Generalized Ulam–Hyers–Rassias Stability Results of Solution for Nonlinear Fractional Differential Problem with Boundary Conditions

'ere are various, not equivalent, definitions of fractional derivatives according to Grunwald Letnikov, Weil, Caputo, and Riemann–Liouville, etc. Ordinary and partial differential order equations (with fractional derivatives of Caputo and Riemann–Liouville) have awakened in recent years with considerable interest both in mathematics and in applications. Let us describe the abstract Cauchy problem:


Introduction and Position of Problem
ere are various, not equivalent, definitions of fractional derivatives according to Grunwald Letnikov, Weil, Caputo, and Riemann-Liouville, etc. Ordinary and partial differential order equations (with fractional derivatives of Caputo and Riemann-Liouville) have awakened in recent years with considerable interest both in mathematics and in applications. Let us describe the abstract Cauchy problem: where the corresponding solutions are represented through the Mittag-Leffler function. In mathematical papers on fractional differential equations, the Riemann-Liouville approach to the concept of a fractional order derivative c ≥ 0 is usually used as follows: (2) e fractional Riemann-Liouville derivative is the left inverse to the corresponding fractional integral, which is a natural generalization of the Cauchy formula for the antiderivative function u(t). e initial conditions, of the initial value problem for ordinary differential equations of fractional order c with fractional derivatives in the Riemann-Liouville form, are given in terms of fractional integrals: To satisfy the physical requirements, Caputo introduced an alternative definition of the fractional differential derivative. It was adopted by Caputo and Mainardi as (4) e advantage of this definition is a more natural solution for the problem of initial conditions for solving integro-differential equations of noninteger orders. e cases of Caputo derivative for 0 < c < 1 was called the regularized fractional derivative of order c.
is paper concerns the existence with Ulam stability for the following equation: For a continuous function u(t) together with boundary conditions, , the functions f: R + × R ⟶ R and g: R ⟶ R are continuous and θ is a continuous decreasing positive function such that 0 < θ(t) ≤ 1, for all t ∈ [0, +∞). D β 0 + is the standard Riemann-Liouville fractional derivative of order β.

Literature Overview
Fractional differential equations, which are often encountered in mathematical modeling of various processes in natural and technical sciences, play an important role in describing many phenomena in physics, bioengineering, and engineering applications. e properties of such equations were investigated in many reviews (among them, we refer [1][2][3][4][5][6]).
Regarding the existence, we mention the work by Zhao and Ge [7], where the authors used the well-known Leray Schauder nonlinear alternative theorem to prove the existence of positive solutions to the problem where f ∈ C([0, +∞) × R, [0, +∞)), 0 ≤ ξ, β < + ∞. Next, Wang et al. [8] extended the above results and discussed the question of existence for solutions of (7) with condition: where 0 ≤ λ, τ < + ∞. Shen et al. [9] considered the existence of solution for boundary value problem of nonlinear multipoint fractional differential equation: SM Ulam in 1940 was the first to raise the question of stability for functional equations. After his lecture, this question became popular for many specialists in mathematical analysis. It became an area of in-depth research (see for more details [10][11][12]). Next, many mathematicians turned in their studies to two types of stability-according to Ulam-Hyers and according to Ulam-Hyers-Rassias. is kind of study has become one of the central and most important in the fields of fractional differential equations. Details of recent advances in Ulam-Hyers sustainability and according to Ulam-Hyers for differential equations can be found in [13,14] and in articles [15][16][17]. However, as far as we know, most authors discussed Ulam stability of some fractional differential problem on bounded/unbounded intervals, while the present paper discusses the existence of solutions and stability in the sense of Ulam-Hyers-Rassias for nonlinear fractional differential equations boundary conditions, for which research is just beginning, please see [18][19][20][21][22][23].

Preliminaries
Here, we present some notations, definitions, auxiliary lemmas concerning fractional calculus, fixed point theorems, and some preliminary concepts of fractional calculus.
Definition 1 (see [4,24]). e Riemann-Liouville fractional integral of order β for a function f is defined as provided the right side is pointwise defined on (0; +∞).

Lemma 3.
Let us define the following space: equipped with the norm en, clearly (E, ‖ · ‖ E ) is a Banach space.

Lemma 4. u is a solution of the problem (5)-(6) if and only if
u satisfies the following integral equation: where Proof. Using Lemma 2, we have By the first and second conditions, we get Consequently, From the third boundary condition, it follows that On the other hand, we have Mathematical Problems in Engineering 3 en, we deduce By substituting the values of c 1 , c 2 , and c 3 in (18), we get the following integral equation: en, we get (16).
Conversely, suppose that (16) is satisfied. To get (5), we use the following appropriate relationships: e present paper is organized as follows. In Section 4, we prove the existence of the solution for problem (5)-(6) in the Banach space. e generalized Ulam-Hyers stable is stated and proved in Section 5. Finally, an illustrative example is given.

Existence Result
In order to prove the existence of the solution for problem (5)-(6), we transform problem (5)-(6) into the fixed point problem Pu � u, where P is an operator defined on ere exist a nonnegative measurable function ψ 1 defined on [0, +∞) and a real constant L > 0 such that: en, problem (5) Proof. If s ≤ t, and s ≤ ξ, we get H(t, s) All other cases of H(t, s) are simple. is completes the proof of Lemma 5.

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Proof. of eorem 1. We shall use Schauder's fixed point theorem, which is divided into three steps.

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Step 1. Let r > 0 such that If u is a continuous function on J, then Pu ∈ C(J). In order to show P(B r ) ⊂ B r , let u ∈ B r , t ∈ R + . en, Mathematical Problems in Engineering erefore, ‖P‖ E ≤ r; thus, P(B r ) ∈ B r .
Step 2. P: B r ⟶ B r is continuous. Let u n be a sequence which converges to u in B r . en, for all t ∈ [0, +∞), So, we conclude that ‖Pu n − Pu‖ E ⟶ 0 as n ⟶ + ∞. Hence, P is a continuous operator on E.
Step 3. We have two claims to verify that P(B r ) is a relatively compact set.
First claim: let I ⊂ J be a compact interval, t 1 , t 2 ∈ I with t 1 < t 2 . en, for any u ∈ B r , we have |f(s, u(s)) + θ(s)g(u(s))|ds Since it is continuous on J × J, we have that H(t, s)/(1 + t β− 1 ) is a uniformly continuous function on the compact set I × I.
For s ≥ t, the function depends only on t, then it is uniformly continuous on I × (J/I).
erefore, we have ∀s ∈ J and t 1 , t 2 ∈ I; the next property holds.
is property, together with (36) and the fact that means that Pu(t)/(1 + t β− 1 ) is equicontinuous on I.
Second claim: in order to achieve (ii) of Lemma 1, we use 6 Mathematical Problems in Engineering lim t⟶+∞ H(t, s) From (39), it is not hard to see, ∀ε > 0, there exists a dependent constant T � T(ε) > 0 such that, for t 1 , t 2 ≥ T and s ∈ J, we have Now, from (36) and (38), the same property holds for us, Lemma 1 implies that P(B r ) is relatively compact. erefore, the operator P has a fixed point on B r . en, from Schauder's fixed point theorem, we conclude that problem (5)-(6) has at least one solution.
Proof. By the equivalence between the operators (I d − P) and F and the assumptions (A 1 ), (A 2 ), we find

Data Availability
No data were used in this study.