[Paper Review] Fractional differential inclusions with a new class of set-valued contractions

Hüseyin Işık|arXiv (Cornell University)|Jul 12, 2018

Fixed Point Theorems Analysis16 references4 citations

TL;DR

This paper introduces a new class of set-valued contractions—generalizing $θ$-contractions—by weakening conditions on the auxiliary function $θ$ and incorporating auxiliary functions. It establishes new fixed point theorems for multivalued mappings in complete metric spaces, which are then applied to prove the existence of solutions for a nonlocal Caputo-type fractional differential inclusion with integral boundary conditions.

ABSTRACT

The aim of this study to investigate the existence of solutions for the following nonlocal integral boundary value problem of Caputo type fractional differential inclusions. To achieve our goals, we take advantage of fixed point theorems for multivalued mappings satisfying a new class of contractive conditions in the setting of complete metric spaces. We derive new fixed point results which extend and improve many results in the literature by means of this new class of contractions. We also supply some examples to support the new theory.

Motivation & Objective

• To develop a new class of multivalued contractions that generalize existing $θ$-contraction frameworks by weakening the conditions on the function $θ$ and introducing auxiliary functions.
• To establish fixed point theorems for multivalued mappings in complete metric spaces under this new contraction condition, extending prior results in the literature.
• To apply the new fixed point theory to prove the existence of solutions for a nonlocal Caputo-type fractional differential inclusion with integral boundary conditions.
• To demonstrate the generality and applicability of the new contraction framework through concrete examples and comparisons with existing results.
• To provide sufficient conditions ensuring the existence of solutions for fractional differential inclusions under nonlocal integral boundary conditions using the developed fixed point theory.

Proposed method

• Introduce a new class of contractions for multivalued mappings by modifying the $θ$-contraction framework, allowing weaker conditions on the function $θ$ and incorporating auxiliary functions.
• Define the generalized Pompeiu-Hausdorff metric $H$ on the space of nonempty, closed, bounded subsets of a complete metric space to measure distances between sets.
• Establish a fixed point theorem for multivalued mappings satisfying the new contraction condition, proving the existence of a fixed point under the new contractive condition.
• Apply the fixed point theorem to the solution operator $\Lambda_F$ associated with the fractional differential inclusion problem, showing that $\Lambda_F$ is a $\vartheta_\rho$-contraction.
• Use the properties of the function $\vartheta(t) = e^{\sqrt{t}}$ and the contraction condition $\vartheta(H(\Lambda_F(x), \Lambda_F(\tilde{x}))) \leq [\vartheta(\rho(\|x-\tilde{x}\|, \dots))]^k$ with $k \in (0,1)$ to verify the contraction condition.
• Verify the contraction condition by bounding the Hausdorff distance $H(\Lambda_F(x), \Lambda_F(\tilde{x}))$ using norms of the multivalued map $F$ and the functions $g_k$, leading to $H(\Lambda_F(x), \Lambda_F(\tilde{x})) \leq e^{-\tau}\|x - \tilde{x}\|$ for some $\tau > 0$.

Experimental results

Research questions

• RQ1 Can the conditions on the function $\theta$ in $\theta$-contractions be weakened while still preserving the existence of fixed points for multivalued mappings?
• RQ2 Can the new class of contractions be used to generalize existing fixed point theorems for multivalued maps in complete metric spaces?
• RQ3 Does the new contraction framework allow for the existence of solutions in fractional differential inclusions with nonlocal integral boundary conditions?
• RQ4 How does the new contraction condition compare in generality and applicability to existing results such as those by Banach, Nadler, Jleli and Samet, and Vetro?
• RQ5 Can the new fixed point theory be effectively applied to prove existence results for nonlinear fractional differential inclusions with complex boundary conditions?

Key findings

• The paper establishes a new class of multivalued contractions that generalize existing $\theta$-contraction results by weakening the conditions on $\theta$ and introducing auxiliary functions.
• A new fixed point theorem is proven for multivalued mappings in complete metric spaces under the new contraction condition, extending results from Banach, Nadler, Jleli and Samet, and Vetro.
• The solution operator $\Lambda_F$ for the fractional differential inclusion is shown to be a $\vartheta_\rho$-contraction, satisfying $\vartheta(H(\Lambda_F(x), \Lambda_F(\tilde{x}))) \leq [\vartheta(\rho(\|x-\tilde{x}\|, \dots))]^k$ with $k = \sqrt{e^{-\tau}} \in (0,1)$.
• For the example problem with $\beta = 6.7$, $T=1$, and $\alpha=0.5$, the constants $\gamma_1 \approx 2.07$, $\gamma_2 \approx 0.5727$, and $\|m\| \approx 0.125$ yield $\gamma_1\|m\| + \gamma_2 \approx 0.83145 \leq e^{-\tau}$ for $\tau \in (0, \frac{1}{6}]$, satisfying the contraction condition.
• The existence of a solution to the nonlocal fractional differential inclusion is guaranteed by the fixed point theorem, as the solution operator $\Lambda_F$ has a fixed point in the complete metric space.
• The theoretical framework is supported by two examples: one general and one specific, where the contraction condition is verified numerically, demonstrating the applicability and strength of the new results.