Integral inequality approach to existence and approximate controllability of higher-order Hilfer fractional stochastic inclusions with impulses and jumps

Abstract This paper presents an analytical investigation of a new class of higher-order Hilfer fractional stochastic differential inclusions with order $\alpha \in (1,2)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:math> and type $\beta \in [0,1)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</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:math> . The proposed framework incorporates Brownian motion, Poisson jump perturbations, impulsive effects, nonlocal fractional initial conditions, and hereditary memory terms. The system is formulated in a separable Hilbert space and governed by multi-valued nonlinear dynamics involving integral perturbations and Clarke sub-differentials, which enable the treatment of discontinuous and nonconvex nonlinearities and provide a unified setting for history-dependent stochastic processes. By establishing appropriate Hilfer-type fractional integral inequalities and generalized Gronwall–Bihari estimates, new a priori bounds for the associated solution operators are derived. These inequality techniques, together with fixed point principles for multi-valued mappings and tools based on the measure of non-compactness, yield sufficient conditions for the existence of mild solutions and ensure the well-posedness of the proposed system. Furthermore, refined inequality estimates for the resolvent family and the corresponding stochastic convolution terms are employed to establish the approximate controllability of the considered inclusions. The controllability analysis is conducted within the same inequality-based framework and relies on structural properties of stochastic semi-groups associated with the underlying fractional dynamics. A constructive example satisfying all imposed hypotheses is provided to illustrate the applicability of the theoretical results and to demonstrate the effectiveness of the proposed approach in handling impulsive actions, nonlocal constraints, and jump perturbations within a higher-order Hilfer fractional stochastic setting. The obtained results extend inequality-based methods to complex stochastic inclusions with memory effects and discontinuities, and contribute to the theory of fractional stochastic systems through a systematic use of integral inequality techniques.