On the nonlinear Hadamard-type integro-differential equation

This paper studies uniqueness of solutions for a nonlinear Hadamard-type integro-differential equation in the Banach space of absolutely continuous functions based on Babenko’s approach and Banach’s contraction principle. We also include two illustrative examples to demonstrate the use of main theorems.


Introduction
The Hadamard fractional integration and differentiation are based on the nth integral of the form [1,2]  where log(·) = log e (·), 0 < a < x < b, and μ ∈ R.
The fractional version of the Hadamard-type integral and derivative are given by In particular, for α = 1, which leads to defining the space X μ [a, b] of those Lebesgue measurable functions u on [a, b] for which x μ-1 u(x) is absolutely integrable [2]: Let AC[a, b] be the set of absolutely continuous functions on [a, b]. Then it follows from [3]  Obviously, The latter is a Banach space under its norm. We further define the space where C μ is the maximum value of the function Then by changing the order of integration. Using by Banach's contraction principle and Babenko's approach [4], with two applicable examples presented to illustrate the main results. It seems impossible to obtain these by any existing integral transforms or analytic local model methods. Babenko's approach treats integral operators like variables in solving differential and integral equations. The method itself is close to the Laplace transform method in the ordinary sense, but it can be used in more cases [5,6], such as dealing with integral or fractional differential equations with distributions whose Laplace transforms do not exist in the classical sense. Furthermore, it works well on certain differential or integral equations whose solutions cannot be achieved by the local model. Clearly, it is always necessary to show convergence of the series obtained as solutions. Recently, Li studied the generalized Abel's integral equations of the first [7] and second kind with variable coefficients by Babenko's technique [8][9][10].
It is well known that fractional calculus [3,11,12] has been an emergent tool which uses fractional differential and integral equations to develop more sophisticated mathematical models that can accurately describe complex systems. There are many definitions of fractional derivatives available in the literature, such as the Riemann-Liouville derivative which played an important role in the development of the theory of fractional analysis. However, the commonly used is the Hadamard fractional derivative (with μ = 0) given by Hadamard in [13]. Butzer et al. [14][15][16] studied various properties of the Hadamardtype derivative which is more generalized than the Hadamard fractional derivative. In particular, Hadamard fractional differential equations with boundary value problems or initial conditions have been investigated by researchers using fixed point theories [17,18]. In 2014, Thiramanus et al. [19] studied the existence and uniqueness of solutions for a fractional boundary value problem involving Hadamard differential equations of order q ∈ (1, 2] and nonlocal fractional integral boundary conditions by fixed point theories. In 2018, Matar [20] obtained the solution of the linear equations with the initial conditions (three terms on the left-hand side at most and a given function on the right) by the parameter technique, and then investigated the existence problems of the corresponding nonlinear types of Hadamard equations using fixed point theorems. Very recently, Ding et al. [21] applied the fixed point index and nonnegative matrices to study the existence of positive solutions for a system of Hadamard-type fractional differential equations with semipositone nonlinearities. In 1967, Caputo [22] introduced another type of fractional derivative which has an advantage over R-L derivative in differential equations since it does not require to define fractional order initial conditions. Jarad et al. [23] defined the Caputo-type modification of the Hadamard fractional derivatives which preserve physically interpretable initial conditions similar to the ones in Caputo fractional derivatives. Gambo et al. [24] further presented the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting with several new results. Adjabi et al. [25] studied Cauchy problems for a differential equation with a left Caputo-Hadamard fractional derivative in spaces of continuously differentiable functions.
There are new studies on fixed point theorems for different operators on metric spaces [26][27][28], as well as their applications in differential and integral equations, existence and uniqueness of solutions for equations [29][30][31]. Palve et al. [32] recently constructed the existence and uniqueness of solutions for the fractional implicit differential equation with boundary condition of the form 1+ is the Hilfer-Hadamard type fractional derivative of order α and type β given by and c 1 , c 2 , c 3 ∈ R with c 1 + c 2 = 0 and c 2 = 0. Li [33] obtained uniqueness of solutions for the coupled system of integral equations (it is a Banach space), based on Babenko's approach and Banach's contraction principle.
Define a mapping T on AC 0 [a, b] by Using inequality (4), we claim that . It remains to prove that T is contractive. Indeed, we derive Therefore T is contractive. This completes the proof of Theorem 2.2.
Example 2 Let a = 1, b = e and μ = 2. Then there is a unique solution for the following nonlinear Hadamard-type integro-differential equation: sin u (t) + cos(sin t) + e t 2 dt, where the constant C is to be determined. Clearly, C 2 = e 2 is the maximum value of the function ( t x ) 2 over the interval [1, e] | sin y 1 -sin y 2 | ≤ x 2 C(1 + x 100 ) |y 1y 2 | ≤ 1 C |y 1y 2 |.