Existence criteria for fractional di ﬀ erential equations using the topological degree method

: In this work, we analyze the fractional order by using the Caputo-Hadamard fractional derivative under the Robin boundary condition. The topological degree method combined with the ﬁxed point methodology produces the desired results. Finally to show how the key ﬁndings may be utilized, applications are presented


Introduction
Fractional calculus is a field of mathematics that expands the notion of differentiation and integration beyond integer orders. These operations are applicable to all real numbers, including non-integer values. Fractional calculus has found practical applications in a variety of physical systems, as can be seen in [2,16,17,20,30,40,41] and some classic books [22,27,29,39].
Based on the literature, fixed point theory has been applied for many years to establish that differential equations have a solution [26,31,35,46,48,49]. Mahwin [32] in their paper made use of the topological degree theory (TDT) to solve integral equations for the first time. Isaia [24] theoretically applied TDT to analyze some integral equations. Use of TDT can also be observed in [14,42,43,47].
To date, a lot of good work with integro-differential equations has been conducted, including the studies described in [7,34]. Zuo et al. [33] derived the following fractional integro-differential equations with impulsive and antiperiodic boundary conditions: D γ ζ(q) + λζ(q) = f (q, ζ(q), Pζ(q), S ζ(q)), q ∈ J δζ(q i ) = I i (ζ i ), i = 1, 2, . . . , m ζ(0) = −ζ (1) where J = J \ {q 1 , q 2 , . . . , q m }, 0 < γ ≤ 1, λ > 0 and D γ is denoted as CFD, 1 < γ ≤ 2. Here f ∈ C(J × R × R × R, R), J = [0, 1] is the integro-differential function and P and S are linear operators: (Pζ)(q) = It is observed from the literature that for last many years, fixed point theory has been used to prove the existence of a solution to the differential equations. However, the use of fixed point theory requires strong conditions, which severely limits its applicability. Also the uniqueness is proved via the Banach contraction principle, which is applied to find a unique solution for the defined problem. It is noticed that most of the work on the topic of fractional differential equations (FDEs) involves either the RL or CFD. While these derivatives are common place in the study of FDEs, the Hadamard fractional derivative (HFD) is another kind of fractional derivatives. This kind of derivative was introduced by Hadamard [21]. This fractional derivative differs from the other ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains the logarithmic function of an arbitrary exponent. In [25], we see the modification of the HFD into a more suitable one called the Caputo-Hadamard fractional derivative (CHFD). Applications of where Hadamard derivative and the Hadamard derivative integral can be found in papers by Butzer et al. [11][12][13]. Other important results dealing with studies on fractional calculus using Hadamard derivatives can be seen in [4,6,8,9,18,19,23,[36][37][38]45].
A paper by Jarad et al. [25], deals with the CHFD by modifying the to be of caputo type. This is familiar with different kinds of boundary conditions like the Neumann boundary condition and the Dirichlet boundary condition. A weighed combination of these boundary conditions is called the Robin boundary condition. It finds its applications in fields such as physics. The Caputo-Hadamard (CH) derivative type of FDEs with boundary value problems are described in [1,3,5,10].
TDT-based existence results for FIDEs with CH-derivatives have the following form: where C H D ν , C H D γ are CH-derivatives of order ν, γ, respectively with 1 < ν ≤ 2, 0 < γ ≤ 1, HD integral of order H I ν i , ν i , i ∈ [1, 2] and f ∈ C(J × R × R × R, R) is the continuous function; P and S are linear operators; In this paper, we determine the existence results via TDT in Section 2. Additionally, we discuss the FIDEs existence results under boundary conditions. An appropriate illustration and conclusion are provided in Sections 4 and 5.

Facts
Here, we shall establish basic results and definitions for our analysis. We shall refer to the notations and results from [15]. Let the Banach space (BS) be X and B ⊂ P(X) be bounded subsets. Let B be a compact set and set B of a space X is compact if and only if it is complete and totally bounded. The value of is the measured value, and the value of (B) is 0. The set is compact, when the value is 0. The larger the value of , the less it is like a compact set.
Definition 2.4. [14,15,24] Let a C ( ) be a class of all -condensing maps F : Therefore, T has a fixed point.

Main results
We shall define some hypotheses: (A2) ∃ constants α, β, χ such that We define k max = sup q∈J q 0 |k(q, s)|ds and h max = sup q∈J 1 0 |h(q, s)|ds. Thus using (A2), Now we prove the existence result: Let h be a continuous function on J; then, we have the following FIDE: have a unique solution given by where, and χ is given by (1.2).
Proof. By Lemma 2.11, Eq (3.1) becomes, By using boundary conditions, we have Solving for k 0 , k 1 we get the following solutions: ) and, . Substituting for k 0 and k 1 we get (3.2).
Let T : Ψ → Ψ given that T = T 1 + T 2 + T 3 . Thus the problem is reduced to finding the fixed points of the operator T .
Proof. Let ζ n be a sequence in Ψ that converges to ζ ∈ Ψ. Let g ζ by continuous; it follows that g ζ n → g ζ . So, T 2 is continuous according to the Lebesgue dominated convergence theorem (LDCT). Hence, Theorem 3.4. Assume that the operator T 3 is continuous and the following growth relation is satisfied: Proof. Let ζ n be a sequence in Ψ that converges to ζ ∈ Ψ. Since g ζ is continuous, it follows that g ζ n → g ζ . Thus by the LDCT, it follows that T 3 is continuous. Hence, Theorem 3.5. Suppose that T 2 is a compact map implying that T 2 is Lipschitz constant zero.
Theorem 3.6. If T 3 is a compact then T 3 is Lipschitz constant zero.
Proof. Let ζ 1 , ζ 2 ∈ Ψ be arbitrary and q ∈ J; then Taking the supremum over all q ∈ J, we have From Theorems 3.1-3.3, We have the following from the definition of T 2 : Thus, We have the following from the definition of T 3 : Thus, Hence, we have, Since (log L) ν Γ(ν + 1) + C T 2 + C T 3 (α + βk max + χh max ) < 1, the FODE (1.1), has a unique solution.

Example
The FIDEs with boundary value conditions are as follows: , By the above parameters in C T 2 and C T 3 , we get (log L) ν Γ(ν + 1) + C T 2 + C T 3 = 3.2498.

Conclusions
This work was performed to investigate the existence and uniqueness of FIDEs with CH derivatives of fractional order by using the Robin boundary condition, TDT and fixed point theorem have been used to accomplish the analysis. The fundamental idea is shown with an efficient example.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.