in this article we decide to define a modified homotopy perturbation for solving non-linear integral equations. almost, all of the papers that was presented to solve non-linear problems by the homotopy method, they used from two non-linear and linear operators. but we convert a non-linear problem to two suitable non-linear operators also we use from appropriate bases functions such as legendre ...

For a wide class of pairs of unbounded selfadjoint operators (A,B) with bounded commutator and with operator (1+A2+B2)−1 being compact we construct aK-theoretical integer invariant ω(A,B), which is continuous, is equal to zero for commuting operators and is equal to one for the pair (x,−i d/dx). Such invariants in the case of bounded operators were constructed by T. Loring [1, 3] for matrices r...

Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most closely imitate finite-dimensional operator theory. In addition, compact operators are important in practice. We prove a spectral theorem for self-adjoint compact operators, which does not use broader discussions of properties of spectra, only using the Cauchy-Schwarz-Bunyakowsky inequality...

Various bounds in the operator order for the following operator transform f̃ (A) := 1 2 [f (A)− f ((m + M) 1H −A)] , where A is a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) ⊆ [m,M ] and f : [m,M ] → C is a continuous function on [m,M ] are given. Applications for the power and logaritmic functions are provided as well.

A general integral formula for the spectral flow of a path of unbounded selfadjoint Fredholm operators subject to certain summability conditions is derived from the interpretation of the spectral flow as a winding number.

This is the second in a series of works devoted to small non-selfadjoint perturbations of selfadjoint semiclassical pseudodifferential operators in dimension 2. As in our previous work, we consider the case when the classical flow of the unperturbed part is periodic. Under the assumption that the flow average of the leading perturbation vanishes identically, we show how to obtain a complete asy...