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A class of nonlinear systems with new boundary conditions: existence of solutions, stability and travelling waves
(Vilnius Gediminas Tech Univ, 2025) Lamamri, Abdelkader; Gouari, Yazid; Dahmani, Zoubir; Rakah, Mahdi; Sarikaya, Mehmet Zeki
In this work, we begin by introducing a new notion of coupled closed fractional boundary conditions to study a class of nonlinear sequential systems of Caputo fractional differential equations. The existence and uniqueness of solutions for the class of systems is proved by applying Banach contraction principle. The existence of at least one solution is then accomplished by applying Schauder fixed point theorem. The Ulam Hyers stability, with a limiting-case example, is also discussed. In a second part of our work, we use the tanh method to obtain a new travelling wave solution for the coupled system of Burgers using time and space Khalil derivatives. By bridging these two aspects, we aim to present an understanding of the system's behaviour.
EXISTENCE RESULTS FOR A HYBRID SYSTEM OF MIXED DIFFERENTIAL EQUATIONS WITH SEQUENTIAL FRACTIONAL DERIVATIVES
(Univ Nis, 2025) Benmehidi, Hammou; Sarikaya, Mehmet Zeki; Dahmani, Zoubir
In this paper, we focus on the study of a hybrid system of sequential type that incorporates both Caputo and Hadamard fractional derivatives. Our approach leverages the fixed point principle to derive novel results concerning the existence and uniqueness of solutions to this system. Additionally, we establish further results by employing Schaefer's fixed point theorem, which allows us to extend the applicability of our findings. To illustrate the practical relevance and application of our theoretical results, we also provide a detailed example at the conclusion of the paper. At the end, an example is given.
Extensions of Simpson's Inequality via Nonnegative Weight Functions and Fractional Operators
(Wiley, 2025) Ogunmez, Hasan; Sarikaya, Mehmet Zeki
In this paper, we present a new version of Simpson-type inequalities for differentiable functions defined on a subinterval of the positive real axis. The approach involves a nonnegative integrable weight function and provides an identity that refines the classical Simpson inequality by incorporating the first derivative of the function. A key aspect of this work is the inclusion of the Riemann-Liouville fractional integral, through which we derive specific inequalities that extend the classical framework. In certain cases, our results reduce to the well-known Simpson inequality, demonstrating the generality and flexibility of the method.MSC2020 Classification: 26A09, 26D10, 26D15, 33E20
Fractional Hermite-Hadamard and Bullen-Type Inequalities on the Discrete Time Scale
(Mdpi, 2026) Alqahtani, Rubayyi T.; Sarikaya, Mehmet Zeki
This paper develops a unified fractional version of the Hermite-Hadamard inequality and Bullen-type inequalities for convex functions defined on discrete time scales. By employing generalized fractional difference operators, the obtained result encompasses and extends previously known discrete formulations, including both the classical case and higher-order variants. Furthermore, we investigate the approximation accuracy of the introduced fractional mean operator. Specifically, we establish explicit error bounds for Lipschitz functions and for functions with convex differences, providing a more comprehensive analysis of the discrete fractional setting.
Generalized Bullen Type Inequalities and Their Applications
(2025) Sarikaya, Mehmet Zeki
This paper presents a novel extension of Bullen-type inequalities for convex functions by leveraging recently established generalized identities. Through rigorous proofs, we derive new inequalities that exhibit strong connections to both the left- and right-hand sides of the Hermite-Hadamard inequalities for Riemann-integrable functions. Additionally, we apply these results to various special means of two positive numbers.
Generalized fractional Hermite-Hadamard type inequalities for products of two co-ordinated convex mappings
(Springer Heidelberg, 2026) Budak, Huseyin; Ali, Muhammad Aamir; Sarikaya, Mehmet Zeki; Abbas, Mujahid
In this paper, we establish some generalized Hermite-Hadamard type inequalities for products of two co-ordinated convex functions by using generalized fractional integrals. The inequalities obtained in this study provide generalizations of some result given in earlier works.
GENERALIZED HERMITE-HADAMARD INTEGRAL INEQUALITIES ON INTERVALS
(Erhan SET, 2025) Sarikaya, Mehmet Zeki
The aim of this study is to derive numerous new generalized inequalities of trapezoid and midpoint types by dividing the closed interval [a, b] into n equal subintervals and utilizing convex functions in conjunction with the Hermite-Hadamard inequality. © 2025, Erhan SET. All rights reserved.
Gompertz Growth in Tumor-Immune Competition: Bifurcations, Multistability, and Chemotherapeutic Implications
(Mdpi, 2026) Alqahtani, Rubayyi T.; Ajbar, Abdelhamid; Sarikaya, Mehmet Zeki
This study investigates the nonlinear dynamics that emerge from the interactions between cancer cells and immune cells within a predator-prey model, wherein cancer cell growth obeys the Gompertz law. A bifurcation analysis allows for the identification of states of dormancy, uncontrolled growth, and multistability with and without chemotherapy. In the absence of chemotherapy, a Gompertz model predicts bistability with low (dormant) and high (active) tumor levels. However, unlike models based on the logistic equation, it shows that a stable tumor-free solution does not exist, consistent with the medical understanding about the risk of remaining disease even with successful treatment. Under chemotherapy, the model demonstrates highly complex dynamics with up to four coexisting stable steady states, stable tumor levels, and chemotherapy-induced oscillations. Parameter continuation studies show that the potency of immune recruitment, rate of cell inactivation, and drug saturation are essential in characterizing transitions among these dynamical regions. The analysis indicates that the choice of growth rate plays a significant role in determining the physiological implications for therapy, suggesting a cure for a model with a logistic growth rate, but merely tumor control for a Gompertzian scenario. Moreover, these results provide insights into optimal chemotherapy dosage to prevent problems associated with bistability and to capitalize on stable dormancy.
Hardy Type Inequalities for Convex Functions
(Prof. Dr. Mehmet Zeki SARIKAYA, 2023) Sarikaya, Mehmet Zeki
In this article, we aim to extend the scope of Hardy type inequalities by exploring their applicability to convex functions. We present various types of new Hardy integral inequalities for convex functions, which can be applied in diverse scenarios. Additionally, we provide several practical applications of these inequalities. © 2024 Elsevier B.V., All rights reserved.
Hermite-Hadamard Inequalities for a New Riemann-Liouville-Type Operator
(Mdpi, 2026) Alqahtani, Rubayyi T.; Sarikaya, Mehmet Zeki; Alharthi, Nadiyah Hussain
We introduce a new class of fractional remainder operators, denoted by R alpha,n xi+ and R alpha,n eta-, which generalize and unify various classical integral identities. Using these operators, we formulate refined Hermite-Hadamard and trapezoidal inequalities for differentiable functions. The novelty of our approach lies in its symmetric kernel structure, which facilitates tighter error bounds. Numerical examples are given, including applications to non-differentiable convex functions, to demonstrate the applicability and limitations of the derived results.
Hermite-Hadamard Inequalities via Gateaux Derivatives in Banach Spaces
(Springer Singapore Pte Ltd, 2026) Sarikaya, Mehmet Zeki
This paper establishes a new framework for Hermite-Hadamard type inequalities by employing Gateaux derivatives and the Bochner integral. By extending the classical scalar results to real Banach spaces, we provide sharp error bounds for trapezoidal and midpoint type functional approximations. Under the assumption that the magnitude of the directional curvature |phi '|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\varphi <^>{\prime }|$$\end{document} is convex, we derive deterministic error estimates with an optimal constant of 1/8. Furthermore, we demonstrate the practical utility of these results through applications in nonlinear optimization search paths, quadratic growth characterization, and the stability analysis of energy potentials in nonlinear elasticity. These results remain valid in infinite-dimensional Banach spaces and therefore extend the classical Hermite-Hadamard framework beyond the usual real-variable setting. An illustrative example on the space & ell;2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell <^>2$$\end{document} demonstrates how the proposed approach provides effective error bounds in infinite-dimensional optimization problems.
Integral bounds for differentiable functions via monotone derivative constraints and their applications
(Springer-Verlag Italia Srl, 2026) Sarikaya, Mehmet Zeki
This paper develops new integral inequalities that bound the deviation of a differentiable function from its mean using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>1$$\end{document}-bounded first and second derivatives. By leveraging monotonicity properties of these bounding functions, we refine classical Ostrowski-type results to obtain sharper and more general estimates. Several corollaries, including constant-bound cases, are also presented with rigorous proofs based primarily on Taylor expansions and integral inequalities.
Lyapunov-Type and Hartman-Wintner-Type Inequalities for a Class of Composite Fractional Integral Operators
(Mdpi, 2026) Alqahtani, Rubayyi T.; Sarikaya, Mehmet Zeki
In this paper, we establish Lyapunov-type and Hartman-Wintner-type integral inequalities for boundary value problems involving a class of composite fractional integral operators. By employing an explicit Green function representation and sharp uniform bounds for the associated kernel, we derive necessary conditions for the existence of nontrivial solutions. A distinctive feature of the obtained results is the higher-order scaling behavior induced by the interaction of left- and right-sided components of the operator, which cannot be observed in classical one-sided fractional models. As applications, we obtain quantitative nonexistence criteria and explicit lower bounds for the principal eigenvalue of the corresponding eigenvalue problem. These estimates extend classical results to a higher-order fractional framework and highlight the influence of the domain geometry on the stability of the solutions.
Milne-Mercer type inequalities for h-convex functions
(University of Nis, 2026) Benaissa, Bouharket; Sarikaya, Mehmet Zeki
In this study, Milne-Mercer type inequalities for h-convex functions involving the generalized fractional integral operators are established. In addition, new results are presented that generalized various inequalities known in the literature. © 2026, University of Nis. All rights reserved.
MILNE-TYPE INEQUALITIES FOR h- CONVEX FUNCTIONS
(Michigan State Univ Press, 2024) Benaissa, Bouharket; Sarikaya, Mehmet Zeki
Milne-type inequalities for h- convex functions involving conformable operators are established. Additionally, new results are presented that generalize various known inequalities.
New (α, β)-order pre-Gruss fractional integral inequalities and applications for CRV
(Univ Nis, Fac Sci Math, 2025) Dahmani, Zoubir; Sarikaya, Mehmet Zeki
In this paper, we employ Riemann-Liouville fractional integrals to derive two primary integral findings concerning the alpha- pre-Gruss inequality and the (alpha, beta)- pre-Gruss inequality. Our results extend existing integral inequalities reported in the literature. Additionally, we discuss some applications of our results for continuous random variables (CRV, for short) with bounded probability density functions. Some new estimates are provided in this context, along with classical results obtained as special cases from our results.
NEW FRACTIONAL INTEGRAL EXTENSIONS FOR INEQUALITIES INVOLVING MONOTONE FUNCTIONS
(Element D.O.O., 2024) Dahmani, Zoubir; Kaddar, Djamal; Sarikaya, Mehmet Zeki
This paper extends classical results on integral inequalities involving monotone functions to the domain of Riemann-Liouville fractional integrals with positive arbitrary order a . By employing a unified framework, our approach provides a more generalized understanding of the interplay between monotonicity and integrability in the case of fractional integration. We review classical results, introduce Riemann-Liouville integrals, and establish the fractional integral extensions. Our main results are presented, with discussions on their applications, contributing to a broader comprehension of this type of inequalities in mathematical analysis and its applications. © 2025 Elsevier B.V., All rights reserved.
New generalizations of fractional Iyengar-type inequalities involving multi-point quadratures and Lp norms
(University of Nis, 2026) Bilişik, Candan Can; Sarikaya, Mehmet Zeki
In this paper, we establish several new generalizations of Iyengar-type integral inequalities within the setting of Riemann-Liouville fractional calculus. By employing a fractional version of the Montgomery identity, we derive sharp estimates for the deviation between the fractional integral of a function and its discrete weighted averages at n arbitrary nodes. We extend these results to functions whose derivatives belong to the Lr [a, b] spaces by utilizing Hölder’s inequality. The established inequalities provide a unified setting that recovers the classical Iyengar-type results as special cases when the fractional order α = 1. Moreover, the influence of the node distribution on the associated error bounds is examined, and several corollaries corresponding to midpoint and trapezoidal-type rules are derived. © 2026, University of Nis. All rights reserved.
NEW GENERALIZATIONS OF HERMITE-HADAMARD TYPE INEQUALITIES
(Erhan SET, 2023) Sarikaya, Mehmet Zeki
In this study, we present a new generalization of the Hermite-Hadamard type inequalities for convex functions using a newly developed generalized an identity, which is rigorously proven. Moreover, we present new inequalities that are closely linked to both the left and right-hand side of the Hermite-Hadamard inequalities for Riemann and Riemann-Liouville fractional integrals. The results of this study build upon previous works and provide additional insights. © 2025 Elsevier B.V., All rights reserved.
New regular perturbation for a sequential random differential problem of Airy type
(Univ Nis, Fac Sci Math, 2024) Fettouch, Houari; Dahmani, Zoubir; Sarikaya, Mehmet Zeki; Beddani, Hamid
In this paper, we study a new problem of random differential equations of Airy type by means of the stochastic mean square theory. A new perturbation problem is introduced and some existence and uniqueness results for stochastic process solutions are established. At the end, an example is discussed in details.

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