Existence of positive solutions for fractional q-difference. equations involving integral boundary conditions with p- Laplacian operator

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1 Invenion Journal of Researh Tehnology in Engineering & Managemen IJRTEM) ISSN: Volume Issue 7 ǁ July 8 ǁ PP 5-5 Exisene of osiive soluions for fraional -differene euaions involving inegral boundary ondiions wih - Lalaian oeraor Yizhu Wang Chengmin Hou * Dearmen of Mahemais Yanbian Universiy Yanji P.R. China) ABSTRACT: The exisene of osiive soluions is onsidered for a fraional -differene euaion wih - Lalaian oeraor in his arile. By emloying he Avery-Henderson fixed oin heorem a new resul is obained for he boundary value roblems. KEY WORDS: osiive soluions; Riemann-Liouville fraional -derivaives; Cauo fraional -derivaives; -Lalaian; Avery-Henderson fixed oin heorem I. INTRODUCTION Fraional alulus is he exension of ineger order alulus o arbirary order alulus. Wih he develomen of fraional alulus fraional differenial euaions have wide aliaions in he modeling of differen hysial and naural siene fields suh as fluid mehanis hemisry onrol sysem hea onduion e. There are many aers onerning fraional differenial euaions wih he -Lalaian oeraor [-5] and fraional differenial euaions wih inegral boundary ondiions [6-]. In [] Yang sudied he following fraional -differene boundary value roblem wih -Lalaian oeraor D D x ))) = f u )) x) = x) = D x) = D x) = where. The exisene resuls for he above boundary value roblem were obained by using he uer and lower soluions mehod assoiaes wih he Shauder fixed oin heorem. In [] Yuan and Yang onsidered a lass of four-oin boundary value roblems of fraional -differene euaions wih -Lalaian oeraor D D x ))) = f u )) P Where D D is he fraional -derivaive of he Riemann-Liouville ye wih. By alying he uer and lower soluions mehod assoiaed wih he Shauder fixed oin heorem he exisene resuls of a leas one osiive soluion for he above fraional -differene boundary value roblem wih -Lalaian oeraor are esablished. Moivaed by he aforemenioned work his work disusses he exisene of osiive soluions for his fraional -differene euaion: x) = x) = ax D x) = D x) = bd x ) Volume Issue 7 5

2 Exisene of osiive soluions for fraional -differene D [ D u ))] f u )) = ) ' D u)) = [ ))] )) D u = D u = D u) D = u) = au) bd u) = g ) u ) d.) where and 5 6. u ) u u =. D is he Cauo fraional - derivaive D is he Riemann-Liouville fraional -derivaive. We will always suose he following ondiions are saisfied: H ) g ) :[] [ ) wih g g ) d and ) L [] H ) ab ) a g) d and b a; g ) d ; H ) f u) :[] ) ) is oninuous. II. BACKGROUND AND DEFINITIONS To show he main resul of his work we give in he following some basi definiions and heorems whih an be found in [4]. Definiion. [] Le and f be a funion defined on []. The fraional -inegral of Riemann- Liouville ye is I f ) and ) f ) = ) s) I f ) ) = f s) d s []. ) Lemma. [] Le hen he following eualiy holds: I D f ) f ) D f ) ). k k = k = k ) Lemma. [4]Le where n =. If D u C[] hen n i) ) = ) i i= I D f f k ) = Theorem.4 Avery-Henderson fixed oin heorem [5]) Le E ) be a Banah sae and P E be a one. Le and be inreasing non-negaive oninuous funionals on P and be a non-negaive oninuous funionals on P wih suh ha for some r and M u) u) u) Volume Issue 7 6

3 Exisene of osiive soluions for fraional -differene and u M u) for all u r ) where r) = { u P : u) r}. Suose ha here exis osiive numbers r r r suh ha If lu) l u) for l and u P r ). T : r ) P is a omleely oninuous oeraor saisfying: C ) Tu) r for all u P r ); C) Tu) r for all u P r ); ) C P r ) and Tu) r for all u P r ) hen T has a leas wo fixed oins u and wih u) r. u suh ha r u) wih u) r and r u) III. PRELIMINARY LEMMAS Lemma. The boundary value roblem.) is euivalen o he following euaion: where u d d s ) ) = ) ) ) d g s d ) = [ a g ) d ] ) [ b g ) d ] [ a ) ] ) ) ) ) s v s d s d = s ) ) ) ) ) v s) = H s ) f u )) d.).).).4) Hs ) = ) ) s )) ) ) s )) s ) ) s is he inverse funion of s ) a.e. s ) = s s s s. =..5) Proof From D [ D u ))] f u )) we ge = D u )) ) f u )) d. ) = ) Volume Issue 7 7

4 Exisene of osiive soluions for fraional -differene ' In view of D u )) [ D = u ))] = we ge = = i.e. D u )) ) f u )) d. ) = ) Condiions D u)) imly ha = = f u d ) ) ) ))..6).7) By use of.6) and.7) we ge In view of.8) we obain Le by use of.9) we ge = D u )) H ) f u )) d. ) D u ) = H ) f u )) d. ) v ) = H ) f u )) d ) u ) = s) v s) d s d d d. ).8).9) Condiions D u) imly ha d = i.e. = u s d d ) ) = ) ) ) hen we have ' ) u ) = s) v s) d s d. ) Condiions D u) = imly ha d = s ) ) ) ). From au ) bd u ) g ) u ) d = we ge d g s d ) = [ a g ) d ] ) [ b g ) d ] [ a ) ] ) ) ) ). s v s d s Therefore we an obain u d d s ) ) = ) ) ) Volume Issue 7 8

5 Exisene of osiive soluions for fraional -differene = g s v s dsd ) [ b g ) d ] [ a ) ] ) ) ) ) s ) ) ) ) s) v s) d ) ). s s v s d s The roof is omlee. Lemma. [6]) The funion H s ) defined by.5) is oninuous on [] [] and saisfy s s) ) ) Hs ) ) ) ) ) for s []. Le E be he real Banah sae C [] wih he maximum norm define he oeraor T : E Tu d d s ) ) = ) ) ) = g s v s dsd ) [ b g ) d ] [ a ) ] ) ) ) ) s ) ) ) ) s) v s) d ) ). s s v s d s E by Lemma. For u C[] wih u ) Tu) ) is non-inreasing and non-negaive. Proof Sine so we ge Tu d d s ) ) = ) ) ) Tu ) d s) v s) d ) s ) ' ) = So Tu) ) is non-inreasing hen we have ) ) = s) v s) d s s) v s) ds ) ). min Tu ) = Tu). We have [] Tu d d s ) ) = ) ) ) Volume Issue 7 9

6 Exisene of osiive soluions for fraional -differene = g s v s dsd ) [ b g ) d ] [ a ) ] ) ) ) ) s s) v s) d s s) v s) ds ) ) ) ) ) g s d [ b g ) d ] [ a g ) d ] [ ) ] ). ) ) ) s a ) ) s) ) The roof is omlee. IV. MAIN RESULTS Theorem 4. Suose ha here exis numbers r r r suh ha f saisfies he following ondiions: ) H f u M ) ) for [] r u[ r ]; k ) for [] u [ r ]; H f u M ) H f u M where ) for [] u [ r ] M ) r = B ) L M ) r = B ) L [ a g ) d ] ) ) Then he roblem.) has a leas wo osiive soluions u and u suh ha r u) wih u) r b a L = s s) d s L = ) g ) d b g ) d [ a g ) d ] ) [ a g ) d ] ) b g) d L = s s) ) d. s [ a g ) d ] ) M ) r = B ) L Volume Issue 7

7 Exisene of osiive soluions for fraional -differene and r u) wih u) r. Proof Define he one P Eby P = u u E min u ) k u [] [] where [ b g ) d ] [ ) ] a k = b g) d For any u P in view of Lemma. we ge k. Tu Tu d d s ) min ) = ) = ) ) [] ) = g s v s dsd ) [ b g ) d ] [ a ) ] ) ) ) ) s s) v s) d s s) v s) ds ) ) ) ) ) g s d [ b g ) d ] [ a g ) d ] ) ) ) s v s d s [ a g ) d ] ) ) k g s v s d sd [ a g ) d ] ) [ b g ) d ] ) s) v s) d s [ a g ) d ] ) = ktu) = k Tu. Therefore T : P P. In view of he Arzela-Asoli heorem we have T : P P is omleely oninuous. We define he funions on he one P: u) = min u ) = u) [] u) = max u ) = u). [] Obviously we have ) = u) u) u). u) = max u ) = u) [] For any u r ) we ge min u ) k u ha is u) k u herefore we obain [] u u). k Volume Issue 7

8 Exisene of osiive soluions for fraional -differene For any u P r ) we ge lu) = l u) for l. In he following we rove ha he ondiions of Theorem. hold. r Firsly le u P r ) ha is u[ rfor ] []. By means of H ) we have k ) v s) = H s ) f u )) d ) s s) ) M d ) M s s) B ) = ) ) where B ) = ) d. So we ge Tu Tu Tu d d s ) ) = min ) = ) = ) ) [] ) = g s v s dsd ) [ b g ) d ] [ a ) ] ) ) ) ) s s) v s) d s s) v s) ds ) ) ) ) [ b g ) d ] [ a g ) d ] [ a ) ] ) ) ) ) s ) ) s) ) M ) s s) B ) b a ) s [ a g ) d ] ) ) M ) s s) B ) s) ) ) ) M B s s) ) d ) ) s d s ) b a M B = s s) ) ds [ a ) ] ) ) M B ) = L = r ). d s Volume Issue 7

9 Exisene of osiive soluions for fraional -differene Seondly le u P r ) ha is u [ r ] for []. By means of H ) we ge So we have M B ) = L = r. ) Finally le u P r ) ha is u [ r ] for []. By means of H ) we ge So we ge ) v s) = H s ) f u )) d ) ) M B ) M d = ) ) Tu) = max Tu ) = Tu) = d [] ) = g s v s dsd [ b g ) d ] [ a ) ] ) ) ) ) s v s d s M ) B ) ) ) ) [ b g ) d ] M ) B ) s) a ) g s d sd [ a ) ] ) [ ) ] ) g) d M B ) [ a g ) d ] ) ) [ b g ) d ] M B ) [ a g ) d ] ) ) ) v s) = H s ) f u )) d ) s s) ) Ms s) B ) M d = ) ). d s. Tu) = max Tu ) = Tu) = d [] = g s v s dsd ) Volume Issue 7

10 Exisene of osiive soluions for fraional -differene Therefore in view of Theorem. we see ha he roblem.) has a leas wo osiive soluions u and u suh ha r u) wih u) r and r u) VI. wih u) r. EXAMPLE In his seion we give a simle examle o exlain he main resul. Examle 5. For he roblem.) le =.8 =. a = 4 b = = g ) = hen we ge = g ) d = g ) d = Le From a dire alulaion we ge f u) M f u) M f u) M.4 for for for In view of Theorem 4. we see ha he aforemenioned roblem has a leas wo osiive soluions u and u suh ha.5 u) wih u) 9 and 9 u) wih u). REFERENCES [] Chen T Liu W: An ani-eriodi boundary value roblem for he fraional differenial euaion wih a -Lalaian oeraor. Al. Mah. Le ) [] Chai G: Posiive soluions for boundary value roblem of fraional differenial euaion wih - Lalaian oeraor. Bound. Value Probl. 8 ) [ b g ) d ] [ a ) ] ) ) ) ) s b M ) s s B s) ) [ ) ] ) ) [ a ) ] ) ) ) ) [ b g ) d ] M B ) = s s ds [ a ) ] ) M B ) = L = r ) [ b g ) d ] [ ) ] a 7 k = =.679. b g) d 58 f u) = 6 u 9) 6. [] u[9] [] u[9] [] u [ ). 58 [] u[ ] 7 [] u[9] [] u[.5]. [] Feng X Feng H Tan H: Exisene and ieraion of osiive soluions for hird-order Surm-Liouville d s Volume Issue 7 4

11 Exisene of osiive soluions for fraional -differene boundary value roblems wih -Lalaian. Al. Mah. Comu ) [4] Li Y Qi A: Posiive soluions for muli-oin boundary value roblems of fraional differenial euaions wih -Lalaian. Mah. Mehods Al. Si ) [5] Han Z Lu H Zhang C: Posiive soluions for eigenvalue roblems of fraional differenial euaion wih generalized -Lalaian. Al. Mah. Comu ) [6] Jankowski T: Posiive soluions o fraional differenial euaions involving Sieljes inegral ondiions. Al. Mah. Comu. 4-4) [7] Nouyas S Eemad S: On he exisene of soluions for fraional differenial inlusions wih sum and inegral boundary ondiions. Al. Mah. Comu ) [8] Cabada A Hamdi Z: Nonlinear fraional differenial euaions wih inegral boundary value ondiions. Al. Mah. Comu ) [9] Gunendi M Yaslan I: Posiive soluions of higher-order nonlinear muli-oin fraional euaions wih inegral boundary ondiions. Fra. Cal. Al. Anal ) [] Cabada A Dimirijevi S Tomovi T Aleksi S: The exisene of a osiive soluion for nonlinear fraional differenial euaions wih inegral boundary value ondiions. Mah. Mehods Al. Si. 6) [] Yang W: Posiive soluion for fraional -differene boundary value roblems wih -Lalaian oeraor. Bull.Malays.Mah.So. 64)95-) [] Yang QYang W: Posiive soluion for -fraional four-oin boundary value roblems wih - Lalaian oeraor. J.Ineual.Al. 484) [] Bashir A Soiris KN LoannisKP: Exisene resuls for nonloal boundary value roblems of nonlinear fraional -differene euaions. Adv.Differ.Eu.4) [4] Rajkovi PM Marinkovi SD Sankovi MS: Fraional inegrals and derivaives in -alulus. Al. Anal. DisreeMah. - 7) [5] Avery Rl Chyan CJ Henderson J: Twin soluions of boundary value roblems for ordinary differenial euaions and finie differene euaions. Comu. Mah. Al ) [6] Yuan C: Mulile Posiive soluions for n-)-ye semi osione onjugae boundary value roblems of nonlinear fraional differenial euaions. Eleron. J. Qual. Theory Differ. Eu. 6 ) Chengmin Hou. Exisene of Posiive Soluions for Fraional -Differene Euaions Involving Inegral Boundary Condiions wih -Lalaian Oeraor. Invenion Journal of Researh Tehnology in Engineering & Managemen IJRTEM) vol. no. 7 5 July Volume Issue 7 5