Topological degree theory and Caputo–Hadamard fractional boundary value problems

We study two hybrid and non-hybrid fractional boundary value problems via the Caputo–Hadamard type derivatives. We seek the existence criteria for these two problems separately. By utilizing the generalized Dhage’s theorem, we derive desired results for an integral structure of solutions for the hybrid problems. Also by considering the special case as a non-hybrid boundary value problem (BVP), we establish other results based on the existing tools in the topological degree theory. In the end of the article, we examine our theoretical results by presenting some numerical examples to show the applicability of the analytical findings.


Introduction
The fractional calculus has always been one of the most widely used branches of mathematics in other applied and computational sciences. This degree of importance is due to the high flexibility of the tools and operators defined in this theory. On this basis, researchers have been using various powerful fractional operators in recent decades to model different types of existing natural processes in the world. In the meantime, because modeling based on fractional operators yields more accurate numerical results than modeling based on integer order operators, different generalizations of these fractional operators have been introduced by numerous mathematicians.
The fractional operators utilized in models of the current paper are the Hadamard and Caputo-Hadamard integration and differentiation operators, respectively. In this regard, one can point to some papers based on these operators; see, for example, [29][30][31][32]. In more recent decades, the attention of researchers has been focused on designing newer fractional hybrid BVPs subject to hybrid or non-hybrid conditions. For more details, see [33][34][35][36][37][38][39]. More precisely, this novel aspect of fractional modeling initiated with a research manuscript proposed by Dhage and Lakshmikantham in 2010 (see [40]). They turned to a new family of differential equation entitled hybrid differential equation and then established some useful existence criteria of extremal solutions by utilizing some basic inequalities [40]. Two years later, Zhao et al. extended their work to fractional type models and formulated a BVP relying on fractional hybrid differential equations [41]. Later, Ullah et al. continued this process and employed a new structure of hybrid fractional modeling in which both boundary conditions are presented in the hybrid framework by follows:

Preliminaries
First, some important and necessary preliminaries on the fractional calculus are recalled in this section. Assume that κ * ≥ 0. The Hadamard fractional integral of y ∈ C R ([a, b]) of order κ * is given by H I 0 a + (y(t)) = y(t) and H I κ * a + (y(t)) = 1 t a (ln t ) (κ * -1) y( ) d whenever the RHS-integral has finite value [49,50]. Note that for each κ * 1 , κ * 2 ∈ R + , we have [50]. It is obvious that H I κ * 1 a + 1 = 1 Γ (κ * 1 +1) (ln t a ) κ * 1 for any t > a by setting κ * 2 = 0 [50]. Now, let n = [κ * ] + 1. The Hadamard fractional derivative of order κ * for a function y : (a, b) → R is introduced by H D κ * a + (y(t)) = 1 Γ (n-κ * ) (t dt t ) n t a (ln t ) (n-κ * -1) y( ) d provided that the RHSintegral has finite value [49,50]. The Caputo-Hadamard fractional derivative of order κ * for y ∈ AC n R ([a, b]) is represented by whenever the RHS-integral has finite value [49,50]. Now assume that y ∈ AC n R ([a, b]) and n -1 < κ * ≤ n. In the monograph [50], it is verified that the general solution of the Caputo-Hadamard differential equation CH D κ * a + (y(t)) = 0 is obtained of the form y(t) = n-1 j=0 m * j (ln t a ) j , and so we have t a 2 + · · · + m * n-1 ln t a n-1 for any t > a. In the following, we review some notions and results on the topological degree theory which are useful throughout the paper. Let B represent the collection of all bounded sets in a Banach space X . The Kuratowski's measure of noncompactness μ : B → R + is defined by μ(B) := inf{ > 0 : B = n j=1 B j and diam(B i ) ≤ for j = 1, . . . , n}, where diam(B j ) = sup{|yy | : y, y ∈ B j } and B is a bounded element of B. It is evident that 0 ≤ μ(B) ≤ diam(B) < +∞ [51,52].
Note that conditions (a5) and (a6) mean that μ is a seminorm. Let B ∈ B be a bounded subset of a Banach space X . We say that a continuous bounded map Φ : The following theorem due to Dhage is utilized for our result related to the mixed Caputo-Hadamard hybrid BVP (1)-(2).

Theorem 5 ([54]
) Let X be a Banach algebra and B be a convex bounded closed nonempty subset of X . Moreover, suppose that three operators Φ 1 , Φ 2 : X → X and Φ 3 : B → X satisfy the following three assumptions: The following theorem due to Isaia is utilized for our result related to the mixed Caputo-Hadamard non-hybrid BVP (3).

Theorem 6 ([53])
Let Φ : X → X be a μ-condensing operator on the Banach space X and assume that Moreover, Φ has at least one fixed point, and the family of all fixed points of Φ belongs to V ρ (0).

Main results
Now, we are ready to derive the desired analytical findings. For this reason, we build a new space as X = {y(t) : y(t) ∈ C R ([1, e])} supplemented with the sup-norm y X = sup t∈ [1,e] |y(t)| and the multiplication action on X by (y · y )(t) = y(t)y (t) for each y, y ∈ X . Then it is easily verified that an ordered triple (X , · X , ·) is a Banach algebra. In the following lemma, we derive an integral structure for the solution of the hybrid BVP (1)-(2).

Lemma 7 Let g ∈ X . Then a functionỹ 0 is a solution for the Caputo-Hadamard hybrid equation
furnished with mixed Hadamard integral hybrid boundary value conditions ]| t=e (5) if and only ifỹ 0 is a solution for the Hadamard integral equation where Proof As a first step, we assume thatỹ 0 is a solution for the hybrid differential equation (4). Then, by properties of the κ * th order Hadamard integral, we seek constants m * 0 , m * 1 ∈ R such that and sõ Thus, In the light of both mixed hybrid boundary conditions given in (5), we obtain By inserting the obtained values m * 0 and m * 1 into (8), we reach The last equation implies thatỹ 0 satisfies the Hadamard integral equation (6), and soỹ 0 is the solution of the mentioned integral equation. In the opposite direction, we can easily confirm thatỹ 0 is a solution for the two-point Caputo-Hadamard hybrid BVP (4)-(5) if y 0 is supposed to be a solution for the Hadamard integral equation (6). This completes the proof.
Now, based on the obtained Hadamard integral equation in the above lemma, we provide an existence criterion for solutions of the mixed Caputo-Hadamard hybrid BVP (1)- (2).

Lemma 9 Let g ∈ X . Then a functionỹ 0 is a solution for two-point Caputo-Hadamard fractional differential equation with mixed Hadamard integral boundary conditions
if and only ifỹ 0 is a solution of the Hadamard fractional integral equation where Q * is given by (7).
Proof The proof is similar to that of Lemma 7 and so is omitted.
We define an operator Φ : X → X by Φy(t) = Φ 1 y(t) + Φ 2 y(t) which splits into two operators Φ 1 : X → X and Φ 2 : X → X as follows: for each y ∈ X and t ∈ [1, e]. In this case, the equivalence of the existence of a solution for Caputo-Hadamard BVP (3) and the existence of a fixed point for operator Φ is obvious. Note that in all the following lemmas, we assume that X is a Banach space with sup-norm · X and two operators Φ 1 and Φ 2 are defined as in (10) and (11).
Proof By assumption, we know thatΥ is continuous on [1, e] × X , and so we conclude that lim n→∞Υ (t, y n ) =Υ (t, y). By invoking the Lebesgue's dominated convergence theorem, we obtain for any t ∈ [1, e]. Hence, we see that Φ 2 y n → Φ 2 y as n → ∞, and so Φ 2 is continuous on V ρ (0). Now, for the sake of the investigation of the growth condition on Φ 2 , we utilize hypothesis (HP6) and obtain which is the desired conclusion.
Proof Consider a bounded subset B ⊂ V ρ (0) in X and take a sequence {y n } belonging to B. Then, by Lemma 11, we have Φ 2 (y n ) X ≤ 1 y n X + 2 < ∞ for each y n ∈ B which yields that Φ 2 (B) is a bounded set. Besides, we verify that {Φ 2 (y n )} is equicontinuous for each y n ∈ B. Take t 1 , t 2 ∈ [1, e] so that t 1 < t 2 . Then, we obtain Evidently, it is seen that the RHS of the inequality approaches 0 (regardless of the choice of y n ∈ B) whenever t 1 → t 2 . Thus, letting t 1 → t 2 , we get |Φ 2 (y n )(t 2 ) -Φ 2 (y n )(t 1 )| → 0 and so {Φ 2 (y n )} is equicontinuous. Taking into account the Arzela-Ascoli theorem, we obtain that Φ 2 (B) is compact. In addition, in view of Proposition 3, Φ 2 is μ-Lipschitz with constant zero.
In this step, it is enough to show that B is a bounded subset of X . For this, select y ∈ B. Then in the light of the growth conditions obtained in Lemmas 10 and 9, we may write implying that the set B is bounded in X . Thus there is a number ρ > 0 such that B ⊂ V ρ (0), and so we have deg(I -λΦ, V ρ (0), 0) = 1, by applying Theorem 6. Finally, under the hypotheses of Theorem 6 due to Isaia, the operator Φ = Φ 1 + Φ 2 has at least one fixed point and the family of fixed points of Φ is bounded in X . This means that the mixed Caputo-Hadamard nonhybrid BVP (3) has at least one solution on [1, e] and the family of solutions is bounded. The proof is finished.
Eventually, we derive a uniqueness criterion for the mixed Caputo-Hadamard nonhybrid BVP (3) in the following theorem.
Proof Let y ∈ X be arbitrary. By Lemma 10 and assumption (HP5), we obtain where Φ 1 : X → X is defined in (10). Furthermore, we have the following estimate: where Φ 2 : X → X is defined in (11). From (12) and (13), we have which yields that Φ = Φ 1 + Φ 2 : X → X is a contraction. By utilizing the Banach contraction principle, it is deduced that the mixed Caputo-Hadamard nonhybrid BVP (3) has a unique solution.
Then y R ≤ 1 2 Φy R ≤ 1 implies that B is a bounded set and so, by Theorem 13, it is deduced that the mixed Caputo-Hadamard nonhybrid BVP (16) has at least one solution y in C R ( [1, e]). In addition,

Conclusion
The fractional calculus has always been one of the most widely used branches of mathematics in other applied and computational sciences. This degree of importance is due to the high flexibility of the tools and operators defined in this theory. On this basis, researchers have been using various powerful fractional operators in recent decades to model different types of existing natural processes in the world. In the current research article, two hybrid and nonhybrid fractional BVPs of Caputo-Hadamard type are addressed. We seek the existence criteria for these two problems separately. We first utilize the generalized Dhage's theorem to derive desired results for an integral structure of solutions for the proposed hybrid BVP (1)- (2). Next, we establish other results for nonhybrid BVP (3) based on some existing notions in the topological degree theory. At the end of the paper, we examine our theoretical results by presenting some numerical examples to show the applicability of the analytical findings.