STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION OF FIRST ORDER WITH VARIABLE DELAYS

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1 Bulletin of Mathematical Analyi and Application ISSN: , URL: Volume 1 Iue 2(218), Page STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION OF FIRST ORDER WITH VARIABLE DELAYS CEMİL TUNÇ, İREM AKBULUT Abtract. We conider a linear integro-differential equation (IDE) of firt order with two variable delay. We contruct new condition guaranteeing the trivial olution of thi IDE i table. The technique of the proof baed on the ue of the fixed-point theory. Our finding generalize and improve ome reult can be found in the literature. 1. Introduction It i well known from the related literature that the IDE are often ued to model ome practical problem in mechanic, phyic, biology, ecology and the other cientific field (ee, e.g. Burton [5], Levin [1], Rahman [14], Wazwaz [24] and their reference). Thi paper deal with the following linear IDE of the firt order with two variable delay dx dt = b(t)x a(t, )x()d, (1.1) where t, t R, x R, R = (, ), r i = [, ) [, ), (i = 1, 2), are continuou differentiable function uch that t r i (t) a t with r = max t {t r 1 (t), t r 2 (t)}, a : [ r, ) [ r, ) R and b : (, ) R are continuou function. In the pat decade, a number of reearche have dealt with qualitative behavior of linear and non-linear IDE by mean of the fixed point theory, the perturbation method, the variation of parameter formula, the Lyapunov function or functional method, etc., (ee Ardjouni and Djoudi [1], Becker and Burton [2], Burton ([3], [4]), Gabi et al. [6], Gözen and Tunç [7], Graef and Tunç [8], Jin and Luo [9], Levin [1], Pi ([11], [12]), Raffoul [13], Tunç ([15], [16],[17],[18], [19]), Tunç and Mohammed [2], Tunç and Tunç ([21], [22],[23]) and their reference). Among thee invetigation, the tability analyi of olution ha been an important topic for IDE with contant or variable delay and without delay. For variou model and kind of IDE, many ignificant reult have been preented, ee, for example, the reference of thi article and thoe regitered therein. Here, 2 Mathematic Subject Claification. 34K2, 45J5. Key word and phrae. Contraction mapping, tability, fixed point, integro-differential equation, variable delay, firt order. c 218 Univeriteti i Prihtinë, Prihtinë, Koovë. Submitted April 4, 218. Publihed May 15, 218. Communicate by Tongxing Li. 19

2 2 C. TUNÇ, İ. AKBULUT we would not give the detail of the work and their application. However, we would like to ummarize here a few related reult on the topic. Firt, in 1963, Levin [1] conidered the linear IDE dx dt = a(t )g(x())d and the author gave ufficient condition guaranteeing that if any olution of the above IDE exit on [, ), then the olution of thi IDE, it firt and econd order derivative tend to zero when t. The proof of the main reult of [1] involve the ue of a uitably choen Lyapunov function. Later, in 24, Burton [4] took into conideration the following non-linear IDE with contant delay of the form dx dt = t r a(t, )g(x())d. In [4], intead of uing a Lyapunov functional, the author tudied aymptotic tability of the above IDE by uing the concept of a contraction mapping in line with the fixed point theory for a cla of equation which ha been comprehenively tudied in the lat fifty year. The reult of [4] look very intereting. Finally, Jin and Luo [9] tudied a calar integro-differential equation (in both the linear and nonlinear cae) etablihing ufficient condition for the exitence, tability and aymptotic tability for the null olution. The IDE examined by the author are given by dx t dt = a(t, )x()d t r(t) and dx t dt = a(t, )g(x())d. t r(t) In [9], in order to have the poibility of applying Banach fixed point theorem, firt IDE i written in an equivalent form and if x(t) = ψ(t) on [ r, ), it i hown that the olution x(t) of firt IDE i bounded on [ r, ) and the trivial olution of the ame IDE i alo table. Under an additional condition, it i proved for the econd IDE that x(t) a t. The nonlinear cae (the econd IDE) look omewhat more complicated but it admit a treatment imilar to that of the former one. The motivation to conider IDE (1.1) and invetigation it ome qualitative propertie come from the paper of Levin [1], Burton [4], Jin and Luo [9] and the ource that found in the reference of thi article. Here, we give new reult on the boundedne, tability, aymptotic tability and ome other propertie of olution of IDE (1.1). The conidered IDE, the reult and aumption to be given here are different from that can be found in the literature and complete that one. Thee are the contribution of thi paper to the literature and it novelty and originality. The following definition may be ueful for reader. Definition 1. The zero olution of IDE (1.1) i aid to be table at t = if, for every, ε >, there exit a δ > uch that ψ : [ r, ] ( δ, δ) implie that x(t) < ε for t r.

3 STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION We can write IDE (1.1) a where G(t, ) = dx dt b(t)x = G(t, t r i (t))(1 r i(t))x(t r i (t)) t a(u, )du with G(t, t r i (t)) = d t G(t, )x()d, (1.2) dt ri(t) t a(u, t r i (t))du. (1.3) Theorem 1. If x(t) i a olution of IDE (1.1) on an interval [, T ) and atifie the initial condition x(t) = ψ(t) for t [ r, ], then x(t) i a olution of IE x(t) = 2 b()d)ψ() [b(u) G(t, u)]x(u)du b() r i() r i() [b(u) G(, u)]x(u)dud [b(u) G(, u)]ψ(u)du [b( r i () G(, r i ())](1 r i())x( r i ())d b()x()d (1.4) on [, T ), where b : [ r, ) R i an arbitrary continuou function. Converely, if a continuou function x(t) i equal to ψ(t) for t [ r, ] and i a olution of IE (1.4) on an interval [, τ), then x(t) i a olution of IDE (1.1) on [, τ). Proof. Multiplying both ide of IDE (1.2) by the factor exp( and integrating from to any t, t [, τ), then we get o that d dt x (t)exp( ( x()exp( = exp( exp( exp( ) = b(t)x(t) G(t, t r i (t))(1 r i(t))x(t r i (t)) [ d dt G(t, )x()d G(, r i ())(1 r i())x( r i ()) d d r i() G(, u)x(u)du]exp(.

4 22 C. TUNÇ, İ. AKBULUT Hence o that x(t)exp( x(t) = d d ψ()exp( = [ G(, r i ())(1 r i())x( r i ()) d d ψ() r i() Hence, we can write that x(t) = 2 b()d)ψ() r i() [ G(, u)x(u)du]exp( G(, r i ())(1 r i())x( r i ()) G(, u)x(u)du]exp( d d r i(). (1.5) [b(u) G(, u)]x(u)dud [b( r i ()) G(, r i ())](1 r i())x( r i ())d b()x()d. (1.6) We will now how that etimate (1.6) i equal to etimate (1.5). In fact, it follow from (1.6) that x(t) = 2 b()d)ψ() d d r i() d d G(, u)x(u)dud r i() b( r i ())(1 r i())x( r i ())d G( r i ())(1 r i())x( r i ())d b()x()d. b(u)x(u)dud

5 STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION Applying integration by part to the econd and third term of (1.6), we get x(t) = 2 = b()d)ψ() b()d) [b(u) G(, u)]x(u)du t r i() b() [b(u) G(, u)]x(u)dud r i() b()d)ψ() [b( r i ()) G(, r i ())](1 r i())x( r i ())d b()x()d b()d) [b(u) G(, u)]x(u)du r i() [b(u) G(t, u)]x(u)du b() [b(u) G(, u)]x(u)dud r i() 2 [b( r i ()) G(, r i ())](1 r i())x( r i ())d b()x()d, which lead to (1.4). Converely, uppoe that there exit a continuou function x(t) which i equal to ψ(t) on [ r, ] and atifie IE (1.4) on an interval [, τ). Then, the function x(t) i differentiable on [, τ) and if we calculate the time derivative of IE (1.4) with the aid of Leibniz rule, then we obtain IDE (1.2). Next, we will define a mapping directly from (1.4). By Theorem 1, a fixed point of that map will be a olution of IE (1.4) and IDE (1.1). To obtain tability of the zero olution of IDE (1.1), we need the mapping defined by IE (1.4) to map bounded function into bounded function. Let (C,. ) be the et of real-valued and bounded continuou function on [ r, ) with the upremum norm. that i, for φ C, φ = up{ φ(t) : t [ r, )}. In other word, we carry out our invetigation in the complete metric pace (C, ρ), where ρ denote the upremum (uniform) metric for φ 1, φ 2 C, and it i defined

6 24 C. TUNÇ, İ. AKBULUT by ρ(φ 1, φ 2 ) = φ 1 φ 2. For a given continuou initial function ψ : [ r, ) R, define the et C ψ C by C ψ = {φ : [ r, ) R φ C, φ(t) = ψ(t) for t [ r, ]}. Let. denote the upremum on [ r, ] or on [ r, ). Finally, note that (C ψ,. ) i itelf a complete metric pace ince C ψ i a cloed ubet of C. Theorem 2. Let b : [ r, ) R be a continuou function and T be a mapping on C ψ defined for φ C ψ by and (T φ)(t) = 2 b()d)ψ() (T φ)(t) = ψ(t) if t [ r, ] r i() [b(u) G(t, u)]φ(u)du b() [b(u) G(, u)]ψ(u)du r i() [b(u) G(, u)]φ(u)dud [b( r i ()) G(, r i ())](1 r i())φ( r i ())d b()φ()d. (1.7) Aume that there exit contant k and α > uch that and 2 b(u) G(t, u) du b()d k (1.8) b() b(u) G(, u) dud r i() b( r i ()) G(, r i ()) (1 r i() d b() d a (1.9) for t, then T : C ψ C ψ. Proof. For φ C ψ, T φ i continuou and agree with ψ on [ r, ] by virtue of

7 STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION the definition of T. In view of (1.8) and (1.9), for t >, we have (T φ)(t) exp(k) ψ() exp(k) r i() b(u) G(, u) φ(u) du α φ. Hence, ubject to the hypothee of Theorem 2, it i clear that (T φ)(t) exp(k) ψ (1 b(u) G(, u) du) α φ <. (1.1) r i() Thu, we can conclude that T φ C ψ. Theorem 3. Let k, α (, 1) and b : [ r, ) R be a continuou function uch that (1.8) and (1.9) hold for t. Then for each continuou function ψ : [ r, ] R, there i an unique continuou function x : [ r, ) R with x(t) = ψ(t) on [ r, ). In addition, x(t) i bounded on [ r, ) and the zero olution of IDE (1.1) i table at t =. Finally, if b()d (1.11) hold a t, then x(t) a t. Proof. We now take into conideration the pace C ψ defined by the continuou initial function ψ : [ r, ] R. For φ, η C ψ in the light of the hypothee of Theorem 3, it can be een that (T φ)(t) (T η)(t) b(u) G(t, u) φ(u) η(u) du b() r i() b( r i () G(, r i ())) 1 r i() φ( r i ()) η( r i ()) d 2 ( 2 b() φ() η() d b(u) G(t, u) du b() r i() b(u) G(, u) φ(u) η(u) dud b(u) G(, u) dud b( r i () G(, r i ())) 1 r i() d b() φ η d. By the definition of T and (1.9), T i a contraction mapping with contraction contant α. By Banach contraction mapping principle, T ha a unique fixed point x in C ψ which i a bounded and continuou function. By Theorem 3, it i a olution

8 26 C. TUNÇ, İ. AKBULUT of IDE (1.1) on [, ). It follow that x i the only bounded and continuou function atifying IDE (1.1) on [, ) and the initial condition. It i clear that the zero olution of IDE (1.1) i table. If x(t) i a olution with the initial function ψ on [ r, ], then, by (1.1), we have (1 α) x exp(k) ψ (1 b(u) G(, u) du). r i() Then, for each ε >, there exit a δ > uch that x(t) < ε for all t r if ψ < δ. Next we prove that the olution of IDE (1.1) tend to zero when (1.11) hold. Firt, we define a ubet C ψ of C ψ by C ψ = {φ : [ r, ) R φ C, φ(t) = ψ(t) for t [ r, ], φ(t) a t }. Since Cψ i a cloed ubet of C ψ and (Cψ, ρ) i complete, then the metric pace (Cψ, ρ) i alo complete. Now we how that (T φ)(t) a t when φ C ψ. By (1.7) and (1.9), we have (T φ)(t) b()d)( ψ() b(u) G(, u) )du α φ [t ri(t),t] I 4 I 5, r i() where t >, I 4 and I 5 denote fourth and fifth, ixth term of (1.7), repectively. We can prove that each of the above term tend to zero a t. In fact, it i eay to ee that the firt term tend to by (1.11) and the econd term approache zero a t ince t r i (t), i = 1, 2. Hence, for each ε >, there exit a M > uch that φ [M ri(m), ) ε 2α ince t r i (t) a t. Then, for t M we can ee that M M I 4 b() b(u) G(, u) dud φ T b() r i() r i() b(u) G(, u) dud φ [M ri(m), ). By (1.11), there exit a τ M uch that φ T < ε 2α for t > τ. Thu, for every ε > there exit a τ > uch that t > τ implie I 4 < ε, that i, I 4 a t. Similarly, we can how that I 5 tend to zero t. Thi yield (T φ)(t) a t, and hence T : Cψ C ψ. Therefore, T i a contraction on Cψ with a unique fixed point x. By Theorem 1, x i a olution of IDE (1.1) on [, ). Hence, x(t) i the only continuou olution of IDE (1.1) agreeing with the initial function ψ. Since x Cψ, then x(t) a t. We now give an example to how the applicability of Theorem 3. Example. Let u conider the following integro -differential equation of firt order with two variable delay, which i a pecial cae of IDE (1.1): x (t) =.2t t 2 1 x(t).2 2 x()d, (1.12) 1 T

9 STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION where r 1 (t) =.385t and r 2 (t) =.476t. When we compare IDE (1.12) with IDE (1.1) and take into conideration the hypothee of Theorem 3, it can eaily be een that and Hence, we have G(t, ) = For the function b(t), we obtain b()d = 1 5 Hence, clearly, it follow that Thu, the hypothei (1.11) hold. We alo have b(t) =.2t t 2 1 a(t, ) = b(u) G(t, u) du = By an eay calculation, we can get t.2.2( t) 2 du = d = 1 1 ln(t2 1). 1 1 ln(t2 1) when t. µ 1 (t) =.2u.2(u t) u 2 1 u 2 1 du.4u.2t =.615t u 2 du 1 = µ 1 (t) µ 2 (t)..615t.4u t u 2 1.2u.615t u t.4u.2t u 2 du 1 =.2ln(u 2 1) t.615t.2t arctan u t.615t =.2ln(t 2 1).2ln( t 2 1).2t arctan t.2t arctan.615t =.2t[arctan.615t arctan t].2ln(t 2 1).2ln( t 2 1). By the ame way, we can calculate µ 2 and obtain µ 2 (t) =.2t[arctan.524t arctan t].2ln(t 2 1).2ln( t 2 1). It i obviou that both of thee function, that i, µ 1 and µ 2 are increaing in [, ). We now need to find that lim t µ 1(t) lim t µ 2 (t).

10 28 C. TUNÇ, İ. AKBULUT If we do the neceary calculation, then lim µ 1(t) = lim [.2[(.615/.615t2 1) (1 1/t 2 )] t t 1/t 2 t 2 (1 1/t 2 ).2 ln( t 2 ( /t 2 ) )] 1 = lim.2[ t 1 1/t ln(1/ )] t 2 =.2[1 1/ ln(.615)] = and lim µ 2(t) = lim [.2[(.524/.524t2 1) (1 1/t 2 )] t t 1/t 2 t 2 (1 1/t 2 ).2 ln( t 2 ( /t 2 ) )] 1 = lim.2[ t 1 1/t ln(1/ )] t 2 =.2[1 1/ ln(.524)] = Hence, we have and b(u) G(t, u) <.7.8 =.15, =.2(2 1/.615).2(2 1/.524) b() b(u) G(, u) dud <.15, r i() b()d) b( r i () G(, r i ())) 1 r i() d <.2[(2 1/.615)].2[(2 1/.524)] = <.1 2 u u 2 1 ) 2 1/.615t 2 d u u 2 1 ) 2 1/.524t 2 d b() d <.3, repectively. Let α = =.7 < 1. Hence, we can conclude that x(t) of IDE (1.12) i bounded on [ r, ) and the zero olution of IDE (1.12) i aymptotically table.

11 STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION Reference [1] Ardjouni, A., Djoudi, A., Stability in nonlinear neutral integro-differential equation with variable delay uing fixed point theory. J. Appl. Math. Comput. 44 (214),no.1-2, [2] Becker, L.C., Burton, T. A., Stability, fixed point and invere of delay. Proc. Roy. Soc. Edinburgh Sect. A 136 (26), no. 2, [3] Burton, T. A., Stability by fixed point theory or Liapunov theory: a comparion. Fixed Point Theory 4 (23), no. 1, [4] Burton, T. A., Fixed point and tability of a non-convolution equation. Proc. Amer. Math. Soc. 132 (24), no. 12, [5] Burton, T.A., Stability by fixed point theory for functional differential equation. Dover Publication, Mineola, New York, 26. [6] Gabi, H., Ardjouni, A., Djoudi, A., Fixed point and tability of a cla of nonlinear delay integro-differential equation with variable delay. Facta Univ. Ser. Math. Inform. 32 (217), no. 1, [7] Gözen, M., Tunç, C., Stability in functional integro-differential equation of econd order with variable delay. J. Math. Fundam. Sci. 49 (217), no. 1, [8] Graef, J. R., Tunç, C., Continuability and boundedne of multi-delay functional integrodifferential equation of the econd order. Rev. R. Acad. Cienc. Exacta F. Nat. Ser. A Math. RACSAM 19 (215), no. 1, [9] Jin, C., Luo, J., Stability of an integro-differential equation. Comput. Math. Appl. 57 (29), no. 7, [1] Levin, J. J., The aymptotic behavior of the olution of a Volterra equation. Proc. Amer. Math. Soc. 14 (1963) [11] Pi, D., Study the tability of olution of functional differential equation via fixed point. Nonlinear Anal.74 (211), no.2, [12] Pi, D., Stability condition of econd order integro-differential equation with variable delay. Abtr. Appl. Anal. 214, Art. ID , 11 pp. [13] Raffoul, Y. N., Stability in neutral nonlinear differential equation with functional delay uing fixed-point theory. Math. Comput. Modelling, 4 (24) no.7-8, [14] Rahman, M., Integral equation and their application. WIT Pre, Southampton, 217. [15] Tunç, C., A note on the qualitative behavior of non-linear Volterra integro-differential equation. J. Egyptian Math. Soc. 24 (216), no. 2, [16] Tunç, C., New tability and boundedne reult to Volterra integro-differential equation with delay. J. Egyptian Math. Soc. 24 (216), no. 2, [17] Tunç, C., Stability and boundedne in Volterra-integro differential equation with delay. Dynam. Sytem Appl.26 (217), no.1, [18] Tunç, C., Qualitative propertie in nonlinear Volterra integro-differential equation with delay. Journal of Taibah Univerity for Science 11 (217), no.2, [19] Tunç, C., On the qualitative behavior of a functional differential equation of econd order. Appl. Appl. Math. 12 (217), no. 2, [2] Tunç, C., Mohammed, S. A., On the tability and intability of functional Volterra integrodifferential equation of firt order. Bull. Math. Anal. Appl. 9 (217), no. 1, [21] Tunç, C., Tunç, O., On the exponential tudy of olution of Volterra integro-differential equation with time lag. Electron. J. Math. Anal. Appl., 6 (1), (218), [22] Tunç, C., Tunç, O., New reult on the tability, integrability and boundedne in Volterra integro-differential equation. Bull. Comput. Appl. Math., 6 (1),(218), [23] Tunç, C., Tunç, O., On behavior of functional Volterra integro-differential equation with multiple time-lag. Journal of Taibah Univerity for Science, 12 (2), (218), [24] Wazwaz, A. M., Linear and nonlinear integral equation. Method and application. Higher Education Pre, Beijing, Springer, Heidelberg, 211. Cemİl Tunç Department of Mathematic Faculty of Science Van Yuzuncu Yil Univerity 658, Van-Turkey addre: cemtunc@yahoo.com

12 3 C. TUNÇ, İ. AKBULUT İrem Akbulut Department of Mathematic Faculty of Education Siirt Univerity 561, Siirt-Turkey addre: