We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositi...

and Applied Analysis 3 Definition 2.2. For f, g ∈ E the order interval 〈f, g〉 is defined as 17 〈 f, g 〉 { h ∈ E : f ≤ h ≤ g. 2.1 Definition 2.3. A subset E ⊂ Π is called order bounded if E is contained in some order interval. Definition 2.4. A coneK is called normal if there exists a positive constant μ such that f, g ∈ V and θ ≺ f ≺ g implies that ‖f‖ ≤ μ‖g‖. We state the two fixed point resul...

This current work is devoted to develop qualitative theory of existence solution some families fractional order differential equations (FODEs). For this purposes we utilize fixed point due Banach and Schauder. Further using transform method (DTM), also compute analytical or semi-analytical results the proposed problems. Also by proper examples demonstrate results.

Abstract In this research study, we investigate the existence and uniqueness of solutions for a coupled multiorder system fractional differential equations involving integro-differential boundary conditions in Riemann–Liouville setting. The presented results are obtained via classical Banach principle along with Leray–Schauder Krasnosel’skiĭ’s fixed-point theorems. Examples included to support ...

This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost operators, differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove result. We start by emphasizing existence a mild solution demonstrate system. In addition, an example is presented principle.

We study a nonlocal evolution equation generalising model introduced by Shigeru Kondo to explain colour patterns on skin of guppy fish. prove the existence stationary solutions using either bifurcation theory or Schauder fixed‐point theorem. also present numerical studies this and show that it exhibits similar those modelled well‐known reaction‐diffusion equations.

where t ∈ T = [ν – β – 1,b + ν – β – 1]Nν–β–1 . βν–2, b∇ν are left and right fractional difference operators, respectively, and φp(s) = |s|p–2s, p > 1. By using the method of upper and lower solution and the Schauder fixed point theorem, we obtain the existence of positive solutions for the above boundary value problem; and applying a monotone iterative technique, we establish iterative schemes...

This paper studies a coupled system of nonlinear fractional differential equation with three-point boundary conditions. Applying the Schauder fixed point theorem, an existence result is proved for the following system Du (t) = f (t, v (t) , Dv (t)) , t ∈ (0, 1) , Dv (t) = g (t, u (t) , Du (t)) , t ∈ (0, 1) , u (0) = 0, Du (1) = δDu (η) , v (0) = 0, Dv (1) = δDv (η) , where α, β, m, n, η, δ, θ s...