AI 中文总结
本文针对经典不动点理论适用性局限，引入修正C类函数，构建广义压缩不等式，在特定条件下证明皮卡迭代收敛到非唯一不动点，给出统一框架、存在性结果与唯一性判别准则，并通过实例验证，为相关理论与应用研究奠定基础。
AI 中文摘要
不动点理论是非线性分析的核心，但经典结果常假设严格压缩性并保证唯一性，限制了对固有非唯一不动点映射的适用性。本文通过引入修正的C类函数，为完备度量空间中的Ciric型映射建立新的非唯一不动点定理。利用含三变量的修正C类函数结合变距函数构建广义压缩不等式，在轨道连续性和轨道完备性下，证明皮卡迭代收敛到不动点，无需全局利普希茨条件或严格压缩性。精心设计的例子说明了该框架的精确性和一般性，严格扩展了经典的Ciric和Chatterjea压缩以及混合$({\Phi},{\psi})$-压缩。主要贡献包括：一个包含经典和广义C类压缩的统一框架、非唯一不动点的存在性结果、唯一性的简单判别准则以及通过实例的实际验证。这些结果为非线性算子理论、迭代方法以及优化和变分问题中的应用提供了坚实基础。
英文摘要
Fixed point theory is central to nonlinear analysis, yet classical results often assume strict contractivity and guarantee uniqueness, limiting applicability to mappings with inherently non-unique fixed points. This paper addresses this gap by introducing a modified C-class function and establishing new non-unique fixed point theorems for Ciric-type mappings in complete metric spaces. We construct a generalized contractive inequality using a three-variable modifed C-class function combined with an altering distance function. Under orbital continuity and orbital completeness, we prove that Picard iteration converges to a fixed point without requiring global Lipschitz conditions or strict contractivity. Carefully designed examples illustrate the sharpness and generality of the framework, strictly extending classical Ciric and Chatterjea contractions as well as hybrid $(Φ,ψ)$-contractions. Our main contributions are: (i) a unified framework encompassing classical and generalized C-class contractions, (ii) existence results for non-unique fixed points, (iii) a simple criterion for uniqueness when desired, and (iv) practical validation through nontrivial examples. These results provide a robust foundation for further research in nonlinear operator theory, iterative methods, and applications in optimization and variational problems.
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