorems 4 4 are also valid if we replace he exisece of lower g differeiable soluio ( g differeiable soluio) o (4) by a upper g soluio ( g soluio) Proof If η is a upper g differeiable soluio for (4) he ) ) η ( s z s [ A η]( ) Similarly If η is a upper g differeiable soluio for (4) we have ) ) Θ ( ) η ( s y Θ ( ) F( s [ Aη ( )] hus η A η ad η Aη Hece A ad A saisfies all hypoheses of Lemma ad Lemma ad herefore A A has a uiue soluio i C 5 Refereces Puri ML Ralescu DA Differeials of fuzzy fucios Joural of Mahemaical Aalysis ad Applicaios 983; 9():55 58 Kaleva O Fuzzy differeial euaios Fuzzy Ses ad Sysems 987; 4(3):3 7 3 Cao CY Flores RH O ew soluios of fuzzy differeial euaios Chaos Solios Fracals 8 Oc; 38(): 9 4 Li J Zhao A Ya J he Cauchy problem of fuzzy differeial euaios uder geeralized differeiabiliy Fuzzy Ses ad Sysems Aug ; : 4 5 Murhy MSN Suresh Kumar G hree poi boudary value problems for hird order fuzzy differeial euaios Joural of he Chugcheog Mahemaical Sociey 6; 9(): 6 Murhy MSN Suresh Kumar G Iiial ad boudary value problems for fuzzy differeial euaios Demosraio Mahemaical 7; 4(4): Murhy MSN Suresh Kumar G O corollabiliy ad observabiliy of fuzzy dyamical marix Lyapuov sysems Advaces i Fuzzy Sysems 8; 8: 6 8 Murhy MSN Suresh Kumar G O observabiliy of fuzzy dyamical marix Lyapuov sysems Kyugpoo Mahemaical Joural 8; 48(3): Murhy MSN Suresh Kumar G Appa Rao BV Prasad KASNV O corollabiliy of fuzzy dyamical marix Lyapuov sysems Aalele Uiversiaii de ves imisoara 3; LI():73 86 Rajumar Pahiaha Seivig ou he poor usig fuzzy decisio maig ools Idia Joural of Sciece ad echology 5 Sep; 8() Doi: 7485/ijs/5/ v8i/796 Azadeh ZJ Adem K Abdul Raza S A adjusable mehod for daa raig based o fuzzy sof ses Idia Joural of Sciece ad echology 5 Sep; 8() Doi: 7485/ ijs/5/v8i/7587 Villamizar-Roa EJ Exisece of soluios o fuzzy differeial euaios wih geeralized Huuhara derivaive via coracive-lie mappig priciples Fuzzy Ses ad Sysems 5; 65: Agarwal RP Boher M O Rega D Peerso A Dyamic euaios o ime scales : A survey J Compu Appl Mah Apr ; 4(-): 6 4 Boher M Peerso A Dyamic euaios o ime scales: A iroducio wih Applicaio Berli: Birhauser Boso 5 Vasavi Ch Suresh Kumar G Mury MSN Geeralized differeiabiliy ad iegrabiliy for fuzzy se-valued fucios o ime scales Sof Compuig 6; (3): Vasavi Ch Suresh Kumar G Mury MSN Fuzzy dyamic euaios o ime scales uder secod ype Huuhara dela derivaive Ieraioal Joural of Chemical Scieces 6; 4(): Harjai J Sadaragai K Geeralized coracios i parially ordered meric spaces ad applicaios o ordiary differeial euaios Noliear Aal Feb ; 7(3-4): Nieo JJ Rodriguez-Lopez R Applicaios of coracive-lie mappig priciples o fuzzy euaios Rev Ma Complu 6; 9(): Vol 9 (5) July 6 wwwidjsorg Idia Joural of Sciece ad echology