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The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

In this work, we investigate a Riemann-Liouville-type impulsive fractional integral boundary value problem. Using the fixed point index, we obtain two existence theorems on positive solutions under some conditions concerning the spectral radius of the relevant linear operator. Our method improves and generalizes some results in the literature.

Abstract To make better decisions on approximation, we may need to increase reliable and useful information on different aspects of approximation. To enhance information about the quality and certainty of approximating the solution of an Apollonius-type quadratic functional equation, we need to measure both the quality and the certainty of the approximation and the maximum errors. To measure the quality of it, we use fuzzy sets, and to achieve its certainty, we use the probability distribution function. To formulate the above problem, we apply the concept of Z-numbers and introduce a special matrix of the form diag ( A , B , C ) $\mathrm{diag}(A, B, C)$ (named the generalized Z-number) where A is a fuzzy time-stamped set, B is the probability distribution function, and C is a degree of reliability of A that is described as a value of A ∗ B $A\ast B$ . Using generalized Z-numbers, we define a novel control function to investigate H–U–R stability to approximate the solution of an Apollonius-type quadratic functional equation with quality and certainty of the approximation.

To make a decision to select a suitable approximation for the solution of a functional inequality, we need reliable information. Two useful information ideas are quality and certainty, and the measure of quality and certainty approximation of the solution of a functional inequality helps us to find the optimum approximation. To measure quality and certainty, we used the idea of the Z-number (Z-N) and we introduced the generalized Z-N (GZ-N) as a diagonal matrix of the form $ diag(X, Y, X\ast Y) $, where $ X $ is a fuzzy set time-stamped, $ Y $ is the probability distribution function and the third part is the fuzzy-random trace of the first and the second subjects. This kind of diagonal matrix allowed us to define a new model of control functions to stabilize our problem. Using stability analysis, we obtained the most suitable approximation for functional inequalities.

The Cohen–Grossberg neural network is studied in the case when the dynamics of the neurons is modeled by a Riemann–Liouville fractional derivative with respect to another function and an appropriate initial condition is set up. Some inequalities about both the quadratic function and the absolute values functions and their fractional derivatives with respect to another function are proved and they are based on an appropriate modification of the Razumikhin method. These inequalities are applied to obtain the bounds of the norms of any solution of the model. In particular, we apply the squared norm and the absolute values norms. These bounds depend significantly on the function applied in the fractional derivative. We study the asymptotic behavior of the solutions of the model. In the case when the function applied in the fractional derivative is increasing without any bound, the norms of the solution of the model approach zero. In the case when the applied function in the fractional derivative is equal to the current time, the studied problem reduces to the model with the classical Riemann–Liouville fractional derivative and the obtained results gives us sufficient conditions for asymptotic behavior of the solutions for the corresponding model. In the case when the function applied in the fractional derivative is bounded, we obtain a finite bound for the solutions of the model. This bound depends on the initial function and the solution does not approach zero. An example is given illustrating the theoretical results.

The main issue we are studying in this paper is that of aggregation maps, which refers to the process of combining various input values into a single output. We apply aggregated special maps on Mittag-Leffler-type functions in one parameter to get diverse approximation errors for fractional-order systems in Hilfer sense using an optimal method. Indeed, making use of various well-known special functions that are initially chosen, we establish a new class of matrix-valued fuzzy controllers to evaluate maximal stability and minimal error. An example is given to illustrate the numerical results by charts and tables.

In this paper, we consider a differential equation system and present a new method based on Clique polynomials (CP-M) to obtain numerical solutions of this system. The system of differential equations is a mathematical model of a new virus called Corona, which causes an infectious disease called COVID-19. By solving this system of equations, we check the transmissibility of the Coronavirus by the CPs method. In particular we turn the system of differential equations into an algebraic system to obtain solutions. Finally, we compare the numerical results obtained by the CPs method with the numerical results of other methods.

In our current work we investigate the following critical quasilinear Schrödinger equation $ -\Delta \Theta+\mathcal V(x)\Theta-\Delta (\Theta^2)\Theta = |\Theta|^{22^*-2}\Theta+\lambda \mathcal K(x)g(\Theta), \ x \ \in \mathbb R^N, $ where $ N\geq 3 $, $ \lambda > 0 $, $ \mathcal V, \ \mathcal K\in C(\mathbb R^N, \mathbb R^+) $ and $ g\in C(\mathbb R, \mathbb R) $ has a quasicritical growth condition. We use the dual approach and the mountain pass theorem to show that the considered problem has a positive solution when $ \lambda $ is a large parameter.

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.

In this paper we study a fourth-order differential equation with Riemann-Stieltjes integral boundary conditions. We consider two cases, namely when the nonlinearity satisfies superlinear growth conditions (we use topological degree to obtain an existence theorem on nontrivial solutions), when the nonlinearity satisfies a one-sided Lipschitz condition (we use the method of upper-lower solutions to obtain extremal solutions).

In this paper we use the fixed point index theory to study the existence of positive radial solutions for a system of boundary value problems with semipositone second order elliptic equations. Some appropriate concave and convex functions are utilized to characterize coupling behaviors of our nonlinearities.

Abstract In this paper, a simple proof of the convergence of the recent iterative algorithm by relaxed ( u , v ) $(u, v)$ -cocoercive mappings due to Imnang (J. Inequal. Appl. 2013:249, 2013) is presented.

The main goal of the paper is to present and study models of multi-agent systems for which the dynamics of the agents are described by a Caputo fractional derivative of variable order and a kernel that depends on an increasing function. Also, the order of the fractional derivative changes at update times. We study a case for which the exchanged information between agents occurs only at initially given update times. Two types of linear variable-order Caputo fractional models are studied. We consider both multi-agent systems without a leader and multi-agent systems with a leader. In the case of multi-agent systems without a leader, two types of models are studied. The main difference between the models is the fractional derivative describing the dynamics of agents. In the first one, a Caputo fractional derivative with respect to another function and with a continuous variable order is applied. In the second one, the applied fractional derivative changes its constant order at each update time. Mittag–Leffler stability via impulsive control is defined, and sufficient conditions are obtained. In the case of the presence of a leader in the multi-agent system, the dynamic of the agents is described by a Caputo fractional derivative with respect to an increasing function and with a constant order that changes at each update time. The leader-following consensus via impulsive control is defined, and sufficient conditions are derived. The theoretical results are illustrated with examples. We show with an example the leader’s influence on the consensus.

The present paper considers a fractional-order smoke epidemic model. We apply fuzzy systems and probability theory to make the best decision on the stability of the smoking epidemic model by using a new class of controllers powered by special functions to effectively generalize Ulam-type stability problems. Evaluation of optimal controllability and maximal stability is the new issue. This different concept of stability not only covers the old concepts but also investigates the optimization of the problem. Finally, we apply a new optimal method for the governing model with the Atangana–Baleanu–Caputo fractional derivative to obtain stability results in Banach spaces.

We apply known special functions from the literature (and these include the Fox H–function, the exponential function, the Mittag-Leffler function, the Gauss Hypergeometric function, the Wright function, the G–function, the Fox–Wright function and the Meijer G–function) and fuzzy sets and distributions to construct a new class of control functions to consider a novel notion of stability to a fractional-order system and the qualified approximation of its solution. This new concept of stability facilitates the obtention of diverse approximations based on the various special functions that are initially chosen and also allows us to investigate maximal stability, so, as a result, enables us to obtain an optimal solution. In particular, in this paper, we use different tools and methods like the Gronwall inequality, the Laplace transform, the approximations of the Mittag-Leffler functions, delayed trigonometric matrices, the alternative fixed point method, and the variation of constants method to establish our results and theory.

In this study, we investigate the concept of antiderivations in Banach algebras and study multi-super-stability of antiderivations in Banach algebras, associated with functional inequalities.

In this paper, we study a class of degenerate unperturbed problems. We first investigate some properties of eigenvalues and eigenfunctions for the strongly degenerate elliptic operator and then obtain two existence theorems of nontrivial solutions when the nonlinearity is a function with an asymptotically condition.

In this study, we consider positivity and other related concepts such as α−convexity and α−monotonicity for discrete fractional operators with exponential kernel. Namely, we consider discrete Δ fractional operators in the Caputo sense and we apply efficient initial conditions to obtain our conclusions. Note positivity results are an important factor for obtaining the composite of double discrete fractional operators having different orders.

In this paper we use the topological degree and the fountain theorem to study the existence of weak solutions for a fractional p-Laplacian equation in a bounded domain. For the nonlinearity f, we consider two situations: (1) the non-resonance case where f is (p−1)-asymptotically linear at infinity; (2) the resonance case where f satisfies the Landesman-Lazer type condition.

We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where $ \left({}_{{c_0}}^{C{F_R}}\nabla^{\theta} \mathtt{F}\right)(t) > -\epsilon\, \Lambda(\theta-1)\, \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1) $ such that $ \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1)\geq 0 $ and $ \epsilon > 0 $. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of $ \epsilon $ and $ \theta $.

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

This paper focuses on obtaining traveling wave solutions of the Fornberg–Whitham model derived from Gilson–Pickering equations, which describe the prorogation of waves in crystal lattice theory and plasma physics by some analytical techniques, i.e., the exp-function method (EFM), the multi-exp function method (MEFM) and the multi hyperbolic tangent method (MHTM). We analyze and compare them to show that MEFM is the optimum method.

In this paper, we apply two different methods, namely, the G′G-expansion method and the G′G2-expansion method to investigate the nonlinear time fractional Harry Dym equation in the Caputo sense and the symmetric regularized long wave equation in the conformable sense. The mentioned nonlinear partial differential equations (NPDEs) arise in diverse physical applications such as ion sound waves in plasma and waves on shallow water surfaces. There exist multiple wave solutions to many NPDEs and researchers are interested in analytical approaches to obtain these multiple wave solutions. The multi-exp-function method (MEFM) formulates a solution algorithm for calculating multiple wave solutions to NPDEs and at the end of paper, we apply the MEFM for calculating multiple wave solutions to the (2 + 1)-dimensional equation.

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy by the process involving Gaussian white noise. We introduce a regularized method to establish an approximate solution. We establish an upper bound on the rate of convergence of the mean integrated squared error. This article is inspired by the article by Tuan and Nane [1].

In this paper, we study an integral boundary value problem involving a Hadamard-type fractional differential equation. Using fixed point theory and upper-lower solutions, we present some sufficient conditions to obtain existence theorems of positive solutions for the problem. Examples are provided to illustrate our results.

The existence of two non trivial solutions for a nonlinear fourth-order boundary value problem is established. The coefficients of the equations are possibly variable and the nonlinearity includes, in particular, the combined effects of concavity and convexity. The approach is based on variational methods.

We apply CRM based on an alternative FPT to investigate the approximation of a Δ-Hilfer FDE. In comparison to the Picard method, we show that the CRM has a better error estimate and economic solution.

Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from (1,2). We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.

This papers presents new maximal element type results for general majorized type maps, and as an application, we consider equilibria for some one-person games.

Some inequalities for generalized proportional Riemann–Liouville fractional derivatives (RLGFDs) of convex functions are proven. As a special case, inequalities for the RLGFDs of the most-applicable Lyapunov functions such as the ones defined as a quadratic function or the ones defined by absolute values were obtained. These Lyapunov functions were combined with a modification of the Razumikhin method to study the stability properties of the Cohen–Grossberg model of neural networks with both time-variable and continuously distributed delays, time-varying coefficients, and RLGFDs. The initial-value problem was set and studied. Upper bounds by exponential functions of the solutions were obtained on intervals excluding the initial time. The asymptotic behavior of the solutions of the model was studied. Some of the obtained theoretical results were applied to a particular example.

We introduce a matrix-valued fractional delay differential system in diverse cases and present Fox type stability results with applications of aggregated special functions. In addition we present an example showing the numerical solutions based on the second type Kudryashov method. Finally, via the method of variation of constants, and some properties of the parametric Mittag–Leffler matrices, we obtain both symmetric random and symmetric fuzzy finite-time stability results for the governing fractional delay model. A numerical example is considered to illustrate applicability of the study.

In recent years, various qualitative investigations of the properties of differential equations with different types of generalizations of Riemann–Liouville fractional derivatives were studied and stability properties were investigated, usually using Lyapunov functions. In the application of Lyapunov functions, we need appropriate inequalities for the fractional derivatives of these functions. In this paper, we consider several Riemann–Liouville types of fractional derivatives and prove inequalities for derivatives of convex Lyapunov functions. In particular, we consider the classical Riemann–Liouville fractional derivative, the Riemann–Liouville fractional derivative with respect to a function, the tempered Riemann–Liouville fractional derivative, and the tempered Riemann–Liouville fractional derivative with respect to a function. We discuss their relations and their basic properties, as well as the connection between them. We prove inequalities for Lyapunov functions from a special class, and this special class of functions is similar to the class of convex functions of many variables. Note that, in the literature, the most common Lyapunov functions are the quadratic ones and the absolute value ones, which are included in the studied class. As a result, special cases of our inequalities include Lyapunov functions given by absolute values, quadratic ones, and exponential ones with the above given four types of fractional derivatives. These results are useful in studying types of stability of the solutions of differential equations with the above-mentioned types of fractional derivatives. To illustrate the application of our inequalities, we define Mittag–Leffler stability in time on an interval excluding the initial time point. Several stability criteria are obtained.

We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability.

The main goal of the paper is to use a generalized proportional Riemann–Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved, such as the fractional derivative of a squared function. Additionally, some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time, the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given, illustrating the meaning of the equilibrium as well as its stability properties.

In this paper, by introducing environmental white noise and telegraph noise, we proposed a stochastic predator–prey model with the Beddington–DeAngelis type functional response and investigated its dynamical behavior. Persistence and extinction are two core contents of population model research, so we analyzed these two properties. The sufficient conditions of the strong persistence in the mean and extinction were established and the threshold between them was obtained. Moreover, we took stability into account and, by means of structuring a suitable Lyapunov function with regime switching, we proved that the stochastic system has a unique stationary distribution. Finally, numerical simulations were used to illustrate our theoretical results.

In this current work, we apply the topological degree and fixed point theorems to investigate the existence, uniqueness, and multiplicity of solutions for a boundary value problem associated with a fractional-order difference equation. Moreover, we provide some appropriate examples to verify our main conclusions.

Abstract In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

Abstract We consider fuzzy sets and generalized triangular norms on positive elements of order commutative C ∗ $C^{*}$ -algebras to study the concept of C ∗ $C^{*}$ -algebra valued normed algebras with uncertainty. Using n-expansively super-homogeneous and ( n , k ) $(n,k)$ -contractively sub-homogeneous control functions, we make stochastic ( Θ , ϒ , Ξ ) $(\Theta,\Upsilon,\Xi )$ -derivations stable and get a better estimated error. We present some numerical examples of control functions and approximations to illustrate the applicability of the main results.

In this article, we consider the existence and uniqueness of solutions for a class of initial value problems of fuzzy Caputo–Katugampola fractional differential equations and the stability of the corresponding fuzzy fractional differential equations. The discussions are based on the hyperbolic function, the Banach fixed point theorem and an inequality property. Two examples are given to illustrate the feasibility of our theoretical results.

Abstract We introduce a class of stochastic matrix control functions and apply them to stabilize pseudo stochastic κ-random operator inequalities in matrix MB-algebras. We obtain an approximation for stochastic κ-random operator inequalities and calculate the maximum error of the estimate.

A nonlinear Riemann–Liouville fractional differential equation with constant delay is studied. Initially, some existence results are proved. Three Ulam type stability concepts are defined and studied. Several sufficient conditions are obtained. Some of the obtained results are illustrated on fractional biological models.

Abstract In this paper, we consider a time-fractional backward problem for the fractional Rayleigh–Stokes equation in a general bounded domain. We propose a fractional Landweber regularization method for solving this problem. Error estimates between the regularized solution and the sought solution are also obtained under some choice rules for both a-priori and a-posterior regularization parameters.

Abstract In this paper, we consider C ∗ $C^{*}$ -algebra valued fuzzy normed spaces. We study the random integral equation ( 1 2 c ) ∫ x − c d x + c d u ( γ , τ , d 0 ) d τ = u ( γ , x , d ) $(\frac{1}{2c})\int _{x-cd}^{x+cd}u(\gamma ,\tau ,d_{0})\,d\tau =u( \gamma ,x,d)$ which is related to the stochastic wave equation. In addition, using a C ∗ $C^{*}$ -algebra valued fuzzy controller function, we consider its C ∗ $C^{*}$ -algebra valued fuzzy Hyers–Ulam stability.

This paper provides an answer if the nonuniform behavior can destroy the structural stability of nonlinear systems. We show that if the linear system $\dot{x}(t)=A(t)x(t)$ admits a nonuniform exponential dichotomy, then the perturbed nonautonomous system $\dot{x}(t)=A(t)x(t)+f(t,x)$ is structurally stable under suitable conditions.

Let P be a complex polynomial. We prove that the associated polynomial matrix-valued function \tildeP is surjective if for each λ ∈ ℂ the polynomial P-λ has at least a simple zero. The null controllability for difference and differential matrix equations is also presented.

We discuss the existence and multiplicity of positive solutions for a class of second order quasilinear equations. To obtain our results we will use the Ekeland variational principle and the Mountain Pass Theorem.

Abstract In this paper, we study a sideways heat equation with a nonlinear source in a bounded domain, in which the Cauchy data at x = X $x = \mathcal {X}$ are given and the solution in 0 ≤ x < X $0 \le x < \mathcal {X}$ is sought. The problem is severely ill-posed in the sense of Hadamard. Based on the fundamental solution to the sideways heat equation, we propose to solve this problem by the filter method of degree α, which generates a well-posed integral equation. Moreover, we show that its solution converges to the exact solution uniformly and strongly in L p ( ω , X ; L 2 ( R ) ) $\mathscr {L}^{p}(\omega,\mathcal {X};\mathscr {L}^{2}(\mathbb {R}))$ , ω ∈ [ 0 , X ) $\omega\in [0,\mathcal {X})$ under a priori assumptions on the exact solution. The proposed regularized method is illustrated by numerical results in the final section.

In this article, we introduce a non-instantaneous impulsive operator associated with the heat semigroup and give some basic properties. We derive an abstract formula for the solutions to non-instantaneous impulsive heat equations. Also we show the existence and uniqueness of the non-homogeneous periodic trajectory.