In this work, we propose a nonlinear fourth‐order system of elliptic partial differential equations as follows: where the constants , , , and satisfy the relation The parameters and are real, and and denote the deposition rates of the incoming particles. The parameters and satisfy where and are the critical values of and , respectively. We investigate the system of radial solutions of the governing coupled PDEs. Since the radial system involves a nonself‐adjoint operator, exhibits a singularity at the origin, and is nonlinear and of higher order, it is difficult to analyze both theoretically and numerically. In this work, we determine the sign of the radial solutions. We apply the monotone iterative method to the associated radial problem to establish the existence of at least one solution in a continuous space. We also compute the parameter ranges for and for which the nonexistence of solutions is guaranteed. To obtain approximate solutions, we develop an iterative scheme and provide numerical results to validate the theoretical findings.
ABSTRACT This paper investigates periodic solutions in the broad sense of evolutionary systems subject to periodic conditions. For the corresponding homogeneous system, normalized fundamental matrices are constructed. For nonhomogeneous linear systems, conditions ensuring the existence and uniqueness of periodic solutions in the broad sense are obtained, and the asymptotic stability of these solutions with respect to the time variable is investigated. The analysis is then extended to nonlinear evolutionary systems of triangular‐block type. Using the contraction mapping principle, sufficient conditions for the existence and uniqueness of periodic solutions in the broad sense are established, and their asymptotic stability is proved. An example is presented to verify the obtained sufficient conditions and illustrate the asymptotic stability of the periodic solution. The results are applicable to a wide class of evolutionary systems subject to periodic conditions.
ABSTRACT The primary objective of this research is to establish and analyze wavelet‐type approximation properties for a newly constructed class of ‐Szász‐Kantorovich operators. This work bridges quantum calculus (‐calculus) and wavelet analysis to enhance function approximation in various mathematical spaces. Furthermore, the results demonstrate that these operators are a general extension of several classical frameworks, including the ‐Szász‐Mirakjan and standard Kantorovich operators. We explore wavelet‐type approximation properties for ‐Szász‐Kantorovich operators by introducing specific unbounded positive sequences. Our construction integrates ‐Szász type operators with wavelet‐based Kantorovich variants to achieve ‐approximation results for . Furthermore, we use the Ditzian‐Totik ‐functional to characterize second‐order Lipschitz spaces and establish fundamental inequalities. Moreover, the numerical illustrations demonstrate that this wavelet‐infused approach is particularly favorable for approximating functions that exhibit both smoothness and rapid growth.
ABSTRACT The ray transform integrates a symmetric rank tensor field on over lines. In the case of , the range characterization of the operator on weighted Sobolev spaces was obtained by Krishnan and Sharafutdinov. Here we obtain a range characterization result in higher‐order weighted Sobolev spaces in two dimensions. Range characterization in the case of is very different from that for , and this allows us to obtain such a result in higher order weighted Sobolev spaces for any real . Nevertheless, our main tool is again the Reshetnyak formula stating that for a solenoidal tensor field .
ABSTRACT In this paper, we explore the convergence rate of the periodic homogenization of the weighted ‐Laplace equations with two different kinds of correctors. The results fill the gap for the case , where the convergence rate is not known yet. We reduce the problem of calculating the convergence rate to the layer and co‐layer estimates of the inhomogeneous ‐Laplace type equations. We concentrate on the case where the corrector is linear on by imposing a special condition on the weight, which simplifies the proof. At last, our results return to convergence rate when , which coincide with the previous results.
