On Ulam Stability of a Generalized Delayed Differential Equation of Fractional Order

We investigate Ulam stability of a general delayed differential equation of a fractional order. We provide formulas showing how to generate the exact solutions of the equation using functions that satisfy it only approximately. Namely, the approximate solution ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} generates the exact solution as a pointwise limit of the sequence Λnϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda ^n\phi $$\end{document} with some integral (possibly, nonlinear) operator Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varLambda $$\end{document}. We estimate the speed of convergence and the distance between those approximate and exact solutions. Moreover, we provide some exemplary calculations, involving the Chebyshev and Bielecki norms and some semigauges, that could help to obtain reasonable outcomes for such estimations in some particular cases. The main tool is the Diaz–Margolis fixed point alternative.


Introduction
The differential equations of a fractional order have numerous applications in the modeling of various physical phenomena and processes in economics, chemistry, aerodynamics, etc. (for further information see [12][13][14][15][16][17][18][19]21]). They are also an excellent tool for the description of hereditary properties of many materials. For details concerning the fractional calculus we refer to [12,15,19,20].
Unfortunately, quite often we only have a description of approximate solutions to such equations and it is very difficult to get any sufficient information on the exact solutions to them. So, the natural question is how big is the difference between those approximate and exact solutions and whether it is possible to generate somehow the exact solutions by the means of those approximate ones. Some convenient tools to study such issues provides the theory of Ulam (often also called the Hyers-Ulam) type stability. It has been motivated by a problem of Ulam, concerning approximate homomorphisms of groups, and an answer to it provided by Hyers [6] (see [3,7,11,22] for more details and references).
The main idea of such stability can be very roughly expressed in the following way: When a function satisfying an equation approximately (in some sense) must be near an exact solution to the equation?
The following definition (cf. [3, p. 119, Ch. 5, Definition 8]) makes that notion a bit more precise (R + stands for the set of all nonnegative reals and C D denotes the family of all functions mapping a set D = ∅ into a set C = ∅). Definition 1. Let A be a nonempty set, (X, d) be a metric space, E ⊂ C ⊂ R + A be nonempty, T be an operator mapping C into R + A and F 1 , F 2 be operators mapping nonempty set D ⊂ X A into X A . We say that the equation is (E, T )-stable provided for any ε ∈ E and ϕ 0 ∈ D with there exists a solution ϕ ∈ D of Eq.
In what follows, R stands for the field of real numbers. Moreover, for every positive integer n, C n (D 1 , D 2 ) always denotes (as usual) the family of all functions from a real interval D 1 into a real interval D 2 , that are n-times continuously differentiable. Analogously, C(D 1 , D 2 ) means the family of all functions from an interval D 1 ⊂ R into an interval D 2 ⊂ R, that are continuous.
for every function η ∈ C n (D, R).
The advantage of the Caputo derivative is that (contrary to the Riemann-Liouville fractional derivative) it does not involve any initial conditions of fractional order while solving differential equations including it.
In this paper, α ∈ (0, 1) and h > 0 are fixed, I ⊂ R is an interval, which has one of the following three forms: are fixed functions satisfying appropriate regularity conditions (specified later).
If y ∈ C(I h , R) and t ∈ I, then we define the function y t ∈ C(H, R) by We consider approximate solutions y ∈ C 1 (I h , R) for the following general delayed fractional differential equation with a fixed function g ∈ C 1 (I h , R). That is we study functions y ∈ C 1 (I h , R) satisfying the inequality with a given function Φ : I → R + satisfying some natural restrictions. Solutions to a particular case of (4) have been investigated in [13] for with a fixed real number γ > 0 and the function f 0 : I × C(H, R) → R, satisfying some regularity conditions. In such a case Eq. (4) is equivalent (cf. [5,13]) to the following integral equation We refer to [13] for results on solutions to (6).
In [1] the authors investigated Ulam's type stability of a simplified version of (4) with the Chebyshev (supremum) norm · ∞ in C(H, R). Namely, the following outcome has been proved in [1, Theorem 2.4] (R 0 + stands for the set of positive reals). Theorem 1. Let τ 0 ∈ R + , κ 0 ∈ (0, 1), functions f 1 : I × C(H, R) → R, λ : I → R + and Φ : I h → R 0 + be continuous, g ∈ C 1 (I h , R) be fixed, and the following three inequalities be valid: where Then there exists a unique function Please note that no regularity conditions on ω in Theorem 1 have been assumed explicitly, but actually from the assumption that y is continuously differentiable and (11) it follows that ω must be continuously differentiable.
In this paper we present a significant generalization of Theorem 1, because we consider Eq. (4), which is much more general than (12). Moreover, we admit a wider range of ways of measuring the distance in C(H, R). Namely, instead of the supremum norm as in (7), we use semigauges depicted in the subsequent Definition 3 (cf., e.g., [2]), which include the cases of various norms, seminorms, quasinorms, semi-quasinorms etc. We provide some suitable examples at the end of this paper.
Furthermore, we provide formulas showing how to generate the exact solutions of the equation from the functions that satisfy it only approximately. Namely, some approximate solutions φ of (4) generate the exact solutions as the pointwise limits of the sequence Λ n φ with an integral operator Λ, given by (26). We estimate the speed of this convergence and the distance between φ and that generated exact solution. In the second part of the paper, we provide some exemplary calculations, involving in particular the Chebyshev and Bielecki norms, that could help to obtain reasonable outcomes for such estimations in several particular cases.
A gauge on A is any semigauge ρ on A such that ρ(x) = 0 for x = 0. Note that if ρ is a semigauge on a real linear space A, x ∈ A and ρ(λx) < ∞ for some λ ∈ R \ {0}, then ρ(ηx) < ∞ for every η ∈ R, because (13) yields ρ(ηx) = |ηλ −1 |ρ(λx). Remark 1. Clearly, the norms and extended norms are gauges. Let us recall that an extended norm, on a real (or complex) linear space X, is a function · : X → [0, +∞] (i.e., · may also take the value +∞) such that, for every scalar α and every x, y ∈ X with x , y ∈ [0, +∞), and the equality x = 0 holds if and only if x is the zero vector.
If Y is a real or complex normed space and S is a nonempty set, then an extended norm in Y S can be defined by: Further, if f is a linear functional on a real or complex linear space X, then the formula: defines a semigauge on X.
Finally, let us recall the Diaz-Margolis fixed point alternative (see [4]), which will be useful in the proof of our main result. To this end we need the following definition.

Definition 4.
An extended metric on a set X = ∅ is a function d : X 2 → [0, +∞] satisfying the following three conditions: In the sequel, given a set X = ∅ and L : X → X, we sometimes write for simplicity Lx := L(x) for x ∈ X. Moreover, as usual, L 0 x := x and L n x := L(L n−1 x) for x ∈ X, n ∈ N (positive integers). The Diaz-Margolis fixed point alternative [4] can be formulated as follows (N 0 := N ∪ {0}).

Theorem 2.
Let d be an extended complete metric on a nonempty set X and L : X → X be contractive with the Lipschitz constant L < 1 (i.e., d(Lx, Ly) ≤ Ld(x, y) for x, y ∈ X with d(x, y) ∈ (0, +∞)). Assume that x ∈ X is such that there exists k ∈ N with d(L k−1 x, L k x) < ∞. Then the sequence (L n x) n∈N converges to a fixed point x * ∈ X of L, x * is the unique fixed point of L in the set X * = {y ∈ X : d(x * , y) < ∞} and Proof. The convergence of L n x to a fixed point x * of L results from [4,Theorem]. The uniqueness of x * follows from the subsequent simple inequality which is true for every fixed point u ∈ X * of L. Also the proof of (14) is a routine, but for the convenience of readers, we present it. So, note that, for each m ∈ N, m ≥ k, Now (14) results from the above inequality and from the fact that, for every n ∈ N 0 with n ≥ k − 1, Remark 2. Assume that k = 1 in Theorem 2. Clearly, in such a case, (14) (with n = 0) implies that d(x, x * ) < +∞, whence x ∈ X * . Let z ∈ X be a fixed point of L with d(x, z) < +∞. Then d(x * , z) ≤ d(x * , x) + d(x, z) < +∞ and consequently z ∈ X * , which means that z = x * , because x * is the unique fixed point of L in X * .

The Main Result
In the sequel, ρ is a semigauge on C(H, R), U ⊂ C(I, R + ) is nonempty and G : U → R I + is given. Next, ξ : I h → I h and f : I × R × C(H, R) → R are such that the function f w : I → R + , given by: is continuous for every w ∈ C(I h , R). Moreover, we assume that there is a nondecreasing sequence (r n ) n∈N in I such that Remark 3. Clearly, if ξ(t) ≤ t for t ∈ I, then any nondecreasing sequence (r n ) n∈N in I, with sup n∈N r n = sup I, fulfils (16). Moreover, if I = [t 0 , b] with some b > t 0 , then (16) holds with the constant sequence r n := b for n ∈ N.
We write I n := [t 0 , r n ], I h n := [t 0 − h, r n ], n ∈ N. Next, g : I → R \ {0} is an arbitrarily fixed function that is continuously differentiable. We define T : C(I, R + ) → C(I, R + ) by We also need the following two hypotheses (with n ∈ N) concerning functions Φ ∈ U.
(H n ) If w ∈ C(I h n , R) is such that |w(s)| ≤ Φ(s) for s ∈ I n and w(s) = 0 for s ∈ H t0 , then ρ(w t ) ≤ GΦ (t) for every t ∈ I n . (H) There is a continuous function L : Examples of functions Φ satisfying the hypotheses, with suitable operators G (cf., e.g., (47)), are given in the further parts of the paper.
Some further examples of semigauges, both nondecreasing and not, can be found in Sect. 6. Now we have all tools to prove the next theorem, which is the main result of this paper.

Theorem 3. Let Φ ∈ U fulfil hypotheses (H)
and (H n ) for all n ∈ N. Let λ : I × R 2 + → R + be such that the functions λ(x, ·, v) and λ(x, v, ·) are nondecreasing for every x ∈ I and v ∈ R + , the function I s → λ s, μΦ(ξ(s)), μ GΦ (s) ∈ R + is locally integrable for each μ ∈ R 0 + , and there is a function K : Assume also that and K is continuous or nondecreasing.
then there is a unique function y ∈ C 1 (I h , R), which fulfils the conditions where K n := sup t∈In K(t) for n ∈ N. Moreover, where, for each w ∈ C(I h , R), Λ : C(I h , R) → C(I h , R) is given by the formula Proof. Let y ∈ C 1 (I h , R) fulfil (21). Fix n ∈ N and denote by X n the space of all continuous functions w : I h n → R such that w(t) = ω(t) for t ∈ H t0 . Define an extended complete metric in X n by The metric may take the infinite value +∞ (i.e., can be an extended metric), because we do not exclude the situation where Φ(t) = 0 for some t ∈ I n and assume that inf ∅ = ∞. The completeness of d n follows from the fact that Φ is continuous on I n and therefore bounded, which means that the convergence in X n , with respect to d n , is actually the uniform convergence on I n (with regard to the natural distance in R).
Let p ∈ (1, ∞) be such that (1 − α)p < 1 and q := p/(p − 1). Then 1/p + 1/q = 1 and p(α − 1) + 1 > 0. Since for every w ∈ C(I h , R) the function f w (defined by (15)) is continuous, the functions are continuous and, by the Hölder inequality, we have is convergent for every t ∈ I. This means that we can define operator Λ n : X n → X n by the formula for each w ∈ X n (see (16)). Take z, w ∈ X n with C zw := d(z, w) < +∞, which means Write and consequently So, (20) and the monotonicity assumptions on λ yield because z(t 0 ) = ω(t 0 ) = w(t 0 ). This means that Since K is continuous or nondecreasing, so K n < 1 and consequently Λ n is contractive on X n . Next, integrating the inequality in (21) from t 0 to t, we get where y n (t) = y(t) for t ∈ I n . That is we have the inequality: d y n , Λ n y n ≤ L n := sup t∈In L(t) < +∞.
Consequently, by Theorem 2 (with k = 1, L = Λ n and L = K n ) and Remark 2, there is a unique fixed point y n ∈ C(I h n , R) of Λ n with d( y n , y n ) < ∞, and d( y n , Λ m n y n ) ≤ Clearly, y n as a fixed point of Λ n is continuously differentiable on the interval [t 0 , t n ). Further, since the uniqueness means in this case that y n (t) = y l (t), n,l ∈ N, l ≤ n, t ∈ I l , we can define y ∈ C(I h , R) by y(t) := y n (t), t ∈ I n , n ∈ N.
It is easily seen that y is continuously differentiable and it is the unique solution to (22) fulfilling because of the uniqueness of y n and the definition of Λ n . Finally, notice that (28) and (29) imply (24).

Remark 5.
Assume that ξ(t) ≤ t for each t ∈ I. Then the points r n can be selected quite arbitrarily and, for a given t ∈ I, we can assume that t = r j with some j ∈ N (see Remark 3). As y in Theorem 3 is unique and its form (given by (24)) in this situation does not depend on the choice of points r n , this means that, for such ξ, (23) can be actually replaced by the following inequality where K(t) := sup s∈[t0,t] K(s) for t ∈ I. Clearly, if K is nondecreasing, then K = K.
Remark 6. Assume that Φ(t) = 0 for each t ∈ I. Then hypothesis (H) holds with L given by

Remark 7.
It can be very difficult to calculate precisely the form of the function T Φ occurring in (23) and (25). Therefore we show below some very easy approaches that might help to obtain some useful estimations of it. Let t ∈ I, m ∈ N, m > 1, and s 1 , . . . , s m ∈ I be such that s 1 := t 0 < s 2 < . . . < s m := t. Write J i := [s i , s i+1 ] for i = 1, . . . , m − 1. Then it is easily seen that where Φ k := sup s∈J k Φ(s) for k = 1, . . . , m − 1. Also, if m > 2, then (32) implies that If Φ is nondecreasing, then Φ k = Φ(s k+1 ) for k = 1, . . . , m − 1 and (34) yields the inequality whence for m = 2 we get the following very simple form while for m = 3 we have which gives the estimation Clearly, we can argue analogously when Φ is nonincreasing, because then Φ k = Φ(s k ). Also, we can choose for estimations (34) and (35) points s k such that Φ is nondecreasing or nonincreasing on each interval J k .