Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps

Let E be a real q-uniformly smooth Banach space which is also uniformly convex (for example, Lp or ?p spaces, 1<p<∞), and K a nonempty closed convex (not necessarily bounded) subset of E. Let be a k-strictly asymptotically pseudocontractive map with a nonempty fixed-point set. It is proved that (I−T) is demiclosed at 0. Furthermore, weak and strong convergence of an averaging iteration method to a fixed point of T are proved.     

Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings

Let E be a real Banach space, K a closed convex nonempty subset of E. Let be m total asymptotically nonexpansive mappings. An iterative sequence for approximation of common fixed points (assuming existence) of T₁,T₂,…,Tm is constructed; necessary and sufficient conditions for the convergence of the scheme to a common fixed point of the mappings are given. Furthermore, in the case that E is uniformly convex, a sufficient condition for convergence of the iteration process to a common fixed point of mappings under our setting is established.     

Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space, K be a closed convex nonempty subset of E which is also a nonexpansive retract with retraction P. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . Let be a sequence in [?,1−?],?∈(0,1), for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E^∗ satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Let E be a real normed linear space, K be a nonempty subset of E and be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., , and there exist x^∗∈F(T) and a strictly increasing function , Φ(0)=0 such that for all x∈K, there exists j(x−x^∗)∈J(x−x^∗) such that

〈Tx−x^∗,j(x−x^∗)〉?‖x−x^∗‖2−Φ(‖x−x^∗‖).     

Iterative approximation of fixed points of nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and be a nonexpansive mapping with F(T):={x∈K:Tx=x}≠∅. For a fixed δ∈(0,1), define by Sx:=(1−δ)x+δTx, ∀x∈K. Assume that {zt} converges strongly to a fixed point z of T as t→0, where zt is the unique element of K which satisfies zt=tu+(1−t)Tzt for arbitrary u∈K. Let {αn} be a real sequence in (0,1) which satisfies the following conditions: ; . For arbitrary x₀∈K, let the sequence {xn} be defined iteratively by

xn+1=αnu+(1−αn)Sxn.     

The generalized Roper-Suffridge extension operator

Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z₁,z₂,…,z_n), z₁∈D, , p_i?1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z₁),f′(z₁)^1/p₂z₂,…,f′(z₁)^1/p_nz_n) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f′(z₁))^1/p_i|_z1=0=1, i=2,…,n. Some other related results are proved.     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

Precise rates in complete moment convergence for ρ-mixing sequences

Let X₁,X₂,… be a strictly stationary sequence of ρ-mixing random variables with mean zeros and positive, finite variances, set Sn=X₁+?+Xn. Suppose that , , where q>2δ+2. We prove that, if for any 0<δ?1, then

     

Strong convergence theorems for uniformly continuous pseudocontractive maps

Let K be a nonempty closed convex subset of a real Banach space E and let be a uniformly continuous pseudocontraction. Fix any u∈K. Let {xn} be defined by the iterative process: x₀∈K, xn+1:=μn(αnTxn+(1−αn)xn)+(1−μn)u. Let δ(?) denote the modulus of continuity of T with pseudo-inverse ?. If and {xn} are bounded then, under some mild conditions on the sequences n{αn} and n{μn}, the strong convergence of {xn} to a fixed point of T is proved. In the special case where T is Lipschitz, it is shown that the boundedness assumptions on and {xn} can be dispensed with.     

New upper bounds on the spectral radius of unicyclic graphs

Let G=(V(G),E(G)) be a unicyclic simple undirected graph with largest vertex degree Δ. Let C_r be the unique cycle of G. The graph G-E(C_r) is a forest of r rooted trees T₁,T₂,…,T_r with root vertices v₁,v₂,…,v_r, respectively. Let

     

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that at all x∈M and at some point x₀∈M, where μ₀>0 is the least eigenvalue of the Laplacian acting on L²-functions on M. Let 2?q?p. Then any q-harmonic map of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism of finite q-energy is constant.     

The generalized Roper-Suffridge extension operator for some biholomorphic mappings

Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C. Let , where 1?p₁?2 (or 0<p₁?2), pj?1, j=2,…,n, are real numbers. In this paper, we prove that

     

On ω-limit sets of ordinary differential equations in Banach spaces

Let X be an infinite-dimensional real Banach space. We classify ω-limit sets of autonomous ordinary differential equations x^′=f(x), x(0)=x₀, where f:X→X is Lipschitz, as being of three types I-III. We denote by SX the class of all sets in X which are ω-limit sets of a solution to (1), for some Lipschitz vector field f and some initial condition x₀∈X. We say that S∈SX is of type I if there exists a Lipschitz function f and a solution x such that S=Ω(x) and . We say that S∈SX is of type II if it has non-empty interior. We say that S∈SX is of type III if it has empty interior and for every solution x (of Eq. (1) where f is Lipschitz) such that S=Ω(x) it holds . Our main results are the following: S is a type I set in SX if and only if S is a closed and separable subset of the topological boundary of an open and connected set U⊂X. Suppose that there exists an open separable and connected set U⊂X such that , then S is a type II set in SX. Every separable Banach space with a Schauder basis contains a type III set. Moreover, in all these results we show that in addition f may be chosen Ck-smooth whenever the underlying Banach space is Ck-smooth.     

Integral representations for a class of operators

Let X be a completely regular Hausdorff space, E Hausdorff a quasi-complete locally convex space and Cb(X,E) all E-valued bounded continuous functions on X with strict topologies βt, , . We prove that a linear continuous mapping T:Cb(X,E)→E arises from a scalar measure μ∈(Cb^′(X),βz)(z=t,∞,τ) if and only if g(T(f))=0 whenever g○f=0 for any f∈Cb(X,E), g∈E^′.     

The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations

Let ∥ · ∥ be the Frobenius norm of matrices. We consider (I) the set S_E of symmetric and generalized centro-symmetric real n × n matrices R_n with some given eigenpairs (λ_j, q_j) (j = 1, 2, … , m) and (II) the element in S_E which minimizes for a given real matrix R^∗. Necessary and sufficient conditions for S_E to be nonempty are presented. A general form of elements in S_E is given and an explicit expression of the minimizer is derived. Finally, a numerical example is reported.     

Some multipoint boundary value problems of Neumann-Dirichlet type involving a multipoint p-Laplace like operator

Let ? and θ be two increasing homeomorphisms from R onto R with ?(0)=0, θ(0)=0. Let be a function satisfying Carathéodory's conditions, and for each i, i=1,2,…,m−2, let , be a continuous function, with , ξi∈(0,1), 0<ξ₁<ξ₂<?<ξm−2<1.In this paper we first prove a suitable continuation lemma of Leray-Schauder type which we use to obtain several existence results for the m-point boundary value problem: