Solvability of Nonlinear Singular Problems for Ordinary Differential Equations - Free Computer, Programming, Mathematics, Technical Books, Lecture Notes and Tutorials

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Solvability of

Nonlinear

Singular Problems for

Ordinary Differential

Equations

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Solvability of Nonlinear Singular Problems for

Ordinary Differential Equations

Irena Rach˚unkov´a, Svatoslav Stanˇek, and Milan Tvrd´y

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Series Editors: Ravi P. Agarwal and Donal O’Regan

Hindawi Publishing Corporation

410 Park Avenue, 15th Floor, #287 pmb, New York, NY 10022, USA Nasr City Free Zone, Cairo 11816, Egypt

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All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without written permission from the publisher.

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Preface

The topic of singular boundary value problems has been of substantial and rapidly grow-ing interest for many scientists and engineers. This book is devoted to sgrow-ingular bound-ary value problems for ordinbound-ary diﬀerential equations. It presents existence theory for a variety of problems having unbounded nonlinearities in regions where their solutions are searched for. The importance of thorough investigation of analytical solvability is emphasized by the fact that numerical simulations of solutions to such problems usually break down near singular points.

The contents of the monograph is mainly based on results obtained by the authors during the last few years. Nevertheless, most of the more advanced results achieved to date in this field can be found here. Besides, some known results are presented in a new way. The selection of topics reflects the particular interests of the authors.

The book is addressed to researchers in related areas, to graduate students or advan-ced undergraduates, and, in particular, to those interested in singular and nonlinear boundary value problems. It can serve as a reference book on the existence theory for singular boundary value problems of ordinary differential equations as well as a textbook for graduate or undergraduate classes. The readers need basic knowledge of real analysis, linear and nonlinear functional analysis, theory of Lebesgue measure and integral, theory of ordinary differential equations (including the Carathédory theory and boundary value problems) on the graduate level.

The monograph deals with boundary value problems which are considered in the frame of the Carathéodory theory. If nonlinearities in differential equations fulfil the Carathéodory conditions, the boundary value problems are called regular, while, if the Carathéodory conditions are not fulfilled on the whole region, the problems are called singular. Two types of singularities are distinguished—time and space ones. For singular boundary value problems, we introduce notions of a solution and of aw-solution. Solu-tions ofnth-order differential equations are understood as functions having absolutely continuous derivatives up to ordern−1 on the whole basic compact interval. On the other hand, w-solutions have these derivatives only locally absolutely continuous on a noncompact subset of the basic interval. The main attention is paid to the existence of solutions of singular problems. The proofs are mostly based on regularization and sequential technique. The impact of our theoretical results is demonstrated by illustrative examples.

Essentially, the book is divided into two parts and four appendices.

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Chapters3–6investigate other higher-order boundary value problems having only space singularities which appear most frequently in literature. They provide existence results for (n,p)-problems, conjugate problems, Sturm-Liouville problems, and Lidstone problems.

Part IIconsists of Chapters7–11and deals with scalar second-order singular bound-ary value problems with one-dimensionalφ-Laplacian. The exposition is focused mainly on Dirichlet and periodic problems which are considered in Chapters7and8, respec-tively.Section 7.1is fundamental for further investigation. The operator representation of the regular Dirichlet problem with φ-Laplacian is derived here and the methods of a priori estimates and lower and upper functions are developed. In Sections 7.2–7.4, three existence principles are presented. These principles together with the principles ofChapter 1 are then specialized to important particular cases and existence theorems and criteria extending and supplementing earlier results are obtained.Section 7.2deals with time singularities,Section 7.3with space singularities, andSection 7.4with mixed singularities, that is, both time and space ones. InChapter 8, we consider the existence of periodic solutions. We start with the method of lower and upper functions and with its relationship to the Leray-Schauder degree inSection 8.1.Section 8.2is devoted to prob-lems with a nonlinearity having an attractive singularity in its first space variable. Sec-tions8.3and8.4deal with problems with strong and weak repulsive space singularities, respectively. An existence theorem for periodic problems with time singularities is given in the last section ofChapter 8. InChapter 9, we study two singular mixed boundary value problems. The latter arises in the theory of shallow membrane caps and we discuss its solvability in dependence on parameters which appear in the diﬀerential equation. In

Chapter 10, we treat problems which may have singularities in space variables. Boundary conditions under discussion are generally nonlinear and nonlocal. We present general principles for solvability of regular and singular nonlocal problems and show some of their applications.Chapter 11is devoted to a class of problems having singularities in space variables. Implementation of a parameter into the equation enables us to prove solvability of problems with three independent (generally nonlocal) boundary condi-tions. We deliver an existence principle and its specialization to the problem with given maximal values for positive solutions.

Appendices give an overview of some basic classical theorems and assertions which are used in Chapters1–11.Appendix Apresents several criteria for uniform integrability or equicontinuity. Some convergence theorems are given inAppendix B. In particular, we recall the Lebesgue dominated convergence theorem, the Fatou lemma, the Vitali convergence theorem for integrable functions, and the Arzel`a-Ascoli theorem and the diagonalization theorem for diﬀerentiable functions.Appendix Ccontains the Schauder fixed point theorem, the Leray-Schauder degree theorem, the Borsuk antipodal theorem, and the Fredholm-type existence theorem.Appendix Dcollects some useful facts from half-linear analysis which are needed inChapter 8.

Acknowledgments

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List of notation

LetJ⊂R, [a,b]⊂R,k∈N,p∈(1,∞),M⊂Rk_{. Then we will write}

(i) L∞(J) for the set of functions essentially bounded and (Lebesgue) measurable

onJ; the corresponding norm is

u∞=sup essu(t):t∈J;

(ii) L1(J) for the set of functions (Lebesgue) integrable onJ; the corresponding norm isu1=J|u(t)|dt;

(iii) Lloc(J) for the set of functions (Lebesgue) integrable on each compact interval I⊂J;

(iv) Lp(J) for the set of functions whosepth powers of modulus are integrable on J; the corresponding norm isup=(_J|u(t)|p_dt)1/ p_;

(v) C(J) andCk_{(J) for the sets of functions continuous on}_J_{and having continuous} kth derivatives onJ, respectively;

(vi) AC(J) and ACk_{(J) for the sets of functions absolutely continuous on} _J _and having absolutely continuouskth derivatives onJ, respectively;

(vii) ACloc(J) andACkloc(J) for the sets of functions absolutely continuous on each compact intervalI ⊂ J and having absolutely continuouskth derivatives on each compact intervalI⊂J, respectively;

(viii) Car([a,b]×M) for the set of functions satisfying the Carath´eodory conditions on [a,b]×M.

IfJ ⊂[a,b] andJ =J, then f ∈Car(J×M) will denote that f ∈Car(I×M) for each compact intervalI⊂J.

IfJ =[a,b], we will simply writeC[a,b] instead ofC([a,b]) and similarly for other types of intervals and other functional sets defined above.

Ifu∈L∞[a,b]∩C[a,b], then max{|u(t)|: t∈[a,b]} = sup ess{|u(t)| :t∈ [a,b]}.

Therefore, the norms inC[a,b] andCk_[a,_{b] will be denoted by}

u∞=maxu(t):t∈[a,b], uCk= k

i=0

_u(i)

∞,

respectively.

Mwill denote the closure ofM,∂Mthe boundary ofM, and meas(M) the Lebesgue measure ofM.

The symbol deg(I−F,Ω) stands for the Leray-Schauder degree ofI−F with respect toΩ, whereIdenotes the identity operator.

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Throughout this text we exploit the following basic theorems listed in appendices. (i) Lebesgue dominated convergence theorem (Theorem B.1).

(ii) Fatou lemma (Theorem B.2).

(iii) Vitali convergence theorem (Theorem B.3). (iv) Arzel`a-Ascoli theorem (Theorem B.5).

(v) Diagonalization theorem (Theorem B.6). (vi) Schauder fixed point theorem (Theorem C.1). (vii) Leray-Schauder degree theorem (Theorem C.2). (viii) Borsuk antipodal theorem (Theorem C.3).

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Consider the boundary value problem

u(n)₌ _f_t,_u,_{. . .}_,_u(n−1) _, _u_∈_B_, _(BVP) wheren ∈ N, [0,T] ⊂ R, and B ⊂ C[0,T]. In what follows, we will investigate the solvability of problem (BVP) on the set [0,T]×A, whereAis a closed subset ofRn_{. If} we impose some additional conditions on solutions of (BVP), for example, if we search for positive or for monotonous solutions, we express this requirement in terms of the set

A =Rn_{and prove the existence of a solution}_u_{such that (u(t),}_{. . .}_,_u(n−1)_(t)) _∈ _A_for t∈ [0,T]. On the other hand, if there are no additional requirements on solutions, we can assumeA=Rn_.

LetM⊂Rn_{. We say that a function} _f _{satisfies the Carath´eodory conditions on the set} [a,b]×M(f ∈Car([a,b]×M)) if

(i) f(·,x0,. . .,xn−1) : [a,b]→Ris measurable for all (x0,. . .,xn−1)∈M,

(ii) f(t,·,. . .,·) :M→Ris continuous for a.e.t∈[a,b],

(iii) for each compact setK ⊂ M, there is a function mK ∈ L1[a,b] such that

|f(t,x0,. . .,xn−1)≤mK(t) for a.e.t∈[a,b] and all (x0,. . .,xn−1)∈K.

IfJ ⊂ [a,b] andJ = J, then f ∈ Car(J×M) means that f ∈ Car(I ×M) for each compact intervalI⊂J.

The classical existence results are based on the assumption

f ∈Car[0,T]×A .

In this case, we will say that problem (BVP) is regular on [0,T]×A. If f /∈Car([0,T]×A), we will say that problem (BVP) is singular on [0,T]×A. The research of singular prob-lems was essentially initiated by Kiguradze in [116,117]. For further development see, for example, the monographs Agarwal [2], Agarwal and O’Regan [12], Agarwal, O’Regan, and Wong [21], O’Regan [150], Kiguradze [118], Kiguradze and Shekhter [120], Mawhin [137], Rach ˚unkov´a, Stanˇek, and Tvrd´y [165], and references therein.

Example 1. In certain problems in fluid dynamics and boundary layer theory (see, e.g., Callegari and Friedman [53], Callegari and Nachman [54, 55]) the second-order diﬀerential equation

u+ψ(t) uλ =0

arose. Hereλ∈(0,∞) andψ∈C(0, 1),ψ /∈L1[0, 1]. This equation is known as the gen-eralized Emden-Fowler equation. Its solvability with the Dirichlet boundary conditions

u(0)=u(1)=0

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Example 2. Consider the fourth-order degenerate parabolic equation

Ut+

|U|μ_U

y y y y=0,

which arises in droplets and thin viscous flows models (see, e.g., Bernis, Peletier, and Williams [39] and Bertozzi, Brenner, Dupont, and Kadanoﬀ[40]). The source-type solu-tions of this equation have the form

U(y,t)=t−buyt−b , b= 1 μ+ 4,

which leads to the study of the third-order ordinary diﬀerential equation

u=btu1−μ

on [−1, 1]. We see that f(t,x)=btx1−μ_{is singular on [−1, 1]}_×_[0,_{∞) if}_{μ >}_1.

Example 3. Similar to the previous example, the sixth-order degenerate equation

Ut−|U|μ_{Uy y y y y} y=0,

which arises in semiconductor models (Bernis [37,38]), leads to the fifth-order ordinary diﬀerential equation

−u(5)₌ t uλ which is singular forλ >0.

Example 4. Consider the nonlinear elliptic partial diﬀerential equation

Δu+g(r,u)=0 onΩ, u|Γ=0,

whereΔis the Laplace operator,Ωis the open unit disk inRn_{centered at the origin,}_Γ is its boundary, andris the radial distance from the origin. When searching for positive radially symmetric solutions to this problem, we get the singular problem of the form

u+n−1

t u+g(t,u)=0, u(0)=0, u(1)=0. (See Berestycki, Lions, and Peletier [36] or Gidas, Ni, and Nirenberg [98].)

Example 5. Assume f ∈Car([0,∞)×R) and consider the regular boundary value prob-lem

u= f(t,u), u(1)=0, u(∞)=0

on the infinite interval [1,∞). We can transform this problem to a finite interval, for example, on [0, 1]. Then we get the singular problem of the form

v+2 tv

₌ 1

t4f

1 t,v

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1

Existence principles for

singular problems

1.1. Formulation of the problem

Forn∈N, [0,T]⊂R,i∈ {0, 1,. . .,n−1}, and a closed setB⊂Ci_[0,_{T], consider the} boundary value problem

u(n)₌ _f_t,_u,_{. . .}_,_u(n−1) _, _(1.1)

u∈B. (1.2)

A decision concerning solvability for singular boundary value problems requires an exact definition of a solution to such problems. Here, we will work with the same definition of a solution both for the regular problems and for the singular ones.

Definition 1.1. A functionu∈ACn−1_[0,_T]_∩_B_{is called a solution of problem (}_1.1_{), (}_1.2₎ if it satisfies the equality

u(n)_(t)₌ _f_t,_u(t),_{. . .}_,_u(n−1)_(t) _{for a.e.}_t_∈_[0,_T].

If problem (1.1), (1.2) is investigated on [0,T]×A, where A = Rn_{, then (u(t),}_{. . .}_, u(n−1)_(t))_∈_A_for_t_∈_[0,_{T] is required.}

In literature, an alternative approach to solvability of singular problems can be found. In that approach, authors search for solutions which are defined as functions whose (n− 1)st derivatives can have discontinuities at some points in [0,T]. Here, we will call them w-solutions. According to Kiguradze [117] or Agarwal and O’Regan [12], we define them as follows. In contrast to our starting setting, to definew-solutions we assume (in general) thatBis a closed subset inCi_[0,_T_{], where}_i_{∈ {0, 1,}_{. . .}_,_n₋_2}.

Definition 1.2. A functionu∈Cn−2_[0,_{T] is a}_{w-solution of problem (}_1.1_{), (}_1.2_{) if there} exists a finite number of pointstν∈[0,T],ν=1, 2,. . .,r, such that ifJ=[0,T]\ {tν}r

ν=1, thenu∈ACnloc−1(J)∩B, and

u(n)_(t)₌ _f_t,_u(t),_{. . .}_,_u(n−1)_(t) _{for a.e.}_t_∈_[0,_T].

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Clearly, each solution is aw-solution and eachw-solution which belongs toACn−1_[0, T] is a solution. While only the existence ofw-solutions was proved in the works cited above, our main goal is to prove the existence of solutions. However, in some cases, we first findw-solutions and then prove that they are also solutions.

When studying the singular problem (1.1), (1.2), we will focus our attention on two types of singularities of the function f.

LetJ ⊂ [0,T]. We say that f :J×A→ Rhas singularities in its time variabletif J=J=[0,T] and

f ∈Car(J×A), f /∈Car[0,T]×A . (1.3) Let D ⊂ A. We say that f : [0,T]×D → R has singularities in its space variables x0,x1,. . .,xn−1, ifD=D=Aand

f ∈Car[0,T]×D , f /∈Car[0,T]×A . (1.4) We will study particular cases of (1.3) and (1.4), which will be described inSection 1.2

andSection 1.3, respectively.

1.2. Singularities in time variable

A functionf has a singularity in its time variablet(in short a time singularity) if, roughly speaking, f is not integrable on [0,T]. Let us define it more precisely. Letk ∈ N,ti ∈ [0,T],i =1,. . .,k,J =[0,T]\ {t1,t2. . .,tk}and let f ∈ Car(J×A). Assume that for eachi∈ {1,. . .,k}, there exists (x0,. . .,xn−1)∈Asuch that

_ti+ε

ti

_f

t,x0,. . .,xn−1 dt= ∞ or

_ti

ti−ε

_f

t,x0,. . .,xn−1 dt= ∞ (1.5)

for any suﬃciently smallε >0. Then f /∈Car ([0,T]×A) and f has singularities in its time variablet, namely, at the valuest =t1,. . .,tk. We will callt1,. . .,tk singular points of f.

Example 1.3. Let fi:Rn_→_R_,_i₌_{1, 2,}_{. . .}_,_{k, be continuous. Then the function}

ft,x0,. . .,xn−1 = k

i=1 1 t−tifi

x0,. . .,xn−1 ,

has singular pointst1,t2,. . .,tk.

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Consider problem (1.1), (1.2) on [0,T]×A. For the sake of simplicity, assume that f has only one time singularity att=t0,t0∈[0,T]. Thus,

J=[0,T]\t0,f ∈Car(J×A) satisfies one of the conditions: (i)

t0

t0−ε

_f

t,x0,. . .,xn−1 dt= ∞, t0∈(0,T],

(ii)

_t0+ε

t0

_f

t,x0,. . .,xn−1 dt= ∞, t0∈[0,T),

(1.6)

for some (x0,. . .,xn−1)∈Aand each suﬃciently smallε >0.

Further, consider a sequence of regular problems:

u(n)(t)= fkt,u(t),. . .,u(n−1)(t) , u∈B, (1.7)

where fk ∈ Car([0,T]×Rn_),_k _∈_N_{. Solutions of problem (}_1.7_{) are understood in the} sense ofDefinition 1.1. The following two theorems deal with the case

Bis a closed subset inCn−2_[0,_T_]. _(1.8)

Theorem 1.4 (first principle for time singularities). Let (1.6) and (1.8) hold. Assume that the conditions

for eachk∈Nand eachx0,. . .,xn−1 ∈A, fk

t,x0,. . .,xn−1 = f

t,x0,. . .,xn−1 a.e. on [0,T]\Δk, whereΔk=

t0−1_k,t0+1 k

∩[0,T];

(1.9)

there exists a bounded setΩ⊂Cn−1_[0,_{T] such that} for eachk∈N, the regular problem (1.7) has a solution uk∈Ω, uk(t),. . .,u_k(n−1)(t) ∈Afort∈[0,T]

(1.10)

are fulfilled. Then,

there exist a functionu∈Cn−2_[0,_{T] and a subsequence}

uk

⊂uk

such that lim→∞uk−uCn−2=0;

(1.11)

lim→∞u(nk−1)(t)=u(n−1)(t) locally uniformly onJ

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Assume, moreover, that

there existψ∈L1[0,T],η >0,0∈N, andλ1,λ2∈ {−1, 1} such that

λ1fkt,uk(t),. . .,u(n_k−1)(t) ≥ψ(t)

for each∈N,≥0, and for a.e.t∈t0−η,t0 ⊂[0,T] provided (1.6)(i) holds

and

λ2fkt,uk(t),. . .,u(n_k−1)(t) ≥ψ(t)

for each∈N,≥0, and for a.e.t∈t0,t0+η⊂[0,T] provided (1.6)(ii) is true.

(1.14)

Thenu∈ACn−1_[0,_T],_u_{is a solution of problem (}_1.1_{), (}_1.2_{) and}

u(t),. . .,u(n−1)_(t) _∈_A _for_t_∈_[0,_T]. Proof

Step 1. Convergence of the sequence of approximate solutions. Condition (1.10) implies that the sequences{u(i)

k }, 0≤i≤n−2, are bounded and equicontinuous on [0,T]. By the Arzel`a-Ascoli theorem, we see that assertion (1.11) is true andu∈B⊂Cn−2_[0,_T_{]. Let}_t0₌_{0. Since}_{u(n−1)

k }is bounded on [0,T], we get, due to (1.9), that for eachτ∈[0,t0) there existkτ∈Nandhτ ∈L1[0,T] such that for each k≥kτ,

_f_k

s,uk(s),. . .,u(n_k−1)(s) ≤hτ(s) for a.e.s∈[0,τ]. (1.15) Hence, by virtue of (1.7), fork≥kτ,t1,t2∈[0,τ], we have

_u(n−1) k

t2 −u(n_k−1)t1 ≤

t2

t1

hτ(s)ds,

which implies that the sequence{u(n_k−1)}is equicontinuous on [0,τ]. The same holds on [τ,T] ifτ ∈ (t0,T] andt0 = T. The Arzel`a-Ascoli theorem implies that for each compact subsetK ⊂J=[0,T]\{t0}, a subsequence of{u(n−1)

k }uniformly converging to u(n−1)_on_K_{can be chosen. Therefore, using the diagonalization theorem, we can choose} a subsequence{uk}satisfying both (1.11) and (1.12).

Step 2. Convergence of the sequence of approximate nonlinearities.

LetV1be the set of allt∈[0,T] such that f(t,·,. . .,·) :Rn→Ris not continuous and letV2be the set of allt∈[0,T] such that (1.9) is not satisfied. Then meas(V1∪V2)= 0. Choose an arbitraryτ∈[0,T]\(V1∪V2). Then there exists0∈Nsuch that for≥0,

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and, by (1.11) and (1.12),

lim →∞fk

τ,uk(τ),. . .,u(nk−1)(τ) = f

τ,u(τ),. . .,u(n−1)_(τ) _.

Hence,

lim →∞fk

t,uk(t),. . .,u(n_k−1)(t) = ft,u(t),. . .,u(n−1)(t) for a.e.t∈[0,T]. (1.16)

Step 3. The functionuis aw-solution of problem (1.1), (1.2).

Lett0=0 and∈N. Choose an arbitraryτ∈[0,t0) and integrate the equality u(n)_k (t)= fkt,uk(t),. . .,u(n_k−1)(t) for a.e.t∈[0,T].

We get

u(n_k−1)(τ)=u(n_k−1)(0) +

τ

0 fk

s,uk(s),. . .,u(n_k−1)(s) ds.

According to (1.15), (1.16), and the Lebesgue dominated convergence theorem on [0,τ], we can deduce (having in mind thatτis arbitrary) that ift0 = 0 the limitusolves the equation

u(n−1)_(t)₌_u(n−1)_{(0) +}

_t

0 f

s,u(s),. . .,u(n−1)_(s) _ds _for_t_∈_0,_t0 _. _(1.17)

Similarly, ift0=T, the limitusolves the equation

u(n−1)(t)=u(n−1)(T)−

T

t f

s,u(s),. . .,u(n−1)(s) ds fort∈t0,T. (1.18)

The equalities (1.17) and (1.18) immediately yield (1.13).

Step 4. The functionuis a solution of problem (1.1), (1.2). Assume, moreover, that (1.14) and (1.6)(i) hold. Since

u_k(n−1)(t)−u_k(n−1)t0−η =

_t

t0−η

fks,uk(s),. . .,u(n_k−1)(s) ds

fort∈(0,t0), we get, due to (1.10), that there is ac∈(0,∞) such that

λ1

t0

t0−η

fk

s,uk(s),. . .,u(nk−1)(s) ds≤c (1.19)

for each ∈ N. By the Fatou lemma, using conditions (1.16), (1.14), and (1.19), we deduce that

ft,u(t),. . .,u(n−1)(t) ∈L1t0−η,t0. Similarly, if condition (1.6)(ii) holds, we deduce that

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Hence,

ft,u(t),. . .,u(n−1)(t) ∈L1t0−η,t0+η∩[0,T] .

Recall that, by (1.12), we have (u(t),. . .,u(n−1)_(t)) _∈ _A_for_t _∈ _J _{and, by (}_1.6_), _f _∈ Car(J ×A). Further, by virtue of (1.10) and (1.11), the functionsu,u,. . .,u(n−2) _are bounded on [0,T] and (1.10), (1.12) imply thatu(n−1)_{is bounded on [0,}_T]\(t0−η,_t0_+η). Hence,

ft,u(t),. . .,u(n−1)(t) ∈L1[0,T]\t0−η,t0+η , which together with the above arguments yields

ft,u(t),. . .,u(n−1)(t) ∈L1[0,T].

Therefore, due to (1.17) and (1.18), we have thatu∈ACn−1_[0,_T_{], that is,}_u_{is a solution} of problem (1.1), (1.2). Finally, sinceAis closed, we get

lim t→t0

u(t),. . .,u(n−1)(t) =ut0 ,. . .,u(n−1)t0 ∈A.

Theorem 1.5 (second principle for time singularities). Let (1.6), (1.8), (1.9), and (1.10) hold. Assume that

there existψ∈L1[0,T], η >0, andλ1,λ2∈ {−1, 1}such that

λ1fk

t,uk(t),. . .,u(nk−1)(t) signu (n−1)

k (t)≥ψ(t)

for each ∈Nand for a.e.t∈t0−η,t0 ⊂[0,T] if (1.6)(i) holds and

λ2fkt,uk(t),. . .,u_k(n−1)(t) signu(n_k−1)(t)≥ψ(t)

for each∈Nand for a.e.t∈t0,t0+η⊂[0,T] if (1.6)(ii) is true.

(1.20)

Then, there exists a functionu ∈ ACn−1_[0,_{T] satisfying (}_1.11_{) and (}_1.12_{) which is a} solution of problem (1.1), (1.2), and (u(t),. . .,u(n−1)_(t))_∈_A_for_t_∈_[0,_T].

Proof . Steps 1–3 are the same as in the proof ofTheorem 1.4and guarantee the existence of aw-solutionuof problem (1.1), (1.2).

Step 4. Arguing as in step 4 of the proof of Theorem 1.4, we see that to showu ∈ ACn−1_[0,_{T], it su}_ﬃ_{ces to prove} _f_(t,_u(t),_{. . .}_,_u(n−1)_(t))_∈_{L1(I0), where}_I0₌_[t0₋_η,_t0₊ η]∩[0,T]. PutM=V1∪V2∪V3, where

V1=

t∈I0: f(t,·,. . .,·) :Rn _→_R_{is not continuous}_,

V2=

t∈I0:tis an isolated zero ofu(n−1)_,

V3=

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(a) Letu(n−1)_(s) ₌ _{0. Assume, for example, sign}_u(n−1)_(s) ₌ _{1. Then, there exists} 0 ∈Nsuch that for each≥0, we have signu_k(n−1)(s)=1 and so, due to (1.9), (1.11), (1.12), ands /∈V1,

lim →∞λ1fk

s,uk(s),. . .,u(nk−1)(s) signu (n−1)

k (s)=λ1f

s,u(s),. . .,u(n−1)_(s) _sign_u(n−1)_(s). (1.21)

If signu(n−1)_(s)_{= −1, we get (}_1.21_{) in the same way.}

(b) Let s be an accumulation point of a set of zeros of u(n−1)_{. Then, there exists} a sequence {sm} ⊂ I0 such that u(n−1)(sm) = 0 and limm→∞sm = s. Since u(n−1) is continuous onI0\ {t0}, we getu(n−1)_(s)₌_{0. Further,}

lim m→∞

u(n−1)_s

m −u(n−1)(s) sm−s =0

and, by virtue ofs /∈V3, we get 0 =u(n)_(s)₌ _f_(s,_u(s),_{. . .}_,_u(n−1)_{(s)). Since}_{s /}_∈_V_{1, we} have by (1.9), (1.11), and (1.12)

lim →∞fk

s,uk(s),. . .,u_k(n−1)(s) signu(n_k−1)(s) = fs,u(s),. . .,u(n−1)(s) lim

→∞signu

(n−1) k (s)=0.

So, we have proved that (1.21) is valid for a.e.s∈I0.

Assume that (1.6)(i) holds andt0−η ≥ 0. Then, by (1.10), there existc > 0 and 0∈Nsuch that for each≥0,

t0

t0−η

λ1fk

s,uk(s),. . .,u(nk−1)(s) signu (n−1)

k (s)ds=λ1

t0

t0−η

_u(n−1) k (s)

_ds

=λ1u(n_k−1)t0 −u(n_k−1)t0−η ≤c,

and hence, due to (1.20) and (1.21), we can use the Fatou lemma to deduce that

λ1ft,u(t),. . .,u(n−1)(t) signu(n−1)(t)∈L1t0−η,t0,

which yieldsf(t,u(t),. . .,u(n−1)_(t)_∈_L1[t0_{−η,t0]. Similarly, if (}_1.6_{)(ii) holds and}_t0+η_≤ T, we deduce that f(t,u(t),. . .,u(n−1)_(t))_∈_L1[t0,_t0₊_η]. Now, we will consider the boundary conditions (1.2) which are characterized by the setB, where

Bis a closed subset inCn−1_[0,_T_]. _(1.22)

Theorem 1.6 (third principle for time singularities). Let (1.6), (1.9), (1.10), and (1.22) hold. Assume that

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Then, there exist a functionu∈Ωand a subsequence{uk} ⊂ {uk}such that lim→∞uk− uCn−1 =0, (u(t),. . .,u(n−1)(t)) ∈Afort∈[0,T] andu∈Cn−1[0,T] is aw-solution of

problem (1.1), (1.2).

If, in addition, (1.20) holds, thenu∈ACn−1_[0,_{T], that is,}_u_{is a solution of problem} (1.1), (1.2).

Proof

Step 1. Convergence of the sequence of approximate solutions{uk}. By (1.10), there is ac >0 such that

_uk

Cn−1≤c for eachk∈N. (1.24)

This implies that sequences{u(i)k }, 0≤i ≤ n−2, are equicontinuous on [0,T]. Let us prove that{un−1

k }is also equicontinuous on [0,T]. Choose an arbitraryε >0. By (1.23), we can findδ0 >0 such that for eachk ∈Nand eacht∈[t0−δ0,t0+δ0]∩[0,T], the inequality

_u(n−1)

k (t)−u(n− 1) k

t0 < ε

holds. Therefore, for eacht1,t2∈[t0−δ0,t0+δ0]∩[0,T], we have

_u(n−1) k

t1 −u(n_k−1)t2 <2ε. (1.25)

Now, lett1,t2∈K, whereK=[0,T]\(t0−δ0,t0+δ0). Put

h(t)=supft,x0,. . .,xn−1 :xi≤c,i=0,. . .,n−1

.

Then,h∈L1(K) and we can findδ1>0 such that

_t1₋_t2_{< δ1} _⇒t2

t1

h(t)dt< ε.

By (1.24), we have|fk(t,uk(t),. . .,u_k(n−1)(t))| ≤h(t) a.e. onKfor each suﬃciently large k∈N. Hence, we get

_t1₋_t2_{< δ1} _⇒_u(n−1) k

t1 −u(n_k−1)t2 < ε. (1.26)

Finally, lett1 ∈(t0−δ0,t0+δ0)∩[0,T],t2 ∈K,t2 > t0+δ0. Putδ =min{δ0,δ1}and assume that|t1−t2|< δ. Then, by (1.25) and (1.26),|u(n−1)

k (t1)−u(n− 1)

k (t2)|<3ε. For t2 < t0−δ0, we argue similarly. So, we have proved that{u(n−1)

k }is equicontinuous on [0,T]. By the Arzel`a-Ascoli theorem, there exists a function u ∈ Ωand a subsequence {uk} ⊂ {uk}such that

lim

→∞uk−uCn−1=0,

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Moreover,u∈B⊂Cn−1_[0,_{T] and, by}_{Theorem 1.4}_,_u_{is a}_{w-solution of problem (}_1.1_), (1.2).

Step 2. If we assume, in addition, that (1.20) holds, then to prove thatu∈ACn−1_[0,_T_], we can argue as in step 4 of the proof ofTheorem 1.5.

1.3. Singularities in space variables

A function f has a singularity in one of its space variables (in short, a space singularity) if f is not continuous in this variable on a region, where f is studied. Motivated by the equation

u+ψ(t)u−λ₌_0,

whereλ ∈ (0,∞), we will consider the following case of discontinuity. LetAi ⊂ Rbe a closed interval and let ci ∈ Ai,Di = Ai\ {ci},i = 0, 1,. . .,n−1. Let us choose

j∈ {0, 1,. . .,n−1}and assume that

lim sup xj→cj,xj∈Dj

_f

t,x0,. . .,xj,. . .,xn−1 = ∞ for a.e.t∈[0,T] and for somexi∈Di, i=0, 1,. . .,n−1, i=j.

(1.27)

If we putA=A0×· · ·×An−1, we see that f is not continuous onA(for a.e.t∈[0,T]).

Consequently, f has a singularity in its space variablexj, namely, at the valuecj. Letube a solution of (1.1), (1.2) and let a pointtu∈[0,T] be such thatu(j)_(tu)₌_{cj. Then,}_tu_is called a singular point corresponding to the solutionu. Now, letube aw-solution of (1.1), (1.2). Assume that a pointtu∈[0,T] is such thatu(n−1)(tu) does not exist oru(j)(tu)=cj. Then,tuis called a singular point corresponding to thew-solutionu.

Example 1.7. Letα∈(0,∞),h1,h2,h3∈L1[0,T],h2=0,h3=0 a.e. on [0,T]. Consider the Dirichlet problem

u+h1(t) +h2(t) u(t) +

h3(t)

_u_(t)α =0, u(0)=u(T)=0. (1.28)

Let u be a solution of (1.28). Then, 0 andT are singular points corresponding to u. Moreover, there exists at least one pointtu ∈(0,T) satisfyingu(tu)=0, which means thattuis also a singular point corresponding tou. Note that (in contrast to the points 0 andT) we do not know the location oftuin (0,T).

In accordance with this example, we will distinguish two types of singular points corresponding to solutions or tow-solutions: singular points of type I, where we know their location in [0,T], and singular points of type II whose location is not known.

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the original singular problem (1.1), (1.2). LetAi⊂R,i=0,. . .,n−1, be closed intervals and letA=A0× · · · ×An−1. Consider problem (1.1), (1.2) on [0,T]×A. Denote

Di=Ai\

ci

, i=0,. . .,n−1.

First, we will assume that f has one singularity at eachxi, namely, at the valuesci∈Ai, i=0,. . .,n−2. Hence, we assume

D=D0× · · · ×Dn−2×An−1,

f ∈Car[0,T]×D satisfies (1.27) for j=0,. . .,n−2. (1.29) In the next two theorems, we work with the notion of uniform integrability which can be found inAppendix A.

Theorem 1.8 (first principle for space singularities). Let (1.8), (1.10), and (1.29) hold. (i) Assume that

for eachk∈N, for a.e.t∈[0,T] and eachx0,. . .,xn−1 ∈D, fkt,x0,. . .,xn−1 = f

t,x0,. . .,xn−1 ifxi−ci≥ 1

k, 0≤i≤n−1.

(1.30)

Then assertion (1.11) is valid.

(ii) If, moreover, the set of singular points

S=s∈[0,T] :u(i)_(s)₌_c

ifori∈ {0,. . .,n−2}

is finite,

then assertion (1.12) is valid forJ=[0,T]\Sand if the sequencefkt,uk(t),. . .,u(n_k−1)(t)

is uniformly integrable on each interval [a,b]⊂J, (1.31) thenu∈AClocn−1(J) is aw-solution of problem (1.1), (1.2).

(iii) If, in addition, there exists a functionψ∈L1[0,T] such that

fkt,uk(t),. . .,u(n_k−1)(t) ≥ψ(t) for a.e.t∈[0,T] and all∈N, thenu∈ACn−1_[0,_T_{] and}_u_{is a solution of problem (}_1.1_{), (}_1.2_).

Proof

Step 1. Convergence of the sequence of approximate solutions.

As in step 1 of the proof ofTheorem 1.4, we derive from (1.10) that (1.11) holds and u ∈B ⊂Cn−2_[0,_{T]. Assume that}_S_{is finite and choose an arbitrary [a,}_b]_⊂ _{J. Then,} there existk0∈Nandh∈L1[0,T] such that for eachk∈N,k≥k0,

_u(i)

k (t)−ci≥ 1

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and, for a.e.t∈[a,b],

_f_k

t,uk(t),. . .,u(n_k−1)(t) =f(t,uk(t),. . .,u(n_k−1)(t) ≤h(t). So, for eachε >0, there existsδ >0 such that the implication

_t2₋_t1_{< δ} _⇒_u(n−1) k

t2 −u(n_k−1)t1 ≤

t2

t1

h(t)dt< ε

is valid fort1,t2∈[a,b],k≥k0. Thus, the sequence{u(n−1)

k }is equicontinuous on [a,b]. By (1.10), the sequence{u(n−1)

k }is bounded on [0,T]. Using the Arzel`a-Ascoli theorem and the diagonalization theorem, we deduce that the subsequence{uk}in (1.11) can be chosen so that it fulfils (1.12).

Step 2. Convergence of the sequence of approximate nonlinearities. Consider the set

V1=

t∈[0,T] : f(t,·,. . .,·) :D →Ris not continuous.

We can see that meas(V1)=0. By (1.30), there existsV2⊂[0,T] such that meas(V2)=0 and for eachk∈N, eacht∈[0,T]\V2, and each (x0,. . .,xn−1)∈D, the equality

fkt,x0,. . .,xn−1 = f

t,x0,. . .,xn−1

holds if|xi−ci| ≥1/k, 0≤i≤n−1. DenoteU=S∪V1∪V2and choose an arbitrary t∈[0,T]\U. By (1.11) and (1.12), there exists0∈Nsuch that for each∈N,≥0,

_u(i)_(t)₋_c

i> _k1, u(i)k(t)−ci≥_k1 fori∈ {0,. . .,n−1}. According to (1.30), we have

fk

t,uk(t),. . .,u(nk−1)(t) = f

t,uk(t),. . .,u(nk−1)(t) and, by (1.11), (1.12),

lim →∞fk

t,uk(t),. . .,u(n_k−1)(t) = ft,u(t),. . .,u(n−1)(t) . (1.32)

Since meas(U)=0, equality (1.32) holds for a.e.t∈[0,T]. Step 3. The functionuis aw-solution of problem (1.1), (1.2).

Choose an arbitrary interval [a,b]⊂J. By virtue of (1.31) and (1.32), we can use the Vitali convergence theorem to show that

ft,u(t),. . .,u(n−1)(t) ∈L1[a,b] and that if we pass to the limit in the sequence

u(n_k−1)(t)=u(n_k−1)(a) +

t

a fk

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we get

u(n−1)_(t)₌_u(n−1)_{(a) +}

t

a f

s,u(s),. . .,u(n−1)_(s) _ds, _t_∈_[a,_b].

Since [a,b]⊂Jis an arbitrary interval, we conclude thatu∈AClocn−1(J) satisfies (1.1) for a.e.t∈[0,T].

Step 4. The functionuis a solution of problem (1.1), (1.2). Let, moreover,

fk

t,uk(t),. . .,u(nk−1)(t) ≥ψ(t) for a.e.t∈[0,T] and all∈N.

Assumption (1.10) yields the existence ofc >0 such that

T

0 fk

t,uk(t),. . .,u(n_k−1)(t) dt=u_k(n−1)(T)−u(n_k−1)(0)≤c.

Therefore, by (1.32) and the Fatou lemma, f(t,u(t),. . .,u(n−1)_(t)) _∈ _L1[0,_T_{] and}_u _∈

ACn−1_[0,_T].

Now we will consider problem (1.1), (1.2) on [0,T]×AprovidedA=A0× · · · ×

An−1andf has space singularities at eachxi, namely, at the valuesci∈Ai,i=0,. . .,n−1. So, we assumeDi=Ai\ {ci},i=0,. . .,n−1,

f ∈Car[0,T]×D satisfies (1.27) for j=0,. . .,n−1, whereD =D0× · · · ×Dn−2×Dn−1.

(1.33)

Theorem 1.9 (second principle for space singularities). Let (1.10), (1.22), (1.30), and (1.33) hold. Assume that the sequence

fkt,uk(t),. . .,u(n_k−1)(t) is uniformly integrable on [0,T]. (1.34)

Then there exist a functionu∈Ωand a subsequence{uk} ⊂ {uk}such that lim→∞uk− uCn−1=0 and (u(t),. . .,u(n−1)(t))∈Afort∈[0,T].

If, moreover, the functionsu(i)₋_ci_{, 0}_≤_i_≤_n₋_{1, have at most a finite number of zeros} in [0,T], thenu∈ACn−1_[0,_{T] is a solution of (}_1.1_{), (}_1.2_).

Proof

Step 1. Convergence of the sequence of approximate solutions.

Assumption (1.34) yields that for eachε > 0, there existsδ > 0 such that for each t1,t2∈[0,T] and eachk∈N, the implication

_t2₋_t1_{< δ}_⇒_u(n−1) k

t2 −u(n_k−1)t1 =

t2

t1

fk

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is valid. Therefore, the sequence{u(n_k−1)}is equicontinuous on [0,T]. This, together with (1.10) and the Arzel`a-Ascoli theorem, guarantees the existence of a subsequence{uk}of {uk}such that

lim

→∞uk−ukCn−1=0.

SinceAis closed inRn_and_B_{is closed in}_Cn−1_[0,_T_{], we get}

u(t),. . .,u(n−1)(t) ∈A fort∈[0,T],u∈B. Step 2. As in step 2 in the proof ofTheorem 1.6, we get that (1.32) is valid.

Step 3. The functionuis a solution of problem (1.1), (1.2). By virtue of (1.7), we have for∈N,

u(n)_k (t)= ft,uk(t),. . .,u(nk−1)(t) for a.e.t∈[0,T],

u(n_k−1)(t)=u(n_k−1)(0) +

t

0 fk

s,uk(s),. . .,u_k(n−1)(s) ds fort∈[0,T].

By (1.32), (1.34), and the Vitali convergence theorem, we can pass to the limit and get

u(n−1)(t)=u(n−1)(0) +

_t

0 f

s,u(s),. . .,u(n−1)(s) ds fort∈[0,T]

with f(t,u(t),. . .,u(n−1)_(t))_∈_L1[0,_{T]. Therefore,}_u_∈_ACn−1_[0,_{T] satisfies (}_1.1_{) a.e. on}

[0,T].

All the above-mentioned existence principles (Theorems1.4–1.6,1.8, and1.9require condition (1.10) and so, in order to apply them, we need global a priori estimates for all approximate solutionsuk and for all their derivativesu(i)k , 1 ≤ i ≤ n−1. We can see in literature that local a priori estimates ofu(n_k−1) can be suﬃcient for the existence of w-solutions (see, e.g., Kiguradze and Shekhter [120]). However, such existence results givew-solutions with, in general, unbounded (n−1)st derivative. Here, our main goal is to prove the existence of solutions. To this purpose, onlyw-solutions, whose (n−1)st derivatives are bounded on the set where they are defined, are useful. Therefore, condition (1.10) appears in all our principles.

Bibliographical notes

The proof ofTheorem 1.4is given in Rach ˚unkov´a, Stanˇek, and Tvrd´y [165]. Theorems

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2

Focal problems

Focal problems have received large attention (see, e.g., Agarwal [2]). This is due to the fact that these types of problems are basic, in the sense that the methods employed in their study are extendable to other types of problems. Here, we will consider thenth order diﬀerential equation with (p,n−p) right focal conditions:

u(i)₍₀₎₌_0, ₀_≤_i_≤_p₋_1, _u(j)_(T₎₌_0, _p_≤ _j_≤_n₋₁ _(2.1)

or with (n−p,p) left focal conditions

u(i)₍₀₎₌_0, _p_≤_i_≤_n₋_1, _u(j)_(T₎₌_0, ₀_≤_j_≤_p₋_1, _(2.2)

wheren∈N,n≥2, andp∈ {1,. . .,n−1}is fixed.

Using the existence principles ofChapter 1, we will investigate both the focal prob-lems with time singularities and the focal probprob-lems with space singularities.

2.1. Time singularities

First, consider a (1,n−1) left focal problem

u(n)= ft,u,. . .,u(n−1) , (2.3) u(n−1)(0)=0, u(i)(T)=0, 0≤i≤n−2. (2.4)

We will assume that

f ∈Car[0,T)×Rn _{has a time singularity at}_t₌_T _(2.5)

and prove the existence result for problem (2.3), (2.4) by means ofTheorem 1.6(third principle for time singularities). Since we impose no additional conditions on solutions of (2.3), (2.4), we have

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Theorem 2.1. Assume (2.5) holds and let

ft,x0,. . .,xn−1 signxn−1≤ −h(t)xn−1+ n−1

j=0

hj(t)xjαj

for a.e.t∈[0,T] and allx0,. . .,xn−1 ∈Rn,

(2.6)

whereαj ∈(0, 1),hj ∈L1[0,T], j =0,. . .,n−1, are nonnegative andh ∈Lloc[0,T) is nonnegative and satisfies

T

T−εh(s)ds= ∞for each suﬃciently smallε >0. (2.7) Then, problem (2.3), (2.4) has a solutionu∈ACn−1_[0,_T].

Proof

Step 1. Approximate regular problems. Fors,ρ∈(0,∞), put

χ(s,ρ)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 ifs∈[0,ρ], 2ρ−s

ρ ifs∈(ρ, 2ρ), 0 ifs≥2ρ.

Further, fork∈N, (x0,. . .,xn−1)∈Rnand for a.e.t∈[0,T], define

fkt,x0,. . .,xn−1 =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ft,x0,. . .,xn−1 ift∈

0,T−1 k

,

0 ift∈

T−1 k,T

,

(2.8)

gkt,x0,. . .,xn−1 =χ

_n₋₁

i=0

_xi_,_ρ_fk

t,x0,. . .,xn−1 . (2.9)

Choose ak∈Nand consider auxiliary approximate regular equations

u(n)= fkt,u,. . .,u(n−1) , (2.10) u(n)₌_g

k

t,u,. . .,u(n−1) _. _(2.11) For a.e.t∈[0,T], define

mk(t)= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ sup _f

t,x0,. . .,xn−1 : n−1

i=0

_xi_≤_2ρ _if_t_≤_T₋1 k,

0 ift > T−1

k.

Then,mk∈L1[0,T] andgk(t,x0,. . .,xn−1)| ≤mk(t) for a.e.t∈[0,T]. Since the

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Step 2. Estimates of approximate solutionsuk. Let us fixk∈Nand assume that

maxu(n_k−1)(t):t∈[0,T]=u(n_k−1)(b)=r >0. By condition (2.4), we haveb∈(0,T] and we can finda∈[0,b) such that

_u(n−1)

k (a)=0, u (n−1)

k (t)>0 fort∈(a,b].

Sinceu(n_k−1)(t)=u(n_k−1)(T−1/k) fort∈[T−1/k,T], we can assume thatb≤T−1/k. By virtue of assumption (2.6), we get for a.e.t∈[a,b],

u(n)_k (t) signu(n_k−1)(t)=χ

_n₋1

i=0

_u(i) k (t),ρ

ft,uk(t),. . .,u(nk−1)(t) signu (n−1) k (t)

≤χ

_n₋₁

i=0

_u(i) k (t),ρ

_n₋₁

j=0

hj(t)u(j)_k (t)αj ≤ n−1

j=0

hj(t)u(j)_k (t)αj,

and hence

_u(n−1) k (t)≤

n−1

j=0

hj(t)u(j)_k (t)αj. (2.12)

Conditions (2.4) yieldu(kj)∞≤rTn−j−1,j=0,. . .,n−2. Integrating inequality (2.12) over [a,b], we obtain

r=u(n_k−1)(b)≤ n−1

j=0

Tαj(n−j−1)_rαj

_T

0 hj(t)dt,

1≤ n−1

j=0

Tαj(n−j−1)_rαj−1_hj

1=:F(r).

We have limx→∞F(x)=0, which implies the existence ofr∗>0 such thatF(x)<1 for all

x≥r∗. Therefore, by (2.1), the estimater < r∗must be true. Sincer∗does not depend onuk(but just onT,hj,αj), we get

_uk

Cn−1< r∗

n−1

j=0

Tn−j−1 for eachk∈N.

If we define

ρ=r∗ n−1

j=0

Tn−j−1_, _Ω₌_x_∈_Cn−1_[0,_{T] :}_xC

n−1≤ρ,

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Step 3. Properties of approximate solutions.

According to (2.6) and (2.8), we get for a.e.t∈[0,T−1/k],

fk

t,uk(t),. . .,u(nk−1)(t) signu (n−1) k (t)≤

n−1

j=0

hj(t)u(j)k (t) αj

<(ρ+ 1) n−1

j=0 hj(t).

Put

ψ(t)= −(ρ+ 1) n−1

j=1

hj(t) for a.e.t∈[0,T].

Thenψ∈L1[0,T],ψ≤0 a.e. on [0,T], and −fk

t,uk(t),. . .,u(n_k−1)(t) signu_k(n−1)(t)≥ψ(t) for a.e.t∈[0,T]. (2.13) Due to (2.7), condition (1.6)(i) witht0=Tis satisfied.

Putλ1 = −1 and choose an arbitraryη ∈ (0,T). Then, by (2.13), we get (1.20). Moreover, condition (2.4) yields (1.22).

Now, let us putvk(t) = u(n_k−1)(t) for t ∈ [0,T]. Then for eachk ∈ N,k > 1/η, the functionvk satisfies (A.20) with h∗ = 0 a.e. on [T −1/k,T]. Since uk ∈ Ω, we can findβ0 ∈(0,ρ) such thatvkfulfils condition (A.18). By (2.6), we get (A.19), where g∗(t)=(ρ+1)n_j₌−01hj(t). Hence, byCriterion A.11, the sequence{vk}is equicontinuous atTfrom the left. Therefore,{u(n−1)

k }satisfies (1.23) witht0 =Tand, byTheorem 1.6, there exists a solutionu∈ACn−1_[0,_{T] of problem (}_2.3_{), (}_2.4_). Example 2.2. Letc∈R,α∈[1,∞). Then the function

ft,x0,. . .,xn−1 = −xn−1 tα +

c √ t

n−1

j=0 x2/3_j

satisfies (2.5) and (2.6), wherehj(t)= |c|/√t,h(t)=1/tα_,_α

j =2/3 for j=0,. . .,n−1. Therefore, the corresponding problem (2.3), (2.4) has a solutionu∈ACn−1_[0,_T].

2.2. Space singularities

LetR−=(−∞, 0) andR+=(0,∞). We study the singular (p,n−p) right focal problem (−1)n−p_u(n)₌ _f_t,_u,_{. . .}_,_u(n−1) _, _(2.14) u(i)(0)=0, 0≤i≤p−1, u(j)_(T)₌_0, _p_≤_j_≤_n₋_1, _(2.15) where f ∈Car([0,T]×D) with

D=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

Rp+1

+ ×R−×R+×R−× · · · ×R+

n

ifn−pis odd, Rp+1

+ ×R−×R+×R−× · · · ×R−

n

(34)

and f may be singular at the value 0 of any of its space variables. Notice that if f is positive, then the singular points corresponding to the solutions of problem (2.14), (2.15) are of type I. The Green function of problemu(n)₌_{0, (}_2.15_{), is presented in Agarwal [}₁_], Agarwal and Usmani [23,24], and Agarwal, O’Regan, and Wong [21].

We introduce the following assumptions:

f ∈Car[0,T]×D) and there exist positive constantsa,rsuch that

a(T−t)r≤ ft,x0,. . .,xn−1 (2.16) for a.e.t∈[0,T] and eachx0,. . .,xn−1 ∈D;

the inequality

ft,x0,. . .,xn−1 ≤h

t, n−1

j=0

_x_j₊n−1 j=0

ωjxj

holds for a.e.t∈[0,T] and eachx0,. . .,xn−1 ∈D, where h∈Car[0,T]×[0,∞) is positive and nondecreasing in the second variable,

ωj:R+ →R+is nonincreasing for 0≤ j≤n−1,

lim sup v→∞

1 v

_T

0 h(t,V v)dt <1, whereV=

⎧ ⎪ ⎨ ⎪ ⎩

Tn₋₁

T−1 ifT=1, n ifT=1,

1

0ωj

tr+n−j dt <∞ for 0≤ j≤n−1.

(2.17)

Substitutingt=T−sin (2.14), (2.15), we get the singular (n−p,p) left focal problem

(−1)p_u(n)₌ _f_s,_u,_{. . .}_,_u(n−1) _, _(2.18) u(i)₍₀₎₌_0, _p_≤_i_≤_n₋_1, _u(j)_(T₎₌_0, ₀_≤_j_≤_p₋_1, _(2.19)

where f∈Car([0,T]×D∗) fulfils

ft,x0,x1,. . .,xn−1 = f

T−t,x0,−x1,. . ., (−1)n−1_x n−1

for a.e.t∈[0,T] and all (x0,. . .,xn−1)∈D∗. Here

D∗= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

R+×R−×R+× · · · ×R−×Rn+−p

n

ifpis even, R+×R−×R+× · · · ×R+×Rn−−p

n

(35)

The corresponding assumptions for problem (2.18), (2.19) have the following form:

f ∈Car[0,T]×D∗ and there exist positive constantsa,rsuch that

atr_≤ _f_t,_x0,_{. . .}_,_x

n−1 (2.20)

for a.e.t∈[0,T] and eachx0,. . .,xn−1 ∈D∗;

the inequality

ft,x0,. . .,xn−1 ≤h

t, n−1

j=0

_x_j₊n−1 j=0

ωjxj

holds for a.e.t∈[0,T] and eachx0,. . .,xn−1 ∈D∗,

where the functionshandωj, 0≤j≤n−1, have the properties given in (2.17).

(2.21)

A priori estimates

Let us choose positive constantsaandrand define the set

B(r,a)=u∈ACn−1[0,T] :ufulfils (2.15) and (2.23), (2.22) where

(−1)n−p_u(n)_(t)_≥_a(T₋_t)r _{for a.e.}_t_∈_[0,_T_]. _(2.23) The next two lemmas are devoted to the study of the setB(r,a). The results obtained in this part will be used in the proofs of existence results for auxiliary regular problems.

Lemma 2.3. There existsc >0 such that the inequalities

u(j)(t)≥ctr+n−j for 0≤j≤p−1, (2.24) (−1)j−p_u(j)_(t)_≥_c(T₋_t)r+n−j _for_p_≤_j_≤_n₋₁ _(2.25) are true fort∈[0,T] and eachu∈B(r,a).

Proof . Put

c= a

(r+ 1)(r+ 2)· · ·(r+n).

Then, integrating inequality (2.23) and using condition (2.15), we get step by step that (2.25) holds on [0,T] and that

Solvability of Nonlinear Singular Problems for Ordinary Differential Equations - Free Computer, Programming, Mathematics, Technical Books, Lecture Notes and Tutorials

Solvability of

Nonlinear

Singular Problems for

Ordinary Differential

Equations

Solvability of Nonlinear Singular Problems for

Ordinary Differential Equations

Contents

Preface

List of notation

1

Existence principles for

singular problems

2

Focal problems