Abstract
In this paper, we consider a fixed point theorem that extends and unifies several existing results in the literature. We apply the proven fixed point results on the existence of solution of ordinary boundary value problems and fractional boundary value problems with integral type boundary conditions in the frame of some Caputo type fractional operators.
Highlights
Introduction and preliminariesIn the last decades, two topics have been densely studied: “fixed point theory” and “fractional differential/integral equations”
We show that T is a generalized θ -h-θ-contraction type mapping
Proof We prove that T is a generalized α-h-θ-contraction mapping
Summary
We reserve the letter G for all functions γ : [0, ∞) → [0, 1) so that γ (tn) → 1 implies tn → 0. Theorem 2.2 On a quadruplet (X, d, T, θ ), if the following assumptions hold: (i) inequality (I3) holds; (ii) T is continuous and forms a triangular θ -orbital admissible;. Theorem 2.6 On a quadruplet (X, d, T, θ ), if the following assumptions hold: (i) inequality (I3)∗ holds; (ii) X is θ -regular and T is continuous and forms triangular θ -orbital admissible;. Definition 2.7 On a structure (X, d, T, θ ), we consider the following inequality: (I6)∗ θ (κ, ζ )θ(d(Tκ, Tζ )) ≤ h(κ, ζ )θ(Q(κ, ζ )) for all κ, ζ ∈ X, where h ∈ A(X), θ ∈ Φ, and Q(κ, ζ ) is defined as in (2.9) under the condition lim n→∞. We consider the uniqueness of the derived fixed point of certain mappings that are mentioned in the theorems above.
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