We investigate an initial-boundary value problem for a quasilinear parabolic equation with inner absorption and nonlinear Neumann boundary condition. We establish, respectively, the conditions on nonlinearity to guarantee that $u(x,t)$ exists globally or blows up at some finite time $t^{*}$. Moreover, an upper bound for $t^{*}$ is derived. Under somewhat more restrictive conditions, a lower bound for $t^{*}$ is also obtained.

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form $\sum \lambda \in I{\alpha }_{\lambda }(z)({\prod }_{j=0}^{n}f{(z+c_j)}^{{\lambda }_j})=R(z,f\circ p)=((a_0(z)+a_1(z)(f\circ p)+\hspace{0.17em}\cdots \hspace{0.17em}+a_s(z)$ $(f\circ p{)}^s)/(b_0(z)+b_1(z)(f\circ p)+\hspace{0.17em}\cdots \hspace{0.17em}+b_t(z)(f\circ p{)}^t))$, where $I$ is a finite set of multi-indexes $\lambda =({\lambda }_{\mathrm{0}},{\lambda }_{\mathrm{1}},\dots ,{\lambda }_n)$, $c_0=0,c_j\in ℂ\setminus \{0\}\hspace{0.17em}(j=\mathrm{1,2},\dots ,n)$ are distinct complex constants, $p(z)$ is a polynomial, and ${\alpha }_{\lambda }(z)(\lambda \in I)$, $a_i(z)(i=\mathrm{0,1},\dots ,s)$, and $b_j(z)(j=\mathrm{0,1},\dots ,t)$ are small meromorphic functions relative to $f(z)$. We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.

This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function on $\mathrm{\Omega }\subset {ℂ}^2\times {ℂ}^2$ that has a dual split Cauchy-Riemann system in dual split quaternions.

We mainly investigate the unicity of meromorphic functions sharing two or three sets with their linear difference polynomials and prove some results.

A comparison theorem on oscillation behavior is firstly established for a class of even-order nonlinear neutral delay difference equations. By using the obtained comparison theorem, two oscillation criteria are derived for the class of even-order nonlinear neutral delay difference equations. Two examples are given to show the effectiveness of the obtained results.

The value distribution of solutions of certain difference equations is investigated. As its applications, we investigate the difference analogue of the Brück conjecture. We obtain some results on entire functions sharing a finite value with their difference operators. Examples are provided to show that our results are the best possible.

This paper is to investigate the Schwarzian type difference equation ${\left[\left({\Delta }^{3}f/\Delta f\right)-\left(3/2\right){\left({\Delta }^{2}f/\Delta f\right)}^2\right]}^k=R\left(z,f\right)=\left(P(z,f)/Q(z,f)\right),$ where $R(z,f)$ is a rational function in $f$ with polynomial coefficients, $P(z,f)$, respectively $Q(z,f)$ are two irreducible polynomials in $f$ of degree $p$, respectively $q$. Relationship between $p$ and $q$ is studied for some special case. Denote $d=\max \left\{p,q\right\}$. Let $f(z)$ be an admissible solution of $(*)$ such that ; then for $s$ (≥2) distinct complex constants ${\alpha }_1,\dots ,{\alpha }_s$, $q+2k{\sum }_{j=1}^s\delta ({\alpha }_j,f)\le 8k.$ In particular, if $N(r,f)=S(r,f)$, then $d+2k{\sum }_{j=1}^s\delta ({\alpha }_j,f)\le 4k.$

This paper proposes the least squares method to estimate the drift parameter for the stochastic differential equations driven by small noises, which is more general than pure jump $\alpha$-stable noises. The asymptotic property of this least squares estimator is studied under some regularity conditions. The asymptotic distribution of the estimator is shown to be the convolution of a stable distribution and a normal distribution, which is completely different from the classical cases.

The characteristic functions of differential-difference polynomials are investigated, and the result can be viewed as a differential-difference analogue of the classic Valiron-Mokhon’ko Theorem in some sense and applied to investigate the deficiencies of some homogeneous or nonhomogeneous differential-difference polynomials. Some special differential-difference polynomials are also investigated and these results on the value distribution can be viewed as differential-difference analogues of some classic results of Hayman and Yang. Examples are given to illustrate our results at the end of this paper.

The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form $({\sum }_{\lambda \in I}‍{\alpha }_{\lambda }(z)({\prod }_{\nu =1}^n‍f(z+c_{\nu }{)}^{l_{\lambda ,\nu }}))/({\sum }_{\mu \in J}‍{\beta }_{\mu }(z)({\prod }_{\nu =1}^n‍f(z+$ $c_{\nu }{)}^{m_{\mu ,\nu }}))=$ $Q(z,f(p(z)))$, where $I=$ $\{\lambda =(l_{\lambda ,1},l_{\lambda ,2},\dots ,l_{\lambda ,n})\mid l_{\lambda ,\nu }\in \mathbb{N}\bigcup ‍\{0\}\text{,}\nu =$ $\mathrm{1,2},\dots ,n\}$ and $J=\{\mu =(m_{\mu ,1},m_{\mu ,2},\dots ,m_{\mu ,n})\mid m_{\mu ,\nu }\in \mathbb{N}\bigcup ‍\{0\}\text{,}\nu =\mathrm{1,2},\dots ,n\}$ are two finite index sets, $c_{\nu }(\nu =\mathrm{1,2},\dots ,n)$ are distinct complex numbers, ${\alpha }_{\lambda }(z)(\lambda \in I)$ and ${\beta }_{\mu }(z)(\mu \in J)$ are small functions relative to $f(z),$ and $Q(z,u)$ is a rational function in $u$ with coefficients which are small functions of $f(z)$, $p(z)=p_k{z}^k+p_{k-1}{z}^{k-1}+\cdots +p_0\in ℂ[z]$ of degree $k\ge 1$. We also give some examples to show that our results are sharp.

We prove a unicity theorem of entire functions that share two distinct small functions with their shifts. The corollary of the theorem confirms the conjecture posed by Li and Gao (2011).

The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper.

Using value distribution theory and maximum modulus principle, the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that our results are sharp.