Definition 1 (see [35], [36].)The left Riemann-Liouville fractional integral and left Caputo fractional derivative of variable order u(t) of H are recalled as

with u(t): J⟶(n − 1, n].

Remark 2 (see [37], [38].)In Equations (5) and (6), by choosing the constant order u, the Caputo fractional derivative and the Riemann-Liouville fractional integral of variable order will be the same conventional of the Caputo fractional derivative and Riemann-Liouville fractional integral of constant order, respectively.

Lemma 3 (see [38].)Suppose α₁, b₁ > 0 and n = [α₁] + 1, then

Lemma 4 (see [39].)Let α₁, α₂ > 0, ψ ∈ L¹(b₁, b₂), and $$. Then, the differential equation

has unique solution and under the conditions n − 1 < α₁ ≤ n, ϖ_j ∈ ℝ.

Moreover,

Remark 5 (see [26], [27].)It is observed that the property of semigroup is not consistent for arbitrary functions u(t), v(t) in the role of variable orders, i.e.,

Example 1. Assume y(t) ≡ 1 for t ∈ J = [0, 3] and u(t) = t/2 and $$ Then,

For t = 2, we write

and accordingly,

As a result, the semigroup property is not valid in the generalized case for the Riemann-Liouville fractional integral of variable order.

Lemma 6 (see [26].)Let u ∈ C(J, (1, 2]), then for

we have

Definition 7 (see [26], [27], [40].)Consider a subset $$ of ℝ:

• (a)

  A generalized interval is either empty, singleton subset, or an interval

• (b)

  The finite set $$ of generalized interval is a partition of $$ whenever every $$ belongs to exactly one of the generalized intervals $$ in $$

• (c)

  $$ is piece-wise constant with respect to $$ of $$ if $$, h is constant on $$

Theorem 8 (Guo-Krasnoselskii fixed point theorem [41]). Assume a cone P and bounded subsets B₁, B₂ of a Banach space E with

and assume a completely continuous operator $$ such that

• (i)

  ∥Ty∥≥∥y∥, y ∈ P∩∂B₁ and ∥Ty∥≤∥y∥, y ∈ P∩∂B₂; or

• (ii)

  ∥Ty∥≤∥y∥, y ∈ P∩∂B₁ and ∥Ty∥≥∥y∥, y ∈ P∩∂B₂.

Then, operator T possesses a fixed point in $$

Lemma 9. Let $$. Then, y ∈ E_j is a solution of the auxiliary BVP of constant order of the Caputo fractional thermostat model

if it fulfills where Green’s function G_j(t, s) is stated as

Proof. Using Lemma 3, there are c₀, c₁ ∈ ℝ such that

In view of Lemma 4,

By y^′(0) = 0, we get that c₁ = 0.. Moreover,

Using the boundary condition  ^CD^α−1y(S_j) + y(S_j−1) = 0, we get

At the end, (26) takes the form

The proof is completed.

Remark10. Notice that for each fixed s ∈ [S_j−1, S_j], ∂G_j/∂t = 0 for t ≤ s and ∂G_j/∂t < 0 for t > s; this turns G_j(t, s) a decreasing function. Therefore

As a result, due to behavior of G_j(t, s) with respect to s, we have

We shall prove that Gj(t, s) meets the following property to ensure the existence of positive solution of the variable order Caputo fractional thermostat model (4) proposed by Lan and Webb in [42]:

(H2): ∃φ : [S_j−1, S_j]⟶[0, ∞) as a measurable mapping, [c, d]⊆[S_j−1, S_j] and ξ ∈ [0, 1] such that

Lemma 11. If $$, then G_j(t, s) > 0 for all t, s ∈ [S_j−1, S_j], and G_j(t, s) satisfies (H2).

Proof. Taking [c, d] = [S_j−1, S_j], then

where

The proof is completed.

Lemma 12. If $$, then G_j(t, s) ≥ 0 for all t, s ∈ [S_j−1, S_j], and G_j(t, s) satisfies (H2).

Proof. Taking [c, d] = [Sj − 1, d] such that S_j−1 ≤ d < S_j, and using the preceding lemma arguments, we obtain

Setting

we get

The proof is completed.

Lemma 13. If $$, then G(t, s) changes sign on [S_j−1, S_j] × [S_j−1, S_j], and G_j(t, s) satisfies (H2).

Proof. Let [c, d] = [S_j−1, d] be such that S_j−1 ≤ d < S_j and $$ We have

where

The proof is completed.

Theorem 14. Let f(s, y(s)) ∈ C([S_j−1, S_j]) × [0, ∞). Assuming one of the below mentioned cases:

• (1)

  f₀ = ∞ and f_∞ = 0 (Sublinear case)

• (2)

  $$ and $$ (Superlinear case),

if $$, then there is at least positive solution of the auxiliary BVP of constant order Caputo fractional thermostat model (22).

Proof. Define T : C([S_j−1, S_j])⟶C([S_j−1, S_j]) as

We now define the cone

where ξ is as in (35).

In the start, we prove that T(P) ⊂ P. Since the functions f and G_j are positive and continuous, it follows that if y ∈ P, then Ty ∈ C([S_j−1, S_j]) and Ty(t) ≥ 0 for all t ∈ [S_j−1, S_j]. We reach to the inequality derived below for a fixed y ∈ P for all t ∈ [S_j−1, S_j] and that G_j(t, s) satisfies (H2)

which leads to the first claim. In the next step, we prove the complete continuity of operator T : P⟶P. Because of the continuous behavior of G_j and f, continuity of T : P⟶P is followed immediately. Let B ⊂ P be bounded. Define

So for all y ∈ B, we have

for all t ∈ [S_j−1, S_j]. This defines the boundedness of T(B). Now, for each y ∈ B and t₁, t₂ ∈ [S_j−1, S_j] such that t₁ < t₂, we write

Tending t₁⟶t₂ implies that the RHS of the above inequality goes to 0 and thus T(B) is equicontinuous. At the end, the Arzela-Ascoli theorem confirms complete continuity of operator T : P⟶P.

Next, assume that (2) is true. As f₀ = ∞, a ρ₁ > 0 exists such that f(t, y) ≥ δ₁y, ∀0 < y ≤ ρ₁, where δ₁ satisfies

Taking y ∈ P with ∥y∥ = ρ₁, then

Let B₁ = {y ∈ C([S_j−1, S_j])\∥y∥<ρ₁}. Hence, we have ∥Ty∥≥∥y∥, y ∈ P∩∂B₁.

Due the continuity of f(t.) on [0, ∞), a function $$ can be defined which is nondecreasing on (0, ∞) by assumption and

Therefore, there exists ρ₂ > ρ₁ > 0 such that $$, where δ₂ satisfies

Define Ω₂ = {y ∈ C([S_j−1, S_j])\∥y∥<ρ₂} and let y ∈ P such that ∥y∥ = ρ₂. Then,

Hence, we have ∥Ty∥≤∥y∥, y ∈ P∩∂B₂. In conclusion, there is at least one positive solution by (i) of Theorem 8 for the auxiliary BVP of constant order Caputo fractional thermostat model (22).

Now, assume that (3) is true. For δ₂ > 0 and by assumption, there is r₁ > 0 with f(t, y) ≤ δ₂y for 0 ≤ y ≤ r₁. Let y ∈ P such that ∥y∥ = r₁. Then,

If we let B₁ = {y ∈ C[S_j−1, S_j]\∥y∥<r₁}, then ∥Ty∥≤∥y∥ for y ∈ P∩∂B₁. Again by assumption, there is r > 0 such that f(t, y) ≥ δ₁u, ∀y ≥ r. Define B₂ = {y ∈ C([S_j−1, S_j])\∥y∥<r₂}, where r₂ = max(2r₁, (r/ξ)). Then, y ∈ P and ∥y∥ = r₂ imply that

and so we obtain

This shows that ∥Ty∥≥∥y∥ for y ∈ P∩∂B₂. In conclusion, it is found at least one positive solution, by (ii) of Theorem 8, for the auxiliary BVP of constant order Caputo fractional thermostat model (22) as $$

Theorem 15. Let the conditions (H1) and (H3) be satisfied and the inequality

holds. If $$, then the thermostat BVP (22) has a unique positive solution in E_j.

Proof. We shall use the Banach contraction principle to prove that T has unique fixed point. For x(t), y(t) ∈ E_j, by Lemma 11 and Lemma 12, we obtain

Consequently by (57), the operator T is a contraction. Hence, by Banach’s contraction principal, T has a unique fixed point $$, which is a unique positive solution of the auxiliary BVP of the constant order Caputo fractional thermostat model (22).

Theorem 16. Assuming (H1)-(H2), the variable order BVP of the main Caputo fractional thermostat model (4) admits at least one solution in C(J, ℝ).

Proof. We know that the solution $$ fulfills the auxiliary BVP of constant order Caputo fractional thermostat model (22) by Theorem 14 for any j ∈ {1, 2, ⋯, n}. Now, the continuous function on [0, S_j] is defined as

is a solution of (20) for t ∈ J_j, j ∈ {1, 2, ⋯, n}.

Therefore, y(t) = y_j(t), t ∈ J_j, j ∈ {1, 2, ⋯, n} solves the variable order BVP of the main Caputo fractional thermostat model (4). The proof is completed.

Definition 17 (see [43].)The variable order Caputo fractional thermostat model (4) is Ulam-Hyers stable if there exists c_f > 0 such that for each ε > 0 and for every solution z ∈ C(J, ℝ) of the following inequality

there exists a solution y ∈ C(J, ℝ) of (4) with

Theorem 18. Assume that the conditions (H1) and (H2) to be held. Then, the Caputo fractional thermostat model (4) is Ulam-Hyers stable.

Proof. Let ε > 0 be an arbitrary number and the function z(t) belonging to C(J, ℝ) satisfies the following inequality

For any j ∈ {1, 2, ⋯, n}, we define the functions z₁(t) ≡ z(t), t ∈ [0, S₁] and for j = 2, 3, ⋯, n:

For any j ∈ {1, 2, ⋯, n} and according to the equality (6), for t ∈ J_j, we get

Taking $$ on both sides of the inequality (62), we obtain

According to Theorem 16, the Caputo fractional thermostat model (4) has a positive solution y ∈ C(J, ℝ) defined by y(t) = y_j(t) for t ∈ J_j, j = 1, 2, ⋯, n, where

and $$ is a positive solution of (22). According to Lemma 9, the integral equation

holds. Let t ∈ J_j, j = 1, 2, ⋯, n. Then, by Eq (66) and (67) we get where

Then,

We obtain, for each t ∈ J_j

Therefore, by Definition 17, the Caputo fractional thermostat model (4) is Ulam-Hyers stable and the proof is completed.

Example 2. Consider the nonlinear function

on (t, y) ∈ [0, 2] × [0, +∞) and

Then, in consistent with (22) and corresponding to the variable order BVP of fractional differential equation

the constant order auxiliary BVPs are

Clearly, f₀ = ∞ and f_∞ = 0.

For j = 1, we get Γ(1.4) − 1 ≈ −0.11274 < 0, we take

and define the cone P₁ = {y\y ∈ C[0, 1], min_{t∈[0, 1]}y(t) ≥ ξ₁∥y∥}. By Theorem 14, it is deduced that that the auxiliary BVP of constant order Caputo fractional thermostat model (75) possesses a positive solution $$.

For j = 2, we get Γ(1.5) − 1 ≈ −0.11377 < 0, we take

and consider the cone P₂ = {y\y ∈ C[1, 2], min_{t∈[1, 2]}y(t) ≥ ξ₂∥y∥}.

According to Theorem 14, the auxiliary BVP of constant order Caputo fractional thermostat model (76) admits a positive solution $$, and by Theorem 16, the BVP for variable order Caputo fractional thermostat model (74) has a solution

where