Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by coupling these space by meas of the duality product.
Generalized conjugation extends classical conjugation to any pair of domains, using an arbitrary coupling function between these spaces. This generalization of conjugation is now being widely used in optima transportation problems, variational analysis and also optimization.
If the coupled spaces are equal, generalized conjugations define order reversing maps of a family of functions into itself. In this case, is natural to ask for the existence of fixed points of the conjugation, that is, functions which are equal to their (generalized) conjugateds. Here we prove that any generalized symmetric conjugation has fixed points. The basic tool of the proof is a variational principle involving the order reversing feature of the conjugation.
As an application of this abstract result, we will extend to real linear topological spaces a fixed-point theorem for Fitzpatrick's functions, previously proved in Banach spaces.
Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exist a convex representation of the operator which is a fixed point of this conjugation.
In this paper, we investigate the complex symmetric structure of generalized weighted composition operators $D_{m,ψ,φ}$ on the weighted Hardy space $H^2(β)$. We obtain explicit conditions for $ D_{m,ψ,φ}$ to be complex symmetric with the conjugation $J_w$. Under the assumption that $ D_{m,ψ,φ}$ is $J_w$-symmetric, some sufficient and necessary conditions for $D_{m,ψ,φ}$ to be Hermitian and normal are given.
We obtain sufficient conditions for existence of unique fixed point of Kannan type mappings on complete metric spaces and on generalized complete metric spaces depended an another function.
In this paper, we study a new iterative method for finding the fixed point of a weak Bregman relatively nonexpansive mapping and the set of solutions of generalized mixed equilibrium problems in Banach spaces.
The aim of this paper is to present some fixed point theorems for generalized contractions by altering distance functions in a complete cone metric spaces endowed with a partial order. We also generalize fixed point theorems of J. Harjani, K. Sadarangani [J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Analysis 72 (2010) 1188-1197] from metric spaces to cone metric spaces.
Let (X, d) be a complete metric space, and let S, T : X to CB(X) be a duality of multi-valued generalized weak contraction mappings or a duality of generalized weak contraction mappings. We discuss the common fixed points and endpoints of the two kinds of multi-valued weak mappings. Our results extend and improve some results given by Daffer and Kaneko (1995), Rouhani and Moradi (2010), and Moradi and Khojasteh (2011).
In this paper, we introduce a commutative mappings satisfying the class of generalized non-expansive mappings which is wider than the class of mappings satisfying the condition (C), so called Condition B gamma, mu. The results obtained in this paper extend and generalized non-expansive mappings and other results in this direction. Different properties and some fixed point results for the mappings are obtained here. Keywords: Non expansive mappings, Condition (C), Condition B gamma mu, Commuting mappings, Fixed point.
In this paper we introduce generalized symmetric Meir-Keeler contractions and prove some coupled fixed point theorems for mixed monotone operators $F:X \times X \rightarrow X$ in partially ordered metric spaces. The obtained results extend, complement and unify some recent coupled fixed point theorems due to Samet [B. Samet, \textit{Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces}, Nonlinear Anal. \textbf{72} (2010), 4508-4517], Bhaskar and Lakshmikantham [T.G. Bhaskar, V. Lakshmikantham, \textit{Fixed point theorems in partially ordere…
In this work, we prove an existence theorem of the Hyers-Ulam stability for the nonlinear Volterra integral equations which improves and generalizes Castro-Ramos theorem by using some weak conditions.
It is shown that a contraction on a Hilbert space is complex symmetric if and only if the values of its characteristic function are all symmetric with respect to a fixed conjugation. Applications are given to the description of complex symmetric contractions with defect indices equal to 2.
In this paper, we consider a wider class of simulation functions and present some coincidence and common fixed point results in metric spaces. Results obtained in this paper extend, generalize and unify some well-known fixed and common fixed point results. Some illustrative examples are presented. We also discuss some applications to system of integral equations and functional equations arising in dynamic programming.
In this paper, we introduce a new class of Bregman generalized $α$-nonexpansive mappings in terms of Bregman distances, and investigate the Ishikawa and Noor iterations for these mappings. We establish weak and strong convergence theorems of Ishikawa and Noor iterative schemes for Bregman generalized $α$-nonexpansive mappings in Banach spaces. Furthermore, we propose an example of our generated mapping and some numerical examples which support our main theorem. Our results are new and improve the recent ones in the literature.
A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the Legendre-Fenchel transform. Motivated by the Hilbert space version of this result and by the well-known result saying that this convex conjugation transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the above-mentioned class of functions. It turns out that this nonlinear …
In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use Sahu([6]) and Thakur([10])'s iterative scheme to establish several convergence theorems in uniformly convex Banach spaces and give an example to show that this scheme converges faster than the scheme in [1]
In this paper, we show that the generalized fixed-point algebra of a proper groupoid dynamical system, under certain assumptions, may be fibered over any locally compact Hausdorff space to which a continuous map exists from the unit space of the underlying groupoid. We will also provide some important examples.
