‎We use of two notions functionally convex (briefly‎, ‎F--convex) and functionally closed (briefly‎, ‎F--closed) in functional analysis and obtain more results‎. ‎We show that if $lbrace A_{alpha} rbrace _{alpha in I}$ is a family $F$--convex subsets with non empty intersection of a Banach space $X$‎, ‎then $bigcup_{alphain I}A_{alpha}$ is F--convex‎. ‎Moreover‎, ‎we introduce new definition o...

and Applied Analysis 3 Remark 1.6. The examples of weak relatively uniformly nonexpansive multivalued mapping can be found in Su 1 and Homaeipour and Razani 2 . Let E be a real Banach space, and let E∗ be the dual space of E. Let f be a bifunction from C × C to R. The equilibrium problem is to find x̂ ∈ C such that fx̂, y ≥ 0, ∀y ∈ C. 1.3 The set of solutions of 1.3 is denoted by EP f . Given a m...

and Applied Analysis 3 It is shown in [19] that if E = H, a real Hilbert space, then ρ(t) = t, for t > 0. In our general setting, throughout this paper we assume that ρ(t) ≤ 2t. Lemma 3. Let E be a real Banach space. Then the following inequality holds: 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + ⟨y, j (x + y)⟩ , ∀x, y ∈ H, j (x + y) ∈ J (x + y) . (10) Lemma 4 (see [20]). Let E be a uniformly convex Banach spac...

In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

and Applied Analysis 3 where J is the normalized duality mapping from E into 2E ∗ . If E = H, a Hilbert space, then (13) reduces to φ(x, y) = ‖x − y‖ 2, for x, y ∈ H. Let E be a reflexive, strictly convex, and smooth Banach space, and letC be a nonempty closed and convex subset ofE. The generalized projectionmapping, introduced byAlber [29], is a mapping Π C : E → C that assigns an arbitrary po...

We introduce a modified Noor iteration scheme generated by an infinite family of strict pseudo-contractive mappings and prove the strong convergence theorems of the scheme in the framework of q−uniformly smooth and strictly convex Banach space. Results shown here are extensions and refinements of previously known results.

and Applied Analysis 3 Lemma 3. Let p > 1 and E be a p-uniformly convex and smooth Banach space. Then, for each x, y ∈ E, φ p (x, y) ≥ c 0 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 p (8) holds, where c 0 is maximum in Remark 2. Proof. Let x, y ∈ E. By Theorem 1, we have ‖x‖ p ≥ 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 p + p⟨x − y, J p y⟩ + c 0 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩 p , (9) where c 0 is maximum in Remark 2. Hence, we get φ p (x, y) = ‖x‖ p − 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 p − p...

We characterize the solution of a broad class convex optimization problems that address reconstruction function from finite number linear measurements. The underlying hypothesis is decomposable as sum components, where each component belongs to its own prescribed Banach space; moreover, problem regularized by penalizing some composite norm solution. establish general conditions for existence an...

begin{abstract} in this paper, we prove some strong and weak convergence of three step random iterative scheme with errors to common random fixed points of three asymptotically nonexpansive nonself random mappings in a real uniformly convex separable banach space. end{abstract}