Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study

: Boundary value problems are very applicable problems for different types of differential equations and stability of solutions, which are an important qualitative question in the theory of differential equations. There are various types of stability, one of which is the so called Ulam-type stability, and it is a special type of data dependence of solutions of differential equations. For boundary value problems, this type of stability requires some additional understanding, and, in connection with this, we discuss the Ulam-Hyers stability for different types of differential equations, such as ordinary differential equations and generalized proportional Caputo fractional differential equations. To propose an appropriate idea of Ulam-type stability, we consider a boundary condition with a parameter, and the value of the parameter depends on the chosen arbitrary solution of the corresponding differential inequality. Several examples are given to illustrate the theoretical considerations.


Introduction
Ulam stability is an important problem investigated in differential equations, which include fractional derivatives.It has applications in optimization, biology, economics, etc., and it is a special type of data dependence of solutions (see, for example, [1][2][3][4][5]).Usually, one first proves the existence and uniqueness of the problem and then one considers a corresponding differential inequality so, in the definition of Ulam-type stability, one assumes that, for any solution of the corresponding differential inequality, there exists a solution of the given problem, such that both solutions are close enough.In the case of an initial value problem (IVP), usually one chooses the initial value of the differential equation, which will depend on an arbitrary chosen solution of the corresponding differential inequality.For example, Ulam-type stability for IVPs was studied in [6], the Caputo fractional differential equation (FDE) was studied in [7], the Darboux problem for a partial FDE was studied in [8], generalized fractional derivatives were studied in [9], fractional Volterra-type integral equations with delay were studied in [10], non-linear delay differential equations with fractional integrable impulses were studied in [11], delay differential equations were studied in [12,13], Caputo FDEs were studied in [14], Caputo FDEs with impulses were studied in [15], Riemann-Liouville FDEs were studied in [16], Riemann-Liouville FDEs with delay were studied in [17], Caputo FDEs with delays were studied in [18], Hadamard FDEs were studied in [19], Hilfer-Katugampola FDEs with impulses were studied in [20], and sequential FDEs were studied in [21] for first-order impulsive fuzzy differential equations.
In the case of boundary value problems (BVP), the situation is more complicated.There are mainly two different types of boundary conditions.One type is with a parameter, and the other type is without a parameter (see, for example [22,23]), and, in this paper, we propose an appropriate idea for Ulam-type stability.To motivate the idea, we study linear boundary value conditions.First, we consider well known ordinary differential equations, and we will illustrate the above ideas on a simple example, and we will set up the boundary condition and the definition of Ulam-Hyers stability.Then, we study, in detail, a linear BVP for nonlinear differential equations with generalized proportional Caputo fractional derivatives(GPFDE).The generalized proportional Caputo fractional derivative was recently introduced (see, [24,25]), and it provides wider possibilities for modeling more complexity of real world problems, and this type of derivative is a generalization of the Caputo fractional derivative.We consider the case when a parameter is involved in the boundary condition and an integral representation of the solution of the studied linear BVP for GPFDE is presented.The existence and uniqueness of the solution of the BVP for GPFDE for any value of the parameter is also studied.In an appropriate way, the Ulam-Hyers stability is defined, and some sufficient conditions are obtained.The main idea is to choose an arbitrary solution of the corresponding differential inequality with a generalized proportional Caputo fractional derivative and then to use a parameter depending on this solution in the given boundary condition.As a special case, we provide some results for linear BVPs for Caputo fractional differential equations.Our theoretical results are illustrated with examples.

Ordinary Differential Equations and Ulam-Hyers Stability
Without loss of generality, we assume the initial time point is 0. Let 0T ≤ ∞.Consider the scalar ordinary differential equation (ODE): where f ∈ C([0, T] × R, R).
Assume that, for any initial value y 0 , the IVP for the ODE (1), ( 5) has a solution on [0, T].
Let us recall the definition for Ulam-Hyers stability.
In the case of the initial value condition, we introduce the following definition: Definition 2. The solution y ∈ C 1 [0, T] of (1), ( 5) with an initial given value x 0 is called modified Ulam-Hyers stable (MUHS) if there exists a constant C f > 0, such that, for each ε > 0 and for each solution Remark 2. If we consider Definition 2 for all initial values x 0 , then one gives a description of Definition 1.
We will use the following integral operator and the integral operator The solution y ∈ C 1 [0, T] of (1), ( 5) is a fixed point of the operator Ω and vice versa.The solution ν ∈ C 1 [0, T] of the ODE (3) is a fixed point of Ω 1 and vice versa.Case 2. Boundary value problem (BVP) for ODE (1).Consider the boundary condition where a, b ∈ R : a + b = 0, µ ∈ R.
and the operator For any value of µ ∈ R the solution y ∈ C 1 [0, T] of BVP (1), ( 8) is a fixed point of the operator Ω and vice versa.The solution ν ∈ C 1 [0, T] of the ODE (3) is a fixed point of the operator Ω 1 iff µ = aν(0) + bν(T) and vice versa.
Let ε = 1.Choose the solution ν(t) = 2e −t of (12) satisfying the initial condition ν(0 Now, let x 0 = −5.Then, the IVP (11), (13) Summarizing, if we have a particular initial value, then we consider a particular solution of the corresponding IVP, and we study MUHS.If we have an arbitrary initial value, then we study UHS.
Consider µ ∈ R as a parameter.Then, the BVP (11), ( 14) has a unique solution x(t) = µe 1.5t   for any given value of the parameter µ.
Choose an arbitrary solution of the differential inequality (12) Summarizing, if we have a particular boundary condition (initially given the value µ), then we consider a particular solution of the corresponding BVP, and we study MUHS.If we have a boundary condition with a parameter (arbitrary value of µ), then we study UHS.
Remark 4. In connection with the above example and the above discussion to study UHS of any type of differential equation, we need a boundary condition with a parameter.Otherwise, we could only study MUHS of a particular solution.

Preliminary Results from Fractional Calculus
Fractional differential equations arise in many investigations in recent years, and they are widely used in dynamical models with chaotic dynamical behavior, quasi-chaotic dynamical systems, the dynamics of complex material or porous media, and random walks with memory (for some physical background and numerical algorithms of fractional partial differential equations, see Chapter 1 [26] and the recent book [27], for example).Recently, there are several types of fractional derivatives defined and studied in the literature.In our paper, we will use the generalized proportional Caputo fractional derivative, which is a generalization of the Caputo fractional derivative (see, [24,25]).
Let ν : [0, T] → R, (T ≤ ∞).The generalized proportional fractional integral is defined by (as long as all integrals are well defined, see [24,25]) and the generalized Caputo proportional fractional derivative is defined by (see, [24,25]) We consider the following classes of functions: Remark 6. Note, in this paper, the space C α,ρ [0, T] is not the Hölder space in the literature.
Assume the following holds: Consider the fractional integral operator Ω : For any type of fractional differential equation, an appropriate integral operator is defined, and its fixed point is called a mild solution (see, for example, Definition 3.1 [28] for fractional neutral evolution equations, [29] for fractional evolution equation, and [30] for Caputo-Hadamard fractional differential equations).We use the ideas in the above mentioned papers to define a mild solution of ( 18), (19).
Definition 5.The fixed point x ∈ C([0, T]) of the fractional integral operator Ω, defined by (20) (if any), is called a mild solution of the IVP for GPFDE (18), (19).19) is a mild solution and vice versa.
In the case of an initial condition (19), we introduce the following definition: Definition 7. The solution x ∈ C α,ρ [0, T] of (18), (19), with an initial given value x 0 , is called modified Ulam-Hyers stable (MUHS) if there exists a constant C f > 0, such that, for each ε > 0, and for each solution ν ∈ C α,ρ [0, T], ν(0) = x 0 , of the PFDI (22), the inequality Remark 8.If we consider Definition 7 for all initial values x 0 , then one gives a description of Definition 6.

Existence and Uniqueness of the Solution of Boundary Value Problem
Consider the following linear generalized proportional Caputo fractional differential equation with the boundary condition where a, b ∈ R : a + be Note that, in the case b = 0 in (26), we obtain an initial condition.We introduce the assumption: (B).The condition a + be ρ−1 ρ T = 0 holds with α ∈ (0, 1), ρ ∈ (0, 1]. Lemma 5. Let F ∈ I α,ρ [0, T] and condition (B) be satisfied.
Then, for any value of the parameter µ ∈ R, the BVP (18), ( 25) has a unique solution: The  Proof.From (28), it follows that x(t) satisfies boundary condition (26).Take the derivative C 0 D α,ρ on both sides of (28), use Corollaries 1 and 2, and obtain Theorem 4. Let condition (B) be satisfied and let the function x ∈ C α,ρ [0, T] be a solution of BVP for GPFDE (18), (26) for a given value of the parameter µ.If the function F ∈ I α,ρ [0, T] where F(t) = f (t, x(t)), then the function x(t) is a mild solution of the same problem.
Corollary 4. If conditions (A) and (B) are satisfied, then any solution x ∈ C α,ρ [0, T] of ( 18), ( 26) for a given value of the parameter µ is a mild solution and vice versa.
We now consider the existence and uniqueness of the solution of the BVP for GPFDE (18), (26) for any value of the parameter µ.Theorem 5. Let the following conditions be fulfilled: 1.The condition (B) is satisfied.2. The condition (A) is satisfied and there exists a constant L > 0, such that, for t ∈ [0, T], x j ∈ R, j = 1, 2, the inequality

The inequality LT
Then, for any given value of the parameter µ, the BVP for GPFDE (18), ( 26) has a unique mild solution.
Remark 10.According to Corollary 4, if the conditions of Theorem 5 are satisfied, then the BVP for GPFDE (18), (26) for a given value of the parameter µ has a unique solution x(t), such that where Ω is defined by (28).

Ulam Type Stability for Boundary Value Problems
Consider the boundary condition (26).This condition depends on µ, which could be a given value, or it could be a parameter.According to Remark 10, the BVP ( 18), (26) has a solution x(t) in both cases.
In the case of an initial given value of µ ∈ R in the boundary condition (26), following the ideas of Definition 7, we could introduce the following definition.
Proof.Let ε > 0 be an arbitrary number and let the function ν ∈ C α,ρ [0, T] be a solution of inequality (22).According to Lemma 6, there exists a function g ∈ I α,ρ [0, T], such that the function ν is a fixed point of the operator Ω 1 defined by (35).Let µ = aν(0) + bν(T) in the boundary conditions (26).According to Theorem 5 and Remark 10, the BVP for GPFDE (18), (26) has a solution x ∈ C α,ρ [0, T] for this particular value of the parameter µ, and it is a fixed point of the operator Ω, defined by (28).
The proof of Theorem 7 is similar to the one in Theorem 6, so we omit it.

Examples
Since Caputo fractional differential equations are studied by many authors, we will start with a simple example with the Caputo fractional derivative (ρ = 1).
For an arbitrary ε > 0, we choose a solution of (40), which satisfies the initial condition (41) with x 0 = 0, for example ν(t) = εt 2 .It satisfies (40 Summarizing, if we have a particular initial value, then we consider a particular solution of the corresponding IVP, and we study MUHS.If we have an arbitrary initial value, then we study UHS. Case 2. We will consider the linear Caputo fractional differential equation (α ∈ (0, 1)) (39) with a boundary condition.
Consider the boundary condition where µ is an arbitrary constant.
Summarizing, the type of boundary condition is very important for Ulam-type stability.If it has a parameter, one might be able to select this parameter, so that PFDI (40) is satisfied, and the BVP has some sort of Ulam-type stability.
In connection with the above example and the above discussion to study UHS of any type of differential equations, we need a boundary condition with a parameter.
In the case when the initially given boundary value problem does not have a parameter, we could only study MUHS of a particular solution.In this case, we have to use only solutions of the corresponding differential inequality, which are satisfying the given boundary condition (see, for example, the given BVPs without any parameter, the definitions of Ulam type stability and the study of the stability of the unique solution in [31] for Riemann-Liouville FDEs, Lemma 4.1 in [32] for nonlinear coupled systems of Riemann-Liouville fractional differential equations, for implicit Caputo FDEs, and [33,34] for Caputo FDEs).Remark 13.The above definition and ideas for Ulam-Hyers stability could be easily used as a basis for definitions of other types of Ulam stability, such as Ulam-Hyers-Rassias stability, generalized Ulam-Hyers-Rassias stability, and for other types of boundary conditions for differential equations with different types of derivatives.

Conclusions
A linear boundary value problem for a generalized proportional Caputo fractional differential equation is considered.The boundary condition depends on a parameter.The existence and uniqueness of the solution depending on this parameter are discussed, and the Ulam-Hyers stability is defined and discussed.Several examples are given to illustrate the theory.The obtained results and the ideas in this paper could be applied to study other types of Ulam stability for various types of fractional differential equations with boundary conditions, i.e., the proposed ideas could be applied to study Ulam-type stability for various types of boundary value problems for other types of fractional equations, such as fractional differential equations of order α ∈ (1, 2), integro-differential fractional equations, and partial differential equations with time fractional derivatives.

) Definition 8 .Remark 9 .Theorem 3 .
The solution (if any) u ∈ C([0, T]) of the fractional integral operator Ω, defined by(28), is called a mild solution of the BVP for GPFDE(18), (26).Both the mild solution and the solution of BVP for GPFDE(18), (26) depend on the parameter µ.Let condition (B) be satisfied and the function x ∈ C([0, T]) be a mild solution of the BVP for GPFDE (18), (26) for a given value of the parameter µ.If x ∈ C α,ρ [0, T], then the function x(t) is a solution of the same problem.
is a mild solution of (23) and vice versa.