The focal boundary value problem for strongly singular higher-order nonlinear functional-differential equations

Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 DOI 10.1186/s13661-014-0277-1 RESEARCH Open Access The focal boundary value problem for strongly singular higher-order nonlinear functional-differential equations Sulkhan Mukhigulashvili1,2* and Bedˇ rich P˚uža2 *Correspondence: smuk[email protected] 1Mathematical Institute, Academy of Sciences of the Czech Republic, Žižkova 22, Brno, 616 62, Czech Republic 2Faculty of Business and Management, Brno University of Technology, Kolejni 2906/4, Brno, 612 00, Czech Republic Abstract The aprioriboundedness principle is proved for the two-point right-focal boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the two-point right-focal problem under consideration are derived from the aprioriboundedness principle. The proof of the aprioriboundedness principle is based on Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear differential equations with argument deviations under the two-point right-focal boundary conditions. MSC: Primary 34B15; 34K10 Keywords: higher order nonlinear functional-differential equations; two-point right-focal boundary value problem; strong singularity; Fredholm property 1 Statement of the main results 1.1 Statement of the problem and the literature survey Consider the functional differential equation u(n)(t)=F(u)(t) (.) with the two-point boundary conditions u(i–)(a)= (i=,...,m), u(j–)(b)= (j=m+,...,n). (.) Heren≥,mistheintegerpartofn/,–∞<a<b<+∞,andtheoperatorFactsfromthe set of (m–)th time continuously differentiable on ]a,b] functions to the set Lloc(]a,b]). By u(i–)(a) we denote the right limit of the function u(i–) at the point a. Theproblemissingularinthesensethatforanarbitraryu∈Cm–(]a,b]) theright-hand side of equation (.) may have nonintegrable singularities at the point a. Throughoutthe paper we use the following notations: R+=[,+∞[; [x]+the positive part of number x,thatis,[x]+=x+|x| ; Lloc(]a,b]) is the space of functions y:]a,b]→R,whichareintegrableon[a+ε,b]for arbitrarily small ε>; ©2015 Mukhigulashvili and P˚uža; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 2 of 21 Lα(]a,b])(L α(]a,b]))isthespaceofintegrable(squareintegrable)withtheweight(t–a)α functions y:]a,b]→Rwith the norm yLα=b a(s–a)αy(s)ds yL α=b a(s–a)αy(s)ds/; L([a,b])=L(]a,b]), L([a,b])=L (]a,b]); M(]a,b]) is the set of measurable functions τ:]a,b]→]a,b];  L α(]a,b]) is the Banach space of y∈Lloc(]a,b]) functions with the norm y L α≡maxt a(s–a)αt sy(ξ)dξds/ :a≤t≤b; Ln(]a,b]) is the Banach space of y∈Lloc(]a,b]) functions with the norm yLn=sup(s–a)m–/ t s(ξ–a)n–my(ξ)dξ:a<s≤t≤b<+∞;  Cn– loc (]a,b])isthespaceoffunctionsy:]a,b]→R,whicharecontinuous(absolutelycontinuous) together with y,y,...,y(n–) on [a+ε,b] for arbitrarily small ε>;  Cn–,m(]a,b]) is the space of functions y∈ Cn– loc (]a,b]) such that b ax(m)(s)ds<+∞;(.) Cm– (]a,b]) is the Banach space of functions y∈Cm– loc (]a,b]) such that limsup t→a|x(i–)(t)| (t–a)m–i+/ <+∞(i=,...,m)(.) with the norm xCm– =m i= sup{|x(i–)(t)| (t–a)m–i+/ :a<t≤b};  Cm– (]a,b]) is the Banach space of functions y∈ Cm– loc (]a,b]) such that conditions (.) and (.)holdwiththenormx Cm– =xCm– +(b a|x(m)(s)|ds)/; Dn(]a,b]×R+)isthesetofsuchfunctionsδ:]a,b]×R+→Ln(]a,b])thatδ(t,·):R+→R+ is nondecreasing for every t∈]a,b], and δ(·,ρ)∈Ln(]a,b]) for any ρ∈R+. Asolutionofproblem(.), (.) is sought in the space  Cn–,m(]a,b]). The principles of the theory of singular boundary value problems were built by Kiguradze in his study []. This theory has been intensively developed and studied with sufficient completeness both for the ordinary differential equations and the functional differential equations (see [–]). But equation (.), even under the boundary condition (.), is not studied in the case when the operator Fhastheform F(x)(t)=m j= pj(t)x(j–)(τj(t))+q(x)(t),wherethesingularities of the functions pj:Lloc(]a,b]) (j=,...,m) are such that the inequalities b a(s–a)n–(–)n–mp(s)+ds<+∞,b a(s–a)n–jpj(s)ds<+∞(.) are notfulfilled (in this case wesay that the linear part of theoperator Fisstrongly singular), the operator qcontinuously acts from Cm– (]a,b]) to L L n–m–(]a,b]), and the inclu- Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 3 of 21 sion supq(x)(t):xCm– ≤ρ∈ L n–m–]a,b] holds. The first step in studying the differential equations with strong singularities was made by Agarwal and Kiguradze in the article [], where the linear ordinary differential equations under conditions (.),inthecasewhenthefunctionspjhave strong singularities at the points aand b, are studied. Also the ordinary differential equations with strong singularities under two-point boundary conditions are studied in the articles [, ]byKiguradze.Inthepapers[–] these results are generalized for a linear differential equation with deviating arguments, i.e.,theAgarwal-Kiguradzetypetheoremsare proved, which guarantee the Fredholm property for the linear differential equation with deviating arguments. In this paper, on the basis of articles [,], we prove the apriori boundedness principle for problem (.), (.) from which several sufficient conditions of the solvability of this problem follow. Nowweintroduce some results from the articles[,]inthissection,whichweneed for this work. Consider the equation u(n)(t)= m  j= pj(t)u(j–)τj(t)+q(t)fora<t<b(.) with q,pj∈Lloc(]a,b]). By hj:]a,b]×]a,b]→R+and fj:[a,b]×M(]a,b])→Cloc(]a,b]×]a,b]) (j=,...,m)we denote the functions and the operator, respectively, defined by the equalities h(t,s)=t s(ξ–a)n–m(–)n–mp(ξ)+dξ, hj(t,s)=t s(ξ–a)n–mpj(ξ)dξ, (.) and fj(c,τj)(t,s)=t s(ξ–a)n–mpj(ξ)τj(ξ) ξ(ξ–c)(m–j)dξ / dξ.(.) Let also k=k+(k∈Z), then k!!=⎧ ⎨ ⎩ fork≤, ·····kfor k≥. Nowwecanintroducethemaintheoremofthepapers[]and[]. Theorem . Letthereexistthenumbersj>,j≥, and γj>(j=,...,m)such that along with B≡m  j= (m–j)m–j+j (m–)!!(m–j+)!! +m–j–(b–a)γjj (m–j–)!!(m–)!!γj<, (.) Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 4 of 21 the conditions (t–a)m–jhj(t,s)≤j,(t–a)m–γj–/fj(a,τj)(t,s)≤j(.) hold for a <t≤s≤b.Then problem (.), (.)is uniquely solvable in the space  Cn–,m(]a,b]). Remark. From Lemma . it is clear that any solution of problem (.), (.)fromthe space  Cn–,m(]a,b])belongsalsotothespace Cm– (]a,b]). Theorem . Let all the conditions of Theorem . be satisfied.Then the unique solution uofproblem(.), (.)for every q ∈ L n–m–(]a,b]) admits the estimate u(m)L≤rq L n–m–, (.) with r=m–(n–m–) (νn–B)(m–)!! ,νm=, νm+ =m+ , and thus constant r >depends only on the numbers j,j,γj(j=,...,m), and a,b,n. Remark . Under the conditions of Theorem .,foreveryq∈ L n–m–(]a,b]), the unique solution uof problem (.), (.) admits the estimate u Cm– ≤rnq L n–m–, (.) with rn=(+m j= (m–j+)–/ (m–j)! )m–(n–m–) (νn–B)(m–)!! . 1.2 Theorems on the solvability of problem (1.1), (1.2) Define the operator P:Cm– (]a,b])×Cm– (]a,b])→Lloc(]a,b]) by the equality P(x,y)(t)= m  j= pj(x)(t)y(j–)τj(t)for a<t≤b, (.) where pj:Cm– (]a,b]) →Lloc(]a,b]) and τj∈M(]a,b]). Also, for any γ>,definetheset Aγby the relation Aγ=x∈ Cm– ]a,b]:x Cm– ≤γ. (.) Now, following the article [] by Kiguradze and Půža, we introduce the following definitions. Definition . Let γand γbe positive numbers. We say that the continuous operator P:Cm– (]a,b])×Cm– (]a,b])→Ln(]a,b])isγ,γconsistentwithboundarycondition(.) if: Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 5 of 21 (i) for any x∈Aγand almost all t∈]a,b],theinequality m  j= pj(x)(t)x(j–)τj(t)≤δt,x Cm– x Cm– (.) holds, where δ∈Dn(]a,b]×R+); (ii) for any x∈Aγand q∈ L n–m–(]a,b]),theequation y(n)(t)= m  j= pj(x)(t)y(j–)τj(t)+q(t) (.) under boundary conditions (.)has the unique solution yin the space  Cn–,m(]a,b]) and y Cm– ≤γq L n–m–. (.) Definition . We say that the operator Pis γconsistent with boundary condition (.) if the operator Pis γ,γconsistent with boundary condition (.) for any γ>. In the sequel it will always be assumed that the operator Fpis defined by the equality Fp(x)(t)=F(x)(t)– m  j= pj(x)(t)x(j–)τj(t)(t), continuously acting from Cm– (]a,b]) to L L n–m–(]a,b]), and  Fp(t,ρ)≡supFp(x)(t):xCm– ≤ρ∈ L n–m–]a,b](.) for each ρ∈[,+∞[. Then the following theorem is valid. Theorem. Let theoperator P be γ,γconsistentwith boundary condition(.),and let there exist a positive number ρ≤γsuch that  Fp·,min{ρ,γ} L n–m– ≤γ γ. (.) Let,moreover,for any λ∈],[, an arbitrary solution x ∈Aγof the equation x(n)(t)=(–λ)P(x,x)(t)+λF(x)(t) (.) under conditions (.)admit the estimate x Cm– ≤ρ. (.) Then problem (.), (.)is solvable in the space  Cn–,m(]a,b]). From Theorem . with ρ=γ, the corollary immediately follows. Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 6 of 21 Corollary. Let the operator P be γ,γconsistent with boundary condition (.), and F(x)(t)– m  j= pj(x)(t)x(j–)τj(t)(t)≤ηt,x Cm– (.) for x∈Aγand almost all t ∈]a,b], and η(·,γ) L n–m– ≤γ γ, (.) where η∈Dn–m–(]a,b]×R+). Then problem (.), (.)is solvable in the space  Cn–,m(]a,b]). Corollary. LettheoperatorPbeγconsistentwithboundarycondition(.),letinequality (.)hold for x ∈ Cm– (]a,b]) and almost all t ∈]a,b], where η(·,ρ)∈ L n–m–(]a,b]) for any ρ∈R+,and limsup ρ→+∞  ρη(·,ρ) L n–m– < γ. (.) Then problem (.), (.)is solvable in the space  Cn–,m(]a,b]). Now define the operators hj:Cm– (]a,b])×]a,b]×]a,b]→Lloc(]a,b]×]a,b]), fj: Cm– (]a,b])×[a,b]×M(]a,b])→Cloc(]a,b]×]a,b]) (j=,...,m)bytheequalities h(x,t,s)=t s(ξ–a)n–m(–)n–mp(x)(ξ)+dξ, hj(x,t,s)=t s(ξ–a)n–mpj(x)(ξ)dξ(j=,...,m), (.) fj(x,c,τj)(t,s)=t s(ξ–a)n–mpj(x)(ξ)τj(ξ) ξ(ξ–c)(m–j)dξ / dξ(.) and the functions αj:[a,b]→R+by the equality αj(t)=(t–a)m–j+/. Theorem . Let the continuous operator P :Cm– (]a,b])×Cm– (]a,b]) →Ln(]a,b]) admitcondition(.)whereδ∈Dn(]a,b]×R+),τj∈M(]a,b]),andletthenumbersγ∈]a,b], lj>,lj>,γj>(j=,...,m)be such that the inequalities (t–a)m–jhj(x,t,s)≤lj,limsup t→a(t–a)m– –γjfj(x,a,τj)(t,s)≤lj(.) fora<t≤s≤b,x Cm– ≤γ,andconditions(.)hold.Let,moreover,theoperatorF and the function η∈Dn–m–(]a,b]×R+)be such that condition (.)and the inequality η(·,γ) L n–m– <γ rn, (.) are fulfilled,where rn=(+m j= (m–j+)–/ (m–j)! )m–(n–m–) (νn–B)(m–)!! .Thenproblem(.),(.)issolvable in the space  Cn–,m(]a,b]). Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 7 of 21 Theorem . Let the operator F and the function ηbe such that conditions (.), (.) hold,and let the continuous operator P :Cm– (]a,b]) ×Cm– (]a,b]) →Ln(]a,b]) admit condition (.), where δ∈Dn(]a,b]×R+). Let,moreover,the measurable functions τj∈ M(]a,b]) and the numbers lj>,lj>,γj>(j=,...,m)be such that the inequalities (t–a)m–jhj(x,t,s)≤lj,limsup t→a(t–a)m– –γjfj(x,a,τj)(t,s)≤lj(.) for a <t≤s≤b,x∈ Cm– (]a,b]), and conditions (.)hold.Then problem (.), (.)is solvable in the space  Cn–,m(]a,b]). Remark. Letγ> ,lettheoperatorsαjpj(j=,...,m)continuouslyactfromthespace Cm– (]a,b]) to the space Ln(]a,b]), let there exist the function δj∈Dn(]a,b]) such that for any x∈Aγ, pj(x)(t)αj(t)≤δjt,x Cm– for a<t≤b, (.) and let there exist constants κ>,ε>suchthat τj(t)–t≤κ(t–a)(j=,...,m)fora<t<a+ε. (.) Then the operator Pdefined by equality (.) continuously acts from Aγto the space Ln(]a,b]), and there exists the function δ∈Dn(]a,b]) such that item (i) of Definition . holds. Now consider the equation with deviating arguments u(n)(t)=ft,uτ(t),uτ(t),...,u(m–)τm(t) for a<t≤b, (.) where–∞<a<b<+∞,f:]a,b]×Rm→RisafunctionsatisfyingthelocalCarathéodory conditions and τj∈M(]a,b]) (j=,...,n–) are measurable functions. Corollary . Let the functions τj∈M(]a,b]) and the numbers κ≥, ε>,lj>,lj>, γj>(j=,...,m)be such that conditions (.), (.), (.)and the inclusions αjpj∈Ln]a,b](j=,...,m) (.) are fulfilled.Let,moreover, ft,xτ(t),xτ(t),...,x(m–)τm(t)–m  j= pj(t)x(j–)τj(t)(t) ≤ηt,x Cm–  for x ∈ Cm– (]a,b]) and almost all t ∈]a,b], where η(·,ρ)∈ L n–m–(]a,b]) for any ρ∈ R+,and let condition (.)hold.Then problem (.), (.)is solvable in the space  Cn–,m(]a,b]). Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 8 of 21 Remark. Conditions (.) do not follow from conditions (.). Now,toillustrateourresults, consideron ]a,b]thesecond-orderfunctional-differential equations u(t)=– λ|u(t)|k (t–a)+k/ uτ(t)+q(x)(t), (.) u(t)=–λ|sinuk(t)| (t–a)uτ(t)+q(x)(t), (.) whereλ,k∈R+thefunctionτ∈M(]a,b]),theoperatorq:Cm– (]a,b])→ L (]a,b])iscontinuous and η(t,ρ)≡supq(x)(t):x Cm– ≤ρ∈ L ]a,b]. Then, from Theorems . and . with n= , the corollary follows. Corollary . Let the function τ∈M(]a,b]), the continuous operator q :Cm– (]a,b]) →  L (]a,b]), and the numbers γ>,λ≥, k>be such that τ(t)–t≤(t–a)/ for a<t≤b, (.) η(t,γ) L ≤–λγ k (+[(b–a)]/) γ, (.) and λ< γk (+[(b–a)]/). (.) Then problem (.), (.)is solvable. Corollary . Let the function τ∈M(]a,b]), the continuous operator q :Cm– (]a,b[) →  L ,(]a,b]), and the number λ≥be such that inequalities (.), (.)and λ< (+[(b–a)]/), (.) hold.Then problem (.), (.)is solvable. 2 Auxiliary propositions 2.1 Lemmas on some properties of the equation x(n)(t)=λ(t) First, we introduce two lemmas without proofs. The first lemma is proved in []. Lemma. Let i∈{,},x∈ Cm– loc (]t,t[) and x(j–)(ti)= (j=,...,m), t tx(m)(s)ds<+∞.(.) Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 9 of 21 Then t ti (x(j–)(s)) (s–ti)m–j+ ds / ≤m–j+ (m–j+)!!t tix(m)(s)ds / (.) for t≤t≤t. This second lemma is a particular case of Lemma . in []. Lemma. If x∈Cn– loc (]a,a]), then for any s,t∈]a,a]the equality (–)n–mt s(ξ–a)n–mx(n)(ξ)x(ξ)dξ=wn(x)(t)–wn(x)(s)+νnt sx(m)(ξ)dξ is valid,where νm=,νm+ =m+ ,wm(x)(t)=m j=(–)m+j–x(m–j)(t)x(t), wm+(x)(t)= m  j= (–)m+j(t–a)x(m+–j)(t)–jx(m–j)(t)x(j–)(t)–t–a x(m)(t). Lemma. Let the numbers a∈]a,b[, t,k∈]a,a[, and εi,k,εi,βk,β∈R+,k∈N,i=m+ ,...,n,be such that lim k→+∞t,k=a,lim k→+∞βk=β,lim k→+∞εi,k=εi.(.) Let,moreover, λ∈ L n–m–]a,a](.) be a nonnegative function,xk∈ Cn–,m(]a,a]) be a solution of the problem x(n)(t)=βkλ(t), (.) x(i–)(t,k)= (i=,...,m), x(j–)(a)=εj,k(j=m+,...,n), (.) and x∈ Cn–,m(]a,a]) be a solution of the problem x(n)(t)=βλ(t), (.) x(i–)(a)= (i=,...,m), x(j–)(a)=εj(j=m+,...,n). (.) Then lim k→+∞x(j–) k(t)=x(j–)(t)(j=,...,n)uniformly in ]a,a]. (.) Proof First, let us prove our lemma under the assumption that there exists the number r> such that the estimates a t,kx(m) k(s)ds≤r,k∈N(.) Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 16 of 21 3Proofs ProofofRemark. Letxbeasolutionofproblem(.),(.),thenfrominequalities(.) (with y=x), by the definition of the norm in the space  Cm– (]a,b]) and estimate (.), estimate (.) immediately follows.  ProofofTheorem. LetδandλbethefunctionsandnumbersappearinginDefinition.. We set η(t)=δ(t,γ)γ+ Fpt,min{ρ,γ},(.) χ(s)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ for≤s≤ρ, –s/ρfor ρ<s<ρ, fors≥ρ, (.) q(x)(t)=χx Cm– Fp(x)(t). (.) From (.) it is clear that the nonnegative functions Fp,ηadmit the inclusion  Fp·,min{ρ,γ},η∈ L n–m–]a,b],(.) and for every x∈Aγ⊂ Cm– (]a,b]) and almost all t∈]a,b], the inequality q(x)(t)≤ Fpt,min{ρ,γ}for a<t≤b(.) holds. LetU:Aγ→En∩ Cm– (]a,b])betheoperatorinLemma.,fromwhichitfollowsthat Uis a continuous operator. On the other hand, from items (i) and (ii) of Definition ., (.)and(.), it is clear that for each x∈Aγ, the conditions y Cm– ≤γ,y(n–)(t)–y(n–)(s)≤t sη(ξ)dξfor a<t<b hold.Thus,inview ofDefinition ., the operator Umaps the ball Aγinto its own subset S(ρ,η). From Lemma . it follows that S(ρ,η)isacompactsubsetoftheballAγ⊂  Cm– (]a,b]), i.e., the operator umapsthe ball Aγinto its own compact subset.Therefore, owing to Schauder’s principle, there exists x∈S(ρ,η)⊂Aγsuch that x(t)=U(x)(t)fora<t≤b. Thus by (.) and notation (.), the function x(x∈Aγ) is a solution of problem (.), (.), where λ=χx Cm– .(.) If γ=ρ, then in view of the condition x∈Aγ,by(.)wehavethatλ=,andthen,in view of (.)and(.), the function xis a solution of problem (.), (.)whichadmits estimate (.). Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 17 of 21 Let us show now that xadmits estimate (.)inthecasewhenρ<γ. Assume the contrary. Then either ρ<x Cm– <ρ,(.) or x Cm– ≥ρ.(.) Ifcondition(.)holds,thenbyvirtueof(.)and(.)wehavethatλ∈],[,whichbythe conditions of our theorem guarantees the validity of estimate (.). But this contradicts (.). Assume now that (.) is fulfilled. Then, by virtue of (.)and(.), we have that λ=. Thereforex∈Aγisasolutionofproblem(.),(.).Thusfromitem(ii)ofDefinition. it is obvious that x≡, because problem (.), (.) has only a trivial solution. But this contradicts condition (.), i.e.,estimate(.) is valid. From estimates (.)and(.)we havethatλ=,andtheninviewof(.)and(.)thefunctionxisasolutionofproblem (.), (.) which admits estimate (.).  Proof of Corollary . First note that in view of condition (.)thereexistssuchγ>ρ thatcondition(.)holds,andinviewofDefinition.the operator Pis γ,γconsistent. On the other hand, from (.) it follows the existence of the number ρsuch that γη(·,ρ) L n–m– <ρfor ρ>ρ.(.) Letxbeasolutionofproblem(.),(.)forsomeλ∈],[.Theny=xisalsoasolution ofproblem(.),(.)whereq(t)=λ(F(x)(t)–P(x,x)(t)).Letnowρ=x Cm– andassume that ρ>ρ(.) holds.Theninviewoftheγ-consistencyoftheoperatorpwithboundaryconditions(.), inequality (.) holds and thus by condition (.)wehave ρ=x Cm– ≤γq(x) L n–m– ≤γη(·,ρ) L n–m–. But the last inequality contradicts (.). Thus assumption (.)isnotvalidandρ≤ρ. Therefore, for any λ∈],[, an arbitrary solution of problem (.), (.) admits estimate (.).ThereforealltheconditionsofTheorem. are fulfilled,fromwhich thesolvability of problem (.), (.) follows.  Proof of Theorem . Let rnbe the constant defined in Remark .. First prove that the operator Pis γ,rnconsistent with boundary conditions (.). From the conditions of our theorem it is obvious that item (i) of Definition . is satisfied. Let now xbe an arbitrary fixedfunctionfromtheset Aγ,andletpj(t)≡pj(x)(t).Thus,inviewof (.)and(.),all theassumptionsofTheorem.aresatisfied,andthenforanyq∈ L n–m–(]a,b])problem Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 18 of 21 (.), (.)hastheuniquesolutiony.AlsoinviewofRemark. there exists the constant rn>  (which depends only on the numbers lj,lj,γj(j=,...,m), and a,b,n)suchthat estimate (.)holdswithγ=rn,i.e., the operator Pis γ,rnconsistent with boundary conditions (.). Therefore all the assumptions of Corollary . are fulfilled, from which the solvability of problem (.), (.) follows.  Proof of Theorem . Let rnbe the constant defined in Remark .. First prove that the operator Pis rnconsistent with boundary conditions (.). From the conditions of our theorem it is obvious that item (i) of Definition . is satisfied. Let now γbe an arbitrary nonnegative number, xbe an arbitrary fixed function from the space Aγ,andlet pj(t)≡pj(x)(t). Then in view of (.)and(.) all the assumptions of Theorem . are satisfied, and thenforany q∈ L n–m–(]a,b]) problem(.),(.)hastheuniquesolution y.AlsoinviewofRemark. there exists the constant rn> (which depends only on the numbers lj,lj,γj(j=,...,m), and a,b,n)suchthatestimate(.)holdswithγ=rn,i.e., theoperatorPisγ,rnconsistentwithboundaryconditions(.)forarbitraryγ>.Thus by Definition ., the operator Pis rnconsistent with boundary conditions (.). ThereforealltheassumptionsofCorollary.arefulfilled,fromwhichthesolvabilityofproblem (.), (.) follows.  Proof of Remark . By Schwarz’s inequality, the definition of the norm y Cm– and inequalities (.) for any x,y∈Aγand z=y–x,wehave pj(y)(t)z(j–)τj(t) =pj(y)(t)z(j–)(t)+pj(y)(t)τj(t) tz(j)(ψ)dψ ≤z Cm– pj(y)(t)αj(t)+  αj(t)τj(t) t(ψ–a)m–jdψ/(.) for a<t≤b. On the other hand, from conditions (.) by Lemma . it is clear that α– j(s)τj(s) s(ξ–a)m–jdξ/ ≤√κ(+κ)mfor s∈]a,a+ε], α– j(s)τj(s) s(ξ–a)m–jdξ/ ≤ε–m+j–/b a(ξ–a)m–jdξ/ =(b–a)m–j+/ m–j+εm–j+/ for s∈]a+ε,b]. Then if we put κ=max ≤j≤m√κ(+κ)m,(b–a)m–j+/ m–j+εm–j+/ ,(.) from (.) by the last estimates and (.), we get the inequality pj(y)(t)z(j–)τj(t)≤z Cm– (+κ)pj(y)(t)αj(t) ≤z Cm– (+κ)δjt,y Cm– (.) Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 19 of 21 for a<t≤b. Analogously, we get that pj(y)(t)–pj(x)(t)x(j–)τj(t)≤x Cm– (+κ)pj(y)(t)–pj(x)(t)αj(t) for a<t≤b.From(.) and the last inequality it is obvious that the operator Pdefined by equality (.) continuously acts from Aγto the space Ln(]a,b]), and item (ii) of Definition . holds with δ(t,ρ)=(+κ)m j= δj(t,ρ).  Proof of Corollary . From conditions (.)and(.), by Remark .,weobtainthat the operator Pdefined by equality (.)withpj(x)(t)=pj(t) continuously acts from Aγ to the space Ln(]a,b]) for any γ>,i.e., continuously acts from  Cm– (]a,b]) to the space Ln(]a,b]). Therefore it is clear that all the conditions of Theorem . would be satisfied with F(x)(t)=ft,xτ(t),xτ(t),...,x(m–)τm(t),δ(t,ρ)=(+κ)m  j= pj(t), where the constant κis defined by equality (.). Thus problem (.), (.)issolvable.  Proof of Corollary. LettheoperatorsF,p:Cm–(]a,b]) →Lloc(]a,b]),and thefunction η:]a,b]×R+→R+be defined by the equalities F(x)(t)=– λ|x(t)|k (t–a)+k/ xτ(t)+q(x)(t), p(x)(t)=– λ|x(t)|k (t–a)+k/ and δ(t,ρ)=λ(τ(t)–a)/ρk (t–a),l=γk λ,l=γk λ, r= –λγ k (+[(b–a)]/),B=λγ k +(b–a)/,γ= .(.) Thenitiseasytoverifythat,inviewof (.)-(.),conditions(.),(.),(.),(.), (.) are satisfied and δ∈Dn(]a,b]×R+). ThusalltheconditionofTheorem.aresatisfied,fromwhichthesolvabilityofproblem (.), (.) follows.  Proof of Corollary . Let the operators F,p:Cm–(]a,b])→Lloc(]a,b]) and the function η:]a,b]×R+→R+be defined by the equalities F(x)(t)=–λ|sinxk(t)| (t–a)xτ(t)+q(x)(t), p(x)(t)=–λ|sinxk(t)| (t–a). Then it is easy to verify that, in view of (.), (.)and(.), all the conditions of Theorem. follow, where δ,l,l,r,B,γare defined by (.)withρ=,γ=,fromwhich the solvability of problem (.), (.) follows.  Mukhigulashvili and P˚uža Boundary Value Problems (2015) 2015:17 Page 20 of 21 Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed to each part of this work equally and read and approved the final manuscript. Acknowledgements The research was supported by Grants FP-S-13-2148 (B P˚uža, S Mukhigulashvili) and RVO: 67985840 (S Mukhigulashvili). Received: 16 May 2014 Accepted: 2 September 2014 References 1. Kiguradze, IT: Some Singular Boundary Value Problems for Ordinary Differential Equations. Tbilisi University Press, Tbilisi (1975) (in Russian) 2. Kiguradze, IT, Shekhter, BL: Singular boundary value problems for second order ordinary differential equations. In: Itogi Nauki i Tekhniki. Ser. 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