Two point fuzzy boundary value problem with eigenvalue parameter contained in the boundary condition

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1 Malaya Journal of Matemat Vol 6 No ttps://doorg/6637/mjm64/ wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton ar Ceylan * and Nat Altınıs ı Astract In ts artcle two pont fuzzy oundary value prolem s defned under te approac generalzed Huuara dfferentalty (gh-dfferentalty We researc te soluton metod of te fuzzy oundary prolem wt te (x λ and χ (x λ wc are defned y te specal procuder We gve operator-teoretcal forasc solutons Φ mulaton construct fundamental solutons and nvestgate some propertes of te egenvalues and correspondng egenfunctons of te consdered fuzzy prolem en results of te proposed metod are llustrated wt a numercal example Keywords gh-dervatve egenvalue fuzzy egenfuncton fuzzy Hlert Space AMS Suject Classfcaton 3E7 34A7 65L5 Department of Matematcs Faculty of Arts and Scences Snop Unversty 57 Snop urey of Matematcs Faculty of Arts and Scences Ondouz Mayıs Unversty 5539 Kurupelt Samsun urey *Correspondng autor: tceylan@snopedutr; anat@omuedutr Artcle Hstory: Receved August 8; Accepted 6 Octoer 8 Department c 8 MJM Contents Introducton 766 Prelmnares Operator-eoretc Formulaton of te Prolem Soluton Metod of te Fuzzy Prolem Concluson 77 References 77 Introducton In ts paper we consder te two pont fuzzy oundary value prolem L d dt L u λ u t [a ] ( wc satsfy te condtons a u a u ( u( λ u ( (3 were a a are nonnegatve trangular fuzzy numers λ > reel numer a a 6 6 and u(t fuzzy functon e topc of fuzzy dfferental equaton (FDE as een rapdly growng n recent years Many prolems suc as populaton economcs dynamcs and pyscs can e modeled wt FDE e fuzzy oundary value prolem dependng on te egenvalue parameter provdes enefts n solvng tese prolems ere are many suggestons to defne a fuzzy dervatve and to study fuzzy dfferental equaton [ ] One of te most well-nown defntons of dfference and dervatve for fuzzy set value functons was gven ye Huuara n [] By usng te H-dervatve Kaleva n [4] started to develop a teory for fuzzy dfferental equatons Many wors ave een done y several auters n teoretcal and appled felds for fuzzy dfferental equatons wt te Huuara dervatve [4 4] It soon appeared tat te soluton of fuzzy dfferental equaton nterpreted y Huuara dervatve as a drawac: t ecame fuzzer as tme

2 wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton 767/773 goes [3] So ere we use gh-dfference and gh-dervatve to solve FDE under muc less restrctve condtons [] e paper s organzed as follows; secton ntroduces te asc concept of fuzzy functon spaces fuzzy Hlert spaces and fuzzy gh- dervatve In secton 3 we defne operator formulaton of te prolem n te adequate fuzzy Hlert spaces In secton 4 we gve a numercal example In secton 5 te results are stated ganed troug te study of te paper Defnton 3 [3] Let [a ] < e a gven famly of non-empty ntervals Assume tat [a ] [a ] for all < ( lm a lm [a ] wenever { } s an n creasng sequence n ( ] convergng to (c < a < for all ( ] Prelmnares In ts secton we gve some concepts and results esdes te essental notatons wc wll e used trougout te paper Let u e a fuzzy suset on R e a mappng u : R [ ] assocatng wt eac real numer t ts grade of memersp u(t In ts paper te concept of fuzzy real numers (fuzzy ntervals s consdered n te sense of Xaoand Zu wc s defned elow: Defnton [] A fuzzy suset u on R s called a fuzzy real numer (fuzzy ntervals wose cut set s denoted y [ u] e [ u] {t : u(t } f t satsfes two axoms: [ u v] [λ u] [ u] [ v] {x y : x [ u] y [ v] } [ u] {λ x : x [ u] } λ d( u v sup {max[ u v u v ]} < were [ u] [u u ] [ v] [v v ] Let R e an nterval We denote y te F (C( ; Rn space of all contnuous fuzzy functons on For u v F (C( ; Rn defne a metrc (N For all < tere exst real numers < u u < suc tat [ u ] s equal to te closed nterval [u u ] e set of all fuzzy real numers (fuzzy ntervals s denoted y F (R FK (R te famly of fuzzy sets of R wose cuts are nonempty compact susets of R If u F (R and u(t wenever t < ten u s called a non-negatve fuzzy real numer and F (R denotes te set of all nonnegatve fuzzy real numers For all u F (R and eac ( ] real numer u s postve e fuzzy real numer r F (R defned y Defnton 4 [3] For u v F (R and λ R te sum u v and te product λ u are defned y for all [ ] Defne d : F (R F (R R {} y te equaton (N ere exsts r R suc tat u(r r(t en te famly [a ] represents te cut sets of a fuzzy real numer u F (R Conversely f [a ] < are te cut sets of a fuzzy numer u F (R ten te condtons ( and (c are satsfed t r t 6 r t follows tat R can e emedded n F (R tat s f r ( ten r F (R satsfes r (t (t r and cut of r s gven y [ r] [r r] ( ] D ( u v supd( u(t v(t t From [3] F (C( ; Rn D s a complete metrc space A fuzzy functon F : F (R s measurale f for all [ ] te set valued mappng F : FK (R defned ye F (t F (t s measurale Let L ( ; R denote te space of Leesque ntegrale functons We denote ye SF te set of all Leesque ntegrale selectons of F : FK (R tat s SF f L ( ; R : f (t F (t ae Defnton [3] An artrary fuzzy numer n te parametrc form s represented y an ordered par of functons (u u wc satsfy te followng requrements ( u s ounded non-decreasng left contnuous functon on ( ] and rgt- contnuous for A fuzzy functon F : F (R s ntegraly ounded f tere exsts an ntegrale functon suc tat x (t for all x F (t A measurale and ntegraly ounded fuzzy functons F : FK (R s sad to e ntegrale over f tere exsts F F (R suc tat Z Z ( u s ounded non-ncreasng left contnuous functon on ( ] and rgt- contnuous for ( F (t dt f (t dt : f SF for all [ ] [7] For measurale functons f : R defne te norm ( u u 767

3 wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton 768/773 ( f L p R ( f (t p dt p p< nfµ( supt f (t p (5 F L ([ ] ; R It can G for f g L [ ] suc tat F e sowed tat f g s Feln-fuzzy nner product space on L [ ] and -cut set of f g s gven ye [ f g] f g f g Z Z f (xg (xdx f (xg (xdx were µ ( s te Leesque measure on and Let L p ( ; R e te Banac space of all measurale functons f : R wt f L p < By F (L p ( ; R p < t s denoted te space of all functons u : F (R suc tat te functon t d( u(t elong to L p ( ; R en Z d p ( u v (dh ([ u] [ v] p d (6 for all [ ] From teorem 6 of [8] t s clear tat f g s a fuzzy real numer So L ([ ] ; R s a fuzzy Hlert space p ( Defnton 8 [] Let u v F (R If tere exst w ten w s called te Huuara F (R suc tat u v w dfference of u and v and t s denoted y u v If u v exsts ts cuts are [ u H v] u v u v s ametrc on F (L p ( ; R and for p < d ( u v d( u v sup dh ([ u] [ v] (3 < s a metrc on F (L ( ; R [7] So for p let L ( ; R e te Banac space of all mea- for all [ ] surale functons f : R wt f L < By F L ( ; R Defnton 9 e generalzed Huuara dfference of two we denote te space of all square-ntegrale fuzzy functons fuzzy numers u v F (R s defned as follows u : F (R suc tat te functon t d( u(t elong to L ( ; R en from ( ( u v w [ u gh v] w or ( v u (w Z d ( u v (dh ([ u] [ v] d s a metrc on F L ( ; R for p Defnton 5 [9] Let X e a vector space over R A Felnfuzzy nner product on X s a mappng : X X F (R suc tat for all vectors x y z X and all r R ave (FIP x y z x z y z (FIP rx y r x y (FIP3 x y y x (FIP4 x 6 x x (t for all t < (FIP5 x x f and only f x Remar 6 Condton (FIP4 n te aove defnton s equvalent to te condton for all (6 x X x x > for eac ( ] were [x x] x x x x see n [9]e vector space X equpped wt a Feln-fuzzy nner product s called a Feln-fuzzy nner space A Feln-fuzzy nner product on X defnes a fuzzy numer p x x x for all x X (4 A fuzzy Hlert space s a complete Feln-fuzzy nner product space wt te fuzzy norm defned y (4 f g (t sup ( ] t Z f g dx Z f g dx and f te H -dfference exsts ten u v u gh v; te u gh v F (R are condtons for of w te exstence w u v and w u v [ ] case ( w ncreasng w decreasngw w wt w u v and w u v [ ] case ( wt w ncreasng w decreasngw w It s easy to sow tat ( and ( are ot vald f and only f w s a crsp numer [] Remar rougout te rest of ts paper we assume tat u gh v F (R and -cut representaton of fuzzy val ued functon f : F (R s expressed y f(t [( f (t ( f (t] t [a ] for eac [ ] Defnton [] Let t and e suc tat t ten te gh-dervatve of a functon f : F (R at t s defned as f (t fgh (t lm Example 7 Consder te lneer space L ([ ] ; R of all square-ntegrale functons Defne In terms of -cuts we ave [ u gh v] mn u v u v max u v u v 768 gh f (t (7 (t F (R satsfyng (7 exsts we say tat f s If fgh generalzed Huuara dfferentale (gh-dfferentale for sort at t

4 wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton 769/773 Defnton [] Let f : [a ] F (R and t wt f (t and f (t ot dfferentale at t We say tat - f s [( gh]-dfferentale at t f fgh (t f (t f (t (8 - f s [( gh]-dfferentale at t f fgh (t f (t f (t (9 for all [ ] Defnton 3 e second generalzed Huuara dervatve of a fuzzy functon f : [a ] F (R at t s defned as f (t fgh (t lm gh f (t (t F (R we say tat f (t s generalzed Huuara f fgh gh dervatve at t (t s [( gh] dfferentale at t Also we say tat fgh f fgh (t ( f (t ( f (t f f e [( gh] ( f (t ( f (t f f e [( gh] (t s [( gh] dfferentale for all [ ] and tat fgh at t f fgh (t ( f (t ( f (t f f e [( gh] ( f (t ( f (t f f e [( gh] In ts secton we concern wt fuzzy egenvalues and egenfunctons of two-pont fuzzy oundary value prolems wt egenvalue parameter contaned n te oundary condton o do ts at frst we need to use fuzzy dervatves So ere we use gh-dfference and gh-dervatve to solve fuzzy prolem [] 3 Operator-eoretc Formulaton of te Prolem Consder te ( (3 egenvalue prolem s s not te usual type of egenvalue prolem ecause te egenvalue appears n te oundary condtons; so we cannot put L dtd and consder prolem as a specal case of L u λ u ecause D te doman of L depends upon λ So we enlarge our defnton of L o do ts we formulate a teoretc approac to te prolem From teorem 6 of [8] we defne a fuzzy Hlert space L ([ ] ; R wt a Feln-fuzzy nner product H Z Z AF λ F f were F D (A en te egenvalues and te f egenfunctons of te prolem ( (3 are defned as te egenvalues and te frst components of te correspondng egenelements of te operator A respectvely 4 Soluton Metod of te Fuzzy Prolem for all [ ] [] f g [ f (t f (t] were F F H and G ( f ( f g (t] [g (t F L ([ ] ; R G H and F G (g (g suc tat f (t f (t g (t g (t L ([ ] ; R and ( f ( f (g (g R Consder te space of two- component fuzzy vectors F H wose frst component s a twce gh-dfferentale fuzzy functon f(t F L ([ ] ; R and wose second com f(x ponent s a fuzzy real numer f F (R Here F and f are poztve fuzzy numers and ter cut sets are respectvely [F F ] [ f (t f (t] and ( f ( f In ts fuzzy Hlert space we construct te lneer operator H wt doman A:H f(t f (t fuzzy asolutely f (t contnuous (see [6] n [ ] ; D (A f a f a f f f (! (t (t f f wc acts y te rule AF A A f f (! f (t f wt F D (A f f( us te prolem ( (3 can e wrtten n te form Defnton 4 Let u : [a ] R F (R e a fuzzy functon and [ u (t λ ] [u (t λ u (t λ ] e te cut representaton of te fuzzy functon u(t for all t [a ] and [ ] If te fuzzy dfferental equaton ( as te nontrval solu tons suc tat u (t λ 6 u (t λ 6 ten λ s egenvalue of ( Consder te egenvalues of te fuzzy oundary value prolem ( (3 Usng te cut sets and [ gh]dfferentalty of u [ gh]-dfferentalty of u ; we get from ( (3 for λ ( > : u (t u (t λ u (4 (t u (t f (tg (tdt ( f (g f (tg (tdt ( f (g 769

5 wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton 77/773 u u u u sn (a cos (t Φ (t λ cos (a cos (a sn (a sn (t (46 (4 ( ( u ( u ( λ ( u ( u ( (43 ( Smlarly we fnd Φ (t λ as sn (a Φ (t λ cos (a cos (t cos (a sn (a sn (t (47 Suppose tat te two lnearly ndependent solutons of u λ u classc dfferental equaton are u (t λ and u (t λ Usng tese solutons te general soluton of te fuzzy dfferental equaton (4 s [ u (t λ ] u (t λ u (t λ Let [χ (x λ ] e a soluton wc s satsfyng were u ( u ( ( ( ( u ( u ( ( u (t λ C ( λ u (t λ C ( λ u (t λ u (t λ C3 ( λ u (t λ C4 ( λ u (t λ ntal condtons of fuzzy dfferental equatons (4 s soluton can e expressed as (x λ e a soluton wc s satsfyng Let Φ u u u u (48 χ (t λ C ( λ u (t λ C ( λ u (t λ χ (t λ C3 ( λ u (t λ C4 ( λ u (t λ (49 (44 and we wrte (48 n (49 suc tat ntal condtons of fuzzy dfferental equatons (4 s soluton can e expressed as χ λ C ( λ cos (a C ( λ sn (a ( (χ λ C3 ( λ cos (a C4 ( λ sn (a ( Φ (t λ C ( λ u (t λ C ( λ u (t λ Φ (t λ C3 ( λ u (t λ C4 ( λ u (t λ (45 From te determnant of te coeffcents matrx of te aove lnear system we get C ( λ and C ( λ suc tat and we wrte (44 n (45 suc tat C sn ( cos ( ( sn ( ( ( cos ( ( C Φ λ C ( λ cos (a C ( λ sn (a (Φ λ C3 ( λ cos (a C4 ( λ sn (a Susttutng ts C and C coeffcents te aove equatons From te determnant of te coeffcents matrx of te n (49 te general soluton s otaned as aove lnear system we get C ( λ and C ( λ suc tat sn ( χ (t λ ( cos ( ( cos (t sn (a C cos (a cos ( cos (a ( sn ( ( sn (t C sn (a (4 Susttutng ts C and C coeffcents te aove equatons n (45 te general soluton s otaned as Smlarly we fnd χ (t λ as 77

6 wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton 77/773 ntal condton fuzzy dfferental equatons (4 en of we fnd Φ (t as sn ( χ (t λ ( cos (t cos ( ( (t λ sn (x Φ (45 cos ( sn (t ( sn ( ( Smlarly let [χ (t λ ] e te soluton wc s satsfyng (4 u ( u λ ( en from (46 (47 and (4 (4 we fnd u ( u ( [ ] W (Φ χ (t λ and W (Φ χ (t λ Wronsan functons as ntal condtons of fuzzy dfferental equatons (4 en we fnd [χ (t λ ] as sn (t ( χ (t λ cos (t sn (t χ (t λ cos (t W Φ χ (t λ Φ (t λ χ (t λ Φ (t λ χ (t λ ( ( cos (! 3 ( sn ( ( (46 From teorem 43 egenvalues of te fuzzy prolem (4(44 are zeros of te functons W (λ and W (λ So we get and W Φ χ (t λ Φ (t λ χ (t λ Φ (t λ χ (t λ ( ( cos (! 3 ( ( sn ( eorem 43 e egenvalues of te fuzzy oundary value prolem ( (3 are concded zeros of te functons W (λ and W (λ [4] (44 ale For W (λ n values correspondng to [ ] (43 ( ( sn cos and frst four values of (48 wt n n tale suc tat Example 44 Consder te two pont fuzzy oundary prolem u( λ u ( (48 3 ale For W (λ n values correspondng to [ ] eorem 4 e Wronsan functons W (Φ χ (t λ and W (Φ χ (t λ are ndependent of varale t for t [a ] were functons Φ χ Φ χ are te soluton of te fuzzy oundary value prolem ( (3 and t s sow tat W (λ W (Φ χ (t λ and W (λ W (Φ χ (t λ for eac [ ] [4] (4 (47 W (λ For eac [ ] f te values satsfyng (47 and (48 equatons compute wt Matla Program ten an nfnte numer of egenvalues suc tat λn (n are otaned So we sow frst four values of (47 wt n n tale suc tat u λ u u( W (λ sn 3 cos were [] [ ] λ > and u(t s [ gh]dfferentale and u (t s [ gh]-dfferentale fuzzy functons (t λ e te soluton wc s satsfyng Let Φ For eac n values and [ ] grapc of W (λ and W (λ functons are gven fgures So f we wrte n and n values n (45 and (46 n (t λ and [χn (t λ ] are equatons ten Φ n (t λ (Φn (t (Φn (t [sn (nt sn (nt] Φ u ( u ( 77

7 wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton 77/773 Fgure W (λ W (λ and W (λ (49 and Egenfunctons and fuzzy nterval of Φ Fgure 3 Φ [χn (t λ ] (χn (t (χn (t (n cos (nt n sn (nt n n ( (n cos (nt n sn (nt n n (4 en for all [ ] (49 are egenfunctons correspondng to λn (n egenvalues satsfyng (47 equaton and (4 are egenfunctons correspondng to λn (n egenvalues satsfyng (48 equaton n (t λ and [χn (t λ ] Consder tat egenvalues of Φ egenfunctons depend on cut So f we cange ten ts egenvalues cange and egenfunctons correspondng to λ cange In partcular we select 7854 n tale and n tale for If we susttute ts values respectvely n (49 and (4 we ave te functons followng fgures for χ and Φ Egenfunctons and Φ Fgure 4 cut of χ 4 functons are gven same grafc wt fgure Also χ and Φ In fgure χ χ χ functons and n fgure 3 Φ Φ Φ functons are seem to ave te same grapc But ecause of egenvalues ave very close to eac oter ts egenfunctons n (t λ and [χn (t λ ] egenfunctons are dfferent Also Φ are fuzzy at certan ntervals see fgures 5 Concluson Egenvalues and egenfunctons of te fuzzy oundary prolem are ntroduced and examned y usng generalzed Huuara dfferentalty concept In ts prolem te egenvalue parameter s contaned n te oundary condton at So we defne lneer operator n fuzzy Hlert space o solve ts prolem we use some ntal value prolems en we gve a numercal example References [] Egenfunctons and fuzzy nterval of χ Fgure χ 77 Y Calco-Cano and H Roman-Flores Comparson etween some approaces to solve fuzzy dfferental equatons Fuzzy Sets Syst 6(

8 wo pont fuzzy oundary value prolem wt egenvalue parameter contaned n te oundary condton 773/773 [] [3] [4] [5] [6] [7] [8] [9] [] [] [] [3] [4] [5] I Sadeq F Moradlou and M Sale On Approxmate caucy equaton n Feln s type fuzzy normed lnear spaces Iranan Journal of Fuzzy Systems (3 563 P Damond and P Kloeden Metrc Spaces of Fuzzy Sets: eory and Applcatons World Scentfc Sngapore H G Çtl and N Altınıs ı On te egenvalues and te egenfunctons of te Sturm-Louvlle fuzzy oundary value prolem J Mat Comput Sc 7( A Kastan and J J Neto A oundary value prolem for second order dfferental equatons Nonlnear Analyss 7( L Gomes and L C Barros Annual Meetng of te Nort Amercan Fuzzy Informaton Processng Socety (NAFIPS R P Agarwal S Arsad D O Regan and V Lupulescu Fuzzy fractonal ntegral equatons under compactness type condton Fractonal Calculus and Appled Analyss 4( B Daray Z Solman and A Ram A note on fuzzy Hlert spaces Journal of Intellgent and Fuzzy Systems 3( A Hasanan A Nazar and M Sael Some propertes of fuzzy Hlert spaces and norm of operators Iranan Journal of Fuzzy Systems 7( 9-57 L M Pur and D Ralescu Dfferental and fuzzy functons J Mat Anal Appl 9( B Bede and L Stefann Generalzed dfferentalty of fuzzy-valued functons Fuzzy Sets and Systems 3( 9-4 A Armand and Z Gouyande Solvng two-pont fuzzy oundary prolem usng varatonal teraton metod Communcatons on Advanced Computatonal Scence wt Applcatons (3 - O Kaleva and S Seala On fuzzy metrc spaces Fuzzy Sets and Systems ( O Kaleva Fuzzy dfferetal equatons Fuzzy Sets and Systems 4( Ceylan and N AltınıS ı Egenvalue prolem wt fuzzy coeffcents of oundary condtons Scolars Journal of Pyscs Mat and Stat 5( ????????? ISSN(P: Malaya Journal of Matemat ISSN(O:3 5666????????? 773