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In this paper, we aim to establish a set of sufficient conditions for the existence of an integral form mild solution and approximate controllability for a class of Sobolev-type Hilfer fractional stochastic differential systems driven by the Rosenblatt process and Poisson jumps. In the proposed control problem, we deal with a system that is under non-instantaneous impulsive effect. The sufficient condition for the existence of a mild solution for the proposed nonlinear control system has been established by using Schauder’s fixed point theorem. The approximate controllability results for the proposed control problem have been established under the consideration that the corresponding linear system is approximate controllable. By utilising stochastic analysis, the theory of resolvent operator, fractional calculus, and the fixed point technique, sufficient conditions have been established. At the end, an example is given to illustrate the abstract results. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence and the asymptotic behavior of positive continuous solutions for a class of nonlinear polyharmonic boundary value problems in the punctured unit ball of $ \mathbb {R}^{n} (n \geqslant 3) $ R n (n ⩾ 3). Our arguments are based on potential theory associated to the polyharmonic operator $ (-\Delta)^{m} $ (− Δ) m , where m is a positive integer less than $ \frac {n}{2} $ n 2 , properties of functions in the Kato class $ K_{m,n} $ K m , n , Karamata regular variation theory tools and the Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of traveling wave solution for temporally discrete Lotka Volterra competitive system with delays. By using the cross iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The obtained results makes up and improves the results of the existence of traveling wave solutions for this systems. [ABSTRACT FROM AUTHOR]

This research considered the fractional controllability problem of internally controlled semilinear hyperbolic systems. In the first method, we show the controllability of the linear system using Hilbert uniqueness method HUM, and the nonlinear fractional controllability problem is solved applying Schauder's fixed point theorem. Secondly, the analytics study is then attempted by employing generalized inverse methods and changed also into a fixed point problem. Hence, we give an approximate approach to find a control that brings the Riemann-Liouville fractional attained position RL D x β z (x , T) (resp. Caputo speed C D t α z (x , T) ) to the required position y 1 d (resp. speed y 2 d ). Finally, computing simulations are used to confirm the obtained results. [ABSTRACT FROM AUTHOR]

This paper aims to study the existence of traveling wave solutions (TWS) for a three-component noncooperative systems with nonlocal diffusion. Our main results reveal that when a threshold ℜ > 1 , there exists a critical wave speed c ∗ > 0. By using sub- and super-solution methods and Schauder's fixed point theorem, we prove that the system admits a nontrivial TWS for each c ≥ c ∗ . Meanwhile, we show that there exists no nontrivial TWS for c < c ∗ by detailed analysis. Finally, we apply our results to a nonlocal diffusive epidemic model with vaccination, and the boundary asymptotic behavior of TWS for the special case is obtained by constructing a suitable Lyapunov functional. Our research provides some insights on how to deal with the problem of TWS for the nonlocal diffusive epidemic models with bilinear incidence, which extends some results in the previous studies. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence. When the threshold R$$ \mathcal{R} $$ is greater than one, we prove that there is a wave speed c∗>0$$ {c}&#x0005E;{\ast }>0 $$ such that the model has a traveling wave solution with speed c>c∗$$ c>{c}&#x0005E;{\ast } $$, and there is no traveling wave solution with speed less than c∗$$ {c}&#x0005E;{\ast } $$. When the threshold R$$ \mathcal{R} $$ is less than or equal to one, we show that there is no nontrivial traveling wave solution. The approaches we use here are Schauder's fixed point theorem with an explicit construction of a pair of upper and lower solutions, the contradictory approach, and the two‐sided Laplace transform. [ABSTRACT FROM AUTHOR]

In this article, the solvability of the pantograph equation of fractional orders under a fractal-fractional feedback control was investigated. This investigation was located in the class of all continuous functions. The necessary conditions for the solvability of that problem and the continuous dependence of the solution on some parameters and the control variable were established with the help of some fixed point theorems. Additionally, the Hyers-Ulam stability of the issue was explored. Finally, some specific problems extended to the corresponding problem with integer orders were illustrated. The theoretical results were supported by numerical simulations and comparisons with existing results in the literature. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder’s and Perov’s fixed-point theorems. Examples are given to validate the theoretical results.

In this article, the solvability of the pantograph equation of fractional orders under a fractal-fractional feedback control was investigated. This investigation was located in the class of all continuous functions. The necessary conditions for the solvability of that problem and the continuous dependence of the solution on some parameters and the control variable were established with the help of some fixed point theorems. Additionally, the Hyers-Ulam stability of the issue was explored. Finally, some specific problems extended to the corresponding problem with integer orders were illustrated. The theoretical results were supported by numerical simulations and comparisons with existing results in the literature.

The purpose of this paper is to study a generalized Riemann–Liouville fractional differential equation and system with nonlocal boundary conditions. Firstly, some properties of the Green function are presented and then Lyapunov-type inequalities for a sequential ψ-Riemann–Liouville fractional boundary value problem are established. Also, the existence and uniqueness of solutions are proved by using Banach and Schauder fixed-point theorems. Furthermore, the existence and uniqueness of solutions to a sequential nonlinear differential system is established by means of Schauder's and Perov's fixed-point theorems. Examples are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder's fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables. [ABSTRACT FROM AUTHOR]

The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using Schauder's fixed point theorem; the uniqueness of solutions is obtained by adding an additional hypothesis and applying Banach's contraction principle. An example is provided to demonstrate the results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder’s fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results.

In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder's fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an elliptic equation defined on R N , N ≥ 1 , and involving the p (u) - Laplacian. When p (u) = p (u (x)) , x ∈ R N , i.e., when p depends on the variable x ∈ R N (through the unknown solution u), we say that we are dealing with the local case of the problem. In this case the p (u) - Laplacian can be considered as a new class of operators with variable exponents. When p (u) = p (α (u)) where α is a scalar function of the unknown solution u, we say that we are dealing with the nonlocal case of the problem. In the present work, the issue of the existence of nontrivial solution in the both cases is addressed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined. [ABSTRACT FROM AUTHOR]

The main motive and, at the same time, the goal of this work is to investigate a revisited Nicholson's blowflies equation that involves a time varying delay and an iterative term. We make use of Schauder's fixed point theorem to tackle the existence of positive periodic solutions and under an additional condition, we apply the Banach contraction principle for establishing the existence, uniqueness and stability results. Finally, we give two examples to illustrate the effectiveness of our main results that are completely new and complement some earlier investigations to some extent. [ABSTRACT FROM AUTHOR]

This study introduces several updated results for the piecewise plant-pathogen-herbivore interactions model with singular-type and nonsingular fractional-order derivatives. A piecewise fractional model has developed to describe the interactions between plants, disease, (insect) herbivores, and their natural enemies. We derive essential findings for the aforementioned problem, specifically regarding the existence and uniqueness of the solution, as well as various forms of Ulam Hyers (U-H) type stability. The necessary results were obtained by utilizing fixed-point theorems established by Schauder and Banach. Additionally, the U-H stabilities were determined based on fundamental principles of nonlinear analysis. To implement the model as an approximate piecewise solution, the Newton Polynomial approximate solution technique is employed. The applicability of the model was validated through numerical simulations both in fractional as well as piecewise fractional format. The motivation of our article is that we have converted the integer order problem to a global piecewise and fractional order model in the sense of Caputo and Atangana-Baleanu operators and investigate it for existence, uniqueness of solution, Stability of solution and approximate solution along with numerical simulation for the validity of our obtained scheme. [ABSTRACT FROM AUTHOR]

We prove an existence result for strong solutions $u\in W^{2,q}\left(\Omega\right) $ of singular semilinear elliptic problems of the form $-\Delta u=g\left( \cdot,u\right) $ in $\Omega,$ $u=\tau$ on $\partial\Omega,$ where $1

In this paper, we are concerned with some new first order differential equation defined on the whole real axis $\mathbb{R}.$ The principal part of the equation involves an operator with variable exponent $p$ depending on the variable $x \in \mathbb{R} $ through the unknown solution while the nonlinear part involves the classical variable exponent $ p(x). $ Such kind of situation is very related to the presence of the variable exponent and has not been treated before. Our existence result of nontrivial solution cannot be reached using standard variational or topological methods of nonlinear analysis and some sophisticated arguments have to be employed.

In this work, we investigate a class of nonlinear combined Sturm-Liouville problems with zero Dirichlet boundary conditions. Using the Karamata regular variation theory and the Schauder fixed point theorem, we prove the existence of a unique positive solution satisfying a precise asymptotic behavior where a competition between singular and non singular terms in the nonlinearity appears. [ABSTRACT FROM AUTHOR]

The current research deals with the exact solutions of the nonlinear partial differential equations having two important difficulties, that is, the coefficient singularities and the stochastic function (white noise). There are four major contributions to contemporary research. One is the mathematical analysis where the explicit a priori estimates for the existence of solutions are constructed by Schauder's fixed point theorem. Secondly, the control of the solution behavior subject to the singular parameter ∊ when ∊ → 0. Thirdly, the impact of noise that is present in the differential equation has been successfully handled in exact solutions. The final contribution is to simulate the exact solutions and explain the plots. [ABSTRACT FROM AUTHOR]

This paper mainly concerns about the traveling wave solution (TWS) for a discrete diffusive epidemic model with asymptomatic carriers. Analysis of the model shows that the minimum wave speed c ∗ exists if a threshold ℜ is greater than one. With the help of sub- and super-solutions, we find that the condition for the existence of TWS is ℜ > 1 and wave speed c > c ∗ . Further, we prove that the TWS connects two different boundary steady states. Through the arguments with Laplace transform, we show there is no TWS for the model if ℜ > 1 and 0 < c < c ∗ or ℜ ≤ 1. [ABSTRACT FROM AUTHOR]

The current research deals with the exact solutions of the nonlinear partial differential equations having two important difficulties, that is, the coefficient singularities and the stochastic function (white noise). There are four major contributions to contemporary research. One is the mathematical analysis where the explicit a priori estimates for the existence of solutions are constructed by Schauder’s fixed point theorem. Secondly, the control of the solution behavior subject to the singular parameter ϵ when ϵ → 0. Thirdly, the impact of noise that is present in the differential equation has been successfully handled in exact solutions. The final contribution is to simulate the exact solutions and explain the plots.

We consider in an infinite horizontal layer the stationary convective flow of a viscous compressible and heat-conductive fluid subject to gravitational force, where the slip boundary condition for the velocity and the boundary condition of the temperature near the hydrostatic distribution are assumed. The existence of stationary solution close to hydrostatic state is obtained in Sobolev spaces as limit of fixed points of some suitable operators. [ABSTRACT FROM AUTHOR]

We prove the existence of at least one nontrivial solution for a third-order boundary value problem with an integral condition under different growth assumptions on the nonlinearity in the equation. The main tool in the proofs is Schauder's fixed point theorem. To compare the applicability of the obtained results, we consider some examples. [ABSTRACT FROM AUTHOR]

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: ut - μΔu + (u · ∇)u + αu + β|u|r-1u + ∇p = F := fg, ∇·u = 0; in bounded domains Ω ⊃ Rd (d = 2, 3) with smooth boundary, where α, β, μ > 0 and r ∊ [1,∞). The CBF equations describe the motion of incompressible uid ows in a saturated porous medium. The inverse problem under our consideration consists of reconstructing the vector-valued velocity function u, the pressure gradient ∇p and the vector-valued function f. We prove the well-posedness result (existence, uniqueness and stability) of an inverse problem for 2D and 3D CBF equations with the final overdetermination condition using Schauder's fixed point theorem for arbitrary smooth initial data. The well-posedness results hold for r ≥ 1 in two dimensions and for r ≥ 3 in three dimensions. The global solvability results available in the literature helped us to obtain the uniqueness and stability results for the model with fast growing nonlinearities. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish an existence theorem for a system of nonlinear fourth-order differential equations with two parameters $ \begin{eqnarray*} \left\{ \begin{array}{rcl} u^{(4)}+A(x)u& = &\lambda f(x, u, v, u'', v''), \ 0 0, \mu > 0 $ are two parameters and $ f, g: [0, 1]\times[0, \infty)\times[0, \infty)\times(-\infty, 0)\times(-\infty, 0) \rightarrow \mathbb{R} $ are two continuous functions satisfy the growth conditions.

$ {equation*} u^{(4)}+A(x)u = \lambda f (x, \ u, \ u''), \ 0 0 $ is a parameter and $ p, q \in \mathbb{L}^{1}[0, 1]. $

Anew class of implicit Hadamard fractional differential equations with Riemann-Stieltjes integral boundary conditions is studied in this research paper. The existence and uniqueness results of the aforesaid problem are investigated using Schauder's fixed point theorem and Banach's contraction mapping principle. A simulative example is given to highlight the acquired outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations. The investigation of the main results depends on Schauder's fixed point theorem and Banach's contraction principle. An illustrative example is given to show the applicability of theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates approximate controllability of semilinear measure driven equations in Hilbert spaces. We focus on a specific category of nonlocal integrodifferential equations. We apply the theory of the resolvent operator in the sense of Grimmer, as well as the fixed point strategy and the theory of the Lebesgue-Stieljes integral, in the context of the space of regulated functions. In light of this, the prevalence of our findings is greater than that which is found in the literature. At last, an example is comprised that exhibits the significance of developed theory. [ABSTRACT FROM AUTHOR]

In this article we study the existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales. The main tool employed here is Schauder’s fixed point theorem. The results obtained here extend the work of Culakova, Hanustiakova and Olach [12]. Two examples are also given to illustrate this work.

In this paper we discuss the solvability of Langevin equations with two Hadamard fractional derivatives. The method of this discussion is to study the solutions of the equivalent Volterra integral equation in terms of Mittag–Leffler functions. The existence and uniqueness results are established by using Schauder's fixed point theorem and Banach's fixed point theorem, respectively. An example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, under some sufficient conditions on the nonlinearity, we established the existence of nonnegative solutions for a hybrid nonlinear differential equation with iterative terms. Our approach is based on the application of Schauder's fixed point theorem for proving the results. Finally, an example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with solvability and algorithms for a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By employing the Banach’s fixed point theorem, Schauder’s fixed point theorem, and FanKKM theorem, we obtain a sufficient condition which guarantees the existence of solutions for the generalized nonlinear variational-like inequality. We introduce also an auxiliary variational-like inequality and, by utilizing the minimax inequality, get the existence and uniqueness of solutions for the auxiliary variational-like inequality, which is used to suggest an iterative algorithm for solving the generalized nonlinear variational-like inequality. Under certain conditions, by means of the auxiliary principle technique, we both establish the existence and uniqueness of solutions for the generalized nonlinear variational-like inequality and discuss the convergence of iterative sequences generated by the iterative algorithm. Our results extend, improve, and unify several known results in the literature.

Abstract Fractional difference equations have become important due to their qualitative properties and applications in discrete modeling. Stability analysis of solutions is one of the most widely used qualitative properties with tremendous applications. In this paper, we investigate the existence and stability results for a class of non-linear Caputo nabla fractional difference equations. To obtain the existence and stability results, we use Schauder’s fixed point theorem, the Banach contraction principle and Krasnoselskii’s fixed point theorem. The analysis of the theoretical results depends on the structure of nabla discrete Mittag-Leffler functions. An example is provided to illustrate the theoretical results.

In this paper, we consider the second order semilinear impulsive differential equations with state-dependent delay. First, we consider a linear second order system and establish the approximate controllability result by using a feedback control. Then, we obtain sufficient conditions for the approximate controllability of the considered system in a separable, reflexive Banach space via properties of the resolvent operator and Schauder's fixed point theorem. Finally, we apply our results to investigate the approximate controllability of the impulsive wave equation with state-dependent delay. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of positive solutions of boundary value problems for fractional differential equations involving the generalized proportional derivatives of Caputo type. The investigation of the proposed BVP depends on the properties of the Green function, the method of upper and lower solutions, and the fixed point theorems due to Schauder and Banach. An example is proposed to clarify the applicability of the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the finite approximate controllability of some Hilfer fractional evolution systems. Using a variational approach and Schauder’s fixed point theorem, we give sufficient conditions for finite approximate controllability of semilinear controlled systems. An example is given to illustrate our theory.

Abstract In this article, we study the existence of positive periodic solutions of second order damped boundary value problem u ″ + p ( t ) u ′ + q ( t ) u = g ( t , u , u ′ ) $u'' + p(t)u'+q(t)u = g(t,u,u')$ , u ( 0 ) = u ( T ) $u(0) = u(T )$ , u ′ ( 0 ) = u ′ ( T ) $u'(0) = u'(T )$ . The main tools are the nonlinear alternative principle of Leray–Schauder and Schauder’s fixed point theorem. We emphasize that the damped term and nonnegative Green’s functions are the key points. We also apply the results to examples for testing. Some recent results in the literature are improved and generalized.

Abstract This paper is devoted to study the existence of at least one continuous mild solution to semilinear fractional differential inclusion with deviated-advanced nonlocal conditions. We develop the results obtained, by exchanging the deviated-advanced nonlocal condition to an integral form. Our results will be accomplished by using the nonlinear Leray-Schauder’s alternative fixed point theorem. The main results are well illustrated with the aid of an example.

Abstract This paper deals with existence, uniqueness, and Hyers–Ulam stability of solutions to a nonlinear coupled implicit switched singular fractional differential system involving Laplace operator ϕp $\phi _{p}$. The proposed problem consists of two kinds of fractional derivatives, that is, Riemann–Liouville fractional derivative of order β and Caputo fractional derivative of order σ, where m−1

Abstract In the present paper, we firstly improve the results on traveling wave solution that were established in (Liu and Weng in J. Differ. Equ. 258:3688–3741, 2015) for a neutral reaction–diffusion equation with quasi-monotone reaction. Secondly, by constructing two auxiliary equations and using Schauder’s Fixed Point Theorem, we further establish the existence and the asymptotic properties of the traveling wave solution for the equation with non-monotone reaction. Two examples are also given as the application of our results.

In this paper, we investigate the existence criteria of at least one positive solution to the three-point boundary value problems with coupled system of Riemann-Liouville type nonlinear fractional order differential equations. The analysis of this study is based on the well-known Schauder’s fixed point theorem. Some new existence and multiplicity results for coupled system of Riemann-Liouville type nonlinear fractional order differential equation with three-point boundary value conditions are obtained.

In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mixed problem for a nonlinear elasticity system with laws of general behavior. The coeffcients of elasticity depends on x meanwhile the density of the volumetric forces depends on the displacement. The main aim of this paper is to apply the Schauder's fixed point theorem and the techniques of topological degree to prove a theorem of the existence and the uniqueness of the solution of the corresponding variational problem. [ABSTRACT FROM AUTHOR]