IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 Existence of Positive Solution fo r Boundary Value Problems S. M . Hussein Departme nt of Mathematics, Education College For Pure Sciences, Unive rsity of Anbar Recei ved in June ,1,2010 Accepted in O ct,19,2010 Abstract This p aper st udies the existence of p ositive solutions for the following boundary value p roblem :- 0 y (b) 0(a)y β - y (a) α bta f(y ) g(t) λy    The solution p rocedure follows using the Fixed p oint theorem and obtains that this p roblem has at least one positive solution .Also,it determines (  ) Eigenvalue which would be needed to find the p ositive solution . Keywords: Positive Solution , Boundary Value Problem , Fixed Point Theorem . Introduction In this p aper 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 p roblem (1.1) arises in the app lied mathematical sciences such as nonlinear diffusion generated by nonlinear sources , thermal ignition of gases and chemical concentrations in biological p roblems ; for examp le see [1] , [2] , [3] . When =1 and f is either sup erlinear 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 p ositive with resp ect to a cone which lies in an annular ty p e 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 p ositive for an op en interval of eigenvalues . Not required in this work that f would be either sup erlinear or sublinear , y et, as in [4] , [5] but as in [7] , the arguments p resented here for obtaining solutions of(1.1)for certain involve concavity p rop erties of solutions, which are emp loyed in defining a cone on which a p ositive integral op erator is defined . A Krasnosel’skii fixed p oint theorem [8] is app lied to y ield p ositive solutions of (1.1) , for  belongs t o an open interval. Section 2 , p resents some p rop erties of Green’s functions that are used in defining a p ositive op erator , also states t he Krasnosel’skii fixed point t heorem . Section 3 , gives an ap p rop riate Banach sp ace and constructs a cone to which we app ly the fixed p oint t heorem yielding solutions of 1 .1 , for an open interval of eigenvalues . 2- S ome Preliminaries In this section , we st ate the above mentioned Krasnosel’skii fixed p oint theorem. We will app ly this fixed p oint theorem to comp letely continuous integral op erator , whose kernal , G (t , s ) , is t he 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 app ly the following fixed p oint t heorem to obt ain solutions of (1.1) , for certain  The orem 1 [8]. Let B a Banach sp ace , and let P be a cone in B . Assume N , K are be KNN0  , and let PN)\K(P:T  op en subsets of B with a completely continuous op erator 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. S oluti ons in The Cone In this section , app ly Theorem 1 to t he eigenvalue p roblem (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 const ruction , 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 t he 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       The orem 2. Assume that conditions (A),(B),(C) and (D) are satisfied .Then , for each  satisfy ing ... (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 op erator 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 p oint 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 t hat T is As a consequence , T : p comp letely 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 KM KyMy (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,λT y(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) Ap p lying (1) of theorem 1 to (3.6) and (3.7) y ields that T has a fixed p oint y (t)  )\(P 12  . As such , y (t) is a desired solution of 1.1 for the given  . Furt her , since G (t , s) > 0 , it follows t hat 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 p reserving , comp letely continuous op erator 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) , sup p ose 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 t hat 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 t his case , if we let 4 = {x  B x < K8} then Ty  y , for y  P 4 ……….(3.12) Thus , in both cases , an app lying of p art (2) of theorem 1 to (3.10),(3.11) and (3.12) y ields that T has a fixed p oint y (t)  )\(P 34  . As such , y (t) is a desired solution of 1.1 for the given  . Furt her , since G (t , s) > 0 , it follows t hat y (t) > 0 for a < t < b . This comp letes 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 Re ferences 1. Kuip er,H. J. (1979). On Positive Solution of nonlinear ellip tic eigenvalue Problems , Rend. M ath. Cire. Palermo , Serie II ,Tom. XX 113-138 . 2. Erbe, L. H.; Hu, S. and Wang, H. (1994) M ultip le Positive Solutions of some boundary value problems , J. M ath. Anal.Ap p l.184: 640–748 . 3. Eloe, P.W.; Henderson, J. and Wong, P.J.Y. Positive Solutions for two – p oint boundary value problems , Dy n. Sy s. Ap p l. , in p ress . 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. M ath. Soc. 120: 743 – 748 . 6. Garaizar, X. (1987). Existence of Positive radial Solutions for Semilinear ellip tic 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 Op erator 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 (