Statement of the problem and the literature survey
Consider the functional differential equation
$$ u^{(n)}(t)=F(u) (t) $$
(1.1)
with the two-point boundary conditions
$$ u^{(i-1)}(a)=0 \quad (i=1, \ldots, m), \qquad u^{(j-1)}(b)=0\quad (j=m+1, \ldots, n). $$
(1.2)
Here \(n\geq2\), m is the integer part of \(n/2\), \(-\infty< a < b < +\infty\), and the operator F acts from the set of \((m-1)\)th time continuously differentiable on \(]a, b]\) functions to the set \(L_{\mathrm{loc}}(]a, b])\). By \(u^{(i-1)}(a)\) we denote the right limit of the function \(u^{(i-1)}\) at the point a.
The problem is singular in the sense that for an arbitrary \(u\in C^{m-1}(]a, b])\) the right-hand side of equation (1.1) may have nonintegrable singularities at the point a. Throughout the paper we use the following notations:
\(R^{+}=[0,+\infty[\); \([x]_{+}\) the positive part of number x, that is, \([x]_{+}=\frac{x+|x|}{2}\);
\(L_{\mathrm{loc}}(]a, b])\) is the space of functions \(y:\,]a, b] \to R\), which are integrable on \([a+\varepsilon, b]\) for arbitrarily small \(\varepsilon>0\);
\(L_{\alpha}(]a, b]) \) (\(L_{\alpha}^{2}(]a, b])\)) is the space of integrable (square integrable) with the weight \((t-a)^{\alpha}\) functions \(y: \,]a, b] \to R\) with the norm
$$\|y\|_{L_{\alpha}}=\int_{a}^{b}(s-a)^{\alpha} \bigl|y(s)\bigr|\,ds \qquad\biggl( \|y\|_{L^{2}_{\alpha}}= \biggl( \int_{a}^{b}(s-a)^{\alpha} y^{2}(s)\,ds \biggr)^{1/2} \biggr); $$
\(L([a, b])= L_{0}(]a, b])\), \(L^{2}([a, b])= L^{2}_{0}(]a, b])\);
\(M(]a, b])\) is the set of measurable functions \(\tau: \,]a, b]\to]a, b]\);
\(\widetilde{L}^{2}_{\alpha}(]a, b])\) is the Banach space of \(y\in L_{\mathrm{loc}}(]a, b])\) functions with the norm
$$\|y\|_{\widetilde{L}^{2}_{\alpha}}\equiv\max \biggl\{ \biggl[\int_{a}^{t}(s-a)^{\alpha} \biggl(\int_{s}^{t}y(\xi)\,d\xi \biggr)^{2}\,ds {\biggr]}^{1/2}: a \leq t\leq b \biggr\} ; $$
\(L_{n}(]a, b])\) is the Banach space of \(y \in L_{\mathrm{loc}}(]a, b])\) functions with the norm
$$\|y\|_{L_{n}}=\sup \biggl\{ (s-a)^{m-1/2}\int_{s}^{t}( \xi-a)^{n-2m}\bigl|y(\xi )\bigr|\,d\xi: a< s\leq t \leq b \biggr\} < +\infty; $$
\(\widetilde{C}^{n-1}_{\mathrm{loc}}(]a, b])\) is the space of functions \(y:\,]a, b]\to R\), which are continuous (absolutely continuous) together with \(y', y'', \ldots, y^{(n-1)}\) on \([a+\varepsilon, b]\) for arbitrarily small \(\varepsilon>0\);
\(\widetilde{C}^{n-1, m}(]a, b])\) is the space of functions \(y\in \widetilde{C}^{n-1}_{\mathrm{loc}}(]a, b])\) such that
$$ \int_{a}^{b}\bigl|x^{(m)}(s)\bigr|^{2}\,ds< +\infty; $$
(1.3)
\(C^{m-1}_{1}(]a, b])\) is the Banach space of functions \(y\in C^{m-1}_{\mathrm{loc}}(]a, b])\) such that
$$ \limsup_{t \to a}\frac{|x^{(i-1)}(t)|}{(t-a)^{m-i+1/2}}< +\infty\quad (i=1,\ldots,m ) $$
(1.4)
with the norm \(\|x\|_{C^{m-1}_{1}}= \sum_{i=1}^{m}\sup \{ \frac{|x^{(i-1)}(t)|}{(t-a)^{m-i+1/2}} : a< t\leq b\}\);
\(\widetilde{C}^{m-1}_{1}(]a, b])\) is the Banach space of functions \(y\in\widetilde{C}^{m-1}_{\mathrm{loc}}(]a, b])\) such that conditions (1.3) and (1.4) hold with the norm \(\|x\|_{\widetilde{C}^{m-1}_{1}}=\|x\|_{C^{m-1}_{1}}+ ( \int_{a}^{b}|x^{(m)}(s)|^{2}\,ds )^{1/2}\);
\(D_{n}(]a, b]\times R^{+})\) is the set of such functions \(\delta:\,]a, b]\times R^{+}\to L_{n}(]a, b])\) that \(\delta(t, \cdot): R^{+} \to R^{+}\) is nondecreasing for every \(t \in\,]a, b]\), and \(\delta(\cdot, \rho)\in L_{n}(]a, b])\) for any \(\rho\in R^{+}\).
A solution of problem (1.1), (1.2) is sought in the space \(\widetilde{C}^{n-1, m}(]a, b])\).
The principles of the theory of singular boundary value problems were built by Kiguradze in his study [1]. This theory has been intensively developed and studied with sufficient completeness both for the ordinary differential equations and the functional differential equations (see [2–28]).
But equation (1.1), even under the boundary condition (1.2), is not studied in the case when the operator F has the form \(F(x)(t)=\sum_{j=1}^{m}p_{j}(t)x^{(j-1)}(\tau_{j}(t))+q(x)(t)\), where the singularities of the functions \(p_{j}:L_{\mathrm{loc}}(]a, b])\) (\(j=2, \ldots,m\)) are such that the inequalities
$$ \int_{a}^{b}(s-a)^{n-1} \bigl[(-1)^{n-m}p_{1}(s)\bigr]_{+}\,ds < +\infty,\qquad \int _{a}^{b}(s-a)^{n-j} \bigl|p_{j}(s)\bigr|\,ds < +\infty $$
(1.5)
are not fulfilled (in this case we say that the linear part of the operator F is strongly singular), the operator q continuously acts from \(C^{m-1}_{1}(]a, b])\) to \(L_{\widetilde {L}^{2}_{2n-2m -2}}(]a, b])\), and the inclusion
$$\sup\bigl\{ q(x) (t): \|x\|_{C^{m-1}_{1}}\leq\rho\bigr\} \in \widetilde{L}^{2}_{2n-2m -2}\bigl(]a, b]\bigr) $$
holds. The first step in studying the differential equations with strong singularities was made by Agarwal and Kiguradze in the article [29], where the linear ordinary differential equations under conditions (1.2), in the case when the functions \(p_{j}\) have strong singularities at the points a and b, are studied. Also the ordinary differential equations with strong singularities under two-point boundary conditions are studied in the articles [30, 31] by Kiguradze. In the papers [32–34] these results are generalized for a linear differential equation with deviating arguments, i.e., the Agarwal-Kiguradze type theorems are proved, which guarantee the Fredholm property for the linear differential equation with deviating arguments. In this paper, on the basis of articles [33, 34], we prove the a priori boundedness principle for problem (1.1), (1.2) from which several sufficient conditions of the solvability of this problem follow.
Now we introduce some results from the articles [33, 34] in this section, which we need for this work. Consider the equation
$$ u^{(n)}(t)=\sum_{j=1}^{m}p_{j}(t)u^{(j-1)} \bigl(\tau_{j}(t)\bigr)+q(t) \quad\mbox{for } a< t<b $$
(1.6)
with \(q, p_{j}\in L_{\mathrm{loc}}(]a, b])\).
By \(h_{j}: \,]a, b]\times\,]a, b] \to R_{+}\) and \(f_{j}: [a, b]\times M(]a, b])\to C_{\mathrm{loc}}(]a, b]\times\,]a, b]) \) (\(j=1, \dots,m\)) we denote the functions and the operator, respectively, defined by the equalities
$$ \begin{aligned} &h_{1}(t, s)= \biggl| \int _{s}^{t}(\xi-a)^{n-2m} \bigl[(-1)^{n-m}p_{1}(\xi )\bigr]_{+}\,d\xi \biggr|, \\ &h_{j}(t, s)= \biggl| \int_{s}^{t}( \xi-a)^{n-2m}p_{j}(\xi)\,d\xi \biggr|, \end{aligned} $$
(1.7)
and
$$ f_{j}(c,\tau_{j}) (t, s)= \biggl|\int _{s}^{t}(\xi-a)^{n-2m}\bigl|p_{j}( \xi)\bigr| \biggl|\int_{\xi}^{\tau_{j}(\xi)}(\xi_{1}-c)^{2(m-j)}\,d\xi_{1} \biggr|^{1/2}\,d\xi \biggr|. $$
(1.8)
Let also \(k=2k_{1}+1\) (\(k_{1}\in Z\)), then
$$k!!= \begin{cases} 1 & \mbox{for } k \leq0, \\ 1 \cdot3 \cdot5 \cdots k & \mbox{for } k \geq1. \end{cases} $$
Now we can introduce the main theorem of the papers [33] and [34].
Theorem 1.1
Let there exist the numbers
\(\ell_{j}>0\), \(\overline{\ell}_{j}\geq0\), and
\(\gamma_{j}>0 \) (\(j=1, \ldots,m\)) such that along with
$$\begin{aligned} B\equiv\sum_{j=1}^{m} \biggl(\frac{(2m-j)2^{2m-j+1} \ell _{j}}{(2m-1)!!(2m-2j+1)!!} + \frac{2^{2m-j-1}(b-a)^{\gamma_{j}} \overline{\ell}_{j}}{(2m-2j-1)!!(2m-3)!!\sqrt{2\gamma_{j}}} \biggr) < 1, \end{aligned}$$
(1.9)
the conditions
$$ (t-a)^{2m-j}h_{j}(t, s)\leq\ell_{j}, \qquad (t-a)^{{m-\gamma _{0j}-1/2}}f_{j}(a, \tau_{j}) (t, s)\leq \overline{\ell}_{j} $$
(1.10)
hold for
\(a< t\leq s\leq b\). Then problem (1.6), (1.2) is uniquely solvable in the space
\(\widetilde{C}^{n-1, m}(]a, b])\).
Remark 1.1
From Lemma 2.5 it is clear that any solution of problem (1.6), (1.2) from the space \(\widetilde{C}^{n-1, m}(]a, b])\) belongs also to the space \(\widetilde{C}_{1}^{m-1}(]a, b])\).
Theorem 1.2
Let all the conditions of Theorem
1.1
be satisfied. Then the unique solution
u
of problem (1.6), (1.2) for every
\(q \in\widetilde{L}^{2}_{2n-2m -2}(]a, b])\)
admits the estimate
$$ \bigl\| u^{(m)}\bigr\| _{L^{2}} \leq r \|q\|_{\widetilde{L}^{2}_{2n-2m -2}}, $$
(1.11)
with
$$r=\frac{2^{m-1}(2n-2m-1)}{(\nu_{n}-B)(2m-1)!!}, \qquad \nu_{2m}=1,\qquad \nu_{2m+1}= \frac{2m+1}{2}, $$
and thus constant
\(r>0\)
depends only on the numbers
\(\ell_{j}\), \(\overline{\ell}_{j}\), \(\gamma_{j}\) (\(j=1,\ldots, m\)), and
a, b, n.
Remark 1.2
Under the conditions of Theorem 1.2, for every \(q \in \widetilde{L}^{2}_{2n-2m -2}(]a, b])\), the unique solution u of problem (1.6), (1.2) admits the estimate
$$ \|u\|_{\widetilde{C}^{m-1}_{1}} \leq r_{n} \|q\|_{\widetilde{L}^{2}_{2n-2m -2}}, $$
(1.12)
with \(r_{n}= (1+\sum_{j=1}^{m}\frac{(2m-2j+1)^{-1/2}}{(m-j)!} )\frac{2^{m-1}(2n-2m-1)}{(\nu_{n}-B)(2m-1)!!}\).
Theorems on the solvability of problem (1.1), (1.2)
Define the operator \(P:C_{1}^{m-1}(]a, b])\times C_{1}^{m-1}(]a, b])\to L_{\mathrm{loc}}(]a, b])\) by the equality
$$ P(x,y) (t)=\sum_{j=1}^{m}p_{j}(x) (t)y^{(j-1)}\bigl(\tau_{j}(t)\bigr) \quad\mbox{for } a< t\leq b, $$
(1.13)
where \(p_{j}:C^{m-1}_{1}(]a, b])\to L_{\mathrm{loc}}(]a, b])\) and \(\tau_{j}\in M(]a, b])\). Also, for any \(\gamma>0\), define the set \(A_{\gamma}\) by the relation
$$ A_{\gamma}=\bigl\{ x\in\widetilde{C}^{m-1}_{1}\bigl(]a, b]\bigr): \|x\|_{\widetilde {C}^{m-1}_{1}}\leq\gamma\bigr\} . $$
(1.14)
Now, following the article [6] by Kiguradze and Půža, we introduce the following definitions.
Definition 1.1
Let \(\gamma_{0}\) and γ be positive numbers. We say that the continuous operator \(P: C^{m-1}_{1}(]a, b])\times C^{m-1}_{1}(]a, b])\to L_{n}(]a, b])\) is \(\gamma_{0}\), γ consistent with boundary condition (1.2) if:
-
(i)
for any \(x\in A_{\gamma_{0}}\) and almost all \(t\in\,]a, b]\), the inequality
$$ \sum_{j=1}^{m}\bigl|p_{j}(x) (t) x^{(j-1)}\bigl(\tau_{j}(t)\bigr)\bigr|\leq\delta\bigl(t, \|x \|_{\widetilde{C}^{m-1}_{1}}\bigr)\|x\|_{\widetilde{C}^{m-1}_{1}} $$
(1.15)
holds, where \(\delta\in D_{n}(]a, b]\times R^{+})\);
-
(ii)
for any \(x\in A_{\gamma_{0}}\) and \(q\in\widetilde{L}^{2}_{2n-2m -2}(]a, b])\), the equation
$$ y^{(n)}(t)=\sum_{j=1}^{m}p_{j}(x) (t)y^{(j-1)}\bigl(\tau_{j}(t)\bigr) +q(t) $$
(1.16)
under boundary conditions (1.2) has the unique solution y in the space \(\widetilde{C}^{n-1, m}(]a, b])\) and
$$ \|y\|_{\widetilde{C}^{m-1}_{1}}\leq\gamma\|q\|_{\widetilde {L}^{2}_{2n-2m -2}}. $$
(1.17)
Definition 1.2
We say that the operator P is γ consistent with boundary condition (1.2) if the operator P is \(\gamma_{0}\), γ consistent with boundary condition (1.2) for any \(\gamma_{0}>0\).
In the sequel it will always be assumed that the operator \(F_{p}\) is defined by the equality
$$F_{p}(x) (t)=\Biggl|F(x) (t)-\sum_{j=1}^{m}p_{j}(x) (t)x^{(j-1)}\bigl(\tau_{j}(t)\bigr) (t)\Biggr|, $$
continuously acting from \(C^{m-1}_{1}(]a, b])\) to \(L_{\widetilde {L}^{2}_{2n-2m -2}}(]a, b])\), and
$$ \widetilde{F}_{p}(t,\rho)\equiv\sup\bigl\{ F_{p}(x) (t): \|x\|_{C^{m-1}_{1}}\leq\rho\bigr\} \in \widetilde{L}^{2}_{2n-2m -2}\bigl(]a, b]\bigr) $$
(1.18)
for each \(\rho\in[0, +\infty[\). Then the following theorem is valid.
Theorem 1.3
Let the operator
P
be
\(\gamma_{0}\), γ
consistent with boundary condition (1.2), and let there exist a positive number
\(\rho _{0}\leq\gamma_{0}\)
such that
$$ \bigl\| \widetilde{F}_{p}\bigl( \cdot, \min\{2 \rho_{0},\gamma_{0}\}\bigr)\bigr\| _{ \widetilde {L}^{2}_{2n-2m -2}}\leq \frac{\gamma_{0}}{\gamma}. $$
(1.19)
Let, moreover, for any
\(\lambda\in\,]0, 1[\), an arbitrary solution
\(x\in A_{\gamma_{0}}\)
of the equation
$$ x^{(n)}(t)=(1-\lambda)P(x, x) (t)+\lambda F(x) (t) $$
(1.20)
under conditions (1.2) admit the estimate
$$ \|x\|_{\widetilde{C}^{m-1}_{1}}\leq\rho_{0}. $$
(1.21)
Then problem (1.1), (1.2) is solvable in the space
\(\widetilde{C}^{n-1, m}(]a, b])\).
From Theorem 1.3 with \(\rho_{0}=\gamma_{0}\), the corollary immediately follows.
Corollary 1.1
Let the operator
P
be
\(\gamma_{0}\), γ
consistent with boundary condition (1.2), and
$$ \Biggl|F(x) (t)-\sum_{j=1}^{m}p_{j}(x) (t)x^{(j-1)}\bigl(\tau_{j}(t)\bigr) (t)\Biggr|\leq\eta \bigl(t, \|x \|_{\widetilde{C}^{m-1}_{1}}\bigr) $$
(1.22)
for
\(x\in A_{\gamma_{0}}\)
and almost all
\(t\in\,]a, b]\), and
$$ \bigl\| \eta(\cdot, \gamma_{0})\bigr\| _{\widetilde{L}^{2}_{2n-2m -2}}\leq \frac{\gamma _{0}}{\gamma}, $$
(1.23)
where
\(\eta\in D_{2n-2m -2}(]a, b]\times R^{+})\). Then problem (1.1), (1.2) is solvable in the space
\(\widetilde{C}^{n-1, m}(]a, b])\).
Corollary 1.2
Let the operator
P
be
γ
consistent with boundary condition (1.2), let inequality (1.22) hold for
\(x\in\widetilde{C}^{m-1}_{1}(]a, b])\)
and almost all
\(t\in\,]a, b]\), where
\(\eta(\cdot, \rho)\in \widetilde{L}^{2}_{2n-2m -2}(]a, b])\)
for any
\(\rho\in R^{+}\), and
$$ \limsup_{\rho\to+\infty}\frac{1}{\rho}\bigl\| \eta(\cdot, \rho) \bigr\| _{\widetilde{L}^{2}_{2n-2m -2}}< \frac{1}{\gamma}. $$
(1.24)
Then problem (1.1), (1.2) is solvable in the space
\(\widetilde{C}^{n-1, m}(]a, b])\).
Now define the operators \(h_{j}:C^{m-1}_{1}(]a, b])\times\,]a, b]\times\,]a, b] \to L_{\mathrm{loc}}(]a, b]\times\,]a, b])\), \(f_{j}: C^{m-1}_{1}(]a, b])\times[a, b]\times M(]a, b])\to C_{\mathrm{loc}}(]a, b]\times\,]a, b])\) (\(j=1, \dots,m\)) by the equalities
$$\begin{aligned}& \begin{aligned} &h_{1}(x, t, s)= \biggl| \int_{s}^{t}( \xi -a)^{n-2m}\bigl[(-1)^{n-m}p_{1}(x) (\xi)\bigr]_{+}d \xi \biggr|, \\ &h_{j}(x, t, s)= \biggl| \int_{s}^{t}( \xi-a)^{n-2m}p_{j}(x) (\xi)\,d\xi \biggr| \quad(j=2, \ldots,m), \end{aligned} \end{aligned}$$
(1.25)
$$\begin{aligned}& f_{j}(x, c,\tau_{j}) (t, s)= \biggl|\int_{s}^{t}( \xi -a)^{n-2m}\bigl|p_{j}(x) (\xi)\bigr| \biggl|\int_{\xi}^{\tau_{j}(\xi)}( \xi_{1}-c)^{2(m-j)}\,d\xi_{1} \biggr|^{1/2}\,d\xi \biggr| \end{aligned}$$
(1.26)
and the functions \(\alpha_{j}:[a, b]\to R_{+}\) by the equality \(\alpha _{j}(t)=(t-a)^{m-j+1/2}\).
Theorem 1.4
Let the continuous operator
\(P: C^{m-1}_{1}(]a, b])\times C^{m-1}_{1}(]a, b])\to L_{n}(]a, b])\)
admit condition (1.15) where
\(\delta\in D_{n}(]a, b]\times R^{+})\), \(\tau_{j}\in M(]a, b])\), and let the numbers
\(\gamma_{0}\in\,]a, b]\), \(l_{j}>0\), \(\overline {l}_{j}>0\), \(\gamma_{j}>0\) (\(j=1,\ldots,m\)) be such that the inequalities
$$ (t-a)^{2m-j}h_{j}(x, t, s)\leq l_{j}, \qquad\limsup_{t\to a}(t-a)^{{m-\frac{1}{2}-\gamma_{j}}}f_{j}(x, a, \tau_{j}) (t, s) \leq \overline{l}_{j} $$
(1.27)
for
\(a< t\leq s\leq b\), \(\|x\|_{\widetilde{C}^{m-1}_{1}}\leq\gamma_{0}\), and conditions (1.9) hold. Let, moreover, the operator
F
and the function
\(\eta\in D_{2n-2m -2}(]a, b]\times R^{+})\)
be such that condition (1.22) and the inequality
$$ \bigl\| \eta(\cdot, \gamma_{0})\bigr\| _{\widetilde{L}^{2}_{2n-2m -2}}< \frac{\gamma_{0}}{r_{n}}, $$
(1.28)
are fulfilled, where
\(r_{n}= (1+\sum_{j=1}^{m}\frac{(2m-2j+1)^{-1/2}}{(m-j)!} )\frac{2^{m-1}(2n-2m-1)}{(\nu_{n}-B)(2m-1)!!}\). Then problem (1.1), (1.2) is solvable in the space
\(\widetilde{C}^{n-1, m}(]a, b])\).
Theorem 1.5
Let the operator
F
and the function
η
be such that conditions (1.22), (1.24) hold, and let the continuous operator
\(P: C^{m-1}_{1}(]a, b])\times C^{m-1}_{1}(]a, b])\to L_{n}(]a, b])\)
admit condition (1.15), where
\(\delta\in D_{n}(]a, b]\times R^{+})\). Let, moreover, the measurable functions
\(\tau_{j}\in M(]a, b])\)
and the numbers
\(l_{ j}>0\), \(\overline{l}_{j}>0\), \(\gamma_{j}>0\) (\(j=1,\ldots,m\)) be such that the inequalities
$$ (t-a)^{2m-j}h_{j}(x, t, s)\leq l_{j}, \qquad \limsup_{t\to a}(t-a)^{{m-\frac{1}{2}-\gamma_{j}}}f_{j}(x, a, \tau_{j}) (t, s) \leq \overline{l}_{j} $$
(1.29)
for
\(a< t\leq s\leq b\), \(x\in\widetilde{C}^{m-1}_{1}(]a, b])\), and conditions (1.9) hold. Then problem (1.1), (1.2) is solvable in the space
\(\widetilde{C}^{n-1, m}(]a, b])\).
Remark 1.3
Let \(\gamma_{0} >0\), let the operators \(\alpha_{j}p_{j}\) (\(j=1,\ldots,m\)) continuously act from the space \(C^{m-1}_{1}(]a, b])\) to the space \(L_{n}(]a, b])\), let there exist the function \(\delta_{j}\in D_{n}(]a, b])\) such that for any \(x\in A_{\gamma_{0}}\),
$$ \bigl|p_{j}(x) (t)\bigr|\alpha_{j}(t)\leq \delta_{j}\bigl(t, \|x\|_{\widetilde {C}^{m-1}_{1}}\bigr) \quad\mbox{for } a< t\leq b, $$
(1.30)
and let there exist constants \(\kappa>0\), \(\varepsilon>0\) such that
$$ \bigl|\tau_{j}(t)-t\bigr|\leq\kappa(t-a) \quad (j=1,\ldots,m) \mbox{ for } a< t<a+\varepsilon. $$
(1.31)
Then the operator P defined by equality (1.13) continuously acts from \(A_{\gamma_{0}}\) to the space \(L_{n}(]a, b])\), and there exists the function \(\delta\in D_{n}(]a, b])\) such that item (i) of Definition 1.1 holds.
Now consider the equation with deviating arguments
$$ u^{(n)} (t)=f\bigl(t, u\bigl(\tau_{1}(t)\bigr), u'\bigl(\tau_{2}(t)\bigr),\ldots, u^{(m-1)}\bigl( \tau _{m}(t)\bigr)\bigr) \quad\mbox{for } a < t \leq b, $$
(1.32)
where \(-\infty< a < b < +\infty\), \(f:\,]a, b]\times R^{m} \to R \) is a function satisfying the local Carathéodory conditions and \(\tau_{j}\in M(]a,b]) \) (\(j=0,\dots, n-1\)) are measurable functions.
Corollary 1.3
Let the functions
\(\tau_{j} \in M(]a, b])\)
and the numbers
\(\kappa\geq 0\), \(\varepsilon>0\), \(l_{j}>0\), \(\overline{l}_{j}>0\), \(\gamma_{j}>0\) (\(j=1,\ldots,m\)) be such that conditions (1.9), (1.10), (1.31) and the inclusions
$$ \alpha_{j}p_{j}\in L_{n}\bigl(]a, b]\bigr) \quad (j=1,\ldots,m) $$
(1.33)
are fulfilled. Let, moreover,
$$\begin{aligned} & \Biggl|f\bigl(t, x\bigl(\tau_{1}(t)\bigr), x'\bigl( \tau_{2}(t)\bigr),\ldots, x^{(m-1)}\bigl(\tau _{m}(t) \bigr)\bigr)-\sum_{j=1}^{m}p_{j}(t)x^{(j-1)} \bigl(\tau_{j}(t)\bigr) (t) \Biggr| \\ &\quad\leq\eta\bigl(t, \|x\|_{\widetilde{C}^{m-1}_{1}}\bigr) \end{aligned}$$
for
\(x\in\widetilde{C}^{m-1}_{1}(]a, b])\)
and almost all
\(t\in\,]a, b]\), where
\(\eta(\cdot, \rho)\in \widetilde{L}^{2}_{2n-2m -2}(]a, b])\)
for any
\(\rho\in R^{+}\), and let condition (1.24) hold. Then problem (1.32), (1.2) is solvable in the space
\(\widetilde{C}^{n-1, m}(]a, b])\).
Remark 1.4
Conditions (1.5) do not follow from conditions (1.33).
Now, to illustrate our results, consider on \(]a, b]\) the second-order functional-differential equations
$$\begin{aligned}& u''(t)=-\frac{\lambda|u(t)|^{k}}{(t-a)^{2+k/2}}u\bigl(\tau(t)\bigr)+q(x) (t), \end{aligned}$$
(1.34)
$$\begin{aligned}& u''(t)=-\frac{\lambda|\sin u^{k}(t)|}{(t-a)^{2}}u\bigl(\tau(t)\bigr)+q(x) (t), \end{aligned}$$
(1.35)
where \(\lambda, k\in R^{+}\) the function \(\tau\in M(]a, b])\), the operator \(q: C^{m-1}_{1}(]a, b])\to\widetilde{L}^{2}_{0}(]a, b])\) is continuous and
$$\eta(t, \rho)\equiv\sup\bigl\{ \bigl|q(x) (t)\bigr|: \|x\|_{\widetilde{C}^{m-1}_{1}}\leq \rho\bigr\} \in\widetilde{L}^{2}_{0}\bigl(]a, b]\bigr). $$
Then, from Theorems 1.4 and 1.5 with \(n=2\), the corollary follows.
Corollary 1.4
Let the function
\(\tau\in M(]a, b])\), the continuous operator
\(q: C^{m-1}_{1}(]a, b])\to\widetilde{L}^{2}_{0}(]a, b])\), and the numbers
\(\gamma_{0}>0\), \(\lambda\geq0\), \(k>0\)
be such that
$$\begin{aligned}& \bigl|\tau(t)-t\bigr|\leq (t-a)^{3/2} \quad \textit{for } a< t \leq b, \end{aligned}$$
(1.36)
$$\begin{aligned}& \bigl\| \eta(t, \gamma_{0})\bigr\| _{\widetilde{L}^{2}_{0}} \leq\frac{1-4\lambda\gamma ^{k}_{0}(1+[4(b-a)]^{1/4})}{2} \gamma_{0}, \end{aligned}$$
(1.37)
and
$$ \lambda<\frac{1}{4\gamma^{k}_{0}(1+[4(b-a)]^{1/4})}. $$
(1.38)
Then problem (1.34), (1.2) is solvable.
Corollary 1.5
Let the function
\(\tau\in M(]a, b])\), the continuous operator
\(q: C^{m-1}_{1}(]a, b[)\to\widetilde{L}^{2}_{0,0}(]a, b])\), and the number
\(\lambda\geq0\)
be such that inequalities (1.24), (1.36) and
$$ \lambda<\frac{1}{4(1+[4(b-a)]^{1/4})}, $$
(1.39)
hold. Then problem (1.35), (1.2) is solvable.