We study a group pursuit–evasion game with phase constraints on time scales under conformable dynamics. Sufficient conditions for capture and evasion are obtained using geometric properties of the initial positions and suitable conditions on the conformable dynamics. We prove the existence of an interception time, give conditions that guarantee capture, and obtain explicit upper bounds for the capture time. We also identify conditions under which the evader can avoid capture. The results extend known pursuit–evasion results on time scales to the conformable setting. Several examples on continuous, discrete, and nonuniform time scales, together with numerical simulations and graphical illustrations, are provided to explain the theoretical results and show the effect of conformable dynamics.
In classical calculus, concavity properties of a function are determined by the sign of its second derivative. In this work, we develop a real-variable framework for studying concavity in the nonlocal setting of Riemann–Liouville fractional derivatives of order 1<α <2 . We show that, under homogeneous boundary conditions, the sign of the fractional derivative alone does not determine global concavity. More precisely, (i) if the fractional derivative is nonnegative, non-decreasing, and nontrivial, then the solution is strictly concave up, recovering a classical-type result; (ii) if the fractional derivative is nonnegative and nontrivial, vanishes at the right endpoint, and satisfies a suitable regularity condition near the left endpoint, then the solution is necessarily strictly concave up near the left endpoint and strictly concave down near the right endpoint. In particular, the second derivative vanishes at some point in the interior. These results reveal a fundamental qualitative difference between classical and fractional derivatives. As an application, we establish the existence of a positive real zero of the Mittag–Leffler function E_α ,α -2(-z) for 1<α <2 . Moreover, denoting by λ _β the first positive real zero of E_α ,β(-z) , we prove the strict ordering λ _α -1<λ _α -2<λ _α . More broadly, our approach provides a direct method for detecting and comparing real zeros of special functions through geometric properties of solutions to fractional differential equations, offering an alternative to traditional techniques based on complex analysis.
Let 𝒱_p(λ ) be the collection of all functions defined in the open unit disc 𝔻 , having a simple pole at z=p , where 0
In this paper, we examine the relationship between demicompact operators and quasi-Fredholm operators. Additionally, we explore how demicompact linear operators relate to the upper B-Browder, Drazin, and B-Quasi-Fredholm spectra.
In this work, a mathematical analysis of a time-dependent coupled system of convection-diffusion-reaction equations with mixed Dirichlet and Neumann boundary conditions that arise in fluidized bed spray granulation is investigated. The system serves as a representation of the temperature and concentration distributions within a fluidized bed. This work establishes the existence and uniqueness of a weak solution for the model equations with linear reaction terms, relying on classical findings. Employing these outcomes in conjunction with Schauder’s fixed point theorem, the work further demonstrates the existence and uniqueness of a solution for the model equations featuring nonlinear reaction terms.
