Positive Solutions for Singular m-point Boundary Value Problems with Sign Changing Nonlinearities Depending on x

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1 Electroic Joural of Qualitative Theory of Differetial Equatios 9, No. 43, -4; Positive Solutios for Sigular m-poit Boudary Value Problems with Sig Chagig Noliearities Depedig o x Ya Ma, Baoqiag Ya Departmet of Mathematics, Shadog Normal Uiversity, Jia, 54, P.R. Chia Abstract Usig the theory of fixed poit theorem i coe, this paper presets the existece of positive solutios for the sigular m-poit boudary value problem x (t) + a(t)f(t,x(t),x (t)) =, < t <, x () =, x() = α i x(ξ i ), where < ξ < ξ < < ξ <,α i [,), i =,,, m, with < α i < ad f may chage sig ad may be sigular at x = ad x =. Keywords: m-poit boudary value problem; Sigularity; Positive solutios; Fixed poit theorem Mathematics subject classificatio: 34B5, 34B. Itroductio The study of multi-poit BVP (boudary value problem) for liear secod-order ordiary differetial equatios was iitiated by Il i ad Moiseev [3-4]. Sice the, may authors studied more geeral oliear multi-poit BVP, for examples [, 5-8], ad refereces therei. I [7], Gupta, Ntouyas, ad Tsamatos cosidered the existece of a C [, ] solutio for the m-poit boudary value problem x (t) = f(t, x(t), x (t)) + e(t), < t <, x () =, x() = a i x(ξ i ), The project is supported by the fud of Natioal Natural Sciece (87), the fud of Shadog Educatio Committee (J7WH8) ad the fud of Shadog Natural Sciece (Y8A6) Correspodig author: yabqc@yahoo.com.c EJQTDE, 9 No. 43, p.

2 where ξ i (, ), i =,,, m, < ξ < ξ < < ξ <, a i R, i =,,, m, have the same sig, a i, e L [, ], f : [, ] R R is a fuctio satisfyig Carathéodory s coditios ad a growth coditio of the form f(t, u, v) p (t) u + q (t) v + r (t) with p, q, r L [, ]. Recetly, usig Leray- Schauder cotiuatio theorem, R.Ma ad Doal O Rega proved the existece of positive solutios of C [, ) solutios for the above BVP, where f : [, ] R R satisfies the Carathéodory s coditios (see [8]). Motivated by the works of [7,8], i this paper, we discuss the equatio x (t) + a(t)f(t, x(t), x (t)) =, < t <, x () =, x() = α i x(ξ i ), where < ξ i <, < ξ < ξ < < ξ <, α i [, ) with < (.) α i < ad f may chage sig ad may be sigular at x = ad x =. Our mai features are as follows. Firstly, the oliearity af possesses sigularity, that is, a(t)f(t, x, x ) may be sigular at t =, t =, x = ad x = ; also the degree of sigularity i x ad x may be arbitrary(i. e., if f cotais x ad, α ad γ may α (x ) γ be big eough). Secodly, f is allowed to chage sig. Fially, we discuss the maximal ad miimal solutios for equatios (.). Some ideas come from [-].. Prelimiaries Now we list the followig coditios for coveiece. (H ) β, a, k C((, ), R + ), F C(R +, R + ), G C(R, R + ), ak L[, ]; (H ) F is bouded o ay iterval [z, + ), z > ; (H 3 ) dy = + ; G(y) ad the followig coditios are satisfied (P ) f C((, ) R + R, R); (P ) < α i <, < ξ i < ad f(t, x, y) k(t)f(x)g(y); (P 3 ) There exists δ > such that f(t, x, y) β(t), y (δ, ); where R + = (, + ), R = (, ), R = (, + ). Lemma. [] Let E be a Baach space, K a coe of E, ad B R = {x E : x < R}, where < r < R. Suppose that F: K B R \B r = K R,r K is a completely cotiuous operator ad the followig coditios are satisfied () F(x) x for ay x K with x = r. () If x λf(x) for ay x K with x = R ad < λ <. The F has a fixed poit i K R,r. EJQTDE, 9 No. 43, p.

3 Let C[, ] = {x : [, ] R x(t) is cotiuous o [, ]} with orm y = max t [,] y(t). The C[, ] is a Baach space. Lemma. Let (H )-(P 3 ) hold. For each give atural umber >, there exists y C[, ] with y (t) such that where y (t) = a(s)f(s, (Ay )(s) +, y (s))ds, t [, ], (.) (Ay)(t) = α y(τ)dτ α ξi i y(τ)dτ i α i y(τ)dτ, t [, ]. Proof. For y P = {y C[, ] : y(t), t [, ]}, defie a operator as follows (T y)(t) = + mi{, a(s)f(s, (Ay)(s) +, mi{y(s), })ds}, t [, ], (.) where > is a atural umber. For y P, we have Let α ξi i y(τ)dτ (Ay)(t) = α y(τ)dτ i α y(τ)dτ α ξi i y(τ)dτ i α i α ξ i α i y(τ)dτ y(τ)dτ α i α i α i α y(τ)dτ i ξ, t [, ]. c(y(t)) = c(y k (t)) = By the equality mi{c, } = c c α i y(τ)dτ y(τ)dτ a(s)f(s, (Ay)(s) +, mi{y(s), })ds, t [, ], a(s)f(s, (Ay k )(s) +, mi{y k(s), })ds, t [, ]. (T y)(t) =, it is easy to kow c(y(t)) c(y(t) +, t [, ]. Let y k, y P with lim k + y k y =. The, there exists a costat h >, such that y k h ad y h. Thus, mi{y k (s), } mi{y(s), }, uiformly for s [, ] as k +. Therefore, (Ay k )(s)+ ((Ay)(s)+ ) for all s [, ] as k +. (P ) implies that {a(s)f(s, (Ay k )(s) +, mi{y k(s), })} {a(s)f(s, (Ay)(s) +, mi{y(s), })}, for s (, ) as k +. By the Lebesgue domiated covergece EJQTDE, 9 No. 43, p. 3

4 theorem (the domiatig fuctio a(s)k(s)f[, + )G[h, ]), we have cy k cy, which yields that T y k T y = c(y k) c(y) c(y k ) + c(y) c(y k) c(y) + c(y k ) c(y) c(y k ) c(y), as k +. Cosequetly, T is a cotiuous operator. Let C be a bouded set i P, i.e., there exists h > such that y h, for ay y C. For ay t, t [, ], t < t, y C, (T y)(t ) (T y)(t ) a(s)f(s, (Ay)(s) + t =, mi{y(s), }ds a(s)f(s, (Ay)(s) + +, mi{y(s), )ds t a(s)f(s, (Ay)(s) +, mi{y(s), )ds a(s)f(s, (Ay)(s) + t, mi{y(s), }ds a(s)f(s, (Ay)(s), mi{y(s), t + }ds a(s)k(s)ds supf[ t, + ) supg[h, ]. Accordig to the absolute cotiuity of the Lebesgue itegral, for ay ɛ >, there exists δ > such that t a(s)k(s)ds < ɛ, t t < δ. Therefore, {T y, y C} is equicotiuous. (T y)(t) = + mi{, a(s)f(s, (Ay)(s) +, mi{y(s), })ds} + a(s)f(s, (Ay)(s) +, mi{y(s), })ds + + a(s) f(s, (Ay)(s) +, mi{y(s), } )ds a(s)k(s)ds sup F[, + )G[h, ], t [, ]. Therefore {T y, y C} is bouded. Hece T is a completely cotiuous operator. By (H 3 ), choose a sufficietly large R > to fit For >, we prove that δ R dy G(y) > a(s)k(s)ds supf[, + ). y(t) λ(t y)(t) = λ +λ mi{, a(s)f(s, (Ay)(s)+, mi{y(s), })ds}, t [, ], (.3) EJQTDE, 9 No. 43, p. 4

5 for ay y P with y = R ad < λ <. I fact, if there exists y P with y = R ad < λ < such that y(t) = λ + λ mi{, a(s)f(s, (Ay)(s) +, mi{y(s), })ds}, t [, ]. (.4) y() = λ. Sice > δ, we have δ < y() <, which implies there exists δ > such that y(t) > δ, t (, δ ). (P 3 ) implies Let t = sup{s [, ] a(s)f(s, (Ay)(s) +, mi{y(s), })ds >, t [, ]. We show that t =. If t <, we have t a(τ)f(τ, (Ay)(τ) +, mi{y(τ), })dτ >, t s}. a(s)f(s, (Ay)(s) +, mi{y(s), })ds >, t (, t ), a(s)f(s, (Ay)(s) +, mi{y(s), })ds =, t = t, y(t) = λ λ a(s)f(s, (Ay)(s) +, mi{y(s), })ds, t (, t ], (.5) y(t ) = λ > δ. (.6) (.6) ad (P 3 ) imply there exists r > such that f(t, x, y) β(t), t (t r, t ). So y(t ) = λ λ t a(s)f(s, (Ay)(s) +, mi{y(s), })ds λ r λ a(s)f(s, (Ay)(s) +, mi{y(s), })ds λ a(s)β(s)ds, t r r a(s)f(s, (Ay)(s) +, mi{y(s), })ds + which is a cotradictio. The, t =. Hece, y(t) = λ λ t r a(s)β(s)ds <, a(s)f(s, (Ay)(s) +, mi{y(s), })ds, t [, ]. (.7) Sice y = R > ad y P, there exists a t (, ) with y(t ) = R < ad a t (, ) such that y(t) < <, t (t, t ], which together with (.7) implies that y(t) = λ λ a(s)f(s, (Ay)(s) +, y(s))ds, t (t, t ]. (.8) Differetiatig (.8) ad usig (H ), we obtai y (t) = λa(t)f(t, (Ay)(t) +, y(t)) a(t)f((ay)(t) + )G(y(t)), t (t, t ]. EJQTDE, 9 No. 43, p. 5

6 Ad the y (t) G(y(t)) a(t)k(t) sup F[(Ay)(t) +, + ) a(t)k(t) sup F[, + ), t (t, t ). (.9) Itegratig for (.9) from t to t, we have The R y(t ) dy G(y) y(t) R which cotradicts y(t ) dy G(y) dy G(y) R t a(s)k(s)ds sup F[, + ), t (t, t ). (.) a(s)k(s)ds supf[ t, + ) a(s)k(s)ds sup F[, + ), dy G(y) > a(s)k(s)ds supf[, + ). Hece(.3) holds. The put r =, Lemma. leads to the desired result. This completes the proof. Lemma.3 [] Let {x (t)} be a ifiite sequece of bouded variatio fuctio o [a, b] ad {x (t )}(t [a, b]) ad {V (x )} be bouded(v (x) deotes the total variatio of x). The there exists a subsequece {x k (t)} of {x (t)}, i j, i j, such that {x k (t)} coverges everywhere to some bouded variatio fuctio x(t) o [a, b]. Lemma.4 [9] (Zor) If X is a partially ordered set i which every chai has a upper boud, the X has a maximal elemet. 3. Mai results Theorem 3. Let (H )-(P 3 ) hold. The the m-poit boudary value problem (.) has at least oe positive solutio. Proof. Put M = mi{y (t) : t [, ξ ]}, (H ) implies γ = sup{m } <. I fact, if γ =, there exists k > N > such that M k ad δ < y k <. (H ) implies ξ The y k (ξ ) <. Set τ = max{γ, δ, y k (t) = a(s)f(s, (Ay k )(s) +, y k (s))ds < a(s)β(s)ds < a(s)β(s)ds, t [, ξ ]. a(s)β(s)ds, which cotradicts to M k. ξ a(s)β(s)}. I the remaider of the proof, assume > τ ). First, we prove there exists a t (, ξ ] with y (t ) = τ. I fact, sice y () = > τ, there exists δ > such that y (t) > τ, t (, δ ). Let t = sup{t s EJQTDE, 9 No. 43, p. 6

7 [, t], y (s) > τ}.the y (t ) = τ. If t > ξ, we have y (t) > τ > δ, t [, ξ ]. (H ) shows that y (t) = a(s)f(s, (Ay )(s) +, y (s))ds a(s)β(s)ds a(s)β(s)ds, t [, ξ ]. The τ < y (ξ ) ξ a(s)β(s)ds < τ, which is a cotradictio. Secod, we prove y (t) τ, t [t, ]. (3.) I fact, if there exists a t (t, ] such that y (t) > τ, ad we choose t, t [t, ], t < t to fit y (t ) = τ, τ < y (t) <, t (t, t ], from (.) < t a(s)f(s, (Ay )(s) +, y (s))ds = y (t ) y (t ) <. This cotradictio implies that (3.) holds. The y (t) a(s)β(s)ds, t [, t ], y (t) τ, t [t, ]. Let W(t) = max{ a(s)β(s)ds, τ}, t (, ). Obviously, W(t) is bouded o [ 3k, 3k ] ad y (t) W(t), t [, ]. ). {y (t)} is equicotiuous o [ 3k, ](k is a atural umber) ad uiformly 3k bouded o [, ]. Notice that (Ay )(t) + = α i > α i α i We kow from (.9) y (t) ξ y (τ)dτ y (τ)dτ α ξi i y (τ)dτ t y (τ)dτ + α i α i α (τ)( ξ) = Θ, t [, ]. i dy t G(y ) a(s)k(s)ds sup F[Θ, + ), t [, ]. (3.) Now (H 3 ) ad (3.) show that ω(t) = if{y (t)} > is bouded o [, ]. O the other had, it follows from (.) ad (3.) that y (t) k(t)a(t) sup F[Θ, + ) supg[ω k, max{τ, W( )}], ( k), (3.3) 3k EJQTDE, 9 No. 43, p. 7

8 where ω k = if{ω(t), t [ 3k, ]}. Thus (3.3) ad the absolute cotiuity of Lebesgue 3k itegral show that {y (t)} is equicotiuous o [ 3k, ]. Now the Arzela-Ascoli 3k theorem guaratees that there exists a subsequece of {y (k) (t)}, which coverges uiformly o [ 3k, ]. Whe k =, there exists a subsequece {y() 3k (t)} of {y (t)}, which coverges uiformly o [ 3, ].Whe k =, there exists a subsequece {y() (t)} of {y() 3 (t)}, which coverges uiformly o [ 6, 5 ]. I geeral, there exists a subsequece {y(k+) (t)} 6 of {y (k) (t)}, which coverges uiformly o [ 3(k + ), ]. The the diagoal 3(k + ) sequece {y (k) k (t)} coverges poitwise i (, ) ad it is easy to verify that {y(k) k (t)} coverges uiformly o ay iterval [c, d] (, ). Without loss of geerality, let {y (k) k (t)} be itself of {y (t)} i the rest. Put y(t) = lim y (t), t (, ). The y(t) is cotiuous o (, ) ad sice y (t) W(t) <, we have y(t), t (, ). 3) Now (3.) shows sup{max{y (t), t [, ]}} < +. We have ad lim sup{ y (s)ds} =, t + lim sup{ y t (s)ds} =, t [, ], (3.4) t (Ay )(t) = α i < α i < +, t [, ]. y (τ)dτ y (τ)dτ α ξi i y (τ)dτ α i y (τ)dτ (3.5) Sice (3.4) ad (3.5) hold, the Fatou theorem of the Lebesgue itegral implies (Ay)(t) < +, for ay fixed t (, ). 4) y(t) satisfies the followig equatio y(t) = a(s)f(s, (Ay)(s), y(s))ds, t (, ). (3.6) Sice y (t) coverges uiformly o [a, b] (, ), (3.4) implies that (Ay )(s) coverges to (Ay)(s) for ay s (, ). For fixed t (, ) ad ay d, < d < t, we have y (t) y (d) = d a(s)f(s, (Ay )(s) +, y (s))ds. (3.7) for all > k. Sice y (s) max{τ, W(d)}, (Ay )(s) + Θ, s [d, t], {(Ay )(s)} ad {y (s)} are bouded ad equicotiuous o [d, t] y(t) y(d) = d a(s)f(s, (Ay)(s), y(s))ds. (3.8) EJQTDE, 9 No. 43, p. 8

9 Puttig t = d i (3.), we have y (d) Lettig ad d +, we obtai Lettig d + i (3.8), we have ad dy d G(y ) a(s)k(s)ds sup F[Θ, + ). (3.9) y( + ) = lim y(d) =. d + y(t) = a(s)f(s, (Ay)(s), y(s))ds, t (, ), (3.) (Ay)() = α i (Ay)(ξ i ). Hece x(t) = (Ay)(t) is a positive solutio of (.). Theorem 3. Suppose that (H )-(P 3 ) hold. The the set of positive solutios of (.) is compact i C [, ]. Proof Let M = {y C[, ]: (Ay)(t) is a positive solutio of equatio (.) }. We show that () M is ot empty; () M is relatively compact(bouded, equicotiuous); (3) M is closed. Obviously, Theorem 3. implies M is ot empty. First, we show that M C[, ] is relatively compact. For ay y M, differetiatig (3.) ad usig (H ), we obtai y (t) = a(t)f(t, (Ay)(t), y(t)) a(t) f(t, (Ay)(t), y(t)) a(t)k(t)f[θ, + )G(y(t)), t (, ), y (t) G(y(t)) a(t)k(t) sup F[(Ay)(t), + ) a(t)k(t) sup F[Θ, + ), t [, ]. (3.) Itegratig for (3.) from to t, we have y(t) dy G(y) a(s)k(s)ds sup F[Θ, + ), t [, ]. (3.) Now (H 3 ) ad (3.) show that for ay y M, there exists K > such that y(t) < K, t [, ]. The M is bouded. For ay y M, we obtai from (3.) y (t) = a(t)f(t, (Ay)(t), y(t)) a(t) f(t, (Ay)(t), y(t)) a(t)k(t)f[θ, + )G(y(t)), t (, ), EJQTDE, 9 No. 43, p. 9

10 ad which yields ad y (t) = a(t)f(t, (Ay)(t), y(t)) a(t) f(t, (Ay)(t), y(t)) a(t)k(t)f[θ, + )G(y(t)), t (, ), y (t) a(t)k(t) sup F[Θ, + ), t (, ), (3.3) G(y(t)) + y (t) a(t)k(t) sup F[Θ, + ), t (, ). (3.4) G(y(t)) + Notice that the rights are always positive i (3.3) ad (3.4). Let I(y(t)) = For ay t, t [, ], itegratig for (3.3) ad (3.4) from t to t,we obtai I(y(t )) I(y(t )) y(t) dy G(y) +. t a(t)k(t)f[θ, + )dt. (3.5) Sice I is uiformly cotiuous o [I(K), ], for ay ɛ >, there is a ɛ > such that I (s ) I (s ) < ɛ, s s < ɛ, s, s [I(K), ]. (3.6) Ad (3.5) guaratees that for ɛ >, there is a δ > such that Now (3.6) ad (3.7) yield that I(y(t )) I(y(t )) < ɛ, t t < δ, t, t [, ]. (3.7) y(t ) y(t ) = I (I(y(t )) I (I(y(t )) < ɛ, t, t [, ], (3.8) which meas that M is equicotiuous. So M is relatively compact. Secod, we show that M is closed. Suppose that {y } M ad lim + max t [,] y (t) y (t) =. Obviously y C[, ] ad lim + (Ay )(t) = (Ay )(t), t [, ]. Moreover, (Ay )(t) = α y (τ)dτ i < α y (τ)dτ i K <, t [, ]. α i For y M, from (3.) we obtai α ξi i y (τ)dτ α i y (τ)dτ (3.9) y (t) = a(s)f(s, (Ay )(s), y (s))ds, t (, ). (3.) For fixed t (, ), there exists < d < t such that y (t) y (d) = a(s)f(s, (Ay )(s), y (s))ds. (3.) d EJQTDE, 9 No. 43, p.

11 Sice y (s) max{τ, W(d)}, (Ay )(s) Θ, s [d, t], the Lebesgue Domiated Covergece Theorem yields that From (3.), we have which yields Itegratig from to d y (t) y (d) = a(s)f(s, (Ay )(s), y (s))ds, t (, ). (3.) d y (t) = a(t)f(t, (Ay )(s), y (s)) a(t)k(t)f[θ, + )G(y (t)), t (, ), y (t) G(y (t)) y (d) Lettig ad d +, we obtai Lettig d + i (3.), we have ad a(t)k(t) sup F[Θ, + ), t (, ). dy d G(y ) a(s)k(s)ds sup F[Θ, + ). (3.3) y ( + ) = lim d + y (d) =. y (t) = a(s)f(s, (Ay )(s), y (s))ds, t (, ), (3.4) (Ay )() = α i (Ay )(ξ i ). The x (t) = (Ay )(t) is a positive solutio of (.). So y M ad M is a closed set. Hece {Ay, y M} C [, ] is compact. Theorem 3.3 Suppose (H )-(P 3 ) hold. The (.) has a miimal positive solutio ad a maximal positive solutio i C [, ]. Proof. Let Ω = {x(t) : x(t) is a C [, ] positive solutio of (.)}. Theorem 3. implies that is oempty. Defie a partially ordered i Ω : x y iff x(t) y(t) for ay t [, ]. We prove oly that ay chai i < Ω, > has a lower boud i Ω. The rest is obtaied from Zor s lemma. Let {x α (t)} be a chai i < Ω, >. Sice C[, ] is a separable Baach space, there exists coutable set at most {x (t)}, which is dese i {x α (t)}. Without loss of geerality, we may assume that {x (t)} {x α (t)}. Put z (t) = mi{x (t), x (t),, x (t)}. Sice {x α (t)} is a chai, z (t) Ω for ay (i fact, z (t) equals oe of x (t)) ad z + (t) z (t) for ay. Put z(t) = lim z (t). We prove m + that z(t) Ω. By Lemma., there exists y (t) (e.g., y (t) may be z (t)), which is a solutio of (Ty)(t) = a(s)f(s, (Ay)(s), y(s))ds t [, ], EJQTDE, 9 No. 43, p.

12 such that z (t) = α ξi i y (τ)dτ y (τ)dτ α i α i y (τ)dτ. (3.) imply that { y } is bouded. From Lemma.3, there exists a subsequece {y k (t)} of {y (t)}, i j, i j, which coverges everywhere o [, ]. Without loss of geerality, let {y k (t)} be itself of {y (t)}. Put y (t) = lim y (t), t [, ]. Use y (t), y (t), ad m + i place of y (t), y(t), ad / i Theorem 3., respectively. A similar argumet to show Theorem 3. yields that y (t) is a solutio of The boudedess of { y } leads to z(t) = lim m + z (t) y(t) = a(s)f(s, (Ay)(s), y (s))ds, t [, ]. = lim [ m + α y (τ)dτ i = α y (τ)dτ i α i α ξi i y (τ)dτ α i y (τ)dτ ξi α i y (τ)dτ. y (τ)dτ] Hece z Ω. By Lemma., for ay x {x α }, there exists {x k } {x } such that x k x. Notice that x k (t) z k (t) z(t), t [, ]. Lettig k +, we have x(t) z(t), t [, ]; i.e.,{x α } has lower boudedess i Ω. Zor s lemma shows that (.) has a miimal C [, ] positive solutio. By a similar proof, we ca get the a maximal C [, ] positive solutio. The proof is complete. Theorem 3.4 Suppose that (H )-(P 3 ) hold, f(t, x, z) is decreasig i x for all (t, z) [, ] R, a()f(, x, z) ad lim f(t, x, y) +. The (.) has a uique positive t solutio i C [, ]. Proof. Assume that x ad x are two positive differet solutios to (.), i.e., there exists t (, ] such that x (t ) x (t ). Without loss of geerality, assume that x (t ) > x (t ). Let ϕ(t) = x (t) x (t) for all t [, ]. Obviously, ϕ C[, ] C (, ] with ϕ(t ) >. Let t = if{ < t < t ϕ(s) > for all s t [t, t ]} ad t = sup{t < t < ϕ(s) > for all s t [t, t]}. It is easy to see that ϕ(t) > for all t (t, t ) ad ϕ has maximum i [t, t ]. Let t satisfyig that ϕ(t ) = max t [t,t ] ϕ(t). There are three cases: () t (t, t ); () t = t = ;(3) t =. () t (t, t ). It is easy to see that ϕ (t ) ad ϕ (t ) =. The ϕ (t ) = x (t ) x (t ) a cotradictio. = a(t )f(t, x (t ), x (t )) + a(t )f(t, x (t ), x (t )) >, EJQTDE, 9 No. 43, p.

13 () t = t =. Sice t = t =, we have α i max{ϕ(ξ i )} > cotradictio to < α i <. α i ϕ(ξ i ) = ϕ(), a (3) t =. Sice t = ad x ad x are solutios, the proof of lemma. implies that there exist x, ad x, such that where x, x < ϕ(), x, x < ϕ() x, (t) = α y, (τ)dτ α ξi i y,(τ)dτ i α i y, (τ)dτ, t [, ], x, (t) = ad α y, (τ)dτ α ξi i y,(τ)dτ i α i y, (t) = a(s)f(s, x, (s) +, y,(s))ds, t [, ], y, (t) = a(s)f(s, x, (s) +, y,(s))ds, t [, ], y, (τ)dτ, t [, ], y, (t), y,(t) for all t [, ]. By a similar proof with above, there exists t (, ] such that x, (t ) x, (t ). Without loss of geerality, assume that x, (t ) > x, (t ). Let ϕ (t) = x, (t) x, (t) for all t [, ]. Obviously, ϕ C[, ] C (, ] with ϕ (t ) >. Let t = if{ < t < t ϕ (s) > for all s t [t, t ]} ad t = sup{t < t < ϕ (s) > for all s t [t, t]}. It is easy to see that ϕ (t) > for all t (t, t ) ad ϕ has maximum i [t, t ]. Let t satisfyig that ϕ(t ) = max t [t,t ] ϕ(t). There are three cases: ) t (t, t ); ) t = t = ; 3) t =. The proof of ) ad ) are similar with () ad (). 3) t =. We have ϕ (t) < ϕ (), t (, ], ϕ () =, ϕ (t ξ) <, t ξ (, ). The lim tξ +ϕ ϕ (t) = lim (t ξ) ϕ () tξ +. t ξ O the other had, sice ϕ () = x, () x, () = a()f(, x, () +, x, ()) + a()f(, x,() +, x, ()) >, a cotradictio. The (.) has at most oe solutio. The proof is complete. Example 3.. I (.), let f(t, x, y) = k(t)[+x γ +(y) σ (y) l(y)], a(t) = t 3, ad k(t) = t, < t <, EJQTDE, 9 No. 43, p. 3

14 where γ >, σ <, ad let F(x) = + x γ, G(y) = + (y) σ (y) l(y). The ad f(t, x, y) k(t)f(x)g(y), δ =, β(t) = k(t), dy G(y) = +. By Theorem 3., (.) at least has a positive solutio ad Corollary 3. implies the set of solutios is compact. Refereces [] K.Deimlig, Noliear Fuctioal Aalysis, Spriger -Verlag, Berli, 985. [] Y.Guo, W.Ge, Positive solutios for three-poit boudary value problems with depedece o the first order derivative, J. Math. Aal. Appl., 9(4), pp9-3. [3] V.A.Il i, E.I.Moiseev, Nolocal boudary value problem of the secod kid for a Sturm-Liouville operator, Differetial Equatio 3 (8)(987), pp [4] V.A.Il i, E.I.Moiseev, Nolocal value problem of the first kid for a Sturm-Liouville operator i its differetial ad fiite differece aspects, Differetial Equatios 3 (7) (987), pp83-8. [5] B.Liu, Positive solutios of a oliear three-poit boudary value problem, Applied Mathematics ad Computatio 3 (), pp-8. [6] R.Ma, Positive solutios for a oliear three-poit boudary value problem, Electro. J.Differetial Equatios 34 (999), pp-8. [7] C.P. Gupta, S.K. Ntouyas, P.Ch. Tsamatos, Solvability of a m-poit boudary value problem for secod order ordiary differetial equatios, J. Math. Aal. Appl. 89 (995) pp [8] R.Ma, Doal O Rega, Solvability of sigular secod order m-poit boudary value problems Joural of Mathematical Aalysis ad Applicatios 3(5), pp4-34. [9] Mario Petrich ad Norma Reilly, Completely regular semigroups, Caadia Mathematical Society series of moographs, Spriger New York 64 (). [] D. Xia et al., Foudatio of Real Variable Fuctio ad Applied Fuctioal Aalysis, Shaghai Sciece ad Techology Press, Shaghai, (987) (i Chiese). [] G.Yag, Miimal Positive Solutios to Some Sigular Secod-Order Differetial Equatios,Computers J. Math. Aal. Appl. 66 (), pp [] G.Yag, Positive solutios of some secod order oliear sigular differetial equatios, Computers Math.Appl. 45 (3), pp [3] G.Yag, Secod order sigular boudary value problems with sig-chagig oliearities o ifiity itervals. Noliear Aalysis Forum. 9 () (4), pp (Received September 9, 8) EJQTDE, 9 No. 43, p. 4