We construct a semigroup of composition operators on a subspace of the Dirichlet space of the upper half-plane. We then determine both the semigroup and spectral properties of the composition semigroup. Finally, we represent the resolvents of the infinitesimal generator as integral operators and obtain their norm and spectra.
Fractional differential equations is a rapidly growing field of mathematical analysis with a wide and robust applicability in several areas of physics and geometry. Picone identity is a powerful tool which has been applied extensively in the study of second order elliptic equations. In this paper we prove some nonlinear anisotropic Picone type identities and give its applications to deriving Sturmian comparison principle and Liouville type results for anisotropic conformable fractional elliptic differential equations and systems.
This paper proposes an explicit numerical scheme based on Delannoy polynomials in conjunction with the tau method for solving the time-fractional diffusion equation involving the Caputo derivative. The proposed method constructs approximate solutions using shifted Delannoy polynomials as basis functions, allowing efficient and accurate treatment of the nonlocal nature of fractional derivatives. The method transforms the time-fractional diffusion problem into a system of algebraic equations, which can be solved explicitly. Several benchmark examples are provided to confirm the efficiency, accuracy, and applicability of the new scheme.
We consider the unsteady problem for the general planar Broadwell model with fourh velocities in a rectangular spatial domain over a finite time interval. We impose a class of non-negative initial and Dirichlet boundary data that are bounded and continuous, along with their first-order partial derivatives. We then prove the existence and uniqueness of a non-negative continuous solution, bounded together with its first-order partial derivatives, to the initial-boundary value problem.
This paper develops a comprehensive theory for variable-exponent Bochner spaces \(L^{p(\cdot)}([0,T];X)\), establishing fundamental results on compact embeddings and maximal regularity with applications to nonlocal evolution equations. We extend the classical Aubin-Lions framework through innovative modular convergence techniques, proving sharp compactness criteria under log-Holder continuity conditions. For time-dependent fractional operators, including the fractional Laplacian \((-\Delta)^{s(t)}\) and Levy-type processes with variable order \(\alpha(t)\), we derive optimal maximal regularity estimates that reveal new connections between exponent functions \(p(t)\) and operator orders. A groundbreaking contribution is our systematic analysis of fractal dimension dynamics in variable-order fractional PDEs, characterizing how evolving regularity \(s(t)\) governs solution behavior. Furthermore, we develop novel functional-analytic tools for stochastic exponents \(p(t,\omega)\), yielding compact embedding results in \(L^{p(\cdot,\omega)}(X)\) spaces and boundedness properties for nonlinear operators. Combining techniques from modular function theory, refined interpolation methods, and stochastic analysis, our work provides powerful new approaches for problems in anomalous diffusion and heterogeneous media. These results significantly advance both the theoretical foundations and practical applications of variable-exponent spaces in modern PDE analysis.
This study examines the multivariable Jensen integral inequality for convexity in coordinates. Using the co-ordinate convex functions, we give some results to develop new fractional Hermite-Hadamard type inequalities.
Inspired by a problem proposal recently published in the journal The Fibonacci Quarterly we offer a generalization consisting of two combinatorial identities involving three complex parameters. These identities turn out to be immensely rich. We demonstrate this by providing basic applications to four different fields: polynomial identities, trigonometric identities, identities involving Horadam numbers, and combinatorial identities. Many of our findings will generalize existing results.
This paper establishes the existence of traveling wave solutions in a Leslie-Gower predator-prey model featuring nonlocal dispersal and multiple time delays in both diffusion and reaction terms. The model captures realistic ecological effects such as spatial movement and delayed species responses. Due to the competitive nature of the interaction, the reaction terms satisfy only a partial monotonicity condition. We establish the existence of traveling waves. This is done by construction upper and lower solutions and developing an iterative scheme whose convergence is ensured by Schauder’s fixed point theorem. The approach is extended to accommodate a relaxed class of super and sub-solutions. Explicit examples, and numerical illustrations are provided.
In this work, we seek conditions for the existence or nonexistence of solutions for nonlinear Riemann-Liouville fractional boundary value problems of order \(\alpha + 2n\), where \(\alpha \in (m-1, m]\) with \(m \geq 3\) and \(m, n \in \mathbb{N}\). The problem’s nonlinearity is continuous and also depends on a positive parameter upon which our constraints are established. Our approach involves constructing a Green’s function by combining the Green’s functions of a lower-order fractional boundary value problem and a right-focal boundary value problem \(n\) times. Leveraging the properties of this Green’s function, we apply Krasnosel’skii’s Fixed Point Theorem to establish our results. Several examples are presented to illustrate the existence and nonexistence regions.
Let \(f(z) = \sum\limits_{k=0}^{\infty} f_k z^k\) be an entire transcendental function, and let \((\lambda_n)\) be a sequence of positive numbers increasing to \(+\infty\). Suppose that the series \(A(z) = \sum\limits_{n=1}^{\infty} a_n f(\lambda_n z)\) is regularly convergent in \(\mathbb{D} = \{ z : |z| < 1 \}\), i.e., \(\mathfrak{M}(r, A) := \sum\limits_{n=1}^{\infty} |a_n| M_f(r \lambda_n) < +\infty\) for all \(r \in [0, 1)\). For a positive function \(l\) continuous on \([0, 1)\), the function \(A\) is said to be of bounded \(l\)–\(\mathfrak{M}\)-index if there exists \(N \in \mathbb{Z}_+\) such that \[\frac{\mathfrak{M}(r, A^{(n)})}{n! \, l^n(r)} \leq \max \left\{ \frac{\mathfrak{M}(r, A^{(k)})}{k! \, l^k(r)} : 0 \leq k \leq N \right\},\] for all \(n \in \mathbb{Z}_+\) and all \(r \in [0, 1)\). The growth of bounded \(l\)–\(\mathfrak{M}\)-index functions is studied. In particular, under the conditions \(a_n \geq 0\) and \(f_k \geq 0\), it is proved that the function \(A\) is of bounded \(l\)–\(\mathfrak{M}\)-index with \(l(r) = p(1 – r)^{-(p+1)}\), \(p > 0\), if and only if \[\lim\limits_{r \uparrow 1} (1 – r)^p \ln \mathfrak{M}(r, A) < +\infty.\] This condition is satisfied if and only if \[\lim\limits_{k \to \infty} k^{-p} \left( \ln^+ (f_k \mu_D(k)) \right)^{p+1} < +\infty,\] where \(\mu_D(k) = \max\{ a_n \lambda_n^k : n \geq 1 \}\).
