f : J × E → E is a given function satisfying some assumptions that will be specified later, and E is a Banach space with norm ∥ u ∥.
Let k be a fixed positive integer which will be specified later, and set (x=(x_{n})_{nin mathbb {Z}}) with (x_{n}=rho(frac {|n|}{k}) v_{n}).
In one word, the nonlinear term f of the above-mentioned boundary value problem does not include Ku, Hu (Ku, Hu are given functions satisfying some assumptions that will be specified later), and boundary conditions are particularly chosen.
maps with compact and convex values, where (overline{B}_{D} := { z inmathbb{R}mid|z| leq D}) is the closed ball, constant D is such that (D = D_{0}+k_{2}+k_{3}), where (D_{0}>0) is a suitable constant which will be specified later, and (k_{2}), (k_{3}) are defined above.
In this paper we investigate the existence of solutions for boundary value problems with fractional order differential equations and nonlinear integral conditions of the form (11). where is the Caputo fractional derivative,,, and are given functions satisfying some assumptions that will be specified later, and is a Banach space with norm.
Let X be a Hilbert space, Y a reflexive Banach space and Λ be a nonempty, closed and convex subset Y. Let (A:Xrightarrow X), (B XtimesLambdarightarrowmathbb{R}), (phi :Xtomathbb{R}) be given maps to be specified later, and (w,fin X) be fixed.
Then, for some symmetric positive definite weighting matrix A to be specified later, the weighted a priori and a posteriori estimation errors are, respectively, defined as [21] e a n A = x n T A v n, and e p n A = x n T A v n + 1. (24).
where 0 < α ≤ 1, D α is the Riemann-Liouville fractional derivative, y ˜ ( t ) = t 1 − α y ( t ), f : ( 0, b ] × B → R is a given function satisfying some assumptions, and ℬ is the phase space that will be specified later. We give the definition of solutions and investigate the existence and continuous dependence of solutions to such equations in the space C 1 − α ( ( a, b ] ).
We define, for positive constants and which will be specified later, a class of functions which consists of all functions satisfying conditions (2.2), (2.3), and (3.12).
