Abstract
We prove two existence theorems for the integrodifferential equation of mixed type:x'(t)=f(t,x(t),∫0tk1(t,s)g(s,x(s))ds,∫0ak2(t,s)h(s,x(s))ds),x(0)=x0, where in the first part of this paperf, g, h, xare functions with values in a Banach spaceEand integrals are taken in the sense of Henstock-Kurzweil (HK). In the second partf, g, h, xare weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functionsf, g, h, xsatisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.
Highlights
The Henstock-Kurzweil integral encompasses the Newton, Riemann, and Lebesgue integrals [1,2,3]
A particular feature of this integral is that integrals of highly oscillating functions such as F (t), where F(t) = t2 sin t−2 on (0, 1] and F(0) = 0, can be defined. This integral was introduced by Henstock and Kurzweil independently in 1957–1958 and has since proved useful in the study of ordinary differential equations [4,5,6,7]
In the paper, [8] Cao defined the Henstock integral in a Banach space, which is a generalization of the Bochner integral
Summary
The Henstock-Kurzweil integral encompasses the Newton, Riemann, and Lebesgue integrals [1,2,3]. A particular feature of this integral is that integrals of highly oscillating functions such as F (t), where F(t) = t2 sin t−2 on (0, 1] and F(0) = 0, can be defined. This integral was introduced by Henstock and Kurzweil independently in 1957–1958 and has since proved useful in the study of ordinary differential equations [4,5,6,7]. In the paper, [8] Cao defined the Henstock integral in a Banach space, which is a generalization of the Bochner integral. In [10], Cichon et al.generalized both concepts of integral introducing the HenstockKurzweil-Pettis integral
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