Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations Involving the Hadamard Derivatives - Nemat Nyamoradi; Sotiris K. Ntouyas; Jessada Tariboon (Mathematics, 2022)
In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: <semantics>HDαx(t)=Af(t,x(t))+<msubsup>∑i=1k</msubsup>Ci<mspace width="3.33333pt"></mspace>HIβigi(t,x(t)),t∈(1,e),</semantics> supplemented with fractional Hadamard boundary conditions: <semantics>HDξx(1)=0,<mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace>HDξx(e)=a<mspace width="3.33333pt"></mspace>HDα−ξ−12(HDξx(t))|t=δ,<mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace>δ∈(1,e),</semantics> where <semantics>1<α≤2</semantics>, <semantics>0<ξ≤12</semantics>, <semantics>a∈(0,∞)</semantics>, <semantics>1<α−ξ<2</semantics>, <semantics>0<βi<1</semantics>, <semantics>A,Ci</semantics>, <semantics>1≤i≤k</semantics>, are real constants, <semantics>HDα</semantics> is the Hadamard fractional derivative of order <semantics>α</semantics> and <semantics>HIβi</semantics> is the Hadamard fractional integral of order <semantics>βi</semantics>. By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration.