On mild solutions of a semilinear mixed Volterra-Fredholm functional integrodifferential evolution nonlocal problem in Banach spaces

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1 MAEMAIA, 16, Volume 3, Number, c Penerbit UM Pre. All right reerved On mild olution of a emilinear mixed Volterra-Fredholm functional integrodifferential evolution nonlocal problem in Banach pace 1 amalendra umar, Rakeh umar and 3 Manoj arnatak 1 Department of Mathematic, SRMS College of Engineering & echnology, Bareilly-431, India,3 Department of Mathematic,indu College, Moradabad 441, India 1 kamlendra.14kumar@gmail.com, rakehnaini1@gmail.com, 3 karnatak.manoj@gmail.com Abtract he exitence, uniquene and continuou dependence on initial data of mild olution of nonlocal mixed Volterra-Fredholm functional integrodifferential equation with delay in Banach pace ha been dicued and proved in the preent paper. he reult are etablihed by uing the emigroup theory and modified verion of Banach contraction theorem. eyword Mixed Volterra-Fredholm functional integrodifferential equation; fixed point; emigroup theory; nonlocal condition. 1 Mathematic Subject Claification 471, 45J5, 47B38. 1 Introduction Byzewki and Acka [1 etablihed the exitence, uniquene and continuou dependence of a mild olution of a emilinear functional differential equation with nonlocal condition of the form du t + Au t = f t, u t,t [, a, dt u + [ g u t1,...., u tp = φ, [ r,, where < t 1 <... < t p a p N, A i the infiniteimal generator of a C emigroup of operator on a general Banach pace, f, g and φ are given function and u t = u t + for t [, a, [ r,. he problem of exitence, uniquene and other qualitative propertie of olution for emilinear differential equation in Banach pace ha been tudied extenively in the literature for lat many year, ee [-5, [6, [7-9, [1. heorem about the exitence, uniquene and tability of olution of differential, integrodifferential equation and functional differential abtract evolution equation with nonlocal condition were tudied by Byzewki [1, 11, Balachandran and Chandraekaran [1, Lin and Liu [13 and Balachandran and Park [14. In the preent paper, we conider emilinear mixed Volterra-Fredholm functional integrodifferential equation of the form dx t dt = Ax t + f t, x t, k t,, x d, h t,, x d, t [,, 1 x t + g x t1,...., x tp t = φ t,t [ r,, where A i the infiniteimal generator of a trongly continuou emigroup of bounded linear operator t,t on X, f, k, h, g and φ are given function atifying ome aumption and x t θ = x t + θ, for θ [ r, and t [,. he objective of thi paper i to improve

2 134 amalendra umar, Rakeh umar and Manoj arnatak the reult of [ and alo generalize the reult of Jain and Dhakne [15. We are finding the reult with le retriction by uing modified verion of Banach contraction principle. he ret of the paper i organied a follow: In ection we give preliminarie and hypothee. In ection 3 we prove exitence and uniquene of olution. In ection 4 we deal with continuou dependence of initial data of mild olution. Finally in ection 5 we give application to illutrate the theory. Preliminarie and hypothee Let X be Banach pace with the norm.. Let C = C [ r,, X, < r < be the Banach pace of all continuou function ψ : [ r, X endowed with the remum norm ψ C = { ψ t : r < t < }. Let B = C [ r,, X, > be the Banach pace of all continuou function x : [ r, X with theremum norm x B = { x t : r t }. For any x B and t [,, we denote x t the element of C given by x t θ = x t + θ, for θ [ r, and φ i given element of C. In thi paper, we aume that there exit poitive contant M 1 uch that t M, for every t [,. Definition.1 A function x Batifying the equation: x t = t [ φ g x t1,..., x tp t + t f, x, k, τ, x τ dτ, h, τ, x τ, t [, x t + g x t1,..., x tp t = φ t, r t, i called a mild olution of the initial value problem 1. Our reult are baed on the modified verion of Banach contraction principle. Lemma.1 [16, p.196 Let X be a Banach pace. Let D be an operator which map the element of X into itelf for which D r i a contraction, where r i a poitive integer. hen D ha a unique fixed point. We make the following aumption: A 1 Let f : [, C X X X uch that for every w B, x, y X and t [,, f., w t, x, y B and there exit a contant L > uch that f t, ψ, x, y f t, φ, u, v L ψ φ C + x u + y v, φ, ψ C, x, y, u, v X. A Let k : [, [, C X uch that for every w B and t [,, k.,., w t B and there exit a contant > uch that k t,, ψ k t,, φ ψ φ C,φ, ψ C. A 3 Let h : [, [, C X uch that for every w B and t [,, h.,., w t B and there exit a contant > uch that h t,, ψ h t,, φ ψ φ C,φ, ψ C.

3 Solution of a mixed Volterra-Fredholm functional integrodifferential in Banach pace 135 A 4 Let g : C p C uch that there exit a contant G uch that g x t1,..., x tp t g yt1,..., y tp t G x y B,t [ r,. 3 Exitence of mild olution heorem 3.1 Conider that the aumption A 1 A 4 are atified. hen the initial value problem 1 ha a unique mild olution x on [ r,. Proof: Let x t be a mild olution of the problem 1. hen it atifie the equivalent integral equation x t = tφ t g x t1,..., x tp + t f, x, k, τ, x τ dτ, h, τ, x τ dτ d, t [,, 3 x t + g x t1,..., x tp t = φ t, r t. 4 Now we rewrite olution of initial value problem 1 a follow: For φ C, define φ B by { [ φ t g xt1,..., x tp t, if r t, φ t = t [ φ g x t1,..., x tp, if t. If y B and x t = y t + φ t,t [ r,, then it i eay to ee that y atifie and y t = t f, y + φ, if and only if x t atifie the equation 3 4. We define the operator F : B B, by k y t = ; t [ r, 5, τ, y τ + φ τ dτ, h, τ, y τ + φ τ, t [, 6 Fyt = {, r t,, t f y + φ, k, τ, y τ + φ τ dτ, h, τ, y τ + φ τ, if t. 7 From the definition of an operator F defined by the equation 7, it i to be noted that the equation 5 6 can be written a y = Fy.

4 136 amalendra umar, Rakeh umar and Manoj arnatak Now we how that F n i a contraction on B for ome poitive integer n. Let y, w B and uing aumption A 1 A 4, we get Fyt Fw t f, y + t φ, k, τ, y τ + φ τ dτ, h, τ, y τ + φ τ dτ f, w + φ, k, τ, w τ + φ τ dτ, h, τ, w τ + φ τ dτ d [ y + φ w + φ C y + τ + φ τ w τ + φ C τ dτ + y τ + φ τ w τ + φ τ C y w C d + ML y w B d + ML y τ w τ C dτd + ML y w B dτd + ML t y w B t + ML y w B + ML y w t B t y w B t + ML y w B + ML y w t B y w B t + ML y w B t + ML y w B t. y τ w τ C y w B y w B t. F y t F w t = F Fyt F Fwt = F y 1 t F w 1 t f, y 1 + t φ, k, τ, y 1τ + φ τ dτ, h, τ, y 1τ + φ τ dτ f, w 1 + φ, k, τ, w 1τ + φ τ dτ, h, τ, w 1τ + φ τ dτ d [ ML y 1 + φ w 1 + φ C + y 1 τ + φ τ w 1τ + φ C τ dτ + y 1τ + φ τ w 1τ + φ C τ ML y 1 w 1 C d + ML + ML y 1τ w 1τ C dτd y 1 w 1 C[ r,,x d + ML + ML τ [ r, + ML τ [ r, y 1τ w 1τ C dτd y 1 w 1 C[ r,τ,x y 1 τ w 1 τ d + ML Fy τ Fw τ d + ML y 1 w 1 C[ r,τ,x y 1 η w 1 η y 1 η w 1 η Fy η Fw η

5 Solution of a mixed Volterra-Fredholm functional integrodifferential in Banach pace ML τ [ r, + ML + ML Fy η Fw η ML1 + + y w B τd t M L y w B [ ML1 + + y w B η ML1 + + y w B ηdτd + τ τ [ r, [ M L y w B d + [ t M L y w B + t3 3! + t3 3! M L y w B [ t M L y w B [ t M L y w B t!. Continuing in thi way, we get! + t 3! d + t + 3! t t + +!! η τ + η dτd τ n F n yt F n [ML1 + + t wt y w n! B. For n large enough, [ML1 + + t n < 1. n! hu there exit a poitive integer nuch that F n i a contraction in B. By virtue of Lemma.1, the operator Fha a unique fixed point ỹ in B. hen x = ỹ + φ i olution of the initial value problem 1. hi complete the proof of the theorem Continuou dependence of a mild olution heorem 4.1 Suppoe that the function f, k, h and gatifie the aumption A 1 A 4. hen for each φ 1, φ Cand for the correponding mild olution x 1, x of the problem dx t t = Ax t + f t, x t, k t,, x d, h t,, x d, t [,, 8 dt x t + g x t1,...., x tp t = φi t,t [ r,, i = 1,. 9

6 138 amalendra umar, Rakeh umar and Manoj arnatak he following inequality x 1 x B [ φ 1 φ C + G + L x 1 x B Me ML1+ i true. Additionally, if M G + L e ML1+ < 1 then x 1 x B Me ML1+ 1 M G + L e ML1+ φ 1 φ C. Proof wa given in paper [17, o we omit detail here. 5 Application o illutrate the application of our reult in ection 3, conider the following emilinear partial functional mixed integrodifferential equation of the form w u, t = w u, t t u + F t, w u, t r, k 1 t, w u, rd, h 1 t, w u, rd, t [,, 1 w, t = w π, t =, t, 11 p w u, t + w u, t i + t = φ u, t, u π, r t, 1 i=1 where < t 1... t p, the function F : [, R R R R i continuou. We aume that the function F, k 1 and h 1 atifying the following condition: For every t [, and u, v, x 1, x, y 1, y R, there exit a contant F t, u, x 1, x F t, v, y 1, y l u v + x 1 y 1 + x y ; k 1 t,, u k 1 t,, v p u v ; h 1 t,, u h 1 t,, v q u v. Let u take X = L [, π. Define the operator A : X X by Az = z with domain D A = {z X : z, z are abolutely continuou,z X andz = z π = }. hen the operator A can be written a Az = n z, z n z n, z D A n=1 where z n u = /π innu, n = 1,,... i the orthogonal et of eigenvector of A and A i the infiniteimal generator of an analytic emigroup t, t and i given by tz = exp n t z, z n z n, z X. n=1

7 Solution of a mixed Volterra-Fredholm functional integrodifferential in Banach pace 139 Now, the analytic emigroup tbeing compact, there exit contant M uch that t M, for each t [,. Define the function f : [, C X X X, a follow f t, ψ, x, yu = F t, ψ r,xu, y u, k t, φu = k 1 t, φ ru, h t, φu = h 1 t, φ ru. For t [,, ψ, φ C, x X and u π. With thee choice of the function the equation 1-1 can be formulated a an abtract mixed integrodifferential equation in Banach pace X : dx t dt = Ax t + f t, x t, x t + g x t1,...., x tp t = φ t,t [ r,. k t,, x d, h t,, x d, t [,, Since, all the hypothee of theorem 3.1 are atified, the theorem 3.1, can be applied to guarantee the exitence of mild olution w u, t = x tu, t [,, u [, π, of emilinear partial integrodifferential equation Concluion In thi paper, the exitence, uniquene and continuou dependence of initial data on a mild olution of emilinear mixed Volterra-Fredholm functional integrodifferential equation with nonlocal condition in general Banach pace are dicued. We apply the concept of emigroup theory and modified verion of Banach contraction theorem. We alo give example to illutrate the theory. Reference [1 Byzewki, L. heorem about the exitence and uniquene of olution of a emilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl : [ Byzewki, L. and Akca,. On mild olution of a emilinear functional differential evolution nonlocal problem. J. Appl. Math. Stochatic Anal : [3 Byzewki, L. and Akca,. Exitence of olution of a emilinear functional evolution nonlocal problem. Nonlinear Anal : [4 Byzewki, L. and Lakhamikantham, V. heorem about the exitence and uniquene of a olution of a nonlocal abtract Cauchy problem in a Banach pace. Appl. Anal : [5 Corduneanu, C. Integral Equation and Stability of Feedback Sytem. New York: Academic Pre

8 14 amalendra umar, Rakeh umar and Manoj arnatak [6 ale, J.. heory of Functional Differential Equation. New York: Springer [7 Ntouya, S.. Initial and boundary value problem for functional differential equation via the topological tranverality method: a urvey. Bull. Greek Math. Soc [8 Ntouya, S.. Nonlocal initial and boundary value problem: A urvey. andbook of Differential Equation: Ordinary Differential Equation, Vol [9 Pazy, A. Semigroup of Linear Operator and Application to Partial Differential Equation. New York: Springer Verlag [1 Quang, Y. Delay Differential Equation with Application in Population Dynamic New York: Academic Pre [11 Byzewki, L. Exitence, uniquene and aymptotic tability of olution of abtract nonlocal Cauchy problem. Dynamic Sytem and Application : [1 Balachandran,. and Chandraekaran, M. Exitence of olution of a delay differential equation with nonlocal condition. Indian J. Pure Appl. Math : [13 Lin, Y and Liu, J.. Semilinear integrodifferential equation with nonlocal Cauchy problem. Nonlinear Anal : [14 Balachandran, and Park, J. Y. Exitence of a mild olution of a functional integrodifferential equation with nonlocal condition. Bull. orean Math. Soc : [15 Jain, R. S. and Dhakne, M. B. On ome qualitative propertie of mild olution of nonlocal emilinear functional differential equation. Demontratio Mathematica 14. XLVII 4. [16 Siddiqi, A.. Functional Analyi with Application. New Delhi: ata McGraw-ill Publihing ltd [17 umar, and umar, R. Nonlocal Cauchy problem for Sobolev type mixed Volterra- Fredholm functional integrodifferential equation. Journal of Phyical Science :

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