Existence and Uniqueness of Solution of Integrodifferential Equation of Finite Delay in Cone Metric Space

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1 Existence and Uniqueness of Solution of Integrodifferential Equation of Finite Delay in Cone Metric Space Rupesh T. More, Shridhar C. Patekar 2,, Vijay B. Patare 3 Department of Mathematics,Arts, Commerce and Science College, Bodwad 425 3, India. 2 Department of Mathematics, Savitribai Phule Pune University, Pune 47, India. 3 Department of Mathematics, Nutan Mahavidyalaya, Sailu 4353, India. Abstract This paper, we study the existence and uniqueness of solution of Volterra integrodifferential equation of finite delay with nonlocal condition in cone metric space. The result is obtained by using the some extensions of Banach s contraction principle in complete cone metric space and also provide example to demonstrate of main result. Keywor: Cone metric Space,Banach Contraction Principle,Volterra integral equation 2 Mathematics Subject Classification:- 47H,primary 45N5,34A2.. INTRODUCTION The purpose of this paper is study the existence and uniqueness of solution of Volterra integrodifferential equation with nonlocal condition in cone metric space of the form: x t = Atxt f t,xt,xt k s,xs, t J =,b. xt = ψt t <..2 x gx = x,.3 where At is a bounded linear operator on a Banach space X with domain DAt, the unknown x takes values in the Banach space X; f : J X X X, k : J X X,g : CJ,X X are appropriate continuous functions and x is given element of X. ψt is a continuous function for t <, lim ψt exists, for t which we denote by ψ = c. if we observed a function xt which is unable to define as solution for t <. Hence, we have to impose some condition, for example the condition.2. We note that, if t <, the problem is reduced to integrodifferential equation x t = Atxt f t,xt,ψt ks, xs, Corresponding author with initial condition x gx = x. Here, it is essential to obtain the solutions of..3 for t < b. H.L. Tidke and R.T. More 4 5 studied model in cone metric space in the form x t = Atxt f t,xt k s,xs, t J =,b.4 x gx = x,.5 we inspired by their work and extend.4.5by adding finite delay. The objective of present paper is to study the existence and uniqueness of solution of the integrodifferential equation..3 in cone metric space The paper is organized as follows: we discuss the preliminaries. we dealt with study of existence and uniqueness of solution of integrodifferential equation with nonlocal condition in cone metric space. Finally we give example to illustrate the application of our result. 2. PRELIMINARIES Let us recall the concepts of the cone metric space and we refer the reader to, 2, 3, 4, 5, 6 for the more details. Let E be a real Banach space and P is a subset of E. Then P is called a cone if and only if,. P is closed, nonempty and P {}; 2. a,b R, a,b, x,y P ax by P; 3. x P and x P x =. For a given cone P E, we define a partial ordering relation with respect to P by x y if and only if y x P. We shall write x < y to indicate that x y but x y, while x << y will stand for y x intp, where intp denotes the interior of P. The cone P is called normal if there is a number K > such that x y implies x K y, for every x,y E. The least positive number satisfying above is called the normal constant of P. 646

2 In the following we always suppose E is a real Banach space, P is a cone in E with intp φ, and is partial ordering with respect to P. Definition 2. Let X be a nonempty set. Suppose that the mapping d : X X E satisfies: d dx,y for all x,y X and dx,y = if and only if x = y; d 2 dx,y = dy,x, for all x,y X; d 3 dx,y dx,z dz,y, for all x,y,z X. Then d is called a cone metric on X and X,d is called a cone metric space. The concept of cone metric space is more general than that of metric space. The following example is a cone metric space, see?. Example 2.2 Let E = R 2, P = {x,y E : x,y }, X = R, and d : X X E such that dx,y = x y,α x y, where α is a constant. Then X,d is a cone metric space. Definition 2.3 Let X be a an ordered space. A function Φ : X X is said to a comparison function if for every x,y X, x y, implies that Φx Φy, Φx x and lim n Φ n x =, for every x X. Example 2.4 Let E = R 2, P = {x,y E : x,y }. It is easy to check that Φ : E E, with Φx,y = ax,ay, for some a, is a comparison function. Also if Φ,Φ 2 are two comparison functions over R, then Φx,y = Φ x,φ 2 y is also a comparison function over E. 3. EXISTENCE AND UNIQUENESS OF SOLUTION Let X is a Banach space with norm. Let B = CJ,X be the Banach space of all continuous functions from J into X endowed with supremum norm x = sup{ xt : t J}. Let P = {x,y : x,y } E = R 2 be a cone and define d f,g = f g,α f g, for every f,g B. Then it is easily seen that B,d is a cone metric space. Definition 3. The function x B satisfies the integral equation case I :for t < case II :for t < b xt = x gx As f s,xs,xs k τ,xτ dτ, 3. xt = x gx As f s,xs,xs k τ,xτ dτ, As f s,xs,xs k τ,xτ dτ, 3.2 is called the mild solution of the equation..3. We need the following lemma for further discussion: Lemma Let X,d be a complete cone metric space, where P is a normal cone with normal constant K. Let f : X X be a function such that there exists a comparison function Φ : P P such that for every x,y X. Then f has a unique fixed point. d f x, f y Φdx,y, We list the following hypotheses for our convenience: H At is a bounded linear operator on X for each t J, the function t At is continuous in the uniform operator topology and hence there exists a constant K such that K = sup At. t J 646

3 H 2 Let Φ : R 2 R 2 be a comparison function. i There exists continuous function p, p 2 : J R such that case I :for t < f t,xt,ψt f t,yt,ψt,α f t,xt,ψt f t,yt,ψt case II :for t < b p tφ dx, y, f t,xt,xt f t,yt,yt,α f t,xt,xt f t,yt,yt H 3 sup t J for every t J and x,y X. ii There exists continuous function q : J R such that for every t J and x,y X. iii There exists a positive constant G such that for every x,y X. { G K p s p 2 s p 2 tφ dx, y, kt,x kt,y,α kt,x kt,y qtφ dx, y, gx gy,α gx gy GΦ dx, y, qτdτ =. Theorem 3.3 Assume that hypotheses H H 3 hold. Then the evolution equation..3 has a unique solution x on J. Proof: The operator F : B B is defined by case I :for t < case II :for t < b Fxt = x gx As f s,xs,ψs k τ,xτ dτ. 3.3 Fxt = x gx By using the hypotheses H H 3, we have As f s,xs,xs As f s,xs,xs k τ,xτ dτ k τ,xτ dτ

4 case I :for t < Fxt Fyt,α Fxt Fyt gx gy As f s,xs,ψs f s,ys,ψs kτ,xτ kτ,yτ dτ, α gx gy α As f s,xs,ψs f s,ys,ψs kτ,xτ kτ,yτ dτ gx gy,α gx gy K f s,xs,ψs f s,ys,ψs,α f s,xs,ψs f s,ys,ψs K kτ,xτ kτ,yτ,α kτ,xτ kτ,yτ dτ GΦ x y,α x y K qτφ x y,α x y GΦ Φ x y,α x y GΦ dx, y Φ { Φ dx, y G { Φ dx, y Φ dx, y, G K p sφ xs ys,α xs ys xτ yτ,α xτ yτ dx, y K p s K K p s p s qτdτ K p s p 2 s dτ qτdτ qτdτ qτdτ

5 Now case II: for t < b Fxt Fyt,α Fxt Fyt gx gy As f s,xs,ψs f s,ys,ψs As f s,xs,xs f s,ys,ys α gx gy α As f s,xs,ψs f s,ys,ψs α α As kτ,xτ kτ,yτ dτ f s,xs,xs f s,ys,ys GΦ dx, y K p 2 sφ GΦ dx, y K dx, y p sφ dx, y kτ,xτ kτ,yτ dτ, kτ,xτ kτ,yτ dτ qτφ qτφ dx, y K p s p 2 s Φ p K s p 2 s Φ dx, y { Φ dx, y G Φ dx, y, K p s p 2 s dτ dx, y qτφ kτ,xτ kτ,yτ dτ dx, y dτ qτφ dx, y dτ dx, y qτdτ dτ 3.6 for every x,y B. This implies that d Fx,Fy Φ dx, y, for every x, y B. Now an application of Lemma 3.2, the operator has a unique point in B. This means that the equation..3 has unique solution. This completes the proof of the Theorem EXAMPLE In this section, we give an example to illustrate the usefulness of our result discussed in previous section. Let us consider the following evolution equation: where dx dt = 2 6 e t xt f t,xt,xt x f t,xt,xt = f t,xt,xt = sxs, t J =,2, 4. 2 x 8 x = x, 4.2 te t xt 9 e t xt, f or t < 2te t xt 9 e t xt, f or t <

6 Therefore, we have At = 2 6 e t, t J kt,xt = txt 2, t,x J X te t xt ψt = 9 e t xt, gx = x 8 x, x X. Now for x,y CJ,X and t J, we have case I :for t < f t,xt,xt f t,yt,yt,α f t,xt,xt f t,yt,yt = te t 9 e t xt xt yt xt,α yt xt yt yt = te t xt yt 9 e t xt yt,α xt yt xt yt te t 9 e t xt yt,α xt yt te t 9 e t x y,α x y te t 9 e t dx,y t Φ dx, y, 4.3 where p t = t, which is continuous function of J into R and a comparison function Φ : R 2 R 2 such that Φ dx, y = dx,y. case II :for t < b f t,xt,xt f t,yt,yt,α f t,xt,xt f t,yt,yt = 2te t 9 e t = 2te t 9 e t xt xt yt xt yt,α yt xt yt xt yt xt yt xt yt,α xt yt 2te t 9 e t xt yt,α xt yt 2te t 9 e t x y,α x y t dx,y 5 Φ, 4.4 where p 2 t = t 5, which is continuous function of J into R and a comparison function Φ : R 2 R 2 such that Φ dx, y = dx,y. 6465

7 Similarly, we can have kt,x kt,y,α kt,x kt,y = txt 2 tyt 2,α txt 2 tyt 2 t xt yt,α xt yt 2 x y,α x y t 2 t 2 dx,y t 2 Φ dx, y, 4.5 where qt = t 2, which is continuous function of J into R and the comparison function Φ defined as above. Also, x y gx gy,α gx gy 8 8 x 8 y,α x y 8 x 8 y 8 x y,α x y 64 x y,α x y 8 dx,y 8 Φ, 4.6 where G = 8, and the comparison function Φ defined as above. Hence the condition H hol with K = 2 6. Moreover, { { sup G K p s p 2 s qτdτ = sup t J t J 8 2 s 6 s 5 τ 2 dτ { = sup t J 8 2 3s 6 s2 4 { = sup t J 8 2 3t t3 } 2 { = } 2 { = } 5 { = 8 2 } 6 5 = 8 7 = Note that we define a partial ordering relation with respect to P by x y if and only if y x P. We shall write x < y to indicate that x y but x y, while x << y will stand for y x intp, where intp denotes the interior of P. Since all the conditions of Theorem 3.3 are satisfied, the problem has a unique solution x on J. REFERENCES M. Abbas and G. Jungck; Common fixed point results for noncommuting mappings without continuity in cone metric spaces, Journal of Mathematical Analysis and Applications, Vol. 34, 28, No., J. Banas; Solutions of a functional integral equation in BCR, International Mathematical Forum, 26, No. 24, H. L. Tidke and M. B. Dhakne, Existence and uniqueness of solution of differential equation of second order in cone metric spaces, Fasciculi Mathematici, Nr. 45, 2, 6466

8 H. L. Tidke and R.T. More, Existence and uniqueness of solution of integrodifferential equation in cone metric spaces, SOP TRANSCATIONS ON APPLIED MATHEMATICS, In press, 24, ISSN Print H. L. Tidke and R.T. More,Existence And Osgood Type Uniqueness Of Mild Solutions Of Nonlinear Integrodifferential Equation With Nonlocal Condition, International Journal of Pure and Applied Mathematics,25,Volume 4 No. 3, ISSN: 3-88 printed version; ISSN: on-line version 6 P. Raja and S. M. Vaezpour; Some extensions of Banach s contraction principle in complete cone metric spaces, Fixed Point Theory and Applications, Volume 28, Article ID , pages. 6467