EXISTENCE AND UNIQUENESS SOLUTION FOR THREE-POINT HADAMARD-TYPE FRACTIONAL VOLTERRA BVP

This article investigates the existence and unique solution of a fractional Volterra boundary value problem of the first sort with Hadamard type and three-point boundary conditions. Our analysis is based on the fixed-point theorem of Krasnoselskii-Zabreiko and the Banach contraction principle. We explored the solution of a Hadamard type boundary value issue with fractional integral boundary conditions, and our conclusions are well demonstrated with examples.


INTRODUCTION
The theory of fractional differential equations and inclusions has received a lot of attention in recent years. It has become an important academic issue because to its numerous applications in the fields of physics, economics, and engineering sciences. Fractional differential equations and inclusions provide appropriate models for addressing real-world situations that cannot be addressed using classical integer-order differential equations (Benchohra et al., 2009;Bai, 2010;Balachandran & Trujillo, 2010;Agarwal et al., 2010;Ahmad, 2010).
Fractional calculus is a branch of mathematics concerned with the study and application of arbitrary order integrals and derivatives. Fractional differential equations are derived from the mathematical modelling of systems and operations encountered in a wide variety of engineering and scientific disciplines, including physics, chemistry, aerodynamics, electrodynamics of complex media, polymer rheology, economics, control theory, signal and image processing, biophysics, and blood flow phenomena, among others (Ishak, 2020;Kilbas & Trujillo, 2003;Guotao et al., 2018;Sial et al., 2021;Ntouyas et al., 2021;Jhanthanam et al, 2019).
The majority of study on this issue has long been recognized to be based on Riemann-Liouville and Caputo-type fractional differential equations. Another type of fractional derivative that appears in the literature alongside Riemann-Liouville and Caputo derivatives is the Hadamard fractional derivative introduced in 1892 ( Chen et al., 2013), which is distinguished from the preceding ones by the presence of a logarithmic function of any exponent in the kernel of the integral. Details and properties of Hadamard fractional derivative and integral can be found in ; Samadi & Ntouyas (2021); Kiataramkul et al. (2021); Benkerrouche et al. (2021).
This study investigated the existence and uniqueness of the following boundary value problem for the Volterra fractional differential equation of the Hadamard type.
Where is the Hadamard derivative of order , :[1,w]× ℝ →ℝ is a continuous function and is a real number.  [q] denotes the integer part of the real number q and log(. ) = log (. ).
Theorem 1. (Ahmad et al., 2017) Let (  Using Lemma 2, the solution of the problem (1) can be written as:

MAIN RESULT
Consider the Banach space Then the BVP (1) has at least one nontrivial solution in [1, ].
Proof: define an operator Ψ: → by basing on this the mapping Ψ is well defined, now we have to proof that there exist a fixed points for the operator Ψ in the Banach space . We split the proof into three steps.
Step2: we prove that Ψ( ) is relatively compact in we set: Then we have: As the right-hand side tends to 0 as 1 → 2 , this guarantees that Ψ(C) is equicontinuous by Arzela´-Ascoli theorem the mapping Ψ is completely continuous on . This completes the proof of Step 2.
Next consider the following boundary value problem Where Let us define an operator : → by Clearly, A is a bounded linear operator, in addition any fixed point of A is a solution of the boundary value problem (6) and vice versa.
Step 3. We now make sure that 1 is not an eigenvalue of A. Suppose that the boundary value problem (6)  It follows from the assumption (7) that Ψ is a contraction. In consequence by Banach's fixed point theorem the operator Ψ has a fixed point which corresponds to the unique solution of problem (1). This completes the proof.

CONCLUSION
In this research paper we have proven the existence and uniqueness of solutions for the Hadamard type Volterra fractional integrodifferential equation with three-point boundary value conditions by selecting 1 < ≤ 2 and optional interval [1, ]. Boundary value conditions have been chosen to contain three different point for which have never been used together with Volterra equation before in any article as far as we know. Existence of solutions have been shown by Krasnol'sk'ii-Zabreiko's fixed point theorem, and uniqueness solutions have been investigated by Banach contraction principal theorem.
The case of fractional integral boundary conditions was discussed, examples have been supported in order to demonstrate all theorems very well.