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This paper is concerned with the existence and multiplicity of weak solutions for discrete Kirchhoff type equations with Dirichlet boundary conditions. We first establish the existence of a nontrivial weak solution and three weak solutions, by using variational methods, mainly based on the mountain pass theorem of A. Ambrosetti and P.H. Rabinowitz and critical point theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the null controllability of a space-time fractional order parabolic equation involving fractional diffusion and Caputo fractional time derivative of orders s (0, 1) and (0, 1]. We prove that the system under consideration is not null controllable. Moreover, we generalize some existing results about the null controllability of fractional order parabolic systems. We introduce the adjoint system of space-time fractional order parabolic equation, which is used to study the duality relationship between observability and controllability in Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper we prove existence theorems for integro - differential equations xΔ(t) = f(t, x(t), ∫ 0 t k(t, s, x(s))Δs), x(0) = x0 t ∊ Ia = [0, a] ∩ T, a ∊ R+, 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 expressed in terms of measures of noncompactness. Moreover, we prove an Ambrosetti type lemma on a time scale. [ABSTRACT FROM AUTHOR]