In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales x Δ ( t ) = f ( t , x ( t ) ) , x ( 0 ) = x 0 , t ∈ I a , $$\matrix{{x^\Delta (t) = f(t,x(t)),} \hfill & {} \hfill \cr {x(0) = x_0 ,} \hfill & {t \in I_a ,} \hfill } $$ where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.

In this paper, we first prove an existence theorem for the integrodifferential equation (*) wheref,k,xare functions with values in a Banach spaceEand the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the setSof all solutions of the problem (*) is compact and connected in (C(I_d,E),ω), where $I_{d} \subset I_{a} $ .

We prove some existence theorems for nonlinear integral equations of the Urysohn type $x(t)=\varphi (t)+\lambda \int _0^a f(t,s,x(s))\, ds$ and Volterra type $x(t)=\varphi (t)+\int _0^tf(t,s,x(s))\, ds$, $t\in I_a=[0,a]$, where $f$ and $\varphi $ are func

In this paper, by using Sadovskii’s fixed point theorem and the properties of the measure of noncompactness, we establish some sufficient conditions for the asymptotic stability results of nonlinear neutral integro-differential equations with variable delays. The results presented in this paper improve and generalize some results in the literature. An example is considered to illustrate our main results.

In this article we prove the existence of solutions to the integrodifferential equation of mixed type \begin{gather*}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),\cr x(0)=x_0, \quad x_0 \in E,\; t \in I_a=[0,a] \cap \mathbb{T},\; a>0, \end{gather*} where \(\mathbb{T}\) denotes a time scale (nonempty closed subset of real numbers \(\mathbb{R}\)), \(I_a\) 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.

In this paper we consider the modified Szasz-Mirakyan-Kantorovich operators for functions \(f\) integrable in the sense of Denjoy-Perron. Moreover, we estimate the rate of pointwise convergence of \(M_nf(x)\) at the Lebesgue-Denjoy points \(x\) of \(f\) .