**1. Introduction**

In this paper, we study the solvability of boundary value problems (BVPs) for the differential equation

$$\mathbf{x}^{\prime\prime\prime} = f(t, \mathbf{x}, \mathbf{x}^{\prime}, \mathbf{x}^{\prime\prime}), t \in (0, 1), \tag{1}$$

with some of the boundary conditions

$$\mathbf{x}(0) = A, \mathbf{x}'(1) = B, \mathbf{x}''(1) = \mathbb{C},\tag{2}$$

$$\mathbf{x}(0) = A, \mathbf{x}'(0) = B, \mathbf{x}''(1) = \mathbb{C},\tag{3}$$

$$\mathbf{x}(0) = A, \mathbf{x}(1) = B, \mathbf{x}^{\prime\prime}(1) = \mathbb{C},\tag{4}$$

$$\mathbf{x}(0) = A, \mathbf{x}'(0) = B, \mathbf{x}'(1) = \mathbb{C},\tag{5}$$

$$\mathbf{x}(1) = A, \mathbf{x}'(0) = B, \mathbf{x}'(1) = \mathbb{C},\tag{6}$$

where *f* : [0, 1] × *Dx* × *Dp* × *Dq* → R, *Dx*, *Dp*, *Dq* ⊆ R, and *A*, *B*, *C* ∈ R.

The solvability of BVPs for third-order differential equations has been investigated by many authors. Here, we will cite papers devoted to two-point BVPs which are mostly with some of the above boundary conditions; in each of these works *A*, *B*, *C* = 0. Such problems for equations of the form

$$\mathbf{x}^{\prime\prime\prime} = f(t, \mathbf{x}), \mathbf{t} \in (0, 1),$$

have been studied by H. Li et al. [1], S. Li [2] (the problem may be singular at *t* = 0 and/or *t* = 1), Z. Liu et al. [3,4], X. Lin and Z. Zhao [5], S. Smirnov [6], Q. Yao and Y. Feng [7]. Moreover, the boundary conditions in References [2,3] are (3), in Reference [4] they are (4), in References [1,5,7] they are (5), and in Reference [6] are

$$\mathbf{x}(0) = \mathbf{x}(1) = 0, \mathbf{x}'(0) = \mathbf{C}.$$

Y. Feng [8] and Y. Feng and S. Liu [9] have considered the equation

$$\mathbf{x}^{\prime\prime\prime} = f(t, \mathbf{x}, \mathbf{x}^{\prime}), t \in (0, 1),$$

with (6) and (5), respectively. Y. Feng [10] and R. Ma and Y. Lu [11] have studied the equations

$$f(t, \mathbf{x}, \mathbf{x}', \mathbf{x}'^{\prime\prime}) = 0 \text{ and } \mathbf{x}'^{\prime\prime} + M\mathbf{x}'' + f(t, \mathbf{x}) = 0, t \in (0, 1).$$

with (5). BVPs for the equation

$$\mathbf{x}^{\prime\prime\prime} = f(t, \mathbf{x}, \mathbf{x}^{\prime}, \mathbf{x}^{\prime\prime}), t \in (0, 1),$$

have been investigated by A. Granas et al. [12], B. Hopkins and N. Kosmatov [13], Y. Li and Y. Li [14]; the boundary conditions in [12] are (5), these in Reference [13] are (2) and (3), and in Reference [14]—(2).

Results guaranteeing positive or non-negative solutions can be found in References [2–4,7–11,13,14], and results that guarantee negative or non-positive ones in References [7,9,10]. The existence of monotone solutions has been studied in References [3,7,9].

As a rule, the main nonlinearity is defined and continuous on a set such that each dependent variable changes in a left- and/or a right-unbounded set; in Reference [13] it is a Carathéodory function on an unbounded set. Besides, the main nonlinearity is monotone with respect to some of the variables in References [1,5], does not change its sign in References [2–4,14] and satisfies Nagumo type growth conditions in Reference [14]. Maximum principles have been used in References [8,10], Green's functions in References [1,2,4,5], and upper and lower solutions in References [1,7–11].

Here, we use a different tool—barrier strips which allow the right side of the equation to be defined and continuous on a bounded subset of its domain and to change its sign.

To prove our existence results we apply a basic existence theorem whose formulation requires the introduction of the BVP

$$\mathbf{x}^{\prime\prime\prime} + a(t)\mathbf{x}^{\prime\prime} + b(t)\mathbf{x}^{\prime} + c(t)\mathbf{x} = f(t, \mathbf{x}, \mathbf{x}^{\prime}, \mathbf{x}^{\prime\prime}),\\t \in (0, 1), \tag{7}$$

$$V\_i(\mathbf{x}) = r\_i, i = 1, 2, 3(i = \overline{1,3} \text{ for short}), \tag{8}$$

where *a*, *b*, *c* ∈ *C*([0, 1], <sup>R</sup>), *f* : [0, 1] × *Dx* × *Dp* × *Dq* → R,

$$V\_i(\mathbf{x}) = \sum\_{j=0}^{2} [a\_{ij}\mathbf{x}^{(j)}(0) + b\_{ij}\mathbf{x}^{(j)}(1)]\_i i = \overline{1,3}\_{\prime}$$

with constants *aij* and *bij* such that ∑2 *<sup>j</sup>*=<sup>0</sup>(*a*2*ij* + *b*2*ij*) > 0, *i* = 1, 3, and *ri* ∈ R, *i* = 1, 3. Next, consider the family of BVPs for

$$\mathbf{x}^{\prime\prime\prime} + a(t)\mathbf{x}^{\prime\prime} + b(t)\mathbf{x}^{\prime} + c(t)\mathbf{x} = \mathbf{g}(t, \mathbf{x}, \mathbf{x}^{\prime}, \mathbf{x}^{\prime\prime}, \boldsymbol{\lambda}),\\t \in (0, 1), \boldsymbol{\lambda} \in [0, 1] \tag{7} \tag{7} \boldsymbol{\lambda}$$

with boundary conditions (8), where *g* is a scalar function defined [0, 1] × *Dx* × *Dp* × *Dq* × [0, 1], and *a*, *b*, *c* are as above. Finally, *BC* denotes the set of functions satisfying boundary conditions (8), and *BC*0 denotes the set of functions satisfying the homogeneous boundary conditions *Vi*(*x*) = 0, *i* = 1, 3. Besides, let *<sup>C</sup>*3*BC*[0, 1] = *C*<sup>3</sup>[0, 1] ∩ *BC* and *<sup>C</sup>*3*BC*0[0, 1] = *C*<sup>3</sup>[0, 1] ∩ *BC*0.

The proofs of our existence results are based on the following theorem. It is a variant of Reference [12] (Chapter I, Theorem 5.1 and Chapter V, Theorem 1.2). Its proof can be found in Reference [15]; see also the similar result in Reference [16] (Theorem 4).

**Lemma 1.** *Suppose:*

*(i) Problem* (7)0*,* (8) *has a unique solution x*0 ∈ *C*<sup>3</sup>[0, 1].


$$\mathbf{L}\_{\hbar}\mathbf{x} = \mathbf{x}^{\prime\prime\prime} + a(t)\mathbf{x}^{\prime\prime} + b(t)\mathbf{x}^{\prime} + c(t)\mathbf{x}.$$

*(iv) Each solution x* ∈ *C*<sup>3</sup>[0, 1] *to family* (7)*<sup>λ</sup>,* (8) *satisfies the bounds*

$$m\_i \le x^{(i)} \le M\_i \text{ for } t \in [0, 1], i = \overline{0, 3}, \dots$$

*where the constants* −∞ < *mi*, *Mi* < <sup>∞</sup>, *i* = 0, 3, *are independent of λ and x. (v) There is a sufficiently small σ* > 0 *such that*

$$[m\_0 - \sigma, M\_0 + \sigma] \subseteq D\_{\ge \prime}[m\_1 - \sigma, M\_1 + \sigma] \subseteq D\_{\not p\_{\prime}}[m\_2 - \sigma, M\_2 + \sigma] \subseteq D\_{\not q\_{\prime}}$$

*and g*(*<sup>t</sup>*, *x*, *p*, *q*, *λ*) *is continuous for* (*t*, *x*, *p*, *q*, *λ*) ∈ [0, 1] × *J* × [0, 1] *where J* = [*<sup>m</sup>*0 − *σ*, *M*0 + *σ*] × [*<sup>m</sup>*1 − *σ*, *M*1 + *σ*] × [*<sup>m</sup>*2 − *σ*, *M*2 + *σ*]; *mi*, *Mi*, *i* = 0, 3, *are as in (iv).*

*Then boundary value problem* (7)*,* (8) *has at least one solution in C*<sup>3</sup>[0, 1].

For us, the equation from (7)*λ* has the form

$$\mathbf{x}^{\prime\prime\prime} = \lambda f(\mathbf{t}, \mathbf{x}, \mathbf{x}^{\prime}, \mathbf{x}^{\prime\prime}). \tag{1}$$

Preparing the application of Lemma 1, we impose conditions which ensure the a priori bounds from *(iv)* for the eventual *C*<sup>3</sup>[0, 1] - solutions of the families of BVPs for (7)*<sup>λ</sup>*, *λ* ∈ [0, 1], with one of the boundary conditions (*k*), *k* = 2, 6.

So, we will say that for some of the BVPs (1), (*k*), *k* = 2, 6, the conditions **(H1)** and **(H2)** hold for a *K* ∈ R (it will be specified later for each problem) if:

**(H1)** There are constants *Fi* , *Li*, *i* = 1, 2, such that

$$F\_2' < F\_1' \le \mathcal{K} \le L\_1' < L\_{2'}' \left[ F\_{2'}' L\_2' \right] \subseteq D\_{q'}$$

$$f(t, \mathbf{x}, p, q) \ge 0 \text{ for } (t, \mathbf{x}, p, q) \in [0, 1] \times D\_{\mathbf{x}} \times D\_{\mathbf{p}} \times [L\_1', L\_2'], \tag{9}$$

$$f(t, \mathbf{x}, p, q) \le 0 \text{ for } (t, \mathbf{x}, p, q) \in [0, 1] \times D\_{\mathbf{x}} \times D\_{p} \times [F\_{2}^{\prime}, F\_{1}^{\prime}].\tag{10}$$

**(H2)** There are constants *Fi*, *Li*, *i* = 1, 2, such that

$$F\_2 < F\_1 \le K \le L\_1 < L\_2, [F\_2, L\_2] \subseteq D\_{q'}$$

$$f(t, x, p, q) \le 0 \text{ for } (t, x, p, q) \in [0, 1] \times D\_x \times D\_p \times [L\_1, L\_2].$$

$$f(t, x, p, q) \ge 0 \text{ for } (t, x, p, q) \in [0, 1] \times D\_x \times D\_p \times [F\_2, F\_1].$$

Besides, we will say that for some of the BVPs (1), (*k*), *k* = 2, 6, the condition **(H3)** holds for constants *mi* ≤ *Mi*, *i* = 0, 2, (they also will be specified later for each problem) if:

**(H3)** [*<sup>m</sup>*0 − *σ*, *M*0 + *σ*] ⊆ *Dx*, [*<sup>m</sup>*1 − *σ*, *M*1 + *σ*] ⊆ *Dp*, [*<sup>m</sup>*2 − *σ*, *M*2 + *σ*] ⊆ *Dq* and *f*(*<sup>t</sup>*, *x*, *p*, *q*) is continuous on the set [0, 1] × *J*, where *J* is as in (*v*) of Lemma 1, and *σ* > 0 is sufficiently small.

In fact, the present paper supplements P. Kelevedjiev and T. Todorov [15] where only conditions **(H2)** and **(H3)** have been used for studying the solvability of various BVPs for (1) with other boundary conditions. Here, **(H1)** is also needed. Now, only **(H1)** guarantees the a priori bounds for *x*--(*t*), *x*-(*t*) and *<sup>x</sup>*(*t*), in this order, for each eventual solution *x* ∈ *C*<sup>3</sup>[0, 1] to the families (1)*<sup>λ</sup>*,(*k*), *k* = 2, 4, and **(H1)** and **(H2)** together guarantee these bounds for the families (1)*<sup>λ</sup>*, (*k*), *k* = 5, 6. As in Reference [15], **(H3)** gives the bounds for *x*---(*t*).

The auxiliary results which guarantee a priori bounds are given in Section 2, and the existence theorems are in Section 3. The ability to use **(H1)** and **(H2)** for studying the existence of solutions with important properties is shown in Appendix A. Examples are given in Section 4.

#### **2. Auxiliary Results**

This part ensures a priori bounds for the eventual *C*<sup>3</sup>[0, 1]-solutions of each family (1)*<sup>λ</sup>*, (*k*), *k* = 2, 6, that is, it ensures the constants *mi*, *Mi*, *i* = 0, 2, from *(iv)* of Lemma 1 and **(H3)**.

**Lemma 2.** *Let x* ∈ *<sup>C</sup>*<sup>3</sup>[*<sup>a</sup>*, *b*] *be a solution to* (1)*<sup>λ</sup>*. *Suppose* **(H1)** *holds with* [0, 1] *replaced by* [*a*, *b*] *and K* = *x*--(*b*). *Then*

$$F\_1' \le \mathfrak{x}''(t) \le L\_1' \text{ on } [a, b].$$

**Proof.** By contradiction, assume that *x*--(*t*) > *L*-1 for some *t* ∈ [*a*, *b*). This means that the set

$$S\_+ = \{ t \in [a, b] : L\_1' < \mathfrak{x}^{\prime\prime}(\mathfrak{t}) \le L\_2' \}.$$

is not empty because *x*--(*t*) is continuous on [*a*, *b*] and *x*--(*b*) ≤ *L*-1. Besides, there is a *γ* ∈ *S*+ such that

$${x^{\prime\prime\prime}(\gamma) < 0}$$

As *x*(*t*) is a *<sup>C</sup>*<sup>3</sup>[*<sup>a</sup>*, *b*]—solution to (1)*<sup>λ</sup>*,

$$\mathfrak{x}^{\prime\prime}(\gamma) = \lambda f(\gamma, \mathfrak{x}(\gamma), \mathfrak{x}'(\gamma), \mathfrak{x}^{\prime\prime}(\gamma)).$$

But, (*<sup>γ</sup>*, *<sup>x</sup>*(*γ*), *x*-(*γ*), *x*--(*γ*)) ∈ *S*+ × *Dx* × *Dp* × (*L*-1, *L*-2] and (9) imply

$$x''''(\eta) \ge 0,$$

a contradiction. Consequently,

$$\mathbf{x}^{\prime\prime}(t) \le L\_1^{\prime} \text{ for } t \in [a, b].$$

Along similar lines, assuming on the contrary that the set

$$S\_- = \{ t \in [a, b] : F\_2' \le x''(t) < F\_1' \} $$

is not empty and using (10), we achieve a contradiction which implies that

$$F\_1' \le \mathbf{x}''(t) \text{ for } t \in [a, b].$$

The proof of the next assertion is virtually the same as that of Lemma 2 and is omitted; it can be found in [15].

**Lemma 3.** *Let x* ∈ *<sup>C</sup>*<sup>3</sup>[*<sup>a</sup>*, *b*] *be a solution to* (1)*<sup>λ</sup>*. *Suppose* **(H2)** *holds with* [0, 1] *replaced by* [*a*, *b*] *and K* = *x*--(*a*). *Then*

$$F\_1 \le x''(t) \le L\_1 \text{ on } [a, b].$$

Let us recall, conditions of type **(H1)** and **(H2)** are called barrier strips, see P. Kelevedjiev [17]. As can we see from Lemmas 2 and 3 they control the behavior of *x*--(*t*) on [*a*, *b*], depending on the sign of *f*(*<sup>t</sup>*, *x*, *x*-, *x*--) the curve of *x*--(*t*) on [*a*, *b*] crosses the strips [*a*, *b*] × [*L*-1, *L*-2], [*a*, *b*] × [*<sup>L</sup>*1, *<sup>L</sup>*2], [*a*, *b*] × [*F*-2, *F*-1] and [*a*, *b*] × [*<sup>F</sup>*2, *<sup>F</sup>*1] not more than once. This property ensures the a priori bounds for *x*--(*t*).

**Lemma 4.** *Let* **(H1)** *hold for K* = *C. Then every solution x* ∈ *C*<sup>3</sup>[0, 1] *to* (1)*<sup>λ</sup>*, (2) *or* (1)*<sup>λ</sup>*, (3) *satisfies the bounds*

$$|\mathbf{x}(t)| \le |A| + |B| + \max\{|F\_1'| , |L\_1'|\} , t \in [0, 1],$$

$$|\mathbf{x}'(t)| \le |B| + \max\{|F\_1'| , |L\_1'|\} , t \in [0, 1].$$

$$F\_1' \le \mathbf{x}''(t) \le L\_{1'}'t \in [0,1]. \tag{11}$$

**Proof.** Let first *x*(*t*) be a solution to (1)*<sup>λ</sup>*, (2). Using Lemma 2 we conclude that (11) is true. Then, according to the mean value theorem, for each *t* ∈ [0, 1) there is a *ξ* ∈ (*t*, 1) such that

$$\mathbf{x}'(1) - \mathbf{x}'(t) = \mathbf{x}''(\xi)(1 - t),$$

which together with (11) gives the bound for |*x*-(*t*)|. Again from the mean value theorem for each *t* ∈ (0, 1] there is an *η* ∈ (0, *t*) with the property

$$\mathbf{x}(t) - \mathbf{x}(0) = \mathbf{x}'(\eta)t\_{\eta}$$

which yields the bound for |*x*(*t*)|. The assertion follows similarly for (1)*<sup>λ</sup>*, (3).

**Lemma 5.** *Let* **(H1)** *hold for K* = *C. Then every solution x* ∈ *C*<sup>3</sup>[0, 1] *to* (1)*<sup>λ</sup>*, (4) *satisfies the bounds*

$$|\mathbf{x}(t)| \le |A| + |B - A| + \max\{|F\_1'| , |L\_1'|\} , \ t \in [0, 1].$$

$$|\mathbf{x}'(t)| \le |B - A| + \max\{|F\_1'| , |L\_1'|\} , \ t \in [0, 1].$$

$$F\_1' \le \mathbf{x}''(t) \le L\_{1'}' \, t \in [0, 1].$$

**Proof.** By Lemma 2, *F*-1 ≤ *x*--(*t*) ≤ *L*-1 on [0, 1]. Clearly, there is a *μ* ∈ (0, 1) for which *x*-(*μ*) = *B* − *A*. Further, for each *t* ∈ [0, *μ*) there is a *ξ* ∈ (*t*, *μ*) such that

$$\mathbf{x}'(\mu) - \mathbf{x}'(t) = \mathbf{x}''(\xi)(\mu - t)\_r$$

from where, using the obtained bounds for *x*--(*t*), we ge<sup>t</sup>

$$|x'(t)| \le |B - A| + \max\{|F\_1'|, |L\_1'|\}, \ t \in [0, \mu].$$

We can proceed analogously to see that the same bound is valid for *t* ∈ [*μ*, 1]. Finally, for each *t* ∈ (0, 1] there is an *η* ∈ (0, *t*) such that

$$\mathbf{x}(t) - \mathbf{x}(0) = \mathbf{x}'(\eta)t\_r$$

which together with the obtained bound for|*x*-(*t*)| yields the bound for |*x*(*t*)|.

**Lemma 6.** *Let* **(H1)** *and* **(H2)** *hold for K* = *C* − *B. Then every solution x* ∈ *C*<sup>3</sup>[0, 1] *to* (1)*<sup>λ</sup>*, (5) *or* (1)*<sup>λ</sup>*, (6) *satisfies the bounds*

$$|\mathbf{x}(t)| \le |A| + |B| + \max\{|F\_1|, |L\_1|, |F\_1'|, |L\_1'|\}, t \in [0, 1],$$

$$|\mathbf{x}'(t)| \le |B| + \max\{|F\_1|, |L\_1|, |F\_1'|, |L\_1'|\}, t \in [0, 1],$$

$$\min\{F\_1, F\_1'\} \le \mathbf{x}''(t) \le \max\{L\_1, L\_1'\}, t \in [0, 1].$$

**Proof.** Let *x*(*t*) be a solution to (1)*<sup>λ</sup>*, (5); the proof is similar for (1)*<sup>λ</sup>*, (6). We know there is a *ν* ∈ (0, 1) for which *x*--(*ν*) = *C* − *B*. Then, applying Lemmas 2 and 3 on the intervals [0, *ν*] and [*ν*, 1], respectively, we ge<sup>t</sup>

$$F\_1' \le \mathbf{x}^{\prime\prime}(t) \le L\_1^{\prime} \text{ on } [0, \nu] \text{ and } F\_1 \le \mathbf{x}^{\prime\prime}(t) \le L\_1 \text{ on } [\nu, 1]$$

and so the bounds for *x*--(*t*) follow. Further, as in the proof of Lemma 4 we establish consecutively the bounds for |*x*-(*t*)| and |*x*(*t*)|.

#### **3. Existence Results**

**Theorem 1.** *Let* **(H1)** *hold for K* = *C and* **(H3)** *hold for*

$$M\_0 = |A| + |B| + \max\{|F\_1'| \, | \, ^\prime L\_1 \} \}, m\_0 = -M\_0.$$

$$M\_1 = |B| + \max\{|F\_1'| \, |L\_1'| \} \, m\_1 = -M\_1, m\_2 = F\_1', M\_2 = L\_1'.$$

*Then each of BVPs* (1)*,* (2) *and* (1)*,* (3) *has at least one solution in C*<sup>3</sup>[0, 1].

**Proof.** We will establish that the assertion is true for problem (1), (2) after checking that the hypotheses of Lemma 1 are fulfilled; it follows similarly and for (1), (3). We easily check that *(i)* holds for (1)0, (2). Clearly, BVP (1), (2) is equivalent to BVP (1)1, (2) and so *(ii)* is satisfied. Since now **L***h* = *x*---, *(iii)* also holds. Next, according to Lemma 4, for each solution *x* ∈ *C*<sup>3</sup>[0, 1] to (1)*<sup>λ</sup>*, (2) we have

$$m\_i \le \mathfrak{x}^{(i)}(t) \le M\_{i\prime}t \in [0,1], i = 0,1,2.$$

Now use that *f* is continuous on [0, 1] × *J* to conclude that there are constants *m*3 and *M*3 such that

$$\forall m\_3 \le \lambda f(t, \mathbf{x}, p, q) \le M\_3 \text{ for } \lambda \in [0, 1] \text{ and } (t, \mathbf{x}, p, q) \in [0, 1] \times I\_\star$$

which together with (*x*(*t*), *x*-(*t*), *x*--(*t*)) ∈ *J* for *t* ∈ [0, 1] and Equation (1)*λ* implies

$$m\_3 \le \mathfrak{x}^{\prime\prime}(t) \le M\_{3\prime}t \in [0,1].$$

These observations imply that *(iv)* holds, too. Finally, the continuity of *f* on the set *J* gives *(v)* and so the assertion is true by Lemma 1.

**Theorem 2.** *Let* **(H1)** *hold for K* = *C and* **(H3)** *hold for*

$$M\_0 = |A| + |B - A| + \max\{|F\_1'|, |L\_1'|\}, \\ m\_0 = -M\_0.$$

$$M\_1 = |B - A| + \max\{|F\_1'|, |L\_1'|\}, \\ m\_1 = -M\_1, \\ m\_2 = F\_1', \\ M\_2 = L\_1'.$$

*Then BVP* (1)*,* (4) *has at least one solution in C*<sup>3</sup>[0, 1].

**Proof.** It follows the lines of the proof of Theorem 1. Now the bounds

$$m\_i \le \mathbf{x}^{(i)}(t) \le M\_{i\prime}t \in [0,1], i = 0,1,2,$$

for each solution *x* ∈ *C*<sup>3</sup>[0, 1] to a (1)*<sup>λ</sup>*, (4) follow from Lemma 5.

**Theorem 3.** *Let* **(H1)** *and* **(H2)** *hold for K* = *C* − *B and* **(H3)** *hold for*

$$M\_0 = |A| + |B| + \max\{|F\_1|, |L\_1|, |F\_1'|, |L\_1'|\}, m\_0 = -M\_0,$$

$$M\_1 = |B| + \max\{|F\_1|, |L\_1|, |F\_1'|, |L\_1'|\}, m\_1 = -M\_{1'}$$

$$m\_2 = \min\{F\_1, F\_1'\}, M\_2 = \max\{L\_1, L\_1'\}.$$

*Then each of BVPs* (1)*,* (5) *and* (1)*,* (6) *has at least one solution in C*<sup>3</sup>[0, 1].

**Proof.** Arguments similar to those in the proof of Theorem 1 yield the assertion. Now the bounds

$$m\_i \le \mathbf{x}^{(i)}(t) \le M\_{i\prime}t \in [0,1], i = 0,1,2,3$$

for each solution *x* ∈ *C*<sup>3</sup>[0, 1] to (1)*<sup>λ</sup>*, (5) and (1)*<sup>λ</sup>*, (6) follow from Lemma 6.
