Volume 31, Issue 3

The system of Brusselator-type reaction-diffusion equations (RDs) on open bounded convex domains 𝒟 ⊂ ℝ d {\mathcal{D}\subset\mathbb{R}^{d}} ( d ≤ 3 ) {(d\leq 3)} with Robin boundary conditions (Rbcs) has been mathematically analyzed. The Faedo–Galerkin approach is used to demonstrate the global existence and uniqueness of a weak solution to the system. The weak solution’s higher regularity findings are constructed under more regular conditions on the initial data. In addition, continuous dependence on the initial conditions has been proved.

In this paper, we introduce the f -operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q -operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

We introduce the concept of quantum integration for interval-valued functions and establish new q -Hermite–Hadamard and q -Hermite–Hadamard–Fejér inequalities for left and right log ⁢ - ⁢ h {\mathrm{log}\text{-}h} -convex interval-valued functions. Our results generalize the known ones in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and interval differential equations. We illustrate our results with examples.

Let ℳ {\mathcal{M}} be a Hilbert C * {\mathrm{C}^{*}} -module. In this paper, we show that there is a one-to-one correspondence between all Hilbert C * {\mathrm{C}^{*}} -module higher derivations { φ n : ℳ → ℳ } n = 0 ∞ {\{\varphi_{n}:\mathcal{M}\rightarrow\mathcal{M}\}_{n=0}^{\infty}} with φ 0 = I {\varphi_{0}=I} satisfying φ n ( 〈 x , y 〉 z ) = ∑ i + j + k = n 〈 φ i ( x ) , φ j ( y ) 〉 φ k ( z )   ( x , y , z ∈ ℳ , n ∈ ℕ ∪ { 0 } ) \varphi_{n}(\langle x,y\rangle z)=\sum_{i+j+k=n}\langle\varphi_{i}(x),\varphi_% {j}(y)\rangle\varphi_{k}(z)\quad(x,y,z\in\mathcal{M},\,n\in\mathbb{N}\cup\{0\}) and all Hilbert C * {\mathrm{C}^{*}} -module derivations { ψ n : ℳ → ℳ } n = 1 ∞ {\{\psi_{n}:\mathcal{M}\rightarrow\mathcal{M}\}_{n=1}^{\infty}} satisfying ψ n ( 〈 x , y 〉 z ) = 〈 ψ n ( x ) , y 〉 z + 〈 x , ψ n ( y ) 〉 z + 〈 x , y 〉 ψ n ( z )   ( x , y , z ∈ ℳ , n ∈ ℕ ) , \psi_{n}(\langle x,y\rangle z)=\langle\psi_{n}(x),y\rangle z+\langle x,\psi_{n% }(y)\rangle z+\langle x,y\rangle\psi_{n}(z)\quad(x,y,z\in\mathcal{M},\,n\in% \mathbb{N}), and we show that for every Hilbert C * {\mathrm{C}^{*}} -module higher derivation { φ n } n = 0 ∞ {\{\varphi_{n}\}_{n=0}^{\infty}} on ℳ {\mathcal{M}} , there exists a unique sequence of Hilbert C * {\mathrm{C}^{*}} -module derivations { ψ n } n = 1 ∞ {\{\psi_{n}\}_{n=1}^{\infty}} on ℳ {\mathcal{M}} such that ψ n = ∑ k = 1 n ( ∑ ∑ j = 1 k r j = n ( - 1 ) k - 1 ⁢ r 1 ⁢ φ r 1 ⁢ φ r 2 ⁢ … ⁢ φ r k ) \psi_{n}=\sum_{k=1}^{n}\biggl{(}\sum_{\sum_{j=1}^{k}r_{j}=n}(-1)^{k-1}~{}r_{1}% \varphi_{r_{1}}\varphi_{r_{2}}\dots\varphi_{r_{k}}\biggr{)} for all positive integers n , where the inner summation is taken over all positive integers r j {r_{j}} with ∑ j = 1 k r j = n {\sum_{j=1}^{k}r_{j}=n} .

The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave.

This paper provides a comparison between the fractal calculus of fractal sets and fractal curves. There are introduced the analogues of the Riemann–Liouville and Caputo integrals and derivatives for fractal curves, which are non-local derivatives. Moreover, the concepts analogous to the fractional Laplace operator to address fractal non-local differential equations on fractal curves are defined. Additionally, in the paper it is introduced the fractal local Mellin transform and fractal non-local transform as tools for solving fractal differential equations. The results are supported with tables and examples to demonstrate the findings.

In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented.

Recently, Freud and Rodriguez proposed a new counting process which is called the Bell–Touchard process and based on the Bell–Touchard probability distribution. This process was developed to solve the problem of rare events hypothesis which is one of the limitations of the Poisson process. In this paper, we consider the discrete degenerate Bell distributions and the degenerate Bell process which are “degenerate versions” of the Bell–Touchard probability distributions and the Bell–Touchard process, respectively. We investigate several properties of the degenerate Bell distribution. We introduce the degenerate Bell process by giving two equivalent definitions and show one method of constructing a new infinite family of degenerate Bell process out of a given infinite family of degenerate Bell process.

In this paper, we investigate several new classes of multi-dimensional almost automorphic type sequences and focus on their applications to various difference equations involving Volterra difference equations. We provide many structural results, illustrative examples and open problems about the notion under consideration.

Let 𝒜 {\mathcal{A}} be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ {\mathcal{H}} which is closed under adjoint operation. Suppose that ϕ : 𝒜 → 𝒜 {\phi:\mathcal{A}\to\mathcal{A}} is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ {\ast} -derivation.

In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic ( 𝔻 {\mathbb{D}} -valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -convexity, 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -strict convexity, and 𝔹 ⁢ ℂ {\mathbb{B}\mathbb{C}} -uniform convexity. Moreover, the basic inequalities such as 𝔻 {\mathbb{D}} -Hölder’s inequality and 𝔻 {\mathbb{D}} -Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties.

Let Φ i {\Phi_{i}} be Young functions and ω i {\omega_{i}} be weights on ℝ d {\mathbb{R}^{d}} , i = 1 , 2 , 3 {i=1,2,3} . A locally integrable function m ⁢ ( ξ , η ) {m(\xi,\eta)} on ℝ d × ℝ d {\mathbb{R}^{d}\times\mathbb{R}^{d}} is said to be a bilinear multiplier on ℝ d {\mathbb{R}^{d}} of type ( Φ 1 , ω 1 ; Φ 2 , ω 2 ; Φ 3 , ω 3 ) {(\Phi_{1},\omega_{1};\Phi_{2},\omega_{2};\Phi_{3},\omega_{3})} if B m ⁢ ( f 1 , f 2 ) ⁢ ( x ) = ∫ ℝ d ∫ ℝ d f 1 ^ ⁢ ( ξ ) ⁢ f 2 ^ ⁢ ( η ) ⁢ m ⁢ ( ξ , η ) ⁢ e 2 ⁢ π ⁢ i ⁢ 〈 ξ + η , x 〉 ⁢ 𝑑 ξ ⁢ 𝑑 η B_{m}(f_{1},f_{2})(x)=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\hat{f_{1}}(% \xi)\hat{f_{2}}(\eta)m(\xi,\eta)e^{2\pi i\langle\xi+\eta,x\rangle}\,d\xi\,d\eta defines a bounded bilinear operator from L ω 1 Φ 1 ⁢ ( ℝ d ) × L ω 2 Φ 2 ⁢ ( ℝ d ) {L^{\Phi_{1}}_{\omega_{1}}(\mathbb{R}^{d})\times L^{\Phi_{2}}_{\omega_{2}}(% \mathbb{R}^{d})} to L ω 3 Φ 3 ⁢ ( ℝ d ) {L^{\Phi_{3}}_{\omega_{3}}(\mathbb{R}^{d})} . We deduce some properties of this class of operators. Moreover, we give the methods to generate bilinear multipliers between weighted Orlicz spaces.

Let ℒ 2 = ( - Δ ) 2 + V 2 {\mathcal{L}_{2}=(-\Delta)^{2}+V^{2}} be the Schrödinger-type operator on ℝ n {\mathbb{R}^{n}} ( n ≥ 5 {n\geq 5} ), let H ℒ 2 1 ⁢ ( ℝ n ) {H^{1}_{\mathcal{L}_{2}}(\mathbb{R}^{n})} be the Hardy space related to ℒ 2 {\mathcal{L}_{2}} , and let BMO θ ⁢ ( ρ ) {\mathrm{BMO}_{\theta}(\rho)} be the BMO-type space introduced by Bongioanni, Harboure and Salinas. In this paper, we investigate the boundedness of commutator [ b , T α , β , j ] {[b,T_{\alpha,\beta,j}]} , which is generated by the Riesz transform T α , β , j = V 2 ⁢ α ⁢ ∇ j ⁡ ℒ 2 - β {T_{\alpha,\beta,j}=V^{2\alpha}\nabla^{j}\mathcal{L}_{2}^{-\beta}} , j = 1 , 2 , 3 {j=1,2,3} , and b ∈ BMO θ ⁢ ( ρ ) {b\in\mathrm{BMO}_{\theta}(\rho)} . Here, 0 < α ≤ 1 - j 4 {0<\alpha\leq 1-\frac{j}{4}} , j 4 < β ≤ 1 {\frac{j}{4}<\beta\leq 1} , β - α = j 4 {\beta-\alpha=\frac{j}{4}} , and the nonnegative potential V belongs to both the reverse Hölder class RH s {\mathrm{RH}_{s}} with s ≥ n 2 {s\geq\frac{n}{2}} and the Gaussian class associated with ( - Δ ) 2 {(-\Delta)^{2}} . The L p {L^{p}} boundedness of [ b , T α , β , j ] {[b,T_{\alpha,\beta,j}]} is obtained, and it is also shown that [ b , T α , β , j ] {[b,T_{\alpha,\beta,j}]} is bounded from H ℒ 2 1 ⁢ ( ℝ n ) {H^{1}_{\mathcal{L}_{2}}(\mathbb{R}^{n})} to weak L 1 ⁢ ( ℝ n ) {L^{1}(\mathbb{R}^{n})} .

It it well known that a Hilbert space V of functions defined on U is a reproducing kernel Hilbert space if and only if for any z ∈ U {z\in U} , in the set V z := { f ∈ V ∣ f ⁢ ( z ) = 1 } {V_{z}:=\{f\in V\mid f(z)=1\}} , if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality K ⁢ ( z , w ) = K ⁢ ( w , z ) ¯ {K(z,w)=\overline{K(w,z)}} does not have to be true without assumptions that K ⁢ ( z , w ) ≠ 0 , K ⁢ ( w , z ) ≠ 0 {K(z,w)\neq 0,K(w,z)\neq 0} . We give sufficient and necessary conditions for a Banach space of functions to be a reproducing kernel Banach space. At the end, we give some examples including ones which show how reproducing kernel Banach spaces can be used to solve extremal problems of Partial Differential Equations Theory.