Existence and monotone iteration of unique solution for tempered fractional differential equations Riemann–Stieltjes integral boundary value problems

The primary objective of this research paper is to investigate two kinds of high-order tempered fractional differential equations integral boundary value problems. By means of the mixed monotone operators fixed point theorems with perturbation and the increasing φ – (h,σ )-concave operators fixed point theorems, we can not only guarantee the existence-uniqueness of solution, but also construct successively sequences for approximating the unique solution. In addition, we demonstrate the effectiveness of the main result by using two examples.

Over the past few decades, more theories and experiments show that a great deal of abnormal phenomena that occur in the applied science and engineering can be described by fractional differential equations; therefore, fractional differential equations arise in lots of fields such as economics, mechanics, physics, chemistry, biological science, etc.; see [1][2][3][4] for example. It is because the fractional order derivatives provide power tools for the description of memory and hereditary characteristics of different processes and materials in many fields.
In addition, in order to increase the power and applicability of the fractional calculus, Caputo and Fabrizio have recently defined a new fractional derivative possessing a singular kernel [5]. Some researchers have used distinct methods for solving some different equations including the Caputo-Fabrizio fractional derivative (see [6][7][8][9][10][11]). For instance, Aydogan and Baleanu [6] investigated the existence of solutions for two highorder fractional differential equations including the Caputo-Fabrizio derivative. In [7], Baleanu et al. extended the fractional Caputo-Fabrizio derivative of order 0 ≤ σ < 1 on C R [0, 1] and investigated two high-order series-type fractional differential equations involving the extended derivation. In [8], a new fractional model for human liver involving Caputo-Fabrizio derivative with exponential kernel was introduced. By using the fractional Caputo-Fabrizio derivative, Aydogan et al. [12] introduced two types of new highorder derivations called CFD and DCF and investigated the existence of solutions for such two types of high-order fractional integro-differential equations. In [13], the authors showed that four fractional integro-differential inclusions had solutions. Also, it has been proved that working with the Caputo-Fabrizio fractional derivative is much better than with other fractional derivatives (the reader can see, for example, [14][15][16][17] and the references therein).
In [18], the author considered a class of boundary value problems of Caputo tempered fractional differential equations as follows: with θ ∈ (0, 1) and a, b, c are real constants. By using the principle of compressed mapping, spectral method, and stability analysis, the existence, uniqueness, structural stability, and numerical analysis of solutions were investigated.
In [19], by using a new mixed monotone operators fixed point theorem with perturbation, the authors studied the existence and uniqueness of positive solution for the following nonlinear fractional differential equation boundary value problem: Based on the theory of μ 0 -positive linear operator and the Banach contraction map principle, Zhang and Zhong in [20] obtained the existence and uniqueness for the following fractional differential equation integral boundary value problem: By means of the reducing method of fractional orders, the upper and lower solutions methods, and the Schauder fixed point theorem, Zhang, Liu, and Wu in [21] investigated the existence of positive solutions for the following fractional differential equations multipoint boundary value problem: where D γ 0 + is the Riemann-Liouville fractional derivative, λ > 0 is a parameter, n -1 < γ ≤ n, ni -1 ≤ γμ i ≤ ni (i = 1, 2, . . . , n -2), μμ n-1 > 0, γμ n-1 ≤ 2, γμ > 1, a j ≥ 0 (j = 1, 2, . . . , p -2) 0 < ξ 1 < ξ 2 < · · · < 1; f : (0, +∞) n → R + is continuous. Inspired by the above-mentioned excellent works, we aim to investigate the existenceuniqueness of solutions for BVP (1.1) and BVP (1.2), respectively. As far as we know, the high-order Riemann-Liouville tempered fractional differential equations integral boundary values problems have seldom been researched up to now. The main features of the present paper are as follows. Firstly, the tempered fractional derivative R 0 D α,λ t is more general than the standard Riemann-Liouville fractional derivative R 0 D α t . For instance, let λ = 0 and g(t, u(t)) ≡ 0, then BVP (1.1) reduces to the BVP in [22]; let λ = 0, b = 0, and a(s) ≡ 0, then BVP (1.2) is a special case in [20]. Secondly, the integral boundary conditions involving tempered fractional derivative are more general cases, which covers the common integral boundary conditions as special cases. For example, let λ = 0, a(t) ≡ 0, b(t) ≡ 1, then the Riemann-Stieltjes integral boundary condition in (1.1) is reduced to a general integral boundary value condition. Finally, BVP (1.1) and BVP (1.2) are more general than those in the above-mentioned literature works. For instance, if b = 0, 2 < α ≤ 3, and λ = 0, BVP (1.2) presented in [23] has more special cases.
In this paper, it is not necessary for the operator to be completely continuous or compact, nor fix the existence of upper and lower solutions. For system (1.1), the existenceuniqueness and monotone iteration of positive solution are obtained by employing a class of sum-type operators fixed point theorems; for system (1.2), the existence and uniqueness of nontrivial solutions are investigated by means of a new fixed point theorem of increasing ϕ -(h, δ)-concave operator, which is defined on a new set P h,δ . Furthermore, our conclusions can not only guarantee the existence-uniqueness of solutions, but also construct successive sequences for approximating the unique solution. In the end, it is worth mentioning that some important properties of the Green's function rely on the parameter λ.
The paper is organized as follows. In Sect. 2, we present some necessary definitions and preliminary results that will be used to prove our main results. In Sects. 3 and 4, we prove the main results about the existence-uniqueness of solutions for the Riemann-Stieltjes integral boundary value problems (1.1) and (1.2), respectively. Finally, two examples are given to illustrate the validity of our main results.

Preliminaries
A nonempty closed convex set P ⊂ E is a cone if it satisfies: is a real Banach space which is partially ordered by a cone P ⊂ E, that is, x ≤ y if and only if yx ∈ P. If x ≤ y and x = y, then we denote x < y or y > x. By θ we denote the zero element of E.
In addition, for given h > θ , we denote by P h the set P h = {x ∈ E | x ∼ h}, in which ∼ is an equivalence relation, i.e., x ∼ y means that there exist λ > 0 and μ > 0 such that λx ≥ y ≥ μx for all x, y ∈ E.

B : P → P is an increasing sub-homogeneous operator. Assume that
(3) the operator equation A(x, x) + Bx = x has a unique solution x * in P h ; (4) for any initial values x 0 , y 0 ∈ P h , constructing successively the sequences x n = A(x n-1 , y n-1 ) + Bx n-1 , y n = A(y n-1 , x n-1 ) + By n-1 , n = 1, 2, . . . , we have x n → x * and y n → x * as n → ∞.

G(t, s)g s, y n-1 (s) ds,
we obtain x n → u * and y n → u * as n → ∞. , it is easy to know that A : P × P → P is a mixed monotone operator and B : P → P is an increasing operator. Again, from (H 3 ), for ∀γ ∈ (0, 1) and u, v ∈ P, we obtain
Finally, we show that condition (I 2 ) of Lemma 2.5 is also satisfied. For ∀u, v ∈ P, from (3.1) and (H 4 ), we get That is, A(u, v) ≥ δ 0 Bu for ∀u, v ∈ P h . Then, the conclusion of Theorem 3.1 follows Lemma 2.5.

Existence-uniqueness of nontrivial solution for BVP (1.2)
In this section, for h > θ , taking another σ ∈ P with θ ≤ σ ≤ h, we define a new set P h,σ = {x ∈ E | x + σ ∈ P h }. Then we can see that h ∈ P h,σ and P h,σ = {x ∈ E | there exist μ > 0 and v > 0 such that μh ≤ x + σ ≤ vh}. If σ = θ , then P h,σ = P h . Remark 4.1 P h ⊆ P h,σ and P h,σ is not a subset of P for some σ , P h and P h,σ are different two sets.

Then the following integral boundary value problem
has a unique positive solution u * in P h with h = e -λt t α-1 . Moreover, for any given ω 0 ∈ P h , constructing successively the sequences we have ω n (t) → u * (t) as n → ∞.
Proof Letting c = 0 and ρ = 1, from Theorem 4.1, we arrive at the conclusions.

Applications
To test our results established in the previous section, we provide two adequate problems.

Conclusion
A fractional derivative (or integral) is a convolution with a power law, and a tempered fractional derivative multiplies that power law kernel by an exponential factor. Furthermore, the tempered fractional differential equations models open up a new kind of possibility for robust mathematical modeling of complex multi-scale problems and anomalous phenomena. Therefore, the emerging theory of tempered fractional calculus provides a sound mathematical basis for applications.