Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations Involving the Hadamard Derivatives

Nemat Nyamoradi, Sotiris K. Ntouyas and Jessada Tariboon ()
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Nemat Nyamoradi: Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran
Sotiris K. Ntouyas: Department of Mathematics, University of Ioannina, 45110 Ioannina, Greece
Jessada Tariboon: Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

Mathematics, 2022, vol. 10, issue 17, 1-15

Abstract: In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: H D α x ( t ) = A f ( t , x ( t ) ) + ∑ i = 1 k C i H I β i g i ( t , x ( t ) ) , t ∈ ( 1 , e ) , supplemented with fractional Hadamard boundary conditions: H D ξ x ( 1 ) = 0 , H D ξ x ( e ) = a H D α − ξ − 1 2 ( H D ξ x ( t ) ) | t = δ , δ ∈ ( 1 , e ) , where 1 < α ≤ 2 , 0 < ξ ≤ 1 2 , a ∈ ( 0 , ∞ ) , 1 < α − ξ < 2 , 0 < β i < 1 , A , C i , 1 ≤ i ≤ k , are real constants, H D α is the Hadamard fractional derivative of order α and H I β i is the Hadamard fractional integral of order β i . By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration.

Keywords: Hadamard fractional derivative; existence and uniqueness; fixed point theorems (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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