Abstract
This paper is concerned with the existence of a unique solution to a nonlinear discrete fractional mixed type sum-difference equation boundary value problem in a Banach space. Under certain suitable nonlinear growth conditions imposed on the nonlinear term, the existence and uniqueness result is established by using the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main result. MSC:26A33, 39A05, 39A10, 39A12.
Highlights
B ∈ R, such that b – a is a nonnegative integer, we define Na = {a, a +, a +, . . .} andNba = {a, a +, . . . , b} throughout this paper
We will consider the existence of a unique solution to the following discrete fractional mixed type sum-difference equation boundary value problem in the Banach space E:
Boundary value problems for differential equations in Banach spaces have been studied by many authors [ – ]
Summary
It is worth noting that, in what follows, for any Banach-valued function u defined on Na, we appeal to the convention k s=k We will consider the existence of a unique solution to the following discrete fractional mixed type sum-difference equation boundary value problem in the Banach space E: Where n – < α ≤ n, n ∈ N , α denotes the discrete Riemann-Liouville fractional difference of order α, f : Nα– × E × E × E → E is continuous, θ represents the zero element of E, α– u(∞) = limt→+∞ α– u(t) = u∞ ∈ E and t (Tu)(t) = k(t, s)u(s + α – ), s=
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