1 Introduction

In this paper we consider a D-dimensional gravitational model with Gauss–Bonnet term and cosmological term \(\Lambda \). The so-called Gauss–Bonnet term appeared in string theory as a first order correction (in \(\alpha '\)) to the effective action [1,2,3,4].

We note that at present the Einstein–Gauss–Bonnet (EGB) gravitational model and its modifications, see [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and the references therein, are intensively studied in cosmology, e.g. for possible explanation of accelerating expansion of the Universe which follow from supernova (type Ia) observational data [29,30,31].

In Ref. [28] we were dealing with the cosmological solutions with diagonal metrics governed by \(n >3\) scale factors depending upon one variable, which is the synchronous time variable. We have restricted ourselves by the solutions with exponential dependence of scale factors and have presented a class of such solutions with two scale factors, governed by two Hubble-like parameters \(H >0\) and \(h < 0\), which correspond to factor spaces of dimensions \(m > 3\) and \(l > 1\), respectively, with \(D = 1 + m + l\) and \((m,l) \ne (6,6), (7,4), (9,3)\). Any of these solutions describes an exponential expansion of 3d subspace with Hubble parameters \(H > 0\) [32] and has a constant volume factor of \((m - 3 + l)\)-dimensional internal space, which implies zero variation of the effective gravitational constant G either in a Jordan or in an Einstein frame [33, 34]; see also [35,36,37] and the references therein. These solutions satisfy the most severe restrictions on variation of G [38]. We have studied the stability of these solutions in a class of cosmological solutions with diagonal metrics by using results of Refs. [24, 26] (see also approach of Ref. [22]) and have shown that all solutions, presented in Ref. [28], are stable. It should be noted that two special solutions for \(D = 22, 28\) and \(\Lambda = 0\) were found earlier in Ref. [21]; in Ref. [24] it was proved that these solutions are stable. Another set of six stable exponential solutions, five in dimensions \(D = 7, 8, 9, 13\) and two for \(D = 14\), were considered earlier in [27].

In this paper we extend the results of Ref. [28] to the case of solutions with three non-coinciding Hubble-like parameters. The structure of the paper is as follows. In Sect. 2 we present a setup. A class of exact cosmological solutions with diagonal metrics is found for certain \(\Lambda \) in Sect. 3. Any of these solutions describes an exponential expansion of 3-dimensional subspace with Hubble parameter H and zero variation of the effective gravitational constant G. In Sect. 4 we prove the stability of the solutions in a class of cosmological solutions with diagonal metrics. Certain examples are presented in Sect. 5.

2 The cosmological model

The action of the model reads

$$\begin{aligned} S = \int _{M} \mathrm{d}^{D}z \sqrt{|g|} \{ \alpha _1 (R[g] - 2 \Lambda ) + \alpha _2 \mathcal{L}_2[g] \}, \end{aligned}$$
(2.1)

where \(g = g_{MN} \mathrm{d}z^{M} \otimes dz^{N}\) is the metric defined on the manifold M, \({\dim M} = D\), \(|g| = |\det (g_{MN})|\), \(\Lambda \) is the cosmological term, R[g] is scalar curvature,

$$\begin{aligned} \mathcal{L}_2[g] = R_{MNPQ} R^{MNPQ} - 4 R_{MN} R^{MN} +R^2 \end{aligned}$$

is the standard Gauss–Bonnet term and \(\alpha _1\), \(\alpha _2\) are nonzero constants.

We consider the manifold

$$\begin{aligned} M = {\mathbb R} \times M_1 \times \cdots \times M_n \end{aligned}$$
(2.2)

with the metric

$$\begin{aligned} g= - \mathrm{d} t \otimes \mathrm{d} t + \sum _{i=1}^{n} B_i e^{2v^i t} \mathrm{d}y^i \otimes \mathrm{d}y^i, \end{aligned}$$
(2.3)

where \(B_i > 0\) are arbitrary constants, \(i = 1, \dots , n\), and \(M_1, \dots , M_n\) are 1-dimensional manifolds (either \( {\mathbb R} \) or \(S^1\)) and \(n > 3\).

The equations of motion for the action (2.1) give us the set of polynomial equations [24]

$$\begin{aligned} E= & {} G_{ij} v^i v^j + 2 \Lambda - \alpha G_{ijkl} v^i v^j v^k v^l = 0, \end{aligned}$$
(2.4)
$$\begin{aligned} Y_i= & {} \left[ 2 G_{ij} v^j - \frac{4}{3} \alpha G_{ijkl} v^j v^k v^l \right] \sum _{i=1}^n v^i - \frac{2}{3} G_{ij} v^i v^j \nonumber \\&+\frac{8}{3} \Lambda = 0, \end{aligned}$$
(2.5)

\(i = 1,\ldots , n\), where \(\alpha = \alpha _2/\alpha _1\). Here

$$\begin{aligned} G_{ij} = \delta _{ij} -1, \quad G_{ijkl} = G_{ij} G_{ik} G_{il} G_{jk} G_{jl} G_{kl}, \end{aligned}$$
(2.6)

are, respectively, the components of two metrics on \( {\mathbb R} ^{n}\) [16, 17]. The first one is a 2-metric and the second one is a Finslerian 4-metric. For \(n > 3\) we get a set of fourth-order polynomial equations.

We note that for \(\Lambda =0\) and \(n > 3\) the set of Eqs. (2.4) and (2.5) has an isotropic solution \(v^1 = \cdots = v^n = H\) only if \(\alpha < 0\) [16, 17]. This solution was generalized in [19] to the case \(\Lambda \ne 0\).

It was shown in [16, 17] that there are no more than three different numbers among \(v^1,\dots ,v^n\) when \(\Lambda =0\). This is valid also for \(\Lambda \ne 0\) if \(\sum _{i = 1}^{n} v^i \ne 0\) [26].

3 Solutions with constant G

In this section we present a class of solutions to the set of equations (2.4), (2.5) of the following form:

$$\begin{aligned} v =(\underbrace{H,H,H}_{``our'' \ space},\underbrace{\overbrace{H, \ldots , H}^{m-3}, \overbrace{h_1, \ldots , h_1}^{k_1}, \overbrace{h_2, \ldots , h_2}^{k_2}}_{internal \ space}). \end{aligned}$$
(3.1)

where H is the Hubble-like parameter corresponding to an m-dimensional factor space with \(m > 2\), \(h_1\) is the Hubble-like parameter corresponding to an \(k_1\)-dimensional factor space with \(k_1 > 1\) and \(h_2\) \((h_2 \ne h_1)\) is the Hubble-like parameter corresponding to an \(k_2\)-dimensional factor space with \(k_2 > 1\). We split the m-dimensional factor space into the product of two subspaces of dimensions 3 and \(m-3\), respectively. The first one is identified with “our” 3d space, while the second one is considered as a subspace of \((m-3 + k_1 + k_2)\)-dimensional internal space.

We put

$$\begin{aligned} H > 0 \end{aligned}$$
(3.2)

for a description of an accelerated expansion of a 3-dimensional subspace (which may describe our Universe) and also put

$$\begin{aligned} (m-3) H + k_1 h_1 + k_2 h_2 = 0 \end{aligned}$$
(3.3)

for a description of a zero variation of the effective gravitational constant G.

We remind the reader that the effective gravitational constant \(G = G_{eff}\) in the Brans–Dicke–Jordan (or simply Jordan) frame [33] (see also [34]) is proportional to the inverse volume scale factor of the internal space; see [35,36,37] and references therein.

Due to (3.1) “our” 3d space expands isotropically with Hubble parameter H, while the \((m -3)\)-dimensional part of the internal space expands isotropically with the same Hubble parameter H too. Here, like in Ref. [28], we consider for cosmological applications (in our epoch) the internal space to be a compact one, i.e. we put in (2.2) \(M_4 = \cdots = M_n = S^1\). We put the internal scale factors corresponding to present time \(t_0\): \(a_j (t_0) = B_j^{1/2} \exp (v^j t_0) \), \(j =4, \ldots , n\), (see (2.3)) to be small enough in comparison with the scale factor of “our” space for \(t = t_0\): \( a (t_0) = B^{1/2} \exp (H t_0) \), where \(B_1 = B_2 = B_3 = B\).

According to the ansatz (3.1), the m-dimensional factor space is expanding with the Hubble parameter \(H >0\), while the \(k_i\)-dimensional factor space is contracting with the Hubble-like parameter \(h_i < 0\), where i is either 1 or 2.

Now we consider the ansatz (3.1) with three Hubble parameters H, \(h_1\) and \(h_2\) which obey the following restrictions:

$$\begin{aligned}&S_1 = m H + k_1 h_1 + k_2 h_2 \ne 0, \quad H \ne h_1,\nonumber \\&\quad \quad H \ne h_2, \quad h_1 \ne h_2. \end{aligned}$$
(3.4)

The first inequality in (3.4) is valid since \(S_1 = 3H > 0\) due to (3.2) and (3.3).

In this case the set of \(n+1\) equations (2.4), (2.5) is equivalent to the set of three equations

$$\begin{aligned} E =0, \quad Y_H = 0, \quad Y_{h_1} = 0, \quad Y_{h_2} = 0, \end{aligned}$$
(3.5)

where

$$\begin{aligned} Y_{H} = Y_{\mu }, \quad Y_{h_1} = Y_{\alpha }, \quad Y_{h_2} = Y_{a}, \end{aligned}$$
(3.6)

for all \(\mu = 1, \dots , m\); \(\alpha = m + 1, \dots , m + k_1\) and \(a = m + k_1 + 1, \dots , n\). These relations follow from the definition of \(Y_{i}\) in (2.5) and the identities [16, 17]

$$\begin{aligned} v_i= & {} G_{ij}v^j = v^i - S_1, \end{aligned}$$
(3.7)
$$\begin{aligned} A_i= & {} G_{ijkl} v^j v^k v^l = S_1^3 + 2 S_3 -3 S_1 S_2 \nonumber \\&+\, 3 (S_2 - S_1^2) v^i + 6 S_1 (v^i)^2 - 6(v^i)^3, \end{aligned}$$
(3.8)

\(i = 1,\ldots , n\), where here and in what follows

$$\begin{aligned} S_k = \sum _{i =1}^n (v^i)^k. \end{aligned}$$
(3.9)

Due to (2.5), (3.7), (3.8) we obtain

$$\begin{aligned} Y_{h_i} - Y_{h_j} = (h_i - h_j) S_1 [2 + 4 \alpha Q_{h_i,h_j}], \end{aligned}$$
(3.10)

where

$$\begin{aligned} Q_{h_i h_j} = S_1^2 - S_2 - 2 S_1 (h_i + h_j) + 2 (h_i^2 + h_i h_j + h_j^2), \end{aligned}$$
(3.11)

\(i \ne j\); \(i,j =0,1,2\) and \(h_0 = H\). Equations (3.4), (3.5) and (3.10) imply

$$\begin{aligned} Q_{h_i h_j} = - \frac{1}{2 \alpha }, \end{aligned}$$
(3.12)

\(i \ne j\) and \(i,j =0,1,2\).

Due to \(S_1 = m H + k_1 h_1 + k_2 h_2 \ne 0\) the set of equations (3.5) is equivalent to the following set of equations:

$$\begin{aligned} E =0, \quad Y_H - Y_{h_1} = 0, \quad Y_{h_1} - Y_{h_2} = 0, \nonumber \\ \quad m H Y_H + k_1 h_1 Y_{h_1} + k_2 h_2 Y_{h_2} = 0. \end{aligned}$$
(3.13)

The last relation in (3.13) may be omitted since \(E = 0\) implies \(Y_{i}h^i = m H Y_H + k_1 h_1 Y_{h_1} + k_2 h_2 Y_{h_2} = 0\) [26]. Using this fact and Eqs. (3.4) and (3.10) we reduce the system (3.13) to the following one:

$$\begin{aligned} E =0, \quad Q_{H h_1} = - \frac{1}{2 \alpha }, \quad Q_{h_1 h_2} = - \frac{1}{2 \alpha }. \end{aligned}$$
(3.14)

Using the identity

$$\begin{aligned} Q_{H h_1} - Q_{h_1 h_2} = (H - h_2) (-S_1 + H + h_1 + h_2), \end{aligned}$$
(3.15)

we reduce the set of equations (3.14) to the equivalent set

$$\begin{aligned} E =0, \quad Q = - \frac{1}{2 \alpha }, \quad H + h_1 + h_2 - S_1 = 0. \end{aligned}$$
(3.16)

Here we put \(Q = Q_{h_1 h_2}\), though other choices, \(Q = Q_{H h_1}\) or \(Q = Q_{H h_2}\), give us equivalent sets of equations. Thus the set of \((n + 1)\) polynomial equations (2.4), (2.5) under ansatz (3.1) and restrictions (3.4) imposed is reduced to a set (3.16) of three polynomial equations (of fourth, second and first orders). This reduction is a special case of the more general prescription from Ref. [20].

Using the condition (3.3) of zero variation of G and the linear equation from (3.16) we obtain for \(k_1 \ne k_2\),

$$\begin{aligned} h_1 = \frac{m + 2 k_2 - 3}{k_2 - k_1} H, \quad h_2 = \frac{m + 2 k_1 - 3}{k_1 - k_2} H. \end{aligned}$$
(3.17)

For \(k_1 = k_2\) we get \(H=0\), which is not appropriate for our consideration.

The substitution of (3.17) into relation \(Q_{h_1 h_2} = - \frac{1}{2 \alpha }\) gives us the following relation:

$$\begin{aligned} \frac{P}{(k_2 - k_1)^2} H^2 = - \frac{1}{2 \alpha }, \end{aligned}$$
(3.18)

for \(k_1 \ne k_2\), where

$$\begin{aligned} P= & {} P(m,k_1,k_2) =- (m + k_1 + k_2 -3)(m (k_1 + k_2 -2) \nonumber \\&+\,k_1 ( 2 k_2 -5 ) + k_2 ( 2 k_1 -5 ) + 6) \ne 0, \end{aligned}$$
(3.19)

which implies

$$\begin{aligned} H = |k_1 - k_2| (- 2 \alpha P)^{-1/2}, \quad \alpha P < 0. \end{aligned}$$
(3.20)

It may be readily verified that

$$\begin{aligned} P = P(m,k_1,k_2) < 0 \end{aligned}$$
(3.21)

for all \(m > 2\), \(k_1 > 1\), \(k_2 > 1\), \(k_1 \ne k_2\) and hence our solutions take place for \(\alpha > 0\).

The substitution of (3.17) into (3.5) gives us

$$\begin{aligned} 2 \Lambda = - F_1 H^2 - F_2 H^4 \end{aligned}$$
(3.22)

where

$$\begin{aligned} F_{1}= & {} \frac{1}{(k_{2}-k_{1})^2} [(k_{1}+k_{2})m^2+(k_{1}^2 + 6k_{1}k_{2}+ k_{2}^2 - 6k_{1} -6k_{2}) m \nonumber \\&-\, 9(k_{1}^2+k_{2}^2 - k_{1} - k_{2})+2(2k_{1}+2k_{2}-3)k_{1}k_{2}] \nonumber \\ \end{aligned}$$
(3.23)

and

$$\begin{aligned} F_{2}= & {} -\frac{3\alpha (m-3+k_{1}+k_{2})}{(k_{2}-k_{1})^4} [(k_{1}+k_{2})(k_{1}+k_{2}-2)m^3\nonumber \\&+\,(k_{1}+k_{2})(k_{1}^2+k_{2}^2 +10k_{1}k_{2}-15(k_{1}+k_{2})+18)m^2 \nonumber \\&-\,(12(k_{1}^3+k_{2}^3)-63(k_{1}^2+k_{2}^2)+54(k_{1}+k_{2})\nonumber \\&-\,2(4(k_{1}^2+k_{2}^2)\nonumber \\&-\,42(k_{1}+k_{2}+16k_{1}k_{2}+63)k_{1}k_{2}))m \nonumber \\&+\,27(k_{1}^3+k_{2}^3)-81(k_{1}^2+k_{2}^2) +54(k_{1} +k_{2}) \nonumber \\&-\,(40(k_{1}^2+k_{2}^2)-16(k_{1}+k_{2}-6)k_{1}k_{2}+162\nonumber \\&-\,153(k_{1}+k_{2}))k_{1}k_{2}]. \end{aligned}$$
(3.24)

Using Eqs. (3.20), (3.22), (3.23), (3.24) we obtain

$$\begin{aligned} \Lambda= & {} \Lambda (m,k_1,k_2) = \frac{1}{8 \alpha P^2} (m + k_1 + k_2 -3) \nonumber \\&\times \,[(k_1 + k_2)(k_1 + k_2 - 2)m^3 \nonumber \\&+\, (k_1^3 + k_2^3 + 11 (k_1^2 k_2 + k_1 k_2^2) - 19 (k_1^2 + k_2^2) \nonumber \\&-\, 22 k_1 k_2 + 18 (k_1 + k_2)) m^2 \nonumber \\&-\,(8(k_1^3 + k_2^3) - 63 (k_1 + k_2)^2 - 8 k_1^2 (k_1 - 11) k_2\nonumber \\&-\, 8 k_2^2 (k_2 - 11) k_1) - 32 k_1^2 k_2^2 + 54 (k_1 + k_2)) m \nonumber \\&-\, ( 9 (k_1^3 + k_2^3) + 45 (k_1^2 + k_2^2) - 54 (k_1 + k_2)\nonumber \\&\quad +\, 8 (k_1^2 + k_2^2) k_1 k_2 \nonumber \\&-\, 16 (k_1 + k_2 -10) k_1^2 k_2^2 - 9 (21 k_1 + 21 k_2 - 26) k_1 k_2 ) ], \nonumber \\ \end{aligned}$$
(3.25)

where \(P = P(m,k_1,k_2)\) is defined in (3.19).

The function \(\Lambda (m,k_1,k_2)\) in (3.25) is symmetric with respect to \(k_1\) and \(k_2\), i.e.

$$\begin{aligned} \Lambda (m,k_1,k_2) = \Lambda (m,k_2,k_1). \end{aligned}$$
(3.26)

For \(k_2 =0\) we get a function \(\Lambda (m,k_1,0) = \Lambda (m,k_1)\), where \(\Lambda (m,k_1)\) was obtained in Ref. [28] for the case of two different Hubble-like parameters.

It may be readily verified that for \(k_1(k) = n_1 k +q_1\) and \(k_2(k) = n_2 k +q_2\), where \(k, n_1>0, q_1, n_2>0, q_2\) are integer numbers, we get

$$\begin{aligned} \Lambda (m,k_1(k),k_2(k)) \rightarrow \frac{1}{8\alpha }, \end{aligned}$$
(3.27)

as \(k \rightarrow + \infty \) for any fixed \(m \ge 3\). We note that the limit (3.27) is positive and does not depend upon m. For fixed integer \(m > 2\) and \(k_2 \ge 1\) we are led to the following limit:

$$\begin{aligned} \Lambda (m,k_1,k_2)\rightarrow & {} \frac{1}{8\alpha (m + 4 k_2 -5)^2} \left[ m^2 - 8(1 - k_2) m - 9 \right. \nonumber \\&\left. - 8 k_2 + 16 k_2^2\right] = \Lambda (m,\infty ,k_2), \end{aligned}$$
(3.28)

as \(k_1 \rightarrow + \infty \) and there is an analogous relation (due to (3.26)) for fixed \(m > 2\), \(k_1 \ge 1\) and \(k_2 \rightarrow + \infty \). It can be easily verified that, for these values of m, \(k_1\) we get \(\Lambda (m,\infty ,k_2) > 0\).

Equations (3.27) and (3.28) may be used in a context of (1 / D)-expansion for large D in the model under consideration; see [25] and the references therein.

4 The proof of stability

Here, as in [28], we have due to (3.3)

$$\begin{aligned} K = K(v) = \sum _{i = 1}^{n} v^i = 3H >0. \end{aligned}$$
(4.1)

Let us put the restriction

$$\begin{aligned} \det (L_{ij}(v)) \ne 0 \end{aligned}$$
(4.2)

on the matrix

$$\begin{aligned} L =(L_{ij}(v)) = (2 G_{ij} - 4 \alpha G_{ijks} v^k v^s). \end{aligned}$$
(4.3)

We recall that, for a general cosmological setup with the metric

$$\begin{aligned} g= - \mathrm{d}t \otimes \mathrm{d}t + \sum _{i=1}^{n} e^{2\beta ^i(t)} \mathrm{d}y^i \otimes \mathrm{d}y^i, \end{aligned}$$
(4.4)

we have the set of equations [24]

$$\begin{aligned} E= & {} G_{ij} h^i h^j + 2 \Lambda - \alpha G_{ijkl} h^i h^j h^k h^l =0, \end{aligned}$$
(4.5)
$$\begin{aligned} Y_i= & {} \frac{\mathrm{d} L_i}{\mathrm{d}t} + \left( \sum _{j=1}^n h^j\right) L_i - \frac{2}{3} (G_{sj} h^s h^j - 4 \Lambda ) = 0, \end{aligned}$$
(4.6)

where \(h^i = \dot{\beta }^i\),

$$\begin{aligned} L_i = L_i(h) = 2 G_{ij} h^j - \frac{4}{3} \alpha G_{ijkl} h^j h^k h^l , \end{aligned}$$
(4.7)

\(i = 1,\ldots , n\).

Due to the results of Ref. [26] a fixed point solution \((h^i(t)) = (v^i)\) (\(i = 1, \dots , n\); \(n >3\)) to Eqs. (4.5), (4.6) obeying restrictions (4.1), (4.2) is stable under perturbations,

$$\begin{aligned} h^i(t) = v^i + \delta h^i(t), \end{aligned}$$
(4.8)

\(i = 1,\ldots , n\), as \(t \rightarrow + \infty \).

In order to prove the stability of solutions we should prove Eq. (4.2). First, we show that for the vector v from (3.1), obeying Eqs. (3.4) the matrix L has a block-diagonal form,

$$\begin{aligned} (L_{ij}) = \mathrm{diag}(L_{\mu \nu }, L_{\alpha \beta }, L_{a b} ), \end{aligned}$$
(4.9)

where here and in what follows: \(\mu , \nu = 1, \dots , m\); \(\alpha , \beta = m + 1, \dots , m + k_1\) and \(a, b = m + k_1 + 1, \dots , n\).

Indeed, denoting \(S_{ij} = G_{ijkl} v^k v^l\) we get from (3.8)

$$\begin{aligned} S_{ij}= & {} \frac{1}{3} \frac{\partial }{\partial v^j} (G_{iskl} v^s v^k v^l) \nonumber \\= & {} S_1^2 - S_2 + 2 (v^i)^2 + 2 (v^j)^2 + 2 v^i v^j - 2 S_1 (v^i + v^j) \nonumber \\&+\, \delta _{ij} (S_2 - S_1^2 + 4 S_1 v^i - 6(v^i)^2). \end{aligned}$$
(4.10)

Here we use the notation \(S_k = \sum _{i =1}^{n} (v^i)^k\) and the identity \( \frac{\partial }{\partial v^j} S_k = k (v^j)^{k - 1}\). It follows from (3.11) and (4.10) that

$$\begin{aligned} S_{H h_1}\equiv & {} S_{\mu \alpha } = S_{ \alpha \mu } = Q_{H h_1}, \end{aligned}$$
(4.11)
$$\begin{aligned} S_{H h_2}\equiv & {} S_{\mu a} = S_{a \mu } = Q_{H h_2}, \end{aligned}$$
(4.12)
$$\begin{aligned} S_{h_1 h_2}\equiv & {} S_{\alpha a} = S_{a \alpha } = Q_{h_1 h_2} \end{aligned}$$
(4.13)

and hence \(L_{\mu \alpha } = L_{ \alpha \mu } = 0\), \(L_{\mu a} = L_{a \mu } = 0\) and \(L_{\alpha a} = L_{a \alpha } = 0\) due to Eq. (3.12). Thus, the matrix \((L_{ij})\) is block-diagonal.

For the other three blocks we have

$$\begin{aligned} L_{\mu \nu }= & {} G_{\mu \nu } (2 + 4 \alpha S_{HH}), \end{aligned}$$
(4.14)
$$\begin{aligned} L_{\alpha \beta }= & {} G_{\alpha \beta } (2 + 4 \alpha S_{h_1 h_1}), \end{aligned}$$
(4.15)
$$\begin{aligned} L_{a b}= & {} G_{a b} (2 + 4 \alpha S_{h_2 h_2}), \end{aligned}$$
(4.16)

where

$$\begin{aligned} S_{h_i h_i} = S_1^2 - S_2 + 6 h_i^2 - 4 S_1 h_i, \end{aligned}$$
(4.17)

\(i = 0,1,2\) and \(h_0 = H\). Here we denote \(S_{HH} = S_{\mu \nu }\), \(\mu \ne \nu \); \(S_{h_1 h_1} = S_{\alpha \beta }\), \(\alpha \ne \beta \) and \(S_{h_1 h_1} = S_{a b}\), \(a \ne b\).

Due to Eqs. (4.9), (4.14), (4.15), (4.16) the matrix (4.9) is invertible if and only if \(m > 1\), \(k_1 > 1\), \(k_2 > 1\) and

$$\begin{aligned} S_{h_i h_i} \ne - \frac{1}{2 \alpha }, \end{aligned}$$
(4.18)

\(i = 0,1,2\).

Now, we prove that inequalities (4.18) are satisfied for the solutions under consideration. Let us suppose that (4.18) is not satisfied for some \(i_0 \in \{0,1,2 \}\), i.e.

$$\begin{aligned} S_{h_{i_0} h_{i_0}} = S_1^2 - S_2 + 6 h_{i_0}^2 - 4 S_1 h_{i_0} = - \frac{1}{2 \alpha }. \end{aligned}$$
(4.19)

Let \(i_1 \in \{0,1,2 \}\) and \(i_1 \ne i_0\). Then using Eqs. (3.11) and (3.12) we get

$$\begin{aligned} Q_{h_{i_0} h_{i_1}} - S_{h_{i_0} h_{i_0}} = 2 (h_{i_1} - h_{i_0}) ( 2 h_{i_0} + h_{i_1} - S_1) = 0, \end{aligned}$$
(4.20)

which implies

$$\begin{aligned} 2 h_{i_0} + h_{i_1} - S_1 = 0. \end{aligned}$$
(4.21)

But due to (3.16)

$$\begin{aligned} h_{i_0} + h_{i_1} + h_{i_2} - S_1 = 0, \end{aligned}$$
(4.22)

where \(i_2 \in \{0,1,2 \}\) and \(i_2 \ne i_0\), \(i_2 \ne i_1\). Subtracting (4.22) from (4.21) we obtain \(h_{i_0} - h_{i_2} = 0\), i.e. \(h_{i_0} = h_{i_2}\). But due to restrictions (3.4) we have \(h_{i_0} \ne h_{i_2}\). We are led to a contradiction, which proves the inequalities (4.18) and hence the matrix L from (4.9) is invertible (\(m > 2\), \(k_1 > 1\), \(k_2 > 1\)), i.e. Eq. (4.2) is obeyed. Thus, the solutions under consideration are stable.

5 Examples

Here we present several examples of stable solutions under consideration.

5.1 The case \(m =3\)

Let us consider the case \(m =3\). From (3.25) we get

$$\begin{aligned} \Lambda= & {} - \frac{1}{4 \alpha }\frac{1}{(k_{1}- 2k_{1}k_{2}+ k_{2})^2 (k_{1}+k_{1})} \nonumber \\&\times \, (3(k_{1}^3+k_{2}^3)-(2(k_{1}^2+k_{2}^2)+(k_{1}+k_{2})(3+2k_{1}k_{2})\nonumber \\&-\,8k_{1}k_{2})k_{1}k_{2}). \end{aligned}$$
(5.1)

For \((m,k_1,k_2) = (3,3,2)\) we have \(P = -70\),

$$\begin{aligned} \Lambda =\frac{213}{980 \alpha } \end{aligned}$$
(5.2)

and

$$\begin{aligned} H = \frac{1}{\sqrt{140 \alpha }}, \quad h_{1}=-4H, \quad h_{2}=6H. \end{aligned}$$
(5.3)

Now we put \((m,k_1,k_2) = (3,4,2)\). We obtain \(P = -120\),

$$\begin{aligned} \Lambda = \frac{21}{100\alpha }, \end{aligned}$$
(5.4)

and

$$\begin{aligned} H =\frac{1}{2\sqrt{15 \alpha }}, \quad h_{1} = -2H, \quad h_{2} = 4H. \end{aligned}$$
(5.5)

According to our analysis from the previous section both solutions are stable.

5.2 Examples for \(m =4\) and \(m= 5\)

Now we present other examples of stable solutions for \(m =4\) and \(m= 5\).

First we put \((m,k_1,k_2) = (4,3,2)\). We find \(P = -102\) and

$$\begin{aligned} \Lambda = \frac{123}{578\alpha }. \end{aligned}$$
(5.6)

In this case we obtain

$$\begin{aligned} H = \frac{1}{\sqrt{204 \alpha }}, \quad h_{1} = -5H, \quad h_{2}=7H. \end{aligned}$$
(5.7)

Now we enlarge the value of m by putting \((m,k_1,k_2) = (5,3,2)\). We find \(P = -140\),

$$\begin{aligned} \Lambda =\frac{589}{2800\alpha } , \end{aligned}$$
(5.8)

and

$$\begin{aligned} H=\frac{1}{\sqrt{280 \alpha }}, \quad h_{1}= - 6H, \quad h_{2} = 8H. \end{aligned}$$
(5.9)

We note that in all examples above \(\Lambda >0\).

6 Conclusions

We have considered the D-dimensional Einstein–Gauss–Bonnet (EGB) model with the \(\Lambda \)-term and two constants \(\alpha _1\) and \(\alpha _2\). By using the ansatz with diagonal cosmological metrics, we have found, for certain \(\Lambda = \Lambda (m,k_1.k_2)\) and \(\alpha = \alpha _2 / \alpha _1 < 0\), a class of solutions with exponential time dependence of three scale factors, governed by three different Hubble-like parameters \(H >0\), \(h_1\) and \(h_2\), corresponding to submanifolds of dimensions \(m > 2\), \(k_1 > 1\), \(k_2 > 1\), respectively, with \( k_1 \ne k_2\) and \(D = 1 + m + k_1 + k_2 \). Here \(m > 2\) is the dimension of the expanding subspace.

Any of these solutions describes an exponential expansion of “our” 3-dimensional subspace with the Hubble parameter \(H > 0\) and anisotropic behaviour of \((m-3+ k_1 + k_2)\)-dimensional internal space: expanding in \((m-3)\) dimensions (with Hubble-like parameter H) and either contracting in \(k_1\) dimensions (with Hubble-like parameter \(h_1\)) and expanding in \(k_2\) dimensions (with Hubble-like parameter \(h_2\)) for \(k_1 > k_2\) or expanding in \(k_1\) dimensions and contracting in \(k_2\) dimensions for \(k_1 < k_2\). Each solution has a constant volume factor of internal space and hence it describes zero variation of the effective gravitational constant G. By using the results of Ref. [26] we have proved that all these solutions are stable as \(t \rightarrow + \infty \). We have presented several examples of stable solutions for \(m = 3,4,5\).