# Solvability of a nonlocal boundary value problem for linear functional differential equations

- Zdeněk Opluštil
^{1}Email author

**2013**:244

https://doi.org/10.1186/1687-1847-2013-244

© Opluštil; licensee Springer. 2013

**Received: **26 April 2013

**Accepted: **29 July 2013

**Published: **14 August 2013

## Abstract

In the paper, the problem on the existence and uniqueness of a solution to the nonlocal problem

is considered, where $\ell :C([a,b];\mathbb{R})\to L([a,b];\mathbb{R})$ and $h:C([a,b];\mathbb{R})\to \mathbb{R}$ are linear bounded operators, $q\in L([a,b];\mathbb{R})$, and $c\in \mathbb{R}$.

**MSC:**34K06, 34K10.

## Keywords

## 1 Introduction and notation

where $\ell :C([a,b];\mathbb{R})\to L([a,b];\mathbb{R})$ and $h:C([a,b];\mathbb{R})\to \mathbb{R}$ are linear bounded operators, $q\in L([a,b];\mathbb{R})$, and $c\in \mathbb{R}$. By a solution to the equation (1), we understand an absolutely continuous function $u:[a,b]\to \mathbb{R}$ satisfying equality (1) almost everywhere on the interval $[a,b]$. A solution to equation (1) satisfying the boundary condition (2) is said to be a solution to problem (1), (2).

*e.g.*, [1–16] and references therein). There is a lot of interesting general results, but only a few efficient conditions are known, namely, in the case where a nonlocal boundary condition is considered. In the present paper, new efficient conditions are found sufficient for the unique solvability of problem (1), (2). An important particular case of the boundary condition (2) is

*e.g.*, in [13, 17–19]. In [20, 21], the first step of our investigation in the general case was done. It is very useful to consider the boundary condition (2) as a nonlocal perturbation of the two-point condition (3). Therefore, we assume throughout the paper that the functional

*h*is defined by the formula

where $\lambda >0$ and ${h}_{0},{h}_{1}\in P{F}_{ab}$. There is no loss of generality in assuming this, because an arbitrary functional *h* can be represented in form (4).

where $p,g\in L([a,b];{\mathbb{R}}_{+})$, $q\in L([a,b];\mathbb{R})$, and $\tau ,\mu :[a,b]\to [a,b]$ are measurable functions. Some sufficient conditions for the validity of the inclusion $\ell \in {\tilde{V}}_{ab}^{-}(h)$, which are part of the conditions for the main results, are given in Section 4.

- 1.
ℝ is the set of all real numbers, ${\mathbb{R}}_{+}=[0,+\mathrm{\infty}[$.

- 2.
$C([a,b];\mathbb{R})$ is the Banach space of continuous functions $v:[a,b]\to \mathbb{R}$ endowed with the norm ${\parallel v\parallel}_{C}=max\{|v(t)|:t\in [a,b]\}$.

- 3.
$\tilde{C}([a,b];D)$, where $D\subseteq \mathbb{R}$, is the set of absolutely continuous functions $v:[a,b]\to D$.

- 4.
$L([a,b];\mathbb{R})$ is the Banach space of Lebesgue integrable functions $p:[a,b]\to \mathbb{R}$ endowed with the norm ${\parallel p\parallel}_{L}={\int}_{a}^{b}|p(s)|\phantom{\rule{0.2em}{0ex}}ds$.

- 5.
$L([a,b];D)=\{p\in L([a,b];\mathbb{R}):p:[a,b]\to D\}$, where $D\subseteq \mathbb{R}$.

- 6.
$C([a,b];D)=\{v\in C([a,b];\mathbb{R}):v:[a,b]\to D\}$, where $D\subseteq \mathbb{R}$.

- 7.
${\mathcal{L}}_{ab}$ is the set of linear bounded operators $\ell :C([a,b];\mathbb{R})\to L([a,b];\mathbb{R})$. ${P}_{ab}$ is the set of operators $\ell \in {\mathcal{L}}_{ab}$, mapping the set $C([a,b];{\mathbb{R}}_{+})$ into the set $L([a,b];{\mathbb{R}}_{+})$.

- 8.
${F}_{ab}$ is the set of linear bounded functionals $h:C([a,b];\mathbb{R})\to \mathbb{R}$. $P{F}_{ab}$ is the set of functionals $h\in {F}_{ab}$ mapping the set $C([a,b];{\mathbb{R}}_{+})$ into the set ${\mathbb{R}}_{+}$.

- 9.
${C}_{h}([a,b];\mathbb{R})=\{v\in C([a,b];\mathbb{R}):v(a)=h(v)\}$, where $h\in {F}_{ab}$.

## 2 Main results

We assume throughout the paper that the following assumptions hold:

(H1) If $h(1)=1$, then the operator *ℓ* is supposed to be ‘nontrivial’ in the sense that the condition $\ell (1)\not\equiv 0$ holds.

(H2) $\tilde{h}\not\equiv 0$, where the functional $\tilde{h}$ is defined by the formula $\tilde{h}(v)=h(v)-v(a)$ for $v\in C([a,b];\mathbb{R})$.

Since we are interested in the unique solvability of problem (1), (2) for every *q* and *c*, both hypotheses (H1) and (H2) are rather natural. Indeed, if $\ell (1)\equiv 0$, then an arbitrary constant function is a solution to problem (1), (2) with $q\equiv 0$ and $c=0$ in the case, where $h(1)=1$. On the other hand, the assumption (H2) guarantees that the boundary condition (2) is not ‘degenerated.’

Before formulation of the main results, we introduce the following definitions.

**Definition 2.1** [22]

is nonpositive on the interval $[a,b]$.

**Definition 2.2** [23]

is nonnegative (resp. nonpositive) on the interval $[a,b]$.

**Remark 2.1** Efficient conditions, guaranteeing the validity of the inclusions $\ell \in {\tilde{V}}_{ab}^{-}(h)$ and $\ell \in {\mathcal{S}}_{ab}(a)$, $\ell \in {\mathcal{S}}_{ab}(b)$, are stated, respectively, in [22] and [23].

### 2.1 Formulation of results

For the sake of transparency, we first formulate all the results; their proofs are postponed till Section 2.2 below.

**Theorem 2.1**

*Assume that there exist operators*${\phi}_{0}\in {\tilde{V}}_{ab}^{-}(h)$

*and*${\phi}_{1}\in {P}_{ab}$

*such that the inequality*

*holds on the set*${C}_{h}([a,b];\mathbb{R})$.

*If*,

*moreover*,

*then problem* (1), (2) *has a unique solution*.

**Corollary 2.1**

*Let*$\ell ={\ell}_{0}-{\ell}_{1}$

*with*${\ell}_{0},{\ell}_{1}\in {P}_{ab}$

*and the relation*$h(1)>1$

*hold*.

*Moreover*,

*there exists*$\epsilon \in [0,1/2]$

*such that*

*Then problem* (1), (2) *has a unique solution*.

**Remark 2.2**Choosing a suitable number

*ε*in Corollary 2.1 and using the results established in [22], we can obtain several efficient conditions, sufficient for the unique solvability of problem (1), (2). However, we do not formulate them in detail. We note only that for $\epsilon =\frac{1}{2}$, the assumption (8) has the form

**Theorem 2.2**

*Let there exist*$\phi \in {\tilde{V}}_{ab}^{-}(\omega )$

*such that the inequality*

*holds on the set*${C}_{h}([a,b];\mathbb{R})$,

*where the functional*

*ω*

*is given by the formula*

*Then problem* (1), (2) *has a unique solution*.

**Theorem 2.3**

*Let*$\ell ={\ell}_{0}-{\ell}_{1}$

*with*${\ell}_{0},{\ell}_{1}\in {P}_{ab}$

*and the relations*

*be fulfilled*.

*Moreover*,

*there exists a function*$\gamma \in \tilde{C}([a,b];]0,+\mathrm{\infty}[)$

*satisfying the conditions*

*where*

*Then problem* (1), (2) *has a unique solution*.

**Theorem 2.4**

*Let*$\ell ={\ell}_{0}-{\ell}_{1}$

*with*${\ell}_{0},{\ell}_{1}\in {P}_{ab}$

*and the relations*

*be fulfilled*.

*Moreover*,

*there exists a function*$\gamma \in \tilde{C}([a,b];]0,+\mathrm{\infty}[)$

*such that condition*(12)

*is satisfied and*

*where*

*Then problem* (1), (2) *has a unique solution*.

**Remark 2.3** The assumption ${h}_{0}(1)\le 1$ appearing in Theorem 2.3 is not supposed in Theorem 2.4. On the other hand, assumption (17) of Theorem 2.4 is stronger than assumption (13) of Theorem 2.3.

**Theorem 2.5**

*Let*$\ell \in {P}_{ab}$,

*the relations*

*hold*,

*and there exists a function*$\gamma \in C([a,b];\mathbb{R})$

*satisfying the conditions*

*Let*,

*moreover*,

*at least one of the following conditions be fulfilled*

- (a)${\int}_{a}^{b}\ell (1)(s)\phantom{\rule{0.2em}{0ex}}ds<{\omega}_{1},$(23)

*where the number*${\omega}_{1}$

*is given by formula*(15);

- (b)$\ell \in {\mathcal{S}}_{ab}(a);$(24)
- (c)$\ell \in {\mathcal{S}}_{ab}(b).$(25)

*Then problem* (1), (2) *has a unique solution*.

**Remark 2.4**If the relation $h(1)\ge 1$ is fulfilled, then the assumption concerning the existence of a function

*γ*in Theorem 2.5 can be omitted. Indeed, since the operator

*ℓ*is supposed to be nontrivial in the case where $h(1)=1$, the function

satisfies conditions (21) and (22).

**Remark 2.5**Define the operator $\phi :C([a,b];\mathbb{R})\to C([a,b];\mathbb{R})$ by setting

*u*is a solution to problem (1), (2), then the function $v\stackrel{\mathrm{def}}{=}\phi (u)$ is a solution to the problem

and vice versa, if *v* is a solution to problem (26), then the function $u\stackrel{\mathrm{def}}{=}\phi (v)$ is a solution to problem (1), (2).

Using this transformation, we can immediately derive other conditions for the unique solvability of problem (1), (2), complementing those stated above. For example, Theorem 2.3 yields.

**Theorem 2.3′**

*Let*$\ell ={\ell}_{0}-{\ell}_{1}$

*with*${\ell}_{0},{\ell}_{1}\in {P}_{ab}$

*and the relations*

*be fulfilled*.

*Let*,

*moreover*,

*there exist a function*$\gamma \in \tilde{C}([a,b];]0,+\mathrm{\infty}[)$

*satisfying the conditions*

*where*

*Then problem* (1), (2) *has a unique solution*.

### 2.2 Proofs

The following lemma is well known from the general theory of boundary value problems for functional differential equations (see, *e.g.*, [15, 24]; in the case, where the operator *ℓ* is strongly bounded, see also [1, 3, 14]).

**Lemma 2.1**

*Problem*(1), (2)

*is uniquely solvable if and only if the corresponding homogeneous problem*

*has only the trivial solution*.

**Remark 2.6** It follows immediately from Definition 2.1 and Lemma 2.1 that under the condition $\ell \in {\tilde{V}}_{ab}^{-}(h)$ problem (1), (2) has a unique solution for every $q\in L([a,b];\mathbb{R})$ and $c\in \mathbb{R}$.

Now, we are in position to prove the main results. According to Lemma 2.1, it is sufficient to show that the homogeneous problem (27), (28) has only the trivial solution.

*Proof of Theorem 2.1*Let

*u*be a solution to problem (27), (28). Then, in view of (6), we get

*α*. It follows from relations (29)-(31) that

which, together with (7) and (32), yields that $\alpha (t)\le 0$ for $t\in [a,b]$. Consequently, condition (35) guarantees $u\equiv 0$, and thus the homogeneous problem (27), (28) has only the trivial solution. □

*Proof of Corollary 2.1* The validity of the corollary follows immediately from Theorem 2.1 with ${\phi}_{0}=\epsilon {\ell}_{0}$ and ${\phi}_{1}=(1-\epsilon ){\ell}_{0}+{\ell}_{1}$. □

*Proof of Theorem 2.2*Let

*u*be a solution to problem (27), (28). Then, in view of (9), we get

*i.e.*,

Consequently, the homogeneous problem (27), (28) has only the trivial solution. □

*Proof of Theorem 2.3*Suppose that problem (27), (28) possesses a nontrivial solution

*u*. According to conditions (11)-(13) and the assumption ${\ell}_{0}\in {P}_{ab}$, Proposition 4.2 guarantees the validity of the inclusion

*u*changes its sign. Put

*i.e.*,

*a*to ${t}_{m}$ and from ${t}_{M}$ to

*b*, in view of (39) and (40), yield

which, together with (11) and (49), contradicts (14).

The contradiction obtained proves that problem (27), (28) has only the trivial solution. □

*Proof of Theorem 2.4*Suppose that problem (27), (28) possesses a nontrivial solution

*u*. According to conditions (12), (16), and (17) and the assumptions ${\ell}_{0}\in {P}_{ab}$ and ${h}_{0}\in P{F}_{ab}$, Proposition 4.2 guarantees the validity of the inclusion

*u*changes its sign. Define the numbers

*M*and

*m*by formulae (38), and choose ${t}_{M},{t}_{m}\in [a,b]$ such that conditions (39) hold. Obviously, (40) is satisfied, and without loss of generality, we can assume that ${t}_{m}<{t}_{M}$. Using conditions (27), (28), (12), and (17), by virtue of (38), (40), (51), and the assumptions ${\ell}_{0}\in {P}_{ab}$ and ${h}_{0}\in P{F}_{ab}$, we get relations (41), (43),

However, we assume that ${\ell}_{1}\in {P}_{ab}$, and thus, it follows from (41) and (43) that inequalities (45) hold.

which, together with (16) and (49), contradicts (18).

The contradiction obtained proves that problem (27), (28) has only the trivial solution. □

*Proof of Theorem 2.5*Let

*u*be a solution to problem (27), (28). We first show that each of assumptions (23), (24), or (25) ensures that

*u*does not change its sign. Indeed, suppose that, on the contrary,

*u*changes its sign. Define the numbers

*M*and

*m*by formulae (38), and choose ${t}_{M},{t}_{m}\in [a,b]$ such that conditions (39) hold. Obviously, (40) is satisfied, and without loss of generality, we can assume that ${t}_{M}<{t}_{m}$.

- (a)Let condition (23) hold. Then the integrations of (27) from
*a*to ${t}_{M}$, from ${t}_{M}$ to ${t}_{m}$, and from ${t}_{m}$ to*b*, in view of (38), (39), and the assumption $\ell \in {P}_{ab}$, result in$M-u(a)={\int}_{a}^{{t}_{M}}\ell (u)(s)\phantom{\rule{0.2em}{0ex}}ds\le M{\int}_{a}^{{t}_{M}}\ell (1)(s)\phantom{\rule{0.2em}{0ex}}ds,$(54)

- (b)
If (24) holds then, in view of Definition 2.2, the assumption $u(a)\ge 0$ (resp. $u(a)<0$) implies $u(t)\ge 0$ (resp. $u(t)\le 0$) for $t\in [a,b]$, which contradicts (40).

- (c)
If (25) holds, then, in view of Definition 2.2, the assumption $u(b)\ge 0$ (resp. $u(b)<0$) implies $u(t)\ge 0$ (resp. $u(t)\le 0$) for $t\in [a,b]$, which contradicts (40).

*u*does not change its sign. We can assume without loss of generality, that the function

*u*is nonnegative. Since $\ell \in {P}_{ab}$, it follows from equation (27) that

The contradiction obtained proves that $u(b)\le 0$, and thus, condition (61) implies $u\equiv 0$. Consequently, the homogeneous problem (27), (28) has only the trivial solution. □

## 3 Differential equations with argument deviations

In this section, we give some corollaries of the main results for the equation with deviating arguments (5). Recall that we suppose that $p,g\in L([a,b];{\mathbb{R}}_{+})$ and $\tau ,\mu :[a,b]\to [a,b]$ are measurable functions. The conditions stated below show that problem (5), (2) is uniquely solvable, provided that either the coefficients *p* and *g* are ‘small’ in a certain sense, or the deviations *τ* and *μ* are ‘close’ to the identities (the functional differential equation (5) is ‘close’ to the ordinary one).

### 3.1 Formulation of results

Theorem 2.1 implies the following.

**Corollary 3.1**

*Let relations*(11)

*be fulfilled*,

*and let the functions*

*p*

*and*

*τ*

*satisfy at least one of the following conditions*:

- (a)${\int}_{a}^{b}p(s)\phantom{\rule{0.2em}{0ex}}ds\le 2(1-{h}_{0}(1))min\{1,\frac{1}{\lambda}\};$
- (b)$0<{h}_{0}(1)<1$, $\tau (t)\ge t$
*for a*.*e*. $t\in [a,b]$,*and*$esssup\{{\int}_{t}^{\tau (t)}p(s)\phantom{\rule{0.2em}{0ex}}ds:t\in [a,b]\}<{\kappa}^{\ast},$(66)

*where*

*Let*,

*moreover*,

*the functions*

*g*

*and*

*μ*

*satisfy at least one of the following conditions*:

- (A)${\int}_{a}^{b}g(s)\phantom{\rule{0.2em}{0ex}}ds<\frac{h(1)-1}{\lambda +{h}_{0}(1)};$(67)
- (B)$g\not\equiv 0$
*and*$esssup\{{\int}_{\mu (t)}^{t}g(s)\phantom{\rule{0.2em}{0ex}}ds:t\in [a,b]\}<{\xi}^{\ast},$(68)

*where*

*Then problem* (5), (2) *has a unique solution*.

From Theorem 2.3, we derive

**Corollary 3.2**

*Let relations*(11)

*be fulfilled and*

*where the number*${\omega}_{1}$

*is given by formula*(15)

*and*

*and*

*Then problem* (5), (2) *has a unique solution*.

Theorem 2.4 yields the following.

**Corollary 3.3**

*Let relations*(16)

*be fulfilled*,

*and*

*where the functions*${\beta}_{0}$, ${\beta}_{1}$, ${\beta}_{2}$,

*and*

*σ*

*are defined by formulae*(73)-(76),

*the number*${\omega}_{2}$

*is given by formula*(19),

*and*

*Then problem* (5), (2) *has a unique solution*.

*i.e.*, the equation

where $p\in L([a,b];{\mathbb{R}}_{+})$, $q\in L([a,b];\mathbb{R})$, and $\tau :[a,b]\to [a,b]$ is a measurable function.

From Theorem 2.1 we can derive the following.

**Corollary 3.4**

*Let relations*(11)

*be fulfilled*,

*and*

*where*

*Then problem* (80), (2) *has a unique solution*.

The next two statements follow from Theorem 2.5.

**Corollary 3.5**

*Let*$p\not\equiv 0$,

*let the relations*

*be fulfilled*,

*and let*

*where*

*Let*,

*moreover*,

*where*

*Then problem* (80), (2) *has a unique solution*.

**Corollary 3.6**

*Let*$p\not\equiv 0$,

*let the relations*

*be fulfilled*, *and let condition* (85) *hold*, *where the number* ${\xi}^{\ast}$ *is defined by formula* (86). *Then problem* (80), (2) *has a unique solution*.

### 3.2 Proofs

*Proof of Corollary 3.1*Let the operators ${\ell}_{0}$ and ${\ell}_{1}$ be defined by the formulae

(see Propositions 4.3 and 4.4).

(see Propositions 4.5 and 4.6).

Consequently, the assumptions of Corollary 2.1 are satisfied with $\epsilon =\frac{1}{2}$. □

*Proof of Corollary 3.2*Let the operators ${\ell}_{0}$ and ${\ell}_{1}$ be defined by formulae (87) and (88), respectively. According to condition (71), there exists $\epsilon >0$ such that

*γ*. It is clear that the function

*γ*satisfies conditions (12) and (13). Using the inclusion $-{\ell}_{1}\in {\tilde{V}}_{ab}^{-}(h)$, we get $\gamma (t)\ge 0$ for $t\in [a,b]$, and thus, equation (90) yields

Therefore, condition (92) yields that $\gamma (t)>0$ for $t\in [a,b]$.

*γ*is a solution to the equation

Now, it is clear that conditions (89), (95), and (96) guarantee the validity of inequality (14).

Consequently, the assumptions of Theorem 2.3 are satisfied. □

*Proof of Corollary 3.3*Let the operators ${\ell}_{0}$ and ${\ell}_{1}$ be defined by formulae (87) and (88), respectively. Condition (78) implies ${A}_{2}<1$. Therefore, according to (77) and (79), Proposition 4.7 guarantees the validity of the inclusion

*γ*satisfying the boundary condition

*γ*satisfies conditions (12) and (17). Using inclusion (97), we get $\gamma (t)\ge 0$ for $t\in [a,b]$, and thus, equation (90) yields the relation (92). Moreover, on account of (16), (92) and the assumption ${h}_{1}\in P{F}_{ab}$, condition (98) implies

Therefore, condition (92) yields that $\gamma (t)>0$ for $t\in [a,b]$.

*γ*is a solution to the equation

Now it is clear that conditions (78), (100), and (101) guarantee the validity of inequality (18).

Consequently, the assumptions of Theorem 2.4 are satisfied. □

*Proof of Corollary 3.4*Let the operator ${\ell}_{0}$ be defined by formula (87), and let ${\ell}_{1}\equiv 0$. It is easy to verify that conditions (81) and (82) yield

(see Propositions 4.3 and 4.6).

Consequently, assumptions of Corollary 2.1 are satisfied with $\epsilon =\frac{1}{3}$. □

*Proof of Corollary 3.5*Let the operator

*ℓ*be defined by the formula

It is clear that $\ell \in {P}_{ab}$. Moreover, condition (85) implies the validity of inclusion (24) (see Proposition 4.8).

Then, by virtue of (105), (106), and the assumptions ${h}_{0},{h}_{1}\in P{F}_{ab}$, it is easy to verify that the function *γ* satisfies conditions (21) and (22).

Consequently, the assumptions of Theorem 2.5 are fulfilled. □

*Proof of Corollary 3.6* Let the operator *ℓ* be defined by formula (102). It is clear that $\ell \in {P}_{ab}$. Moreover, condition (85) implies the validity of inclusion (24) (see Proposition 4.8).

Consequently, by virtue of Remark 2.4, the assumptions of Theorem 2.5 are satisfied. □

## 4 On the set ${\tilde{V}}_{ab}^{-}(h)$

In this section, we give some sufficient conditions guaranteeing the inclusions $\ell \in {\tilde{V}}_{ab}^{-}(h)$, $\ell \in {\mathcal{S}}_{ab}(a)$, and $\ell \in {\mathcal{S}}_{ab}(b)$, which are stated in [22, 23]. We first formulate rather general results.

**Proposition 4.1** [[22], Cor. 4.1]

*Let*$\ell \in {P}_{ab}$

*be a*

*b*-

*Volterra operator*,

*and let the functional*

*h*

*be defined by formula*(4),

*where*$\lambda >0$

*and*${h}_{0},{h}_{1}\in P{F}_{ab}$

*are such that inequalities*(11)

*are fulfilled*.

*If there exists a function*$\gamma \in \tilde{C}([a,b];]0,+\mathrm{\infty}[)$

*satisfying*

*then* $\ell \in {\tilde{V}}_{ab}^{-}(h)$.

**Proposition 4.2** [[22], Thms. 3.2 and 4.3]

*Let*$-\ell \in {P}_{ab}$,

*and let the functional*

*h*

*be defined by formula*(4),

*where*$\lambda >0$

*and*${h}_{0},{h}_{1}\in P{F}_{ab}$

*are such that inequalities*(11)

*are fulfilled*.

*Then*$\ell \in {\tilde{V}}_{ab}^{-}(h)$

*if and only if there exists a function*$\gamma \in \tilde{C}([a,b];]0,+\mathrm{\infty}[)$

*satisfying*

Choosing suitable functions *γ* in the propositions stated above, we can derive several efficient conditions sufficient for the validity of the inclusion $\ell \in {\tilde{V}}_{ab}^{-}(h)$. These conditions are not formulated here in detail; we present, however, some of their corollaries for ‘operators with argument deviations,’ which are used in the proofs of the results stated in Section 3.

**Proposition 4.3** [[22], Cor. 5.3]

*Let*$p\in L([a,b];{\mathbb{R}}_{+})$, $\tau :[a,b]\to [a,b]$

*be a measurable function*,

*and let the functional*

*h*

*be defined by formula*(4),

*where*$\lambda >0$

*and*${h}_{0},{h}_{1}\in P{F}_{ab}$

*are such that inequalities*(11)

*are fulfilled*.

*If*

*then the operator* *ℓ*, *defined by formula* (102), *belongs to the set* ${\tilde{V}}_{ab}^{-}(h)$.

**Proposition 4.4** [[22], Thm. 5.3(c)]

*Let*$p\in L([a,b];{\mathbb{R}}_{+})$, $\tau :[a,b]\to [a,b]$

*be a measurable function*,

*and let the functional*

*h*

*be defined by formula*(4),

*where*$\lambda >0$

*and*${h}_{0},{h}_{1}\in P{F}_{ab}$

*are such that the inequalities*

*are fulfilled*.

*Assume that*$\tau (t)\ge t$

*for a*.

*e*. $t\in [a,b]$,

*and inequality*(66)

*holds*,

*where*

*Then the operator* *ℓ*, *defined by formula* (102), *belongs to the set* ${\tilde{V}}_{ab}^{-}(h)$.

**Proposition 4.5** [[22], Rem. 4.3]

*Let*$g\in L([a,b];{\mathbb{R}}_{+})$, $\mu :[a,b]\to [a,b]$

*be a measurable function*,

*and let the functional*

*h*

*be defined by formula*(4),

*where*$\lambda >0$

*and*${h}_{0},{h}_{1}\in P{F}_{ab}$

*are such that inequalities*(11)

*are fulfilled*.

*If*,

*moreover*,

*inequality*(67)

*is satisfied*,

*then the operator*

*ℓ*,

*defined by the formula*

*belongs to the set* ${\tilde{V}}_{ab}^{-}(h)$.

**Proposition 4.6** [[22], Cor. 5.2]

*Let* $g\in L([a,b];{\mathbb{R}}_{+})$, $\mu :[a,b]\to [a,b]$ *be a measurable function*, *and let the functional* *h* *be defined by formula* (4), *where* $\lambda >0$ *and* ${h}_{0},{h}_{1}\in P{F}_{ab}$ *are such that inequalities* (11) *are fulfilled*. *If*, *moreover*, $g\not\equiv 0$ *and inequality* (68) *is satisfied*, *where the number* ${\xi}^{\ast}$ *is given by formula* (69), *then the operator* *ℓ*, *defined by formula* (107), *belongs to the set* ${\tilde{V}}_{ab}^{-}(h)$.

**Proposition 4.7** [[22], Thm. 5.7]

*Let*$g\in L([a,b];{\mathbb{R}}_{+})$, $\mu :[a,b]\to [a,b]$

*be a measurable function*,

*and let the functional*

*h*

*be defined by formula*(4),

*where*$\lambda >0$

*and*${h}_{0},{h}_{1}\in P{F}_{ab}$

*are such that inequalities*(11)

*are fulfilled*.

*If*,

*moreover*,

*inequalities*(70)

*and*

*are satisfied*, *where the functions* ${\beta}_{0}$ *and* ${\beta}_{1}$ *are defined by formulae* (73), (74), *and* (76), *then the operator* *ℓ*, *defined by formula* (107), *belongs to the set* ${\tilde{V}}_{ab}^{-}(h)$.

The last statement concerns the set ${\mathcal{S}}_{ab}(a)$.

**Proposition 4.8** [[23], Thm. 1.9]

*Let* $p\in L([a,b];{\mathbb{R}}_{+})$, $p\not\equiv 0$, *be such that inequality* (85) *is satisfied*, *where the number* ${\xi}^{\ast}$ *is defined by formula* (86). *Then the operator* *ℓ*, *defined by formula* (102), *belongs to the set* ${\mathcal{S}}_{ab}(a)$.

## Author’s contributions

The author read and approved the final manuscript.

## Declarations

### Acknowledgements

Published results were supported by the project Popularization of BUT R&D results and support systematic collaboration with Czech students CZ.1.07/2.3.00/35.0004 and by Grant No. FSI-S-11-3 ‘Modern methods of mathematical problem modelling in engineering.’

## Authors’ Affiliations

## References

- Azbelev NV, Maksimov VP, Rakhmatullina LF:
*Introduction to the Theory of Functional Differential Equations*. Nauka, Moscow; 1991. (in Russian)MATHGoogle Scholar - Hale J:
*Theory of Functional Differential Equations*. Springer, Berlin; 1977.View ArticleMATHGoogle Scholar - Schwabik Š, Tvrdý M, Vejvoda O:
*Differential and Integral Equations: Boundary Value Problems and Adjoints*. Academia, Prague; 1979.MATHGoogle Scholar - Kolmanovskii V, Myshkis A:
*Introduction to the Theory and Applications of Functional Differential Equations*. Kluwer Academic, Dordrecht; 1999.View ArticleMATHGoogle Scholar - Kiguradze I, Sokhadze Z: On the global solvability of the Cauchy problem for singular functional differential equations.
*Georgian Math. J.*1997, 4(4):355–373. 10.1023/A:1022994513010MathSciNetView ArticleMATHGoogle Scholar - Kiguradze IT, Půža B: Teoremy tipa Konti-Opiala dlja nelinejnych funkcional’nych defferentsial’nych uravnenij.
*Differ. Uravn.*1997, 33(2):185–194. (in Russian)MathSciNetGoogle Scholar - Kiguradze I, Půža B: On boundary value problems for functional differential equations.
*Mem. Differ. Equ. Math. Phys.*1997, 12: 106–113.MathSciNetMATHGoogle Scholar - Dilnaya N, Rontó A: Multistage iterations and solvability of linear Cauchy problems.
*Miskolc Math. Notes*2003, 4(2):89–102.MathSciNetMATHGoogle Scholar - Ronto AN:Exact solvability conditions of the Cauchy problem for systems of linear first-order functional differential equations determined by $({\sigma}_{1},{\sigma}_{2},\dots ,{\sigma}_{n};\tau )$-positive operators.
*Ukr. Math. J.*2003, 55(11):1853–1884.MathSciNetView ArticleGoogle Scholar - Kiguradze I, Sokhadze Z: On the uniqueness of a solution to the Cauchy problem for functional differential equations.
*Differ. Equ.*1995, 31(12):1947–1958.MathSciNetMATHGoogle Scholar - Kiguradze I, Půža B Folia Facult. Scien. Natur. Univ. Masar. Brunensis. In
*Boundary Value Problems for Systems of Linear Functional Differential Equations*. Masaryk University, Brno; 2003.Google Scholar - Hakl R, Lomtatidze A, Půža B: On a boundary value problem for first order scalar functional differential equations.
*Nonlinear Anal.*2003, 53(3–4):391–405. 10.1016/S0362-546X(02)00305-XMathSciNetView ArticleMATHGoogle Scholar - Hakl R, Lomtatidze A, Šremr J Folia Facult. Scien. Natur. Univ. Masar. Brunensis. In
*Some Boundary Value Problems for First Order Scalar Functional Differential Equations*. Masaryk University, Brno; 2002.Google Scholar - Kiguradze I, Půža B: On boundary value problems for systems of linear functional differential equations.
*Czechoslov. Math. J.*1997, 47: 341–373. 10.1023/A:1022829931363View ArticleMathSciNetMATHGoogle Scholar - Hakl R, Lomtatidze A, Stavroulakis IP: On a boundary value problem for scalar linear functional differential equations.
*Abstr. Appl. Anal.*2004, 2004(1):45–67. 10.1155/S1085337504309061MathSciNetView ArticleMATHGoogle Scholar - Hakl R, Kiguradze I, Půža B: Upper and lower solutions of boundary value problems for functional differential equations and theorems on functional differential inequalities.
*Georgian Math. J.*2000, 7(3):489–512.MathSciNetMATHGoogle Scholar - Hakl R, Lomtatidze A, Šremr J: Solvability and unique solvability of a periodic type boundary value problems for first order scalar functional differential equations.
*Georgian Math. J.*2002, 9(3):525–547.MathSciNetMATHGoogle Scholar - Hakl R, Lomtatidze A, Šremr J: Solvability of a periodic type boundary value problem for first order scalar functional differential equations.
*Arch. Math.*2004, 40(1):89–109.MathSciNetMATHGoogle Scholar - Hakl R, Lomtatidze A, Šremr J: On a periodic type boundary value problem for first order linear functional differential equations.
*Neliniini Koliv.*2002, 5(3):416–433.MathSciNetMATHGoogle Scholar - Lomtatidze A, Opluštil Z, Šremr J: On a nonlocal boundary value problem for first order linear functional differential equations.
*Mem. Differ. Equ. Math. Phys.*2007, 41: 69–85.MathSciNetMATHGoogle Scholar - Lomtatidze A, Opluštil Z, Šremr J: Solvability conditions for a nonlocal boundary value problem for linear functional differential equations.
*Fasc. Math.*2009, 41: 81–96.MathSciNetMATHGoogle Scholar - Lomtatidze A, Opluštil Z, Šremr J: Nonpositive solutions to a certain functional differential inequality.
*Nonlinear Oscil.*2009, 12(4):447–591. 10.1007/s11072-010-0087-zMathSciNetView ArticleMATHGoogle Scholar - Hakl R, Lomtatidze A, Půža B: On nonnegative solutions of first order scalar functional differential equations.
*Mem. Differ. Equ. Math. Phys.*2001, 23: 51–84.MathSciNetMATHGoogle Scholar - Bravyi E: A note on the Fredholm property of boundary value problems for linear functional differential equations.
*Mem. Differ. Equ. Math. Phys.*2000, 20: 133–135.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.