 Advances in Pure Mathematics, 2012, 2, 109-113 http://dx.doi.org/10.4236/apm.2012.22015 Published Online March 2012 (http://www.SciRP.org/journal/apm) Uniformly Stable Positive Monotonic Solution of a Nonlocal Cauchy Problem A. M. A. El-Sayed1, E. M. Hamdallah1, Kh. W. Elkadeky2 1Faculty of Science, Alexandria University, Alexandria, Egypt 2Faculty of Science, Garyounis University, Benghazi, Libya Email: {amasayed, emanhamdalla}@hotmail.com, k-welkadeky@yahoo.com Received October 9, 2011; revised December 7, 2011; accepted December 30, 2011 ABSTRACT In this paper, we study the existence of a uniformly stable positive monotonic solution for the nonlocal Cauchy problem = ,, 0, tft xtt T =1 jj j bx with the nonlocal condition where 1 = ,x m 0,0, .aT j Keywords: Nonlocal Cauchy Problem; Local and Global Existence Nondecreasing Positive Solution; Continuous Dependence; Lyapunov Uniformly Stability 1. Introduction Problems with non-local conditions have been extensi- vely studied by several authors in the last two decades. The reader is referred to (see [1-14] and [15-18]) and re- ferences therein. Here we are concerned with the nonlocal Cauchy pro- blem = ,, 0,, tftx tt T =1 0. mm j jj b (1) 1 =1 = , 0,0,, and jj j bxxa T (2) Let be the class of all continuous functions de- fined on 0,, <TT with the norm [0, ] sup tT =, . xtx X Let be the class of all continuous functions de- fined on Y 0,, <tT T with the equivalent norm 0 [0, ] sup Nt t tT =, , extxY ,, and mN where is positive arbitrary. 0 =max, =1,2, j tj Here we firstly study, in , the local existence of the solution of the problem (1)-(2) and the continuous de- pendence of the parameter , will be proved. Secondly, we study, in , the global existence and Lyapunov uniform stability of the solution of the pro- blem (1)-(2). 2. Integral Equation Representation Consider the nonlocal Cauchy problem (1)-(2). Y Let :0, TR R is continuous and satisfies the Lipschitz condition ,,, >0, for all , ftxftykxyk xy R 100 =1 = ,d ,, d mt j j j (3) Lemma 2.1. The solution of the nonlocal Cauchy problem (1)-(2) can be expressed by the integral equation tBxbfsxssfsxs s (4) 1 =1 = m j j Bb 0 =0 , d t . where Proof. Integrating the Equation (1), we obtain .tx fsxss = (5) t Let in (5), we obtain 0 =0, d, j j xfsxss 0 =1=1 =1 = 0 , d. mmm j jj jj jjj bxbxbfsx ss 10 =1 0= ,d mj j j (6) and (7) Substitute from (2) into (7), we obtain .Bxbfsxs s (8) C opyright © 2012 SciRes. APM  A. M. A. EL-SAYED ET AL. 110 Substitute from (8) into (5), we obtain 100 =1 = ,d mt j j j ,d. tBxb fsxss fsxss Corolla ry 2 .1. The solution of the integral Equation (4) is nondecreasing. Proof. Let be a solution of the integral Equation (4), then for we have 12 <,tt d, d , d 1 11 00 =1 2 100 =1 2 =, < ,d = , mt j j j mt j j j tBx bfsxss Bxbfsxss xt fsxss fsxs s which proves that the solution of the integral Equa- tion (4) is nondecreasing. Corolla ry 2. 2. Let be satisfies (3). The solution of the integral Equation (4) is positive for ,taT. Proof. Let be a solution of the integral Equation (4), and , for 1>0x , taT d, < j , we have 00 , d , t j sxssf sx s st 00 =1 =1 , d ,d. mm t j jj jj bfsxssbfsxss and Multiplying by 1 =1 = m j j Bb 00 =1 =1 0 , d , d = , d mm t j jj jj t Bbfsxs sBb fsxs s fsxs s , we obtain and the solution of the integral Equation (4) is po- sitive for ,taT. This complete the proof. ■ 3. Local Existence of Solution Theorem 3.1. Let be satisfies the Lipschitz condition. If 1 =1 <1 m j j Tk Bb then the nonlocal Cauchy problem (1)-(2) has a unique nondecreasing positive so- lution. Proof. Define the operator :0, 0,TC TC T 100 =1 = ,d ,d mt j j j TxtBxbfsxssfs xss by . (9) Let ,0,yC T 00 00 =1 00 =,d,d,d ,d = ,,d,, d, mm tt jj jj jj mt j fsxs sfsxs sBbfsys sfsys s Bbfsxs fsyssfsxs fsys s , then =1 TxtTytBb =1 j j 00 =1 00 =1 =1 =1 d d dd sup sup 1 t j j t j jtI tI j mm jj jj bxsysskx sy ss kB bxtytskxtyts kTBbxy kTxykTBbxyKxy m j m TxtTytk B but if =1 =1 <1, m j j KkTB b then we get ,TxTyK xy which proves that the map :0, 0,TC TC T is con- traction. Applying the Banach contraction fixed point theorem we deduce that the integral Equation (4) has a unique solution 0, CT. To complete the proof, we prove that the integral Equation (4) satisfies nonlocal problem (1)-(2). Differentiating (4), we get =, . tftxt = (10) t Let in (4), we obtain 10 =1 0 = ,d , d, mj jj j j Bxbfsxs s fsxs s 1 = . mx then =1 jj j bx Copyright © 2012 SciRes. APM  A. M. A. EL-SAYED ET AL. 111 This implies that there exist a unique nondecreasing positive solution 0, CT of the nonlocal Cauchy problem (1)-(2), This complete the proof. ■ 4. Continuous Dependence of the Solution Consider the nonlocal Cauchy problem 1 =1 = ,, = , and m jj j j xtftxt t Pbx x 0, , 0,0, . T a T 1 if Definition 4.1. The solution of the nonlocal Cauchy problem (1)-(2) continuously dependence on 11 >0, ()>0, suchthat <, then < xx xt xt where t P is the solution of the nonlocal Cauchy pro- blem . Now we have the following theorem Theorem 4.1. The solution of the nonlocal Cauchy problem (1)-(2) continuously dependence on 1 . , Proof. Let txt P 11 00 =1 =1 0 =, d,d ,, d mm jj jj jj t are the solutions of (1)-(2) and respectively. Then we can get txtBxxBbfsxssBbfsxss fsxs fsxs s 11 00 =1 00 =1 11 0 =1 ,, d ,,d d d sup sup d sup mt j j j mt j jtI tI j mj jtI tI j 11 txtBxxBbf sxsf sxssf sxsf sxss BxkBbxsxsskxsxss BxxkBbxtxts k x 0d sup t xt xts 11 11 =1 j j=1 1 mm j xBx xkTBbxxkTxxBx xk TBbj xx 1 11 . mm x x >0 11 =1 =1 11 11 j j j j kTB bxxBxxxxkTB bB Therefore, for such that 11 < xx, we can find 1 =1 =1 1m j j kT BbB such that xx , which complete the proof theo- rem. 5. Global Exist ence of Solution Theorem 5.1. Let be satisfies the Lipschitz condition, then the nonlocal Cauchy problem (1)-(2) has a unique nondecreasing positive solution. Proof. Define the operator 00 :, ,TCtT CtT by the Equation (9). Let 0 ,,yCtT 000 0 =,d ,d , d , d d, mm tt jj j TxtTytBbfsxs sfsxs sBbjfsyssfsyss s s , then =1 =1 00 =1 = ,, d ,, jj mt j j j Bbfsxs fsys sfsxs fsy 0 00 =1 dd mtt j j TxtTyt kBbxsysskxsyss 0 000 00 =1 || dd mtt Nt tNt t j j TxtTytkBbexsyskexsyss Nt t e Copyright © 2012 SciRes. APM  A. M. A. EL-SAYED ET AL. 112 ()( )( ) 0 000 0 0 =1 ()() () 00 0 0 =1 |()()| d |()()| d mt Nt tNt tNstNst j j tNt tNstNst m j j eTxtTytkBbee xsyss keexs yss kBb x y 0 00 0 =1 0 =1 =1 dd 1 | 1 tt Nt sNt s Nt tNtNt m j k m m Nt tNt Nt j j j eskxyes ee e kBbx ykxy NN kk Bbeee xyB NN 1 j bxy where =1 =1. m j j b N1< k KB N Choose large enough such that , then , TxTyK xy therefor the map 00 Applying the Banach contraction fixed point theorem we deduce that the integral Equation (4) has a unique solution :,TCt ,T CtT is contraction. 0, Ct T =, . . To complete the proof, we prove that the integral Equation (4) satisfies nonlocal problem (1)-(2). Differentiating (4), we get tftxt = (11) Let t in (4), we obtain 10 =1 0 = , mj jj j j , d d, Bxb fsxs fsxs s s 1 = . jj bx x then =1 m j This implies that there exist a unique nondecreasing positive solution 0 ,CtT of the nonlocal Cauchy problem (1)-(2), This complete the proof. ■ 6. Lyapunov Uniform Stability of the Solution Consider here the nonlocal Cauchy problem 0 10 =1 = ,, ,, = , and 0,,. m jj j j xtftxt t tT Pbxxa tT Definition 6.1. The solution of the nonlocal Cauchy problem (1)-(2) is uniform stable, if >0, >0, such that 11 <, then <.xx xtxt where t P is the solution of the nonlocal Cauchy pro- blem . Now we have the following theorem Theorem 6.1. The solution of the nonlocal Cauchy problem (1)-(2) is uniformly stable. , Proof. Let txt P 11 000 =1 =1 =,d,d ,, d mm t jj jj jj are the solutions of (1)-(2) and respectively. Then we can get txtBxxB bfsxssB bfsxssfsxsfsxss 11 00 =1 0 11 00 =1 ,,d,,d d d mt j j j mtt j j txtBxxBbfsxsf sxssf sxsf sxss Bx xkBbxsxsskxsxs s 0 00 000 11 0 =1 000 d d mt t tNt tNs tNs t j j Nt tNs tNs t BxxkBbeextxts keextxts 0 Nt t N t extxte Copyright © 2012 SciRes. APM  A. M. A. EL-SAYED ET AL. 113 0 11 00 =1 0 11 =1 0 11 =1 11 =1 d 1 1 mtt Nt s j j Nt t d 1 Nt s tNt m j j mNt tNt Nt j j m j j xBxxkBbxxeskxx ee BxxkBbx xkx k BxxBb eee N k Bx xBb N e s e x NN x x xx 1 11 11 =1 m j j k xB Bb N xx Therefore, 11 <,xx <,xx which com- plete the proof of theorem. REFERENCES [1] B. Ahmad and J. J. Nieto, “Existence of Solution for Non- local Boundary Value Problems of Higher-Order Nonlin- ear Fractional Differential Equations,” Abstract and Ap- plied Analysis, Vol. 2009, 2009, pp. 1-9. doi:10.1155/2009/494720 [2] M. Benchohra, J. J. Nieto and A. Ouahab, “Second-Order Boundary Value Problem with Integral Boundary Condi- tions,” Boundary Value Problems, Vol. 2011, 2011, pp. 1-9. doi:10.1155/2011/260309 [3] A. Boucherif, “First-Order Differential Inclusions with Nonlocal Initial Conditions,” Applied Mathematics Letters, Vol. 15, No. 4, 2002, pp. 409-414. doi:10.1016/S0893-9659(01)00151-3 [4] A. Boucherif, “Nonlocal Cauchy Problems for First-Order Multivalued Differential Equations,” Electronic Journal of Differential Equations, Vol. 2002, No. 47, 2002, pp. 1-9. [5] A. Boucherif and R. Precup, “On The Nonlocal Initial Value Problem for First Order Differential Equations,” Fixed Point Theory, Vol. 4, No 2, 2003, pp. 205-212. [6] M. Benchohra, E. P. Gatsori and S. K. Ntouyas, “Exis- tence Results for Seme-Linear Integrodifferential Inclu- sions with Nonlocal Conditions,” Rocky Mountain Jour- nal of Mathematics, Vol. 34, No. 3, 2004, pp. 833-848. doi:10.1216/rmjm/1181069830 [7] Y. K. Chang and J. J. Nieto, “Existence of Solutions for Impulsive Neutral Integro-Differential Inclsions with Non- local Initial Conditions via Fractional Operators, Numeri- cal Functional Analysis and Optimization, Vol. 30, No. 3- 4, 2009, pp. 227-244. doi:10.1080/01630560902841146 [8] R. F. Curtain and A. J. Pritchard, “Functional Analysis in Modern Applied Mathematics,” Academic Press, New York, 1977. [9] A. M. A. El-Sayed and Sh. A. Abd El-Salam, “On the Stability of a Fractional-Order Differential Equation with Nonlocal Initial Conditin,” Electronic Journal of Differ- ential Equations, Vol. 2009, No. 29, 2008, pp. 1-8. [10] A. M. A. El-Sayed and Kh. W. Elkadeky, “Caratheodory Theorem for a Nonlocal Problem of the Differential Equa- tion =, ftx ,” Alexandria Journal of Mathematics, Vol. 1, No. 2, 2010, pp. 8-14. [11] A. M. A. El-Sayed and Kh. W. Elkadeky, “Solutions of a Class of Nonlocal Problems for the Differential Inclusion , tF txt ,” Applied Mathematics and Informa- tion Sciences, Vol. 5, No. 2, 2011, pp. 413-421. [12] A. M. A. El-Sayed, E. M. Hamdallah and Kh. W. El- kadeky, “Solutions of a Class of Deviated-Advanced Non- local Problem for the Differential Inclusion , tFtxt,” Abstract and Applied Analysis, Vol. 2011, 2011, pp. 1-9. doi:10.1155/2011/476392 [13] E. Gatsoi, S. K. Ntouyas and Y. G. Sficas, “On a Nonlo- cal Cauchy Problem for Differential Inclusions,” Abstract and Applied Analysis, Vol. 2004, 2004, pp. 425-434. doi:10.1155/S108533750430610X [14] K. Goebel and W. A. Kirk, “Topics in Metric Fixed Point Theory,” Cambridge University Press, Cambridge, 1990. doi:10.1017/CBO9780511526152 [15] A. Lasota and Z. Opial, “An Application of the Kaku- tani-Ky-Fan Theorem in the Theory of Ordinary Differ- ential Equations,” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astoronom. Phys, Vol. 13, 1955, pp. 781-786. [16] R. Ma, “Existence and Uniqueness of Solutions to First- Order Three-Point Boundary Value Problems,” Applied Mathematics Letters, Vol. 15, No. 2 2002, pp. 211-216. doi:10.1016/S0893-9659(01)00120-3 [17] S. K. Ntouyas, “Nonlocal Initial and Boundary Value Problems: A Survey,” In: A. Canada, P. Drabek and A. Fonda, Eds., Hand Book of Differential Equations, Vol. II Elsevier, New York, 2005. [18] A. Vlez-Santiago, “Quasi-Linear Boundary Value Problems with Generalized Nonlocal Boundary Conditions, Nonlinear Analysis,” Theory, Methods an d App licatio ns, Vol. 74, 2011, pp. 4601-4621. doi:10.1016/j.na.2011.03.064 Copyright © 2012 SciRes. APM
|
