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This paper delves into the exploration of value sharing in connection with a transcendental entire function f and its differential-difference polynomials. Motivated by some recent results of Dong and Liu (Math Slovaca 67(3):691–700, 2017), we present here some results that can be viewed as the differential-difference analogue of the Brück Conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we have exhaustively studied the uniqueness of meromorphic function sharing five small functions in k-punctured complex plane, under the aegis of truncated sharing. Most importantly, by reducing the lower bound of the truncated sharing, our result drastically improved one of the main theorems of Gui et al. (AIMS Math 5(6):7438–7457, 2020). [ABSTRACT FROM AUTHOR]

Fermat type functional equation with four terms f (z) n + g (z) n + h (z) n + k (z) n = 1 is difficult to solve completely even if n = 2 , 3 , in which the certain type of the above equation is also interesting and significant. In this paper, we first to consider the Bi-Fermat type quadratic partial differential equation f (z 1 , z 2) 2 + ∂ f (z 1 , z 2) ∂ z 1 2 + g (z 1 , z 2) 2 + ∂ g (z 1 , z 2) ∂ z 1 2 = 1 in C 2 . In addition, we consider the Bi-Fermat type cubic difference equation f (z) 3 + g (z) 3 + f (z + c) 3 + g (z + c) 3 = 1 in C and obtain partial meromorphic solutions on the above equation. [ABSTRACT FROM AUTHOR]

In this paper, we study value distribution of meromorphic functions concerning differences and mainly prove the following result: Let f be a transcendental meromorphic function of 1 ≤ ρ (f) < ∞ , let c be a nonzero constant, n a positive integer, and let P, Q be two polynomials. If max λ (f - P) , λ 1 f < ρ (f) and Δ c n f ≢ 0 , then we have (i) δ (Q , Δ c n f) = 0 and λ (Δ c n f - Q) = ρ (f) , for Δ c n P ≢ Q ; (ii) δ (Q , Δ c n f) = 1 and λ (Δ c n f - Q) < ρ (f) , for Δ c n P ≡ Q . The results obtained in this paper extend and improve some results due to Chen-Shon[J Math Anal Appl 2008], [Sci China Ser A 2009], Liu[Rocky Mountain J Math 2011], Cui-Yang[Acta Math Sci Ser B 2013], Chen[Complex Var Elliptic Equ 2013], Wang-Liu-Fang[Acta Math. Sinica (Chinese Ser) 2016]. [ABSTRACT FROM AUTHOR]

The existence of entire solutions of certain types of nonlinear homogeneous differential equations are investigated. We also find explicit forms of entire solutions of two types of nonlinear monomial differential equations, which are connected to trigonometric identities. Some further questions about nonlinear homogeneous differential equations are posed. [ABSTRACT FROM AUTHOR]

This study is devoted to exploring the existence and the precise form of finite-order transcendental entire solutions of second-order trinomial partial differential-difference equations L(f)2+2hL(f)f(z1+c1,z2+c2)+f(z1+c1,z2+c2)2=eg(z1,z2)L{(f)}^{2}+2hL(f)f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})+f{\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})}^{2}={e}^{g\left({z}_{1},{z}_{2})} and L˜(f)2+2hL˜(f)(f(z1+c1,z2+c2)−f(z1,z2))+(f(z1+c1,z2+c2)−f(z1,z2))2=eg(z1,z2),\tilde{L}{(f)}^{2}+2h\tilde{L}(f)(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))+{(f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})-f\left({z}_{1},{z}_{2}))}^{2}={e}^{g\left({z}_{1},{z}_{2})}, where L(f)L(f) and L˜(f)\tilde{L}(f) are defined in (2.1) and (2.2), respectively, and g(z)g\left(z) is a polynomial in C2{{\mathbb{C}}}^{2}. Our results are the extensions of some of the previous results of Liu et al. Also, we exhibit a series of examples to explain that the forms of transcendental entire solutions of finite-order in our results are precise.

With the help of the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we mainly give a description of the existence and the forms of the transcendental entire solutions of several product-type complex partial differential equations systems (a1fz1+a2fz2)(a3gz1+a4gz2)=eh1,(a1gz1+a2gz2)(a3fz1+a4fz2)=eh2,\left\{\begin{array}{l}({a}_{1}{f}_{{z}_{1}}+{a}_{2}{f}_{{z}_{2}})({a}_{3}{g}_{{z}_{1}}+{a}_{4}{g}_{{z}_{2}})={e}^{{h}_{1}},\\ ({a}_{1}{g}_{{z}_{1}}+{a}_{2}{g}_{{z}_{2}})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{f}_{{z}_{2}})={e}^{{h}_{2}},\end{array}\right. and (a1fz1+a2gz1)(a3fz2+a4gz2)=eh1,(a1fz2+a2gz2)(a3fz1+a4gz1)=eh2,\left\{\begin{array}{l}({a}_{1}{f}_{{z}_{1}}+{a}_{2}{g}_{{z}_{1}})({a}_{3}{f}_{{z}_{2}}+{a}_{4}{g}_{{z}_{2}})={e}^{{h}_{1}},\\ ({a}_{1}{f}_{{z}_{2}}+{a}_{2}{g}_{{z}_{2}})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{g}_{{z}_{1}})={e}^{{h}_{2}},\end{array}\right. where fzt=∂ft∂zt{f}_{{z}_{t}}=\frac{\partial {f}_{t}}{\partial {z}_{t}}, gzt=∂gt∂zt{g}_{{z}_{t}}=\frac{\partial {g}_{t}}{\partial {z}_{t}}, t=1,2t=1,2; aj∈C,j=1,2,3,4{a}_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2,3,4, and h1{h}_{1} and h2{h}_{2} are the polynomials in C2{{\mathbb{C}}}^{2}. More importantly, we list some examples to explain that our results about the forms of solutions of such systems are precise to some extent.

This paper deals with the extension of Hayman Conjecture to q-shift differential polynomials generated by meromorphic functions. We prove that the differential polynomials assumes every finite value infinitely often when sharing two Borel exceptional Values with appropriate conditions. Few examples are given to prove that the conditions are inevitable. [ABSTRACT FROM AUTHOR]

In this paper we are focusing on finding the transcendental entire solution of Fermat-type trinomial and binomial equations, by restricting the hyper-order to be less than one. As the hyper-order is a crucial parameter that characterizes the growth of entire functions, it will be interesting to investigate this unexplored domain, as far as practible, with certain restriction on hyper order. Our results are the improvements of previous results reported in recent papers [12], [13]. We have provided a series of examples to demonstrate and validate the effectiveness of our proposed solutions.

In this article, we study the unicity of meromorphic functions concerning small functions and derivatives-differences. The results obtained in this article extend and improve some results of Chen et al. [Uniqueness problems on difference operators of meromorphic functions] and Chen and Huang [Uniqueness of meromorphic functions concerning their derivatives and shifts with partially shared values].

Our goal in this paper is to establish some difference analogue of second main theorems for holomorphic curves into projective varieties intersecting arbitrary families of c-periodical hypersurfaces (fixed or moving) with truncated counting functions in various cases. Our results generalize and improve the previous results in this topic. [ABSTRACT FROM AUTHOR]

In this paper, we focus on investigating the entire solutions to one certain type of non-linear binomial differential equations with respect to several problems posed by Gundersen and Yang. We also illustrate the exponential polynomials solutions to this equation. Some examples are used to illustrate our results. [ABSTRACT FROM AUTHOR]

We design an Algorithm to fabricate universal holomorphic maps between any two complex Euclidean spaces, within preassigned transcendental growth rate. As by-products, universal holomorphic maps from C n to C P m ( n ⩽ m ) and to complex tori having slow growth are obtained. We take inspiration from Oka manifolds theory, Nevanlinna theory, and hypercyclic operators theory. [ABSTRACT FROM AUTHOR]

This article is devoted to exploring the solutions of several systems of the first-order partial differential difference equations (PDDEs) with product type u(z+c)[α1u(z)+β1uz1+γ1uz2+α2v(z)+β2vz1+γ2vz2]=1,v(z+c)[α1v(z)+β1vz1+γ1vz2+α2u(z)+β2uz1+γ2uz2]=1,\left\{\begin{array}{l}u\left(z+c){[}{\alpha }_{1}u\left(z)+{\beta }_{1}{u}_{{z}_{1}}+{\gamma }_{1}{u}_{{z}_{2}}+{\alpha }_{2}v\left(z)+{\beta }_{2}{v}_{{z}_{1}}+{\gamma }_{2}{v}_{{z}_{2}}]=1,\\ v\left(z+c){[}{\alpha }_{1}v\left(z)+{\beta }_{1}{v}_{{z}_{1}}+{\gamma }_{1}{v}_{{z}_{2}}+{\alpha }_{2}u\left(z)+{\beta }_{2}{u}_{{z}_{1}}+{\gamma }_{2}{u}_{{z}_{2}}]=1,\end{array}\right. where c=(c1,c2)∈C2c=\left({c}_{1},{c}_{2})\in {{\mathbb{C}}}^{2}, αj,βj,γj∈C,j=1,2{\alpha }_{j},{\beta }_{j},{\gamma }_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2. Our theorems about the forms of the transcendental solutions for these systems of PDDEs are some improvements and generalization of the previous results given by Xu, Cao and Liu. Moreover, we give some examples to explain that the forms of solutions of our theorems are precise to some extent.

In this article, we prove two results on the value distribution of meromorphic functions. Using the theorem of Yamanoi, the first result gives a precise estimation of the relationship between the characteristic function of a meromorphic function and its kkth derivative in a concise form. This result extends and improves some results of Shan, Singh, Gopalakrishna, Edrei, Weitsman, Yang, Wu and Wu, etc. The second result answers a conjecture posed by C. C. Yang. This conjecture turned to be false by a counter-example, but it will be true with an additional condition.

The objective of this study is to determine whether the existence of shared sets S containing both meromorphic (entire) functions and their higher derivatives, as well as powers of meromorphic functions and their differential polynomials, could result in uniqueness. The main focus is on determining the precise solutions to various differential equations. This problem is being studied in a broader context, specifically in set sharing. [ABSTRACT FROM AUTHOR]

In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [2]. [ABSTRACT FROM AUTHOR]

In this article some results on the value distribution theory of analytic functions defined in angles of C , due mainly to B. Ja. Levin and A. Pfluger, will be extended to the more general situation where the functions are defined in angles of C \ { 0 } . More precisely, angles S (θ 1 , θ 2) with vertex at the origin will be considered and where a singularity at zero is allowed. An special class of these functions are those of completely regular growth for which it is proved a basic result which yields an expression of the density of its zeros in terms of the indicator function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and specific form of finite order transcendental entire solutions of certain equations including a Fermat-type functional first-order linear difference equation in C n , n ⩾ 2 and a kth order partial differential difference equation in C 2 . The paper builds upon the previous works of Xu and Cao (Mediterr J Math 15:1–14, 2018; Mediterr J Math 17:1–4, 2020) and Haldar (Mediterr J Math 20: 50, 2023) whose results are extended and further developed in this study. We exhibit several examples to demonstrate the precision and applicability of our results to illustrate how our findings can be utilized in different scenarios or problem contexts. Towards the end of the paper, in the last section, we discuss some relevant questions that have emerged from one of the examples in the paper which also suggest potential directions for further research. [ABSTRACT FROM AUTHOR]

The equation f (z) n + g (z) n = 1 , n ∈ N can be regarded as the Fermat Diophantine equation over the function field. In this paper we study the characterization of entire solutions of some system of Fermat type functional equations by taking e g 1 (z) and e g 2 (z) in the right hand side of each equation, where g 1 (z) and g 2 (z) are polynomials in C n . Our results extend and generalize some recent results. Moreover, some examples have been exhibited to show that our results are precise to some extent. [ABSTRACT FROM AUTHOR]

We investigate the following two types of nonlinear differential-difference equations L (z , f) + H (z , f) = ∑ k = 1 r α k (z) e β k z ; L (z , f) + H (z , f) = ∑ k = 1 r F k (z) , where α 1 , … , α r are meromorphic functions of order < 1 , and F 1 , … , F r are periodic transcendental entire functions, and L, H are defined by L (z , f) = ∑ k = 1 p a k (z) f (m k) (z + τ k) ≢ 0 , H (z , f) = ∑ k = 1 q b k (z) [ f (n k) (z + ζ k) ] s k with small meromorphic coefficients a i , b j. By introducing a new method, we obtain the exact forms of the solutions of these two equations under certain growth conditions. [ABSTRACT FROM AUTHOR]

This paper employs Nevanlinna value distribution theory in several complex variables to explore the characteristics and form of finite as well as infinite order solutions of Circular-type nonlinear partial differential-difference equations in C n . The results obtained in this paper contribute to the improvement and generalization of some recent results. In addition, illustrative examples are provided to validate the conclusions drawn from the main results. Moreover, we investigate the infinite-order solutions of Circular-type functional equations in C n . [ABSTRACT FROM AUTHOR]

In the paper, we investigate the uniqueness problem of a meromorphic function and its difference operator to the most general setting via two shared set problems and thus improve a recent result of Chen–Chen (Bull Malays Math Sci Soc 35(3): 765-774, 2012). [ABSTRACT FROM AUTHOR]

In this paper, we answer the questions posed by Gundersen and Yang about the entire solutions of a class of nonlinear homogeneous binomial differential equations and obtain explicit forms of all the entire solutions of this type of differential equations. Moreover, we provide some examples to demonstrate that the equation solutions we obtained are accurate. [ABSTRACT FROM AUTHOR]