Exact controllability for nonlinear Hilfer fractional Volterra-Fredholm integrodifferential equations $$(1< \varrho < 2)$$ with infinite delay based on the measure of noncompactness

Abstract In this work, we investigate higher-order Hilfer fractional Volterra-Fredholm integrodifferential (HOHFVFI) equations of order $$1&lt;\varrho &lt;2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>ϱ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and type $$\delta \in [0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> with infinite delay. We establish the existence of mild solutions in the presence of infinite delay by using the cosine family and the Laplace transform representations together with a measure of noncompactness and a fixed point argument. Then, we derive exact controllability results for the considered HOHFVFI system with infinite delay. To clarify the proposed structure, we provide an illustrative example to demonstrate the applicability of the theoretical findings.