Yan Sun * 1 Introduction
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1 Sun Boundry Vlue Problems 22, 22:86 hp:// R E S E A R C H Open Access Posiive soluions of Surm-Liouville boundry vlue problems for singulr nonliner second-order impulsive inegro-differenil equion in Bnch spces Yn Sun Correspondence: ysun@shnu.edu.cn; sunyn@fudn.edu.cn; ysun88@sin.com.cn School of Mhemicl Sciences, Fudn Universiy, Shnghi, 2433, P.R. Chin Deprmen of Mhemics, Shnghi Norml Universiy, Shnghi, 2234, P.R. Chin Absrc In his work, we invesige he exisence of posiive soluions of Surm-Liouville boundry vlue problems for singulr nonliner second-order impulsive inegro differenil equion in rel Bnch spce. Some new exisence resuls of posiive soluions re esblished by pplying fixed-poin index heory ogeher wih comprison heorem. Some discussions nd n exmple re given o demonsre he pplicions of our min resuls. MSC: 34B5; 34B25; 45J5 Keywords: mesure of non-comprison; posiive soluion; boundry vlue problem; impulsive inegro-differenil equion Inroducion In his pper, we sudy he exisence of posiive soluions o second-order singulr nonliner impulsive inegro-differenil equion of he form: y + hf, y, y, Ty, Sy =, J, k, y =k = I k y k, k =,...,m, y =k = I k y k, y k, k =,...,m, αy βy =, γ y + δy =,. where α, β, γ, δ, = βγ + αγ + αδ >,I = [, ], J =,, < < 2 < < m <, J = J \, 2,..., m }, J = [, ], J =, ], J k = k, k+ ], k =,...,m,j m = m,], f C[J P P P P, P], nd P is posiive cone in E. θ is zero elemen of E, I k C[P, P], I k C[P, P], nd Ty= K, sys ds, Sy = H, sys ds,.2 22 Sun; licensee Springer. This is n Open Access ricle disribued under he erms of he Creive Commons Aribuion License hp://creivecommons.org/licenses/by/2., which permis unresriced use, disribuion, nd reproducion in ny medium, provided he originl work is properly cied.
2 Sun Boundry Vlue Problems 22, 22:86 Pge 2 of 8 hp:// in which K C[D, J], D =, s J J : s}, H C[J J, J], nd K = mxk, s:, s D}, H = mxh, s:, s D}. y =k nd y =k denoe he jump of y ndy = k, i.e., y =k = y + k y k, y =k = y + k y k, where y k +, y k + ndy k, y k represen he righ-hnd limi nd lef-hnd limi of y nd y = k,respecively.h CJ, R + ndmybesingulr = nd/or =. Boundry vlue problems for impulsive differenil equions rise from mny nonliner problems in sciences, such s physics, populion dynmics, bioechnology, nd economics ec. see [, 2, 4 4, 6 8]. As i is well known h impulsive differenil equions conin jumps nd/or impulses which re min chrcerisic feure in compuionl biology. Over he ps 5 yers, significn dvnce hs been chieved in heory of impulsive differenil equions. However, he corresponding heory of impulsive inegro-differenil equions in Bnch spces does no develop rpidly. Recenly, Guo [5 8] esblished he exisence of soluion, muliple soluions nd exreml soluions for nonliner impulsive inegro-differenil equions wih nonsingulr rgumen in Bnch spces. The min ools of Guo [5 8] re he Schuder fixed-poin heorem, fixed-poin index heory, upper nd lower soluions ogeher wih he monoone ierive echnique, respecively. The condiions of he Kurowski mesure of non-compcness in Guo [5 8] ply n imporn role in he proof of he resuls. Bu ll kinds of compcness ype condiions is difficul o verify in bsrc spces. As resul, i is n ineresing nd imporn problem o remove or wek compcness ype condiions. Inspired nd moived grely by he bove works, he im of he pper is o consider he exisence of posiive soluions for he boundry vlue problem. undersimpler condiions. The min resuls of problem. re obined by mking use of fixed-poin index heory nd fixed-poin heorem. More specificlly, in he proof of hese heorems, we consruc specil cone for sric se conrcion operor. Our min resuls in essence improve nd generlize he corresponding resuls of Guo [5 8]. Moreover, our mehod is differen from hose in Guo [5 8]. Theresofhepperisorgnizedsfollows:InSecion2, we presen some known resuls nd inroduce condiions o be used in he nex secion. The min heorem formuled nd proved in Secion 3. Finlly,inSecion4, some discussions nd n exmple for singulr nonliner inegro-differenil equions re presened o demonsre he pplicion of he min resuls. 2 Preliminries nd lemms In his secion, we shll se some necessry definiions nd preliminries resuls. Definiion 2. Le E be rel Bnch spce. A nonempy closed se P E is clled cone if i sisfies he following wo condiions: x P, λ >implies λx P; 2 x P, x P implies x =. A cone is sid solid if i conins inerior poins, P θ.aconep is clled o be genering if E = P P, i.e., every elemen y E cnberepresenedinheformy = x z,wherex, z P.
3 Sun Boundry Vlue Problems 22, 22:86 Pge 3 of 8 hp:// AconeP in E induces pril ordering in E given by u v if v u P.Ifu vnd u v, we wrie u < v;ifconep is solid nd v u P,wewrieu v. Definiion 2.2 AconeP E is sid o be norml if here exiss posiive consn N such h x + y N, x, y P, x =, y =. Definiion 2.3 Le E be meric spce nd S be bounded subse of E. Themesureof non-compcness ϒSofS is defined by ϒS=infδ >:S dmisfiniecoverbysesofdimeer δ}. Definiion 2.4 An operor B : D E is sid o be compleely coninuous if i is coninuous nd compc. B is clled k-se-conrcion k if i is coninuous, bounded nd ϒBS kϒs for ny bounded se S D,whereϒS denoes he mesure of noncompcness of S. A k-se-conrcion is clled sric-se conrcion if k <.AnoperorB is sid o be condensing if i is coninuous, bounded, nd ϒBS < ϒS for ny bounded se S D wih ϒS>. Obviously, if B is sric-se conrcion, hen B is condensing mpping, nd if operor B is compleely coninuous, hen B is sric-se conrcion. I is well known h y C 2, C[, ] is soluion of he problem.ifndonlyif x C[, ] is soluion of he following nonliner inegrl equion: y= G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k I k yk, y k + δi k yk, y k γ I k yk ]}, where G, s= γ + δ γ β + αs, s, β + αγ + δ γ s, s, 2. where = γβ + αγ + αδ >. In wh follows, we wrie J =[, ], J k = k, k ]k =,2,...,m, J m = m,]. By mking use of 2., we cn prove h G, s hs he following properies. Proposiion 2. G, s Gs, s α + δα + β,, s [, ]. Proposiion 2.2 σ G, s G, s,, s [, b] [, ],where, ],b [ m,nd β + α <σ = min α + β, δ + bγ } <. 2.2 γ + δ
4 Sun Boundry Vlue Problems 22, 22:86 Pge 4 of 8 hp:// Le PC[J, E] = y : y is mp from J ino E such h y is coninuous k,lef coninuous = k nd y k + exiss for k =,...,m} nd PC[J, P] =y PC[J, E] :y θ}. IisesyoverifyPC[J, E] is Bnch spce wih norm y PC = sup J y. Obviously, PC[J, P] is cone in Bnch spce PC[J, E]. Le PC [J, E] =y : y is mp from J ino E such h y exis nd is coninuous k, y lef coninuous = k,ndy k +, y k exisfork =,...,m}, PC [J, P] =y PC [J, E] :y θ, y θ}. IisesyoseehPC [J, E] isbnchspcewihhe norm y = mx y PC, y PC }. Evidenly, y y PC + y PC nd PC [J, P] iscone in Bnch spce PC [J, E]. For ny y PC [J, E], by mking use of he men vlue heorem y k y k h hcoy : k h < < k } h >, obviously we see h y k exissnd y k =lim h y k y k h. h Le K = y PC[J, P]:y σ y, [, b]}. For ny < r < R <+,lek r = y K : y < r}, K r = y K : y = r}, K r,r = y K : r y R}. Ampy PC [J, E] C 2 [J, E] is clled nonnegive soluion of problem.ify θ, y θ for J nd y sisfiesproblem.. An operor y PC [J, E] C 2 [J, E] is clled posiive soluion of problem. ify is nonnegive soluion of problem. nd y θ. For convenience nd simpliciy in he following discussion, we denoe f ν = lim inf min 4 y i ν [,b] f ν = lim sup mx 4 [,b] y i ν f, y, y 2, y 3, y 4 4 y, i f, y, y 2, y 3, y 4 4 y, i where ν denoe or. To esblish he exisence of muliple posiive soluions in E of problem., le us lis he following ssumpions, which will snd hroughou he pper: H f CJ P 4, P, h CJ, P nd f, y, y 2, y 3, y b i y i, 2.3 where nd b i relebesgueinegrblefuncionlsonj i =,2,3,4ndsisfying < shs ds, Gs, shs ds, 2 hs b i s+k b 3 s+h b 4 s ds < 2 G, shs ds <+, α + βγ + δ H 2 I k CP, P, I k CP, P nd here exis posiive consns c k, c k nd c k k =,...,m sisfying, m ck + c k + c k m + α + β [ ] γ + δ c k + c k + γ ck < 4 2.4
5 Sun Boundry Vlue Problems 22, 22:86 Pge 5 of 8 hp:// such h Ik y ck y, Ik y, y 2 c k y + c k y 2, J, H 3 for ny bounded se B i E, i =,2,3,4, f, B, B 2, B 3, B 4 nd I k B ogeher wih I k B, B 2 re relively compc ses, H 4 f > m, H 5 f < m,where m = mx σ α + βγ + δ Gs, shs ds, G, shs ds, } hss ds. 2.5 We shll reduce problem.oninegrlequionine. To his end, we firs consider operor A : K PC[J, E]definedby Ay= G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β γ k +δ I k yk, y k γ m I k yk }. 2.6 Lemm 2. y PC [J, E] C 2 [J, E] is soluion of problem.ifndonlyify PC [J, E] is soluion of he following impulsive inegrl equion: y= G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ] 2.7 i.e., y is fixed poin of operor A defined by 2.6inPC [J, E]. Proof Firs suppose h y PC [J, E] is soluion of problem.. I is esy o see by he inegrion of problem.h y =y f s, ys, y s, Tys, Sys ds + < k < [ y k + y k ] = y f s, ys, y s, Tys, Sys ds I k yk, y k, J. 2.8 < k <
6 Sun Boundry Vlue Problems 22, 22:86 Pge 6 of 8 hp:// Inegregin,wege y=y + y sf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ], J. 2.9 < k < Leing =in2.8nd2.9, we find h y = y f s, ys, y s, Tys, Sys ds m I k yk, y k. 2. y = y + y sf s, ys, y s, Tys, Sys ds m [ + Ik yk k I k yk, y k ]. 2. Since y = β α y, y = δ y γ We ge y = α δ + γ f s, ys, y s, Tys, Sys ds f s, ys, y s, Tys, Sys ds sf s, ys, y s, Tys, Sys ds m I k yk, y k }. 2.2 m [ + γ k +δ I k yk, y k γ I k yk ]}, J. 2.3 Subsiuing 2.2 nd2.3 ino2.9, we obin y= α + β γ s+δ f s, ys, y s, Tys, Sys ds m [ γ Ik yk γ k +δ I k yk, y k ]} γ sf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k <
7 Sun Boundry Vlue Problems 22, 22:86 Pge 7 of 8 hp:// = [ α + β γ s+δ s ] f s, ys, y s, Tys, Sys ds + α + β γ s+δ f s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ] = G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ]. Conversely, if y PC [J, E] is soluion of he inegrl equion 2.7. Evidenly, y =k = I k y k k =,...,m. For = k, direc differeniion of he inegrl equion 2.7implies y = γ + α αs + βf s, ys, y s, Tys, Sys ds γ s+δ f s, ys, y s, Tys, Sys ds I k yk, y k } α < k < m [ γ Ik yk γ k +δ I k yk, y k ] nd y = f, y, y, Ty, Sy. So y C 2 [J, E] nd y =k = I k y k, y k k =,...,m. I is esy o verify h αy βy = nd γ y + δy =. The proof is complee. Thnks o 2., we know h G, s= γ β + αs, s, αγ + δ γ s, s. In he following, le w = sup,s J, s G, s. ForB PC [J, E], we denoe B = y : y B} PC[J, E], B=y:y B} E nd B =y :y B} E J. Lemm 2.2 [2] Le D PC [J, E] be bounded se. Suppose h D is equiconinuous on ech J k k =,...,m. Then ϒ PC D=mx sup J ϒ D, sup ϒ D }, J
8 Sun Boundry Vlue Problems 22, 22:86 Pge 8 of 8 hp:// where ϒ nd ϒ PC denoe he Kurowski mesures of noncompcness of bounded ses in EndPC [J, E], respecively. Lemm 2.3 [5] Le H PC[J, E] be bounded equiconinuous, hen ϒH is coninuous on J nd } ϒ y d : y H ϒ H d. J J Lemm 2.4 [5] H PC [J, E] is relively compc if nd only if ech elemen y H nd y H re uniformly bounded nd equiconinuous on ech J k k =,...,m. Lemm 2.5 [5] Le E be Bnch spce nd H C[J, E] if H is counble nd here exiss ϕ L[J, R + ] such h y ϕ, J, y H. Then ϒy:y H} is inegrble on J, nd } ϒ y d : y H 2 ϒ y:y H } d. J J Lemm 2.6 AK K. Proof For ny y K, from Proposiion 2. nd 2.6, we obin Ay Gs, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k ] m γ I k yk }. On he oher hnd, for ny [, b], by 2.6 nd Proposiion 2.2,weknowh Ay = G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ] σ Gs, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ]} σ Ay. Hence, AK K.
9 Sun Boundry Vlue Problems 22, 22:86 Pge 9 of 8 hp:// Lemm 2.7 Suppose h H ndh 3 hold. Then A : K K is compleely coninuous. Proof Firsly, we show h A : K K is coninuous. Assume h y n, y K nd y n y, y n y n. Since f CJ P P P P, P, I k CJ, P nd I k CJ, P, hen lim f, y n, y n,ty n, Sy n f, y, y,ty, Sy =, 2.4 lim I k yn k I k y k =, k =,...,m, n n nd lim I k yn k, y n k I k y k, y k =, k =,...,m. 2.5 n Thus, for ny J, from he Lebesgue domined convergence heorem ogeher wih 2.4nd2.5, we know h Ay n Ay Gs, shs f s, y n s, y n s, Ty ns, Sy n s f s, y s, y s, Ty s, Sy s ds + [ Ik yn k I k y k < k < k I k yn k, y n k I k y k, y k ] + m α + β [ γ k +δ I k yn k, y n k I k y k, y k γ I k yn k I k y k ] Gs, shs f s, yn s, y n s, Ty ns, Sy n s f s, y s, y s, Ty s, Sy s ds + [ I k yn k I k y k < k < + k I k yn k, y n k I k y k, y k ] + m α + β [ γ k +δ Ik yn k, y n k I k y k, y k + γ Ik yn k I k y k ] sn. Hence, A : K K is coninuous.
10 Sun Boundry Vlue Problems 22, 22:86 hp:// Pgeof8 Le B K be ny bounded se, hen here exiss posiive consn R such h y R.Thus,fornyy B,,, we know h Ay = γ αs + βhsf s, ys, y s, Tys, Sys ds + α γ s+δ hsf s, ys, y s, Tys, Sys } ds I k yk, y k + α < k < m [ γ k +δ I k yk, y k γ I k yk ]. Therefore, Ay γ αs + βhs f s, ys, y s, Tys, Sys ds + α γ s+δ hs f s, ys, y s, Tys, Sys } ds + < k < I k yk, y k + α m [ γ k +δ I k yk, y k + γ Ik yk ] γ αs + βhs s 2 + b i s+k b 3 s+h b 4 s y ds + α γ s+δ hs s 2 + b i s+k b 3 s+h b 4 s ys } ds m + c k + c k ys + α m [ γ k +δ c k + c ] k + γ ck ys 2 γ αs + βhs s+ b i s+k b 3 s+h b 4 s R ds 2 } + α γ s+δ hs s+ b i s+k b 3 s+h b 4 s ds + m c k + c k R + α m [ γ k +δ c k + c ] k + γ ck R, 2.6 R
11 Sun Boundry Vlue Problems 22, 22:86 hp:// Pgeof8 J, k, k =,...,m.le ψ= γ + α 2 αs + βhs s+ b i s+k b 3 s+h b 4 s R ds 2 γ s+δ hs s+ b i s+k b 3 s+h b 4 s ds, m M = c k + c k R + α m [ γ k +δ c k + c ] k + γ ck R. Inegring ψ from o nd exchnging inegrl sequence, hen ψ d = s 2 αγ + δ γ shs s+ b i s + K b 3 s+h b 4 s R d ds γ s β + αshs s 2 + b i s+k b 3 s+h b 4 s R d ds 2 α sγ + δ γ shs s+ b i s+k b 3 s+h b 4 s ds γ 2 β + αshs s+ b i s+k b 3 s+h b 4 s 2 Gs, shs s+ b i s+k b 3 s+h b 4 s < R R R ds R ds Thus, by H nd2.7, we hve ψ L J. Hence, for ny 2 ndforll y E,from2.6, we know h Ay Ay = Ay d ψ+m d. 2.8 From 2.7, 2.8, nd he bsoluely coninuiy of inegrl funcion, we see h AB is equiconinuous. On he oher hnd, for ny y B nd J,weknowh Ay = G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ]}
12 Sun Boundry Vlue Problems 22, 22:86 hp:// Pge2of8 + 2 Gs, shs s+ b i s+k b 3 s+h b 4 s m c k + c k R + α R ds m [ γ k +δ c k + c ] k + γ ck R <+. Therefore, AB is uniformly bounded. By virue of Lemm 2.3 nd H 3, we know h ϒ Ay:y B } = ϒ G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ] Gs, shsϒ f s, ys, y s, Tys, Sys ds + [ ϒ Ik yk + k ϒ I k yk, y k ] =. < k < + m α + β [ γ k +δ ϒ I k yk, y k + γϒ I k yk ]} So, ϒAB =. Therefore, A is compc. To sum up, he conclusion of Lemm 2.7 follows. The min ools of he pper re he following well-known fixed-poin index heorems see [2 4]. Lemm 2.8 Le A : K K be compleely coninuous mpping nd Ay yfory K r. Thus, we hve he following conclusions: i If y Ay for y K r,henia, K r, K=. ii If y Ay for y K r,henia, K r, K=. 3 Min resuls In his secion, we esblish he exisence of posiive soluions for problem.by mking use of Lemm 2.8. Theorem 3. Suppose h H -H 4 hold. Then problem. hs les one posiive soluion. Proof From H 4, here exiss ε >suchhf > m + ε nd lso here exiss r >such h for ny < 4 y i r nd [, b], we hve f, y, y 2, y 3, y 4 m + ε 4 y i. 3.
13 Sun Boundry Vlue Problems 22, 22:86 hp:// Pge3of8 Se K r = y K : y < r}. Then for ny y K r K,byvirueof3., we know h Ay = G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ]} σ σ m + ε G, shsf s, ys, y s, Tys, Sys ds Gs, shsm + ε ys + y s + Tys + Sys ds Gs, shs ds ys mσ Gs, shs ds ys = r. Ay = G, shsf s, ys, y s, Tys, Sys ds I k yk, y k < k < + α m [ γ k +δ I k yk, y k γ I k yk ]} m + ε m G, shsf s, ys, y s, Tys, Sys ds G, shsm + ε ys + y s + Tys + Sys ds So Ay r. Therefore, G, shs ds ys G, shs ds ys = r. ia, K r K, K=. 3.2 Le R > mx4m,2r}.thenk R is bounded open subses in E,ndsofornyy K R K nd J,weobin Ay α + βγ + δ hss+b s ys + b2 s y s + b 3 s Tys + b 4 s Sys ds + [ Ik yk + k Ik yk, y k ] < k < + m α + β [ γ k +δ Ik yk, y k + γ Ik yk ]
14 Sun Boundry Vlue Problems 22, 22:86 hp:// Pge4of8 α + βγ + δ hss ds + [ 2 ] α + βγ + δ hs b i s+k b 3 s+h b 4 s ds y m + c m [ ] } k + c k + c k + γ + δ c k + c k + γ ck y m + 2 y + 4 y < 4 R + 2 R + 4 R = R, Ay γ αs + βhsf s, ys, y s, Tys, Sys ds + α γ s+δ hsf s, ys, y s, Tys, Sys } ds I k yk, y k + α < k < m [ γ k +δ I k yk, y k γ I k yk ] α + βγ + δ hsf s, ys, y s, Tys, Sys ds + I k yk, y k + α < k < m [ γ k +δ Ik yk, y k + γ I k yk ] α + βγ + δ hs s+b s ys + b2 s y s + b 3 s Tys + b 4 s Sys ds + [ Ik yk + k Ik yk, y k ] < k < + m α + β [ γ k +δ Ik yk, y k + γ Ik yk ] α + βγ + δ hss ds + [ 2 ] α + βγ + δ hs b i s+k b 3 s+h b 4 s ds y Hence, Ay < R. Therefore, m + c m [ ] } k + c k + c k + γ + δ c k + c k + γ ck y m + 2 y + 4 y < 4 R + 2 R + 4 R = R. ia, K R K, K=. 3.3
15 Sun Boundry Vlue Problems 22, 22:86 hp:// Pge5of8 From 3.2nd3.3, we ge i A,K R K \ K r K, K = ia, K R K, K ia, K r K, K=. Therefore, A hs les one fixed poin on K R K \ K r K. Consequenly, problem. hs les one posiive soluion. Theorem 3.2 Suppose h H H 3 ndh 5 re sisfied. Then problem. hs les one posiive soluion. Proof From H 5, we cn choose ε >suchhf > m + ε nd lso here exiss R > such h for ny 4 y i R nd [, b], we hve f, y, y 2, y 3, y 4 m + ε 4 y i. 3.4 Le R > R.Byvirueof3.4, we know h σ f, y, y, Ty, Sy m + ε y + y + Ty + Sy. 3.5 Se K R = y K : y < R }. Then for ny y K R K,byvirueof3.5, we know h Ay = G, shsf s, ys, y s, Tys, Sys ds + [ Ik yk k I k yk, y k ] < k < + m α + β [ γ k +δ I k yk, y k γ I k yk ]} σ σ m + ε mσ Ay = G, shsf s, ys, y s, Tys, Sys ds Gs, shsm + ε y + y s + Tys + Sys ds Gs, shs ds ys Gs, shs ds ys = R. G, shsf s, ys, y s, Tys, Sys ds I k yk, y k < k < + α m [ γ k +δ I k yk, y k γ I k yk ]} G, shsf s, ys, y s, Tys, Sys ds
16 Sun Boundry Vlue Problems 22, 22:86 hp:// Pge6of8 m + ε m G, shsm + ε y + y + Tys + Sys ds So, Ay r. Therefore, G, shs ds ys G, shs ds ys = r. ia, K R K, K=. 3.6 Byhesmemehodsheselecionofr in Theorem 3., we cn obin r < R sisfying ia, K r K, K =. 3.7 According o 3.6nd3.7, we ge i A,K R K \ K r K, K = ia, K R K, K ia, K r K, K=. Therefore, A hs les one fixed poin on K R K \ K r K. Consequenly, problem. hs les one posiive soluion. The proof is complee. 4 Concerned resuls nd pplicions In his secion, we del wih specil cse of he problem.. The mehod is jus similrowhwehvedoneinsecion3, so we omi he proof of some min resuls of he secion. Cse F, x, x = f, x, x, Ax, Bx is reed in he following heorem. Under he cse, he problem. reduces o he following boundry vlue problems: y +hf, y, y = θ, J, k, y =k = I k y k, y =k = I k y k, y k, k =,...,m, αy βy =, γ y + δy =, 4. where F CJ P P, P, h CJ. Theorem 4. Assume h H 2 holds, nd he following condiions re sisfied: C F CJ P 2, P, h CJ, P nd F, y, y b i y i, where nd b i re Lebesgue inegrble funcionls on J i =,2 nd sisfying < shs ds, Gs, shs ds, G, shs ds <+,
17 Sun Boundry Vlue Problems 22, 22:86 hp:// Pge7of8 2 hs b i s+k b 3 s+h b 4 s ds < α + βγ + δ, 2 C 2 for ny bounded se B i E i =,2, F, B, B 2 nd I k B ogeher wih I k B, B 2 re relively compc ses. C 3 lim y + y 2 F,y,y 2 y + y 2 > m, where m is defined by 2.4. Then he problem 4. hs les one posiive soluion. Theorem 4.2 Assume h H 2 ndc C 2 hold, nd he following condiion is sisfied: C 4 lim y + y 2 F,y,y 2 y + y 2 > m, where m is defined by 2.4. Then he problem 4.hs les one posiive soluion. To illusre how our min resuls cn be used in prcice, we presen n exmple. Exmple 4. Consider he following boundry vlue problem for sclr second-order impulsive inegro-differenil equion: y = π sin ln y+ e s yds+ e 2s yds 72 + y+ e s ys ds+, e 2s ys ds 2, y = = 3 2 y3 3, y = = 2 y3 3 +y 3, 3 y 3 +y 3 2 αy βy =, γ y + δy = Conclusion The problem 4.2 hs les one posiive soluion y. For J,,leh= π 72, F, y, y,ty, Sy = sin ln y + e s yds+ e 2s yds + y + e s ys ds +. e 2s ys ds 2 Choose =sin. By simple compuion, we know h shs ds = π 8 sin2 2 <+, G, shs ds π 36 <+. Gs, shs ds = π 2,7 <+, Then condiions H H 4 re sisfied. Therefore, by Theorem 3., heproblem4.2 hs les one posiive soluion. Remrk 5. In [2], by requiring h f sisfies some noncompc mesure condiions nd P is norml cone, Guo esblished he exisence of posiive soluions for iniil vlue problem.inhepper,weimposesomewekercondiiononf,weobinheposiive soluion of he problem..
18 Sun Boundry Vlue Problems 22, 22:86 hp:// Pge8of8 Remrk 5.2 For he specil cse when he problem. hs no singulriies nd J =[,], our resuls sill hold. Obviously, our heorems generlize nd improve he resuls in [9 2]. Compeing ineress The uhor declres h she hs no compeing ineress. Acknowledgemens The uhor is very greful o Professor Lishn Liu nd Professor R. P. Agrwl for heir mking mny vluble commens. The uhor would like o express her hnks o he edior of he journl nd he nonymous referees for heir crefully reding of he firs drf of he mnuscrip nd mking mny helpful commens nd suggesions which improved he presenion of he pper. The uhor ws suppored finncilly by he Foundion of Shnghi Municipl Educion Commission Grn Nos. DYL25. Received: 5 June 22 Acceped: 23 July 22 Published: 6 Augus 22 References. Wu, CZ, Teo, KL, Zhou, Y, Yn, WY: An opiml conrol problem involving impulsive inegro-differenil sysems. Opim. Mehods Sofw. 22, Wu, CZ, Teo, KL, Zhou, Y, Yn, WY: Solving n idenificion problem s n impulsive opiml prmeer selecion problem. Compu. Mh. Appl. 5, Guo, D, Lkshmiknhm, V: Nonliner Problems in Absrc Cones. Acdemic Press, Boson Guo, D: Boundry vlue problems for impulsive inegro-differenil equions on unbounded domins in Bnch spce. Appl. Mh. Compu. 99, Guo, D: Exisence of posiive soluions for nh order nonliner impulsive singulr inegro-differenil equions in Bnch spce. Nonliner Anl. 68, Guo, D: Exisence of soluions for nh order impulsive inegro-differenil equions in Bnch spce. Nonliner Anl. 44, Guo, D: Exreml soluions for nh order impulsive inegro-differenil equions on he hlf-line in Bnch spce. Nonliner Anl. 65, Guo, D:Muliple posiivesoluions for nh order impulsive inegro-differenil equions in Bnch spce. Nonliner Anl. 6, Liu, L, Xu, Y, Wu, Y: On unique soluion of n iniil vlue problem for nonliner firs-order impulsive inegro-differenil equions of Volerr ype in Bnch spces. Dyn. Conin. Discree Impuls. Sys., Ser. A Mh. Anl. 3, Xu, Y: A globl soluions of iniil vlue problems for second order impulsive inegro differenil equions in Bnch spces. Ac Mh. Sin., Chin. Ser. 25, Liu, L, Wu, Y, Zhng, X: On well-posedness of n iniil vlue problem for nonliner second order impulsive inegro differenil equions of Volerr ype in Bnch spces. J. Mh. Anl. Appl. 37, Zhng, X, Liu, L: Iniil vlue problem for nonliner second order impulsive inegro differenil equions of mixed ype in Bnch spces. Nonliner Anl. 64, Guo, D: Exisence of soluions for nh-order impulsive inegro-differenil equions in Bnch spce. Nonliner Anl. 47, Liu, L, Wu, C, Guo, F: A unique soluion of iniil vlue problems for firs order impulsive inegro-differenil equions of mixed ype in Bnch spces. J. Mh. Anl. Appl. 275, Liu, L: Ierive mehod for soluion nd coupled qusi-soluions of nonliner inegro differenil equions of mixed ype in Bnch spces. Nonliner Anl. 42, Liu, L: The soluions of nonliner inegro-differenil equions of mixed ype in Bnch spces. Ac Mh. Sin., Chin. Ser. 38, Zhng, X, Feng, M, Ge, W: Exisence of soluions of boundry vlue problems wih inegrl boundry condiions for second-order impulsive inegro-differenil equions in Bnch spces. J. Compu. Appl. Mh. 233, Feng, M, Png, H: A clss of hree-poin boundry-vlue problems for second-order impulsive inegro-differenil equions in Bnch spces. Nonliner Anl. 7, doi:.86/ Cie his ricle s: Sun: Posiive soluions of Surm-Liouville boundry vlue problems for singulr nonliner second-order impulsive inegro-differenil equion in Bnch spces. Boundry Vlue Problems 22 22:86.
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