This paper investigates a nonlinear initial value problem involving the φ–Caputo fractional derivative within the framework of a Banach space. Fractional differential equations of this type have gained significant attention due to their ability to model complex dynamical systems with memory and hereditary properties. To establish the existence of mild solutions for the considered problem, we employ the Meir–Keeler fixed point theorem in combination with the concept of measure of noncompactness. This approach allows us to derive sufficient conditions under which the problem admits at least one mild solution, extending the applicability of fractional analysis to abstract Banach space settings. The theoretical framework developed herein provides a general methodology for analyzing nonlinear fractional differential equations, and the results are applicable to a wide class of nonlinear operators. Furthermore, to illustrate the practical relevance and applicability of the theoretical findings, a representative example is presented, demonstrating how the derived conditions ensure solvability in a concrete scenario. Overall, the paper contributes to the understanding of nonlinear φ–Caputo fractional differential equations by offering both rigorous existence results and a clear methodological approach that can be adapted to related problems in fractional calculus and functional analysis. The results provide a foundation for further studies on qualitative behavior, stability, and numerical approximations of such equations.

To understand the complexity of the operator chain rule, we must first revisit the most fundamental product rule. In a commutative setting, the derivative of a product fg is simply:
　(fg)' = f'g + fg' = f'g + g'f.　Because the order does not matter, we can rearrange the terms freely. However, when the chain rule is exposed to a non-commutative operator situation, the order becomes sacrosanct. For two operators A and B, the derivative of their product is strictly:
(AB)' = A'B + A B',
where A' must be located on the left of B, and B' must be located on the right of A. This rigid preservation of order is essential to all non-commutative calculus, and it is definitely true when differentiating the exponential eA, which is an infinite product of operator A.
In this article, the 'Geometry of Variation in Operator Calculus' is studied for the cases when unbounded operators in a Banach space are included. The standard chain rule, when confronted with a non-commutative operator situation, undergoes a fundamental transformation. This evolution is deeply rooted in the structural relationship between operators. Starting from the generalized Baker–Campbell–Hausdorff (BCH) formula based on the logarithmic representation of operators [1-7], which describes the non-commutative multiplication of exponentials, we can derive the precise dynamics of operator variations. Ultimately, this necessitates the generalization of the classical chain rule into what we know as the non-commutative Duhamel’s identity.

References:
[1] Y. Iwata, Methods Funct. Anal. Topology 23 1 (2017) 26-36.
[2] Y. Iwata, Methods Funct. Anal. Topology 25 2 (2019), 142-15
[3] Y. Iwata, Adv. Math. Phys. Vol. 2020, Article ID 3989572.
[4] Y. Iwata, Math. Meth. Appl. Sci. 9002, 2023.
[5] Y. Iwata,, Mathematics 8 (2020) 747
[6] Y. Iwata, Chaos, Solitons & Fractals: X 13 (2024) 100119.
[7] Y. Iwata, arXiv:2203.00378

\textbf{Introduction.} Fixed point theory is central to nonlinear analysis and to the study of differential and fractional models arising in applied sciences. Generalized nonexpansive-type mappings have attracted increasing attention because they allow the treatment of problems beyond classical contractions. In this paper, we develop a new iterative framework for approximating fixed points of mappings satisfying the $(B_{\gamma,\mu,\eta})$condition, which extends several existing operator classes and iterative algorithms.

\textbf{Methods.} We propose a novel iterative scheme for mappings defined on Banach spaces under the $(B_{\gamma,\mu,\eta})$ condition. By employing tools from nonlinear functional analysis, we study the behavior of the generated sequence and establish sufficient conditions that ensure both weak and strong convergence of the iteration to a fixed point of the underlying mapping.

\textbf{Results.} The main results guarantee convergence under general $(B_{\gamma,\mu,\eta})$ assumptions, thereby broadening the applicability of classical fixed-point methods. The practicality of the scheme is demonstrated through an application to a Caputo-type fractional-order model describing the interaction between tumor stem cell proliferation and cellular crowding. In this setting, fractional derivatives incorporate memory effects in tumor dynamics, and the proposed iteration efficiently approximates equilibrium states of the model.

\textbf{Conclusions.} The developed algorithm provides a robust computational tool for generalized fixed-point problems and contributes to the mathematical modeling of cancer stem cell behavior in fractional biological systems.

According to the Riemann mapping theorem, for an arbitrary simply connected domain $B\subset \mathbb{C}$ and a point $a\in B$, there exists a unique regular function $f(z)$ that univalently maps the given domain onto an open disk centered at the origin such that $f(a)=0$ and $f'(a)>0$. The radius of the resulting disk is defined as the conformal radius of the domain $B$ at the point $a$. In the context of multiply connected domains, the concept of an inner radius serves as a crucial analogue, typically expressed through the Green function of the domain. This research focuses on the extremal problems associated with the products of inner radii for systems of non-overlapping domains, which remains a significant topic in geometric function theory. The study yielded new upper estimates for the products of inner radii of mutually non-overlapping domains containing fixed points on the unit circle. Taking into account these estimates we derived analytical formulas for the potential generated by a system of point sources located in specified configurations. These formulas allow for a formal description of the relationship between the positions of the points $a_k$ and the maximum possible values of the inner radii products. The obtained estimates extend existing theorems in the field of univalent functions and potential theory. These findings have potential applications in mathematical physics, particularly in problems related to electrostatics and fluid dynamics where Green’s functions are fundamental.

This talk investigates a system of nonlinear wave equations exhibiting time-varying delay and logarithmic nonlinearity. We first prove the
local existence and uniqueness of solutions using semigroup theory. Our main result establishes that solutions with negative initial energy undergo
finite-time blow-up, generalizing the scalar case. Our objective throughout this paper is to provide a comprehensive analysis
of the following coupled nonlinear wave system. This system models
the interaction of two nonlinear wave fields subject to internal damping and
logarithmic-type source terms. The nonlinear damping functions g(ut) and g(vt)
contribute to energy dissipation, whereas the coupled source terms may act as
energy generators, depending on the size and structure of the solution.
The strong coupling between u and v, together with the presence of logarithmic
nonlinearities, leads to a delicate competition between damping and source
effects. As a result, solutions may exhibit different qualitative behaviors, including
global existence with energy decay or finite-time blow-up for certain classes
of initial data. We introduce the functional setting, state the main assumptions, and establish the
well-posedness of the problem by proving the existence and uniqueness of weak solutions using semigroup theory and monotone operator techniques. In Section 3, we study the finite-time blow-up of solutions with negative initial energy by constructing an appropriate Lyapunov functional and employing nonlinear differential inequalities together with the concavity method.

This paper analyzes the real dynamical system generated by the quadratic iteration x_n+1 = x_n²- 2. with particular attention to the distribution and behavior of rational and irrational initial values. The map admits exactly two real fixed points, 2 and -1, and all orbits with x₀ outside [-2,2] diverge monotonically to +infinity . For x₀ inside [-2,2], the orbit remains confined to this interval, and its structure is examined through the full backward‑iteration tree defined by x= + or - √2. This construction yields two countable dense subsets of [-2,2] consisting of all rational and irrational preimages of the fixed points. Their forward orbits converge to 2 and -1, respectively, and the nested‑radical representation provides a complete ordering of these preimage sets. Beyond these convergent families, the backward‑iteration framework produces uncountably many additional dense subsets of [-2,2], each arising from a distinct irrational seed whose orbit is not a preimage of either fixed point. These sets are pairwise disjoint and contain both rational and irrational elements, yet none of their forward orbits converge or diverge; instead, they remain perpetually in [-2,2] while exhibiting non‑periodic, non‑convergent behavior. The resulting decomposition of [-2,2] into countably many convergent branches and uncountably many non‑convergent branches highlights the intricate, fractal‑like structure inherent in this quadratic map.

This paper seeks to develop an integration theory based on the classical Henstock–Kurzweil integral ([4]) for abstract measure spaces. More specifically, the objective is to define an HK-type integral in measure spaces that admit a fractal structure, which are characterized by their recursive behavior. These structures were introduced by M.A. Sánchez-Granero and further explored by J.F. Gálvez-Rodríguez to construct measures within such frameworks ([1], [2]).

First, we define the classes of partitions of the space, which are formed by the elements of the fractal structure, and set up the concept of gauge as a function γ : X → N. In order to guarantee the existence of these partitions, an analogous result to Cousin’s Lemma for compact intervals of R^m is proved. Furthermore, the relationship between the classical theory and this fractal integral in R^m is demonstrated by utilizing a natural fractal structure
defined on compact intervals.

Secondly, we recover fundamental properties of the classical HK-integral, such as Cauchy’s Criterion and Henstock’s Lemma, by assuming additional conditions on the underlying fractal structure. Additionally, we prove that the Monotone and Dominated Convergence Theorems hold for this fractal HK-type integral.

Finally, an application of this integral in Linearly Ordered Topological Spaces (LOTS) ([3]) is provided. We present the construction of a fractal structure in a second-countable LOTS that satisfies desirable properties for this integration theory, leading to a version of the Fundamental Theorem of Calculus for LOTS.

The classical Hardy–Littlewood–Sobolev theory establishes fundamental mapping properties for integral operators of potential type. It provides precise boundedness conditions for Riesz potentials acting between Lebesgue spaces, based on the order of the potential, the space dimension, and the integrability exponent. The present research develops a methodological framework for extending this theory beyond its classical limits, as demonstrated through a series of recently obtained results on Riesz potential-type operators. In particular, boundedness is studied not only in classical Lebesgue spaces but also in their modern extensions, such as grand Lebesgue spaces. These spaces form a refined scale for describing integrability, especially for functions with borderline singularities. A key focus lies in the case where the classical Sobolev condition is violated. It is shown that under certain parameters, the Riesz potential-type operator with a power–logarithmic kernel remains bounded from an $L^p$ space to a generalized Hölder space, even when the order of the potential exceeds the classical critical exponent. In this setting, the image of an integrable function is proven to possess a quantified generalized Hölder smoothness. The fundamental methodological contribution of this research is the development of a mapping theory for generalized function spaces, namely, grand Lebesgue spaces and generalized Hölder spaces. The techniques used include spectral analysis via Fourier–Laplace multipliers, Zygmund-type estimates for the continuity modulus, and the properties of grand Lebesgue spaces. This talk aims to outline how such a methodology paves the way for new, more general forms of the Hardy–Littlewood–Sobolev theory, with potential implications for problems in mathematical physics and fractional calculus.

In this talk we analyze the existence of positive solutions for an h-Riemann–Liouville fractional differential equation (E) that involves a positive parameter and a singular nonlinearity that changes sign, subject to nonlocal boundary conditions (BC) incorporating Riemann–Stieltjes integrals and h-Riemann-Liouville fractional derivatives of various orders. Because the nonlinearity may take negative values, the problem is referred to as a semipositone fractional boundary value problem. The h-Riemann–Liouville fractional derivative extends several well-known fractional derivatives, including the classical Riemann–Liouville derivative when h(t)=t, the Hadamard derivative when h(t)=ln t, as well as other related fractional operators. We establish intervals for the parameter for which problem (E), (BC) admits at least one positive solution. We begin by deriving the Green function corresponding to the problem and examining several of its key properties. Afterwards, through a suitable change of variables, we reformulate the original problem into an equivalent one. An operator is then constructed in a suitable Banach space, and its fixed points correspond precisely to the solutions of the equivalent problem. Our main results are obtained by applying the Guo–Krasnosel’skii fixed point theorem of cone expansion and compression of norm type. In the end, we present examples that highlight the usefulness of the established results.

Measles, a highly contagious disease, remains a significant public health challenge despite the availability of effective vaccines whose spread dynamics are influenced by population density, contact rates, and vaccination coverage. Mathematical modeling is essential for forecasting outbreaks and evaluating intervention strategies. Thus, we aimed to analyze the threshold dynamics of measles in Algeria.

This study used the SEIRV+D (Susceptible–Exposed–Infected-Recovered–Vaccinated+Deceased) model, which tracks six key groups, to analyze measles transmission. We performed equilibrium analysis, the Disease-Free Equilibrium (DFE) and derived Endemic Equilibrium (EE). The stability of these equilibria was analyzed through the basic reproduction number ($\mathcal{R}_0$). Stability analysis confirms that the DFE is locally asymptotically stable when $\mathcal{R}_0<1$, while the EE becomes locally asymptotically stable when $\mathcal{R}_0>1$, and the DFE is globally asymptotically stable when $\mathcal{R}_0<1$ and unstable if $\mathcal{R}_0>1$.

Furthermore, the detailed bifurcation analysis reveals the occurrence of a forward bifurcation at $\mathcal{R}_0=1$, and sensitivity analysis helped identify the parameters that most influence the basic reproduction number $\mathcal{R}_0$, highlighting the critical role of vaccination-related factors. Numerical simulations for Algeria demonstrate that increasing the vaccination rate from suboptimal coverage to the WHO-recommended threshold of $95 \%$ significantly reduces both the cumulative number of infections and measles-attributable deaths. The model clearly shows the direct relationship between vaccination coverage, the magnitude of $\mathcal{R}_0$, and the long-term disease burden. The main results show that measles elimination in Algeria was achieved through modest vaccination coverage.