A xed point approach to the stability of a nonlinear volterra integrodierential equation with delay

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1 Haettepe Journal of Mathematis and Statistis Volume 47 (3) (218), A xed point approah to the stability of a nonlinear volterra integrodierential equation with delay Rahim Shah and Akbar Zada Abstrat By using a xed point method, we prove the HyersUlamRassias stability and the HyersUlam stability of a nonlinear Volterra integrodifferential equation with delay. Two examples are presented to support the usability of our results. Keywords: Hyers-Ulam-Rassias stability, Hyers-Ulam stability, Volterra integrodierential equation with delay, xed point approah. 2 AMS Classiation: 47H1, 45J5, 45M1. Reeived : Aepted : Doi : /HJMS Introdution In 194, Ulam posed the following problem related to the stability of funtional equations: Under what onditions does there exist an additive mapping near an approximately additive mapping? see for more detail [15]. One year later, Hyers [8] gave an answer to the problem of Ulam for the ase of funtional equation for homomorphism between the Banah spaes. In 1978, Rassias [13] proved the existene of unique linear mapping near approximate additive mapping that provides generalization of the Hyers result. Jung [9] applied the xed point method to the investigation of Volterra integral equation by using the idea of Cadariu and Radu in [2]. S. M. Jung proved that if a ontinuous funtion v : I C is suh that v(t) G(ξ, v(ξ))dξ φ(t) Department of Mathematis, University of Peshawar, Peshawar 25, Pakistan safeer_rahim@yahoo.om Department of Mathematis, University of Peshawar, Peshawar 25, Pakistan zadababo@yahoo.om, akbarzada@uop.edu.pk

2 616 for all t I, then there exists a unique ontinuous funtion v : I C and a onstant K > suh that v (t) = G(ξ, v (ξ))dξ and d (v(t), v (t)) Lφ(t), for all t I, it is important to obtain a preise L beause it is lear that L will lead us to the error between the atual solution v (t) and the approximate solution v(t). In 213, Jung et al. proved that if g : I R, h: I R, G: I R and φ: I R are suiently smooth funtions and if a ontinuously dierentiable funtion v : I R satises the perturbed Volterra integrodierential equation v (t) g(t)v(t) h(t) K(t, η)v(η)dη φ(t), for some t I, then there exists a unique solution v : I R of the Volterra integrodifferential equation suh that v (t) g(t)v(t) h(t) d(v(t), v (t)) exp K(t, η)v(η)dη =, b ς g(η)dη φ(ς) exp t g(η)dη dς, for all t I. If the reader wishes more details, we reommend [1, 3, 4, 6, 7, 12, 14, 16]. The main purpose of the paper is to investigate the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of following nonlinear Volterra integrodierential equation with delay: (1.1) v (t) = g(t, v(t), v(α(t))) k(t, s, v(s), v(α(s)))ds, for all t I = [, T ], where the funtion g(t, v(t), v(α(t))) is ontinuous funtion with respet to variables t and v on I R R, k(t, s, v(t), v(α(t))) is ontinuous with respet to variables t, s and v on I I R R, β is any onstant and α : [, T ] [, T ] is a ontinuous delay funtion with α(t) t Denition. If for eah ontinuously dierentiable funtion v(t) satisfying v (t) g(t, v(t), v(α(t))) k(t, s, v(s), v(α(s)))ds φ(t), for some φ: [, T ] (, ), there exists a solution v (t) of the Volterra integrodierential equations with delay (1.1) and a onstant K > (independent of v(t) and v (t)) with v(t) v (t) Kφ(t), for all t I, then we an say that the equation (1.1) is Hyers-Ulam-Rassias stable on I. If φ(t) is onstant funtion then we say that the equation (1.1) has Hyers-Ulam stability on I. For a nonempty set Y, the generalized metri on Y is dened as follow: 1.2. Denition. A funtion d : Y Y [, ] is alled a generalized metri on Y if and only if for all u, v, w Y d satises the following onditions: (1) d(u, v) = if and only if u = v. (2) d(u, v) = d(v, u). (3) d(u, w) d(u, v) d(v, w). Now, we are going to introdue one of the most ruial result of xed point theory that will play an important role in proving our main results.

3 Theorem. ([5]) Let (Y, d) be a generalized omplete metri spae. Assume that Θ : Y Y is a stritly ontrative operator with L < 1 as Lipshitz onstant. If there exists a nonnegative integer k suh that d(θ k1 u, Θ k u) < for some u Y, then the following onditions are true: The sequene Θ n u onverges to a xed point u of Θ; u is the unique xed point of Θ in Y = v Y d(θ k u, v) < ; If v Y, then d(v, u ) 1 d(θv, v). 1 L 2. Hyers-Ulam-Rassias stability In this setion, we prove the Hyers-Ulam-Rassias stability of the nonlinear Volterra integrodierential equation with delay (1.1) Theorem. Let I = [, T ] be a losed and bounded interval for a given T > and let N, L g and L k be nonnegative onstants with < NL g N 2 L k < 1. Assume that g : I R R R is a ontinuous funtion that satises a Lipshitz ondition (2.1) g(t, v 1, v 1(α(t))) g(t, v 2, v 2(α(t))) L g v 1 v 2, for all t I and v 1, v 2 R. Let k : I I R R R be a ontinuous funtion whih satises a Liphitz ondition (2.2) k(t, s, v 1, v 1(α(s))) k(t, s, v 2, v 2(α(s)) L k v 1 v 2, for all t, s I, v 1, v 2 R. If v : I R a ontinuously dierentiable funtion satises (2.3) v (t) g(t, v(t), v(α(t))) k(t, s, v(s), v(α(s)))ds φ(t), for all t I, where φ: I (, ) is a ontinuous funtion with (2.4) φ(ζ)dζ Nφ(t) for all t I, then there exists a unique ontinuous funtion v : I R suh that (2.5) v (t) = and (2.6) v(t) v (t) for all t I. g(ζ, v (ζ), v (α(ζ)))dζ N 1 (NL g N 2 L k ) φ(t) k(t, ζ, v (ζ), v (α(ζ))dζds Proof. Let Y be the set of all real valued ontinuous funtions on losed and bounded interval I. For all u, w Y, we set (2.7) d(u, w) = infk [, ] : u(t) w(t) Kφ(t), for all t I. The metri spae (Y, d) is a omplete generalized metri spae, see [1]. Consider the operator Θ : Y Y dened by (2.8) (Θu)(t) = v() g(ζ, u(ζ), u(α(ζ)))dζ k(t, ζ, u(ζ), u(α(ζ)))dζds

4 618 for all t I. We show that the operator Θ is stritly ontrative. Let K uw [, ] be a onstant with d(u, w) K uw for u, w Y. By (2.7), we an write (2.9) u(t) w(t) K uwφ(t) for all t I. From inequalities (2.1)), (2.2), (2.4), (2.8) and (2.9) it follows that for all t I we have (Θu)(t) (Θw)(t) = g(ζ, u(ζ), u(α(ζ))) g(ζ, w(ζ), w(α(ζ))) dζ k(t, ζ, u(ζ), u(α(ζ))) k(t, ζ, w(ζ), w(α(ζ))) g(ζ, u(ζ), u(α(ζ))) g(ζ, w(ζ), w(α(ζ))) dζ dζ ds k(t, ζ, u(ζ), u(α(ζ))) k(t, ζ, w(ζ), w(α(ζ))) dζ ds t L g u(ζ) w(ζ) dζ Lh L gk uw φ(ζ)dζ L k K uw K uwφ(t) ( NL g N 2 L k ), u(ζ) w(ζ) dζ ds φ(ζ) dζ ds i.e. d(θu, Θw) K uwφ(t)(nl g N 2 L k ). Hene, we may onlude that d(θu, Θw) (NL g N 2 L k )d(u, w) for any u, w Y, where < NL g N 2 L k < 1. It follows from (2.8) that for any arbitrary w Y, there exists a onstant K [, ] with Θw (t) w (t) = v() g(ζ, w (ζ), w (α(ζ))) dζ k(t, ζ, u (ζ), u (α(ζ))) dζ ds w Kφ(t), for all t I. Sine g(ζ, w (ζ), w (α(ζ))), k(t, ζ, u (ζ), u (α(ζ))) and w are bounded and min t I φ(t) >. Thus, (2.7) implies that d(θw, w ) <. So, aording to Theorem 1.3, there exists a ontinuous funtion v : I R in a way that Θ n w v in (Y, d) and Θv = v, i.e., v satises (2.5) for all t I. Sine we know that w and w are bounded on losed interval I for any w Y and min t I φ(t) >, then there exists a onstant K w [, ] suh that d(w (t), w(t)) K wφ(t) for any t I. We have w (t) w(t) < for any w Y. Therefore, we get that w Y d(w, w) < is equal to Y. From Theorem 1.3, we onlude that v, given by (2.5), is the unique ontinuous funtion. Again from (2.3), we get (2.1) φ(t) v (t) g(t, v(t), v(α(t))) k(t, s, v(s), v(α(s)))ds φ(t) for all t I. By integrating eah term of inequality (2.1) from to t, we get v(t) v() g(ζ, v(ζ), v(α(ζ))) dζ k(t, ζ, v(ζ), v(α(ζ))) dζ ds

5 619 φ(ζ) dζ, for all t I. From (2.4) and (2.8), we get (2.11) v(t) (Θv)(t) φ(ζ) dζ Nφ(t) for all t I, whih implies that d(v, Θv) N. Next by making the use of Theorem 1.3 and inequality (2.11), we onlude that d(v, v ) 1 1 (NL g N 2 L k ) d(θv, v) N 1 (NL g N 2 L k ). Consequently, this yields the inequality (2.6) for all t I. In the above Theorem 2.1 we examined the Hyers-Ulam-Rassias stability of (1.1) on a losed and bounded interval. Now, we are going to show that Theorem 2.1 is also valid for the ase of unbounded interval Theorem. Suppose that I denote either R or [, ) or (, T ] for a given nonnegative real number T. Let L g, L k and N be positive onstants with < NL g N 2 L k < 1 and α: I R be a ontinuous delay funtion suh that α(t) t for all t I. Assume that g : I R R R is a ontinuous funtion satisfying ondition (2.1) for all t I, v 1, v 2 R. If a ontinuously dierentiable funtion v : I R satises inequality (2.3) for all t I, where φ: I (, ) is a ontinuous funtion satisfying ondition (2.4) for all t I, then there exists a unique ontinuous funtion v : I R whih satises (2.5) and (2.6) for eah t I. Proof. First we assume that I = R and we are going to show that v is a ontinuous funtion. For any n N, we dene the interval I n = [ n, n]. In aordane with Theorem 2.1, there exists a unique ontinuous funtion v n : I n R in suh a way that (2.12) v n(t) = v() and (2.13) v(t) v n(t) g(ζ, v n(ζ), v n(α(ζ)))dζ N φ(t) for all t I. 1 (NL g N 2 L k ) The uniqueness of the funtion v n implies that if t I n, then (2.14) v n(t) = v n1(t) = v n2(t) =. For t R, dene n(t) N as n(t) = minn N t I n. Next, we dene a funtion v : R R by (2.15) v (t) = v n(t) (t), k(t, ζ, v n(ζ), v n(α(ζ)) dζ ds and laim that v is ontinuous. For any t 1 R we take the integer n 1 = n(t 1). Then, t 1 belongs to the interior of the interval I n1 1 and there exists positive ε > suh that v (t) = v n1 1(t) for all t with t 1 ε < t < t 1 ε. Sine v n1 1 is ontinuous at t 1, v is ontinuous at t 1 for t 1 R. Now, we prove that the ontinuous funtion v satises (2.5) and (2.7) for all t R. Assume that n(t) is an integer for any t R. Then, by the use of (2.12) and (2.15), we

6 62 have t I n(t) and v (t) = v n(t) (t) = v() = v() g(ζ, v n(ζ), v n(α(ζ))) dζ k(t, ζ, v n(ζ), v n(α(ζ)) dζ ds g(ζ, v (ζ), v (α(ζ))) dζ k(t, ζ, v (ζ), v (α(ζ))) dζ ds. where the last equality holds true beause n(ζ) n(t) for all ζ I n(t) and from equations (2.14) and (2.15) we get that v n(t) (ζ) = v n(ζ) (ζ) = v (ζ). Sine t I n(t) for all t R, so from (2.13) and (2.15), we have N v(t) v (t) v(t) v n(t) (t) 1 (NL g N 2 L k ) φ(t), for any t R. Finally, we are going to prove that v is unique. To do this we onsider another ontinuous funtion u : R R whih satises (2.5) and (2.7), with u instead of v, for all t R. Let t R be an arbitrary number. Sine the restritions v In(t) and u In(t) satises (2.5) and (2.7) for eah t I n(t), the uniqueness of v n(t) = v In(t) suggest that v (t) = v In(t) (t) = u In(t) (t) = u (t). We an prove similarly for the ases I = (, T ] and I = [, ). 3. Hyers-Ulam stability In this setion, we prove the Hyers-Ulam stability of the nonlinear Volterra integrodierential equation with delay (1.1) Theorem. Let I = [, T ] be a non-degenerated interval, L g and L k be nonnegative onstants suh that < T L g T 2 L 2 k < 1. Assume that g : I R R R is ontinuous funtion whih satises the Lipshitz ondition (2.1) and k : I I R R R is a ontinuous funtion whih satises the Liphitz ondition (2.2). If for some σ and a ontinuously dierentiable funtion v : I R we have (3.1) v (t) g(t, v(t), v(α(t))) k(t, s, v(s), v(α(s))) ds σ for all t I, then there exists a unique ontinuous funtion v : I R satisfying the equation (2.5) and T (3.2) v(t) v (t) 1 (T L g T 2 L σ, for all t I. 2 k) Proof. Let Y be the set of all real valued ontinuous funtions on losed and bounded interval I. For all u, w Y, we dene a metri on Y by (3.3) d(u, w) = inf K [, ]: u(t) w(t) K for all t I. The metri spae (Y, d) is a omplete generalized metri spae, see [1]. Consider the operator Θ: Y Y dened by (3.4) (Θu)(t) = v() g(ζ, u(ζ), u(α(ζ)))dζ k(t, ζ, u(ζ), u(α(ζ))) dζ ds

7 621 for all t I and for all u Y. Next, we need to hek that the operator Θ is stritly ontrative on the set Y. Suppose that K uw [, ] be a onstant with d(u, w) K uw for any u, w Y. We have, (3.5) u(t) w(t) K uw, for all t I. By making the use of (2.1), (2.2), (3.4) and (3.5), we get (Θu)(t) (Θw)(t) = g(ζ, u(ζ), u(α(ζ))) g(ζ, w(ζ), w(α(ζ))) dζ k(t, ζ, u(ζ), u(α(ζ))) k(t, ζ, w(ζ), w(α(ζ))) g(ζ, u(ζ), u(α(ζ))) g(ζ, w(ζ), w(α(ζ))) dζ dζ ds k(t, ζ, u(ζ), u(α(ζ))) k(t, ζ, w(ζ), w(α(ζ))) dζ ds t L g u(ζ) w(ζ) dζ Lk L gk uwt L k K uw T 2 K uw(t L g T L k), for all t I, u(ζ) w(ζ) dζ ds i.e., d(θu, Θw) K uw(t L g T 2 L 2 k). Hene, we may onlude that d(θu, Θw) (T L g T 2 L 2 k)d(u, w) for any u, w Y, where < T L g T 2 L 2 k < 1. Suppose w Y be arbitrary, there exists a onstant K [, ] with Θw (t) w (t) = v() K, for all t I. g(ζ, w (ζ), w (α(ζ)))dζ k(t, ζ, u (ζ), u (α(ζ)))dζds w Sine g(ζ, w (ζ), w (α(ζ))), k(t, ζ, u (ζ), u (α(ζ))) and w are bounded. Thus, equation (3.3) implies that d(θw, w ) <. So, aording to Theorem 1.3, there exists a ontinuous funtion v : I R in a way that Θ n w v in (Y, d) and Θv = v, i.e., v satises (2.5) for all t I. As in the proof of Theorem 2.1, it an be verify easily that w Y d(w, w) < is equal to Y. From Theorem 1.3, we onlude that v, given by equation (2.5), is the unique ontinuous funtion. Again from equation (2.3), we get (3.6) σ v (t) g(t, v(t), v(α(t))) k(t, s, v(s), v(α(s))) ds σ, for all t I. By integrating eah term of inequality (3.6) from to t, then we get v(t) v() g(ζ, v (ζ), v (α(ζ)))dζ k(t, ζ, v (ζ), v (α(ζ))) dζ ds σ T

8 622 for all t I. So, it satises that v Θv σt. Finally, Theorem 1.3 together with (2.11) implies that T d(v, v ) 1 (T L g T 2 L 2 k) d(θv, v) T 1 (T L g T 2 L 2 k) σ. Now we present two examples whih indiate how our theorems an be applied to onrete problems Example. Let a > 1, q (, ) and p are arbitrary but xed onstants. Consider the lass of equations (3.7) u (t) = g(t) 1 2 for λ < λ(t s) p (u(s) u ( α(s) f(s) ) ds, t [, T ] (q ln a)2, where f(t) and g(t) are any ontinuous funtions. Here T p k(t, s, u(s), u(α(s))) = 1 2 λ(t s)p (u(s) u(α(s)) f(s)). Clearly k(t, s, u 1(s), u 1(α(s))) k(t, s, u 2(s), u 2(α(s))) 1 2 λ (t s)p ( u 1(s) u 2(s) u 1(α(s) u 2(α(s)) )) λt p u 1 u 2. Let v : I R be suh that v (t) g(t) 1 (λ(t s) p (v(s) v(α(s)) f(s)))ds 2 σ(t) = aqt, t [, T ]. Clearly, t σ(t)dt = a qt dt = 1 q ln a aqt = 1 q ln a σ(t) for all t [, T ]. Theorem 2.1 ensures the existene of a unique ontinuous funtion v : I R that solves (3.7) and v(t) v(t) q ln a (q ln a) 2 T p λ aqt, t [, T ] Example. Consider the above lass of problems for λ < 2. Let for some σ > T and p2 v : I R we have v (t) g(t) 1 2 λ(t s) p (v(s) v(α(s)) f(s))ds σ, t [, T ]. In the light of Theorem 3.1, there exists a unique ontinuous funtion v : I R that solves (3.7) for λ < 2 and T p2 v(t) v(t) 2T σ, t [, T ]. 2 T p2 λ Conlusion. In this manusript, we investigated the HyersUlamRassias stability and HyersUlam stability of a nonlinear Volterra integrodierential equation with delay by using a xed point method.

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