Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups

Al-Omari, Ahmad; Rashid, Mohammad H. M.

doi:10.3390/axioms15050350

Open AccessArticle

Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups

by

Ahmad Al-Omari

^1,*

and

Mohammad H. M. Rashid

²

¹

Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan

²

Department of Mathematics & Statistics, Faculty of Science, Mutah University, Alkarak 61710, Jordan

^*

Author to whom correspondence should be addressed.

Axioms 2026, 15(5), 350; https://doi.org/10.3390/axioms15050350

Submission received: 5 April 2026 / Revised: 30 April 2026 / Accepted: 3 May 2026 / Published: 8 May 2026

(This article belongs to the Special Issue Delay Differential Equations: Theory, Control and Applications)

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Abstract

We investigate the local well-posedness of semilinear fractional integro-differential equations in Banach spaces with the Hadamard fractional derivative. The equation is

{}^{H}D_{t}^{β} u (t) = A u (t) + φ (t, u (t), \int_{1}^{t} K (t, s) ρ (s, u (s)) d s), u (1) = u_{0},

where A generates a compact

C_{0}

semigroup. Using Schauder’s fixed point theorem, we prove local existence under linear growth conditions. Uniqueness is obtained via Banach’s contraction principle under Lipschitz assumptions. The main contribution is a detailed theorem for non-Lipschitz nonlinearities satisfying Carathéodory conditions and Osgood-type growth, where we prove the existence and additional regularity of mild solutions. An illustrative example with Lipschitz nonlinearities is provided.

Keywords: Hadamard fractional derivative; mild solutions; compact semigroup; global existence; blow-up; fractional integro-differential equations; fixed-point theorems

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MDPI and ACS Style

Al-Omari, A.; Rashid, M.H.M. Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups. Axioms 2026, 15, 350. https://doi.org/10.3390/axioms15050350

AMA Style

Al-Omari A, Rashid MHM. Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups. Axioms. 2026; 15(5):350. https://doi.org/10.3390/axioms15050350

Chicago/Turabian Style

Al-Omari, Ahmad, and Mohammad H. M. Rashid. 2026. "Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups" Axioms 15, no. 5: 350. https://doi.org/10.3390/axioms15050350

APA Style

Al-Omari, A., & Rashid, M. H. M. (2026). Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups. Axioms, 15(5), 350. https://doi.org/10.3390/axioms15050350

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Local Existence and Regularity of Mild Solutions for Hadamard Fractional Semilinear Integro-Differential Equations with Compact Semigroups

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