Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative

The aim of this manuscript is to handle the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equations involving a ξ -Hilfer derivative. The used fractional operator is generated by the kernel of the kind k(θ , s) = ξ (θ ) – ξ (s) and the operator of differentiation Dξ = ( 1 ξ ′(θ ) d dθ ). The existence and uniqueness of solutions are established for the considered system. Our perspective relies on the properties of the generalized Hilfer derivative and the implementation of Krasnoselskii’s fixed point approach and Banach’s contraction principle with respect to the Bielecki norm to obtain the uniqueness of solution on a bounded domain in a Banach space. Besides, we discuss the Ulam–Hyers stability criteria for the main fractional system. Finally, some examples are given to illustrate the viability of the main theories.


Introduction
Fractional calculus (FC) was introduced at the end of the seventeenth century as a branch of mathematical analysis that deals with the examinations of various possibilities to term real (or complex) number powers of the integration (and differentiation) operators. FC is the generalization of ordinary calculus concerned with operations of integration (and differentiation) of noninteger order.
2), then our current results cover the results of Asawasamrit et al. [34] for Hilfer nonlocal BVP.
The main contribution of the current work is to determine the equivalent fractional integral equation to ξ -Hilfer type FDEs (1.1)-(1.2) and to explore the existence and uniqueness results. Further, we discuss the Ulam-Hyers stability result to such equations. Observe that, with the above discussions, problem (1.1)-(1.2) not just incorporates the previously specified BVPs in the literature, yet additionally nontrivially extends the status to a more comprehensive class of nonlocal BVPs, i.e., for various values of 2 and ξ , our considered problem covers the problems referenced in Remark 1.1. Consequently, problem (1.1)-(1.2) studied in this paper is novel and is the first to investigate fractional nonlocal problems of ξ -Hilfer type.
Here is a brief outline of the paper. Section 2 provides the definitions and preliminary facts that we will need for our forthcoming analysis. In Sect. 3, we prove the existence, uniqueness, and UH stability results for problem (1.1)- (1.2). Two examples are given in Sect. 5. This work closes with a conclusion.

Preliminaries
In this section, we give some notions regarding the fractional integrals and derivatives with respect to another function ξ . For more details, we refer to [1][2][3]5].
Let us now conclude this section by recalling the following fixed point theorems.

Main result
The next lemma transacts with a linear form associated with problem (1.1)-(1.2).
then the function z ∈ C is a solution of the linear-type problem if and only if Proof The first equation of (3.2) can be written as Applying the operator I 1 ;ξ a + , we get By Lemma 2.5 and setting I Differentiation of (3.4) with the fact that D k ξ I σ ,ξ a + = I σ -k,ξ a + for k = 0, 1, . . . , n -1, σ > k (see [12]) leads to From the boundary conditions of (3.2), we obtain c 1 = z a + p(z) and It follows that Therefore, Substituting the values of c 1 , c 2 into (3.4), we get Note that The converse follows by direct calculation with the aid of the results in Lemmas 2.4, 2.5. This finishes the proof.
To follow up, we need the following assumptions. (G1) F : J × R → R and p : C → R are continuous. (G2) There exist constants L 1 , L 2 > 0 such that and p(z)p(z) ≤ L 2 |z -z|, z, z ∈ C.
(G3) There exist positive functions ϕ, φ with bounds ϕ and φ , respectively, such that: For simplicity, we denote In what follows, we present the needful lemma that represents the equivalent solution to problem (1.1)-(1.2). Similarly, we obtain

continuous. A function z(ϑ) solves system (1.1)-(1.2) if and only if it is a fixed point of the operator H : C → C defined by
Hence which implies that Hz ≤ R, i.e., HB R ⊆ B R . Now, we show that H is a contraction. Let z, y ∈ C. Then, for every ϑ ∈ J , Also note that Using the above arguments, we get As < 1, we deduce that H is a contraction. Hence, Theorem 2.8 shows that BVP (1.1)-(1.2) has a unique solution. This completes the proof.
Remark 3. 4 We would like to point out that the strong condition < 1 can be removed if we use the well-known Bielecki norm.
In fact, just like the discussion in Theorem 3.3, we only prove that H defined as before is a contraction on C via the Bielecki norm. Given z, y ∈ C and ϑ ∈ J , using (G2) and Lemma 2.7, we have (Hy)(ϑ) -(Hz)(ϑ) denotes the Bielecki-type norm on the Banach space C. Thus, we obtain Hy -Hz θ ≤ L 2 ν * + L 1 1 Taking θ > 0 large enough such that it follows that This means that H is a contraction with respect to the Bielecki norm. Hence, Theorem 2.8 shows that BVP (1.1)-(1.2) has a unique solution.
Proof By assumption (G3), we can fix The proof will be split into numerous steps.
Step 2: H 2 is a contraction map on B ρ . Due to the contractility of H as in Theorem 3.3, H 2 is a contraction map too.
Step 3: H 1 is completely continuous on B ρ . From the continuity of F(·, z(·)) it follows that H 1 is continuous. Since we get H 1 z ≤ p, which emphasizes that H 1 uniformly bounded on B ρ . Finally, we prove the compactness of H 1 .
For z ∈ B ρ and ϑ ∈ J , we can estimate the operator derivative as follows: where we used the fact Hence, for each ϑ 1 , ϑ 2 ∈ J with a < ϑ 1 < ϑ 2 < b and for z ∈ B ρ , we get where (ϑ 2ϑ 1 ) tends to zero independent of z. So, H 1 is equicontinuous. In light of the previous arguments along with the Arzela-Ascoli theorem, we derive that H 1 is compact on B ρ . Thus, the hypotheses of Theorem 2.9 hold. So there exists at least one solution of (1.1)-(1.2) on J .
Remark 3.6 In Theorem 3.5, we can exchange the roles of the operators H 1 and H 2 to obtain a second result by replacing (3.14) with the following condition: 1 ( 1 + 1) < 1.
Proof Let ε > 0 and z ∈ C satisfy inequality (4.1), and let z ∈ C be the unique solution of the following problem: