Application of Fixed Point Theorem of Convex-power Operators to Nonlinear Volterra Type Integral Equations

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1 Ieraioal Mahemaical Forum, Vol 9, 4, o 9, HIKRI Ld, wwwm-hikaricom h://dxdoiorg/988/imf4333 licaio of Fixed Poi Theorem of Covex-ower Oeraors o Noliear Volerra Tye Iegral Equaios Ya Chao-dog Huaiyi Normal Uiversiy, Huai a, Jiagsu, 33, Chia Coyrigh 4 Ya Chao-dog This is a oe access aricle disribued uder he Creaive Commos ribuio Licese, which ermis uresriced use, disribuio, ad reroducio i ay medium, rovided he origial work is roerly cied bsrac This aer firsly geeralizes a kid of ew oeraor, ie covex-ower codesig oeraor, which was obaied by Su Jigxia i aer [], ad defies a ew class of oeraor, ie P covex-ower codesig oeraor i locally covex sace lso, a ew fixed oi heorem of his ew oeraor is roved Fially we aly he resuls obaied o ivesigae he exisece of soluios for oliear Volerra ye iegral equaios i locally covex saces Keywor: locally covex saces; measure of o-comacess; P covex-ower codesig oeraor; Volerra ye iegral equaios; fixed oi heorem Iroducio The heory of differeial equaios, iegral equaios ad iegraldiffereial equaio i absrac sace, develoig i he firs half of he weieh Ceury, is a very acive research area Combiig he heory of differeial equaios ad fucioal aalysis, usig heory ad meho of

2 48 Ya Chao-dog fucioal aalysis o sudy differeial equaios, iegral equaios ad iegral-differeial equaio i absrac sace, i rovides a owerful ool for sudyig he exisece of soluios o oliear roblems i chemisry, hysics, biology, ecoomics ad oher sysems rese, mos of hese researches are coceraed o sudyig he exisece of soluios for differeial equaios, iegral equaios ad iegral-differeial equaios i Baach sace Thus, i has achieved a richer ad more exhausive resul Noe worhily, here are much fewer sudies abou hese roblems relaively i locally covex saces, such as [, 3, 4, 5, 7] s everyoe kows, locally covex sace is a class of much wider absrac sace ad is heory is oe of he mai coes of oliear fucioal aalysis Paer [] defies a ew class of oeraors i Baach sace, amely he covex ower codesig oeraor I also obais he ew fixed oi heorem of he oeraor, which will be alied o solve he exisece of global mild soluio ad osiive mild soluio, a class wih o-comacly, semi-grouedly ad semi-liearly develoed Equaio i sace E of Baach Isired by he lieraure above, he aer firsly geeralizes a kid of ew oeraor, ie covex-ower codesig oeraor, which was obaied by Su Jigxia i aer[], he i defies a ew class of oeraor, ie P covex-ower codesig oeraor i locally covex sace lso, a ew fixed oi heorem of his ew oeraor is roved Fially, we aly he resuls obaied o ivesigae he exisece of soluios for oliear Volerra ye iegral equaios i locally covex saces Defiiio ad Lemma Before rovidig he mai resul, we eed o iroduce some basic facs abou locally covex saces We give he defiiios as followig Paer [8] iroduced Kuraowski s measure of o-comacess ad basic roeries of he bouded se i he Baach sace Defiiio Le E be a Baach sace ad S is a bouded se i E, he α S = if{ δ > S ca be exressed as a fiie umber of ses ad S U S i i= =, makig diam S δ} i

3 licaio of fixed oi heorem 49 as he Kuraowski s measure of o-comacess, o-comacess measure for shor, his idicaes he diameer of S i for diam S i Obviously, α S < + Paer [] gives Kuraowski s measure of o-comacess, deermied by he half of is rage i locally covex sace Defiiio Le X be a locally covex sace, whose oology geeraed by hese mi-orm family P { } α, oed by X, P, Ω is a bouded se i X, called Ω ca be exressed as a μ = if d > α makig diam = α Γ fiie collecio uio : Ω= Ω j= α Ω d, j =,, L, as Ω s o-comacess measure abou half of he rage of j U Ω j α Here, diamα Ω j meas Ω j s diameer, deermied by he half of he rage of α Obviously, α μ Ω < Paer [3] exlais i deail abou Kuraowski s measure of o-comacess i he locally covex sace s we all kow, he famous Schauder fixed oi heorem is a imora coclusio, exremely widely alied This coclusio, however, requires oeraor o be comleely coiuous, which is a very hard codiio To weake his codiio, we have roosed codesig oeraor coce We ur he codiio from he comleely coiuous oeraor codesig oeraors o codesig oeraor, which defied by codesig oeraors wih o-comac measure i he Schauder fixed oi heorem Defiiio 3 Le E be real Baach sace, D is a bouded se of oeraors i E, oeraor : D E is called codesig oeraor If : D E is coiuous ad bouded, ad for ay o-relaively comac bouded se S i D, here is always α S < α S Paer [] roosed he ew coce of covex-ower codesig oeraor ad basically geeralizes i We also aly he Sadovskii fixed oi heorem

4 4 Ya Chao-dog o codesig oeraors furher Firs, we give you a mark Le E be real Baach sace, D E is a covex closed se, : D D, x D For ay give S D, le, x S S, x, x, S = co{ S, x }, =,3, L 3 Defiiio 4 Le E be real Baach sace, D E is a covex closed se, : D D, we calls he covex-ower codesig oeraor If is coiuous ad bouded, ad i has x D ad osiive ieger, allowig for ay o- relaive comacess bouded ses S D wih, x, α S < α S, of which x S is defied as 3, Noe ccordig o he defiiio, if α x S = α S, S is a relaively comac se i E Clearly, he codesig oeraor mus be cohesive by covex ower Wha s more, aer [] esablishes a ew fixed oi heorem abou he ewly defied covex-ower codesig oeraor, amely he followig lemma 6 ad Lemma 7 Lemma [ ] Le D be a oemy bouded closed covex se of he Baach sace, : D D is covex-ower codesig oeraor, here mus be fixed ois of i D Lemma 3 [ ] Le E be real Baach sace, D E is a bouded covex closed se, : D D is coiuous If here is x D, k <, as well as a, osiive ieger, makig α x S < kα S, S D, here mus be fixed ois of i D Like he oaio ad defiiios of covex-ower codesig oeraor i Baach sace of i he aer [], we give he defiiio of P covex-ower codesig oeraor i locally covex sace Le X, P be comlee Hausdorff locally covex sace, D E is a covex closed se, : D D, x D, for ay give S D, le, x S S, x, x, S = co{ S, x }, =,3, L4

5 licaio of fixed oi heorem 4 Defiiio 5 If X, P is comlee Hausdorff locally covex sace, D E is a covex closed se, : D D, so is P covex-ower codesig If is coiuous ad bouded, ad here is osiive ieger, which makes ay o- x D ad a P relaive comacess of bouded ses, S D have x, α S < α S x ad S i i are defied by 3 Wih he covex Baach sace ower codesig oeraors fixed oi heorem obaied by aer [], amely Lemma 6, we ca ge P covex-ower codesig oeraor s fixed oi heorem i locally covex saces Because hey are roved similarly, he rogress is omied here Lemma 4 Le D be a o-emy bouded closed covex se i a comlee Hausdorff locally covex saces X, P, if : D D is P covex-ower codesig oeraor, here mus be fixed oi of i D Exisece of Volerra Iegral Equaios i Locally Covex Saces Noliear This secio examies he exisece of y = h + g, y, J = [, a] Noliear Volerra iegral equaios i locally covex saces mog hem, h C X, f C J X, X, X, P is a comlee Hausdorff locally covex sace, R >, mark D = {, R : s a}, g C D, R, C X is a collecio of coiuous images of all slaves from J o X If, P, { c} P is he family of semi- Fa of C X Here, u = max ad C X is locally covex saces by he family of c J semi- orm They are oed wih α, α, which meas he measure of c

6 4 Ya Chao-dog o-comacess of X, C X resecively Lemma [ 3] a Le H be a bouded se of C X suα H α H J α J Of i, H = { x x H}, H J = U{ x x H} J c, he H, b Le H be a bouded equicoiuous se i C X, we ca coclude ha α H = suα H = α H J c J 3 Similar o he roof of Lemma i he aer [], i is easy o rove he followig lemma Lemma Le B be a bouded equicoiuous subse i C X, u C, so he same as co H, u } X { Lemma 3 Le B C X be a bouded ad equicoiuous, he we + ca coclude ha α C R, ad α { u B} α { u B}, J Lemma 4 For ay give R >, f i he J BR of which B R = { x E : x R, P} is bouded equicoiuous, H C X is bouded ad uiformly coiuous, D = {, R : s a}, g C D, R, we ca coclude ha g, H is also he equicoiuous se i C X Wih he above lemma ad obaied fixed oi heorem i P covex-ower codesig oeraor, Lemma 3, we ca give he exisece resuls of iegral equaio

7 licaio of fixed oi heorem 43 Theorem Le f mee: H For ay give R >, f is i he same row i J BR, P, B R = { x X : x R}, ad here is a coiuous fucio a ad real umbers b > makig a a u + b, s u X, 4 ad M a < Here, M = max{ g, :, D} H If here is a cosa L >, makig for ay bouded equicoiuous se B ad J i C X, as well as P, here is α f, Lα The he iegral equaio has a leas oe exise soluio Proof: Le iegral oeraors : C E C E as follows = h + g,, J 5 I is easy o rove : C E C E coiuous ad bouded, ad is soluio of iegral equaio is equal o he fixed oi of oeraor equaios Le r [ + ] a c h Mab a s, mark B r = { u C E : c u r}, he for ay u Br, we ca coclude ha c h + g, r c c a c + h + M a u + b h + Mr a Mab Wih he codiio H ad he defiiio of The, c u r, or u B r Therefore, B r Br : is coiuous ad bouded

8 44 Ya Chao-dog Furher is he evidece of B beig he equicoiuous ses i r C E I fac, u Br a, we ca coclude by he defiiio of ha h h + g, g, d h h + [ g, g, ] d + g, d * * h h + g, g, a r + b + M a r + b * Of which, a = su{ a J} d kow by he coiuiy of h ad g,, whe, he righ side of he formula above e o zero, so B is equicoiuous Le F = co, i is obvious ha r maed o F, ad is coiuous bouded Kow by Lemma 4, F is equicoiuous bouded ses i C E The we rove ha: : F F is covex-ower codesig oeraor Take u F, we ca rove here is osiive ieger, which for ay B r o-relaively comac se, u B F, α < α is righ For ay, B F, we ca coclude ha u Br is also bouded equicoiuous, from he defiiio of u ad Lemma, Lemma 4 So for ay fixed =,, L, wih Lemma, we kow ha, u, u α = maxα, =,,L 6 J Because of B beig a bouded equicoiuous se i F ad Lemma, we kow ha α = maxα ad wih he codiio H ad J

9 licaio of fixed oi heorem 45 Lemma 4, we also kow ha g,,, s s is he equicoiuous ses i C E So added wih he codiio H ad Lemma 3, we ca coclude ha, u α = α == α{ h + = α{ g, } g, } α g, g, α MLα 7, The due o he degree of coiuiy ad cosisecy of u = B, Lemma ad Lemma 4, we ca kow ha, u g,, }, s is equicoiuous oo Wih Lemma 3 ad codiio H ad 7, we coclude ha, u, u α = α h + g,, u } d, u = α g,, u } d α g,, u g, Lα, u ML α, u, u }, u } MLsα = M L α 8! ML k k, u k k Suose k, α = M L α, J k! J, k +, u k, u α = α g,, u } d, he for

10 46 Ya Chao-dog α g, k, u, u } g, Lα k, u M Lα k, u, u } k k + k k s k + k + M L α = M L α k! k +! 9 ML Therefore, by he iducio shows ha for ay =,, L, here is, u a α M L α! Thus by he 6,, we kow ha α, u a = maxα M L α J!, u a Wih M L α!, we kow ha here mus be a a osiive ieger o make M L α <, which roves : F F is a! P covex-ower codesig oeraor ccordig o Lemma 4, here is a fixed oi u * of i F, which is he soluio of u *, he iegral equaio i he C E Thus he roof is comleed Refereces [] Su Jigxia, Zhag Xiaoya Covex-ower codesig oeraor's fixed oi heorem ad is licaios of bsrac Semiliear Evoluio Equaios [J], ca Mahemaica Siica,5, [] Polewczak J Ordiary differeial equaios o closed subses of locally covex sace wih alicaios o fixed oi heorems [J], J Mah al l

11 licaio of fixed oi heorem 47 [3] Shi Hogbo, Zhu Jiag The Exisece of Noliear Volerra Iegral Equaios i Locally Covex Saces [J] ca alysis Fucioalis licaa, 6, [4] Toshio Yuasa Differeial equaios i a locally covex sace via he measure of orecomacess [J], J Mah al l [5] O Rega D Iegral equaios i reflexive Baach saces ad weak oologies [J], Proc mer Mah Soc996, [6] O Rega D Exisece resuls for oliear iegral equaios [J], J Mah al l [7] O Rega D Some fixed oi heorems for coceraive maigs bewee locally covex liear oological saces [J], Noliear al996, [8] Guo Daju Noliear Fucioal alysis d Ediio M], Jia: Sha Dog Sciece ad Techology Press, [9] Guo Fei, Liu Lisha, Wu Yoghog, Siew Peg-Foo Global soluios of iiial value roblems for oliear secod-order imulsive iegro-differeial of mixed ye i Baach Saces [J], Noliear al [] Schaefer HH, Wolff MP, Toological Vecor saces [M], Berli: Sriger, 999 Received: December, 3