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In this paper, we prove the existence of weak solutions to Neumann boundary value problems for a nonlinear p(x)-elliptic equation of the form -div a(x,u,∇u)=b(x)| u |p(x)-2u+λH(x,u, ∇u). We established the existence result by using the topological degree introduced by Berkovits.

The focus of this paper, is to establish the existence of a weak solution for a problem that involves p(x)-Laplacian. We achieve this by utilizing the topological degree theory, which is based on a class of demi-continuous operators of generalized (𝑆+) type, as presented in [6], in conjunction with the theory of variable-exponent Sobolev spaces. Additionally, we provide a numerical example to verify and validate the theoretical results.

This paper focuses on investigating the existence and uniqueness of solutions for a novel category of Ψ-Hilfer-type fractional differential equations. We employ the approach of topological degree theory for condensing maps to establish the existence of solutions. Additionally, to deal with the uniqueness, we utilize Banach's contraction principle. To demonstrate the practical implications of our theoretical findings, we present an illustrative example.

The aim of this manuscript is to investigate the existence and uniqueness of solutions for a class of nonlinear ψ-Caputo fractional pantograph differential equations with damping and nonlocal conditions. The proofs are based on results from topological degree theory for condensing maps, combined with the technique of measures of noncompactness. As an application, a nontrivial example is presented to illustrate the theoretical results.

Many real-world phenomena exhibit multi-step behavior, demanding mathematical models capable of capturing complex interactions between distinct processes. While fractional-order models have been successfully applied to various systems, their inherent smoothness often limits their ability to accurately represent systems with discontinuous changes or abrupt transitions. This paper introduces a novel framework for analyzing nonlinear fractional evolution control systems using piecewise hybrid derivatives with respect to a nondecreasing function W (ι) . Building upon the theoretical foundations of piecewise hybrid derivatives, we establish sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions, leveraging topological degree theory and functional analysis. Our results significantly improve upon existing theoretical understanding by providing less restrictive conditions for stability compared with standard fixed-point theorems. Furthermore, we demonstrate the applicability of our framework through a simulation of breast cancer disease dynamics, illustrating the impact of piecewise hybrid derivatives on the model’s behavior and highlighting advantages over traditional modeling approaches that fail to capture the multi-step nature of the disease. This research provides robust modeling and analysis tools for systems exhibiting multi-step behavior across diverse fields, including engineering, physics, and biology.

The purpose of this article is to study a singular, discontinuous elliptic problem involving the fractional p-Laplace operator with a Dirichlet boundary condition in the context of fractional Sobolev spaces. Under suitable conditions, the existence of weak solutions to the problem is established via topological degree theory.

We establish the existence of multiple solutions for singular quasilinear Lane–Emden type systems with a precise sign information: two unique solutions of opposite constant sign and a nodal solution with at least components of opposite constant sign. For sign-coupled systems, these components are of changing and synchronized sign. The approach combines sub-supersolutions method and Leray–Schauder topological degree involving perturbation argument.

The purpose of this paper is mainly to investigate the existence of weak solution of the stationary Kirchhoff type equations driven by the fractional p(x)-Laplacian operator with discontinuous nonlinearities for a class of elliptic Dirichlet boundary value problems. By using the topological degree based on the abstract Hammerstein equation, we conduct our existence analysis. The fractional Sobolev space with variable exponent provides an effective functional framework for these situations.

In this paper, we consider the second-order three point boundary value problem on time scales with integral boundary conditions on a half-line. We will use the upper and lower solution method along with the Schauder’s fixed point theorem to establish the existence of at least one solution which lies between pairs of unbounded upper and lower solutions. Further, by assuming two pairs of unbounded upper and lower solutions, the Nagumo condition on the nonlinear term involved in the first-order derivative, we will establish the existence of multiple unbounded solutions on an infinite interval by using the topological degree theory. The results of this paper extend the results of Akcan and Çetin (2018), Akcan and Hamal (2014), Eloe, Kaufmann and Tisdell (2006), and generalize the results of Lian and Geng (2011). Examples are included to illustrate the validation of the results.

In this paper, we study the existence and uniqueness of solution (EUS) as well as Hyers-Ulam stability for a coupled system of FDEs in Caputo’s sense with nonlinear p -Laplacian operator. For this purpose, the suggested coupled system is transferred to an integral system with the help of four Green functions G α 1 ( t , s ) , G β 1 ( t , s ) , G α 2 ( t , s ) , G β 2 ( t , s ) . Then using topological degree theory and Leray-Schauder’s-type fixed point theorem, existence and uniqueness results are proved. An illustrative and expressive example is given as an application of the results.