Approximation and numerical methods for Volterra and Fredholm integral equations for functions with values in L-spaces

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1 Approximion nd numericl mehods for Volerr nd Fredholm inegrl equions for funcions wih vlues in L-spces Vir Bbenko Deprmen of Mhemics, The Universiy of Uh, Sl Lke Ciy, UT, 842, USA Absrc We consider Volerr nd Fredholm inegrl equions for funcions wih vlues in L-spces. This includes corresponding problems for se-vlued funcions, fuzzy-vlued funcions nd mny ohers. We ge iniil resuls in he pproximion of funcions wih vlues in L-spces by dped liner posiive operors nd in priculr by piecewise liner funcions. Besides we ge he error esimes of rpezoidl qudrure formuls. We prove heorems of exisence nd uniqueness of he soluion for such equions nd sugges some lgorihms for finding pproxime soluions. We pply menioned bove resuls from pproximion heory for nlysis of convergence of suggesed lgorihms. Keywords: Volerr nd Fredholm inegrl equions, L-spce, se-vlued nd fuzzy-vlued funcions, pproximion, lgorihms. Inroducion Fredholm nd Volerr inegrl equions for single vlued funcions form clssic subjec in pure nd pplied mhemics. Such inegrl equions hve mny imporn pplicions in biology, physics, nd engineering see, for exmple, [6], [4], [9], nd he references herein. A wide vriey of quesions led o inegrl equions for funcions wih vlues h re compc nd convex ses in finie or infinie dimensionl spces, or h re fuzzy ses see [7] nd [3]. In his pper we consider generlized concep, h of n L-spce, h encompsses ll of hese s specil cses. In priculr, we invesige he exisence nd uniqueness of soluions of liner Fredholm inegrl equions of he second kind nd liner Volerr inegrl equions, for funcions wih vlues in L-spces. In his ricle we ge iniil resuls on pproximion of funcions wih vlues in L-spces by dped liner posiive operors nd in priculr by piecewise liner funcions. Besides we ge he error esimes of rpezoidl qudrure formuls. For known resuls on pproximion nd qudrure formuls for se-vlued nd fuzzy-vlued funcions we refer he reder o [8], [], [4] nd references herein. We use he resuls on piecewise liner pproximion nd error esimes of qudrure formuls for convergence nlysis nd o esime he re of convergence of offered numericl lgorihms of soluion of inegrl equions. Despie he lrge number of ppers devoed o numericl mehods for he soluion of inegrl equions, we do no know of ny work conneced wih inegrl equions for funcions wih vlues in he spce of ses or fuzzy ses of dimensions h re greer hn one. One of he purposes of his pper is o fill his gp. We show h some exising mehods for he soluion of rel-vlued inegrl equions cn be doped o he soluion of inegrl equions in L-spces. These re mehods h hve relively low order of ccurcy such s collocion mehod bsed on piecewise liner inerpolion nd qudrure formul mehod bsed on he rpezoidl rule. Aemps o dp mehods of higher ccurcy o soluions of inegrl equions in L-spces mee difficulies, becuse pproximion heory nd qudrure formuls heory for funcions wih vlues in L-spces re no sufficienly developed ye. Corresponding uhor Emil ddress: ver.bbenko@gmil.com Vir Bbenko Preprin submied o Journl of Approximion Theory Sepember 7, 205

2 The pper is orgnized s follows. In Secion 2 we lis some preliminry resuls h re used in he reminder of he pper. In priculr we develop clculus of funcions wih vlues in L-spces. In Secion 3 we discus some problems of pproximion heory for funcions wih vlues in L-spces nd obin errors of qudrure formuls. We show exisence nd uniqueness of he soluions of hese inegrl equions in Secion 4, nd in Secion 5 we describe wo lgorihms for heir pproximion. Anlysis of convergence of such mehods is presened in Secion 6. In Secion 7 we give illusrive exmples. We end he pper wih some conclusions in Secion Preliminry resuls. Clculus of funcions wih vlues in L-spces 2.. L-spces The following definiion ws inroduced by Vhrmeev in [9]: Definiion. A complee seprble meric spce X wih meric δ is sid o be n L spce if in X operions of ddiion of elemens nd heir muliplicion wih rel numbers re defined, nd he following xioms re sisfied: A. x, y X x + y = y + x; A2. x, y, z X x + y + z = x + y + z; A3. θ X x X x + θ = x where θ is clled zero elemen in X; A4. x, y X λ R λx + y = λx + λy; A5. x X λ, µ R λµx = λµ x; A6. x X x = x, 0 x = θ; A7. x, y X λ R δλx, λy = λ δx, y; A8. x, y, u, v X δx + y, u + v δx, u + δy, v Exmples of L-spces. Any Bnch spce Y, Y over he field of rel numbers endowed wih he meric δx, y = x y Y is n L-spce. 2. Le KR n be he se of ll nonempy nd compc subses of R n nd le K c R n KR n be he subse of convex ses. We define he required operions nd he Husdorff meric on KR n s follows: Definiion 2. For A, B KR n, nd α R: A + B := {x + y : x A, y B} Minkowski Sum, αa := {αx : x A}, { δ h A, B = mx sup inf x y, sup inf x A y B where is he Eucliden norm in R n. x B y A } x y Husdorf f Disnce, Wih hese operions nd meric, KR n nd is subspce K c R n re complee, seprble meric spces see [8] nd since he xioms A-A8 hold, hese spces re L-spces. 3. The se of ll closed bounded subses of given Bnch spce, endowed wih he Husdorff meric is n L-spce. 4. Any qusiliner normed spce Y definiion see in [2] is n L-spce. 5. Consider see, e.g., [7] he clss of fuzzy ses E n consising of funcions u : R n [0, ] such h u is norml, i.e. here exiss n x 0 R n such h ux 0 = ; 2

3 b u is fuzzy convex, i.e. for ny x, y R n nd 0 λ, uλx + λy min{ux, uy}; c u is upper semiconinuous; d he closure of {x R n : ux > 0}, denoed by [u] 0, is compc. For ech 0 < α, he α-level se [u] α of fuzzy se u is defined s [u] α = {x R n : ux α}. The ddiion u + v nd sclr muliplicion cu, c R \ {0}, on E n re defined, in erms of α-level ses, by [u + v] α = [u] α + [v] α, [cu] α = c[u] α for ech 0 < α. Define lso 0 u by he equliy [0 u] α = {θ} here θ = 0,..., 0 R n. One of he possible merics in E n is defined in he following wy /p d p u, v = δ[u] α, [v] α dα p, p <. 0 Then he spce E n, d p is see [7, Theorem 3] complee seprble meric spce nd herefore n L-spce Inegrls of funcions wih vlues in L-spces We need he following noion of convex elemens in L-spces: Definiion 3. An elemen x X is convex if λ, µ 0 λx + µx = λ + µx. Remrk. Noe h if he elemen x is convex, hen i follows from A7 - A8 h λ, µ R δλx, µx λ µ δx, θ. 2. Le X c be se of ll convex elemens of given L-spce X. Remrk 2. X c is closed subse of X. We lso need he definiion of convexifying operor see [9] which we give in somewh modified form. Definiion 4. Le X be n L-spce. The operor P : X X c is clled convexifying operor if. x, y X δp x, P y δx, y; 2. P P = P ; 3. P αx + βy = αp x + βp y, x, y X, α, β R. Exmples of convexifying operors:. The ideniy operor in he spce K c R n is convexifying operor. 2. The operor, h on he spce KR n is defined by he formul P A = co A, is convexifying operor. Here by he co A we denoe he convex hull of se A. 3. The ideniy operor in he spce E n, d p is convexifying operor. Below we discuss L-spce X wih some fixed convexifying operor P. Denoe he imge of he elemen x X for he mpping P : X X s x : P x = x, x X. If ll elemens of n L-spce X re convex in he sense of Definiion, hen we choose he ideniy operor s he convexifying operor. Nex we define he Riemnnin inegrl for funcion f : [, b] X, where X is n L-spce. We gin follow Vhrmeev [9] for his purpose. Le f := f. 3

4 Definiion 5. The mpping f : [, b] X is clled wekly bounded, if δθ, f cons nd wekly coninuous, if f : [, b] X is coninuous. Remrk 3. Noe h if funcion f : [, b] X is coninuous hen f is wekly coninuous. If ll elemens x X re convex i.e. P = Id, hen he conceps of coninuiy nd wek coninuiy coincide. We need he noion of sepwise mpping from [, b] o n L-spce X. Definiion 6. The mpping f : [, b] X is clled sepwise, if here exiss se {x k } n X nd priion = 0 < <... < n = b of he inervl [, b], such h f = x k for k < < k. Definiion 7. The Riemnnin inegrl of sepwise mpping f : [, b] X is n elemen of he spce X h is defined by he following equliy: n fd = k+ k x k. Definiion 8. We sy h wekly bounded mpping f : [, b] X is inegrble in he Riemnnin sense if here exis sequence {f k } of sepwise mppings from [, b] o X, such h δ f, fk d 0, s k, 2.2 where is regulr Riemnnin inegrl for rel-vlued funcions. { } b I follows from 2.2 h he sequence f kd is Cuchy sequence nd hus we cn use he following definiion. Definiion 9. Le f : [, b] X be inegrble in he Riemnnin sense nd le {f k } be sequence of sepwise mppings such h 2.2 holds. Then he Riemnnin inegrl of f is he limi fd = lim k f k d. As described in [9] he Riemnnin inegrl for funcion f : [, b] X hs he following properies:. If f nd g re inegrble, hen α, β R he liner combinion αf +βg is inegrble, nd moreover αf + βgd = α fd + β gd. 2. If f nd g re inegrble, hen he funcion δ f, g is inegrble in he Riemnnin sense nd b δ fd, gd δ f, g d. 3. If f is inegrble in he Riemnnin sense, hen f = P f is lso inegrble nd fd = P fd = fd. The following heorem see [9], [2] gurnees h we cn consider he inegrls which rise below s Riemnnin inegrls. Theorem. A wekly bounded mpping f : [, b] X is inegrble in he Riemnnin sense if nd only if i is wekly coninuous lmos everywhere on [, b]. 4

5 2.4. Hukuhr ype derivive In his subsecion we define he Hukuhr ype derivive see definiion of Hukuhr derivive for se-vlued funcions in [0] of funcion wih vlues in n L-spce, nd prove necessry properies conneced wih his noion. Suppose X consiss only of convex elemens. Definiion 0. We sy h n elemen z X is he Hukuhr ype difference of elemens x, y X, if x = y + z. We denoe his difference by z = x h y. For f : [, b] X we define Hukuhr ype derivive. Definiion. If, b, nd for ll smll enough h > 0 here exis differences f + h h f nd f h f h, nd boh limis exis nd re equl o ech oher h f + h f f lim = lim h 0 + h h 0 + h f h h hen funcion is differenible poin in Hukuhr s sense if = or = b hen here exiss only one limi nd he Hukuhr ype derivive is defined by f + h f D H f = lim. h 0 + h h The following nlog of fundmenl heorem of Clculus holds: Theorem 2. For ny funcion F : [, b] X c h hs coninuous Hukuhr ype derivive on [, b] he equliy holds. F = F + D H F sds, [, b], Proof. For ny coninuous funcion f : [, b] X c nd ny m X c, we cn define F = fsds + m, m X c. 2.3 Nex we find D H F. I follows from F + h = F + +h fsds h for h > 0 he difference F + h h F is defined nd F + h h F = Now we show h lim h 0 + h We hve δ h +h +h +h fsds. fsds = f. 2.4 fsds, f = δ h +h fsds, h +h fds h +h δfs, fds. 5

6 Se ε > 0. Since f is coninuous he poin, here exis σ > 0, such h for s, + σ, we hve δfs, f < ε. This implies h for h < σ we hve +h δ fsds, f +h εds = ε h h which proves equliy 2.4. Similrly, F lim h 0 + h F h h Thus, D H F = f. From 2.3 we hve h = lim fsds = f. h 0 + h h F = m + D H F sds = F + D H F sds. 3. Adpion of clssicl pproximion operors o funcions wih vlues in L-spces As usul denoe by C[, b] he spce of coninuous funcions f : [, b] R wih he norm f C[,b] = mx{ f : [, b]}. Le X be n L-spce. Denoe by C[, b], X he se of ll coninuous funcions ϕ : [, b] X. This se endowed wih he meric ρϕ, ψ = mx [,b] δϕ, ψ = δϕ, ψ C[,b] is complee meric spce see, e.g., [2]. For funcion f C[, b], X we define he modulus of coninuiy by ωf, = sup δf, f, [0, b ]., [,b] Noe h ωf, 0, 0. If ωf, M 3. hen we sy h funcion f sisfies he Lipschiz condiion wih consn M. Denoe by ω f, he les concve up mjorn of he funcion ωf,. I is well known see [2] h he following inequliies hold ωf, ω f, 2ωf,. I will be convenien for us o give n esimion of pproximion in erms of funcion ω f,. 3.. Adpion of clssicl pproximion operors 3... Generl definiions nd esimions Mny clssicl pproximion operors for rel-vlued funcions re defined by he following scheme. Le he se of funcions λ = {λ 0,..., λ N } C[0, ] be given, such h λ i 0 nd N λ i. i=0 Le lso he se of poins ξ = {ξ 0,..., ξ N } [0, ] be given. For ech funcion f C[0, ] we define Λ λ,ξ [f] = N fξ k λ k

7 In his definiion funcion f C[0, ] cn be replced by funcion f C[0, ], X. We ge he operor h is defined on he se of ll coninuous funcions wih vlues in L-spces. Thus he definiion 3.2 we use for boh funcions f C[0, ] nd for f C[0, ], X. The following heorem gives generl esimion of pproximion of he funcion f C[0, ], X c by he operors of he form 3.2. Theorem 3. Suppose h for funcion g = Λ λ,ξ [g]. 3.3 Then for ny funcion f C[0, ], X c nd for ny [0, ] he following inequliy holds δf, Λ λ,ξ [f] ω f, Λ λ,ξ [ 2 ] 2. Proof. We hve N δf, Λ λ,ξ [f] = δ fλ k, N fξ k λ k N δf, fξ k λ k N ω f, ξ k λ k. By using he fc h funcion ω nd squre roo re concve up funcions nd using Jensen s inequliy we hve δf, Λ λ,ξ [f] ω f, N N ξk 2 λ k ω f, ξ k 2 λ k N = ω f, ξk 2λ k 2 N ξ k λ k + 2. Due o 3.3 N ξ k λ k =, nd hus we hve δf, Λ λ,ξ [f] ω f, Λ λ,ξ [ 2 ] Bernsein Operor Le now N λ k = k N k, ξ k = k, k = 0,..., N. k N We obin he nlog of Bernsein Operor for funcions wih vlues in L-spce N k N B N [f] = f k N k. N k For his operor condiions of he Theorem 3 hold. Moreover, B N [ 2 ] = 2 N. Thus he following heorem holds. Theorem 4. For ny f C[0, ], X c nd for ny [0, ] δf, B N [f] ω f,, N nd herefore ρf, B N [f] ω f, 2 N. 7

8 3..3. Schoenberg Operor For inegers N, k > 0, we consider he sequence of knos = { j } N+k j= k k =... = 0 = 0 < <... < N =... = N+k =. Le = mx { j+ j }. Le ξ j,k := j j+k j= k,...,n+k k, k j N, nd le b j,k := j+k+ j [ j,..., j+k+ ] k + be he normlized B-splines. For f C[0, ], X c le S N,k [f] = N j= k S N,k [f] = lim y S N,k [f]y. fξ j,k b j,k, 0 <, For rel-vlued funcions he operor S N,k ws inroduced by Schoenberg in [8]. The normlized B-splines form priion of he uniy N j= k b j,k =, nd he Schoenberg operor reproduces liner funcionssee [5], [5], i.e. N j= k ξ j,k b j,k =. For he funcion g = 2, he error E = S N,k [g] g sisfies [5], [5] 0 E = N j= k ξ 2 j,kb j,k 2 min { } 2 k +, 2k 2 nd besides h for N, k, [0, ] he following poinwise esimion holds see [5] E min{2, k N }. N + k Tking ino ccoun Theorem 3 we obin Theorem 5. For ny f C[0, ], X c { δf, S N,k [f] ω f, min 2k, } k + 2. nd for ny [0, ] δf, S N,k [f] ω f, min{2, k N }. N + k Modified Schoenberg Operor Consider modificion of Schoenberg operor h we obin when 0 if j = k,..., 0, j j = N if j =,..., N, if j = N +,..., N + k 8

9 nd ξ jk = j for j = k,..., N + k. The obined operor hs he form S N,k [f] = N j= k f j b j,k nd S N,k [f] = lim SN,k [f]. Nex we ge he esimion of he pproximion by such n operor. Theorem 6. For ny f C[0, ], X c ρ f, S N,k [f] 2ω f, k +. 2N Proof. For ny [0, ] we hve δ f, S N,k [f] δ f, S N,k [f] + δ S N,k [f], S N,k [f]. 3.4 Due o he definiion of ξ j,k for ny j = k,..., N we hve j ξ j,k k+ N δ S N,k [f], S N,k [f] δ N j= k N j= k ω f, k+ 2N fξ j,k b j,k, N j= k f j b j,k. Thus δ fξ j,k, f j b j,k N ω f, j ξ j,k b j,k j= k. We use Theorem 5 o esime firs erm nd inequliy 3.5 o esime second erm 3.4. We hve δ f, S N,k [f] ω f, k+ 2 N + ω f, k+ 2N 2ω f, k k+ 2 2N = 2ω f, k+ 4N + k+ 3 2ω f, k+ 2N Piecewise-liner inerpolion Define n operor P N h ssigns o funcion f he funcion P N [f] = 3.5 N f k l k, 3.6 where k = + k b N, k = 0,..., N, nd k / k k if [ k, k ] l k = k+ / k+ k if [ k, k+ ] 0 else. 3.7 In he cse = 0 nd b = his operor sisfies condiions of he Theorem 3. Moreover, if [ k, k ], hen P N [ 2 ] 2 = k k = Therefore we obin he following heorem N k + k N N 2. Theorem 7. If f C[0, ], X c, hen for [ k, k ], k =,..., N δf, P N [f] ω f, N k + k N k k = ω f,, 3.8 N nd hus ρf, P N [f] ω f,. 2N 9

10 In priculr, if f sisfies he Lipschiz condiion 3. wih consn M, hen ρf, P N [f] M 2N. 3.9 In order o obin esimions of he error of piecewise-liner inerpolion i is no necessry o use generl Theorem 3. To illusre h we presen he following semen Theorem 8. If f C[, b], X c, hen ρf, P N [f] ω f, b. 3.0 N Moreover, for [ k, k ], k =,..., N δf, P N [f] ω f, 2 k k, 3. k k nd herefore ρf, P N [f] ω f, b N If ωf, M, 0, hen ρf, P N [f] M b 2N. 3.3 Remrk 4. Esimion 3. in he cse = 0, b = for = k + k /2 coincides wih esimion 3.8. For he res of k, k he esimion 3. is beer hn 3.8. Proof. For [ k, k ] P N [f] = k k k f k + k k k f k. Consequenly, using A8 nd A7 we hve δf, P N [f] = δ k k k f + k k k f, k k k f k + k k k f k k k k δf, f k + k k k δf, f k k k k ωf, k + k k k ωf, k ω 3.4 f, b n. Therefore δf, P N [f] ω f, b N nd ρf, PN [f] ω f, b N. The inequliy 3.0 is proved. Using 3.4 nd pplying Jensen s inequliy we hve δf, P N [f] k k k ω f, k + k k k ω f, k ω f, 2 k k. k k The inequliy 3. is proved. Inequliies 3.2 nd 3.3 re now obvious. The following heorem gives he esimion of he error of pproximion by piecewise-liner funcions for such f h D H f C[, b], X. Theorem 9. Suppose he funcion f : [, b] X c hs he Hukuhr ype derivive D H f on he inervl [, b]. Then if D H f C[, b], X c, we hve for ny k =,..., N nd ny [ k, k ] δf, P N [f] 2 k k k k 2 2N 0 ωf, 2udu, 3.5 0

11 nd ρf, P N [f] 2 2N 0 ωf, 2udu. 3.6 In priculr, if D H f sisfies he Lipschiz condiion wih consn M, hen ρ f, P N [f] Mb 2 8N Proof. For [ k, k ] using Theorem 2 we hve δf, P N [f] = δ k k k f k + k D H fudu + k k k f, k k k f k + k k k f + k D H fudu δ k k k k D H fudu, k k k k D H fvdv = δ k k k k D H fudu, k k k k D H f k k u + k k k k k k k δ D H fu, D H f k k u + k k k du k k k k ω f, k k u + k k k u du = k k k k ω f, k k k u du = 2 k k k k 2 2N ωf, 2udu. 0 du Inequliy 3.5 is proved. Inequliy 3.6 holds since k k k k 2 4. Inequliy 3.7 now is obvious Esimion of he reminder of Trpezoidl Qudrure Formul Nex we obin n esimion of he reminder of he rpezoidl qudrure formul fd b N 2 f N 0 + f k + 2 f N. k= Such esimes re well known for rel vlued funcions. For f C[, b], X se Noe h R N f = δ Therefore P N [f]d = R N f = δ fd, b N 2 f N 0 + f k + 2 f N. 3.8 k= P N [ f]d = b N fd, P N [ f]d 2 f N 0 + f k + 2 f N. k= δ f, P N [ f]d mx b δ f, P N [ f]b = b ρ f, P N [ f]. Noe lso h due o he Propery of convexifying operor we hve for ny funcion f C[, b], X he following: ω f, ωf,, [0, b ]. Therefore from Theorems 8 nd 9 we hve

12 Theorem 0. Le f C[, b], X. Then R N f b ω f, b. N If f sisfies he Lipschiz condiion 3. wih consn M, hen R N f Mb 2. N If funcion f is such h f hs coninuous derivive D H f on [, b] hen R N f b 2 2N 0 ω f, 2udu. In priculr, if D H f sisfies he condiion of 3., hen R N f Mb 3 8N Exisence nd Uniqueness of soluion of inegrl equions 4.. Fredholm equion We consider Fredholm equion of he second kind ϕ = λ K, sϕsds + f, 4. where ϕ : [, b] X is he unknown funcion, f : [, b] X is known coninuous funcion, he kernel K, s, s [, b] is known rel-vlued funcion, λ is fixed prmeer. Theorem. Suppose K, s sisfies he following condiions. K, s is bounded, i.e. K, s M for ll, s [, b]; 2. K, s is coninuous every poin, s [, b] [, b] where s. Le f C[, b], X. Then for ny λ such h λ < Mb he equion 4. hs unique soluion ϕ C[, b], X. Proof. Consider he operor A : C[, b], X C[, b], X defined by Aϕ = λ K, sϕsds + f, ϕ C[, b], X. We prove firs h under he condiions on he K, s his operor mps C[, b], X ino C[, b], X. For his i is enough o show h operor Bϕ := K, sϕsds mps C[, b], X ino C[, b], X. To do his consider δbϕ, Bϕ, where < b. We hve δbϕ, Bϕ b = δ = δ K, sϕsds, K, sϕsds K, s ϕsds, K, s ϕsds δk, s ϕs, K, s ϕsds. 2

13 From his, using propery 2., we obin δbϕ, Bϕ K, s K, s δ ϕs, θds mx δ ϕs, θ s [,b] K, s K, s ds = ρ ϕ, θ K, s K, s ds. If K, s sisfies condiions nd 2 hen for ny ε > 0 here exiss σ > 0 such h, [, b] < σ K, s K, s ds < ε From his nd he bove esime i follows h he funcion Bϕ is uniformly coninuous, nd herefore, Bϕ C[, b], X. Nex we show h he operor A is conrcive operor. We hve δaϕ, Aψ = δ λ K, sϕsds + f, λ b K, sψsds + f λ δ K, sϕsds, b K, sψsds λ K, s δϕs, ψsds Thus λ K, s ds mx δϕs, ψs s [,b] λ Mb ρϕ, ψ. ρaϕ, Aψ λ Mb ρϕ, ψ. This mens h he mpping A is conrcive if λ < Mb. This conrcive mpping hs unique fixed poin which implies h he equion 4. hs unique soluion in he spce C[, b], X Volerr equion We consider now he Volerr inegrl equion ϕ = K, sϕsds + f, 4.2 where ϕ : [, b] X is gin n unknown funcion, nd f : [, b] X is known coninuous funcion. The kernel K, s is known rel-vlued funcion defined for, s [, b] [, b] such h s. Below we ssume h K, s is defined on ll squre [, b] [, b], nd K, s = 0, if s >. Since Volerr inegrl equion is priculr cse of Fredholm inegrl equion, we cn use Theorem o gurnee exisence nd uniqueness of he soluion of he equion 4.2 for he kernel K, s such h M < b. However, for Volerr equions we cn prove he following more generl heorem wihou he resricion on M. Theorem 2. Le K, s be coninuous in he domin {, s [, b] [, b] : s } nd le f C[, b], X. Then he equion 4.2 hs unique soluion ϕ C[, b], X. Proof. To prove his heorem we dop he mehod known for Volerr equions for rel-vlued funcions see, e.g., []. Consider he operor A : C[, b], X C[, b], X defined by Aϕ = K, sϕsds + f. Since equion 4.2 is priculr cse of equion 4., i is cler h his operor mps he spce C[, b], X ino C[, b], X.. 3

14 We prove h some ineger power of his operor is conrcive operor. We need o inroduce some ddiionl noions. Le K, s = K, s, K N, s = Noe h if K, s M hen see [, p. 449] K N, uku, sdu, N >. K N, s M N b N. 4.3 N! I is esily seen h for ny N N A N ϕ = K N, sϕsds + N Using 4.4 nd 4.3 we obin δa N ϕ, A N ψ δ K N, sϕsds, K N, sψsds Therefore, k= ρa N ϕ, A N ψ M N b N ρϕ, ψ. N! K N k, sfsds + f. 4.4 δ K N, sϕs, K N, sψs ds K N, s δϕs, ψsds M N b N N! mx δϕs, ψs. sb If N is sufficienly lrge, hen b N N! M N <. Fix such n N. We hve shown h he operor A N is conrcive. Using he generlized conrcive mpping principle see [] we obin h A hs unique fixed poin nd h herefore he equion 4.2 hs unique soluion ϕ C[, b], X. 5. Algorihms for pproxime soluion In his secion we describe lgorihms for he pproxime soluion of Fredholm nd Volerr inegrl equions for funcions wih vlues in L-spces. We dop well-known mehods for inegrl equions for rel-vlued funcions, specificlly collocion mehod see for exmple [3, ch. 3] in Subsecion 5. nd 5.2, nd qudrure formuls mehods see [3], [4] in Secion Fredholm equion We sr wih he Fredholm inegrl equion 4.. Le n N. Choose se of knos = 0 < <... < n = b nd se of coninuous rel-vlued funcions γ 0, γ,..., γ n defined in [, b]. Suppose h hese funcions sisfy he following inerpolion condiions γ k j = δ k,j, j, k = 0,,..., n. Noe h we cn use crdinl Lgrnge inerpolion polynomils s well s crdinl inerpolion splines s such funcions. For exmple we cn ke γ k = l k, where l k is defined by 3.7, k = + k b n. We look for soluion of 4. in he form ϕ n = n ϕ k γ k, ϕ k X, k = 0,..., n. 5. 4

15 Subsiuing in he equion 4. nd seing = j gives ϕ j = λ n ϕ k K j, sγ k sds + f j, j = 0,..., n, 5.2 where f j = f j. Se j,k = K j, sγ k sds. The sysem cn be rewrien in he form ϕ j = f j + λ n ϕ k j,k, j = 0,..., n. 5.3 Under some ddiionl ssumpions we cn solve his sysem by he mehod of consecuive pproximions. We illusre his in deil for γ k = l k, k = 0,,..., n. Choose n iniil pproximion {x 0,..., x n } o he soluion of his sysem 5.3. For j = 0,..., n se Le x 0 j = x j, x m+ j = f j + λ X n+ = X... X. }{{} n+ imes n x m k j,k, m = 0,,.... Consider he operor B : X n+ X n+, which is defined ccording o he rule y j = f j + λ n x k j,k, j = 0,..., n. Le meric in he spce X n+ be defined by ρ x 0,..., x n, y 0,..., y n := n δx j, y j. j=0 Wih his meric he spce X n+ is complee. We hve ρbx 0,..., x n, By 0,..., y n = n The ls inequliy holds since n j,k n j=0 n j=0 λ K j, s l k sds M δ δ f j + λ n x k j,k, f j + λ n ỹ k j,k λ n x k j,k, λ n ỹ k j,k n j=0 n n δ x j, ỹ j j,k λ δx j, y j n j,k j=0 λ Mb ρ x 0,..., x n, y 0,..., y n. n l k sds = Mb. Therefore, if λ < Mb he operor B is conrcive, nd consequenly he sequence {xm 0,..., x m n } m=0 converges o he soluion ϕ 0,..., ϕ n of he sysem 5.3 s m. The funcion of he form 5. where {ϕ 0,..., ϕ n } is he soluion of he sysem 5.3 is our pproximion of he soluion of he equion 4.. 5

16 5.2. Volerr equion Becuse Volerr Equions cn be considered s specil cse of Fredholm equions he mehod described in he preceding secion cn be pplied for heir soluion. However, in he cse of X = X c nd supp γ k [ k, k+ ] he resuling liner sysem becomes ringulr nd cn be solved explicily, nd rher simply here suppγ is he suppor of funcion γ. Consider he Volerr Equion 4.2 under he ssumpion h K, s is nonnegive. As before we look for soluion of he form 5.. As bove, le j,k = j K j, sγ k sds. Assume h γ k 0, k = 0,..., n. Then j,k 0 if k j nd j,k = 0 if k > j. Subsiuing 5. in 4.2 nd evluing j gives he ringulr sysem ϕ j = j j,k ϕ k + f j, j = 0,..., n, h define ϕ 0,..., ϕ n. This sysem cn be rewrien s h j ϕ j j,j ϕ j = ϕ k j,k + f j. We ssume h funcions γ k re uniformly bounded in n nd n is sufficienly lrge so h 0 j,j < for j = 0,..., n. Wih h ssumpion nd using he fc h ϕ j re convex, we obin ϕ j h j,j ϕ j = j,j ϕ j. Thus we obin he explici recursion ϕ 0 = f 0, ϕ j = j,j j ϕ k j,k + f j, j =,..., n Nysröm Mehod for Fredholm Equions Consider gin he Fredholm equion 4.. For rel vlued funcions he following pproch o finding pproxime soluion of such equions is well known see for exmple [3, ch. 4]. Le qudrure formul be given: n gsds p j g j, g C[, b], X, j= where < 2 <... < n b, p j R. Using his qudrure formul we pproxime he inegrl in 4. nd obin new equion: n ϕ = λ p j K, j ϕ j + f. 5.5 j= Evlue 5.5 k, k =,..., n: n ϕ k = λ p j K k, j ϕ j + f k. j= This is he sysem of equions wih unknown ϕ k = ϕ k, which we rewrie in he form n ϕ k = λ b kj ϕ j + fk n, j =,..., n, 5.6 j= where b kj = p j K k, j. Under some ddiionl ssumpions we cn solve 5.6 using mehod of consecuive pproximions, nlogously o he soluion of he sysem 5.3. The soluion ϕ,..., ϕ n of he sysem 5.6 we cn use s pproxime vlues of he soluion of he equion 4. poins,..., n. 6

17 5.4. Nysröm Mehod for Volerr Equions Consider now he Volerr equion 4.2. We describe qudrure formuls mehod for he pproxime soluion of he liner Volerr equion 4.2 bsed on he rpezoidl rule. We ssume h kernel K, s is nonnegive. Since we cn consider Volerr equions s priculr cse of Fredholm equions, he mehod described in secion 5.3 cn be pplied nd for heir soluions oo. However, if we ssume h ll elemens of he spce X re convex we cn obin n explici soluion of our sysem. For simpliciy le = 0, b =, i = i/n, i = 0,,..., n. Once we pply he rpezoidl rule for pproxime clculion of he inegrl k 0 K k, sϕsds in 4.2 nd omi he reminder, we hve he following { } ϕ k = f k + k n 2 K k, 0 ϕ 0 + K k, i ϕ i + 2 K k, k ϕ k, k =, 2,..., n, here 0 i= i= := 0 wih ϕ 0 = f 0. Se n K k, i = c ki nd noe h for n lrge enough we hve 0 c ki <. We obin explici recursive formul h is nlogous o 5.4: ϕ 0 = f 0, ϕ j = 2 c jj 2 c j j0ϕ 0 + i= c ji ϕ i + f j, j =,..., n. I seems possible o use for he pproxime soluion of Volerr ype equions, s well s for Fredholm ype equions for funcions wih vlues in L-spces, qudrure formuls of higher ccurcy see for exmple [4, ch.7]. An nlysis of such mehods requires ddiionl ools h hve no ye been developed for funcions wih vlues in L-spces, nd even for se-vlued funcions. We will explore his subjec in fuure work. 6. Convergence Anlysis In Subsecions 6. nd 6.2 we ssume h ll elemens of he spce X re convex. 6.. Convergence of lgorihm of pproxime soluion of Fredholm equions Recll h in Secion 5. he elemens ϕ 0,..., ϕ n were defined s soluion of he sysem 5.3. As pproxime soluion of he Fredholm equion 4. we consider he funcion 5. wih γ k = l k. Theorem 3. Le K, s nd f sisfy he condiion of Theorem, nd λ < Mb. Le ϕ be he soluion of he equion 4. nd le ϕ n, n N be defined by 5.. Then ρϕ, ϕ n = mx b δϕ, ϕn 0, s n. 6. If for ny s [, b] he kernel K, s sisfies he Lipschiz condiion for wih consn M h does no depend on s nd f sisfies in [, b] he Lipschiz condiion wih consn M 2, hen here exiss consn C such h for ny n mx δϕ, b ϕn C n. 6.2 Moreover, if for ny s [, b] he kernel K, s is coninuously differenible wih respec o, nd K,s sisfies he Lipschiz condiion wih consn M 3 h does no depend on s, f hs Hukuhr ype derivive D H f in [, b] nd D H f sisfies he Lipschiz condiion 3. wih consn M 4, hen here exiss consn C 2 such h for ny n mx δϕ, b ϕn C 2 n

18 Proof. Recll h he operor P N [f] is defined by 3.6. Using 5. nd 5.2 we hve [ n ] P n [ϕ n b ] = λp n ϕ k K, sl ksds + P n [f] = λ n n ϕ k K j, sl k sds l j + P n [f] j=0 = λ [ ] b n n K j, sl j ϕ k l k sds + P n [f] where P n, [K], s = j=0 = λ P n,[k], sϕ n sds + P n [f], n K j, sl j. j=0 Since P n [ϕ n ] = ϕ n piecewise liner funcion h inerpoles piecewise liner funcion he knos coincides wih he inerpoled funcion, we obin h he funcion ϕ n solves he following equion ϕ n = λ P n, [K], sϕ n sds + P n [f]. 6.4 Le ϕ be he soluion of he equion 4. nd le ϕ n solves equion 6.4. We esime he disnce beween ϕ n nd ϕ. We hve δϕ, ϕ n b λ δ K, sϕsds, b P n,[k], sϕ n sds + δf, P n [f] λ δ K, sϕsds, b P n,[k], sϕsds + + λ δ P n,[k], sϕsds, b P n,[k], sϕ n sds + δf, P n [f]. Therefore, mx δϕ, b ϕn λ mx δ b b P n,[k], sϕsds, b P n,[k], sϕ n sds + + λ mx δ b b K, sϕsds, b P n,[k], sϕsds λ mx P n,[k], s δϕs, ϕ n sds b + λ mx b + mx b δf, P n[f] K, s P n,[k], s δϕs, θds + mx b δf, P n[f]. Noe h if K, s M, hen P n, [K], s M s well. Therefore mx δϕ, b ϕn λ Mb mx δϕ, ϕ n b + λ mx b K, s P n,[k], s ds mx b + mx b δf, P n[f]. From he esimion bove we obin h if λ < Mb hen δϕ, θ λ Mb mx δϕ, b ϕn λ mx b I is esily seen h K, s P n,[k], s ds mx sb δϕs, θ + mx b δf, P n[f]. 6.5 mx b K, s P n, [K], s ds 0, s n

19 Besides h see Theorem 8 mx δf, P n[f] 0, s n. b From he ls wo relions nd from 6.5 we see h ρϕ, ϕ n = mx b δf, P n[f] 0, s n. We hve proved he relion 6.. Le us prove now he relion 6.2. Suppose h for ny s [, b] he kernel K, s sisfies he Lipschiz condiion for wih consn M h does no depend on s. I follows from rel-vlued nlog of Theorem 8 h mx b K, s P n, [K], s ds b 2 M n If f sisfies in [, b] he Lipschiz condiion wih consn M 2, hen i follows from Theorem 8 h mx δf, P n[f] M 2b. 6.8 b 2n Using 6.7 nd 6.8 we obin from 6.5 h here exiss consn C such h for ny n mx δϕ, b ϕn C n. We hve proved he relion 6.2. Suppose now h for ny s [, b] he kernel K, s is coninuously differenible wih respec o, nd K, s/ sisfies he Lipschiz condiion wih consn M 3 h does no depend on s, hen see, e.g., [3, p.60] mx b K, s P n, [K], s ds M 3b 3 8n If he Hukuhr ype derivive of f sisfies he Lipschiz condiion 3. wih consn M 4 hen due o Theorem 8 mx δf, P n[f] M 4b 2 b 8n Using 6.9 nd 6.0 we obin from 6.5 h here exiss consn C 2 such h for ny n mx δϕ, b ϕn C 2 n 2. We hve proved he relion Volerr equion Since Volerr equion is priculr cse of Fredholm equion, we cn pply he firs semen of Theorem 3 o obin he convergence of he mehod presened in Secion 5.2 under he ssumpion M < /b. However, in he cse of Volerr equion wih nonnegive kernel, we cn elimine he resricion. We obin Theorem 4. Le he kernel K, s of he equion 4.2 be nonnegive, coninuous nd K, s M in he domin 0 s b. Le ϕ be he soluion of he equion 4.2 nd le ϕ n, n N, be he pproxime soluion 5.. Then ρϕ, ϕ n = mx b δϕ, ϕn 0, s n. 9

20 If in ddiion he kernel K, s sisfies he Lipschiz condiion wih consn G in for ny s in he domin s b, nd f sisfies he Lipschiz condiion wih consn G 2 in [, b], hen here exiss consn G 3 such h for ny n N 2 mx b δϕ, ϕn G 3 n. 6. Proof. We use noions from he previous secion, king ino ccoun h K, s = 0 if s > nd λ =. As in Secion 6. we hve h ϕ nd ϕ n sisfy 4. nd 6.4. We would like o esime he disnce beween ϕ n nd ϕ. Se Since P n,;n [K], s := P n, [K] N, s. mx P n,[k], s mx K, s we hve,sb,sb P n,;n [K], s M N b N. 6.2 Consider he operors Aϕ = K, sϕsds + f, ϕ C[, b], X, A n ϕ = P n,[k], sϕsds + P n [f], ϕ C[, b], X. Le A N ϕ be defined by 4.4 nd le A N n ϕ = P n,;n [K], sϕsds + N k= P n,;n k [K], sp n [f]sds + P n [f]. Since ϕ is fixed poin of he operor A, hen ϕ is fixed poin of operor A N, for ll N. Therefore, ϕ sisfies he equion ϕ = K N, sϕsds + N k= Similrly since ϕ n is fixed poin of he operor A n ϕ n = P n,;n [K], sϕ n sds + K N k, sfsds + f. 6.3 N k= Nex we esime δϕ, ϕ n. Using 6.3 nd 6.4 we hve: P n,;n k [K], sp n [f]sds + P n [f]. 6.4 mx δϕ, b ϕn mx b δk N, sϕs, P n,;n [K], sϕ n sds + N mx δk N k, sfs, P n,;n k [K], sp n [f]sds For N we hve k= b + mx δf, P n[f] b =: N + N N k + 0. k= 6.5 N = mx b δk N, sϕs, P n,;n [K], sϕ n sds mx b δk N, sϕs, K N, sϕ n sds + mx b δk N, sϕ n s, P n,;n [K], sϕ n sds mx b K N, sδϕs, ϕ n sds + mx b K N, s P n,;n [K], s δϕ n s, θds. 20

21 Using 4.3 we obin K N, sδϕs, ϕ n sds M N b N mx b N! 2 mx b δϕ, ϕn, b mx b δϕ, ϕn if N is sufficienly lrge. We fix such n N below. Therefore, we obin from 6.5 if 2 mx b δϕ, ϕn Furher, mx b mx b mx b N K N, s P n,;n [K], s ds mx sb δϕn s, θ + N k k= K N, s P n,;n [K], s ds mx sb δϕn s, θ 0, n, K N, s P n,;n [K], s ds 0, n. 6.7 We prove 6.7 by inducion in N. The relion 6.6 is he bse cse of he inducion. We ssume h for some N N, N >, mx b Using 6.2 we hve mx b K N, s P n,;n [K], s ds 0, n. 6.8 K N, s P n,;n [K], s ds = mx b mx b + mx b Mb mx b + M N b N mx ub K N, uku, sdu P n,;n [K], up n, [K]u, sdu ds K N, u P n,;n [K], u Ku, sduds+ P n,;n [K], u Ku, s P n, [K]u, s duds K N, u P n,;n [K], u du Ku, s P n,[k]u, s ds. The firs erm ends o zero by 6.8, nd he second erm ends o zero by 6.6. Therefore 6.7 holds. Noe h from he bove esimions i follows h for ny N here exiss consn C > 0 independen on n, such h mx b K N, s P n,;n [K], s ds C mx ub Le us esime N k. Using 4.3 nd 6.9 we hve N k = mx b δk N k, sfs, P n,;n k [K], sp n [f]sds mx b δk N k, sfs, K N k, sp n [f]sds + mx b δk N k, sp n [f]s, P n,;n k [K], sp n [f]sds mx b K N k, sδfs, P n [f]sds + mx K N k, s P n,;n k [K], s δp n [f]s, θds b M N b N N! + C mx b mx δfs, P n[f]s sb K, s P n,[k], s ds mx δp n[f]s, θ. sb 2 Ku, s P n, [K]u, s ds. 6.9

22 Tking ino ccoun 6.6 nd Theorem 8 we obin h for ny k =,..., N N k 0, n. Summrizing he obined esimes we hve 2 mx δϕ, b ϕn mx b K N, s P n,;n [K], s ds mx sb δϕn s, θ N k + 0 0, n. + N k= We hve proved he firs smen of he Theorem 4. I follows from 6.6 nd esimions for N k, k =,..., N, h here exiss C > 0 such h 2 mx b δϕ, ϕn mx b K, s P n, [K], s ds mx sb δϕn s, θ +C mx sb δfs, P n[f]s Assume h he kernel K, s sisfies he Lipschiz condiion wih consn G in for ny s in he domin s b. Assume lso h f sisfies he Lipschiz condiion wih consn G 2 in [, b]. Nex we prove h here exis consn G 4 such h for ny n mx b K, s P n, [K], s ds G 4 n. 6.2 ] nd for ny s [, b] we hve For ny [, + b n K, s P n, [K], s 2M. Besides h, if s > + 2 b n K, s P n, [K], s = 0. Therefore for such K, s P n, [K], s ds 4M b n. Furher, for ny [ + b n K, s P n, [K], s G b. 2n For ny s [ b n, + ] b n we hve K, s P n, [K], s 2M., b] nd for ny s [ ], b n we hve Finlly, if s > + b n, hen K, s P n,[k], s = 0. Therefore, for [ + b n, b] K, s P n, [K], s ds G b 2 Seing G 4 = Gb Mb we obin 6.2. We obin from Theorem 8 h 2n + 4M b n. mx δfs, P n[f]s G 2 sb n 6.22 if f sisfies he Lipschiz condiion in [, b] wih consn G 2. Using 6.20, 6.2 nd 6.22 we obin h here exis consn G 3 such h 6. holds nd hus we hve proved he second semen of he heorem. 22

23 6.3. Error nlysis for qudrure mehods We presen here only heorem h gives n error nlysis of he mehod bsed on rpezoidl qudrure formul for pproxime soluion of Fredholm equion. Le ϕ be n exc soluion of he equion 4. nd le ϕ n j, j = 0,,..., n be such h ϕ n j = λ b n Se ε n = mx 0jn δϕ j, ϕ n j. [ n 2 K j, 0 ϕ n 0 + K j, i ϕ n i + 2 K j, n ϕ n n i= ] + f j. Theorem 5.. Le K, s nd f be coninuous, K, s M in he domin [, b] [, b] nd λ < Mb. Then ε n 0, s n If for ny s [, b] he kernel K, s sisfies he Lipschiz condiion for wih consn M h does no depend on s nd f sisfies in [, b] he Lipschiz condiion wih consn M 2, hen here exiss consn C such h for ny n ε n C n. Proof. I is obvious h here exis consn C 2 > 0 such h 6.24 ε n C 2 mx R n K, ϕ definiion of R n K, ϕ see in 3.8. If K, s nd f sisfy he condiion of he semen of he heorem hen ϕ s well s K, ϕ re coninuous. If K, s nd f sisfy he condiion of he semen 2 of he heorem hen K, ϕ sisfies in [, b] he Lipschiz condiion wih some consn M 3. Now he semens of Theorem 5 follow from Theorem Numericl Exmples In his secion we discus numericl exmples for se-vlued funcions, i.e. funcions wih vlues in L-spce of convex, compc subses of R n : K c R n. We consider firs n iniil vlue problem D H X = λx + A, X = X 0, where D H X is Hukuhr derivive see [0], λ : [, b] R + nd A : [, b] K c R n re coninuous funcions, X 0 K c R n. This iniil vlue problem cn be rewrien in he form of he Volerr inegrl equion X = λsxsds + F, where F = X 0 + I is well known see for exmple [6] h soluion of his equion hs he form X = e λsds X 0 + Ase s λτdτ ds. Exmple. We consider he equion 7. wih λs = s nd F = [, ] [0, ] = [, 0] [0, ] + [0, ] {0}, [0, ]. Asds

24 Figure : The Husdorff disnce beween exc soluion nd pproxime soluion The exc soluion of he equion 7. is [ ] [ ] X = e 2 /2, e 2 s 2 /2 ds 0, e 2 /2. 0 We used MATLAB o implemen he lgorihm for he pproxime soluion of Volerr inegrl equions presened in Secion 5.2. We plo see Figure he Husdorff disnce beween he exc soluion nd he pproxime soluion. The ime-sep is /50. Exmple 2. Here we consider he Volerr inegrl equion 4.2 wih = 0, b =. Le K, s = s nd [ ] [ ] f = 0, 3 0, , [0, ]. 3 The exc soluion is ϕ = [0, ] [0, + ]. We plo he Husdorff disnce he error beween he exc soluion nd he soluion obined by boh collocion nd qudrure lgorihms h were described in Subsecions 5.2 nd 5.4. Here, s well s for ll oher exmples below we used MATLAB for implemenion. The ime-sep is /4. As one cn see from he corresponding picure see Fig. 2, he collocion mehods gives beer pproximion hen he qudrure mehod in his cse. Exmple 3. Le in he equion 4.2 K, s = e s, [0, ] nd f = [ 0, e + α cos α sin ] [ 0, e + α sin + α cos α ]. The exc soluion hs he following form ϕ = [ 0, e + α cos ] [ 0, e + α sin ]. This is he exmple of he problem for which qudrure mehod gives beer pproximion hn collocion mehod see Fig. 2b. Exmple 4. This is he exmple of pproxime soluion of Fredholm equion 4. wih = 0, b =, λ = by collocion lgorihm. We use kernel K, s = e +s nd f = [0, e + 2 /ee ] [0, e /e], [0, ]. 24

25 Exmple 2. Kernel K, s = s b Exmple 3. Kernel K, s = e s Figure 2: Exmple of qudrure nd collocion lgorihms for Volerr Equions Exmple 4. Kernel K, s = s sin 4 3. Error for 8, 6, 32 nd 64 knos. b Exmple 5. Kernel K, s = e +s. Error for 8, 6 nd 32 knos. Figure 3: Exmple of collocion lgorihm for Fredholm Equion. Noe we use here logrihmic scle for he y-xis. The exc soluion of he problem in his cse hs he following form ϕ = [ 0, e + ] [0, ] nd s i ws in ll previous exmples we plo he Husdorff disnce he error beween he exc soluion nd he soluion obined by described in Secion 5. lgorihm, hough here o plo resuls we use logrihmic scle for he y-xis see Fig. 3. We plo error for 8, 6, 32 nd 64 knos. Exmple 5. This is noher exmple of collocion lgorihm for Fredholm Equion. This ime we use kernel K, s = s sin 4 3 nd f = [ 0, ] 2 sin4 3 The exc soluion hs he form ϕ = [0, ] [0, + ]. Resuls for 8, 6 nd 32 knos see on Fig. 3b. [ 0, + 5 ] 6 sin4 3, [0, ]. 25

26 8. Discussion We hve shown h mny principles nd conceps governing single vlued inegrl equions rnsfer o he more generl cse of funcions wih vlues in L-spces, priculrly for se-vlued funcions, nd funcions whose vlues re fuzzy ses. The lgorihms we discussed dop he collocion mehod for he pproxime soluion of inegrl equions nd use piecewise-liner funcions. These lgorihms converge he re of O/n if he funcions defining he problem hve smoohness of order, nd converge he re of O/n 2 if funcions hve smoohness of order 2. In fuure work we hope o obin mehods of he pproxime soluion of inegrl equions h will use lernive mehods of pproximion nd will converge fser for funcions of greer smoohness. We lso pln o invesige he soluion of more generl, nonliner inegrl equions, nd inegrl nd differenil equion problems involving funcions of more hn one independen vrible, wih vlues in L-spces. Acknowledgmens The uhor would like o hnk Peer Alfeld nd Elen Cherkev for mny helpful discussions, dvices nd commens h grely improved he mnuscrip. References [] Ansssiou G.A. Fuzzy Mhemics: Approximion Theory. Sudies in Fuzziness nd Sof Compuing, 25 Springer, 200. [2] Aseev S.M. Qusiliner operors nd heir pplicion in he heory of mulivlued mppings. Proceedings of he Seklov Insiue of Mhemics 67, Russin 985; Zbl [3] Akinson, K. E. Cmbridge monogrphs on pplied nd compuionl mhemics: The numericl soluion of inegrl equions of he second kind. Cmbridge Universiy Press, 997. [4] Bbenko V.F., Bbenko V., Polischuk M.V. On Opiml Recovery of Inegrls of Se-Vlued Funcions. 204 rxiv: v. [5] Beuel, L., Gonsk, H., Kcso, D. On vriion-diminishing Schoenberg operors: new quniive semens. Monogrfis de l Acdemi de Ciencis de Zrgoz [6] Cordunenu C. Inegrl equions nd pplicions. Cmbridge universiy press, Cmbridge, 99. [7] Dimond, P., Kloeden, P. Meric spces of fuzzy ses. Fuzzy ses nd sysems [8] Dyn, N., Frkhi, E., Mokhov, A. Approximion of se-vlued funcions: Adpion of clssicl pproximion operors. Hckensck: Imperil College Press, 204. [9] Guo, D., Lkshmiknhm, V., Liu, X. Nonliner Inegrl Equions in Absrc Spces. Kluwer Acdemic Publishers. In Mhemics nd Is Applicions [0] Hukuhr, M. Inegrion des Applicions Mesurbles don l Vleur es un Compc Convexe. Funkcilj Ekvcioj, [] Kolmogorov A.N., Fomin S.V. Elemens of he Theory of Funcions nd Funcionl Anlysis. Dover Books on Mhemics, Dover Publicions, 999. [2] Korneichuk N.P. Exc Consns in Approximion Theory. Moskv: Nuk, 987; English rnsl. in Encyclopedi Mh. Appl. 38, Cmbridge Univ. Press, Cmbridge 99. [3] Lkshmiknhm, V., Mohpr R.N. Theory of fuzzy differenil equions nd inclusions. Series in Mhemicl Anlysis nd Applicions. Tylor nd Frncis Inc.,

27 [4] Linz, P. Anlyicl nd Numericl mehods for Volerr equions. SIAM, Phildelphi, 985. [5] Mrdsen, M. J. On uniform spline pproximion, J. Approx. Theory [6] Plonikov, V. A., Plonikov, A. V., Viyuk, A. N. Differenil equions wih mulivlued righ-hnd side. Asympoic mehods. Asroprin, 999. [7] D Pro, G., Innelli, M. Volerr inegrodifferenil equions in Bnch spces nd pplicions. Pimn reserch noes in mhemics series. Longmn Scienific nd Technicl, 989. [8] Schoenberg, I. J. On spline funcions, wih supplemen by T.N.E. Greville. In: Inequliies Proc. Symposium Wrigh-Person Air Force Bse, Augus 965, ed. by O.Shish New York: Acd. Press 967. [9] Vhrmeev, S.A. Inegrion in L-spces. Book: Applied Mhemics nd Mhemicl Sofwre of Compuers, M.: MSU Publisher, Russin. 27