 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,xtftxtt T=1 jjjbx with the nonlocal condition where 1= ,xm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,,xtftxtt T=1 0.mmjjjb (1) 1=1 = , 0,0,, andjj jbxxa T (2) Let X be the class of all continuous functions de- fined on 0,, 0, for all , ftxftykxykxy R 100=1= ,d ,, dmtjjj (3) Lemma 2.1. The solution of the nonlocal Cauchy problem (1)-(2) can be expressed by the integral equation tBxbfsxssfsxs sx (4) 1=1= mjjBb 0=0 , dt. where Proof. Integrating the Equation (1), we obtain x.tx fsxss= (5) tLet jin (5), we obtain  0=0, d, jjxxfsxss 0=1=1 =1= 0 , d.mmmjjj jjjjjbxbxbfsx ss 10=10= ,dmjjj (6) and (7) Substitute from (2) into (7), we obtain .Bxbfsxs s (8) xCopyright © 2012 SciRes. APM A. M. A. EL-SAYED ET AL. 110 Substitute from (8) into (5), we obtain  100=1= ,dmtjjj ,d.xtBxb fsxssfsxss Corolla ry 2 .1. The solution of the integral Equation (4) is nondecreasing. Proof. Let x be a solution of the integral Equation (4), then for we have 12<,ttd, d, d 11100=12100=12=, < ,d = ,mtjjjmtjjjxtBx bfsxssBxbfsxssxtfsxssfsxs s which proves that the solution x of the integral Equa- tion (4) is nondecreasing. Corolla ry 2. 2. Let f be satisfies (3). The solution of the integral Equation (4) is positive for ,taT. Proof. Let x be a solution of the integral Equation (4), and , for 1>0x, taTd, 0, ()>0, suchthat <, then < xxxt xt   xwhere 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 1x. , Proof. Let xtxtP 11 00=1 =10=, d,d ,, d mmjjjjjjt are the solutions of (1)-(2) and respectively. Then we can get  xtxtBxxBbfsxssBbfsxssfsxs fsxs s     11 00=100=111 0=1,, d ,,d d dsup sup dsupmtjjjmtjjtI tIjmjjtI tIj11txtBxxBbf sxsf sxssf sxsf sxssBxkBbxsxsskxsxssBxxkBbxtxts k    xx 0dsup txt xts11 11=1jj=11mmjxxBx xkTBbxxkTxxBx xk  TBbj xx  111.mm x x  >011=1 =111 11j jj jkTB bxxBxxxxkTB bB        Therefore, for  such that 11< xx, we can find 1=1=1 1mjjkT BbB such that xx, which complete the proof theo- rem. 5. Global Exist ence of Solution Theorem 5.1. Let f 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). xLet 0,,yCtT 000 0=,d ,d , d , d d,mmttjjjTxtTytBbfsxs sfsxs sBbjfsyssfsysss s   , then =1 =100=1 = ,, d ,,jjmtjjjBbfsxs fsys sfsxs fsy   000=1ddmttjjTxtTyt kBbxsysskxsyss    000000=1|| ddmttNt tNt tjjTxtTytkBbexsyskexsyss  Nt te Copyright © 2012 SciRes. APM A. M. A. EL-SAYED ET AL. 112  ()( )( )0000 00=1()() ()00 00=1 |()()| d |()()| d mtNt tNt tNstNstjjtNt tNstNstmjjeTxtTytkBbee xsysskeexs ysskBb x y     0000=10=1 =1 dd1 | 1 ttNt sNt sNt tNtNtmjkm mNt tNt Ntjj jeskxyesee ekBbx ykxyNNkkBbeee xyBNN     1jbxy where =1=1.mjjbN1< kKBN Choose large enough such that K, then , TxTyK xy therefor the map 00Applying the Banach contraction fixed point theorem we deduce that the integral Equation (4) has a unique solution :,TCt ,T CtT is contraction. 0,xCt T=, .. To complete the proof, we prove that the integral Equation (4) satisfies nonlocal problem (1)-(2). Differentiating (4), we get xtftxt= (11) Let jt in (4), we obtain 10=10 = , mjjjjj, dd, xBxbfsxsfsxs ss1 = .jjbx x then =1mj This implies that there exist a unique nondecreasing positive solution 0x,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 010=1 = ,, ,, = , and 0,,.mjj jjxtftxt t tTPbxxa 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 xwhere 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 xtxtP 11 000=1 =1=,d,d ,, dmmtjjjjjj are the solutions of (1)-(2) and respectively. Then we can get txtBxxB bfsxssB bfsxssfsxsfsxss   x   11 00=1011 00=1,,d,,d d dmtjjjmttjjtxtBxxBbfsxsf sxssf sxsf sxssBx xkBbxsxsskxsxs s   x     000 00011 0=1000 d dmtt tNt tNs tNs tjjNt tNs tNs tBxxkBbeextxtskeextxts   0 Nt t Ntextxte  Copyright © 2012 SciRes. APM A. M. A. EL-SAYED ET AL. 113 011 00=1011=1011=111=1 d 1 1mttNt sjjNt t d1Nt sNtNtmjjmNt tNt NtjjmjjxxBxxkBbxxeskxxeeBxxkBbx xkxkBxxBb eeeNkBx xBbN e sexNNx x    xx 11111=1mjjkxxB BbN xx  Therefore, 11<,xx <,xx which com- plete the proof of theorem. REFERENCES  B. Ahmad and J. J. 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