Mthemtic Morvic Vol. 13-1 2009, 95 102 Set Integrl Equtions in Metric Spces Ion Tişe Abstrct. Let P cp,cvr n be the fmily of ll nonempty compct, convex subsets of R n. We consider the following set integrl equtions: 1 2 Xt = Xt = Z b Z t Kt, s, Xs d s + X 0 Kt, s, Xs d s + X 0, where K : [, b] [, b] P cp,cvr n P cp,cvr n nd X 0 P cp,cvr n. The purpose of the pper is to study the existence nd dt dependence of the solutions of the set integrl equtions 1 nd 2, by using fixed point pproch. Our results generlize nd extend the results given in [2]. For other similr results see [3] nd [4]. 1. Introduction Let R n be the rel n-dimensionl Euclidin spce nd P cp,cv R n be the fmily of ll nonempty compct, convex subset of R n endowed with the Pompeiu-Husdorff metric H. It is well-known tht P cp,cv R n, H is complete metric spce. We consider the following set integrl equtions: 1 2 Xt = Xt = Kt, s, Xs d s + X 0, Kt, s, Xs d s + X 0, where K : [, b] [, b] P cp,cv R n P cp,cv R n. A solution of 1 or 2 mens continuous function X : [, b] P cp,cv R n which stisfies 1 respectively 2 for ech t [, b]. The purpose of the rticle is to study the existence nd dt dependence of the solutions of the eqution 1 nd 2. The pproch is bsed on the well-known Bnch-Cccioppoli contrction principle. Our results generlize 2000 Mthemtics Subject Clssifiction. Primry: 47H10. Key words nd phrses. Fixed point, set integrl eqution, Pompeiu-Husdorff metric. 95 c 2009 Mthemtic Morvic
96 Set Integrl Equtions in Metric Spces nd extend the results given in [2]. For other similr results see [3], [5] nd [4]. The pper is orgnized s follows. Next section, Preliminries, contins some bsic nottions nd notions used through the pper. Third section presents existence nd the dt dependence results of the solution for the equtions 1 nd 2. 2. Preliminries The im of this section is to present some notions nd symbols used in the pper. Let us define the following generlized functionls: D : P R n P R n R +, DA, B = inf{d, b A, b B}. D is clled the gp functionl between A nd B. In prticulr, if x 0 X then Dx 0, B := D{x 0 }, B. ρ : P R n P R n R + {+ }, ρ is clled the generlized excess functionl. ρa, B = sup{d, B A}. H : P R n P R n R + {+ }, HA, B = mx{ρa, B, ρb, A}. H is the generlized Pompeiu-Husdorff functionl. It is known tht P cp,cv R n, H is complete metric spce [1]. Lemm 2.1 [6]. Let X be Bnch spce. Then HA + C, B + D HA, B + HC, D, for A, B, C, D P X. Proof. Let ε > 0. From the definition of H it follows tht there exists + c A + C such tht D + c, B + D HA + C, B + D ε or exists b + d B + D such tht Db + d, A + C HA + C, B + D ε. Let us consider the first cse. For, c we get b B, d D such tht: b HA, B + ε 2, c d HC, D + ε 2. Then HA+C, B +D ε D+c, B +D +c b+d we obtin tht HA + C, B + D ε HA, B + HC, D + ε, proving the desired inequlity. 3. Min results We consider on C[, b], P cp,cv R n the metric: H X, Y := mx HXt, Y t. t [,b] The pir C[, b], P cp,cv R n, H forms complete metric spce. Our first result is n existence nd uniqueness theorem for the solution of the eqution 1.
Ion Tişe 97 Theorem 3.1. Let K : [, b] [, b] P cp,cv R n P cp,cv R n be multivlued opertor. Suppose tht: i K is continuous on [, b] [, b] P cp,cv R n nd X 0 P cp,cv R n, ii Kt, s, is Lipschitz, i.e., there exists L K 0 such tht: HKt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd for ll t, s [, b], iii L K b < 1. Then the set integrl eqution 1 Xt = hs unique solution. Kt, s, Xs d s + X 0 Proof. Consider the opertor: Γ : P cp,cv R n P cp,cv R n defined for ech t [, b] by ΓXt = Kt, s, Xs d s + X 0. We need to verify the contrction condition for Γ. H ΓXt, ΓY t = = H Kt, s, Xs d s + X 0, HX 0, X 0 + H Kt, s, Xs d s, H Kt, s, Xs, Kt, s, Y s d s L K H Xs, Y s d s. Tking the mximum for t [, b], then we hve: mx t [,b] H ΓXt, ΓY t L K Kt, s, Y s d s + X 0 Kt, s, Y s d s mx H Xs, Y s d s t [,b] H ΓX, ΓY LK b H X, Y, for ll t [, b], nd X, Y C [, b], P cp,cv R n. Thus, the integrl opertor Γ is Lipschitz with constnt L Γ = L K b < 1. From the contrction principle we get the result. A dt dependence result for the solution of eqution 1 is:
98 Set Integrl Equtions in Metric Spces Theorem 3.2. Let K 1, K 2 : [, b] [, b] P cp,cv R n P cp,cv R n, be continuous. Consider the following set equtions: 3 4 Xt = Y t = K 1 t, s, Xs d s + X 0 K 2 t, s, Y s d s + Y 0 Suppose: i there exists L K 0 such tht H Kt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd t, s [, b], with L K b < 1 denote by X the unique solution of the eqution 3; ii there exists η 1, η 2 > 0 such tht: H K 1 t, s, U, K 2 t, s, U η 1, for ll t, s, U [, b] [, b] P cp,cv R n, nd b HX 0, Y 0 η 2 ; iii there exists Y solution of the eqution 4. Then H X, Y η 2 + η 1 b 1 L K b. Proof. We hve: HX t, Y t = = H K 1 t, s, X s d s + X 0, H K 1 t, s, X s d s, H K 1 t, s, X s d s, + H K 1 t, s, Y s d s, + K 2 t, s, Y s d s + Y 0 K 2 t, s, Y s d s + HX 0, Y 0 K 1 t, s, Y s d s + H K 1 t, s, X s, K 1 t, s, Y s d s+ K 2 t, s, Y s d s + η 2 H K 1 t, s, Y s, K 2 t, s, Y s d s + η 2 H K 1 t, s, X s, K 1 t, s, Y s d s + η 1 d s + η 2.
Ion Tişe 99 By tking the mximum for t [, b], then we hve: mx H X t, Y t t [,b] LK b H X t, Y t + η 1 b + η 2 mx t [,b] L K b mx t [,b] H X t, Y t + η 1 b + η 2 mx H X t, Y t η 2 + η 1 b t [,b] 1 L K b H X, Y η 2 + η 1 b 1 L K b. We will prove now n existence result nd dt dependence result for the solution of the eqution 2. We consider on C [, b], P cp,cv R n the metric: H B X, Y := mx t [,b] [HXt, Y te τt ], with rbitrry τ > 0. The pir C[, b], P cp,cv R n, H B forms complete metric spce. Theorem 3.3. Let K : [, b] [, b] P cp,cv R n P cp,cv R n be n opertor. Suppose tht: i K is continuous on [, b] [, b] P cp,cv R n nd X 0 P cp,cv R n, ii Kt, s, is Lipschitz, i.e., there exists L K 0 such tht H Kt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd t, s [, b]. Then the set integrl eqution 2 Xt = hs unique solution. Kt, s, Xs d s + X 0 Proof. Consider the opertor Γ : P cp,cv R n P cp,cv R n defined for ech t [, b] by ΓXt = Kt, s, Xs d s + X 0.
100 Set Integrl Equtions in Metric Spces We will prove the contrction condition for Γ. H ΓXt, ΓY t = = H Kt, s, Xs d s + X 0, HX 0, X 0 + H Kt, s, Xs d s, Kt, s, Y s d s + X 0 H Kt, s, Xs, Kt, s, Y s d s = L K H Xs, Y s e τs e τs d s L K H B X, Y e τs d s = = L K τ HB X, Y e τt 1 L K τ HB X, Y e τt. Then we hve: Kt, s, Y s d s L K H Xs, Y s d s = H ΓXt, ΓY t e τt L K τ HB X, Y H B L K ΓX, ΓY τ HB X, Y, for ll t [, b], X, Y C [, b], P cp,cv R n, τ > 0. Hence, we cn pply the Bnch contrction principle for Γ, since by choosing τ > L K, we get L Γ := L K τ < 1. By the contrction principle, the proof is complete. Remrk 3.1. Theorem 3.3 in this pper is specil cse of Hmmerstein s equlity. Generl solution of this equlity is given in [7]. A dt dependence result is: Theorem 3.4. Let K 1, K 2 : [, b] [, b] P cp,cv R n P cp,cv R n be continuous. Consider the following set equtions: 5 6 Xt = Y t = K 1 t, s, Xs d s + X 0 K 2 t, s, Y s d s + Y 0 Suppose: i H Kt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd t, s [, b], where L K 0 denote by X the unique solution of the eqution 5; ii there exists η 1, η 2 > 0, such tht H K 1 t, s, U, K 2 t, s, U η 1, for ll t, s, U [, b] [, b] P cp,cv R n nd HX 0, Y 0 η 2 ;
Ion Tişe 101 iii There exists Y solution of the eqution 6. Then Proof. We hve: H B X, Y η 2 + η 1 b 1 L K τ e τb where τ > L K. HX t, Y t = = H K 1 t, s, X s d s + X 0, H K 1 t, s, X s d s, H K 1 t, s, X s d s, + H K 1 t, s, Y s d s, + K 2 t, s, Y s d s + Y 0 K 2 t, s, Y s d s + HX 0, Y 0 K 1 t, s, Y s d s + H K 1 t, s, X s, K 1 t, s, Y s d s+ K 2 t, s, Y s d s + η 2 H K 1 t, s, Y s, K 2 t, s, Y s d s + η 2 H K 1 t, s, X s, K 1 t, s, Y s d s + L K H X s, Y s e τs e τs d s + By tking the mximum for t [, b], we hve: mx H X t, Y t e τt e τt t [,b] mx t [,b] L K H X s, Y s e τs e τs ds + H B X, Y e τb = L K H B X, Y L K τ HB X, Y e τt 1 + η 1 b + η 2 L K τ HB X, Y e τb + η 1 b + η 2, H B X, Y η 2 + η 1 b 1 L K τ e τb. η 1 d s + η 2 η 1 d s + η 2. η 1 d s + η 2, e τs d s + η 1 b + η 2 =
102 Set Integrl Equtions in Metric Spces For τ > L K, we hve H B X, Y η 2 + η 1 b 1 L K τ e τb. References [1] J.P. Aubin, H. Frnkowsk, Set-Vlued Anlysis, Birkhuser, Bsel, 1990. [2] A.J. Brndo, F.S. De Blsi, F. Iervillino, Uniqueness nd Existence Theorems for Differentil Equtions with Compct Convex Vlued Solutions, Boll. Unione Mt. Itl., IV. Ser. 31970, 47-54. [3] F.S. de Blsi, Semifixed sets of mps in hyperspces with ppliction to set differentil equtions, Set-Vlued Anl., 142006, No. 3, 263-272. [4] G.N. Glnis, T.G. Bhskr, V. Lkshmiknthm nd P.K. Plmides, Set vlued functions in Fréchet spces: continuity, Hukuhr differentibility nd pplictions to set differentil equtions, Nonliner Anl., 61 2005, No. 4, 559-575. [5] V. Lkshmiknthm, T. Gnn Bhskr nd J. Vsundhr Devi, Theory of Set Differentil Equtions in Metric Spces, Cmbridge Scientific Publishers, 2006. [6] A. Petruşel, G. Petruşel, G. Moț, Topics in Nonliner Anlysis nd Applictions to Mthemticl Economics, House of the Book Science, Cluj-Npoc, 2007. [7] M.R. Tskovic, A Chrcteriztion of the Clss of Contrction Type Mppings, Kobe Journl of Mthemtics, 2 1985, 45-55. Deprtment of Applied Mthemtics, Bbeş-Bolyi University Kogălnicenu 1, 400084 Cluj-Npoc, Romni E-mil ddress: ti_cmeli@yhoo.com