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This paper aims to study the approximate controllability of semilinear population dynamics system with diffusion using semigroup theory. The semilinear population dynamical model with the nonlocal birth process is transformed into a standard abstract semilinear control system by identifying the state, control, and the corresponding function spaces. The state and control spaces are assumed to be Hilbert spaces. The semigroup theory is developed from the properties of the population operators and Laplacian operators. Then the approximate controllability results of the system are obtained using C0$$ {C}_0 $$‐semigroup approach and some other simple conditions on the nonlinear term and operators involved in the model. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the regularity in time of solutions for nonlinear fractional reaction–subdiffusion equations (FrRSEs). By constituting regularity estimates of resolvent operator and employing fixed point theorems, we establish some results on the global existence and regularity in time of solutions to FrRSE in two different cases of nonlinear perturbations, namely, sublinear and superlinear cases. As an application of these results, several results on the existence and regularity of solutions in an inverse source problem governed by FrRSE with the additional measurements given at terminal time will be shown. [ABSTRACT FROM AUTHOR]

In this paper, using sunny nonexpansive retractions, which are different from the metric projection in Banach spaces, a new type of study regarding the iterative methods in view of two quasi‐nonexpansive nonself mappings is presented. We also give the convergence analysis for the proposed method in the background of uniformly convex Banach spaces. Moreover, we apply our results to find solutions of common zeros of accretive operators, convexly constrained least square problems, and convex minimization problems. Furthermore, we also discuss novel applications of these methods to differential problems, image deblurring, and signal recovering problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence of extreme solutions and their properties for a general σ$$ \sigma $$‐Hessian equation involving a nonlinear operator. By introducing a suitable growth condition and developing a iterative technique, some new results on existence and asymptotic estimates of minimum and maximum solutions are derived. Moreover, we also establish the iterative sequences that converge uniformly to the extreme solutions. [ABSTRACT FROM AUTHOR]

Let V$$ V $$ be a finite set, E⊂2V$$ E\subset {2}^V $$ be a set of hyperedges, and w:E→(0,∞)$$ w:E\to \left(0,\infty \right) $$ be an edge weight. On the (wighted) hypergraph G=(V,E,w)$$ G=\left(V,E,w\right) $$, we can define a multivalued nonlinear operator LG,p$$ {L}_{G,p} $$ (p∈[1,∞)$$ p\in \left[1,\infty \right) $$) as the subdifferential of a convex function on ℝV$$ {\mathbb{R}}^V $$, which is called "hypergraph p$$ p $$‐Laplacian." In this article, we first introduce an inequality for this operator LG,p$$ {L}_{G,p} $$, which resembles the Poincaré–Wirtinger inequality in PDEs. Next, we consider an ordinary differential equation on ℝV$$ {\mathbb{R}}^V $$ governed by LG,p$$ {L}_{G,p} $$, which is referred as "heat" equation on the graph and used to study the geometric structure of the hypergraph in recent researches. With the aid of the Poincaré–Wirtinger type inequality, we can discuss the existence and the large time behavior of solutions to the ODE by procedures similar to those for the standard heat equation in PDEs with the zero Neumann boundary condition. [ABSTRACT FROM AUTHOR]

Lp$$ {L}^p $$ estimates for the fractional Φ$$ \Phi $$‐Laplacian operator defined in bounded domains are established, where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with Moser's iteration, we prove that any weak solution for fractional Φ$$ \Phi $$‐Laplacian operator defined in bounded domains belongs to L∞(Ω)$$ {L}^{\infty}\left(\Omega \right) $$, under appropriate hypotheses on the N$$ N $$‐function Φ$$ \Phi $$. Using the Orlicz space and taking into account the fractional setting for our problem, the main results are stated for a huge class of nonlinear operators and nonlinearities. [ABSTRACT FROM AUTHOR]

In this article, we consider the nonlinear problem ∂u∂t=Au+V(x)um+p−2+λuqinΩ×(0,T),u(x,0)=u0(x)≥0inΩ,|∇u|p−2∂u∂ν=β(x)up−1on∂Ω×(0,T).$$ \left\{\begin{array}{cc}\frac{\partial u}{\partial t}=\mathbf{A}u+V(x){u}^{m+p-2}+\lambda {u}^q\kern0.30em & \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)={u}_0(x)\ge 0\kern0.30em & \mathrm{in}\kern0.30em \Omega, \\ {}{\left|\nabla u\right|}^{p-2}\frac{\partial u}{\partial \nu }=\beta (x){u}^{p-1}\kern0.30em & \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right).\end{array}\right. $$where Au=div(mum−1|∇u|p−2∇u)$$ \mathbf{A}u=\operatorname{div}\left(m{u}^{m-1}{\left|\nabla u\right|}^{p-2}\nabla u\right) $$ is the doubly nonlinear operator. Here, Ω⊂ℝN$$ \Omega \subset {\mathbb{R}}^N $$ is a bounded domain with smooth boundary, m>0,10,λ∈ℝ,q>0$$ m>0,1<p<N,V\in {L}_{loc}^1\left(\Omega \right),m+p-2>0,\lambda \in \mathbb{R},q>0 $$ and β∈Lloc1(∂Ω)$$ \beta \in {L}_{loc}^1\left(\mathrm{\partial \Omega}\right) $$. We establish some sufficient conditions on the functions V(x),β(x)$$ V(x),\beta (x) $$, and the exponents m+p$$ m+p $$ and q$$ q $$, so that the above problem has no positive solutions. Furthermore, various concrete potentials V(x)$$ V(x) $$ are taken into account to demonstrate applications of our main result. [ABSTRACT FROM AUTHOR]

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic N‐function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework of capacities in fully anisotropic Orlicz–Sobolev spaces is developed and the capacitary characterization of a bounded measure is given. Moreover, we give an example of an anisotropic Young function Φ, such that |ξ|p≲Φ(ξ)≲|ξ|plogα(1+|ξ|), with arbitrary p ≥ 1, α > 0, but so irregularly growing that the Orlicz–Sobolev‐type space generated by Φ indispensably requires fully anisotropic tools to be handled. [ABSTRACT FROM AUTHOR]