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

The a priori boundedness principle is proved for the two-point right-focal boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the two-point right-focal problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear differential equations with argument deviations under the two-point right-focal boundary conditions.

Statement of the problem and the literature survey

Consider the functional differential equation

with the two-point boundary conditions

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

\(L([a, b])= L_{0}(]a, b])\), \(L^{2}([a, b])= L^{2}_{0}(]a, b])\);

\(M(]a, b])\) 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

\(L_{n}(]a, b])\) is the Banach space of \(y \in L_{\mathrm{loc}}(]a, b])\) functions with the norm

\(\widetilde{C}^{n-1}_{\mathrm{loc}}(]a, b])\) 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

\(C^{m-1}_{1}(]a, b])\) is the Banach space of functions \(y\in C^{m-1}_{\mathrm{loc}}(]a, b])\) such that

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

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

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

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

and

Let also \(k=2k_{1}+1\) (\(k_{1}\in Z\)), then

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

the conditions

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

with

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

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

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

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:

1. (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 \(\delta\in D_{n}(]a, b]\times R^{+})\);

2. (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

continuously acting from \(C^{m-1}_{1}(]a, b])\) to \(L_{\widetilde {L}^{2}_{2n-2m -2}}(]a, b])\), and

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

Let, moreover, for any \(\lambda\in\,]0, 1[\), an arbitrary solution \(x\in A_{\gamma_{0}}\) of the equation

under conditions (1.2) admit the estimate

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

for \(x\in A_{\gamma_{0}}\) and almost all \(t\in\,]a, b]\), and

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

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

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

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

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

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 \(x\in A_{\gamma_{0}}\),

and let there exist constants \(\kappa>0\), \(\varepsilon>0\) such that

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

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

are fulfilled. Let, moreover,

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

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

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

and

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

hold. Then problem (1.35), (1.2) is solvable.

Lemmas on some properties of the equation \(x^{(n)}(t)=\lambda(t)\)

First, we introduce two lemmas without proofs. The first lemma is proved in [29].

Lemma 2.1

Let \(i\in\{1,2\}\), \(x\in\widetilde{C}^{m-1}_{\mathrm{loc}}(]t_{0}, t_{1}[)\) and

Then

for \(t_{0} \leq t \leq t_{1}\).

This second lemma is a particular case of Lemma 4.1 in [35].

Lemma 2.2

If \(x \in C^{n-1}_{\mathrm{loc}}(]a, a_{1}])\), then for any \(s, t \in\,]a, a_{1}]\) the equality

is valid, where \(\nu_{2m}=1\), \(\nu_{2m+1}=\frac{2m+1}{2}\), \(w_{2m}(x)(t)=\sum_{j=1}^{m} (-1)^{m+j-1} x^{(2m-j)}(t)x(t)\),

Lemma 2.3

Let the numbers \(a_{1}\in\,]a, b[\), \(t_{0,k}\in\,]a, a_{1}[\), and \(\varepsilon_{i,k}, \varepsilon_{i}, \beta_{k}, \beta\in R^{+}\), \(k\in N\), \(i=m+1,\ldots,n\), be such that

Let, moreover,

be a nonnegative function, \(x_{k}\in\widetilde{C}^{n-1, m}(]a, a_{1}])\) be a solution of the problem

and \(x\in\widetilde{C}^{n-1, m}(]a, a_{1}])\) be a solution of the problem

Then

Proof

First, let us prove our lemma under the assumption that there exists the number \(r_{1}>0\) such that the estimates

hold. Now, suppose that \(t_{1}, \dots, t_{n}\) are such numbers that \(t_{0 k}< t_{1} < \cdots< t_{n}< a_{1} \) (\(k\in N\)), and \(g_{i}\) are the polynomials of \((n-1)\)th degree satisfying the conditions \(g_{j}(t_{j})=1\), \(g_{j}(t_{i})=0 \) (\(i\neq j\); \(i,j=1, \ldots, n\)). Then if \(x_{k}\) is a solution of problem (2.5), (2.6), and x is a solution of problem (2.7), (2.8), for the solution \(x - x_{k}\) of the equation \(\frac{d^{n}(x(t)-x_{k}(t))}{d t^{n}}=(\beta-\beta _{k})\lambda(t)\), the representation

is valid. On the other hand, in view of inequality (2.10), the identities

by Schwarz’s inequality yield

where \(r_{2}=\frac{r_{1}}{(m-i)!\sqrt{2m-2i+1}}\). By virtue of the Arzela-Ascoli lemma and (2.12), the sequence \(\{ x_{k}\}^{+\infty}_{k=1}\) contains a subsequence \(\{x_{k_{l}}\}^{+\infty}_{l=1}\) which is uniformly convergent in \(]a, a_{1}]\). Suppose \(\lim_{l \to+\infty} x_{k_{l}} (t)=x_{0}(t)\). Thus from (2.11) by (2.3) the existence of such \(r_{3}>0\) that

follows, and then, without loss of generality, we can assume that

Then, by virtue of (2.3), (2.11) and (2.13), we have

From the last two relations by (2.10) it is clear that \(x^{(n)}=x^{(n)}_{0}\) and \(x_{0}\in\widetilde{C}^{n-1, m}(]a, a_{1}])\), i.e., the function \(x_{0}\in\widetilde{C}^{n-1, m}(]a, a_{1}])\) is a solution of problem (2.7), (2.8). In view of (2.4) all the conditions of Theorem 1.1 are fulfilled, thus problem (2.7), (2.8) is uniquely solvable in the space \(\widetilde{C}^{n-1, m}(]a, a_{1}])\) and \(x=x_{0}\). Therefore from (2.13) it follows that

Now suppose that relations (2.9) are not fulfilled. Then there exist \(\delta\in\,]0, \frac{a_{1}-a}{2}[\), \(\varepsilon>0\), and the increasing sequence of natural numbers \(\{k_{l}\}^{+\infty}_{l=1}\) such that

By virtue of the Arzela-Ascoli lemma and condition (2.10), the sequence \(\{x^{(j-1)}_{k_{l}}\}^{+\infty}_{l=1} \) (\(j=1, \ldots, m\)), without loss of generality, can be assumed to be uniformly converging in \(]a+\delta, a_{1}]\). Then, in view of what we have shown above, equality (2.14) holds. But this contradicts condition (2.15). Thus (2.9) holds if conditions (2.10) are fulfilled.

Now assume that conditions (2.10) are not fulfilled. Then there exists the subsequence \(\{t_{0,k_{l}}\}_{l=1}^{+\infty}\)of the sequence \(\{t_{0,k}\}_{k=1}^{+\infty}\) such that

Suppose that \(\widetilde{\beta_{l}}= (\int_{t_{0,k}}^{a_{1}} |x_{k_{l}}^{(m)}(s)|^{2}\,ds )^{-1}\) and \(v_{l}(t)=x_{k_{l}}(t)\widetilde{\beta_{l}}\). Thus in view of (2.16) and our notations,

From the first part of our lemma by (2.17) it follows that there exists limit \(\lim_{l\to+\infty}v_{l}(t)\equiv v_{0}(t)\), and \(v_{0}\) is a solution of the corresponding homogeneous problem (2.18), (2.19). Thus \(v_{0}\equiv0\). On the other hand, from (2.17) it is clear that \(\int_{t_{0,k_{l}}}^{a_{1}} |v_{0}^{(m)}(s)|^{2}\,ds=1\), which contradicts with \(v_{0}\equiv0\). Thus our assumption is invalid and (2.10) holds. □

Lemma 2.4

Let \(a < a_{1} < b\), \(\varepsilon_{j}\in R^{+}\) and \(\lambda\in\widetilde{L}^{2}_{2n-2m-2}(]a, a_{1}]) \) be a nonnegative function. Then, for the solution \(x\in\widetilde {C}^{n-1, m}(]a, a_{1}])\) of problem (2.7), (2.8) with \(\beta=1\), the estimate

is valid, where

Proof

Suppose that \(x_{k}\) is a solution of problem (2.5), (2.6) with \(\beta_{k}=1\), \(\varepsilon_{j,k}=\varepsilon_{j}\). Then, in view of Lemma 2.3, relations (2.9) hold. On the other hand, by Lemma 2.2 we get

Now, on the basis of Lemma 2.1, Schwarz’s and Young’s inequalities, we get

Thus from (2.22) by the definition of numbers \(\nu_{n}\) it immediately follows that the estimate

By means of (2.9), from the last inequality, (2.20) and (2.21) follow. □

Lemmas on the Banach space \(\widetilde{C}^{m-1}_{1}(]a, b])\)

Definition 2.1

Let \(\rho\in R^{+}\) and the function \(\eta\in L_{\mathrm{loc}}(]a, b])\) be nonnegative. Then \(S(\rho, \eta)\) is a set of such \(y \in C^{n-1}_{\mathrm{loc}}(]a, b]) \) that

and

Lemma 2.5

Let, for the function \(y\in\widetilde{C}^{n-1, m}(]a, b])\), conditions (2.25) be satisfied. Then \(y\in\widetilde {C}^{m-1}_{1}(]a, b])\) and the estimates

\(a < t \leq b\), \(i=1,\dots,m\).

Proof

First note that in view of inclusion \(y\in\widetilde{C}^{n-1, m}(]a, b])\), the equality

on \([a , b]\), for \(i=1,\ldots,l\), \(l = 1,\ldots, n\), holds, where: (1) \(c\in[a, b]\) if \(l \leq m\); (2) \(c\in \,]a, b]\) if \(l > m\); and there exists \(r>0\) such that

Equality (2.27), with \(l=m\), \(c=a\), by conditions (2.25), (2.28) and Schwarz’s inequality yields (2.26). From (2.26) and (2.28) it is clear that \(y\in\widetilde{C}^{m}_{1}(]a, b])\). □

Lemma 2.6

Let \(\rho\in R^{+}\), and let \(\eta\in\widetilde {L}^{2}_{2n-2m -2}(]a, b])\) be a nonnegative function. Then \(S(\rho,\eta)\) is a compact subset of the space \(\widetilde {C}^{m-1}_{1}(]a, b])\).

Proof

Condition (2.24) yields the inequality \(|y^{(n)}(t)|\leq \eta (t)\). Thus there exists such function \(\eta_{1} \in\widetilde {L}^{2}_{2n-2m -2}(]a, b])\) that

where

From Theorem 1.1 it follows that problem (2.29), (2.25) has the unique solution \(y\in C^{n-1, m}(]a,b])\), i.e., there exists \(r>0\) such that inequality (2.28) holds.

For any \(y\in S(\rho,\eta)\), from equality (2.27) with \(l=n\), by (2.23), (2.29) and (2.30), we get

where \(\gamma_{i}(t)= \rho_{i}+\frac{1}{(n-i)!} |\int_{c}^{t}(t-s)^{n-i}\eta (s)\,ds | \) (\(i=1,\ldots, n\)), and \(\rho_{i} \in R^{+}\). Let now \(y_{k}\in S(\rho,\eta) \) (\(k\in N\)). By virtue of the Arzela-Ascoli lemma and conditions (2.24), (2.31), the sequence \(\{y_{k}\}_{k=1}^{+\infty}\) contains a subsequence \(\{y_{k_{\ell}}\}_{\ell=1}^{+\infty}\) such that \(\{y^{(i-1)}_{k_{\ell }}\}_{\ell=1}^{+\infty} \) (\(i=1,\ldots,n\)) are uniformly convergent on \(]a, b]\). Thus, without loss of generality, we can assume that \(\{y^{(i-1)}_{k}\}_{k=1}^{+\infty} \) (\(i=1,\ldots,n\)) are uniformly convergent on \(]a, b]\). Let \(\lim_{k\to+\infty}y_{k}(t)=y_{0}(t)\), then \(y_{0} \in\widetilde{C}^{n-1}_{\mathrm{loc}}(]a, b])\) and

From (2.32), in view of the inclusions \(y_{k} \in S(\rho,\eta)\), it immediately follows that

and

From (2.33)-(2.35) it is clear that \(y_{0}\in S(\rho,\eta)\). To finish the proof, we must show that

and

Let \(x_{k}=y_{0}-y_{k}\) and \(a_{1}\in\,]a, b]\). Then it is clear that \(x_{k}\in S(\rho', \eta')\), where \(\rho'= 2\rho\), \(\eta'=2\eta\). Thus, for any \(x_{k}\), there exists \(\eta_{k}\in\widetilde{L}^{2}_{2n-2m -2}(]a, b])\) such that

where

On the other hand, from (2.26) with \(y=x_{k}\), in view of (2.39) we get

for \(i=1,\dots,m\).

Let now \(w_{n}\) be the operator defined in Lemma 2.2 and \(\Theta_{1}\) be a function defined by (2.21) with \(\lambda=\eta_{k}\). Then conditions (2.32) yield

and from the definition of the norm \(\|\cdot\|_{\widetilde{L}_{\alpha }^{2} }\), (2.40) and (2.42) it follows that for any \(\varepsilon> 0\), we can choose \(a_{1}\in\,]a, \min\{a+1, b\}[\) and \(k_{0}\in N\) such that

By using Lemma 2.4 for \(x_{k}\), in view of (2.43) and (2.41), we get respectively

and

Also, in view of (2.32), without loss of generality, we can assume that

and

From (2.44)-(2.47), equality (2.36) immediately follows.

Let now \(y\in S(\rho, \eta)\) and \(y_{k}=\delta_{k} y\), where \(\lim_{k\to+\infty} \delta_{k}=0\). Then by (2.32) it is clear that \(y_{0}\equiv0\), and then from (2.36) it follows that \(y \in \widetilde{C}^{m-1}_{1}(]a, b])\), i.e., the inclusion (2.37) holds. □

Now we introduce one lemma which is proved in [33].

Lemma 2.7

Let \(\tau\in M(]a, b])\), \(\alpha, \beta, k_{1} \geq0\), and let there exist \(\delta\in\,]0, b-a[\) such that

Then

for \(a < t \leq a+\delta\).

Lemmas on the solutions of auxiliary problems

Throughout this section we assume that the 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), and the operator \(q:C^{m-1}_{1}(]a, b])\to\widetilde{L}^{2}_{2n-2m -2}(]a, b])\) is continuous.

Consider, for any \(x\in\widetilde{C}^{m-1}_{1}(]a, b])\subset C^{m-1}_{1}(]a, b])\), the nonhomogeneous equation

and the corresponding homogeneous equation

and let \(E^{n}\) be a set of the solutions of problem (2.49), (2.25).

From inequality (1.17) of item (ii) of Definition 1.1, it follows that for any \(x \in A_{\gamma_{0}}\) boundary problem (2.49), (2.25) has the unique solution y in the space \(\widetilde{C}^{n-1, m}(]a, b])\) such that \(y\in\widetilde {C}^{m-1}_{1}(]a, b])\). Thus \(E^{n}\cap\widetilde{C}^{m-1}_{1}(]a, b])\neq \emptyset\), and there exists the operator \(U:A_{\gamma_{0}} \to E^{n}\cap\widetilde{C}^{m-1}_{1}(]a, b])\) defined by the equality

Lemma 2.8

\(U: A_{\gamma_{0}} \to E^{n}\cap\widetilde{C}^{m-1}_{1}(]a, b])\) is a continuous operator.

Proof

Let \(x_{k}\in A_{\gamma_{0}}\) and \(y_{k}(t)=U(x_{k})(t) \) (\(k=1,2\)), \(y=y_{2}-y_{1}\), and let the operator P be defined by (1.13). Then

where \(q_{0}(x_{1}, x_{2})(t)=P(x_{2}, y_{1})(t)-P(x_{1}, y_{1})(t)+q(x_{2})(t)-q(x_{1})(t)\). Hence, by item (ii) of Definition 1.1 we have

Since the operators P and q are continuous, this estimate implies the continuity of the operator U. □

Proof of Remark 1.2

Let x be a solution of problem (1.6), (1.2), then from inequalities (2.26) (with \(y=x\)), by the definition of the norm in the space \(\widetilde{C}_{1}^{m-1}(]a, b])\) and estimate (1.11), estimate (1.12) immediately follows. □

Proof of Theorem 1.3

Let δ and λ be the functions and numbers appearing in Definition 1.1. We set

From (1.18) it is clear that the nonnegative functions \(\widetilde{F}_{p}\), η admit the inclusion

and for every \(x\in A_{\gamma_{0}} \subset\widetilde{C}^{m-1}_{1}(]a, b])\) and almost all \(t\in\,]a, b]\), the inequality

holds.

Let \(U:A_{\gamma_{0}} \to E^{n}\cap\widetilde{C}^{m-1}_{1}(]a, b])\) be the operator in Lemma 2.8, from which it follows that U is a continuous operator. On the other hand, from items (i) and (ii) of Definition 1.1, (1.19) and (3.5), it is clear that for each \(x\in A_{\gamma_{0}}\), the conditions

hold. Thus, in view of Definition 2.1, the operator U maps the ball \(A_{\gamma_{0}}\) into its own subset \(S(\rho_{1}, \eta)\). From Lemma 2.2 it follows that \(S(\rho_{1}, \eta)\) is a compact subset of the ball \(A_{\gamma_{0}}\subset\widetilde{C}^{m-1}_{1}(]a, b])\), i.e., the operator u maps the ball \(A_{\gamma_{0}}\) into its own compact subset. Therefore, owing to Schauder’s principle, there exists \(x\in S(\rho_{1}, \eta)\subset A_{\gamma_{0}}\) such that

Thus by (2.49) and notation (3.3), the function x (\(x\in A_{\gamma_{0}}\)) is a solution of problem (1.20), (1.2), where

If \(\gamma_{0}=\rho_{0}\), then in view of the condition \(x\in A_{\gamma _{0}}\), by (3.2) we have that \(\lambda=1\), and then, in view of (2.49) and (3.3), the function x is a solution of problem (1.1), (1.2) which admits estimate (1.21).

Let us show now that x admits estimate (1.21) in the case when \(\rho_{0}<\gamma_{0}\). Assume the contrary. Then either

or

If condition (3.7) holds, then by virtue of (3.2) and (3.6) we have that \(\lambda\in\,]0,1[\), which by the conditions of our theorem guarantees the validity of estimate (1.21). But this contradicts (3.7).

Assume now that (3.8) is fulfilled. Then, by virtue of (3.2) and (3.6), we have that \(\lambda=0\). Therefore \(x\in A_{\gamma_{0}}\) is a solution of problem (2.50), (1.2). Thus from item (ii) of Definition 1.1 it is obvious that \(x\equiv 0\), because problem (2.50), (1.2) has only a trivial solution. But this contradicts condition (3.8), i.e., estimate (1.21) is valid. From estimates (1.21) and (3.2) we have that \(\lambda=1\), and then in view of (2.49) and (3.3) the function x is a solution of problem (1.1), (1.2) which admits estimate (1.21). □

Proof of Corollary 1.2

First note that in view of condition (1.24) there exists such \(\gamma_{0} > 2\rho_{0}\) that condition (1.19) holds, and in view of Definition 1.2 the operator P is \(\gamma_{0}\), γ consistent.

On the other hand, from (1.24) it follows the existence of the number \(\rho_{0}\) such that

Let x be a solution of problem (1.20), (1.2) for some \(\lambda\in\,]0, 1[\). Then \(y=x\) is also a solution of problem (1.16), (1.2) where \(q(t)=\lambda (F(x)(t)-P(x, x)(t) )\). Let now \(\rho=\|x\|_{{\widetilde{C}_{1}^{m-1}}}\) and assume that

holds. Then in view of the γ-consistency of the operator p with boundary conditions (1.2), inequality (1.17) holds and thus by condition (1.22) we have

But the last inequality contradicts (3.9). Thus assumption (3.10) is not valid and \(\rho\leq\rho_{0}\). Therefore, for any \(\lambda \in\,]0, 1[\), an arbitrary solution of problem (1.20), (1.2) admits estimate (1.21). Therefore all the conditions of Theorem 1.3 are fulfilled, from which the solvability of problem (1.1), (1.2) follows. □

Proof of Theorem 1.4

Let \(r_{n}\) be the constant defined in Remark 1.2. First prove that the operator P is \(\gamma_{0}\), \(r_{n}\) consistent with boundary conditions (1.2). From the conditions of our theorem it is obvious that item (i) of Definition 1.1 is satisfied. Let now x be an arbitrary fixed function from the set \(A_{\gamma _{0}}\), and let \(p_{j}(t)\equiv p_{j}(x)(t)\). Thus, in view of (1.27) and (1.9), all the assumptions of Theorem 1.1 are satisfied, and then for any \(q\in\widetilde{L}^{2}_{2n-2m -2}(]a, b])\) problem (1.16), (1.2) has the unique solution y. Also in view of Remark 1.2 there exists the constant \(r_{n}>0\) (which depends only on the numbers \(l_{j}\), \(\overline{l}_{j}\), \(\gamma_{j}\) (\(j=1,\ldots, m\)), and a, b, n) such that estimate (1.17) holds with \(\gamma=r_{n}\), i.e., the operator P is \(\gamma_{0}\), \(r_{n}\) consistent with boundary conditions (1.2). Therefore all the assumptions of Corollary 1.1 are fulfilled, from which the solvability of problem (1.1), (1.2) follows. □

Proof of Theorem 1.5

Let \(r_{n}\) be the constant defined in Remark 1.2. First prove that the operator P is \(r_{n}\) consistent with boundary conditions (1.2). From the conditions of our theorem it is obvious that item (i) of Definition 1.1 is satisfied. Let now \(\gamma_{0}\) be an arbitrary nonnegative number, x be an arbitrary fixed function from the space \(A_{\gamma_{0}}\), and let \(p_{j}(t)\equiv p_{j}(x)(t)\). Then in view of (1.29) and (1.9) all the assumptions of Theorem 1.1 are satisfied, and then for any \(q\in\widetilde{L}^{2}_{2n-2m -2}(]a, b])\) problem (1.16), (1.2) has the unique solution y. Also in view of Remark 1.2 there exists the constant \(r_{n}>0\) (which depends only on the numbers \(l_{j}\), \(\overline{l}_{j}\), \(\gamma_{j}\) (\(j=1,\ldots, m\)), and a, b, n) such that estimate (1.17) holds with \(\gamma=r_{n}\), i.e., the operator P is \(\gamma_{0}\), \(r_{n}\) consistent with boundary conditions (1.2) for arbitrary \(\gamma _{0}>0\). Thus by Definition 1.1, the operator P is \(r_{n}\) consistent with boundary conditions (1.2). Therefore all the assumptions of Corollary 1.2 are fulfilled, from which the solvability of problem (1.1), (1.2) follows. □

Proof of Remark 1.3

By Schwarz’s inequality, the definition of the norm \(\|y\|_{\widetilde {C}^{m-1}_{1}}\) and inequalities (2.2) for any \(x, y\in A_{\gamma _{0}}\) and \(z=y-x\), we have

for \(a< t\leq b\). On the other hand, from conditions (1.31) by Lemma 2.7 it is clear that

Then if we put

from (3.11) by the last estimates and (1.30), we get the inequality

for \(a< t \leq b\). Analogously, we get that

for \(a< t\leq b\). From (3.13) and the last inequality it is obvious that the operator P defined by equality (1.13) continuously acts from \(A_{\gamma_{0}}\) to the space \(L_{n}(]a, b])\), and item (ii) of Definition 1.1 holds with \(\delta(t, \rho)=(1+\kappa_{0})\sum_{j=1}^{m}\delta_{j}(t, \rho)\). □

Proof of Corollary 1.3

From conditions (1.33) and (1.31), by Remark 1.3, we obtain that the operator P defined by equality (1.13) with \(p_{j}(x)(t)=p_{j}(t)\) continuously acts from \(A_{\gamma_{0}}\) to the space \(L_{n}(]a, b])\) for any \(\gamma_{0}>0\), i.e., continuously acts from \(\widetilde {C}^{m-1}_{1}(]a, b])\) to the space \(L_{n}(]a, b])\).

Therefore it is clear that all the conditions of Theorem 1.5 would be satisfied with

where the constant \(\kappa_{0}\) is defined by equality (3.12). Thus problem (1.32), (1.2) is solvable. □

Proof of Corollary 1.4

Let the operators \(F, p_{1}:C^{m-1}(]a, b])\to L_{\mathrm{loc}}(]a, b])\), and the function \(\eta:\,]a, b]\times R^{+}\to R^{+}\) be defined by the equalities

and

Then it is easy to verify that, in view of (1.36)-(1.38), conditions (1.9), (1.15), (1.22), (1.27), (1.28) are satisfied and \(\delta\in D_{n}(]a, b]\times R^{+})\).

Thus all the condition of Theorem 1.4 are satisfied, from which the solvability of problem (1.34), (1.2) follows. □

Proof of Corollary 1.5

Let the operators \(F, p_{1}:C^{m-1}(]a, b])\to L_{\mathrm{loc}}(]a, b])\) and the function \(\eta:\,]a, b]\times R^{+}\to R^{+}\) be defined by the equalities

Then it is easy to verify that, in view of (1.24), (1.36) and (1.39), all the conditions of Theorem 1.5 follow, where δ, \(l_{1}\), \(\overline{l}_{1}\), \(r_{2}\), B, \(\gamma_{1} \) are defined by (3.14) with \(\rho=1\), \(\gamma_{0}=1\), from which the solvability of problem (1.34), (1.2) follows. □

The research was supported by Grants FP-S-13-2148 (B Půža, S Mukhigulashvili) and RVO: 67985840 (S Mukhigulashvili).

Affiliations

1. Mathematical Institute, Academy of Sciences of the Czech Republic, Žižkova 22, Brno, 616 62, Czech Republic

   • Sulkhan Mukhigulashvili

2. Faculty of Business and Management, Brno University of Technology, Kolejni 2906/4, Brno, 612 00, Czech Republic

   • Sulkhan Mukhigulashvili
   •  & Bedřich Půža

Corresponding author

Correspondence to Sulkhan Mukhigulashvili.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of this work equally and read and approved the final manuscript.

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