Abstract
This article is devoted to studying a nonhomogeneous boundary value problem involving Stieltjes integral for a more general form of the fractional q-difference equation with p(t)-Laplacian operator. Here p(t)-Laplacian operator is nonstandard growth, which has been used more widely than the constant growth operator. By using fixed point theorems of φ – (h, e)-concave operators some conditions, which guarantee the existence of a unique positive solution, are derived. Moreover, we can construct an iterative scheme to approximate the unique solution. At last, two examples are given to illustrate the validity of our theoretical results.
Highlights
Fractional calculus, appeared at the beginning of twentieth century, has provided many hot topics of research in many disciplines such as biological sciences, engineering, aerodynamics and communications
The widespread applications of fractional q-calculus have lead to a new development direction of fractional q-difference equations, which has exhibited adamantine incorporation to application in fluid mechanics and quantum calculus
Kinds of fixed point theorems have been used to deal with various fractional q-difference equation boundary value problems; see [1, 3, 6, 7, 10,11,12, 15, 16, 18, 19] for instance
Summary
Fractional calculus, appeared at the beginning of twentieth century, has provided many hot topics of research in many disciplines such as biological sciences, engineering, aerodynamics and communications (see [3,4,5, 13] for example). Where 0 < γ < 1, 2 < α < 3, φ : R → R is a generalized p-Laplacian operator, which includes two cases: φ(u) = u and φ(u) = |u|p−2u, p 1 They gave the existence of positive solutions by some fixed point theorems in cones. Different from the above-mentioned works, in this article, we discuss the following nonhomogeneous two-point boundary value problem of a fractional q-difference equation containing p(t)-Laplacian operator: Dqβ φp(t) Dqαu(t) − g(t) + μf t, u(t) = 0, 0 < t < 1, u(0) = 0, (Dqu)(0) = 0, (Dqu)(1) − λ[u] = γ, Dqαu(t)|t=0 = 0,.
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