Integrodifferential Equations on Time Scales with Henstock-Kurzweil-Pettis Delta Integrals

We prove existence theorems for integro-differential equations ð ‘¥ Δ ∫ ( ð ‘¡ ) = ð ‘“ ( ð ‘¡ , ð ‘¥ ( ð ‘¡ ) , ð ‘¡ 0 𠑘 ( ð ‘¡ , ð ‘ , ð ‘¥ ( ð ‘ ) ) Δ ð ‘ ) , ð ‘¥ ( 0 ) = ð ‘¥ 0 , ð ‘¡ ∈ ð ¼ ð ‘Ž = [ 0 , ð ‘Ž ] ∩ 𠑇 , ð ‘Ž ∈ ð ‘… + , where 𠑇 denotes a time scale (nonempty closed subset of real numbers ð ‘… ), and ð ¼ ð ‘Ž is a time scale interval. The functions ð ‘“ , 𠑘 are weakly-weakly sequentially continuous with values in a Banach space ð ¸ , and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions ð ‘“ and 𠑘 satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.