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.     

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.     

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

     

Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space

Let E be a real Banach space. Let K be a nonempty closed and convex subset of E, a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with sequence {kn}_n?0⊂[1,+∞), limn→∞kn=1 such that F(T)≠∅. Let {αn}_n?0⊂[0,1] be such that _∑n?0αn=∞, and _∑n?0αn(kn−1)<∞. Suppose {xn}_n?0 is iteratively defined by xn+1=(1−αn)xn+αnTnxn, n?0, and suppose there exists a strictly increasing continuous function , ?(0)=0 such that 〈Tnx−x^∗,j(x−x^∗)〉?kn‖x−x^∗‖2−?(‖x−x^∗‖), ∀x∈K. It is proved that {xn}_n?0 converges strongly to x^∗∈F(T). It is also proved that the sequence of iteration {xn} defined by xn+1=anxn+bnTnxn+cnun, n?0 (where {un}_n?0 is a bounded sequence in K and {an}_n?0, {bn}_n?0, {cn}_n?0 are sequences in [0,1] satisfying appropriate conditions) converges strongly to a fixed point of T.     

Generalized almost periodic and ergodic solutions of linear differential equations on the half-line in Banach spaces

The Bohl-Bohr-Amerio-Kadets theorem states that the indefinite integral y=Pφ of an almost periodic (ap) is again ap if y is bounded and the Banach space X does not contain a subspace isomorphic to c₀. This is here generalized in several directions: Instead of it holds also for φ defined only on a half-line , instead of ap functions abstract classes with suitable properties are admissible, can be weakened to φ in some “mean” class , then ; here contains all f∈L¹_loc with in for all h>0 (usually strictly); furthermore, instead of boundedness of y mean boundedness, y in some , or in , ergodic functions, suffices. The Loomis-Doss result on the almost periodicity of a bounded Ψ for which all differences Ψ(t+h)−Ψ(t) are ap for h>0 is extended analogously, also to higher order differences. Studying “difference spaces” in this connection, we obtain decompositions of the form: Any bounded measurable function is the sum of a bounded ergodic function and the indefinite integral of a bounded ergodic function. The Bohr-Neugebauer result on the almost periodicity of bounded solutions y of linear differential equations P(D)y=φ of degree m with ap φ is extended similarly for ; then provided, for example, y is in some with U=L^∞ or is totally ergodic and, for the half-line, Reλ?0 for all eigenvalues P(λ)=0. Analogous results hold for systems of linear differential equations. Special case: φ bounded and Pφ ergodic implies Pφ bounded. If all Reλ>0, there exists a unique solution y growing not too fast; this y is in if , for quite general .     

Strong convergence theorems for maximal monotone mappings in Banach spaces

Let E be a uniformly convex and 2-uniformly smooth real Banach space with dual E^∗. Let be a Lipschitz continuous monotone mapping with A⁻¹(0)≠∅. For given u,x₁∈E, let {xn} be generated by the algorithm xn+1:=βnu+(1−βn)(xn−αnAJxn), n?1, where J is the normalized duality mapping from E into E^∗ and {λn} and {θn} are real sequences in (0,1) satisfying certain conditions. Then it is proved that, under some mild conditions, {xn} converges strongly to x^∗∈E where Jx^∗∈A⁻¹(0). Finally, we apply our convergence theorems to the convex minimization problems.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Isomorphisms between unital C-algebras

It is shown that every almost linear bijection of a unital C^∗-algebra A onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C^∗-algebra A of real rank zero onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C^∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C^∗-algebra isomorphisms between unital C^∗-algebras.     

Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings

Suppose that K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E. Let be two nonself asymptotically nonexpansive mappings with sequences {kn},{ln}⊂[1,∞), limn→∞kn=1, limn→∞ln=1, , respectively. Suppose {xn} is generated iteratively by

     

Solution of a tridiagonal operator equation

Let H be a separable Hilbert space with an orthonormal basis {e_n/n∈N}, T be a bounded tridiagonal operator on H and T_n be its truncation on span ({e₁, e₂, … , e_n}). We study the operator equation Tx = y through its finite dimensional truncations T_nxⁿ = y_n. It is shown that if and are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence . We also give sufficient conditions for the boundedness of and in terms of the entries of the matrix of T.     

Markov interlacing property for exponential polynomials

Let Un be an extended Tchebycheff system on the real line. Given a point , where x₁<?<xn, we denote by the polynomial from Un, which has zeros x₁,…,xn. (It is uniquely determined up to multiplication by a constant.) The system Un has the Markov interlacing property (M) if the assumption that and interlace implies that the zeros of and interlace strictly, unless . We formulate a general condition which ensures the validity of the property (M) for polynomials from Un. We also prove that the condition is satisfied for some known systems, including exponential polynomials and . As a corollary we obtain that property (M) holds true for Müntz polynomials , too.     

A hypercyclicity criterion with applications

A continuous linear operator on a topological vector space X is called hypercyclic if there is x∈X such that the orbit {Tnx}_n?0 is dense in X. We establish a criterion for hypercyclicity, and study some applications. In particular, we establish hypercyclic left-multipliers on the space L(X,Y) of continuous linear operators between X and Y, provided with the topology of uniform convergence on bounded sets, for some spaces X,Y of holomorphic functions.     

A generalization of Barbashin-Krasovski theorem

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a function V(t,x), a(‖x‖)?V(t,x)?b(‖x‖) for some a,b∈K, such that for some c∈K. In this paper we prove that if f(t,x) is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , t∈[−T,T], where , uniformly for (t,x)∈[−T,T]×BH.     

Subordination results for classes of analytic functions related to conic domains defined by a fractional operator

Let a fractional operator (n∈N₀={0,1,2,…}, 0?α<1, λ?0) be defined by

     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .