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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]

In this paper, we prove the existence of solutions to a nonlinear singular transport equation (ie, transport equation with unbounded collision frequency and unbounded collision operator) with vacuum boundary conditions in bounded spatial domain on Lp‐spaces with 1 ≤ p<+∞. This problem was already considered in6,8,9 under the hypothesis that the collision frequency σ(·) and the collision operator are bounded. In this work, we show that these hypotheses are not necessary; it suffices to assume that σ(·) is locally bounded, and the collision operator is bounded between Xpσ (a weighted space) and Xp (cf Section 2). Although the analysis for p∈(1,+∞) is standard in the sense that it uses the Schauder fixed point theorem, the compactness of the involved operator is not easy to derive. However, the analysis in the case p=1 uses the concept of Dunford‐Pettis operators and a new version of the Darbo fixed point theorem for a measure of weak noncompactness introduced in the paper. [ABSTRACT FROM AUTHOR]

The concern of this paper is to continue the investigation of convergence properties of nonlinear approximation operators, which are defined by Karsli. In details, the paper centers around Urysohn‐type nonlinear counterpart of the Bernstein operators. As a continuation of the study of Karsli, the present paper is devoted to obtain Voronovskaya‐type theorems for the Urysohn‐type nonlinear Bernstein operators. [ABSTRACT FROM AUTHOR]

In this paper, impulsive Lasota-Wazewska model with infinite delay is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence, which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by Liapunov functional. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we, consider an n‐th order nonlinear difference equation with parameter dependence. An exhaustive study of the related Green's function is done. The exact expression of the function is given. The range of parameter for which either it has constant sign or it changes sign is obtained. Some existence results for the nonlinear problem are deduced by using the classical Krasnosel'skii's fixed point theorem on cones and fixed point index theory. [ABSTRACT FROM AUTHOR]

In this paper, we consider a generalized predator‐prey system with prey‐taxis under the Neumann boundary condition. We investigate the local and global asymptotical stability of constant steady states (including trivial, semitrivial, and interior constant steady states). On the basis of a priori estimate and the fixed‐point index theory, several sufficient conditions for the nonexistence/existence of nonconstant positive solutions are given. [ABSTRACT FROM AUTHOR]

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin-Bona-Mahony equation, Burger's equation, Fornberg-Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of a non‐autonomous porous elastic systems with nonlinear damping and sources terms. By employing nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We also prove the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the sources terms. Finally, we prove the upper‐semicontinuity of pullback attractors with respect to non‐autonomous perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a family of nonlinear integral operators of Urysohn‐type and study the pointwise convergence of the family at characteristic points of L1−function. The kernel Kλ(x,t,u(t)) depends on the positive parameter λ changing on the set of numbers with the accumulation point at infinity and Kλ(x,t,u(t)) is an entire analytic function of variable u, which is a bounded function belonging to L1(R). [ABSTRACT FROM AUTHOR]

In this paper, sufficient conditions are established for the existence and uniqueness of global solutions to stochastic impulsive systems with expectations in the nonlinear terms. The maximal interval and the estimate of mild solutions are also discussed. These results are obtained by using the fixed point theorem, interval partition, and Lyapunov-like technique. Finally, examples are given to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we are dealing with an analytical study of a singular fractional order nonlinear differential equation with fractional integral and differential boundary conditions and ϕp‐operator, for existence and stability results. Our problem is based on two types of fractional order derivatives, that is, Caputo factional derivative of order σ and Riemann‐Liouville derivative of order β, where m − 1  <  σ,β  ≤  m, and m  ∈  {3,4,5,...}. The suggested problem will be converted into an equivalent integral form by the help of Green function. After the proofs for these properties, some classical fixed point theorems are employed for the existence of positive solution (EPS). For application of the results, an expressive example is included. [ABSTRACT FROM AUTHOR]

In this paper, we construct the pullback exponential attractors for evolution processes in which the difference of 2 solutions lacks the smoothing property. To do this, by the uniform squeezing property of the corresponding discrete process, we add the points to the pullback attractor such that every new set of it has the finite fractal dimension and pullback exponentially attracts every bounded subset of the phase space. As the applications, we establish the existence of pullback exponential attractors for non‐autonomous reaction‐diffusion equation without any restriction on the growing order of nonlinear term and non‐autonomous strongly damped wave equation in R n with critical nonlinearity. [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).$$ \left\{\begin{array}{cc}\frac{\partial u}{\partial t}&amp;#x0003D;\mathbf{A}u&amp;#x0002B;V(x){u}&amp;#x0005E;{m&amp;#x0002B;p-2}&amp;#x0002B;\lambda {u}&amp;#x0005E;q\kern0.30em &amp; \mathrm{in}\kern0.30em \Omega \times \left(0,T\right),\\ {}u\left(x,0\right)&amp;#x0003D;{u}_0(x)\ge 0\kern0.30em &amp; \mathrm{in}\kern0.30em \Omega, \\ {}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;{p-2}\frac{\partial u}{\partial \nu }&amp;#x0003D;\beta (x){u}&amp;#x0005E;{p-1}\kern0.30em &amp; \mathrm{on}\kern0.3em \mathrm{\partial \Omega}\times \left(0,T\right).\end{array}\right. $$where Au=div(mum−1|∇u|p−2∇u)$$ \mathbf{A}u&amp;#x0003D;\operatorname{div}\left(m{u}&amp;#x0005E;{m-1}{\left&amp;#x0007C;\nabla u\right&amp;#x0007C;}&amp;#x0005E;{p-2}\nabla u\right) $$ is the doubly nonlinear operator. Here, Ω⊂ℝN$$ \Omega \subset {\mathbb{R}}&amp;#x0005E;N $$ is a bounded domain with smooth boundary, m>0,10,λ∈ℝ,q>0$$ m&gt;0,1&lt;p&lt;N,V\in {L}_{loc}&amp;#x0005E;1\left(\Omega \right),m&amp;#x0002B;p-2&gt;0,\lambda \in \mathbb{R},q&gt;0 $$ and β∈Lloc1(∂Ω)$$ \beta \in {L}_{loc}&amp;#x0005E;1\left(\mathrm{\partial \Omega}\right) $$. We establish some sufficient conditions on the functions V(x),β(x)$$ V(x),\beta (x) $$, and the exponents m+p$$ m&amp;#x0002B;p $$ 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]

By studying a weakly singular integral whose kernel involves Mittag‐Leffler functions, we obtain some new Gronwall‐type integral inequalities. Applying these inequalities and fixed point theorems, existence and uniqueness of positive solution of initial value problem to nonlinear fractional differential equation with Caputo‐like counterpart hyper‐Bessel operators are established. [ABSTRACT FROM AUTHOR]

We discuss the existence of solutions for a class of nonlinear equations driven by the fractional p‐Laplacian operator of the form (−Δ)psu+λsV(x,λ)|u|p−2u=f(x,u)inRN,where 0ps and λ is a positive parameter. By combining variational techniques with a version of the mountain pass theorem without Palais‐Small (PS) condition, we establish the existence of nontrivial solutions for the above equation under certain appropriate assumptions on nonlinearity and weight functions. [ABSTRACT FROM AUTHOR]

In this article, we prove the Liouville‐type theorem for stable solutions of weighted p‐Laplace–type Grushin equations 1−divG(a(z)|∇Gu|p−2∇Gu)=h(z)eu,z=(x,y)∈RN=RN1×RN2 and 2divG(a(z)|∇Gu|p−2∇Gu)=h(z)u−q,z=(x,y)∈RN=RN1×RN2,where p ≥ 2, q>0 and a(z),h(z)∈Lloc1(RN) are nonnegative functions satisfying a(z)≤C1‖z‖Gb and h(z)≥C2‖z‖Gθ as ‖z‖G ≥ R0 with p−Nγqc(p,Nγ,b,θ) in. Here, μ0 and qc are new exponents, which are always larger than the classical critical ones and depend on the parameters p,b and θ. Nγ=N1+(1+γ)N2 is the homogeneous dimension of RN. [ABSTRACT FROM AUTHOR]

A generalized non‐linear nonautonomous model for the haematopoiesis (cell production) with several delays and an oscillating circulation loss rate is studied. We prove a fixed point theorem in abstract cones, from which different results on existence and uniqueness of positive almost periodic solutions are deduced. Moreover, some criteria are given to guarantee that the obtained positive almost periodic solution is globally exponentially stable. [ABSTRACT FROM AUTHOR]

By application of Green's function and some fixed-point theorems, that is, Leray-Schauder alternative principle and Schauder's fixed-point theorem, we establish two newexistence results of positive periodic solutions for nonlinear fourthorder singular differential equation, which extend and improve significantly existing results in the literature. [ABSTRACT FROM AUTHOR]