Integrodifferential equations of mixed type on time scales with Delta-HK and Delta-HKP integrals

[摘要] In this article we prove the existence of solutions to the integrodifferential equation of mixed type $$ \displaylines{ x^\Delta (t)=f \Big( t,x(t), \int_0^t k_1 (t,s)g(s,x(s)) \Delta s, \int_0^a k_2(t,s)h(s,x(s)) \Delta s \Big),\\ x(0)=x_0, \quad x_0 \in E,\; t \in I_a=[0,a] \cap \mathbb{T},\; a>0, }$$ where\(\mathbb{T}\)denotes a time scale (nonempty closed subset of realnumbers\(\mathbb{R}\)), Ia is a time scale interval. In the first part of this paper functions f,g,h are Caratheodory functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil delta integrals, which generalizes the Henstock-Kurzweil integrals. In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis delta integrals. Additionally, functions f, g, h satisfy some boundary conditions and conditions expressed in terms of measures of noncompactness.

[学科分类] 数学（综合）

[关键词] Integrodifferential equations;nonlinear Volterra integral equation;time scales;Henstock-Kurzweil delta integral;HL delta integral;Banach space;Henstock-Kurzweil-Pettis delta integral;fixed point;measure of noncompactness;Caratheodory solutions;pseudo-solution