Georgian Mathematical Journal Volume 10 (2003), Number 1, 165–191

STURM–LIOUVILLE AND FOCAL HIGHER ORDER BVPs WITH SINGULARITIES IN PHASE VARIABLES

IRENA RACHŮNKOVÁ AND SVATOSLAV STANĚK

Abstract. The paper deals with the existence of solutions for singular higher order differential equations with Sturm–Liouville or (p, n − p) right focal boundary conditions or the (n − p, p) left focal boundary conditions. Righthand sides of differential equations may be singular in the zero values of all their phase variables. The proofs are based on the regularization and sequential techniques.

2000 Mathematics Subject Classification: 34B15, 34B16, 34B18. Key words and phrases: Singular higher order differential equation, Sturm–Liouville boundary conditions, focal boundary conditions, existence, regularization, sequential technique.

1. Introduction

Let T be a positive constant, J = [0, T] and ℝ_{−} = (−∞, 0), ℝ_{+} = (0, ∞), ℝ_{0} = ℝ \ {0}.

We will consider two types of singular boundary value problems for nth order differential equations, where n ≥ 2. The first one is the singular Sturm–Liouville boundary value problem (BVP for short)

−x^{n}(t) = f(t, x(t), …, x^{(n−1)}(t)),  (1.1)

\begin{matrix}x^{(i)}(0) = 0, 0 ≤ i ≤ n − 3, \\ αx^{(n−2)}(0) − βx^{(n−1)}(0) = 0, \\ γx^{(n−2)}(T) + δx^{(n−1)}(T) = 0, \end{matrix}  (1.2)

where α, γ > 0, β, δ ≥ 0 and f satisfies the local Carathéodory conditions on J × D (f ∈ Car(J × D)) with

D = ℝ_{+}^{n−1} × ℝ_{0}.

The second one is the singular (p, n − p) right focal BVP

(−1)^{n−p}x^{(n)}(t) = f(t, x(t), …, x^{(n−1)}(t)),  (1.3)
x^{(i)}(0) = 0, 0 ≤ i ≤ p − 1, x^{(i)}(T) = 0, p ≤ i ≤ n − 1,  (1.4)

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

where p ∈ ℕ is fixed, 1 ≤ p ≤ n − 1, and f ∈ Car(J × X) with

X = \begin{cases}\underbrace{ℝ_{+}^{p+1} × ℝ_{−} × ℝ_{+} × ℝ_{−} × ⋯ × ℝ_{+}}_{n} & if n − p is odd \\ \underbrace{ℝ_{+}^{p+1} × ℝ_{−} × ℝ_{+} × ℝ_{−} × ⋯ × ℝ_{−}}_{n} & if n − p is even \end{cases} .

In both cases the function f(t, x_{0}, …, x_{n−1}) may be singular at the points x_{i} = 0, 0 ≤ i ≤ n − 1, of all its phase variables x_{0}, …, x_{n−1}.

The aim of this paper is to give conditions for the existence of solutions to problems (1.1), (1.2) and (1.3), (1.4).

Definition 1.1. By a solution of BVP (1.1), (1.2) we understand a function x ∈ AC^{n−1}(J) which is positive on (0, T], satisfies conditions (1.2) and for a.e. t ∈ J fulfils (1.1).

Similarly, by a solution of BVP (1.3), (1.4) we understand a function x ∈ AC^{n−1}(J) which is positive on (0, T], satisfies conditions (1.4) and for a.e. t ∈ J fulfils (1.3).

From now on, ∥x∥ = max{|x(t)| : 0 ≤ t ≤ T }, ∥x∥_{L} = ∫_{0}^{T} |x(t)| dt and

∥x∥_{∞} = ess max {|x(t)| : 0 ≤ t ≤ T } stand for the norm in C^{0}(J), L_{1}(J) and L_{∞}(J), respectively. For a subset Ω of a Banach space, cl(Ω) and ∂Ω stand for the closure and the boundary of Ω, respectively. Finally, for any measurable set M, μ(M) denotes the Lebesgue measure of M.

The fact that a BVP is singular means that the right-hand side f of the considered differential equation does not fulfil the Carathéodory conditions on a domain where we seek for solutions, i.e. on J × cl(D) if we work with equation (1.1) or on J × cl(X) if we study equation (1.3). In singular problems the Carathéodory conditions can be broken in the time variable t or in the phase variables or in the both types of variables. The first type of singularities where f need not be integrable on J for fixed phase variables was studied by many authors. For BVPs of n-th order differential equations such problems were considered for the first time by Kiguradze in [16]. The second type of singularities, i.e., the case where f is unbounded in some values of its phase variables x_{0}, x_{1}, …, x_{n−1} for fixed t ∈ J was mainly solved for BVPs of second order differential equations, but during the last decade papers dealing with higher order BVPs having singularities in phase variables have appeared as well. We can refer to the papers [2]–[7], [9]–[14] and [19]–[23]. Some of them (see [3]–[7]) concern higher order singular Sturm–Liouville or right focal BVPs.

In this paper we extend results in the cited papers to the case of a general Carathéodory right-hand side f which may depend on higher derivatives up to order n − 1 and which may have singularities in all its phase variables. The proofs are based on the construction of a proper sequence of regular problems and in limiting processes. The correctness of such processes is warrantable by the Lebesgue dominated convergence theorem in the case of problem (1.3), (1.4).

As regards problem (1.1), (1.2) note that conditions (1.2) imply that for any solution x of this problem its derivative x^{(n−1)} is a sign-changing function on J. Therefore this derivative goes through the singularity of f somewhere inside J, which makes it impossible to find a Lebesgue integrable majorant function for any auxiliary sequence of regular functions {f_{m}} relevant to problem (1.1), (1.2). This implies that in this case instead of the Lebesgue theorem the Vitali convergence theorem is used.

The proofs of the existence results for the auxiliary regular BVPs considered in Section 3 are based on the Nonlinear Fredholm Alternative (see e.g. [17], Theorem 4 or [21], p. 25) which we formulate in the form convenient for the application to the problems mentioned above. In particular, we consider the differential equation

x^{(n)}(t) + ∑_{i=0}^{n−1} a_{i}(t)x^{(i)}(t) = h(t, x(t), …, x^{(n−1)}(t))  (1.5)

and the corresponding homogeneous equation

x^{(n)}(t) + ∑_{i=0}^{n−1} a_{i}(t)x^{(i)}(t) = 0,  (1.6)

where a_{i} ∈ L_{1}(J), 0 ≤ i ≤ n − 1, h ∈ Car(J × ℝ^{n}). Further we deal with the boundary conditions

L_{j}(x) = r_{j}, 1 ≤ j ≤ n,  (1.7)

with r_{j} ∈ ℝ and continuous linear functionals L_{j} : C^{n−1}(J) → ℝ, 1 ≤ j ≤ n.

Definition 1.2. By a solution of BVP (1.5), (1.7) we understand a function x ∈ AC^{n−1}(J) which satisfies conditions (1.7) and for a.e. t ∈ J fulfils (1.5).

Theorem 1.3 (Nonlinear Fredholm Alternative). Let problem (1.6), (1.7) has only a trivial solution and there exist a function g ∈ L_{1}(J) such that

|h(t, x_{0}, …, x_{n−1})| ≤ g(t) for a.e. t ∈ J and all x_{0}, …, x_{n−1} ∈ ℝ.

Then problem (1.5), (1.7) has a solution.

The following assumptions will be used in the study of problem (1.1), (1.2):

(H_{1}) f ∈ Car(J × D) and there exist positive constants ε, r such that

εt^{r} ≤ f(t, x_{0}, …, x_{n−1})
for a.e. t ∈ J and each (x_{0}, …, x_{n−1}) ∈ D;

(H_{2}) For a.e. t ∈ J and for each (x_{0}, …, x_{n−1}) ∈ D,

f(t, x_{0}, …, x_{n−1}) ≤ ϕ(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x_{i}|) + ∑_{i=0}^{n−1} h_{i}(t)|x_{i}|^{α_{i}},  (1.8)

where α_{i} ∈ (0, 1), ϕ, h_{i} ∈ L_{1}(J), q_{i} ∈ L_{∞}(J) are nonnegative, ω_{i} : ℝ_{+} → ℝ_{+} are nonincreasing, 0 ≤ i ≤ n − 1, and

∫_{0}^{T} ω_{n−1}(t^{r+1}) dt < ∞, ∫_{0}^{T} ω_{i}(t^{n−i−1}) dt < ∞ for 0 ≤ i ≤ n − 2;  (1.9)

(H_{3}) For a.e. t ∈ J and for each (x_{0}, …, x_{n−1}) ∈ D, the function f satisfies (1.8) where α_{i} ∈ (0, 1), ϕ, h_{i}, q_{n−2} ∈ L_{1}(J), q_{j}, q_{n−1} ∈ L_{∞}(J) are nonnegative, ω_{i} : ℝ_{+} → ℝ_{+} are nonincreasing, 0 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 3, and^{1}

∫_{0}^{T} ω_{n−1}(t^{r+1}) dt < ∞, ∫_{0}^{T} ω_{j}(t^{n−j−2}) dt < ∞ for 0 ≤ j ≤ n − 3. (1.10)

In the study of problem (1.3), (1.4) we will work with the assumptions:

(H_{4}) f ∈ Car(J × X) and there exist positive constants ε, r such that

ε(T − t)^{r} ≤ f(t, x_{0}, …, x_{n−1})

for a.e. t ∈ J and each (x_{0}, …, x_{n−1}) ∈ X; (H_{5}) For a.e. t ∈ J and for each (x_{0}, …, x_{n−1}) ∈ X, the function f satisfies

(1.8) where α_{i} ∈ (0, 1), ϕ, h_{i} ∈ L_{1}(J), q_{i} ∈ L_{∞}(J) are nonnegative, ω_{i} : ℝ_{+} → ℝ_{+} are nonincreasing, 0 ≤ i ≤ n − 1, and

∫_{0}^{T} ω_{i}(t^{r+n−i}) dt < ∞ for 0 ≤ i ≤ n − 1.  (1.11)

Remark 1.4. Since ω_{i} : ℝ_{+} → ℝ_{+} in (H_{2}) are nonincreasing, assumption (1.9) implies that

∫_{0}^{V} ω_{n−1}(t^{r+1}) dt < ∞, ∫_{0}^{V} ω_{i}(t^{n−i−1}) dt < ∞, 0 ≤ i ≤ n − 2

for each V ∈ ℝ_{+}. The same is true for all integrals into (1.10) and (1.11).

Remark 1.5. After substituting t = T − s in (1.3), (1.4), we get the singular (n − p, p) left focal BVP

(−1)^{p}x^{(n)}(s) = f̃(s, x(s), …, x^{(n−1)}(s)),  (1.12)
x^{(i)}(0) = 0, p ≤ i ≤ n − 1, x^{(i)}(T) = 0, 0 ≤ i ≤ p − 1,  (1.13)

1Throughout the paper conditions and statements depending on j with 0 ≤ j ≤ n − 3 are realized only in the case where n ≥ 3.

where p is fixed, 1 ≤ p ≤ n−1, and f̃ ∈ Car(J×Y) fulfils f̃(s, x_{0}, x_{1}, …, x_{n−1}) = f(T − s, x_{0}, −x_{1}, …, (−1)^{n−1}x_{n−1}). Here

Y = \begin{cases}\underbrace{ℝ_{+} × ℝ_{−} × ℝ_{+} × ⋯ × ℝ_{−} × ℝ_{+}^{n−p}}_{n} & if p is even, \\ \underbrace{ℝ_{+} × ℝ_{−} × ℝ_{+} × ⋯ × ℝ_{+} × ℝ_{−}^{n−p}}_{n} & if p is odd. \end{cases}

By a solution of BVP (1.12), (1.13) we understand a function x ∈ AC^{n−1}(J) which is positive on [0, T), satisfies conditions (1.13) and for a.e. s ∈ J fulfils (1.12).

The corresponding assumptions for problem (1.12), (1.13) have the form:

(H_{6}) f̃ ∈ Car(J × Y) and there exist positive constants ε, r such that

εs^{r} ≤ f̃(s, x_{0}, …, x_{n−1})
for a.e. s ∈ J and each (x_{0}, …, x_{n−1}) ∈ Y;

(H_{7}) For a.e. s ∈ J and for each (x_{0}, …, x_{n−1}) ∈ Y the function f̃ satisfies

f̃(s, x_{0}, …, x_{n−1}) ≤ ϕ(s) + ∑_{i=0}^{n−1} q_{i}(s)ω_{i}(|x_{i}|) + ∑_{i=0}^{n−1} h_{i}(s)|x_{i}|^{α_{i}},

where α_{i} ∈ (0, 1), ϕ, h_{i} ∈ L_{1}(J), q_{i} ∈ L_{∞}(J) are nonnegative, ω_{i} : ℝ_{+} → ℝ_{+} are nonincreasing, 0 ≤ i ≤ n − 1, and ω_{i} fulfil (1.11).

2. Green’s Functions and a Priori Estimates

2.1. Problem (1.1), (1.2). From now on, G(t, s) denotes the Green’s function of the BVP

−x′′(t) = 0,  (2.1)
αx(0) − βx′(0) = 0, γx(T) + δx′(T) = 0.  (2.2)

Then (see, e.g., [1])

G(t, s) = \begin{cases}\frac{1}{d} (β + αs)(δ + γ(T − t)) & for 0 ≤ s ≤ t ≤ T, \\ \frac{1}{d} (β + αt)(δ + γ(T − s)) & for 0 ≤ t < s ≤ T, \end{cases}  (2.3)

where

d = αγT + αδ + βγ > 0.

Let us choose positive constants ε and r and define the set

A(r, ε) = {x ∈ AC^{n−1}(J) : x fulfils (1.2) and (2.4)}

where

−x^{(n)}(t) ≥ εt^{r} for a.e. t ∈ J.  (2.4)

170 IRENA RACHŮNKOVÁ AND SVATOSLAV STANĚK

2.1.1. Problem (1.1), (1.2) with min{β, δ} = 0. In this subsection we assume that at least one constant of β and δ appearing in (1.2) is equal to zero.

Lemma 2.1. Let x ∈ A(r, ε) and set

A = \frac{ε}{(r + 1)(r + 2)} (\frac{T}{2} )^{r+1}.  (2.5)

Then x^{(n−1)} is decreasing on J,

\begin{aligned}x^{(n−1)}(t) & ≥ \frac{ε}{r + 1}(ξ − t)^{r+1} for t ∈ [0, ξ], \\ x^{(n−1)}(t) & < −\frac{ε}{r + 1}(t − ξ)^{r+1} for t ∈ (ξ, T], \end{aligned}  (2.6)

where ξ ∈ (0, T) is the unique zero of x^{(n−1)},

x^{(n−2)}(t) ≥ \begin{cases}At & for t ∈ [0, \frac{T}{2} ] \\ A(T − t) & for t ∈ (\frac{T}{2} , T ] \end{cases}  (2.7)

and

x^{(j)}(t) ≥ \frac{A}{4(n − j − 1)!}t^{n−j−1} for t ∈ J, 0 ≤ j ≤ n − 3.  (2.8)

Proof. From the equality

x^{(n−2)}(t) = − ∫_{0}^{T} G(t, s)x^{(n)}(s) ds, t ∈ J,

(2.3) and (2.4) we deduce that

x^{(n−2)}(0) = − ∫_{0}^{T} G(0, s)x^{(n)}(s) ds = −\frac{β}{d} ∫_{0}^{T}(δ + γ(T − s))x^{(n)}(s) ds
≥ \frac{εβγ}{d} ∫_{0}^{T}(T − s)s^{r} ds ≥ 0,  (2.9)
x^{(n−2)}(T) = − ∫_{0}^{T} G(T, s)x^{(n)}(s) ds = −\frac{δ}{d} ∫_{0}^{T}(β + αs)x^{(n)}(s) ds
≥ \frac{εαδ}{d} ∫_{0}^{T} s^{r+1} ds ≥ 0,  (2.10)
x^{(n−1)}(0) = − ∫_{0}^{T} \frac{∂G(t, s)}{∂t} |_{t=0}x^{(n)}(s) ds = −\frac{α}{d} ∫_{0}^{T}(δ + γ(T − s))x^{(n)}(s) ds

≥ \frac{εαγ}{d} ∫_{0}^{T}(T − s)s^{r} ds > 0

and

x^{(n−1)}(T) = − ∫_{0}^{T} \frac{∂G(t, s)}{∂t}|_{t=T} x^{(n)}(s) ds = \frac{γ}{d} ∫_{0}^{T}(β + αs)x^{(n)}(s) ds

≤ −\frac{εαγ}{d} ∫_{0}^{T} s^{r+1} ds < 0.

Since x^{(n−1)} is decreasing on J by (2.4) and x^{(n−1)}(0) > 0, x^{(n−1)}(T) < 0, we see that x^{(n−1)} has a unique zero ξ ∈ (0, T). Then

−x^{(n−1)}(t) = ∫_{t}^{ξ} x^{(n)}(s) ds ≤ −ε ∫_{t}^{ξ} s^{r} ds = −\frac{ε}{r + 1}(ξ^{r+1} − t^{r+1}), t ∈ [0, ξ],

and so

x^{(n−1)}(t) ≥ \frac{ε}{r + 1}(ξ − t)^{r+1}, t ∈ [0, ξ],

since ξ^{r+1} − t^{r+1} ≥ (ξ − t)^{r+1} for t ∈ [0, ξ]. Analogously, using the inequality t^{r+1} − ξ^{r+1} > (t − ξ)^{r+1} for t ∈ (ξ, T], we obtain

x^{(n−1)}(t) = ∫_{ξ}^{t}x^{(n)}(s) ds ≤ −ε ∫_{ξ}^{t}s^{r} ds = −\frac{ε}{r + 1}(t^{r+1} − ξ^{r+1}) < −\frac{ε}{r + 1}(t − ξ)^{r+1}

for t ∈ (ξ, T]. We have proved that (2.6) holds.

We are going to verify (2.7). From the assumption min{β, δ} = 0, (2.9) and (2.10) we have x^{(n−2)}(0) ≥ 0, x^{(n−2)}(T) ≥ 0 and min{x^{(n−2)}(0), x^{(n−2)}(T)} = 0. Moreover, x^{(n−2)} is concave on J, which follows from (2.4). Hence to prove (2.7) it suffices to show that

x^{(n−2)}(\frac{T}{2} ) ≥ \frac{AT}{2} ,  (2.11)

where A is given by (2.5). From (2.6) it follows that

x^{(n−2)}(t) = x^{(n−2)}(0) + ∫_{0}^{t} x^{(n−1)}(s) ds ≥ \frac{ε}{r + 1} ∫_{0}^{t} (ξ − s)^{r+1} ds

= \frac{ε}{(r + 1)(r + 2)} (ξ^{r+2} − (ξ − t)^{r+2}), t ∈ [0, ξ],

−x^{(n−2)}(t) = −x^{(n−2)}(T) + ∫_{t}^{T} x^{(n−1)}(s) ds < −\frac{ε}{r + 1} ∫_{t}^{T}(s − ξ)^{r+1} ds

= −\frac{ε}{(r + 1)(r + 2)} ((T − ξ)^{r+2} − (t − ξ)^{r+2}), t ∈ (ξ, T],

and since ξ^{r+2} − (ξ − t)^{r+2} ≥ t^{r+2} for t ∈ [0, ξ] and (T − ξ)^{r+2} − (t − ξ)^{r+2} ≥ (T − t)^{r+2} for t ∈ (ξ, T], we have

x^{(n−2)}(t) ≥ \frac{ε}{(r + 1)(r + 2)}t^{r+2} for t ∈ [0, ξ],  (2.12)
x^{(n−2)}(t) > \frac{ε}{(r + 1)(r + 2)}(T − t)^{r+2} for t ∈ (ξ, T].  (2.13)

We know that max{x^{(n−2)}(t) : t ∈ J} = x^{(n−2)}(ξ). If ξ ≥ T/2, then (2.12) gives (2.11) and if ξ < T/2 then from (2.13) we obtain (2.11) as well.

It remains to prove (2.8). By (2.7) and x^{(n−3)}(0) = 0, we have

x^{(n−3)}(t) = ∫_{0}^{t} x^{(n−2)}(s) ds ≥ A ∫_{0}^{t} s ds = \frac{A}{2}t^{2}

for t ∈ [0, T/2]. Hence x^{(n−3)}(T/2) ≥ (A/2)(T/2)^{2} and since x^{(n−3)} is increasing on J and (t/2)^{2} ≤ (T/2)^{2} for t ∈ J, we see that

x^{(n−3)}(t) ≥ \frac{A}{4 · 2!} t^{2}, t ∈ J.

Then using the equalities

x^{(i)}(t) = ∫_{0}^{t} x^{(i+1)}(s) ds, t ∈ J, 0 ≤ i ≤ n − 4,

we can verify that inequalities (2.8) are satisfied. □

Lemma 2.2. For 0 ≤ i ≤ n−1 let ϕ̂, h_{i} ∈ L_{1}(J), q_{i} ∈ L_{∞}(J) be nonnegative, ω_{i} : ℝ_{+} → ℝ_{+} be nonincreasing and satisfy (1.9) and α_{i} ∈ (0, 1). Then there exists a positive constant M̂ such that for each x ∈ A(r, ε) satisfying

−x^{(n)}(t) ≤ ϕ̂(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x^{(i)}(t)|) + ∑_{i=0}^{n−1} h_{i}(t)|x^{(i)}(t)|^{α_{i}}  (2.14)

for a.e. t ∈ J, the estimates

∥x^{(i)}∥ ≤ M̂ for 0 ≤ i ≤ n − 1  (2.15)

are valid.

Proof. Let x ∈ A(r, ε) satisfy inequalities (2.14) a.e. on J. By Lemma 2.1, x^{(n−1)} has a unique zero ξ ∈ (0, T) and x satisfies inequalities (2.6)–(2.8) with A given by (2.5). From x^{(n−2)}(0) = (β/α)x^{(n−1)}(0) ≥ 0 (see (1.2)) it follows that

|x^{(n−2)}(t)| ≤ \frac{β}{α}x^{(n−1)}(0) + ∫_{0}^{t} |x^{(n−1)}(s)| ds ≤ (T + \frac{β}{α})∥x^{(n−1)}∥, t ∈ J.

Hence

∥x^{(n−2)}∥ ≤ (T + \frac{β}{α})∥x^{(n−1)}∥

STURM–LIOUVILLE AND FOCAL HIGHER ORDER BVPs 173

and then the equalities

x^{(j)}(t) = \frac{1}{(n − j − 3)!} ∫_{0}^{t} (t − s)^{n−j−3}x^{(n−2)}(s) ds, t ∈ J, 0 ≤ j ≤ n − 3,

give

∥x^{(j)}∥ ≤ \frac{T^{n−j−2}}{(n − j − 3)!}∥x^{(n−2)}∥

≤ \frac{T^{n−j−2}}{(n − j − 3)!} (T + \frac{β}{α})∥x^{(n−1)}∥, 0 ≤ j ≤ n − 3.

Setting

V = (T + \frac{β}{α}) max\{1, V_{1}\},  (2.16)

where

V_{1} = max \{\frac{T^{n−j−2}}{(n − j − 3)!} : 0 ≤ j ≤ n − 3\},

we see that

∥x^{(j)}∥ ≤ V ∥x^{(n−1)}∥ for 0 ≤ j ≤ n − 2.  (2.17)

Now (2.14) yields

|x^{(n−1)}(t)| = | ∫_{ξ}^{t} x^{(n)}(s) ds|

≤ ∫_{0}^{T} [ϕ̂(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x^{(i)}(t)|) + ∑_{i=0}^{n−1} h_{i}(t)|x^{(i)}(t)|^{α_{i}}] dt

≤ ∥ϕ̂∥_{L} + ∑_{i=0}^{n−1} ∥q_{i}∥_{∞} ∫_{0}^{T} ω_{i}(|x^{(i)}(t)|) dt + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}V^{α_{i}}∥x^{(n−1)}∥^{α_{i}}.  (2.18)

Set K = \sqrt[r+1]{ε/(r + 1)} and R_{j} = \sqrt[n−j−1]{A/(4(n − j − 1)!)}, 0 ≤ j ≤ n − 3. Since (cf. (2.6)–(2.8))

∫_{0}^{T} ω_{n−1}(|x^{(n−1)}(t)|) dt

≤ ∫_{0}^{ξ} ω_{n−1}(\frac{ε}{r + 1} (ξ − t)^{r+1}) dt + ∫_{ξ}^{T} ω_{n−1}(\frac{ε}{r + 1} (t − ξ)^{r+1}) dt

= \frac{1}{K} [ ∫_{0}^{Kξ}ω_{n−1}(t^{r+1}) dt + ∫_{0}^{K(T −ξ)}ω_{n−1}(t^{r+1}) dt] ≤ \frac{2}{K} ∫_{0}^{KT}ω_{n−1}(t^{r+1}) dt, (2.19)

∫_{0}^{T} ω_{n−2}(|x^{(n−2)}(t)|) dt

≤ ∫_{0}^{T/2}ω_{n−2}(At) dt + ∫_{T/2}^{T} ω_{n−2}(A(T − t)) dt = \frac{2}{A} ∫_{0}^{(AT)/2}ω_{n−2}(t) dt

and (for 0 ≤ j ≤ n − 3)

∫_{0}^{T} ω_{j}(|x^{(j)}(t)|) dt ≤ ∫_{0}^{T} ω_{j}(\frac{A}{4(n − j − 1)!}t^{n−j−1}) dt = \frac{1}{R_{j}} ∫_{0}^{R_{j}T}ω_{j}(t^{n−j−1}) dt,

we deduce from (1.9) and Remark 1.4 that there is a positive constant Q independent of x such that

∑_{i=0}^{n−1} ∥g_{i}∥_{∞} ∫_{0}^{T} ω_{i}(|x^{(i)}(t)|) dt ≤ Q.

Then (2.1.1) yields

∥x^{(n−1)}∥ ≤ ∥ϕ̂∥_{L} + Q + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}V^{α_{i}}∥x^{(n−1)}∥^{α_{i}}.  (2.20)

Setting z(u) = (∥ϕ̂∥_{L} + Q)/u + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}V^{α_{i}}u^{α_{i}−1} for u ∈ (0, ∞), we have

lim_{u→∞} z(u) = 0, and so there is a positive constant P such that z(u) < 1 for all u ≥ P. Then from (2.20) it follows that ∥x^{(n−1)}∥ ≤ P and, by (2.17), ∥x^{(j)}∥ ≤ V P for 0 ≤ j ≤ n−2. Hence (2.15) is true with M̂ = max{P, V P }. □

2.1.2. Problem (1.1), (1.2) with min{β, δ} > 0. Throughout this subsection we assume that the constants β and δ in (1.2) are positive.

Lemma 2.3. Let x ∈ A(r, ε) and set

B = \frac{ε}{d} min \{βγ ∫_{0}^{T}(T − s)s^{r} ds, αδ ∫_{0}^{T} s^{r+1} ds\} > 0.  (2.21)

Then x^{(n−1)} is decreasing on J, satisfies inequalities (2.6) where ξ ∈ (0, T) is its unique zero,

x^{(n−2)}(t) ≥ B for t ∈ J  (2.22)

and

x^{(j)}(t) ≥ \frac{B}{(n − j − 2)!} t^{n−j−2} for t ∈ J, 0 ≤ j ≤ n − 3.  (2.23)

Proof. The properties of x^{(n−1)} follow immediately from Lemma 2.1 and its proof. Next, by (2.9) and (2.10),

x^{(n−2)}(0) ≥ \frac{εβγ}{d} ∫_{0}^{T}(T − s)s^{r} ds > 0, x^{(n−2)}(T) ≥ \frac{εαδ}{d} ∫_{0}^{T} s^{r+1} ds > 0.

Since x^{(n−2)} is concave on J, we see that x^{(n−2)}(t) ≥ B for t ∈ J. Now from the last inequality and the equalities x^{(j)}(0) = 0, 0 ≤ j ≤ n − 3 we obtain the validity of (2.23). □

Lemma 2.4. For 0 ≤ i ≤ n − 1 and 0 ≤ j ≤ n − 3, let ϕ̂, h_{i}, q_{n−2} ∈ L_{1}(J), q_{j}, q_{n−1} ∈ L_{∞}(J) be nonnegative, ω_{i} : ℝ_{+} → ℝ_{+} be nonincreasing and satisfying (1.10) and α_{i} ∈ (0, 1). Then there exists a positive constant V̂ such that

∥x^{(i)}∥ ≤ V̂ for 0 ≤ i ≤ n − 1  (2.24)

whenever x ∈ A(r, ε) satisfies inequality (2.14) for a.e. t ∈ J.

Proof. Let x ∈ A(r, ε) satisfy inequality (2.14) for a.e. t ∈ J. By Lemma 2.3, inequalities (2.6), (2.22) and (2.23) are true, where ξ ∈ (0, T) is the unique zero of x^{(n−1)} and B is given by (2.21). From x^{(n−2)}(0) = (β/α)x^{(n−1)}(0) (see (1.2)) and using the same procedure as in the proof of Lemma 2.2 we see that (2.17) holds, where L is defined by (2.16).

Since x^{(n−2)}(t) ≥ B for t ∈ J and x^{(j)}(0) = 0, 0 ≤ j ≤ n − 3, we have

x^{(j)}(t) ≥ \frac{B}{(n − j − 2)!}t^{n−j−2} for t ∈ J, 0 ≤ j ≤ n − 3.  (2.25)

Hence

ω_{n−2}(x^{(n−2)}(t)) ≤ ω_{n−2}(B), t ∈ J,

and

∫_{0}^{T} ω_{j}(x^{(j)}(t)) dt ≤ ∫_{0}^{T} ω_{j}(\frac{B}{(n − j − 2)!}t^{n−j−2}) dt = \frac{1}{m_{j}} ∫_{0}^{m_{j}T}ω_{j}(t^{n−j−2}) dt

for 0 ≤ j ≤ n − 3, where m_{j} = \sqrt[n−j−2]{B/(n − j − 2)!}. Then (see (2.14), (2.17) and (2.1.1))

|x^{(n−1)}(t)| = | ∫_{ξ}^{t} x^{(n)}(s) ds|

≤ ∫_{0}^{T} [ϕ̂(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x^{(i)}(t)|) + ∑_{i=0}^{n−1} h_{i}(t)|x^{(i)}(t)|^{α_{i}}] dt

≤ ∥ϕ̂∥_{L} + ∑_{i=0}^{n−3} \frac{∥q_{i}∥_{∞}}{m_{i}} ∫_{0}^{m_{i}T}ω_{i}(t^{n−i−2}) dt + ∥q_{n−2}∥_{L}ω_{n−2}(B)

+\frac{2∥q_{n−1}∥_{∞}}{K} ∫_{0}^{KT}ω_{n−1}(t^{r+1}) dt + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}L^{α_{i}}∥x^{(n−1)}∥^{α_{i}},

where K = \sqrt[r+1]{ε/(r + 1)}. Consequently,

∥x^{(n−1)}∥ ≤ D_{∗} + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}L^{α_{i}}∥x^{(n−1)}∥^{α_{i}},  (2.26)

where

D_{∗} = ∥ϕ̂∥_{L} + ∑_{i=0}^{n−3} \frac{∥q_{i}∥_{∞}}{m_{i}} ∫_{0}^{m_{i}T}ω_{i}(t^{n−i−2}) dt

+ ∥q_{n−2}∥_{L}ω_{n−2}(B) + \frac{2∥q_{n−1}∥_{∞}}{K} ∫_{0}^{KT}ω_{n−1}(t^{r+1}) dt

is independent of x. Since lim_{u→∞}(D_{∗}/u + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}L^{α_{i}}u^{α_{i}−1}) = 0, there is a

positive constant P_{∗} such that D_{∗}/u + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}L^{α_{i}}u^{α_{i}−1} < 1 for u ∈ [P_{∗}, ∞).

Therefore (2.26) gives ∥x^{(n−1)}∥ ≤ P_{∗}. Now (2.17) leads to (2.24) with V̂ = max{P_{∗}, LP_{∗}}. □

2.2. Problem (1.3), (1.4). Let us choose positive constants ε and r and define the set

B(r, ε) = {x ∈ AC^{n−1}(J) : x fulfils (1.4) and (2.28)},  (2.27)

where

(−1)^{n−p}x^{(n)}(t) ≥ ε(T − t)^{r} for a.e. t ∈ J.  (2.28)

Section 2.2 is devoted to the study of the set B(r, ε). The properties of B(r, ε) obtained here (Lemmas 2.5–2.7) will be used in the proof of Theorem 4.3.

Lemma 2.5. There exists c > 0 such that the inequalities

\begin{aligned}x^{(i)}(t) & ≥ ct^{r+n−i} for 0 ≤ i ≤ p − 1, \\ (−1)^{i−p}x^{(i)}(t) & ≥ c(T − t)^{r+n−i} for p ≤ i ≤ n − 1 \end{aligned}  (2.29)

are true for t ∈ J and each x ∈ B(r, ε).

Proof. Let us put

c = \frac{ε}{(r + 1)(r + 2) … (r + n)} .

Then, using (1.4) and integrating (2.28), we get step by step that (2.29) holds on J for p ≤ i ≤ n − 1 and that

x^{(p−1)}(t) ≥ c(T^{r+n−p+1} − (T − t)^{r+n−p+1}) for t ∈ J.  (2.30)

Put r + n − p + 1 = ν and consider the function φ(t) = T^{ν} − (T − t)^{ν} − t^{ν} on J. Since ν > 2, φ(0) = φ(T) = 0 and φ is concave on J, we have φ > 0 on (0, T) and thus

T^{r+n−p+1} − (T − t)^{r+n−p+1} > t^{r+n−p+1}

holds on (0, T), which together with (2.30) yields

x^{(p−1)}(t) ≥ ct^{r+n−p+1} for t ∈ J.  (2.31)

Now, using (1.4) again and integrating (2.31), we successively obtain (2.29) for 0 ≤ i ≤ p − 1 and t ∈ J. □

Lemma 2.6. Let α_{i} ∈ (0, 1), ϕ^{∗}, h_{i} ∈ L_{1}(J), q_{i} ∈ L_{∞}(J), 0 ≤ i ≤ n − 1. Further, suppose that ω_{i} : ℝ_{+} → ℝ_{+} are nonincreasing and fulfil (1.11). Then there exists r^{∗} > 0 such that for each function x ∈ B(r, ε) satisfying

(−1)^{n−p}x^{(n)}(t) ≤ ϕ^{∗}(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x^{(i)}(t)|) + ∑_{i=0}^{n−1} h_{i}(t)|x^{(i)}(t)|^{α_{i}} (2.32)

for a.e. t ∈ J, the estimate

∥x^{(n−1)}∥ < r^{∗}  (2.33)

is valid.

Proof. Having a function x ∈ B(r, ε) which satisfies (2.32), we put ∥x^{(n−1)}∥ = ρ. Then we integrate the inequality

|x^{(n−1)}(t)| ≤ ρ for t ∈ J,

and due to (1.4), we successively get

∥x^{(i)}∥ ≤ ρT^{n−i−1}, 0 ≤ i ≤ n − 2.  (2.34)

Further, we integrate (2.32) on [t, T] ⊂ J and in view of (2.34) we see that the inequality

ρ ≤ ∥ϕ^{∗}∥_{L} + ∑_{i=0}^{n−1} ∥q_{i}∥_{∞} ∫_{0}^{T} ω_{i}(|x^{(i)}(t)|) dt + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}(ρT^{n−i−1})^{α_{i}}  (2.35)

holds. In order to find r^{∗} fulfilling (2.33) we need to estimate the integrals

∫_{0}^{T} ω_{i}(|x^{(i)}(t)|) dt, 0 ≤ i ≤ n − 1.

To this end we distinguish two cases.

Case (α). Let 0 ≤ i ≤ p − 1. Then, by Lemma 2.5, there exists c > 0 such that

∫_{0}^{T} ω_{i}(|x^{(i)}(t)|) dt ≤ ∫_{0}^{T} ω_{i}(ct^{r+n−i}) dt = ∫_{0}^{T} ω_{i}((c_{i}s)^{r+n−i}) ds,

where c_{i}^{r+n−i} = c. Therefore, having in mind Remark 1.4, we conclude that

∫_{0}^{T} ω_{i}(|x^{(i)}(t)|) dt ≤ C_{i},  (2.36)

with

C_{i} = \frac{1}{c_{i}} ∫_{0}^{c_{i}T}ω_{i}(t^{r+n−i}) dt ∈ ℝ_{+}.  (2.37)

Case (β). Let p ≤ i ≤ n − 1. Then by virtue of Lemma 2.5 and (2.37) we get

∫_{0}^{T} ω_{i}(|x^{(i)}(t)|) dt ≤ ∫_{0}^{T} ω_{i}(c(T − t)^{r+n−i}) dt = ∫_{0}^{T} ω_{i}(ct^{r+n−i}) dt = C_{i},

i.e., (2.36) holds for p ≤ i ≤ n − 1 as well.

After inserting (2.36) into (2.35), we obtain

ρ ≤ ∥ϕ^{∗}∥_{L} + ∑_{i=0}^{n−1} ∥q_{i}∥_{∞}C_{i} + ∑_{i=0}^{n−1} ∥h_{i}∥_{L}(ρT^{n−i−1})^{α_{i}}.  (2.38)

Now, suppose that r^{∗} fulfilling (2.33) does not exist. Then we can find a sequence of functions {x_{m}} such that x_{m} ∈ B(r, ε) satisfy (2.32) for m ∈ ℕ and

lim_{m→∞} ∥x_{m}^{(n−1)}∥ = ∞.

If we put ∥x_{m}^{(n−1)}∥ = ρ_{m}, then ρ_{m} satisfy (2.38) for m ∈ ℕ, which yields

1 ≤ \frac{1}{ρ_{m}} (A + ∑_{i=0}^{n−1} B_{i}ρ_{m}^{α_{i}}),  (2.39)

where

A = ∥ϕ^{∗}∥_{L} + ∑_{i=0}^{n−1} ∥q_{i}∥_{∞}C_{i}, B_{i} = (T^{n−i−1})^{α_{i}}∥h_{i}∥_{L} for 0 ≤ i ≤ n − 1.

Since α_{i} ∈ (0, 1) for 0 ≤ i ≤ n − 1, inequality (2.39) implies

1 ≤ lim_{m→∞} \frac{1}{ρ_{m}} (A + ∑_{i=0}^{n−1} B_{i}ρ_{m}^{α_{i}}) = 0,

a contradiction. Therefore a positive constant r^{∗} satisfying (2.33) must exist. □

Lemma 2.7. Suppose that ω_{i} : ℝ_{+} → ℝ_{+}, 0 ≤ i ≤ n − 1, are nonincreasing and fulfil (1.11). Then for each η > 0 there exists δ > 0 such that the condition

|t_{2} − t_{1}| < δ ⟹ | ∫_{t_{1}}^{t_{2}} ω_{i}(|x^{(i)}(t)|) dt| < η  (2.40)

holds for all t_{1}, t_{2} ∈ J, x ∈ B(r, ε), 0 ≤ i ≤ n − 1.

Proof. Let us choose t_{1}, t_{2} ∈ J and x ∈ B(r, ε). Similarly to the proof of Lemma 2.6 we consider two cases.

Case (α). Let 0 ≤ i ≤ p − 1. Then by Lemma 2.5 there is a positive constant c such that if we put c_{i}^{r+n−i} = c, we have

| ∫_{t_{1}}^{t_{2}} ω_{i}(|x^{(i)}(t)|) dt| ≤ | ∫_{t_{1}}^{t_{2}} ω_{i}(ct^{r+n−i}) dt| = | ∫_{t_{1}}^{t_{2}} ω_{i}(c_{i}s)^{r+n−i}ds|.

Therefore we conclude that

| ∫_{t_{1}}^{t_{2}} ω_{i}(|x^{(i)}(t)|) dt| ≤ \frac{1}{c_{i}} | ∫_{c_{i}t_{1}}^{c_{i}t_{2}}ω_{i}(t^{r+n−i})) dt|.  (2.41)

Case (β). Let p ≤ i ≤ n − 1. Then Lemma 2.5 yields

| ∫_{t_{1}}^{t_{2}} ω_{i}(|x^{(i)}(t)|) dt| ≤ | ∫_{t_{1}}^{t_{2}} ω_{i}(c(T − t)^{r+n−i}) dt| = | ∫_{T −t_{1}}^{T −t_{2}}ω_{i}(ct^{r+n−i}) dt|,

and so

| ∫_{t_{1}}^{t_{2}} ω_{i}(|x^{(i)}(t)|) dt| ≤ \frac{1}{c_{i}} | ∫_{c_{i}(T −t_{1})}^{c_{i}(T −t_{2})}ω_{i}(t^{r+n−i})) dt|  (2.42)

with c_{i} given by Case (α).

Now, let us choose arbitrary η > 0 and i ∈ {0, …, p − 1}, j ∈ {p, …, n − 1}. Then, according to Remark 1.4, there exists δ > 0 such that

|t_{2} − t_{1}| < δ ⟹ \frac{1}{c_{i}} | ∫_{c_{i}t_{1}}^{c_{i}t_{2}}ω_{i}(t^{r+n−i}) dt| < η,

|t_{2} − t_{1}| < δ ⟹ \frac{1}{c_{j}} | ∫_{c_{j}(T −t_{1})}^{c_{j}(T −t_{2})}ω_{j}(t^{r+n−j}) dt| < η

is valid for all t_{1}, t_{2} ∈ J. So inequalities (2.41) and (2.42) imply that (2.40) is true for all t_{1}, t_{2} ∈ J, x ∈ B(r, ε), 0 ≤ i ≤ n − 1. □

3. Auxiliary Regular BVPs

The aim of this section is to prove the existence of solutions to the auxiliary regular BVPs corresponding to singular problems (1.1), (1.2) and (1.3), (1.4). Constructions of regular BVPs are based on a priori estimates obtained in Section 2. The existence of their solutions will be proved by means of the Nonlinear Fredholm Alternative.

3.1. Problem (1.1), (1.2). For any positive constant S and each m ∈ ℕ, define ϱ_{m,S} ∈ C^{0}(ℝ), τ_{S} ∈ C^{0}(ℝ) and f_{m,S} ∈ Car(J × ℝ^{n}) by

ϱ_{m,S}(u) = \begin{cases}\frac{1}{m} & for |u| < \frac{1}{m}, \\ |u| & for \frac{1}{m} ≤ |u| ≤ S + 1, \\ S + 1 & for |u| > S + 1, \end{cases}

τ_{S}(u) = \begin{cases}u & for |u| ≤ S + 1, \\ (S + 1) sgn u & for |u| > S + 1 \end{cases}

and

f_{m,S}(t, x_{0}, …, x_{n−2}, x_{n−1})

= \begin{cases}f(t, ϱ_{m,S}(x_{0}), …, ϱ_{m,S}(x_{n−2}), τ_{S}(x_{n−1})) \\ for (t, x_{0}, …, x_{n−2}, x_{n−1}) ∈ J ×ℝ^{n−1}×(( − ∞, −\frac{1}{m}] ⋃ [\frac{1}{m}, ∞)) \\ \frac{m}{2} [f_{m,S}(t, x_{0}, …, x_{n−2}, \frac{1}{m})(x_{n−1} + \frac{1}{m})+ \\ +f_{m,S}(t, x_{0}, …, x_{n−2}, −\frac{1}{m}))(\frac{1}{m} − x_{n−1})] \\ for (t, x_{0}, …, x_{n−2}, x_{n−1}) ∈ J × ℝ^{n−1} × ( − \frac{1}{m}, \frac{1}{m}). \end{cases}

Then we have under assumption (H_{1}) that

ε t^{r} ≤ f_{m,S}(t, x_{0}, …, x_{n−1}) for a.e. t ∈ J and each (x_{0}, …, x_{n−1}) ∈ ℝ^{n} (3.1)

and under assumption (H_{2}) or (H_{3}) that

f_{m,S}(t, x_{0}, …, x_{n−1}) ≤ ϕ̂(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x_{i}|) + ∑_{i=0}^{n−1} h_{i}(t)|x_{i}|^{α_{i}}  (3.2)

for a.e. t ∈ J and each (x_{0}, …, x_{n−1}) ∈ ℝ_{0}^{n}, where

ϕ̂(t) = ϕ(t) + ∑_{i=1}^{n−1} q_{i}(t)ω_{i}(1) + ∑_{i=1}^{n−1} h_{i}(t)  (3.3)

since ω_{j}(ϱ_{m,S}(u)) ≤ ω_{j}(1) + ω_{j}(|u|), ω_{n−1}(|τ_{S}(u)|) ≤ ω_{n−1}(1) + ω_{n−1}(|u|), (ϱ_{m,S}(u))^{α_{j}} ≤ 1 + |u|^{α_{i}}, |τ_{S}(u)|^{α_{n−1}} ≤ 1 + |u|^{α_{n−1}} for u ∈ ℝ and 0 ≤ j ≤ n − 2.

3.1.1. Problem (1.1), (1.2) with min{β, δ} = 0. Let assumptions (H_{1}) and (H_{2}) be satisfied and let M̂ be the positive constant given by Lemma 2.2 with ϕ̂ defined by (3.3). Consider the auxiliary family of regular differential equations

−x^{(n)}(t) = f_{m,M̂}(t, x(t), …, x^{(n−1)}(t))  (3.4)

depending on m ∈ ℕ.

Lemma 3.1. Let assumptions (H_{1}) and (H_{2}) be satisfied. Then, for each m ∈ ℕ, BVP (3.4), (1.2) has a solution x_{m} ∈ A(r, ε) and

∥x_{m}^{(i)}∥ ≤ M̂ for 0 ≤ i ≤ n − 1.  (3.5)

Proof. Fix m ∈ ℕ and set

g_{m}(t) = sup {f_{m,M̂}(t, x_{0}, …, x_{n−1}) : (x_{0}, …, x_{n−1}) ∈ ℝ^{n}}

= sup \{f(t, x_{0}, …, x_{n−1}) : \frac{1}{m} ≤ x_{i} ≤ M̂ + 1 for

0 ≤ i ≤ n − 2, \frac{1}{m} ≤ |x_{n−1}| ≤ M̂ + 1\}.

Since f ∈ Car(J × D) we see that g_{m} ∈ L_{1}(J). Using the fact that the problem −x^{(n)}(t) = 0, (1.2) has only the trivial solution, the Nonlinear Fredholm Alternative guarantees the existence of a solution x_{m} of BVP (3.4), (1.2). Besides, (3.1) and (3.2) with S = M̂ give

εt^{r} ≤ −x_{m}^{(n)}(t) ≤ ϕ̂(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x_{m}^{(i)}(t)|) + ∑_{i=0}^{n−1} h_{i}(t)|x_{m}^{(i)}(t)|^{α_{i}}

for a.e. t ∈ J. Therefore x_{m} ∈ A(r, ε) and ∥x_{m}^{(i)}∥ ≤ M̂ for 0 ≤ i ≤ n − 1 by Lemmas 2.1 and 2.2. □

Lemma 3.2. Let assumptions (H_{1}) and (H_{2}) be satisfied and let x_{m} be a solution of BVP (3.4), (1.2), m ∈ ℕ. Then the sequence

{f_{m,M̂}(t, x_{m}(t), …, x_{m}^{(n−1)}(t))} ⊂ L_{1}(J)  (3.6)

is uniformly absolutely continuous on J.

Proof. By Lemmas 2.1 and 3.1 we have (for m ∈ ℕ)

\begin{aligned}x_{m}^{(n−1)}(t) & ≥ \frac{ε}{r + 1} (ξ_{m} − t)^{r+1} for t ∈ [0, ξ_{m}], \\ x_{m}^{(n−1)}(t) & < −\frac{ε}{r + 1}(t − ξ_{m})^{r+1} for t ∈ (ξ_{m}, T], \end{aligned}  (3.7)

where ξ_{m} ∈ (0, T) is the unique zero of x_{m}^{(n−1)},

x_{m}^{(n−2)}(t) ≥ \begin{cases}At & for t ∈ [0, \frac{T}{2} ] \\ A(T − t) & for t ∈ (\frac{T}{2} , T ] \end{cases}  (3.8)

and

x_{m}^{(j)}(t) ≥ \frac{A}{4(n − j − 1)!} t^{n−j−1} for t ∈ J, 0 ≤ j ≤ n − 3,  (3.9)

where A is defined by (2.5). Moreover, by Lemma 3.1,

∥x_{m}^{(j)}∥ ≤ M̂ for m ∈ ℕ, 0 ≤ j ≤ n − 1.  (3.10)

Since

0 ≤ f_{m,M̂}(t, x_{m}(t), …, x_{m}^{(n−1)}(t)) ≤ ϕ̂(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x_{m}^{(i)}(t)|) + ∑_{i=0}^{n−1} h_{i}(t)M̂^{α_{i}}

for a.e. t ∈ J and each m ∈ ℕ by (3.2), we see from the properties of the functions ϕ, q_{i} and h_{i}, 0 ≤ i ≤ n − 1, given in (H_{2}) that to prove the assertion of our lemma it suffices to show that the sequences

{ω_{i}(|x_{m}^{(i)}(t)|)}, 0 ≤ i ≤ n − 1,

are uniformly absolutely continuous on J.

Let 0 ≤ i ≤ n − 3. Then

ω_{i}(|x_{m}^{(i)}(t)|) ≤ ω_{i}(\frac{A}{4(n − i − 1)!} t^{n−i−1}), t ∈ (0, T], m ∈ ℕ,

which follows from (3.9) since ω_{i} is nonincreasing on ℝ_{+}. Moreover, (1.9) implies that the functions ω_{i}(\frac{A}{4(n−i−1)!} t^{n−i−1}) belong to the class L_{1}(J). Hence

{ω_{i}(|x_{m}^{(i)}(t)|)} is uniformly absolutely continuous on J for 0 ≤ i ≤ n − 3. Analogously, (3.8) gives

ω_{n−2}(|x_{m}^{(n−2)}(t)|) ≤ ω_{n−2}(φ(t)), t ∈ (0, T), m ∈ ℕ,

where

φ(t) = \begin{cases}At & for t ∈ [0, \frac{T}{2} ], \\ A(T − t) & for t ∈ (\frac{T}{2} , T ]. \end{cases}

Since ω_{n−2}(φ(t)) ∈ L_{1}(J) which follows from the assumption ∫_{0}^{V} ω_{n−2}(t) dt < ∞

for each V ∈ ℝ_{+} (see Remark 1.4), the sequence {ω_{n−2}(|x_{m}^{(n−2)}(t)|)} is uniformly absolutely continuous on J.

It remains to verify the uniform absolute continuity on J of the sequence {ω_{n−1}(|x_{m}^{(n−1)}(t)|)}. Let {(a_{j}, b_{j})}_{j∈𝕁} be a sequence of a mutually disjoint intervals (a_{j}, b_{j}) ⊂ J. Set

𝕁_{m}^{1} = {j : j ∈ 𝕁, (a_{j}, b_{j}) ⊂ (0, ξ_{m})}, 𝕁_{m}^{2} = {j : j ∈ 𝕁, (a_{j}, b_{j}) ⊂ (ξ_{m}, T)}

for m ∈ ℕ and set κ = \sqrt[r+1]{ε/(r + 1)}. Then for j ∈ 𝕁_{m}^{1} and k ∈ 𝕁_{m}^{2} we have (see (3.7))

∫_{a_{j}}^{b_{j}} ω_{n−1}(|x_{m}^{(n−1)}(t)|) dt≤∫_{a_{j}}^{b_{j}}ω_{n−1}((κ(ξ_{m} − t))^{r+1}) dt= \frac{1}{κ} ∫_{κ(ξ_{m}−b_{j})}^{κ(ξ_{m}−a_{j})}ω_{n−1}(t^{r+1}) dt,

∫_{a_{k}}^{b_{k}} ω_{n−1}(|x_{m}^{(n−1)}(t)|) dt≤∫_{a_{k}}^{b_{k}} ω_{n−1}((κ(t − ξ_{m}))^{r+1}) dt= \frac{1}{κ} ∫_{κ(a_{k}−ξ_{m})}^{κ(b_{k}−ξ_{m})}ω_{n−1}(t^{r+1}) dt.

If {j_{0}} = 𝕁 \ (𝕁_{m}^{1} ∪ 𝕁_{m}^{2}), that is a_{j_{0}} < ξ_{m} < b_{j_{0}}, then

∫_{a_{j_{0}}}^{b_{j_{0}}}ω_{n−1}(|x_{m}^{(n−1)}(t)|) dt

≤ ∫_{a_{j_{0}}}^{ξ_{m}}ω_{n−1}((κ(ξ_{m} − t))^{r+1}) dt + ∫_{ξ_{m}}^{b_{j_{0}}}ω_{n−1}((κ(t − ξ_{m}))^{r+1}) dt

= \frac{1}{κ} [ ∫_{0}^{κ(ξ_{m}−a_{j_{0}})}ω_{n−1}(t^{r+1}) dt + ∫_{0}^{κ(b_{j_{0}}−ξ_{m})}ω_{n−1}(t^{r+1}) dt].

Hence

∑_{j∈𝕁} ∫_{a_{j}}^{b_{j}} ω_{n−1}(|x_{m}^{(n−1)}(t)|) dt

≤ \frac{1}{κ} [ ∑_{j∈𝕁_{m}^{1}} ∫_{κ(ξ_{m}−b_{j})}^{κ(ξ_{m}−a_{j})}ω_{n−1}(t^{r+1}) dt + ∑_{j∈𝕁_{m}^{2}} ∫_{κ(a_{j}−ξ_{m})}^{κ(b_{j}−ξ_{m})}ω_{n−1}(t^{r+1}) dt + E],

where

E = \begin{cases}0 & if 𝕁 = 𝕁_{m}^{1} ∪ 𝕁_{m}^{2} \\ ∫_{0}^{κ(ξ_{m}−a_{j_{0}})}ω_{n−1}(t^{r+1}) dt + ∫_{0}^{κ(b_{j_{0}}−ξ_{m})}ω_{n−1}(t^{r+1}) dt & if \{j_{0}\} = 𝕁 \ (𝕁_{m}^{1} ∪ 𝕁_{m}^{2}) \end{cases} .

Since

∑_{j∈𝕁_{m}^{1}} [κ(ξ_{m} − a_{j}) − κ(ξ_{m} − b_{j})] + ∑_{j∈𝕁_{m}^{2}} [κ(b_{j} − ξ_{m}) − κ(a_{j} − ξ_{m})] + c

= κ ∑_{j∈𝕁}(b_{j} − a_{j}),

where

c = \begin{cases}0 & if 𝕁 = 𝕁_{m}^{1} ∪ 𝕁_{m}^{2} \\ κ(b_{j_{0}} − a_{j_{0}}) & if \{j_{0}\} = 𝕁 \ (𝕁_{m}^{1} ∪ 𝕁_{m}^{2}) \end{cases} ,

we se that

∑_{j∈𝕁} ∫_{a_{j}}^{b_{j}} ω_{n−1}(|x_{m}^{(n−1)}(t)|) dt ≤ ∫_{M_{1}} ω_{n−1}(t^{r+1}) dt + ∫_{M_{2}} ω_{n−1}(t^{r+1}) dt

where μ(M_{i}) ≤ κ ∑_{j∈𝕁}(b_{j} −a_{j}) for i = 1, 2. Hence {ω_{n−1}(|x_{m}^{(n−1)}(t)|)} is uniformly

absolutely continuous on J, which follows from the fact that ∫_{0}^{κT}ω_{n−1}(t^{r+1}) dt< ∞

by (H_{2}) and Remark 1.4. □

3.1.2. Problem (1.1), (1.2) with min{β, δ} > 0. Let assumptions (H_{1}) and (H_{3}) be satisfied and V̂ be the positive constant given in Lemma 2.4 with ϕ̂ defined by (3.3). Consider the family of regular differential equations

−x^{(n)}(t) = f_{m,V̂}(t, x(t), …, x^{(n−1)}(t))  (3.11)

depending on m ∈ ℕ.

Lemma 3.3. Let assumptions (H_{1}) and (H_{3}) be satisfied. Then, for each m ∈ ℕ, BVP (3.11), (1.2) has a solution x_{m} ∈ A(r, ε) and

∥x_{m}^{(i)}∥ ≤ V̂ for 0 ≤ i ≤ n − 1.  (3.12)

Proof. Fix m ∈ ℕ. To prove the existence of a solution x_{m} of BVP (3.11), (1.2) we can argue as in the proof of Lemma 3.1. The fact that x_{m} ∈ A(r, ε) and x_{m} satisfies (3.12) now follows from Lemmas 2.3 and 2.4. □

Lemma 3.4. Let assumptions (H_{1}) and (H_{3}) be satisfied and let x_{m} be a solution of BVP (3.11), (1.2), m ∈ ℕ. Then the sequence

{f_{m,V̂}(t, x_{m}(t), …, x_{m}^{(n−1)}(t))} ⊂ L_{1}(J)

is uniformly absolutely continuous on J.

Proof. By Lemma 2.3 the inequalities (3.7) are satisfied for each m ∈ ℕ, where ξ_{m} is the unique zero of x^{(n−1)} and (for m ∈ ℕ)

x_{m}^{(n−2)}(t) ≥ B for t ∈ J,  (3.13)
x_{m}^{(j)}(t) ≥ \frac{B}{(n − j − 2)!}t^{n−j−2} for t ∈ J, 0 ≤ j ≤ n − 3,  (3.14)

where B is the positive constant defined by (2.21). Moreover, by Lemma 2.4,

∥x_{m}^{(j)}∥ ≤ V̂ for m ∈ ℕ, 0 ≤ j ≤ n − 1.  (3.15)

From the inequalities (for a.e. t ∈ J and each m ∈ ℕ)

0 ≤ f_{m,V̂}(t, x_{m}(t), …, x_{m}^{(n−1)}(t)) ≤ ϕ̂(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x_{m}^{(i)}(t)|) + ∑_{i=0}^{n−1} h_{i}(t)V̂^{α_{i}}

which follow from (3.2) and (3.15), from the properties of the function ϕ, q_{j} and h_{j}, 0 ≤ j ≤ n − 1, given in (H_{3}) and finally from (3.13), we see that to prove our lemma it suffices to verify that the sequences

{ω_{n−1}(|x_{m}^{(n−1)}(t)|)}  (3.16)

and

{ω_{j}(|x_{m}^{(j)}(t)|)}, 0 ≤ j ≤ n − 3,  (3.17)

are uniformly absolutely continuous on J. The uniform absolute continuity on J of sequence (3.16) was proved in the proof of Lemma 3.2 and for sequences (3.17) this fact follows immediately from the inequalities (see (3.14)) ω_{j}(|x_{m}^{(j)}(t)|) ≤ ω_{j}(\frac{B}{(n−j−2)!}t^{n−j−2}) for t ∈ (0, T] and m ∈ ℕ since (1.10) and Remark 1.4 imply that ω_{j}(\frac{B}{(n−j−2)!}t^{n−j−2}) ∈ L_{1}(J) for 0 ≤ j ≤ n − 3. □

3.2. Problem (1.3), (1.4). We present an existence principle for (p, n − p) right focal BVPs which are regular. Assume that (H_{4}) and (H_{5}) are satisfied. Put

ϕ^{∗}(t) = ϕ(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(1) + ∑_{i=0}^{n−1} h_{i}(t) for a.e. t ∈ J.

Then ϕ^{∗} ∈ L_{1}(J) and, by Lemma 2.6, a positive constant r^{∗} satisfying (2.33) can be found. For m ∈ ℕ, 0 ≤ i ≤ n − 1, x ∈ ℝ, put

ρ_{i} = 1 + r^{∗}T^{n−i−1}  (3.18)

and

σ_{i}(\frac{1}{m}, x) = \begin{cases}\frac{1}{m} sgn x & for |x| < \frac{1}{m}, \\ x & for \frac{1}{m} ≤ |x| ≤ ρ_{i}, \\ ρ_{i} sgn x & for ρ_{i} < |x|. \end{cases}

Extend f onto J × ℝ_{0}^{n} as an even function in each its phase variable x_{i}, 0 ≤ i ≤ n − 1, and for a.e. t ∈ J and for all (x_{0}, …, x_{n−1}) ∈ ℝ^{n} define auxiliary functions

f_{m}(t, x_{0}, …, x_{n−1}) = f(t, σ_{0}(\frac{1}{m}, x_{0}), …, σ_{n−1}(\frac{1}{m}, x_{n−1})).  (3.19)

In such a way we get the family of differential equations

(−1)^{n−p}x^{(n)}(t) = f_{m}(t, x(t), …, x^{(n−1)}(t))  (3.20)

depending on the parameter m ∈ ℕ. Now, we will study BVPs (3.20), (1.4).

Lemma 3.5. Let assumptions (H_{4}) and (H_{5}) be satisfied, let B(r, ε) be given by (2.27) and r^{∗} be from Lemma 2.6. Let f_{m}, m ∈ ℕ, be defined by (3.19). Then, for each m ∈ ℕ, BVP (3.20), (1.4) has a solution u_{m} ∈ B(r, ε) such that

∥u_{m}^{n−1}∥ < r^{∗}.  (3.21)

Proof. Fix an arbitrary m ∈ ℕ. (H_{4}) and (3.19) yield f_{m} ∈ Car(J × ℝ^{n}). Now, put

g_{m}(t) = sup \{|f(t, x_{0}, …, x_{n−1})| : \frac{1}{m} ≤ |x_{i}| ≤ ρ_{i}, 0 ≤ i ≤ n − 1\},

where ρ_{i}, 0 ≤ i ≤ n − 1, are given by (3.18). We see that g_{m} ∈ L_{1}(J) and |f_{m}(t, x_{0}, …, x_{n−1})| ≤ g_{m}(t) for a.e. t ∈ J and all (x_{0}, …, x_{n−1}) ∈ ℝ^{n}. Since the problem (−1)^{n−p}x^{(n)}(t) = 0, (1.4) has only a trivial solution, the Nonlinear Fredholm Alternative implies that (3.20), (1.4) has a solution u_{m}.

Further, by virtue of (H_{4}) and (H_{5}), we see that for a.e. t ∈ J and all (x_{0}, …, x_{n−1}) ∈ ℝ^{n} the inequalities

ε(T − t)^{r} ≤ f_{m}(t, x_{0}, …, x_{n−1}),  (3.22)
f_{m}(t, x_{0}, …, x_{n−1}) ≤ ϕ^{∗}(t) + ∑_{i=0}^{n−1} q_{i}(t)ω_{i}(|x_{i}|) + ∑_{i=0}^{n−1} h_{i}(t)|x_{i}|^{α_{i}}  (3.23)

are true. Note that inequality (3.23) follows from the relations

|σ_{i}(\frac{1}{m}, x_{i})|^{α_{i}} ≤ (\frac{1}{m})^{α_{i}} + |x_{i}|^{α_{i}} ≤ 1 + |x_{i}|^{α_{i}},

and

ω_{i}(|σ_{i}(\frac{1}{m}, x_{i})|) ≤ ω_{i}(ρ_{i}) + ω_{i}(|x_{i}|) ≤ ω_{i}(1) + ω_{i}(|x_{i}|)

which are valid for 0 ≤ i ≤ n − 1. In view of (3.22) we have u_{m} ∈ B(r, ε) and therefore using (3.23) and Lemma 2.6, we get (3.21). □

4. Main Results, Examples

4.1. Sturm–Liouville boundary value problems.

Theorem 4.1. Let assumptions (H_{1}) and (H_{2}) be satisfied and let min{β, δ} = 0 in (1.2). Then BVP (1.1), (1.2) has a solution.

Proof. By Lemma 3.1 there is a solution x_{m} of BVP (3.4), (1.2) for each m ∈ ℕ. Consider the sequence {x_{m}}. Then inequalities (3.7)–(3.10) are satisfied by Lemmas 2.1 and 3.1, where ξ_{m} ∈ (0, T) denotes the unique zero of x_{m}^{(n−1)}, A is given by (2.5) and M̂ is a positive constant. Moreover,

{f_{m,M̂}(t, x_{m}(t), …, x_{m}^{(n−1)}(t))}

is uniformly absolutely continuous on J by Lemma 3.2, which implies that {x_{m}^{(n−1)}(t)} is equicontinuous on J. Without loss of generality we can assume that {x_{m}} is convergent in C^{n−1}(J) and {ξ_{m}} is convergent in ℝ. Let lim_{m→∞} x_{m} = x, lim_{m→∞} ξ_{m} = ξ. Then x satisfies the boundary conditions (1.2) and from (3.7)– (3.9) we deduce that

\begin{aligned}x^{(n−1)}(t) & ≥ \frac{ε}{r + 1} (ξ − t)^{r+1} for t ∈ [0, ξ], \\ x^{(n−1)}(t) & ≤ −\frac{ε}{r + 1} (ξ − t)^{r+1} for t ∈ (ξ, T], \end{aligned}  (4.1)

x^{(n−2)}(t) ≥ \begin{cases}At & for t ∈ [0, \frac{T}{2} ], \\ A(T − t) & for t ∈ (\frac{T}{2} , T ] \end{cases}

and

x^{(j)}(t) ≥ \frac{A}{4(n − j − 1)!} t^{n−j−1}, t ∈ J, 0 ≤ j ≤ n − 3.

From the construction of the auxiliary functions f_{m,M̂} ∈ Car(J × ℝ^{n}) it follows that there exists of a set U ⊂ J, μ(U) = 0, such that f_{m,M̂}(t, ·, …, ·) is continuous on ℝ^{n} for t ∈ J \ U and m ∈ ℕ. Hence

lim_{m→∞} f_{m,M̂}(t, x_{m}(t), …, x_{m}^{(n−1)}(t)) = f(t, x(t), …, x^{(n−1)}(t))

for t ∈ J \ (U ∪ {0, ξ, T }). Now the Vitali’s convergence theorem gives

f(t, x(t), …, x^{(n−1)}(t)) ∈ L_{1}(J)

and

lim_{m→∞}∫_{0}^{t}f_{m,M̂}(s, x_{m}(s), …, x_{m}^{(n−1)}(s)) ds =∫_{0}^{t}f(s, x(s), …, x^{(n−1)}(s)) ds, t ∈ J.

Taking the limit as m → ∞ in the equalities

x_{m}^{(n−1)}(t) = x_{m}^{(n−1)}(0) + ∫_{0}^{t} f_{m,M̂}(s, x_{m}(s), …, x_{m}^{(n−1)}(s)) ds, t ∈ J, m ∈ ℕ,

we obtain

x^{(n−1)}(t) = x^{(n−1)}(0) + ∫_{0}^{t} f(s, x(s), …, x^{(n−1)}(s)) ds, t ∈ J.

Consequently, x ∈ AC^{n−1}(J) and x satisfies (1.1) a.e. on J. We have proved that x is a solution of BVP (1.1), (1.2). □

Theorem 4.2. Let assumptions (H_{1}) and (H_{3}) be satisfied and let min{β, δ} > 0 in (1.2). Then BVP (1.1), (1.2) has a solution.

Proof. By Lemma 3.3 there exists a solution x_{m} of BVP (3.11), (1.2) for each m ∈ ℕ. Consider the sequence {x_{m}}. Then inequalities (3.7) and (3.13)–(3.15) are satisfied for each m ∈ ℕ, which follows from Lemmas 2.3 and 2.4, where ξ_{m} ∈ (0, T) is the unique zero of x_{m}^{(n−1)} and B, V̂ are positive constants independent of x_{m}. Moreover, by Lemma 3.4, the sequence {f_{m,V̂}(t, x_{m}, …, x_{m}^{(n−1)}(t))} is uniformly absolutely continuous on J, which yields that {x_{m}^{(n−1)}(t)} is equicontinuous on J. Without loss of generality we can assume that {x_{m}} and {ξ_{m}} are convergent in C^{n−1}(J) and ℝ, respectively. Let lim_{m→∞} x_{m} = x, lim_{m→∞} ξ_{m} = ξ. Then x satisfies the boundary conditions (1.2) and from (3.7), (3.13) and (3.14) we conclude that (4.1) holds and

x^{(n−2)}(t) ≥ B, t ∈ J,

x^{(j)}(t) ≥ \frac{B}{(n − j − 2)!} t^{n−j−2}, t ∈ J, 0 ≤ j ≤ n − 3.

The next part of the proof is the same as that of the proof of Theorem 4.1 and therefore it is omitted. □

4.2. Focal boundary value problems. First, we consider the singular (p, n − p) right focal BVP (1.3), (1.4) with 1 ≤ p ≤ n − 1.

Theorem 4.3. Let assumptions (H_{4}) and (H_{5}) be satisfied. Then there exists a solution of BVP (1.3), (1.4).

Proof. Define for m ∈ ℕ functions f_{m} by (3.19). According to Lemma 3.5 there is a positive number r^{∗} such that for each m ∈ ℕ problem (3.20), (1.4) has a solution u_{m} ∈ B(r, ε) satisfying (3.21). By Lemma 2.5 there exists c > 0 such that for m ∈ ℕ and t ∈ J we have

\begin{aligned}u_{m}^{(i)}(t) & ≥ ct^{r+n−i} for 0 ≤ i ≤ p − 1, \\ (−1)^{i−p}u_{m}^{(i)}(t) & ≥ c(T − t)^{r+n−i} for p ≤ i ≤ n − 1. \end{aligned}  (4.2)

Conditions (1.4) and (3.21) yield

∥u_{m}^{(i)}∥ < r^{∗}T^{n−i−1} < ρ_{i}, 0 ≤ i ≤ n − 1.  (4.3)

Moreover, by virtue of (3.23) we have for 0 ≤ t_{1} ≤ t_{2} ≤ T

|u_{m}^{(n−1)}(t_{2}) − u_{m}^{(n−1)}(t_{1})| ≤ ∫_{t_{1}}^{t_{2}} h(t) dt + ∑_{i=0}^{n−1} ∥q_{i}∥_{∞} ∫_{t_{1}}^{t_{2}} ω_{i}(|u_{m}^{(i)}(t)|) dt, (4.4)

where

h(t) = ϕ^{∗}(t) + ∑_{i=0}^{n−1} ρ_{i}^{α_{i}}h_{i}(t), h ∈ L_{1}(J).

Since u_{m} ∈ B(r, ε), we can use Lemma 2.7 and conclude that the sequence {u_{m}^{(n−1)}} is equicontinuous on J. Estimates (4.3) mean that {u_{m}} is bounded in C^{n−1}(J). Thus, by the Arzelà–Ascoli theorem, we can choose a subsequence, which is denoted by {u_{k}} and which converges in C^{n−1}(J) to a function u ∈ C^{n−1}(J). Clearly, u satisfies (1.4). Letting k → ∞ and using (4.2), we get

u^{(i)}(t) ≥ ct^{r+n−i} for 0 ≤ i ≤ p − 1,
(−1)^{i−p}u^{(i)}(t) ≥ c(T − t)^{r+n−i} for p ≤ i ≤ n − 1.

This yields that

u^{(i)}(t) > 0 on (0, T] for 0 ≤ i ≤ p − 1,
(−1)^{i−p}u^{(i)}(t) > 0 mboxon [0, T) for p ≤ i ≤ n − 1.

(4.5)

Finally, let us show that u ∈ AC^{n−1}(J) and that u fulfils (1.3) a.e. on J. Consider the sequence of equalities

u_{k}^{(n−1)}(t) = u_{k}^{(n−1)}(0) + ∫_{0}^{t} f_{k}(s, u_{k}(s), …, u_{k}^{(n−1)}(s)) ds for t ∈ J.  (4.6)

Denote the set of all t ∈ J such that f(t, ·, …, ·) : X → ℝ is not continuous by U. Then μ(U) = 0 and, by virtue of (4.5),

lim_{k→∞} f_{k}(t, u_{k}(t), …, u_{k}^{(n−1)}(t)) = f(t, u(t), …, u^{(n−1)}(t))

for all t ∈ J \ (U ∪ {0, T }). Further, using (H_{4}), (H_{5}) and (4.2), we see that for all k ∈ ℕ and a.e. t ∈ J, the inequalities

|f_{k}(t, u_{k}(t), …, u_{k}^{(n−1)}(t))| ≤ g(t)

are satisfied, where

g(t) = ϕ^{∗}(t) + ∑_{i=0}^{n−1} ρ_{i}^{α_{i}}h_{i}(t) + ∑_{i=0}^{n−1} ∥q_{i}∥_{∞}ω_{i}^{∗}(t)

and

ω_{i}^{∗}(t) = \begin{cases}ω_{i}(ct^{r+n−i}) & for 0 ≤ i ≤ p − 1, \\ ω_{i}(c(T − t)^{r+n−i}) & for p ≤ i ≤ n − 1. \end{cases}

Since g ∈ L_{1}(J), we can use the Lebesgue dominated convergence theorem by which f(t, u(t), …, u^{(n−1)}(t)) ∈ L_{1}(J) and letting k → ∞ in (4.6) we have that

u^{(n−1)}(t) = u^{(n−1)}(0) + ∫_{0}^{t} f(s, u(s), …, u^{(n−1)}(s)) ds for t ∈ J

is valid, i.e., u ∈ AC^{n−1}(J) and u satisfies (1.3) a.e. on J. □

Theorem 4.3 together with Remark 1.5 yields the existence result for the singular (n − p, p) left focal BVP (1.12), (1.13) with 1 ≤ p ≤ n − 1.

Theorem 4.4. Let assumptions (H_{6}) and (H_{7}) be satisfied. Then there exists a solution of BVP (1.12), (1.13).

For the continuous function f in equations (1.1) and (1.3) we get immediately from Theorems 4.1 and 4.3 and our previous considerations the following corollaries. Similar results could be obtained from Theorems 4.2 and 4.4.

Corollary 4.5. Let f ∈ C^{0}(J ×D) satisfy assumptions (H_{1}) and (H_{2}) and let min{β, δ} = 0 in (1.2). Then there exists a solution x of BVP (1.1), (1.2) such that x ∈ AC^{n−1}(J) ∩ C^{n}(J \ {0, T, ξ}) and (1.1) holds for each t ∈ J \ {0, T, ξ} where ξ ∈ (0, T) is the unique zero of x^{(n−1)} in J.

Corollary 4.6. Let f ∈ C^{0}(J ×X) satisfy assumptions (H_{4}) and (H_{5}). Then BVP (1.3), (1.4) has a solution x such that x ∈ AC^{n−1}(J) ∩ C^{n}(J \ {0, T }) and (1.3) holds for each t ∈ J \ {0, T }.

Example 4.7. Let n ∈ ℕ, n ≥ 2, γ, α_{i} ∈ (0, 1), h_{i} ∈ L_{1}(J) be nonnegative and β_{i}, c_{i} ∈ ℝ_{+}, 0 ≤ i ≤ n − 1. Consider the differential equation

−x^{(n)}(t) = (\frac{t}{T − t})^{γ} + ∑_{i=0}^{n−1} \frac{c_{i}}{|x^{(i)}(t)|^{β_{i}}} + ∑_{i=0}^{n−1} h_{i}(t)|x^{(i)}(t)|^{α_{i}}.  (4.7)

If we set r = γ, ε = T^{−γ} and ω_{i}(z) = z^{−β_{i}}, 0 ≤ i ≤ n − 1, then assumptions (H_{1}) and (H_{2}) are satisfied for

β_{n−1} ∈ (0, \frac{1}{1 + γ} ), β_{i} ∈ (0, \frac{1}{n − i − 1}), 0 ≤ i ≤ n − 2,  (4.8)

and assumptions (H_{1}) and (H_{3}) are satisfied for

β_{n−1} ∈ (0, \frac{1}{1 + γ} ), β_{n−2} ∈ ℝ_{+}, β_{j} ∈ (0, \frac{1}{n − j − 2}), 0 ≤ j ≤ n − 3. (4.9)

Hence problem (4.7), (1.2) with min{β, δ} = 0 has a solution for β_{i} satisfying (4.8) by Theorem 4.1 and the solvability of problem (4.7), (1.2) with min{β, δ} > 0 for β_{i} satisfying (4.9) is guaranteed by Theorem 4.2.

Example 4.8. Let n ∈ ℕ, n ≥ 2 and 1 ≤ p ≤ n−1. Consider the differential equation

(−1)^{n−p}x^{(n)}(t) = (\frac{T − t}{t})^{γ} + ∑_{i=0}^{n−1} \frac{c_{i}}{|x^{(i)}(t)|^{β_{i}}} + ∑_{i=0}^{n−1} h_{i}(t)|x^{(i)}(t)|^{α_{i}}, (4.10)

where (for 0 ≤ i ≤ n − 1)

γ, α_{i} ∈ (0, 1), c_{i} ∈ ℝ_{+}, β_{i} ∈ (0, \frac{1}{n + γ − i}),

h_{i} ∈ L_{1}(J) is nonnegative.

(4.11)

Then we can see that if we put r = γ, ε = T^{−γ} and ω_{i}(z) = z^{−β_{i}}, 0 ≤ i ≤ n − 1, assumptions (H_{4}) and (H_{5}) are satisfied. Hence, by Theorem 4.3, problem (4.10), (1.4) has a solution.

Similarly, having the differential equation

(−1)^{p}x^{(n)}(s) = (\frac{s}{T − s})^{γ} + ∑_{i=0}^{n−1} \frac{c_{i}}{|x^{(i)}(s)|^{β_{i}}} + ∑_{i=0}^{n−1} h_{i}(s)|x^{(i)}(s)|^{α_{i}}  (4.12)

and assuming (4.11), we can easily check that (H_{6}) and (H_{7}) are fulfilled, which yields, by Theorem 4.4, that problem (4.12), (1.4) is solvable.

Acknowledgement

This work was supported by Grant No. 201/01/1451 of the Grant Agency of the Czech Republic and by the Council of Czech Government J14/98 153100011.

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(Received 15.09.2002)

Authors’ address:

Department of Mathematical Analysis Faculty of Science Palacký University, Tomkova 40, 779 00 Olomouc Czech Republic E-mail: rachunko@risc.upol.cz

stanek@risc.upol.cz