Existence and Ulam stability of solutions for Caputo-Hadamard fractional differential equations

In this paper, we study the existence of solutions for fractional differential equations with the Caputo-Hadamard fractional derivative of order α ∈ ( 1,2 ] . The uniqueness result is proved via Banach’s contraction mapping principle and the existence results are established by using the Schauder’s ﬁxed point theorem. Furthermore, the Ulam-Hyers and Ulam-Hyers-Rassias stability of the proposed equation is employed. Some examples are given to illustrate the results


Introduction
Fractional differential equations are considered as active field of research due to their applications in various areas including dynamics, computer science, and biological sciences. For instance see ( [19,24,31,32,34,37]). The existence theory is one of the most important topics in fractional calculus. Researchers have obtained many results about the existence and uniqueness solutions for the initial and boundary value problems of fractional differential equations in the sense of Riemann-Liouville and Caputo fractional derivatives, see ( [1,3,4,5,6,7,14,16,17,23,26,27,28,29,38]). Recently, several scientists have been interested in Hadamard-type fractional differential equations. The Hadamard fractional derivative is a specific type of fractional derivative assigned to Hadamard in 1892 [11]. This fractional derivative differs from the Riemann-Liouville and Caputo fractional derivatives in the sense that the kernel of the integral contains a logarithmic function of arbitrary exponent. The existence and uniqueness of mild solutions of boundary value problem for Caputo-Hadamard fractional differential equations with integral and anti-periodic conditions investigated by [20]. Rezapour et al. [35] investigated the existence results for solutions of a new class of the fractional boundary value problem in the Caputo-Hadamard settings. Abbas et al. [2] proved some existence results for a class of Caputo-Hadamard fractional differential equations, the results are based on Mönch's fixed point theorem associated with the technique of measure of non-compactness.
Liu et al. [22] dealt with the existence of solutions of the boundary value problems for nonlinear fractional differential equations.
where C D α −1 denote the Caputo fractional derivatives of orders α, with 0 < α < 1, D β 0+ is the Riemann-Liouville fractional derivative with 1 < β 2 and α + β > 2, and b > 0 is a constant real number. Wang et al. [39] employed the upper and lower solution method along with the fixed point theorem (FPT) of a cone to investigate the existence and uniqueness of a positive solution for Recently, more researchers are interested in applying the Ulam-Hyers stability see ( [8,9,10,15,40]). Murad et al. [30] studied the existence, Ulam-Hyers and Ulam-Hyers-Rassias theorems of solutions to a differential equation of mixed Caputo-Riemann fractional derivatives. Muniyappan and Rajan [25] discussed Ulam-Hyers and Ulam-Hyers-Rassias stability for the fractional differential equation with boundary condition D α y(t) = f(t, y(t)), 0 < α 1, where D α is Caputo fractional derivative of order α. Liu et al. [21] researched the stability of generalized Liouville-Caputo fractional differential equations in Ulam-Hyers sense. Patil et al. [33] concerned the existence and uniqueness of positive solutions to the fractional differential equation.
C D α 0+ y(t) + f(t, y(t)) = 0, 0 < x < 1, with nonlocal integral boundary conditions where 1 < α 2, C D α 0+ is the Caputo fractional derivative of order α, f : [0, 1] × R + −→ R + , and g : Motivated by the above work and the researches going on in this direction, in this paper, we study existence and uniqueness of solution of Caputo-Hadamard fractional differential equation.
with the boundary condition y(1) = y (1), where CH D α 1+ Caputo-Hadamard fractional derivative, with 1 < α 2 and f : [1, e] × R → R is continuous function. We apply Schauder and Banach-fixed-point theorems to obtain the existence and the uniqueness of solution for the (1.1)-(1.2) under certain hypotheses. Furthermore, some stability theorems such as Ulam-Hyers and Ulam-Hyers-Rassias stability are proved. A few examples are presented as an application to illustrate the main results.

Definition 2.3. ([19])
The Hadamard fractional integral of order α ∈ R for a continuous function f is defined as The Hadamard derivative of fractional order α ∈ R for a continuous function f is defined as where n − 1 < α < n ,n = [a] + 1 where [α] denotes the integer part of the real number α Lemma 2.5. [18] Let α > 0, and n = [α] + 1. If y ∈ AC n δ [a, b], then the differential equation CH D α y(t) = 0, has solutions and the following formula holds: Definition 2.6. [36] The equation (1.1) is Ulam-Hyers stable if there exists a real number c f > 0 such that for each ε > 0 and for each solution z ∈ C 1 (J, R) of the inequality Definition 2.7. [36] The equation (1.1) is Ulam-Hyers-Rassias stable with respect to ϕ ∈ C 1 (J, R + ) if there exists a real number c f > 0 such that for each ε > 0 and for each solution z ∈ C(J, R) of the inequality there exists a solution y ∈ C 1 (J, R) of equation (1.1) with holds, where u 0 is a non-negative constant. Then Proof. Applying Lemma 2.5, we can reduce the problem (1.1)-(1.2) to an equivalent integral equation to find c 0 and c 1 , from the first boundary condition y(1) = y (1), we get f(s, y(s)) ds s , by using the condition y(e) = e 1 y(t) dt t , the result is by using Fubini's theorem, the following is obtained f(s, y(s)) ds s , this implies that f(s, y(s)) ds s , and this complete the poof.

Existence and Uniqueness of Solutions
Let C = C(J, R) denote the Banach space of all continuous functions from J to R, with the norm defined by y = sup{|y(t)| , t ∈ J}.
(H2) There exists constant k > 0, such that First result is based on Banach contraction principle. For the sake of convenience, we set the notation: Proof. Define the operator B : C(J, R) → C(J, R) by , and show that BG r ⊂ G r , when G r = {y ∈ C(J, R) : ||y|| r}, and M = sup |f(t, 0)|, for t ∈ G r , we have |f(s, y(s))| ds s , Therefore, BG r ⊂ G r . Now to that B is a contraction mapping, let y 1 , y 2 ∈ G r and for each t ∈ J, we obtain |f(s, y 1 (s)) − f(s, y 2 (s))| ds s , 2(log(e) + 1)   Therefore, it follows from condition λk 1, that B is a contraction operator. Thus we conclude by Banach contraction mapping principle that operator B has a unique fixed point, which is the unique solution to the problem (1.1) and (1.2).
Next, the result is based on Schauder's fixed point theorem. Proof. : This proof will be presented in four steps.
Step 1: We will show that B is continuous. Assume that {y n } be a sequence such that y n → y in C(J, R), then for each t ∈ J we have |f(s, y n (s)) − f(s, y(s))| ds s , Then according to Lebesgue dominated convergence theorem this implies that ||B(y n )(t) − B(y)(t)|| ∞ → 0 as n → ∞.
Step2: B maps the bounded sets into the bounded sets in C(J, R). For any d > 0, such that H d = {y ∈ C : ||y|| ∞ d}. It is obvious that H d is closed, convex subset of C(J, R). Suppose that y ∈ C then for each t ∈ J, we have  (log e s ) α Γ (α + 1) Thus, ||B(y)(t)|| ∞ L, for some constant L.
Step 3: B maps C(J, R) into an equicontinuous set of C(J, R).
Let y ∈ C(J, R) and t 1 , t 2 ∈ J with t 1 < t 2 , then |f(s, y(s))| ds s |f(s, y(s))| ds s , As t 1 → t 2 , the right-hand side of the above inequality tends to zero.
Step 4: Now, we should show that B is prior bound. Let u = {y ∈ C(J, R) : y = ϕBy, for some 0 < ϕ < 1}, we need to show that the set u is bounded. Let y ∈ u and for each t ∈ J, we have |f(s, y(s))| ds s , which implies that u is a bounded set. By Schauder's fixed point theorem, B must have at least one fixed point which is a solution of (1.1) and (1.2).

Stability Theorems
In the following theorems, we will prove the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for equation (  Proof. For ε > 0, and each solution w ∈ C(J, R) of the inequality Let y ∈ C(J, R) be the unique solution of boundary value problem (1.1)-(1.2). Then y(t) is given by f(s, y(s)) ds s .

Example 5.2.
Consider the following problem of Caputo-Hadamard fractional differential equation

Conclusion
In this paper, we studied existence and uniqueness of solutions of Caputo-Hadamard fractional differential equations with boundary conditions. Our results are based on some classical fixed point theorems such as Banach contraction mapping principle and Schauder fixed point theorems. At last, we have presented two examples for the illustration of main results.