in this paper the bagley-torvik equation as a prototype fractional differential equation with two derivatives is investigated by means of homotopy perturbation method. the results reveal that the present method is very effective and accurate.

In this notes we talk about the Baire category theorem and its consequences: the BanachSteinhaus theorem, the open mapping theorem and the closed graph theorem.

In this paper we consider coloring problems on graphs and other combinatorial structures standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or measure. To end, show that produced using certain powerful techniques from finite combinatorics computer science. First, prove efficient distributed algorithms (...

S. Banach pointed out that the graph of generic (in sense Baire category) element $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if measure theoretic dual holds, i.e., all but Haar null many Christensen) elements have We answer this question in affirmative. call $f \in \text{Homeo}([0,1]^d)$ singular it takes a suitable set full to nullset, and strongly is almost everywhere differenti...

We study continuous descent methods for minimizing convex functions defined on general Banach spaces and prove that most of them (in the sense of Baire category) converge.

We review basics concerning metric spaces from a contemporary viewpoint, and prove the important Baire category theorem, for both complete metric spaces and locally compact Hausdorff [1] spaces.

We show that an infinite-dimensional complete linear space X has: • a dense hereditarily Baire Hamel basis if |X| ≤ c; • a dense non-meager Hamel basis if |X| = κ = 2 for some cardinal κ. According to Corollary 3.4 of [BDHMP] each infinite-dimensional separable Banach space X has a non-meager Hamel basis. This is a special case of Theorem3.3 of [BDHMP], asserting that an infinite-dimensional Ba...

We investigate the existence of well-ordered sequences of Baire 1 functions on separable metric spaces. Any set F of real valued functions defined on an arbitrary set X is partially ordered by the pointwise order, that is f ≤ g iff f(x) ≤ g(x) for all x ∈ X. In other words put f < g iff f(x) ≤ g(x) for all x ∈ X and f(x) 6= g(x) for at least one x ∈ X. Our aim will be to investigate the possibl...

We solve a number of questions pertaining to the dynamics linear operators on Hilbert spaces, sometimes by using Baire category arguments and by constructing explicit examples. In particular, we prove following results. (i) A typical hypercyclic operator is not topologically mixing, has no eigenvalues admits non-trivial invariant measure, but densely distributionally chaotic. (ii) {upp...