Volume 45, Issue 4

In this paper, we propose a novel concept of modified intuitionistic fuzzy double controlled metric spaces. Unlike traditional intuitionistic fuzzy bb-metric spaces, this framework integrates two non-comparable functions within the triangle inequality. To demonstrate the relevance and applicability of our results, we include illustrative examples and practical applications.

This paper focuses on establishing the existence of a capacity solution for the following anisotropic elliptic-parabolic system with variable exponent: { u t − ∑ i = 1 d [ | u x i | p i ⁢ ( x ) − 2 ⁢ u x i ( 1 + | u | ) γ ⁢ ( p i ⁢ ( x ) − 1 ) ] x i = Θ ⁢ ( u ) ⁢ | ∇ v | 2 in ⁢ Q T = Ω × ( 0 , T ) , div ⁡ ( Θ ⁢ ( u ) ⁢ ∇ v ) = 0 in ⁢ Q T , u = 0 , v = v 0 on ⁢ Γ = ∂ Ω × ( 0 , T ) , u ⁢ ( ⋅ , 0 ) = u 0 in ⁢ Ω , \left\{\begin{aligned} {}u_{t}-\mathop{\sum_{i=1}^{d}}\biggl{[}\frac{\lvert u_{x_{i}}\rvert^{p_{i}(x)-2}u_{x_{i}}}{(1+\lvert u\rvert)^{\gamma(p_{i}(x)-1)}}\biggr{]}_{x_{i}}&=\Theta(u)\lvert\nabla v\rvert^{2}&&\text{in}\ Q_{T}=\Omega\times(0,T),\\ \operatorname{div}(\Theta(u)\nabla v)&=0&&\text{in}\ Q_{T},\\ u=0,\quad v&=v_{0}&&\text{on}\ \Gamma=\partial\Omega\times(0,T),\\ u(\,{\cdot}\,,0)&=u_{0}&&\text{in}\ \Omega,\end{aligned}\right. where Ω is an open bounded subset of R d \mathbb{R}^{d} with d > 2 d>2 , Q T Q_{T} is the cylinder Ω × ( 0 , T ) \Omega\times(0,T) with T > 0 T>0 , 0 < γ < 1 0<\gamma<1 , p i ∈ C ⁢ ( Ω ̄ ) p_{i}\in\mathcal{C}(\overline{\Omega}) for all i = 1 , … , d i=1,\ldots,d , Θ ∈ C ⁢ ( R ) \Theta\in\mathcal{C}(\mathbb{R}) , u 0 ∈ L 2 ⁢ ( Ω ) u_{0}\in L^{2}(\Omega) , and v 0 ∈ L 2 ⁢ ( 0 , T ; H 1 ⁢ ( Ω ) ) ∩ L ∞ ⁢ ( Q T ) v_{0}\in L^{2}(0,T;H^{1}(\Omega))\cap L^{\infty}(Q_{T}) .

We consider two Finsler p -Laplacian overdetermined problems in a bounded domain with smooth boundary, and show the existence and non-existence of solution based on comparison with radial solutions. By imposing some conditions we show that the solution of the overdetermined problem exists only when the domain is a Finsler ball.

Consider the energy E α ⁢ [ Σ ] = ∫ Σ | p | α ⁢ d Σ E_{\alpha}[\Sigma]=\int_{\Sigma}\lvert p\rvert^{\alpha}\,d\Sigma , where Σ is a surface in Euclidean space R 3 \mathbb{R}^{3} and α ∈ R \alpha\in\mathbb{R} . We prove that planes and spheres are the only stationary surfaces for E α E_{\alpha} with constant Gauss curvature. We also characterize these surfaces assuming that a principal curvature is constant or that the mean curvature is constant.

Let ( M 1 , g ) (M_{1},g) and ( M 2 , h ) (M_{2},h) be two Hermitian manifolds. Then the doubly warped product (abbreviated as DWP) Hermitian manifold of ( M 1 , g ) (M_{1},g) and ( M 2 , h ) (M_{2},h) is the product manifold M 1 × M 2 M_{1}\times M_{2} endowed with the warped product Hermitian metric G = f 2 2 ⁢ g + f 1 2 ⁢ h G=f^{2}_{2}g+f^{2}_{1}h , where f 1 f_{1} and f 2 f_{2} are positive smooth functions on M 1 M_{1} and M 2 M_{2} , respectively. In this paper, the formula of altered Chern scalar curvature of DWP-Hermitian manifold is derived. The necessary and sufficient conditions for a DWP-Hermitian manifold to be Kähler-like is obtained. It is also proved that a compact Kähler-like DWP-Hermitian manifold is Chern flat when it has constant holomorphic sectional curvature or constant holomorphic bisectional curvature.

In the work, the author derives an integral representation of the Gauss hypergeometric functions F 1 2 ⁢ ( a - 1 2 , a ; a + 1 2 ; z ) {{}_{2}F_{1}(a-\frac{1}{2},a;a+\frac{1}{2};z)} by three approaches, applies the integral representation to give integral representations of several functions involving the inverse tangent function and including the Wilf function, and find out several combinatorial identities.