Selections of set-valued functions satisfying the general linear inclusion

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1 J. Fixed Poit Theory Al. 18 ( DOI /s Published olie October 17, 2015 Joural of Fixed Poit Theory 2015 The Author(s ad Alicatios This article is ublished with oe access at Srigerlik.com Selectios of set-valued fuctios satisfyig the geeral liear iclusio Adrzej Smajdor ad Joaa Szczawińska Abstract. Alyig the classical Baach fixed oit theorem we rove that a set-valued fuctio satisfyig a geeral liear fuctioal iclusio admits a uique selectio fulfillig the corresodig fuctioal equatio. We also adot the method of the roof for ivestigatig the Rassias stability of geeral liear equatio. Mathematics Subject Classificatio. 54C65, 54C60, 39B82. Keywords. Set-valued fuctio, selectio, stability. 1. Itroductio Let (Y, be a real ormed sace. We deote by (Y the family of all oemty subsets of Y ad by cl(y, ccl(y, c(y ad c(y we deote collectios of all closed, covex closed, comact ad comlete members of (Y, resectively. The umber diam A := su{ a b : a, b A} is said to be the diameter of A (Y. We say that a set-valued fuctio F : K (Y (a s.v. fuctio for abbreviatio is with bouded diameter if the fuctio K x diam F (x R is bouded. I [5], the authors roved that if S is a commutative semigrou with zero ad (Y, is a real Baach sace, the F : S ccl(y isasubadditive s.v. fuctio; i.e., F (x + y F (x+f(y, x, y S, with bouded diameter admits a uique additive selectio. Poa i [11] roved that if K is a covex coe i a real vector sace X (i.e., t 1 K + t 2 K K for all t 1,t 2 0 ad F : K ccl(y (where (Y, is a real Baach sace is a s.v. fuctio with bouded diameter fulfillig the iclusio F (αx + βy αf (x+βf(y, x, y K, for α, β > 0, α + β 1, the there exists exactly oe additive selectio of F.

2 134 A. Smajdor ad J. Szczawińska JFPTA Nikodem ad Poa i [9] ad Piszczek i [10] roved the followig theorem. Theorem 1.1. Let K be a covex coe i a real vector sace X, (Y, a real Baach sace ad α,β,,q > 0. Cosider a s.v. fuctio F : K ccl(y with bouded diameter fulfillig the iclusio αf (x+βf(y F (x + qy, x, y K. If α+β <1, the there exists a uique selectio f : K Y of F satisfyig the equatio αf(x+βf(y =f(x + qy, x, y K. If α + β>1, the F is sigle valued. The case of + q = 1 was ivestigated by Poa i [13], Ioa ad Poa i [7] ad recetly, by Smajdor ad Szczawińska i [14]. I this aer, we determie the coditios for which a s.v. fuctio F : K (Y satisfyig the iclusio αf (x+βf(y F (x + qy, x, y K, (1.1 where α <ad diam F (x M x, x K (for some M>0 admits a selectio satisfyig the corresodig fuctioal equatio. It is easy to check that if F satisfies the oosite iclusio, the it is sigle valued. Theorem 2.1 ad also the method of its roof are used for the ivestigatio of the Rassias stability of geeral liear fuctioal equatio. 2. The mai theorem Let (X, ad (Y, be real ormed saces ad let K be a oemty subset of X. Cosider a s.v. fuctio F : K (Y. A fuctio f : K Y is a selectio of the s.v. fuctio F if ad oly if f(x F (x, x K. Let Sel(F := {f : K Y : f(x F (x, x K}. It is easy to check that if there exists a costat M>0such that diam F (x M x for all x K, the the fuctio { } f(x g(x d(f,g := su,x K\{0}, f,g Sel(F, x is a metric i Sel(F. Moreover, if F (x is comlete for every x K, the metric sace (Sel(F,d is comlete. Obviously, the covergece i the sace (Sel(F,d imlies the oitwise covergece o the set K. Theorem 2.1. Let (X, ad (Y, be real ormed saces ad let K X be such that 0 K. Assume that, q > 0 ad α, β R are fixed ad oe of the followig coditios holds: (i α <ad K K, (ii β <qad K qk.

3 Vol. 18 (2016 Selectios of set-valued fuctios 135 Cosider a s.v. fuctio F : K c(y such that 0 F (0 ad for some ositive costat M. If diam F (x M x, x K, αf (x+βf(y F (x + qy, x, y K, x + qy K, (2.1 the there exists a uique selectio f : K Y of F such that αf(x+βf(y =f(x + qy, x, y K, x + qy K. Proof. Assume that α <ad K K. Sice diam F (0 = 0 ad 0 F (0, F (0 = {0}. Puttig y = 0 i (2.1, we obtai αf F (x, x K. (2.2 Let T (g(x := αg, x K, g Sel(G. By (2.2, T (g Sel(F, g Sel(F. Moreover, for every g 1,g 2 Sel(F, d ( T (g 1, T (g 2 { ( g1 x } g2 = α su,x K\{0} x { ( = α x ( g1 su x } g2 x,x K\{0} α d(g 1,g 2. Sice α <, the ma T : Sel(F Sel(F is cotractive i the comlete metric sace (Sel(F,d, so by the Baach theorem, it has a uique fixed oit f ad lim T (g =f for each g Sel(F. Hece f : K Y is the uique selectio of the s.v. fuctio F such that f(x =αf, x K. Fix g Sel(F ad x, y K such that x + qy K. The x, y x + qy, K. By (2.1, Hece αg αg + βg + βg ( y,g ( y g ( + qy x + qy F. ( ( x + qy x + qy diam F M x + qy.

4 136 A. Smajdor ad J. Szczawińska JFPTA Thus αt (g(x+βt (g(y T(g(x + qy α M x + qy for every x, y K such that x + qy K. Proceedig by iductio, we get αt (g(x+βt (g(y T (g(x + qy ( α M x + qy for every N ad all x, y K with x+qy K. Lettig, we obtai αf(x+βf(y =f(x + qy, x, y K, x + qy K. Let X ad Y be real vector saces ad K a covex coe i X. It is easy to check that if f : K Y satisfies the equatio αf(x+βf(y =f(x + qy, x, y K, ad f(0 = 0, the f is additive; i.e., f(x+f(y =f(x + y, x, y K. Corollary 2.2. Let, q > 0 ad α <(or β <q. Assume that (X, ad (Y, are ormed saces, K a covex coe i X ad F : K c(y a s.v. fuctio such that 0 F (0 ad for some costat M>0. If diam F (x M x, x K, αf (x+βf(y F (x + qy, x, y K, the the uique selectio f : K Y of F fulfillig the equatio is additive. αf(x+βf(y =f(x + qy, x, y K, Remark 2.3. If α = ad β = q, the above corollary is ot true. Proof. Let K = [0, + ad let F : K c(r be a s.v. fuctio defied by ad diam F (x = x, x K. The F (x = [0,x], x K, F (x + y =F (x+f (y, x, y K, ad for every a [0, 1], the fuctio f(x =ax, x K, is a additive selectio of F.

5 Vol. 18 (2016 Selectios of set-valued fuctios Stability results I the first art of this sectio we reset a alicatio of Theorem 2.1 to the ivestigatio of the Rassias stability of the geeral liear fuctioal equatio αf(x+βf(y =f(x + qy. (3.1 For defiitio ad more results i the Rassias stability theory see, for istace, [6, 8]. The geeral liear equatio was cosidered by several authors (for examle, [1, 2, 12]. A set-valued aroach ca be foud i [3]. Gajda showed i [4] (see also [8, Theorem 2.6] that for ε>0 there is a fuctio f : R R satisfyig the iequality f(x+f(y f(x + y ε( x + y, x, y R, while there is o costat M 0 ad o additive fuctio f 0 : R R such that f(x f 0 (x M x, x R. We aly the method used i the roof of Theorem 2.1 to obtai the stability result for equatio (3.1. We will deote by R + ad Q + the set of all oegative real ad ratioal umbers, resectively. Theorem 3.1. Let K be a covex coe i a real ormed sace (X,, (Y, a real Baach sace ad ε: R + R + R + a R + -homogeous fuctio. Assume that, q > 0, α, β R are fixed ad α <. If a fuctio f : K Y satisfies αf(x+βf(y f(x + qy ε( x, y, x, y K, the there exists a uique fuctio f 0 : K Y fulfillig the fuctioal equatio αf 0 (x+βf 0 (y =f 0 (x + qy, x, y K, ad such that f(x f 0 (x f(0 M x, x K, for some M>0. Moreover, ε(1, 0 f(x f 0 (x f(0 x, x K, α ad f 0 is a additive fuctio. I articular, if ε(1, 0=0, the f(x =f 0 (x f(0, x K. Proof. Let g : K Y be give by g(x =f(x f(0, x K. It is easy to check that g(0 = 0 ad αg(x+βg(y g(x + qy ε( x, y, x, y K. (3.2 Hece, for all x, y K, αg(x+βg(y g(x + qy+ε( x, y B, (3.3

6 138 A. Smajdor ad J. Szczawińska JFPTA where B deotes the uit closed ball i the sace Y. Settig y = 0 ad relacig x by x/ i (3.3, we obtai ( ( x x αg g(x+ε, 0 ε(1, 0 B = g(x+ x B, x K. Cosider a s.v. fuctio G: K cl(y give by The ad αg G(x =g(x+ diam G(x = = αg g(x+ ε(1, 0 α 2ε(1, 0 α ε(1, 0 + α α ε(1, 0 x B, x K. x, x K, x B ( 1+ α α x B = G(x for all x K. The idea of the roof is the same as before so we oly give a sketch. The fuctio T : Sel(G Sel(G, give by T (h(x := αh, x K, h Sel(G, is cotractio with the costat α /. By the Baach theorem, there exists a uique fuctio f 0 Sel(G such that f 0 (x =αf 0 (x, x K, ad lim T (g =f 0. By the defiitio of G, f 0 (x g(x ε(1, 0 α x, x K. Sice g satisfies (3.2 ad ε is ositively homogeeous, αt (g(x+βt(g(y T (g(x + qy α ε( x, y for all x, y K. Proceedig by iductio, we get αt (g(x+βt (g(y T (g(x + qy for every x, y K ad 1. It follows that αf 0 (x+βf 0 (y =f 0 (x + qy, x, y K. ( α ε( x, y Sice f 0 (0 = 0, f 0 is additive. For the ed assume that f 1 : K Y is a solutio of equatio (3.1 such that f 1 (x g(x M x, x K,

7 Vol. 18 (2016 Selectios of set-valued fuctios 139 for some ositive M. The f 1 (0 = 0, αf 1 = f 1 (x, x K ad It is easy to check that f 1 (x f 0 (x f 1 (x f 0 (x ( M + ( ( α M + Hece f 1 = f 0, which eds the roof. ε(1, 0 x, x K. α ε(1, 0 x, x K, N. α I the secod art of this sectio, we aly Theorem 2.1 to obtai a stability result for equatio (3.1. Theorem 3.2. Let K be a covex coe i a real ormed sace (X,, (Y, a real Baach sace ad, q > 0, α, β R fixed costats such that α <. Cosider η 0 ad a fuctio f : K Y satisfyig the iequality αf(x+βf(y f(x + qy η ( x + y, x, y K. If there exists a costat λ>0 such that (1+ λα x + (1 + λβ y λ x + qy, x, y K, (3.4 the there exists a uique fuctio f 0 : K Y fulfillig the fuctioal equatio αf 0 (x+βf 0 (y =f 0 (x + qy, x, y K, ad such that f(x f 0 (x f(0 M x, x K, for some M>0. Moreover, f(x f 0 (x f(0 λη x, x K, ad f 0 is a additive fuctio. Proof. Let g : K Y be give by g(x =f(x f(0, x K. As i the roof of Theorem 3.1, g(0 = 0 ad αg(x+βg(y g(x + qy η ( x + y, x, y K. I articular, for all x, y K, αg(x+βg(y g(x + qy+η( x + y B, where B is the uit closed ball i Y. Defie a s.v. fuctio G: K cl(y as follows: G(x =g(x+λη x B, x K. The 0 G(0, diam G(x =2ηλ x, x K, ad αg(x+βg(y G(x + qy, x, y K.

8 140 A. Smajdor ad J. Szczawińska JFPTA Ideed, let x, y K be fixed. By coditio (3.4, αg(x+βg(y =αg(x+βg(y+η ( λα x + λβ y B g(x + qy+η ( x + y B + η ( λα x + λβ y B g(x + qy+ηλ x + qy B = G(x + qy. By Theorem 2.1, there exists exactly oe selectio f 0 : K Y of the s.v. fuctio G satisfyig equatio (3.1. Hece f(x f 0 (x f(0 λη x, x K. Sice f 0 (0 = 0, f 0 must be additive. The roof that f 0 is the oly solutio of equatio (3.1 such that for some M>0 rus as before. f(x f 0 (x f(0 M x, x K, Examle. Let α <ad β <q. Every covex coe K { (x 1,...,x R : x 1 0 x 0 x 1 0 x 0 } satisfies coditio (3.4 with λ max ad with the orm i R give by { } 1 α, 1 q β (x 1,...,x = x x, (x 1,...,x R. 4. The selectio theorem I this sectio we reset a certai cosequece of Theorem 2.1 for the case of a comact-valued s.v. fuctio F satisfyig the iclusio (1.1 ad such that the fuctio x diam F (x mas bouded sets oto bouded oes. Lemma 4.1. Let, q > 0 ad α, β R be fixed. Assume that X ad Y are real vector saces, K a covex coe i X ad K a collectio of oemty subsets of Y closed uder itersectios of chais. Let F : K Kbe a s.v. fuctio fulfillig the iclusio (1.1 ad 0 F (0. There exists a miimal s.v. fuctio F 0 : K Ksuch that (i F 0 (x F (x, x K, (ii 0 F 0 (0, (iii αf 0 (x+βf 0 (y F 0 (x + qy, x, y K. Proof. Defie the set F of all s.v. fuctios H : K Ksuch that H(x F (x, x K, 0 H(0 ad αh(x+βh(y H(x + qy, x, y K.

9 Vol. 18 (2016 Selectios of set-valued fuctios 141 The set F is oemty, because F F, ad artially ordered by the relatio H 1 H 2 H 1 (x H 2 (x, x K, H 1,H 2 F. It is eough to rove that there is a miimal elemet i F. Let C be a chai i F. Put H 0 (x = {H(x :H C}, x K. Sice K is closed uder itersectios, H 0 (x K, x K. Obviously, 0 H 0 (0 ad H 0 (x H(x, x K. Fix ow x, y K. For every H C, hece αh 0 (x+βh 0 (y αh(x+βh(y H(x + qy, αh 0 (x+βh 0 (y H 0 (x + qy. Therefore, the s.v. fuctio H 0 : K Kbelogs to F ad H 0 H, H C. By the Kuratowski Zor lemma, there exists a miimal elemet F 0 i the set F. Lemma 4.2. Let, q > 0 ad α, β R be fixed. Assume that X ad Y are real vector saces, K a covex coe i X ad K a family of oemty subsets of Y such that αk Kad λk K, λ Q +. Cosider a s.v. fuctio F : K K. If F 0 : K Kis a miimal s.v. fuctio such that (i F 0 (x F (x, x K, (ii 0 F 0 (0, (iii αf 0 (x+βf 0 (y F 0 (x + qy, x, y K, the F 0 is sueradditive ad Q + -homogeeous; i.e., F 0 (x+f 0 (y F 0 (x + y, x, y K, ad F 0 (λx =λf 0 (x, x K, λ Q +. Proof. Puttig y = 0 i (iii ad by (ii, αf 0 F 0 (x, x K. Defie H(x =αf 0, x K. The H(x K, H(x F 0 (x F (x, x K ad 0 αf 0 (0 = H(0. Observe that the s.v. fuctio H : K Ksatisfies the iclusio (iii. Ideed, let x, y K be fixed. The ( ( ( x y αh(x+βh(y =α αf 0 + βf 0 αf 0 + qy = H(x + qy. Thus, by the miimality of F 0, F 0 (x =H(x, x K; i.e., F 0 (x =αf 0, x K.

10 142 A. Smajdor ad J. Szczawińska JFPTA Similarly we get F 0 (y =βf 0 ( y q, y K. Hece, by (iii, F 0 (x+f 0 (y =αf 0 ( y + βf 0 F 0 (x + y q for every x, y K; i.e., F 0 is sueradditive. I articular, F 0 F 0 (x, x K, N. Fix ow N ad cosider the s.v. fuctio K x F 0 K. We have 0 F 0 (0 ad αf 0 F 0 F 0 (x F (x, x K, ( y ( x + qy + βf 0 F 0, x, y K. Therefore, F 0 = F 0 (x, x K, by the miimality of F 0. Cosequetly, F 0 ( λx = λf0 (x, x K, λ Q +. Remark 4.3. Let (X, be a real ormed sace ad K X a coe. Assume that f : K R is Q + -homogeeous. The fuctio f mas bouded subsets of K oto bouded sets if ad oly if there exists a ositive costat M such that f(x M x, x K. Proof. Assume that f : K R is Q + -homogeeous (i articular, f(0 = 0 ad mas bouded subsets of K oto bouded sets. Let M = su{ f(x : x K x 1}. Fix x K \{0}. There exists a decreasig sequece (λ such that lim λ = 1 ad λ x Q +, N. Sice x λ x = 1 < 1, N, λ we have ( x f M, N. λ x Cosequetly f(x M x. The oosite imlicatio is obvious.

11 Vol. 18 (2016 Selectios of set-valued fuctios 143 Theorem 4.4. Let, q > 0, α, β R ad α <. Assume that (X, ad (Y, are real ormed saces, K is a covex coe i X ad F : K c(y a s.v. fuctio such that 0 F (0 ad αf (x+βf(y F (x + qy, x, y K. If K x diam F (x R mas bouded sets oto bouded sets, the there exists a uique selectio f : K Y of F fulfillig the equatio The selectio f is additive. αf(x+βf(y =f(x + qy, x, y K. Proof. The family c(y is closed uder itersectios of chais ad λ c(y c(y, λ R. By Lemma 4.1, there exists a miimal s.v. fuctio F 0 : K c(y such that F 0 (x F (x, x K, 0 F 0 (0 ad αf 0 (x+βf 0 (y F 0 (x + qy, x, y K. Lemma 4.2 shows ow that F 0 is sueradditive ad Q + -homogeeous. Sice diam F 0 (x diam F (x, x K, the fuctio K x diam F 0 (x R mas bouded sets oto bouded sets ad it is Q + -homogeeous. Hece, by Remark 4.3, there exists a ositive costat M such that diam F 0 (x M x, x K. Cosequetly, by Theorem 2.1, the s.v. fuctio F 0 admits a uique selectio f 0 : K Y such that αf 0 (x+βf 0 (y =f 0 (x + qy, x, y K. Sice f 0 (0 = 0 ad f 0 satisfies the above equatio, f 0 must be additive. Assume that f 1,f 2 : K Y are selectios of F such that They are additive ad Hece, for every N, αf i (x+βf i (y =f i (x + qy, x, y K, i =1, 2. f i (x =αf i, x K, i =1, 2. f i (x =α f i, x K, i =1, 2. Fix x K arbitrary. Sice K z diam F (z R mas bouded sets oto bouded sets, there exists a costat M such that diam F (y M, y 2 x.

12 144 A. Smajdor ad J. Szczawińska JFPTA For every N there exists a umber q (1, 2 such that q Q +. Obviously, q x 2 x, N. Hece, for every N, f 1 (x f 2 (x = α f 1 f 2 Lettig + eds the roof. ( ( = α q x q x f 1 q f 2 q = 1 ( α f 1 (q x f 2 (q x q ( ( α α diam F (q x M. Refereces [1] C. Badea, The geeral liear equatio is stable. Noliear Fuct. Aal. Al. 10 (2005, [2] J. Brzdȩk ad A. Pietrzyk, A ote o stability of the geeral liear equatio. Aequatioes Math. 75 (2008, [3] J. Brzdȩk, D. Poa ad B. Xu, Selectios of the set-valued mas satisfyig a liear iclusios i sigle variable via Hyers-Ulam stability. Noliear Aal. 74 (2011, [4] Z. Gajda, O stability of additive maigs. It. J. Math. Math. Sci. 14 (1991, [5] Z. Gajda ad R. Ger, Subadditive multifuctios ad Hyers-Ulam stability. Numer. Math. 80 (1987, [6] D. R. Hyers, G. Isac ad Th. M. Rassias, Stability of Fuctioal Equatios i Several Variables. Birkhäuser Bosto, Bosto, [7] D. Ioa ad D. Poa, O selectios of geeralized covex set-valued mas. Aequatioes Math. 88 (2014, [8] S. M. Jug, Hyers-Ulam-Rassias Stability of Fuctioal Equatios i Noliear Aalysis. Sriger Otimizatio ad Its Alicatios 48, Sriger, New York, [9] K. Nikodem ad D. Poa, O selectios of geeral liear iclusios. Publ. Math. Debrece 75 (2009, [10] M. Piszczek, O selectios of set-valued iclusios i a sigle variable with alicatios to several variables. Results Math. 64 (2013, [11] D. Poa, Additive selectios of (α, β-subadditive set valued mas. Glas. Mat. Ser. III 36 (2001, [12] D. Poa, A stability result for geeral liear iclusio. Noliear Fuct. Aal. A. 3 (2004, [13] D. Poa, A roerty of a fuctioal iclusio coected with Hyers-Ulam stability. J. Math. Iequal. 4 (2009,

13 Vol. 18 (2016 Selectios of set-valued fuctios 145 [14] A. Smajdor ad J. Szczawińska, Selectios of geeralized covex set-valued fuctios with bouded diameter. Fixed Poit Theory, to aear. Adrzej Smajdor Istitute of Mathematics Pedagogical Uiversity Podchor ażych 2 PL Kraków Polad asmajdor@u.krakow.l Joaa Szczawińska Istitute of Mathematics Pedagogical Uiversity Podchor ażych 2 PL Kraków Polad jszczaw@u.krakow.l Oe Access This article is distributed uder the terms of the Creative Commos Attributio 4.0 Iteratioal Licese (htt://creativecommos.org/liceses/by/4.0/, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided you give aroriate credit to the origial author(s ad the source, rovide a lik to the Creative Commos licese, ad idicate if chages were made.

Research Article Approximate Riesz Algebra-Valued Derivations

Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of