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In this paper we prove an existence theorem for the Cauchy problem $$x^{\text{'}}\left(t\right)=f(t,x\left(t\right)),x\left(0\right)=x_0,t\in I_{\alpha }=[0,\alpha ]$$ using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function $f$ are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function $f$ satisfies some conditions expressed in terms of measures of weak noncompactness.

In this paper we prove existence theorems for integro - differential equations $x^{\Delta }\left(t\right)=f(t,x\left(t\right),\int {₀}^{t}k(t,s,x\left(s\right))\Delta s)$, t ∈ Iₐ = [0,a] ∩ T, a ∈ R₊, x(0) = x₀ where T denotes a time scale (nonempty closed subset of real numbers R), Iₐ is a time scale interval. Functions f,k are Carathéodory functions with values in a Banach space E and the integral is taken in the sense of Henstock-Kurzweil delta integral, which generalizes the Henstock-Kurzweil integral. Additionally, functions f and k satisfy some boundary conditions and conditions...

We prove some existence theorems for nonlinear integral equations of the Urysohn type $x\left(t\right)=\phi \left(t\right)+\lambda {\int }_0^{a}f(t,s,x\left(s\right))ds$ and Volterra type $x\left(t\right)=\phi \left(t\right)+{\int }_0^{t}f(t,s,x\left(s\right))ds$, $t\in I_a=[0,a]$, where f and φ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

We prove an existence theorems for the nonlinear integral equation $$x\left(t\right)=f\left(t\right)+{\int }_0^a{k}_1(t,s)x\left(s\right)ds+{\int }_0^a{k}_2(t,s)g(s,x\left(s\right))ds,t\in I_a=[0,a],a\in {ℝ}_{+},$$ where $f,g,x$ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

In the paper, we prove the existence of solutions and Carathéodory’s type solutions of the dynamic Cauchy problem $x^{\Delta }\left(t\right)=f(t,x\left(t\right))$, t ∈ T, x(0) = x₀, where T denotes an unbounded time scale (a nonempty closed subset of R and such that there exists a sequence (xₙ) in T and xₙ → ∞) and f is continuous or satisfies Carathéodory’s conditions and some conditions expressed in terms of measures of noncompactness. The Sadovskii fixed point theorem and Ambrosetti’s lemma are used to prove the main result. The results presented...

In this paper, we present the existence result for Carathéodory type solutions for the nonlinear Sturm- Liouville boundary value problem (SLBVP) in Banach spaces on an arbitrary time scale. For this purpose, we introduce an equivalent integral operator to the SLBVP by means of Green’s function on an appropriate set. By imposing the regularity conditions expressed in terms of Kuratowski measure of noncompactness, we prove the existence of the fixed points of the equivalent integral operator. Mönch’s...