Research Article Triple Positive Solutions of a Nonlocal Boundary Value Problem for Singular Differential Equations with p-laplacian

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1 Abtract and Applied Analyi Volume 23, Article ID 63672, 7 page Reearch Article Triple Poitive Solution of a Nonlocal Boundary Value Problem for Singular Differential Equation with p-laplacian Jufang Wang, Changlong Yu, and Yanping Guo College of Science, Hebei Univerity of Science and Technology, Shijiazhuang, Hebei 58, China Correpondence hould be addreed to Jufang Wang; wangjufang98@26.com and Changlong Yu; changlongyu@26.com Received 22 October 22; Revied 3 January 23; Accepted 26 January 23 Academic Editor: Bahir Ahmad Copyright 23 Jufang Wang et al. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. We etablih the exitence of triple poitive olution of an m-point boundary value problem for the nonlinear ingular econdorder differential equation of mixed type with a p-laplacian operator by Leggett-William fixed point theorem. At lat, we give an example to demontrate the ue of the main reult of thi paper. The concluion in thi paper eentially extend and improve the known reult.. Introduction The exitence and multiplicity of poitive olution for differential equation boundary value problem (BVP) with the p- Laplacian operator ubject to Dirichlet, Sturm-Liouville, or nonlinear boundary value condition have been extenively invetigated in recent year; ee [ ] and the reference therein. Particularly, the following differential equation with one-dimenional p-laplacian (φ p (u )) +q(t) f (t, u) =, t () have been tudied ubject to different kind of boundary condition; ee [ 4] and the reference therein. The method mainly depend on Krannoel kii fixed point theorem, upper and lower olution technique, Leggett-William fixed point theorem, and ome new fixed point theorem in cone, and o forth. Recently, in [9], Kong et al. have tudied the exitence of triple poitive olution for the following BVP: (φ p (u )) (t) +q(t) f and tudied the exitence of triple poitive olution for the following BVP: (φ p (u )) (t) +q(t) f (u(t),u (t), (Tu)(t), (Su)(t)) =, t, u () =βu (η), u () =g(u ()). Firtly, we confirm that the mitake which have been pointed out in [] exit. At the ame time, we think that the value of M deigned in Theorem 3. in [] i not uitable, ince the proof need the condition φ q (M) M, but in fact thi condition doe not alway hold. Motivated by the work above, in thi paper, we will tudy the following more extenive econd-order m-point BVP: (φ p (u )) (t) +q(t) f (3) (u(t),u (t), (Tu)(t), (Su)(t)) =, t, (2) (u(t),u (t), (Tu)(t), (Su)(t)) =, t, u () =, u() =g(u ()). More recently, in [], Hu and Ma have pointed out that the equivalent integral equation of BVP (2) i wrong in[9] φ p (u ()) = m 2 β i φ p (u ( )), u () =g(u ()), (4)

2 2 Abtract and Applied Analyi where φ p () = p 2 (p > ) i an increaing function, (φ p ) () = φ q (), /p + /q = ; β i <, i =,2,..., m 2, m 2 β i <,and<η <η 2 < <η m 2 ; T and S are two linear operator defined by t Tu (t) = k (t, ) u () d, Su (t) = h (t, ) u () d, u C [, ], in which k C[D, R + ], h C[D,R + ], D = {(t, ) R 2 : t }, D = {(t, ) R 2 :,t }, R + = [, + ), R=(,+ ), k = max{k(t, ) : (t, ) D}, and h = max{h(t, ) : (t, ) D }. Obviouly, when m=3,bvp(4)reducetobvp(3), and when m=3, β =,BVP(4)reducetoBVP(2), o BVP (2) and BVP (3)arepecialcaeofBVP(4). Throughout thi paper, we alway uppoe the following condition hold: (C )f C(R + R R + R +,(,+ )); (C 2 ) q(t) C([, ], R + ) may be ingular at t=,and < q(t)dt < +, o it i eay to ee that there exit a contant M>uch that < q(t)dt < φ p(m); (C 3 )g : R R + i nonincreaing and continuou, and g(v) V for V R. 2. Preliminary Reult In thi ection, we firtly preent ome definition, theorem, and lemma, which will be needed in the proof of the main reult. Definition. Let E be a real Banach pace. A nonempty cloed convex et P Eicalled a cone if it atifie the following two condition: (i) x P, λ implie λx P; (ii) x P, x P implie x=. Definition 2. Given a cone P in a real Banach pace E, a continuou map α i called a concave (rep., convex) functional on P if and only if, for all x, y P and t, it hold: α(tx + ( t)y) tα(x) + ( t)α(y), (rep., α(tx + ( t)y) tα(x) + ( t)α(y)). We conider the Banach pace E = C [, ] equipped with norm u = max t { u(t), u (t) },where u = max t u(t). We denote, for any fixed contant a, b, r, C + [,]={u C[,]:u(t),t [,]}, P = {u E u(t) i concave and nonincreaing on [, ]}, (5) P r ={u P: u <r}, P(α, a, b) = {u P : a α(u), u < b}. It eay to ee that P i a cone in E. Theorem 3 (Leggett-William). Let A : P c P c be a completely continuou map and let α be a nonnegative continuou concave functional on P with α(u) u for any u P c.suppoethereexitcontanta, b, andd with <a< b<d cuch that (i) {u P(α,b,d) : α(u) > b} =φand α(au) > b for all u P(α, b, d); (ii) Au < a for all u P a ; (iii) α(au) > b for all u P(α,b,c)with Au > d. Then A ha at leat three fixed point u, u 2,andu 3 atifying u <a, b<α(u 2), u 3 >a, α(u 3)<b. Lemma 4. Suppoe y C [, ] with (φ p (y )) L [, ] atifie φ p (y ()) = (φ p (y )) (t), m 2 t, β i φ p (y ( )), y () =g(y ()). Then, y(t) i concave and nonincreaing on [, ], thati, y P. Proof. Since (φ p (y )) (t), we know that φ p (y ) i nonincreaing, that i, y (t) i nonincreaing, which mean y(t) iconcave.attheametime,wehavey (t) y (), o y () = φ q ( m 2 β iφ p (y ( ))) φ q ( m 2 β iφ p (y ())) = φ q ( m 2 β i)y (), namelyy (). Then,y (t) ; thati to ay, y(t) i nonincreaing. So y(t) y() = g(y ()). Above all, y P.Thicompletetheproof. Lemma 5. Let y C[, ] and (φ p (u )) L [, ],then,bvp φ p (u ()) = (φ p (u )) (t) = y(t), m 2 ha a unique olution u (t) = φ q ( m 2 t β t, β i φ p (u ( )), u () =g(u ()), +g( φ q ( m 2 β where β= m 2 β i. y (r) dr + y (r) dr) d (6) (7) (8) y (r) dr + y (r) dr)), (9)

3 Abtract and Applied Analyi 3 Define the operator A:P Eby (Au)(t) = φ q ( m 2 t β + )d +g( φ q ( m 2 β +φ q ( m 2 β =2φ q ( m 2 β =2 (Au) () + ) + ) =2 (Au). () + )). () Obviouly, A i well defined and u Ei a olution of BVP (4) if and only if u i a fixed point of A. Lemma 6. A:P Pi completely continuou. Proof. It i imilar to the proof of Lemma 2.2 in [9]. Lemma 7. For any u P,oneha Au 2 (Au), Au 2 (Au). Proof. From (), we obtain (Au) = φ q ( m 2 β + )d +g( φ q ( m 2 β φ q ( m 2 β + )) + )d Since Au = max{ Au, (Au) },owehave Au 2 (Au),whichcompletetheproof. 3. Main Reult For any δ (, min{η, /2}), we define a nonnegative continuou concave function α : P R + by α(u) = min δ t ( δ) u(t). Obviouly, the following two concluion hold: α (u) =u( δ) u, α(au) =Au( δ), The main reult of thi paper i following. u P. (2) Theorem 8. Let m = min t q(t) and β= m 2 β i <. Suppoe (C ), (C 2 ),and(c 3 ) hold. Suppoe further that there exit number δ (, min{η, /2}), a, b, c, and d uch that <a<b m δd/m < d c,and (H ) f(u, V, w, l) ( β)φ p (a/2m), for (u, V,w,l) [, a] [ a, ] [, k a] [, h a]; (H 2 ) f(u, V, w, l) ( β)φ p (c/2m), for (u, V,w,l) [, c] [ c, ] [, k c] [, h c]; (H 3 ) f(u, V,w,l) > φ p (b/δl), for(u, V,w,l) [b,d] [ d, ] [, k d] [, h d],wherel=φ q ( δ q(t)dt); (H 4 ) min (u,v,w,l) J f(u, V,w,l)φ p (M/(2m )) δ q(t)dt max (u,v,w,l) J f(u, V,w,l) q(t)dt, wherej=[,c] [ c, ] [, k c] [, h c]. Then, BVP (4) ha at leat three poitive olution u, u 2,and u 3 uch that u u 3 <a, b< min δ t<( δ) u 2 (t), >a, min δ t<( δ) u 3 (t) <b. (3)

4 4 Abtract and Applied Analyi Proof. We divide the proof into three tep. Step. We prove AP c P c, AP a P a ; that i, (ii) of Theorem 3. ByLemma 6, wehaveap c P,o u P c, we get u(t) c, c u (t), (Tu)(t) k c, (Su)(t) h c.fort [,]and by (H 2 ) (Au) = φ q ( m 2 β + )d +g( φ q ( m 2 β φ q ( m 2 β + )) + )d +φ q ( m 2 β =2φ q ( m 2 β + ) + ) 2φ q ( β ) 2φ q (φ p ( c 2M )) φ q ( dr) c. (Au) =φ q ( m 2 β + ) φ q ( m 2 β + ) φ q ( β ) φ q (φ p ( c 2M )) φ q ( dr) c. (4) Hence, Au < c and AP c P c.similarly,weobtainap a P a. Step 2. We how {u P(α, b, d) :α(u) >b} =φ, (5) α (Au) >b, u P(α, b, d), (6) that i, (i) of Theorem 3. Let u = (b + d)/2, thenu P(α, b, d), α(u) = (b + d)/2 > b. Hence,(5) hold.foranyu P(α,b,d),wehave b u(t) d, d u (t), (Tu)(t) k d, (Su)(t) h d, t [, δ],oby(h 3 ),wehave α (Au) = min (Au)(t) = (Au)( δ) t [δ, δ] = φ q ( m 2 δ β + )d +g( φ q ( m 2 β δφ q ( m 2 β + )) δ + ) +g( φ q ( m 2 β + ))

5 Abtract and Applied Analyi 5 δ δφ q ( ) δφ q (φ p ( b δ δl )) φ q ( dr) =b. (7) Hence (6)hold. Step 3. We how that α(au) > b for all u P(α, b, c) with Au > d, that i, (iii) of Theorem 3. If u P(α,b,c) with Au > d, weobtain u(t) c, c u (t), (Tu)(t) k c, (Su)(t) h c,for any t [, ],andoby(h 4 ),wehave φ p ( M δ ) 2m Furthermore, we have. φ p ( M δ ) 2m +φ p ( M 2m ) β (8) δφ q ( m 2 β δφ q (( m 2 β δ + ) + ) (φ p ( M )) ) 2m = 2δm M φ q ( m 2 β + ) m 2 + m 2 β Therefore, by Lemma7, wehave α (Au) = (Au)( δ). (9) = 2δm M (Au) () = 2δm M (Au) δm M > δm M d b. (Au) (2) = φ q ( m 2 δ β + )d +g( φ q ( m 2 β + )) Hence, by Theorem 3, thereultoftheorem 8 hold. Thi complete the proof of Theorem Example Conider the following BVP: ( u u ) (t) +q(t) f(u(t),u (t), (Tu)(t), (Su)(t)) =, t,

6 6 Abtract and Applied Analyi u () u () = 4 u ( 2 5 ) u ( 2 5 ) + 4 u ( 2 ) u ( 2 ), u () =g(u ()), (2) If u 296, 296 V, w, l 296,then f (u, V,w,l) > + >φ p ( b δl )= (26) So (H 3 ) i atified. For any (u, V, w, l) [, 6] [ 6, ] [, 6] [, 6],wehave where { t /2, t 6 q (t) = 25, { 87 { 54 t , 6 25 t. u + 2+in V 3 w + + l, { u, V, w,l, f (u, V,w,l) = u+ 2+in V 3 w + { + l, { u, V, w,l. g (V) ={ V/2, V, V 2, V <. (22) Proof. Since M = 2 and m = /3, β = /2, δ = 9/25, a=/2, b=, d = 296, c = 6, k(t, ) = and h(t, ) =,thenwecanobtain<a<b (m δd)/m < d c,and 9/25 L= 6/25 q (t) dt = t /2 dt = 8 5, φ p ( a 2M )=φ p ( /2 2 2 )= 32, φ p ( c 2M )=φ p ( ) = 32, φ p ( b δl )=φ 5 p ( (9/25) 8/5 )= = Next, we how that (H ) (H 4 ) are atified. If u /2, /2 V, w, l /2,then f (u, V,w,l) < <( β)φ p ( a 2M )= 64. (23) (24) So (H ) i atified. If u 6, 6 V, w, l 6,then f (u, V,w,l) < <( β)φ p ( c (25) 2M ) = 32. So (H 2 ) i atified. min f (u, V,w,l), max f (u, V,w,l) , φ p ( M 2m ) = 2 5 2, q (t) dt = δ q (t) dt = 8 5, (27) Hence, it eay to know that (H 4 ) i atified. SobyTheorem 8,weconcludethattheBVP(2)hathree poitive olution u, u 2,andu 3 atifying u < 2, < min δ t<( δ) u 2 (t), Acknowledgment min δ t<( δ) u 3 (t) <. u 3 > 2 (28) Thi paper i upported by the Natural Science Foundation of China (945) and (22), the Natural Science Foundation of Hebei Province (A29664) and (A2282) and the Foundation of Hebei Univerity of Science and Technology (XL2757). Reference [] J. Wang, The exitence of poitive olution for the onedimenional p-laplacian, Proceeding of the American Mathematical Society,vol.25,no.8,pp ,997. [2] L. Kong and J. Wang, Multiple poitive olution for the onedimenional p-laplacian, Nonlinear Analyi,vol.42,no.8,pp , 2. [3] X. He and W. Ge, Twin poitive olution for the onedimenional p-laplacian boundary value problem, Nonlinear Analyi, vol. 56, no. 7, pp , 24. [4]D.Zhao,H.Wang,andW.Ge, Exitenceoftriplepoitive olution to a cla of p-laplacian boundary value problem, Mathematical Analyi and Application,vol.328,no. 2,pp ,27. [5] A. Lakmeche and A. Hammoudi, Multiple poitive olution of the one-dimenional p-laplacian, JournalofMathematical Analyi and Application,vol.37,no.,pp.43 49,26. [6] D.-X. Ma, Exitence and iteration of poitive olution for a three-point boundary value problem with a p-laplacian operator, Applied Mathematic & Computing, vol. 25, no. -2, pp , 27.

7 Abtract and Applied Analyi 7 [7] H. Feng and W. Ge, Exitence of three poitive olution for m-point boundary-value problem with one-dimenional p- Laplacian, Nonlinear Analyi, vol.68,no.7,pp , 28. [8] Y.Guo,C.Yu,andJ.Wang, Exitenceofthreepoitiveolution for m-point boundary value problem on infinite interval, Nonlinear Analyi,vol.7,no.3-4,pp ,29. [9] D. Kong, L. Liu, and Y. Wu, Triple poitive olution of a boundary value problem for nonlinear ingular econdorder differential equation of mixed type with p-laplacian, Computer & Mathematic with Application, vol.58,no.7,pp , 29. [] J.-X. Hu and D.-X. Ma, Triple poitive olution of a boundary value problem for econd order three-point differential equation with p-laplacian operator, Applied Mathematic and Computing, vol. 36, no. -2, pp , 2.

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