SOBOLEV TYPE NEUTRAL INTEGRO-DIFFERENTIAL SYSTEMS INVOLVING (k, ψ) – HILFER FRACTIONAL DERIVATIVE AND NONLOCAL CONDITIONS: TRAJECTORY CONTROLLABILITY

In this paper, we discuss the trajectory controllability of a new class of Sobolev-type nonlocal \((k,\psi )-\) $(k,\psi )-$ Hilfer fractional neutral integro-differential equations in Hilbert space. By mean of two operators \({_{k}}\mathcal {S}{_{\psi }}^{\alpha ,\beta }\) ${}_k\mathcal{S}{{}_{\psi }}^{\alpha ,\beta }$ , \(\ {_{k}\mathcal {T}}_{\psi }^{\alpha }\) ${{}_k\mathcal{T}}_{\psi }^{\alpha }$ and \(k-\) $k-$ Wright function, the definition of mild solutions for the studied problem was given. The results are obtained by using ideas from semigroup theory, \((k,\psi )-\) $(k,\psi )-$ fractional calculus theory, fractional power of operators, and Gronwall’s inequality. To the end, an example is given to illustrate the abstract results.