PPubMed5 Jan 2015Cited 3×

An Existence Theorem for Fractional <i> q </i> -Difference Inclusions with Nonlocal Substrip Type Boundary Conditions

Ahmed Alsaedi, Sotiris K. Ntouyas, Bashir Ahmad

Abstract

By employing a nonlinear alternative for contractive maps, we investigate the existence of solutions for a boundary value problem of fractional q-difference inclusions with nonlocal substrip type boundary conditions. The main result is illustrated with the aid of an example.

§ The Valyu brief

The paper, in five minutes

Read from the full text via Valyu Contents

The paper proves, via a contractive‑mapping fixed‑point theorem, that a fractional q‑difference inclusion with nonlocal substrip boundary conditions has at least one solution.

What’s new

i.Establishes an existence theorem for fractional q‑difference inclusions with nonlocal substrip type boundary conditions
ii.Applies the nonlinear alternative for contractive maps to a multivalued operator framework
iii.Provides an illustrative example confirming the theoretical conditions
iv.Extends previous single‑valued results to the multivalued inclusion setting

How they did it

The authors decompose the operator associated with the boundary value problem into a contraction (F1) and a compact, upper‑semicontinuous multivalued map (F2). Using the nonlinear alternative for contractive mappings (a fixed‑point theorem), they verify the required hypotheses and obtain existence of a solution.

What they found

Theorem 10 guarantees at least one solution of the problem under hypotheses H1–H4
An explicit example with ν=3/2, q=1/2, b=1/5, ω=1/4, δ=3/4 satisfies all conditions, yielding ϑ≈0.191667, μ₀≈1.3990, k₀≈3.9564, ||p||=43/48, ψ(||x||)=1 and M>1.7026

Caveats

✕Only existence (no uniqueness or explicit solution) is proved
✕Results rely on restrictive hypotheses H1–H4 (e.g., L¹‑Carathéodory multivalued map, Lipschitz constant l with lk₀<1)
✕Applicable only for order 1<ν≤2 and 0<q<1
✕The illustrative example is limited to specific parameter choices

§ The Valyu brief

Reading the full paper and taking notes. This takes a few seconds…

§ Ask this paper

Ask a question about this paper

Valyu reads the full text and answers from what the paper actually says.

§ Related work in other archives

An Existence Theorem for Fractional <i> q </i> -Difference Inclusions with Nonlocal Substrip Type Boundary Conditions

The paper, in five minutes

Ask a question about this paper

Similar papers in all four archives

The paper, in five minutes

Similar papers in all four archives