Optimum solution of ( k , ℷ ) $(k,\gimel )$ -Hilfer FDEs by A $\mathcal{A}$ -condensing operators and the incorporated measure of noncompactness

Abstract

Abstract The notion of A $\mathcal{A}$ -condensing operators via the measure of noncompactness is proposed, which retains the existing classes of condensing operators. Results concerning the existence of the best proximity point (pair) of cyclic (noncyclic) A $\mathcal{A}$ -condensing operators along with the coupled best proximity-point theorem for cyclic A $\mathcal{A}$ -condensing operators have been formulated. An application to a ( k , ℷ ) $(k, \gimel )$ -Hilfer fractional differential equation of order 2 < p < 3 $2< p<3$ , type q ∈ [ 0 , 1 ] $q\in [0,1]$ satistfying some boundary conditions is presented. The paper is the first to investigate the optimum solution of such a generalized fractional differential equation. The hypothesis involved in the investigation is independent of the incorporated measure of noncompactness, thereby making our result better in application than that present in the literature. Moreover, added numerical examples validate the theoretical conclusions.

Keywords

• Best proximity point (pair)
• Measure of noncompactness
• Hilfer fractional differential equation