In this study, the existence and uniqueness of the solution of a class of nonlinear finite-part singular integral equations with Carleman shift preserving orientation has been investigated in the generalized Holder space () Γ ϕ H. 1. Introduction Many important problems of engineering mechanics like elasticity, plasticity, and fracture mechanics and aerodynamics can be reduced to the solution o...

We construct an explicit solution of the Cauchy initial value problem for the onedimensional Schrödinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg–Weyl group N (3) in a certain special case...

In this paper, an attempt is made to present an extension of Darbo's theorem, and its applicationto study the solvability of a functional integral equation of Volterra type.

In this article, we apply two new fixed point theorems to investigate the existence of mild solutions for a nonlocal fractional Cauchy problem with an integral initial condition in Banach spaces.

where we use the notation dxI for (1.4) dxI = dxi1 ∧ dxi2 ∧ ... ∧ dxik for I = {i1, i2, ..., ik} with i1 < i2 < ... < ik. So ΩX is a free module over C ∞(X) generated by dxI . Obviously, Ω k X = 0 for k > n and ⊕ΩX is a graded ring (noncommutative without multiplicative identity) with multiplication defined by the wedge product (1.5) ∧ : (ω1, ω2)→ ω1 ∧ ω2. Note that (1.6) ω1 ∧ ω2 = (−1)12ω2 ∧ ω...

If A is a Lipschitz function, i.e., || A |L < °°, then %A makes a very significant example of non-convolution type singular integral operators. The problem of L -boundedness of the Cauchy transform was raised and solved when || A |L is small by A. P. Calderόn in relation to the Dirichlet problem on Lipschitz domains [Call, Cal2]. Since then, it has been a central problem in the theory of singul...

In this article, we consider a Cauchy problem of heat equation, that is, determining the unknown temperature and heat flux at an inaccessible boundary from scattered temperature measurements on an accessible boundary or in some interior locations. This method is similar to the boundary control approach proposed by Leevan Ling and Tomoya Takeuchi in [1] where the authors considered a Cauchy prob...