IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VO L.24 (1) 2011 Existence of Positive Solution for Boundary Value Problems S. M . Hussein Department of Mathematics, Education College For Pure Sciences, University of Anbar Received in June,1,2010 Accepted in Oct,19,2010 Abstract This paper studies the existence of positive solutions for the following boundary value problem :- 0 y(b) 0(a)y β - y(a) α bta f(y) g(t) λy    The solution procedure follows using the Fixed point theorem and obtains that this problem has at least one positive solution .Also,it determines (  ) Eigenvalue which would be needed to find the positive solution . Keywords: Positive Solution , Boundary Value Problem , Fixed Point Theorem . Introduction In this paper we shall consider the second - order boundary value problem (BVP) The following conditions will be assumed throughout :- A- f : [0 , )  [0 , ) is continuous , B- g : [0 , 1]  [0 , ) is continuous and does not vanish identically on any subinterval , C- x f(x) Limf 0x 0    and x f(x) Limf x    exist , D-  ,  such that  and  are not both zero and Z =    > 0 , and E- a ≥ 0 , b ≤1 . The boundary value problem (1.1) arises in the applied mathematical sciences such as nonlinear diffusion generated by nonlinear sources , thermal ignition of gases and chemical concentrations in biological problems ; for example see [1] , [2] , [3] . When =1 and f is either superlinear that is (f 0 = 0 and f  = ) or f is sublinear that is (f 0 =  and f  = 0 ) , 1.1).........( 0 y(b) 0(a)y β - y(a) α bta f(y) g(t) λy         IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 Erbe and Wang [5] obtained solutions that are positive with respect to a cone which lies in an annular type region .The methods of [5] were then extended to higher order BVP in [4] . For the case  =1, = 0, =1,  = 0, Johnny Henderson and Haiyan Wang [7] obtained solutions that are positive for an open interval of eigenvalues . Not required in this work that f would be either superlinear or sublinear , yet, as in [4] , [5] but as in [7] , the arguments presented here for obtaining solutions of(1.1)for certain involve concavity properties of solutions, which are employed in defining a cone on which a positive integral operator is defined . A Krasnosel’skii fixed point theorem [8] is applied to yield positive solutions of (1.1) , for  belongs to an open interval. Section 2 , presents some properties of Green’s functions that are used in defining a positive operator , also states the Krasnosel’skii fixed point theorem . Section 3 , gives an appropriate Banach space and constructs a cone to which we apply the fixed point theorem yielding solutions of 1 .1 , for an open interval of eigenvalues . 2- Some Preliminaries In this section , we state the above mentioned Krasnosel’skii fixed point theorem. We will apply this fixed point theorem to completely continuous integral operator , whose kernal , G (t , s ) , is the Green’s function for - y = 0  y(a) -  y (a) = 0 0 y(b)  Is ………..(2.1)          btsa )t -1 ( ) βαs ( Z 1 bsta ) s-1 ( ) βαt ( Z 1 s)G(t, from which G(t , s) > 0 on ( 0 , 1 )  ( 0 , 1 ) , ……….(2.2) G(t , s)  G(s , s) = ) s -1 ( ) βs α ( Z 1  , a  t  b , a  s  b , ……(2.3) and it is shown in [5] that :- G(t , s)  M G(s , s) = M ) s -1 ( ) βs α ( Z 1  , 4 12b t 4 12a    , a  s  b , …(2.4) Where          β)4(α 4βα , 4 1 minM We shall apply the following fixed point theorem to obtain solutions of (1.1) , for certain  Theorem 1 [8]. Let B a Banach space , and let P be a cone in B . Assume N , K are be KNN0  , and let PN)\K(P:T  open subsets of B with a completely continuous operator such that , either 1-  Tu    u  , u  P  N , and  Tu    u  , u  P  K , or 2-  Tu    u  , u  P  N , and  Tu    u  , u  P  K IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 . N)\K(P Then T has a fixed point in 3. Solutions in The Cone In this section , apply Theorem 1 to the eigenvalue problem (1.1 ) . Note that y(t) is a solution of (1.1) if , and only if , y(t) =  ds f(y(s)) g(s) s) ,(t G b a  , a  t  b For our construction , let B = C[a , b] , with norm , x(t)Supx bt a   Define a cone P by :             xMx(t)min, b][a,on 0 x(t): BxP 4 12b t 4 12a          β)4(α 4βα , 4 1 minM Where Also , let the number h[a,b] be defined by ...(3.1).......... ds g(s) s)G(t,maxds g(s) s)G(h, 4 12b 4 12a 4 12b 4 12a       Theorem 2. Assume that conditions (A),(B),(C) and (D) are satisfied .Then , for each  satisfying ... (3.2)…… .      b a 0 4 1)(2b 4 1)(2a ds)f g(s) s)G(s,( 1 λ ds)f g(s) s)G(h,(M 4 there exists at least one solution of (1.1) in P . Proof. Let  be given as in (3.2) . Now , let  > 0 be chosen such that .(3.3).......... ε)ds)(f g(s) s)G(s,( 1 λ ε)ds)(f g(s) s)G(h,(M 4 b a 0 4 1)(2b 4 1)(2a        Define an integral operator T : P  B by Ty(t) = ds f(y(s)) g(s) s) ,(t G b a  , y  P ………(3.4) We seek a fixed point of T in the cone P. From (2.2), we note that , for y  P, Ty(t)  0 on [a,b] . Also , for y  P, we have from (2.3) that Ty(t) =  ds f(y(s)) g(s) s) ,(t G b a  IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 Ty    ds f(y(s)) g(s) s) , (sG b a  ………(3.5) Now , if y  P , we have by (2.4) and (3.5) , TyM ds f(y(s)) g(s) s) , (sGλ M ds f(y(s)) g(s) s) ,(t GλminTy(t)min b a b a4 12b t 4 12a 4 12b t 4 12a           p . In addition , standard arguments show that T is As a consequence , T : p completely continuous. Now, turning to f0 , there exist an K 1 > 0 such that f(x)  (f0 + ) x , for 0 < x  K1. y  P such that  y  = K1 , we have from (2.3) and (3.3) So , by choosing y y y(s) ε)(f ds g(s) s)G(s,λ ds y(s) ε)(f g(s) s)G(s,λ ds f(y(s)) g(s) s)G(s,λTy(t) b a 0 b a 0 b a        Consequently , yTy  . So , if we set 1 = {x  B x < K1} then Ty   y  , for y P  1 . ……….(3.6) Next , considering f , there exist an K2 > 0 such that f (x)  (f - ) x ,for all x > K2 . Let K3 = max {2K1 , }M K 2 and let 2 = { x  B  x < K3} If y  P with y  = K3 , then 23 4 12b t 4 12a KMKyMy(t)min     , and we have from (3.1) and (3.3) that y y ε)(f ds g(s) s)G(h, M λ ds y(s) ε)(f g(s) s)G(h,λ ds f(y(s)) g(s) s)G(h,λ ds f(y(s)) g(s) s)G(h,λTy(h) 4 1)(2b 4 1)(2a 4 1)(2b 4 1)(2a 4 1)(2b 4 1)(2a b a                  IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 Thus , yTy  . Hence , Ty   y  , for y  P  2 ……….(3.7) Applying (1) of theorem 1 to (3.6) and (3.7) yields that T has a fixed point y(t)  )\(P 12  . As such , y(t) is a desired solution of 1.1 for the given  . Further , since G (t , s) > 0 , it follows that y (t) > 0 for a < t 0 be chosen such that 3.9).........( ε)ds)(f s)g(s)G(s,( 1 λ ε)ds)(f g(s) s)G(h,(M 1 b a 4 1)(2b 4 1)(2a 0        Let T be the cone preserving , completely continuous operator that was defined by(3.4). Beginning with f0 , there exists an K 4 > 0 such that f(x)  (f0 - ) x , for 0 < x  K4. y  P such that  y  = K4 , we have from (3.1) and (3.9) so , for So y y ε)(f ds g(s) s),G(h λ M ds y(s) ε)(f g(s) s),G(h λ ds f(y(s)) g(s) s),G(h λ ds f(y(s)) g(s) s)G(h,λTy(h) 4 1)(2b 4 1)(2a 0 4 1)(2b 4 1)(2a 0 4 1)(2b 4 1)(2a b a                Thus , yTy  . So , if we let 3 = {x  B x < K4} then Ty   y  for y  P3 ……. (3.10) It remains to consider f , there exists an K5 > 0 such that f (x)  (f + ) x, for all x > K5 . There are the two cases , (a) f is bounded , and (b) f is unbounded . For case (a) , suppose K6 > 0 is such that f(x)  K6 , for all 0 < x <  . IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 Let K7 = max {2K4 , K6  b a ds} f(y(s)) g(s) s)G(s,λ . Then , for y  P with y  = K7 we have from (2.3) and (3.2) so that yTy  . So if 4 = {x  B x < K7} then Ty   y  , for y  P 4 ……….(3.11) For case (b) , let K8 > max {2K4 , K5 } be such that f(x)  f(K8) , for 0 < x  K8 . By choosing y  P such that y  = K8 and we have from (2.3),( 3.2 ) and (3.9 ) But    b a b a 8 yε)(f ds g(s) s)G(s, λε)K(f ds g(s) s)G(s, λ Therefore    b a yε)(f ds g(s) s)G(s, λ Ty(t) and so yTy  . For this case , if we let 4 = {x  B x < K8} then Ty   y  , for y  P 4 ……….(3.12) Thus , in both cases , an applying of part (2) of theorem 1 to (3.10),(3.11) and (3.12) yields that T has a fixed point y(t)  )\(P 34  . As such , y(t) is a desired solution of 1.1 for the given  . Further , since G (t , s) > 0 , it follows that y (t) > 0 for a < t < b . This completes the proof of the theorem .          b a 8 b a 8 b a b a ε)K(f ds g(s) s)G(s, λ ds )f(K g(s) s)G(s, λ ds f(y(s)) g(s) s)G(s, λ ds f(y(s)) g(s) s)G(t,λ Ty(t) y ds g(s) s)G(s,K λ ds} f(y(s)) g(s) s)G(t,λ Ty(t) b a 6 b a      IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 References 1. Kuiper,H. J. (1979). On Positive Solution of nonlinear elliptic eigenvalue Problems , Rend. Math. Cire. Palermo , Serie II ,Tom. XX 113-138 . 2. Erbe, L. H.; Hu, S. and Wang, H. (1994) Multiple Positive Solutions of some boundary value problems , J. M ath. Anal.Appl.184: 640–748 . 3. Eloe, P.W.; Henderson, J. and Wong, P.J.Y. Positive Solutions for two – point boundary value problems , Dyn. Sys. Appl. , in press . 4. Eloe, P. W. and Henderson, J. (1995). Positive Solutions for Higher Order Differential Equations , Elec. J. Diff. Equ. 3 :1 – 8 . 5. Erbe, L. H. and Wang, H. (1994). On The Existence of Positive Solution of Ordinary Differential Equations , Proc. Amer. Math. Soc. 120: 743 – 748 . 6. Garaizar, X. (1987). Existence of Positive radial Solutions for Semilinear elliptic Problems in the annulus , J. Diff. Equ. 70: 69 – 72 . 7. Henderson, J. and Wang, H. (1997). Positive Solutions for Nonlinear Eigenvalue Problems , J. M ath. Anal. APPl. 208: 252 – 259 . 8. Krasnoseelskii, M. A.(1964). Positive Solutions of Operator Equations, Noordhoff, Groning. 2011) 1( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد وجود الحلول الموجبة لمسائل القیم الحدودیة صالح محمد حسین جامعة االنبار،كلیة التربیة للعلوم الصرفة ،قسم الریاضیات 2010حزیران 1استلم البحث في 2010تشرین االول 19قبل البحث في الخالصة -: االتیةالحلول الموجبة للمسألة الحدودیة وجوددرس هذا البحث 0 y(b) 0(a)y β - y(a) α bta f(y) g(t) λy     )قـیم المعلمـة تـم تحدیـدو اموجبـ اواحد حال األقلأن هذه المسألة تمتلك على ت إلىنظریة النقطة الثابتة وتوصل استخدمم . لمسألة الحدودیةل حلول موجبة وجدتعندها التي (