Searcharxiv⌕ Search

arXiv subjects

Mustafa Avci

Publications and source records attributed to Mustafa Avci.

12 recordsLinked to original sources
arXiv2026

A Topological Approach to Singular Double-Phase Equations with Variable Exponents

In the present paper, we study a singular double phase variable exponent Dirichlet problem in the setting of a new Musielak-Orlicz Sobolev space with the nonlinearity (the external source) having gradient dependence (so-called convection terms). We apply a topological existence result incorporating the Leray-Schauder degree and homotopy mapping together to prove the existence of at least one nontrivial solution.

math.AP↗
arXiv2025

On the existence, uniqueness and stability of solutions of SDEs with state-dependent variable exponent

We study a time-inhomogeneous nonlinear SDE with drift and diffusion governed by state-dependent variable exponents. This framework generalizes models like the geometric Brownian motion (GBM) and the constant elasticity of variance (CEV), offering flexibility to capture complex dynamics while posing analytical challenges. Using a fixed-point approach, we prove existence and uniqueness, analyze higher-order moments, derive asymptotic estimates, and assess stability. Finally, we illustrate an application where the Poisson equation admits a probabilistic representation via a time-homogeneous nonlinear SDE with state-dependent variable exponents.

math.PR↗
arXiv2025

Stochastic representation of solutions for the parabolic Cauchy problem with variable exponent coefficients

In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic representation, then verify it satisfies the Cauchy problem. The corresponding SDE, on the other hand, allows the drift and diffusion coefficients to respond nonlinearly to the current state through the state-dependent variable exponents, and thus, extends the expressive power of classical SDEs to better capture complex dynamics. To validate our theoretical framework, we conduct comprehensive numerical experiments comparing finite difference solutions (Crank-Nicolson on logarithmic grids) with Monte Carlo simulations of the SDE.

math.AP↗
arXiv2025

On the geometric Brownian motion with state-dependent variable exponent diffusion term

We propose a new stochastic model involving state-dependent variable exponent $p(\cdot)$ which allows modeling of systems where noise intensity adapts to the current state. This new flexible theoretical framework generalizes both the geometric Brownian motion (GBM) and the Constant-Elasticity-of-Variance (CEV) models. We prove an existence-uniqueness theorem. We obtain an upper-bound approximation for the model-to-model pathwise error between our model and the GBM model as well as test its accuracy through analytical and numerical error estimates. A detailed comparison of the It\^o and Stratonovich interpretations for the proposed model is presented in the Appendix.

math.AP↗
arXiv2025

Existence results for the Cox-Ingersoll-Ross model with variable exponent diffusion

This study proposes a new stochastic model where the diffusion coefficient involves a state-dependent variable exponent function $p(\cdot)$. This new theoretically flexible framework generalizes the classical Cox-Ingersol-Ross model. The existence, uniqueness, and higher moment properties of solutions are analyzed. The validity and efficiency of the model is illustrated with numerical experiments. Finally, a detailed analysis for It\^o vs Stratonovich interpretations of the proposed model is provided.

math.PR↗
arXiv2025

Three solutions for a double phase variable exponent Kirchhoff problem

In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a double-phase operator that governs anisotropic and heterogeneous diffusion associated with the energy functional, as well as encapsulating two different types of elliptic behavior within the same framework. To tackle the problem, we obtain regularity results for the corresponding energy functional, which makes the problem suitable for the application of a well-known critical point result by Bonanno and Marano. We introduce an $n$-dimensional vector inequality, not covered in the literature, which provides a key auxiliary tool for establishing essential regularity properties of the energy functional such as $C^1$-smoothness, the $(S_+)$-condition, and sequential weak lower semicontinuity.

math.AP↗
arXiv2025

Variational and nonvariational solutions for double phase variable exponent problems

In this article, we examine two double-phase variable exponent problems, each formulated within a distinct framework. The first problem is non-variational, as the nonlinear term may depend on the gradient of the solution. The first main result establishes an existence property from the nonlinear monotone operator theory given by Browder and Minty. The second problem is set up within a variational framework, where we employ a well-known critical point result by Bonanno and Chinn\`{\i}. In both cases, we demonstrate the existence of at least one nontrivial solution. To illustrate the practical application of the main results, we provide examples for each problem.

math.AP↗
arXiv2025

Monotone operator methods for a class of nonlocal multi-phase variable exponent problems

In this paper, we study a class of nonlocal multi-phase variable exponent problems within the framework of a newly introduced Musielak-Orlicz Sobolev space. We consider two problems, each distinguished by the type of nonlinearity it includes. To establish the existence of at least one nontrivial solution for each problem, we employ two different monotone operator methods.

math.AP↗
arXiv2017

Existence results to a nonlinear p(k)-Laplacian difference equation

In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.

math.AP↗