In this paper, we study the existence and uniqueness of solution of inhomogeneous semilinear evolution equation 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

More, R. T.

Tidke, Haribhau Laxman

English

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ISSN 2347-1921 1044 | P a g e                                                     F e b r u a r y  2 0 ,  2 0 1 4  Existence And Uniqueness Of Solution Of Inhomogeneous  Semilinear Evolution Equation With Nonlocal Condition1H. L. Tidke, 2R. T. More 1Department of Matheamtics,  North Maharasjhtra University, Jalgaon-425 001, India tharibhau@gmail.com 2 Department of Mathematics,  Arts, Commerce and Science College, Bodwad, Jalgaon-425 310, India  rupeshmore9@yahoo.com    ABSTRACT In this paper, we study the existence and uniqueness of solution of inhomogeneous semilinear evolution equation 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.  Keywords: Inhomogeneous semilinear evolution equation; Cone metric space; Fixed point theorem; Nonlocal condition. Mathematics Subject  Classification: 34L30, 58D25, 65J08, 47J35, 74H10, 34B10.                       Council for Innovative Research Peer Review Research Publishing System Journal: Journal of Advances in Matheamtics Vol 6, No.3 editor@cirworld.com www.cirworld.com, member.cirworld.com       ISSN 2347-1921 1045 | P a g e                                                     F e b r u a r y  2 0 ,  2 0 1 4  1. 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: ][0,=)),(,()()(=)( bJttxtftxtAtx                                                                     (1) ,=)((0) 0xxgx                                                                                                                 (2) where )(tA  is a bounded linear operator on a Banach space X  with domain ))(( tAD , the unknown )(x  takes values in the Banach space X ; XXJf : , XXJCg ),(:  are appropriate continuous functions and 0x  is given element of X . Many authors have been studied the problems of existence, uniqueness, continuation and other properties of solutions of these type or special forms of the equations (1)–(2) are studied by different techniques, for example, see [3, 4, 5, 8, 10, 13] and the references given therein. The objective of the present paper is to study the existence and uniqueness of solution of the evolution equation (1.1)–(1.2) under the conditions in respect of the cone metric space and fixed point theory. Hence we extend and improve some results reported in [2, 8, 10, 11, 13]. The paper is organized as follows: Section 2, we discuss the preliminaries. Section 3, we dealt with study of existence and uniqueness of solution of inhomogeneous evolution equation with nonlocal condition in cone metric space. Finally in Section 4, 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 [1, 6, 7, 9, 12] 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,       1.  P  is closed, nonempty and {0}P ;     2.  ba, , 0, ba , Pyx ,    Pbyax  ;     3.  Px  and Px    0=x .  For a given cone EP , we define a partial ordering relation   with respect to P  by yx   if and only if Pxy  . We shall write yx <  to indicate that yx   but yx  , while yx <<  will stand for intPxy  , where intP  denotes the interior of P . The cone P  is called normal if there is a number 0>K  such that yx 0  implies  yKx  , for every Eyx , . 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 , and   is partial ordering with respect to P . Definition 2.1   Let X  be a nonempty set. Suppose that the mapping EXXd :  satisfies:     1.  ),(0 yxd  for all Xyx ,  and 0=),( yxd  if and only if yx = ;     2.  ),(=),( xydyxd , for all Xyx , ;     3.  ),(),(),( yzdzxdyxd  , for all Xzyx ,, .  Then d  is called a cone metric on X  and ),( dX  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 [11].  Example 2.2 Let 2= E  , 0},:),{(=  yxEyxP , =X , and EXXd :  such that |)||,(|=),( yxyxyxd   , where 0  is a constant. Then ),( dX  is a cone metric space.  ISSN 2347-1921 1046 | P a g e                                                     F e b r u a r y  2 0 ,  2 0 1 4  Definition 2.3   Let X  be a an ordered space. A function XX  :  is said to a comparison function if for every Xyx , , yx  , implies that )()( yx  , xx  )(  and 0=)(lim  xnn  , for every Xx .  Example 2.4  Let 2= E  , 0},:),{(=  yxEyxP . It is easy to check that EE : , with ),(=),( ayaxyx , for some (0,1)a  is a comparison function. Also if 21,  are two comparison functions over  , then ))(),((=),( 21 yxyx   is also a comparison function over E .  3.  Existence and uniqueness of solution  Let X  is a Banach space with norm  . Let ),(= XJCB  be the Banach space of all continuous functions from J  into X  endowed with supremum norm  }.:)({sup= Jttxx     Let 2=0},:),{(= EyxyxP   be a cone and define ),(=),(    gfgfgfd  , for every Bgf , . Then it is easily seen that ),( dB  is a cone metric space. Definition 3.1 The function Bx  satisfies the integral equation   JtdssxsfsAxgxtxt  ,))(,()()(=)( 00  (3) is called the solution of the evolution equation (1)–(2). We need the following lemma for further discussion: Lemma 3.2 [11]  Let ),( dX  be a complete cone metric space, where P  is a normal cone with normal constant K . Let XXf :  be a function such that there exists a comparison function PP :  such that  )),,(())(),(( yxdyfxfd   for every Xyx , . Then f  has a unique fixed point. We list the following hypotheses for our convenience: (H1) )(tA  is a bounded linear operator on X  for each Jt , the function )(tAt  is continuous in the uniform operator topology and hence there exists a constant K  such that   .)(sup=  tAKJt (H2) There exist continuous function Jp :  and a comparison function 22:    such that  )),,(()()),(),(,),(),(( yxdtpytfxtfytfxtf     and for positive constant G , such that  )),,(())()(,)()(( yxdGygxgygxg     for every Jt  and Xyx , . (H3)  1.=])([sup0dssKpGtJt    Theorem 3.3  Assume that hypotheses )()( 31 HH   hold. Then the evolution equation (1.1)–(1.2) has a unique solution x  on J .  Proof: The operator BBF :  is defined by                       .,))(,()()(=)(00 JtdssxsfsAxgxtFxt                                                     (4) ISSN 2347-1921 1047 | P a g e                                                     F e b r u a r y  2 0 ,  2 0 1 4  By using the hypotheses )()( 31 HH  , we have  ))()(,)()((  tFytFxtFytFx   ,))(,())(,()()()((0dssysfsxsfsAygxgt    )))(,())(,()()()(0dssysfsxsfsAygxgt       ))()(,)()((  ygxgygxg    dssysfsxsfsysfsxsfKt)))(,())(,(,))(,())(,((0              dssysxsysxsKpyxyxGt))()(,)()(()(),(0   dssKpyxyxyxyxGt)(),(),(0   dssKpyxdyxdGt)()),(()),((0  ])())[,((0dssKpGyxdt    )),,(( yxd                                                                                         (5) for every Byx , . This implies that )),((),( yxdFyFxd  , for every Byx  , . Now an application of Lemma 3.2, the operator has a unique point in B . This means that the equation (1)–(2) has unique solution. This completes the proof of the Theorem 3.3. 4.  Application 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:  ,[0,1],=,))()(1(9)()(8140= XxJttxetxtetxedtdxttt                                                               (6) ,=8(0) 0xxxx                                                                        (7)  Therefore, we have                           ,8140=)( tetA    XJxttxetxtetxtftt),(,))()(1(9)(=))(,(           .,8=)( Xxxxxg  Now for ),(, XJCyx   and Jt , we have  ))(1)()(1)(,)(1)()(1)((9=))()(,)()(( tytytxtxtytytxtxetetFytFxtFytFxtt      )))())(1((1)()(,))())(1((1)()((9= tytxtytxtytxtytxetett  ISSN 2347-1921 1048 | P a g e                                                     F e b r u a r y  2 0 ,  2 0 1 4      ))()(,)()((9 tytxtytxetett      ),(9  yxyxetett      ),(9yxdetett     )),,((10yxdt                                                                         (8)  where 10=)(ttp , which is continuous function of J  into   and a comparison function 22:    such that ),(=)),(( yxdyxd . Similarly, we can have            )))(8(8,))(8(88())()(,)()((yxyxyxyxygxgygxg                   ),(648 yxyx                                                                           ),(81   yxyx                                                                          )),,((81yxd                                                                     (9) where 81=G , and the comparison function   defined as above. Hence the condition )( 1H  holds with 8140=K . Moreover,   ]10814081[sup=])([sup00dssdsspKGtJttJt                ]20814081[sup=2tJt ]8781[sup=2tJt 1.=]8781[=                                                                                 (10) Since all the conditions of Theorem 3.3 are satisfied, the problem (6)–(7) has a unique solution x  on J . REFERENCES [1] Abbas, M. and Jungck, G. 2008.  Common fixed point results for noncommuting mappings without continuity in cone metric spaces, Journal of Mathematical Analysis and Applications, Vol. 341, No.1, 416-420. [2] Banas, J. 2006.  Solutions of a functional integral equation in BC , International Mathematical Forum, 1 (2006), No. 24, 1181-1194. [3]  Burton, T. A. 1983.  Volterra Integral and Differential Equations, Academic Press, New York, (1983). ISSN 2347-1921 1049 | P a g e                                                     F e b r u a r y  2 0 ,  2 0 1 4  [4] Coddington, E. A. 2003. An Introduction to Ordinary Differential Equations, Prentice-Hall of India, Private Limited, New Delhi, (2003). [5] Deo, S. G., Lakshmikantham, V. and Raghavendra , V. 2003.  Text Book of Ordinary Differential Equations, Tata McGraw-Hill Publishing Company Limited, New Delhi,   Edition, (2003). [6] Huang, L. G. and Zhang, X. 2007. Cone metric spaces and fixed point theorems of contractive mappings, Journal of Mathematical Analysis and Applications, Vol. 332, (2007), No.2, 1468-1476. [7] Ilic, D. and Rakocevic, V. 2008. Common fixed points for maps on cone metric space, Journal of Mathematical Analysis and Applications, Vol. 341, (2008), No.2, 876-882. [8] Karoui, A. 2005. On the existence of continuous solutions of nonlinear integral equations, Applied Mathematics Letters , 18(2005), 299-305. [9] Kwong, M. K, 2008. On Krasnoselskii’s cone fixed point theorems, Fixed Point Theory and Applications, Volume 2008, Article ID 164537, 18 pages. [10] Pazy, A. 1983. Semigroups of Linear Operators and applications to Partial Differential Equations, Springer-Verlag, New York, (1983). [11] Raja,  P. and Vaezpour, S. M. 2008. Some extensions of Banach’s contraction principle in complete cone metric spaces, Fixed Point Theory and Applications, Volume 2008, Article ID 768294, 11pages. [12] Rezapour, Sh. and Hamlbarani, R. 2008.  Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings", Journal of Mathematical Analysis and Applications, Vol. 345, (2008), No.2, 719-724. [13] Tidke,  H. L. and Dhakne, M. B. 2010. Existence and uniqueness of solution of differential equation of second order in cone metric spaces, Fasciculi Mathematici, Nr. 45, (2010), 121-131. 

Existence and uniqueness of solution of inhomogeneous semilinear evolution equation with nonlocal condition

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Existence and uniqueness of solution of inhomogeneous semilinear evolution equation with nonlocal condition

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