Stability of a Volterra Integral Equation on Time Scales 7 1 0 2 Alaa E. Hamza1, and Ahmed G. Ghallab2 n 1 Department of Mathematics, Faculty of Science, Cairo University,Giza, Egypt. a E-mail: [email protected] J 5 2 Department of Mathematics, Faculty of Science, Fayoum University,Fayoum,Egypt. ] S E-mail: [email protected] D . h Abstract t a In this paper, we study Hyers-Ulam stability for integral equation of m Volterratypeintimescalesetting. Moreoverwestudythestabilityofthe [ consideredequationinHyers-Ulam-Rassiassense. Ourtechniquedepends on successive approximation method, and we use time scale variant of 1 inductionprincipletoshowthat(1.1) isstableon unboundeddomainsin v Hyers-Ulam-Rassias sense. 7 1 2 1 Introduction 1 0 In 1940,S. M. Ulam gavea wide range oftalks atthe Mathematics Club of the . 1 UniversityofWisconsin,in whichhe discussedanumber ofimportantunsolved 0 problems. One of them was the following question: 7 1 Let G1 be a group and let G2 be a group endowed with a metric d. Given : ǫ>0, does there exist a δ >0 such that if a mapping h:G →G satisfies the v 1 2 inequality i X d(h(xy),h(x)h(y))<δ, r for all x,y ∈G , can we find a homomorphism θ :G →G such that a 1 1 2 d(h(x),θ(x)) <ǫ, for all x∈G ? 1 This problem was solved by Hyers for approximately additive mappings on Banach spaces [3]. Rassias generalized, in his work [11], the result obtained by Hyers. Since then the stability of many functional, differential, integral equations have been investigated, see [4], [7], [8], and references there in. In this paper we shallconsider the non-homogeneousvolterraintegralequa- tion of the first kind t x(t)=f(t)+ k(t,s)x(s)∆s, t∈IT :=[a,b]T, (1.1) Z a 1 where f ∈Crd(IT,R), k ∈Crd(IT×IT,R) and x is the unknown function. First, we introduce the basic definitions that will be used through out this paper. Definition 1.1. The integral equation (1.1) is said to be has Hyers-Ulam sta- bility on IT if for any ε>0 and each ψ ∈Crd(IT,R) satisfying t |ψ(t)−f(t)− k(t,s)ψ(s)∆s|<ε, ∀ t∈IT; Z a then there exists a solution ϕ of equation (1.1) and a constant C ≥0 such that |ϕ(t)−ψ(t)|≤Cε, ∀ t∈IT. The constant C is called Hyers-Ulam stability constant for equation (1.1). Definition 1.2. The integral equation (1.1) is said to be has Hyers-Ulam- Rassias stability, with respect to ω, on IT if for each ψ ∈Crd(IT,R) satisfying t |ψ(t)−f(t)− k(t,s)ψ(s)∆s|<ω(t), ∀ t∈IT; Z a for some fixed ω ∈ Crd(IT,[0,∞)), then there exists a solution ϕ of equation (1.1) and a constant C >0 such that |ϕ(t)−ψ(t)|≤Cω(t), ∀ t∈IT. we shall investigate Hyers-Ulam stability and Hyers-Ulam-Rassias stability ofintegralequation(1.1)onbothboundedandunboundedtimescalesintervals. 2 Hyers-Ulam stability In this section we investigate Hyers-Ulam stability of equation on IT := [a,b]T by using iterative technique. Theorem 2.1. The integral equation (1.1) has Hyers-Ulam stability on IT := [a,b]T. Proof. For given ε>0 and each ψ ∈Crd(IT,R) satisfying t |ψ(t)−f(t)− k(t,s)ψ(s)∆s|<ε, ∀ t∈IT, Z a we consider the recurrence relation t ψn(t):=f(t)+ k(t,s)ψn−1(s)∆s, n=1,2,3,... (2.1) Z a 2 for t ∈ IT with ψ0(t) = ψ(t). We prove that {ψn(t)}n∈N converges uniformly to the unique solutionof Equation(1.1) on IT. We write ψn(t) as a telescoping sum n ψn(t)=ψ0(t)+ [ψi(t)−ψi−1(t)], Xi=1 so ∞ lim ψn(t)=ψ0(t)+ [ψi(t)−ψi−1(t)], ∀ t∈IT. (2.2) n→∞ Xi=1 Using mathematical induction we prove the following estimate (t−a)i−1 |ψi(t)−ψi−1(t)|≤εMi−1 (i−1)! , ∀ t∈IT. (2.3) For i=1 we have |ψ (t)−ψ(t)|<ε. 1 So the estimate (2.3) holds for i=1. Assume that the estimate (2.3) is true for i=n≥1. We have t |ψn+1(t)−ψn(t)|≤ |k(t,s)||ψn(s)−ψn−1(s)|∆s Z a t (s−a)n−1 ≤M εMn−1 ds Z (n−1)! a (t−a)n ≤εMn , n! hence the estimate (2.3) it valid for i = n+1. This shows that the estimate (2.3) is true for all i≥1 on IT. See that (t−a)i−1 |ψi(t)−ψi−1(t)|≤εMi−1 (i−1)! (b−a)i−1 ≤εMi−1 , (i−1)! and ∞ (b−a)i−1 ∞ [(M(b−a)]i εMi−1 = ε =εeM(b−a). (i−1)! i! Xi=1 Xi=0 Applying Weierstrass M-Test, we conclude that the infinite series ∞ [ψi(t)−ψi−1(t)] Xi=1 converges uniformly on t ∈ IT. Thus from (2.2), the sequence {ψn(t)}n∈N con- vergesuniformly onIT to someϕ(t)∈Crd(IT,R). Next, we showthat the limit 3 ofthe sequenceϕ(t)is theexactsolutionof (??). Forallt∈IT andeachn≥1, we have t t t k(t,s)ψ (s)− k(t,s)ϕ(s)∆s ≤M |ψ (s)−ϕ(s)|∆s. n n (cid:12)Za Za (cid:12) Za (cid:12) (cid:12) (cid:12) (cid:12) Taking the limits as n → ∞ we see that the right hand side of the above inequality tends to zero and so t t lim k(t,s)ψn(s)∆s= k(t,s)ϕ(s)∆s, ∀ t∈IT. n→∞Z Z a a By letting n → ∞ on both sides of (2.1), we conclude that ϕ(t) is the exact solutionof (??)onIT. ThenthereexistsanumberN suchthat|ψN(t)−ϕ(t)|≤ ε. Thus |ψ−ϕ|≤|ψ(t)−ψ (t)|+|ψ (t)−ϕ(t)| N N ≤|ψ(t)−ψ1(t)|+|ψ1(t)−ψ2(t)|+···+|ψn−1(t)−ψN(t)|+|ψN(t)−ϕ(t)| N ≤ |ψi−1(t)−ψi(t)|+|ψN(t)−ϕ(t)| Xi=1 N (b−a)i−1 ≤ εMi−1 +|ψ (t)−ϕ(t)| N (i−1)! Xi=1 ≤εeM(b−a)+ε=ε(1+eM(b−a))ε≤Cε. which completes the proof. Remark 2.2. We can find an estimate on the difference of two approximate solutions of the integral equation (1.1). Let ψ and ψ are two different approx- 1 2 imate solutions to (1.1) that is for some ε1,ε2 >0, and for all t∈IT t ψ (t)−f(t)− k(t,s)ψ (s)∆s ≤ε , (2.4) 1 1 1 (cid:12) Za (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and t ψ (t)−f(t)− k(t,s)ψ (s)∆s ≤ε . (2.5) 2 2 2 (cid:12) Za (cid:12) (cid:12) (cid:12) So (cid:12) (cid:12) |ψ1(t)−ψ2(t)|≤(ε1+ε2)eM(t,a), ∀ t∈IT. If ψ is an exact solution of equation (1.1), then we have ε =0. 1 1 Proof. Adding the two inequalities (2.4), (2.5) and making use of |α|−|β| ≤ |α−β|≤|α|+|β|, we get t ψ (t)−ψ (t)− k(t,s)[ψ (s)−ψ (s)]∆s ≤ε +ε . 1 2 1 2 1 2 (cid:12) Za (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 t |ψ (t)−ψ (t)|− k(t,s)[ψ (s)−ψ (s)] ≤ε +ε 1 2 1 2 1 2 (cid:12)Za (cid:12) (cid:12) (cid:12) for all t∈IT where ε:=ε1+ε(cid:12)2. (cid:12) Put ξ(t):=|ψ1(t)−ψ2(t)|, ∀ t∈IT, then t ξ(t)≤ε+ |k(t,s)|ξ(s)∆s Z a t ≤ε+ Mξ(s)∆s Z a t M ≤ε+e (t,a) ε ∆s, M Za eM(σ(s),a) where we make an application of Gr¨onwall’s inequality in the last step. By Theorem we have t M t 1 ∆ 1 ∆s=− ∆s= 1− , Za eM(σ(s),a) Za heM(s,a)i (cid:16) eM(t,a)(cid:17) thus ξ(t)≤ε+ε [eM(t,a)−1]=ε eM(t,a), ∀ t∈IT. 3 Hyers-Ulam-Rassias Stability In this section we investigate a result concerning Hyers-Ulam-Rassias stability of equation (1.1) on both IT :=[a,b]T and unbounded interval [a,∞)T. Theorem 3.1. Assume ψ ∈Crd(IT,R) satisfying t |ψ(t)−f(t)− k(t,s)ψ(s)∆s|<ω(t), ∀ t∈IT, Z a for some fixed ω ∈Crd(IT,R+) for which there exists a constant P ∈(0,1) such that t ω(s)∆s≤Pω(t), ∀ t∈IT. Z a Then there exist a unique solution ϕ of Equation (??) such that M |ϕ(t)−ψ(t)|≤ 1+ ·ω(t), ∀ t∈IT. 1−P (cid:16) (cid:17) 5 Proof. Consider the following iterative scheme t ψn(t):=f(t)+ k(t,s)ψn−1(s)∆s, n=1,2,3,... (3.1) Z a for t ∈IT with ψ0(t)= ψ(t). By mathematical induction, it is easy to see that the following estimate |ψn(t)−ψn−1(t)|≤MPn−1ω(t), (3.2) holds for each n ∈ N and all t ∈ IT. By the same argument as in Theorem 2.1 we prove that the sequence ψn(t)n∈N converges uniformly on IT to the unique solution, ϕ, of the integral equation (1.1). Then there exists a positive integer N such that |ψN(t)−ϕ(t)|≤w(t), t∈IT. Hence |ψ−ϕ|≤|ψ(t)−ψ (t)|+|ψ (t)−ϕ(t)| N N ≤|ψ(t)−ψ1(t)|+|ψ1(t)−ψ2(t)|+···+|ψn−1(t)−ψN(t)|+|ψN(t)−ϕ(t)| N ≤ |ψk−1(t)−ψk(t)|+|ψN(t)−ϕ(t)| kX=1 N ≤ MPk−1ω(t)+|ψ (t)−ϕ(t)| N kX=1 N ≤ MPk−1ω(t)+|ψ (t)−ϕ(t)| N kX=1 ∞ ≤ MPk−1ω(t)+ω(t) kX=1 1 M ≤M · ω(t)+ω(t)= 1+ ω(t), 1−P 1−P (cid:16) (cid:17) which shows that (1.1) has Hyers-Ulam-Rassiasstability on IT. Theorem 3.2. Assume that for a family of statements A(t), t ∈ [t0,∞)T the following conditions holds 1. A(t ) is true. 0 2. for each right-scattered t∈[t0,∞)T we have A(t)⇒A(σ(t)). 3. for each right-dense t ∈ [t0,∞)T there is a neighborhood U such that A(t)⇒A(s) for all s∈U,s>t. 4. for each left-dense t∈[t0,∞)T one has A(s) for all s with s<t⇒A(t). Then A(t) is true for all t∈[t0,∞)T. Next, we prove that the integral equation (1.1) has Hyers-Ulam-Rassias on unbounded domains. 6 Theorem 3.3. Consider the integral equation (??) with IT := [a,∞)T. Let f ∈ Crd([a,∞)T,R) and k(t,.) ∈ Crd([a,∞)T,R) for some fixed t ∈ [a,∞)T. Assume ψ ∈Crd(IT,R) satisfying t ψ(t)−f(t)− k(t,s)ψ(s)∆s <ω(t), t∈IT; (3.3) (cid:12) Za (cid:12) (cid:12) (cid:12) where ω ∈Crd((cid:12)[a,∞)T,R+) with the property (cid:12) t ω(τ)∆τ ≤λω(t), ∀ t∈[a,∞)T. (3.4) Z a forλ∈(0,1). Thentheintegralequation (1.1)hasHyers-Ulam-Rassiasstability, with respect to ω, on [a,∞)T. Proof. We apply the time scale mathematical induction in [a,∞)T on the fol- lowing statements A(r): the integral equation (1.1) t x(t)=f(t)+ k(t,s)x(s)∆s, Z a has Hyers-Ulam-Rassiasstability, with respect to ω, on [a,r]T. I. A(a) is trivially true. II.LetrbearightscatteredpointandthatA(r)holds. Thatmeansequation (1.1)hasHyers-Ulam-Rassiasstability,withrespecttoω,on[a,r]T,i.e. foreach ψ :[a,r]T →R satisfying t ψ(t)−f(t)− k(t,s)ψ(s)∆s <ω(t), t∈[a,r]T; (cid:12) Za (cid:12) (cid:12) (cid:12) where ω ∈Cr(cid:12)d([a,r]T,R+), then there exist a(cid:12) unique solution to equation (1.1) ϕr :[a,r]T →R such that |ϕr(t)−ψ(t)|≤C1ω(t), t∈[a,r]T. We want to prove that A(σ(r)) is true. Assume that the function ψ satisfies t ψ(t)−f(t)− k(t,s)ψ(s)∆s <ω(t), t∈[r,σ(r)]T. (cid:12) Zr (cid:12) (cid:12) (cid:12) Define th(cid:12)e mapping ϕσ(r) :[a,σ(r)]T →(cid:12)R such that ϕ (t)= ϕr(t), t∈[a,r]T; σ(r) (cid:26) f(σ(r))+µ(r)k(σ(r),r)ϕr(r), t=σ(r). It is clear that ϕσ(r) is a solution of (1.1) on [a,σ(r)]T. Moreover,on we have |ϕ (t)−ψ(t)|= |ϕr(t)−ψ(t)|, t∈[a,r]T; σ(r) (cid:26) |f(σ(r))+µ(r)k(σ(r),r)ϕr(r)−ψ(σ(r))|, t=σ(r). 7 See that |ϕ (σ(r))−ψ(σ(r))| =|f(σ(r))+µ(r)k(σ(r),r)ϕ (r)−µ(r)k(σ(r),r)ψ(r) σ(r) r +µ(r)k(σ(r),r)ψ(r)−ψ(σ(r))| ≤|f(σ(r))+µ(r)k(σ(r),r)ψ(r)−ψ(σ(r))| +|µ(r)k(σ(r),r)||ϕ (r)−ψ(r)| r ≤ω(σ(r))+MC µ(r)ω(r). 1 So we have |ϕ (t)−ψ(t)|≤ C1ω(t), t∈[a,r]T; σ(r) (cid:26) ω(σ(r))+MC1µ(r)ω(r), t=σ(r). III. Let r∈[a,∞)T be right-dense and Ur be a neighborhood of r. Assume A(r) is true, i.e. for each ψ :[a,r]T →R satisfying t ψ(t)−f(t)− k(t,s)ψ(s)∆s <ω(t), for t∈[a,r]T, (cid:12) Za (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) where ω ∈Crd([a,r]T,R+), then there exist a unique solution to equation (1.1) ϕr :[a,r]T →R such that |ϕr(t)−ψ(t)|≤C1ω(t), for t∈[a,r]T. We show that A(τ) is true for all τ ∈Ur∩(r,∞)T. For τ >r assume that the function ψ satisfies t ψ(t)−f(t)− k(t,s)ψ(s)∆s <ω(t), for t∈[r,τ]T. (cid:12) Zr (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) By Theorem for each τ ∈U , τ >r, the integral equation r t x(t)=f(t)+ k(t,s)x(s)∆s, for t∈[r,τ]T, Z r has exactly on solution ϕτ(·). Therefore the mapping ξτ : [a,τ]T → R defined by ξ (t)= ϕr(t), t∈[a,r]T; s (cid:26) ϕτ(t), t∈[r,τ]T. is a solution of the integral equation t x(t)=f(t)+ k(t,s)x(s)∆s, for t∈[a,τ]T. Z a We have |ξ (t)−ψ(t)|= |ϕr(t)−ψ(t)|, t∈[a,r]T; s (cid:26) |ϕs(t)−ψ(t)|, t∈[r,s]T. 8 For t∈[r,s]T, see that t |ϕ (t)−ψ(t)|= f(t)+ k(t,τ)ϕ (τ)∆τ s s (cid:12) Zr (cid:12) (cid:12) t t −ψ(t)+ k(t,τ)ψ(τ)∆τ − k(t,τ)ψ(τ)∆τ Zr Zr (cid:12) (cid:12) t t (cid:12) ≤|f(t)+ k(t,τ)ψ(τ)∆τ −ψ(t)|+ |k(t,τ)||ϕ (τ)−ψ(τ)|∆τ s Z Z r r t ≤C ω(t)+M ω(τ)∆τ 1 Z r ≤C ω(t)+MPω(t)=(C +MP)ω(t). 1 1 IV. Let r ∈ (a,∞)T be left-dense such that A(s) is true for all s < r. We prove that A(r) by the same argument as in (III). By the induction principle the statement A(t) holds for all t ∈ [a,∞)T, that means the integral equation (1.1) has Hyers Ulam Rassias stability on t∈[a,∞)T. Now we givean example to showthat Hyers Ulam stability of volterraInte- gralequation(1.1)notnecessarilyholdsonunboundedintervalforgeneraltime scale. Example 3.4. The integral dynamic equation t x(t)=1+5 x(s)∆s, t∈[0,∞)T, Z 0 has exactly one solution x(t) = e (t,0), also we have x(t) = 0 as approximate 5 solution. 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