EXISTENCE OF A UNIQUE SOLUTION OF A NONLINEAR FUNCTIONAL INTEGRAL EQUATION

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1 Noliear Fuctioal Aalysis ad Applicatios Vol. 2, No , pp Copyright c 215 Kyugam Uiversity Press KUPress EXISTENCE OF A UNIQUE SOLUTION OF A NONLINEAR FUNCTIONAL INTEGRAL EQUATION Pham Hog Dah 1, Le Thi Phuog Ngoc 2, Huyh Thi Hoag Dug 3 ad Nguye Thah Log 4 1 Departmet of Mathematics, Uiversity of Ecoomics of HoChiMih City 59C Nguye Dih Chieu Str., Dist. 3, Ho Chi Mih City, Vietam hogdah282@gmail.com 2 Nha Trag Educatioal College 1 Nguye Chah Str., Nha Trag City, Vietam goc1966@gmail.com 3 Departmet of Mathematics, Uiversity of Architecture of HoChiMih City 196 Pasteur Str., Dist.3, Ho Chi Mih City, Vietam dughth198@gmail.com 4 Departmet of Mathematics ad Computer Sciece, Uiversity of Natural Sciece Vietam Natioal Uiversity Ho Chi Mih City 227 Nguye Va Cu Str., Dist. 5, Ho Chi Mih City, Vietam logt2@gmail.com Abstract. I this paper, we cosider a oliear fuctioal itegral equatio with variable delays. Usig tools of fuctioal aalysis ad Baach s fixed poit theorem i a Fréchet space, the existece of a uique solutio for the above equatio is proved. Notrivial examples are also give to illustrate our result. 1. Itroductio I this paper, we cosider the followig oliear fuctioal itegral equatio with variable delays xt = V t, xt, µ 1 t V 1 t, s, xθ1 s,..., xθ p s, V 2 [xs ds, V 2 [xs = µ s V 2 s, r, x θ 1 r,..., x θ q r dr, Received September 18, 214. Revised November 24, Mathematics Subject Classificatio: 47H1, 45G1, 47N2, 65J15. Keywords: Baach s fixed poit theorem, oliear fuctioal itegral equatio, cotractio mappig.

2 11 P. H. Dah, L. T. P. Ngoc, H. T. H. Dug ad N. T. Log t R +, where E is a Baach space, V : R + E 2 E; V 1 : µ1 E p+1 E; V 2 : µ2 E q E are supposed to be cotiuous ad µi = {t, s R 2 + : s µ i t}, the fuctios µ 1, µ 2, θ i, θ j CR + ; R + are cotiuous, µ 1 t, µ 2 t, θ i t, θ j t [, t, i = 1,..., p; j = 1,..., q. It is well kow that itegral equatios have wide applicatios i egieerig, mechaics, physics, ecoomics, optimizatio, vehicular traffic, biology, queuig theory ad so o. The theory of itegral equatios is rapidly developig with the help of tools i fuctioal aalysis, topology ad fixed-poit theory see [1 - [11 ad the refereces give therei. Applyig a fixed poit theorems ad givig the suitable assumptios, Dhage ad Ntouyas [3, Puraras [11 also obtaied some results o the existece of solutios to the followig oliear fuctioal itegral equatio xt = Qt + µt kt, sfs, xθsds + σt vt, sgs, xηsds, 1.2 t [, 1, where E = R, µt t; σt t; θt t; ηt t, for all t [, 1. Some more geeral equatios tha 1.2 were also studied i [11. Usig the techique of the measure of ocompactess ad the Darbo fixed poit theorem, Z. Liu et al. [6 have proved the existece ad asymptotic stability of solutios for the equatio t xt = f t, xt, ut, s, xas, xbs ds, t R +. I [2, usig a fixed poit theorem of Krasosel skii, Avramescu ad Vladimirescu have proved the existece of asymptotically stable solutios to the equatio ut = qt + t Kt, s, usds + Gt, s, usds, t R +, where fuctios give with real values satisfyig suitable coditios. I case the Baach space E is arbitrary, the existece of asymptotically stable solutios of equatio t xt = qt + ft, xt + + G t, s, xs, V s t, s, xs, s G 1 t, s, r, xr dr V 1 t, s, r, xr dr t R +, have bee proved i [8, by usig the fixed poit theorem of Krasosel skii type. ds, ds

3 A oliear fuctioal itegral equatio 111 Recetly, i [1, the existece ad uiqueess of a solutio of the followig system is proved [ m Xijk x f i x = a ijk Ψ x, f j R ijk x, f j tdt + b ijk f j S ijk x k=1 j=1 + g i x, i = 1,...,, x Ω = [ b, b, where a ijk, b ijk are the give real costats; R ijk, S ijk, X ijk : Ω Ω, g i : Ω R, Ψ : Ω R 2 R are the give cotiuous fuctios ad f i : Ω R are ukow fuctios. The mai tool used here is Baach s fixed poit theorem, it is applied i a suitable space, that is Baach space X = CΩ; R of cotiuous fuctios f : Ω R, with respect to the orm f X = sup i=1 f ix, f = f 1,..., f X. x Ω Motivated by the problems i the above metioed works, we study the existece ad uiqueess of a solutio for 1.1. This paper cosists of three sectios. I sectio 2, we preset the mai result. Fially, the illustrated examples are give i sectio 3. The mai tool employed here is Baach s fixed poit theorem i Fréchet space, with a suitable choice of a umerable family of semiorms. 2. Existece of solutios Let X = CR + ; E be the space of all cotiuous fuctios o R + to E which equipped with the umerable family of semiorms x = sup xt, 1. t [, The X, is complete i the metric dx, y = x y =1 2, 1 + x y ad X is the Fréchet space. I X we also cosider the family of semiorms defied by x = sup e ht xt, 1, t where h > is arbitrary umber, which is equivalet to, sice e h x x x, x X, 1. Based o the costruct of such X,, the followig lemma is valid, it is useful to prove existece of a uique solutio for 1.1. Lemma 2.1. [1 Let X, be a Fréchet space ad let Φ : X X be a L cotractio o X with respect to a family of semiorms equivalet with. The Φ has a uique fixed poit i X.

4 112 P. H. Dah, L. T. P. Ngoc, H. T. H. Dug ad N. T. Log The details of the proof ca be foud i Appedix of [9. I order to establish the existece result for 1.1, we eed the followig assumptios. A 1 The fuctios µ 1, µ 2, θ i, θ j CR + ; R + are cotiuous such that µ 1 t, µ 2 t, θ i t, θ j t [, t, for all t R +, i = 1,..., p, j = 1,...q. A 2 There exist a costat L [, 1 ad a cotiuous fuctio ω : R + R + such that V t; x, y V t; x, ȳ L x x + ω t y ȳ, for all t; x, y, t; x, ȳ R + E 2. A 3 There exists a cotiuous fuctio ω 1 : µ1 R + such that V 1 t, s, x 1,..., x p, y V 1 t, s, x 1,..., x p, ȳ p ω 1 t, s x i x i + y ȳ, i=1 for all t, s, x 1,..., x p, y, t, s, x 1,..., x p, ȳ µ1 E p+1. A 4 There exists a cotiuous fuctio ω 2 : µ1 R + such that V 2 s, r, x 1,..., x q V 2 s, r, x 1,..., x q ω 2 s, r q for all s, r, x 1,..., x q, s, r, x 1,..., x q µ2 E q. The we have the followig theorem. j=1 x j x j, Theorem 2.2. Let A 1 -A 4 hold. The 1.1 has a uique solutio x o R +. Moreover, give x X = CR + ; E, cosider the sequece {x k } be defied by x k t = V t, x k 1 t, µ 1 t V 1 t, s, xk 1 θ 1 s,..., x k 1 θ p s, V 2 [x k 1 s ds, V 2 [x k 1 s = µ 2 s V 2 s, r, x k 1 θ 1 r,..., x k 1 θ q r dr, t R +, k = 1, 2, The sequece {x k } coverges i X to the solutio x with error estimatio x k x x 1 x 1 L L k, k, N, 2.2 where L, < L < 1 is a costast depedig oly o. Proof. First, we rewrite the equatio 1.1 as follows xt = Φxt, t R +, 2.3

5 A oliear fuctioal itegral equatio 113 where Φxt = V t, xt, µ 1 t V 1 t, s, xθ1 s,..., xθ p s, V 2 [xs ds, V 2 [xs = µ 2 s V 2 s, r, x θ 1 r,..., x θ q r dr, t, x R + X. 2.4 By the assumptios A 2 -A 4, for all x, x X, for all t R +, put y = x x, we have Φxt Φ xt L yt + ω t p +ω t q j=1 µ1 t i=1 ω 1 t, s yθ i s ds ω 2 s, r y θ j r dr. µ1 t ω 1 t, sds µ 2 s 2.5 Let N be fixed. For all t [,, with h > to be chose later, Φ has the followig property where ad Φxt Φ xt e ht L yt e ht + ω p ω 1 i=1 e ht t yθ is ds + ω q ω 1 ω 2 j=1 e ht t ds s y θ j r dr L y + ω ω 1 p i=1 I1 i Estimatig I 1 i I 1 i ω = sup {ω t : t }, ω 1 = sup {ω 1 t, s : t, s 1 }, + ω ω 1 ω 2 q j=1 I2 j, ω 2 = sup {ω 2 s, r : s, r 2 }, 1 = {t, s : s µ 1 t, t }, 2 = {s, r : r µ 2 s, s }, I 1 i I 2 j yθ is ds, i = 1,..., p, ds s y θ j r dr, j = 1,..., q. yθ is ds. yθ is ds ehθis e hθis yθ i s ds e ht t ehθis y ds e ht t ehs y ds = e ht 1 h e h t 1 y = 1 h 1 e h t y 1 h y. Estimatig I 2 j I 2 j ds s y θ j r dr. We have ds s y θ j r dr e ht t ds t y θ j r dr e ht t y θ j r dr

6 114 P. H. Dah, L. T. P. Ngoc, H. T. H. Dug ad N. T. Log Similarly Thus Cosequetly This implies that Φxt Φ xt e ht Φx Φ x t e ht y θ j r dr 1 y h. 2.1 I 2 j h y. [ L + 1 ω ω 1 p + q ω 2 y h. [ L + 1 ω ω 1 p + q ω 2 y h = L x x, 2.11 where L = L + 1 h ω ω 1 p + q ω 2. Choosig h such that L = L + 1 h ω ω 1 p + q ω 2 < 1, 2.12 the we have < L < 1, so Φ is a L cotractio operator o the Fréchet space X,, applyig Lemma 2.1, 2.3 has a uique solutio x = x. O the other had, by the operator Φ is a L cotractio, we obtai x k+p x k x 1 x L k 1 L, k, p, N This implies that for all p, N, we have lim x k+p x k k + =, which meas that {x k } is a Cauchy sequece, by the fact that is equivalet with. The space X, is complete, so {x k } coverges to a poit x of X. It is obviously that x is a uique fixed poit of Φ. Passig to the limit i 2.13 as p + for fixed k, 2.2 follows. Theorem 2.2 is proved. 3. The examples Let us illustrate the results obtaied by meas of the examples. Example 3.1. Let E = C[, 1; R be the Baach space of all cotiuous fuctios u : [, 1 R with the orm u E = u = sup uη, u E. η 1 The, for all x X = CR + ; E, for ay t R +, xt is a elemet of E ad we deote xtη = xt, η, η 1.

7 A oliear fuctioal itegral equatio 115 Cosider 1.1 i form xt = V t, xt, µ 1 t V 1 t, s, xθ1 s,..., xθ p s, V 2 [xs ds, dr, t R +, V 2 [xs = µ 2 s V 2 s, r, x θ 1 r,..., x θ q r 3.1 where µ i t = µ i t, < µ i 1, i = 1, 2; θ i t = θ i t, < θ i 1, i = 1,..., p; θ j t = ˆθ j t, < ˆθ i 1, j = 1,..., q. Givig the cotiuous fuctios V, V 1, V 2 as follows. i Fuctio V : R + E 2 E, V t, x, yη = 1 k 1 Z t, η + k 1 xη + e t yη, ii for all η 1, t, x, y R + E 2 with Z t, η = 1 η+e ad k t 1 is give costat such that < k 1 < 1. Fuctio V 1 : 1 E p+1 E, 1 = {t, s R 2 + : s µ 1 t}, V 1 t, s, x 1,..., x p, yη = e 2s Z t, η [ p i=1 si π Z θ i s, η x iη + e t yη, for all η 1, t, s, x 1,..., x p, y 1 E p+1. iii Fuctio V 2 : 2 E q E, 2 = {s, r R 2 + : r µ 2 s}, V 2 s, r, x 1,..., x q η = e 2r Z s, η q j=1 si for all η 1, s, r, x 1,..., x q 2 E q. 2π Z θ j r, η x jη, We ca prove that A 1 -A 4 hold. It is easy to see that A 1 holds. Assumptio A 2 holds, for all t; x, y, t; x, ȳ R + E 2, V t; x, y V t; x, ȳ k 1 x x + ω t y ȳ, with ω t = e t, L = k 1.

8 116 P. H. Dah, L. T. P. Ngoc, H. T. H. Dug ad N. T. Log Assumptio A 3 holds, for all t, s, x 1,..., x p, y, t, s, x 1,..., x p, ȳ 1 E p+1, 1 = {t, s R 2 + : s µ 1 t}, η [, 1, i which V 1 t, s, x 1,..., x p, yη V 1 t, s, x 1,..., x p, ȳη e 2s 1 [ p πe θ i s η + e t + η x i η x i η + e t yη ȳη [ i=1 2πe t s p x i x i + y ȳ i=1 [ p = ω 1 t, s x i x i + y ȳ, i=1 ω 1 t, s = 2πe t s. Assumptio A 4 holds, for all s, r, x 1,..., x q, s, r, x 1,..., x q 2 E q, 2 = {s, r R 2 + : r µ 2 s}, η [, 1, V 2 s, r, x 1,..., x q η V 2 s, r, x 1,..., x q η e 2r 1 q e s + η 2πη + eˆθ j r x j η x j η j=1 4πe s r q x j x j j=1 = ω 2 s, r q x j x j, j=1 with ω 2 s, r = 4πe s r. The, Theorems 2.2 holds for 3.1. For more details, it is ot difficult to show that 3.1 has a uique solutio x = Z. Example 3.2. Let E = R N, cosider the followig system of equatios x i t = U i t, x 1 t,..., x N t, µ 1 t W 1 t, s, x 1 θ 1 s,..., x N θ 1 s ds,..., µ 1 t W N t, s, x 1 θ 1 s,..., x N θ 1 s ds, 3.2 i = 1,..., N, t R +, where the cotiuous fuctios U i, W i are defied by U i : R + R N+1 R, i = 1,..., N; W i : 1 R N R, i = 1,..., N; 1 = {t, s R 2 + : s µ 1 t}. Suppose that

9 A oliear fuctioal itegral equatio 117 Ã1 The fuctios µ 1, θ 1 CR + ; R + are cotiuous such that µ 1 t t, θ 1 t t, for all t R +, we rewrite 3.2 as follows x i t = U i t, xt, µ 1 t W 1 t, s, xθ 1 s ds,..., µ 1 t W N t, s, xθ 1 s ds, i = 1,..., N, t R +, where x = x 1,..., x N X = CR + ; R N. 3.3 We defie two vector fuctios V : R + R 2N R N, V 1 : 1 R N R N as follows: V t, x, y = U 1 t, x, y,..., U N t, x, y, t, x, y R + R 2N, V 1 t, s, x = W 1 t, s, x,..., W N t, s, x, t, s, x 1 R N. The, system 3.3 becomes xt = V Suppose that t, xt, µ1 t V 1 t, s, xθ 1 s ds Φxt, t. 3.4 Ã2 There exist a costat L [, 1 ad a cotiuous fuctio ω : R + R + such that U i t; x, y U i t; x, ȳ L x x + ω t y ȳ, for all t; x, y, t; x, ȳ R + R 2N, for all i = 1,..., N; ad is a orm i R N defied by x = max 1 i N x i, x = x 1,..., x N R N. Ã3 There exists a cotiuous fuctio ω 1 : µ1 R + such that W i t, s, x W i t, s, x ω 1 t, s x x, for all t, s, x, t, s, x 1 R N. Note that CR + ; R N is the Fréchet space which equipped with the umerable family of semiorms x = sup xt, 1. t [, I CR + ; R N we also cosider the family of semiorms defied by x = sup e ht xt, 1, t ad h > is arbitrary umber, which is equivalet to, sice e h x x x, x CR + ; R N, 1.

10 118 P. H. Dah, L. T. P. Ngoc, H. T. H. Dug ad N. T. Log With the choose of the suitable parameter h > h > sufficietly large, we will get L [, 1 such that, the operator Φ is a L cotractio o CR + ; R N. Similarly, the we have the followig result. Propositio 3.3. Let Ã1-Ã3 hold. The 3.2 has a uique solutio y CR + ; R N. Furthermore, give y CR + ; R N, cosider the sequece {y k } be defied by y k t = V t, y k 1 t, µ 1 t V 1 t, s, y k 1 θ 1 s ds 3.5 Φy k 1 t, t R +, k = 1, 2,. The sequece {y k } coverges to y i CR + ; R N with error estimatio y k y 1 y y L k 1 L, k, N, 3.6 where L, < L < 1 is a costast depedig oly o. Ackowledgmets: The authors wish to express their sicere thaks to the referees ad the Editor for their valuable commets. Refereces [1 C. Avramescu, Some remarks o a fixed poit theorem of Krasosel skii, Electroic J. Qualitative Theory of Diff. Equat., 5 23, [2 C. Avramescu ad C. Vladimirescu, A existece result of asymptotically stable solutios for a itegral equatio of mixed type, Electroic J. Qualitative Theory of Diff. Equat., 25 25, 1 6. [3 B.C. Dhage ad S.K. Ntouyas, Existece results for oliear fuctioal itegral equatios via a fixed poit theorem of Krasoselskii-Schaefer type, Nol. Studies, 9 22, [4 Pham Hog Dah, Le Thi Phuog Ngoc ad Nguye Thah Log, Solvability ad asymptotically stable of a mixed fuctioal itegral equatio i N variables, Noliear Fuct. Aal. Appl., , [5 S. Lag, Aalysis II, Addiso-Wesley, Readig, Mass., Califoria Lodo, [6 Zeqig Liu, Shi Mi Kag ad Jeog Sheok Ume, Solvability ad asymptotic stability of a oliear fuctioal-itegral equatio, Applied Math. Letters, , [7 Le Thi Phuog Ngoc ad Nguye Thah Log, O a fixed poit theorem of Krasosel skii type ad applicatio to itegral equatios, Fixed Poit Theory ad Appl., 26 26, Article ID 3847, 24 pages. [8 Le Thi Phuog Ngoc ad Nguye Thah Log, Existece of asymptotically stable solutios for a oliear fuctioal itegral equatio with values i a geeral Baach space, Noliear Aal. TMA., ,

11 A oliear fuctioal itegral equatio 119 [9 Le Thi Phuog Ngoc ad Nguye Thah Log, Solvig a system of oliear itegral equatios ad existece of asymptotically stable solutios, Diff. Equa. Appl., , [1 Le Thi Phuog Ngoc, Huyh Thi Hoag Dug, Pham Hog Dah ad Nguye Thah Log, Liear approximatio ad asymptotic expasio associated with the system of oliear fuctioal equatios, Demostratio Math., , [11 I.K. Puraras, A ote o the existece of solutios to some oliear fuctioal itegral equatios, Electroic J. Qualitative Theory of Diff. Equat., 17 26, 1 24.