Weak convergence of inner superposition operators

This paper provides a sufficient condition to guarantee the stability of weak limits under nonlinear operators acting on vector-valued Lebesgue spaces. This nonlinear framework places the weak convergence in perspective. Such an approach allows short and insightful proofs of important results in Functional Analysis such as: weak convergence in L ∞ implies strong convergence in L p for all 1 ≤ p < ∞, weak convergence in L 1 v.s. strong convergence in L 1 and the Brezis-Lieb theorem. The final goal is to use this framework as a strategy to grapple with a nonlinear weak spectral problem on W 1,p .

Anais Da Academia Brasileira De Ciencias, 2003

In this paper, we prove that if a Nemytskii operator maps Lp( , E) into Lq( , F), for p, q greater than 1, E, F separable Banach spaces and F reflexive, then a sequence that converge weakly and a.e. is sent to a weakly convergent sequence. We give a counterexample proving that if q = 1 and p is greater than 1 we may not have weak sequential continuity of such operator. However, we prove that if p = q = 1, then a weakly convergent sequence that converges a.e. is mapped into a weakly convergent sequence by a Nemytskii operator. We show an application of the weak continuity of the Nemytskii operators by solving a nonlinear functional equation on W1,p( ), providing the weak continuity of some kind of resolvent operator associated to it and getting a regularity result for such solution.

2012

In order to generalize the Bolzano-Weierstrass Theorem, a weaker notion of convergence is introduced. The results presented are in the domain of real Hilbert spaces.

Applications of Mathematics

Journal of Mathematical Analysis and Applications, 2008

We establish some properties of the superposition operator which are associated with monotonicity. Those properties are expressed in terms of the notion of degree of decrease or degree of increase. An application of the obtained results to the study of solvability of a quadratic Volterra integral equation is also derived.

Hacettepe Journal of Mathematics and Statistics, 2018

Let us recall that an operator T : E → F, between two Banach lattices, is said to be weak* Dunford-Pettis (resp. weak almost limited) if fn (T xn) → 0 whenever (xn) converges weakly to 0 in E and (fn) converges weak* to 0 in F (resp. fn (T xn) → 0 for all weakly null sequences (xn) ⊂ E and all weak* null sequences (fn) ⊂ F with pairwise disjoint terms). In this note, we state some sufficient conditions for an operator R : G → E(resp. S : F → G), between Banach lattices, under which the product T R (resp. ST) is weak* Dunford-Pettis whenever T : E → F is an order bounded weak almost limited operator. As a consequence, we establish the coincidence of the above two classes of operators on order bounded operators, under a suitable lattice operations' sequential continuity of the spaces (resp. their duals) between which the operators are defined. We also look at the order structure of the vector space of weak almost limited operators between Banach lattices.

Journal of Universal Mathematics, 2021

Many investigations have been made about of Non-Newtonian calculus and superposition operators until today. Non-Newtonian superposition operator was defined by Sağır and Erdoğan in [9]. In this study, we have defined *- boundedness and *-locally boundedness of operator. We have proved that the non-Newtonian superposition operator $_{N}P_{f}:c_{_{0,\alpha }}\rightarrow \ell _{1,\beta }$ is *-locally bounded if and only if f satisfies the condition (NA₂′). Then we have shown that the necessary and sufficient conditions for the *-boundedness of $% _{N}P_{f}:c_{_{0,\alpha }}\rightarrow \ell _{1,\beta }$ . Finally, the similar results have been also obtained for $_{N}P_{f}:c_{\alpha }\rightarrow \ell _{1,\beta }$ .

Mathematische Nachrichten, 2011

2017

In this paper, we consider the nonlinear superposition operator F in l p spaces of sequences (1 ≤ p ≤ ∞), generated by the function f (s, u) = a(s) + arctan u or f (s, u) = a(s) − arctan u. We find out the Rhodius spectra σ R (F) and the Neuberger spectra σ N (F) of these operators and finally the radii of these spectra. The superposition operator generated by the function f (s, u) = a(s) ± arccot u appears to be a special case of above mentioned operator.

We discuss the notions of strong convergence and weak convergence in n-inner product spaces and study the relation between them. In particular, we show that the strong convergence implies the weak convergence and disprove the converse through a counterexample , by invoking an analogue of Parseval's identity in n-inner product spaces.

International Journal of Mathematics Trends and Technology, 2021

The concern of present article is to study approximation properties of generalized form of Post-Widder operators. The modified operators conserve polynomial function αs(v) = v s , s ∈ N. we find some error in estimation of these operators with help of different approximation tools like usual, weighted and exponential modulus of continuity. The rate of convergence of these operators is also shown by numerical table and graphically using mathematica.

Glasgow Mathematical Journal, 2008

We study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.

Journal of Functional Analysis, 1979

Proceedings of the American Mathematical Society, 1974

Let N ( X ) N(X) be the set { F ε X \{ F\varepsilon X : there exists a weakly unconditionally converging series Σ x n \Sigma {x_n} in X X such that F = σ ( X , X ′ ) − lim n Σ i = 1 n J x i } F = \sigma (X,X’) - {\lim _n}\Sigma _{i = 1}^nJ{x_i}\} . Representation theorems for the unconditionally converging operators (map weakly unconditionally converging series into unconditionally converging series) are developed by using the σ ( X ′ , N ( X ) ) \sigma (X’,N(X)) topology of X ′ X’ .