Study of Existence and uniqueness of solution of abstract nonlinear differential equation of finite delay Rupesh T. More and Vijay B. Patare Department of Mathematics, Arts, Commerce and Science College, Bodwad, Jalgaon-425 3, India rupeshmore82@gmail.com Department of Mathematics, Nutan Mahavidyalaya,Sailu, Sailu-4353, India vijaypatare3@gmail.com Abstract In this paper, we study the existence and uniqueness of solution of differential 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. Introduction The purpose of this paper is study the existence and uniqueness of solution of inhomogeneous semilinear evolution equation with nonlocal condition in cone metric space of the form: x t = Axt + f t, xt, xt, t J = [, b. xt = ψt t <..2 x + gx = x,.3 where A is an infintesimal generator of strongly continuous semigroup of bounded linear operator Tt in X with domain DA, the unknown x takes values in the Banach space X; f : J X 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 which t 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 = Axt + ft, xt, ψt with initial condition x + gx = x...3 for t < b. Here, it is essential to obtain the solutions of 28
The objective of the present paper is to study the existence and uniqueness of solution of the evolution equation..3 under the conditions in respect of the cone metric space and fixed point theory. Hence we extend and improve some results reported in [6. The paper is organized as follows: we discuss the preliminaries. we dealt with study of existence and uniqueness of solution of inhomogeneous evolution equation with nonlocal condition in cone metric space. 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. 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 [?. 29
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 df, 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 xt = T t[x gx + xt = T t[x gx + +T t[x gx + T t sfs, xs, xs ds, 3. T t sfs, xs, xs ds is called the mild solution of the evolution equation..3. T t sfs, xs, xs ds, 3.2 We need the following lemma for further discussion: Lemma 3.2 [5 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 dfx, fy Φdx, y, for every x, y X. Then f has a unique fixed point. 22
We list the following hypotheses for our convenience: H A is an infintesimal generator of strongly continuous semigroup of bounded linear operator Tt in X for each t J, and hence there exists a constant K such that K = sup T t. t J H 2 Let Φ : R 2 R 2 be a comparison function. i There exists continuous function p, p 2 : J R + such that ft, xt, ψt ft, yt, ψt, α ft, xt, ψt ft, yt, ψt p tφ dx, y, ft, xt, xt ft, yt, yt, α ft, xt, xt ft, yt, yt for every t J and x, y X. [ H 3 sup KG + t J p 2 tφ dx, y, gx gy, α gx gy GΦ dx, y, K[p s + p 2 sds =. Theorem 3.3 Assume that hypotheses H H 3 hold. Then the evolution equation..2 has a unique solution x on J. Proof: The operator F : B B is defined by F xt = T t[x gx + F xt = T t[x gx + T t sfs, xs, xs ds, 3.3 T t sfs, xs, xs ds 22
+T t[x gx + T t sfs, xs, xs ds, 3.4 By using the hypotheses H H 3, we have F xt F yt, α F xt F yt T t gx gy + T t s fs, xs, ψs fs, ys, ψs ds, α T t gx gy + α T t s fs, xs, ψs fs, ys, ψs ds T t gx gy, α gx gy + K fs, xs, ψs fs, ys, ψs, α fs, xs, ψs fs, ys, ψs ds KGΦ x y, α x y + Kp sφ xs ys, α xs ys ds KGΦ x y, α x y + Φ x y, α x y KGΦ dx, y + Φ [ Φ dx, y KG + [ Φ dx, y Φ dx, y, KG + dx, y Kp sds Kp sds K[p s + p 2 sds By using the hypotheses H H 3, we have F xt F yt, α F xt F yt T t gx gy + T t s [ fs, xs, ψs fs, ys, ψs ds + α T t gx gy + α T t s [ fs, xs, ψs fs, ys, ψs ds + T t gx gy, α gx gy + Kp sds 3.5 fs, xs, xs fs, ys, ys ds, fs, xs, xs fs, ys, ys ds K fs, xs, ψs fs, ys, ψs, α fs, xs, ψs fs, ys, ψs ds 222
+ K fs, xs, xs fs, ys, ys, α fs, xs, xs fs, ys, ys ds KGΦ x y, α x y KGΦ dx, y + Φ [ Φ dx, y KG + Φ dx, y dx, y [ + Φ x y, α x y [ K[p s + p 2 sds Kp s + p 2 sds + Kp sds + Kp s + p 2 sds Kp 2 sds 3.6 for every x, y B. This implies that d F x, F y 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..2 has unique solution. This completes the proof of the Theorem 3.3. 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., 46-42. [2 J. Banas; Solutions of a functional integral equation in BCR +, International Mathematical Forum, 26, No. 24, 8-94. [3 H. L. Tidke and R.T. More, Existence and uniqueness of solution of integrodifferential equation in cone metric spaces, SOP TRANSCATIONS ON APPLIED MATHEMAT- ICS, In press, 24, ISSN Print2373-8472. [4 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. 3,437-46 ISSN: 3-88 printed version; ISSN: 34-3395 on-line version [5 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 768294, pages. [6 H. L. Tidke and R.T. More,On an abstract nonlinear differential equations with nonlocal condition, Proceeding of National Conference on Recent Applications of Mathematical Tools in Science and Technology RAMT-24 organized by Department of Physics, Chemistry & Mathematics, Government College of Engineering, Amravati during May 8-9, 24, Session-III, 47-5. 223