Hadamard-Type Fractional Integro-Differential Problem: A Note on Some Asymptotic Behavior of Solutions

Abstract

As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integro-differential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and two whereas the nonlinearities may contain fractional derivatives of an order between zero and one as well as some non-local terms. Under some reasonable conditions, we prove that solutions are asymptotic to logarithmic functions. Our approach is based on a generalized version of Bihari–LaSalle inequality, which we prove. In addition, several manipulations and crucial estimates have been used. An example supporting our findings is provided.

Keywords

• asymptotic behavior
• fractional differential equation
• Hadamard fractional derivative