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In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

We investigate the dynamics of a delay differential equation obtained by perturbing a vector field f : R n → R n , admitting a stable periodic orbit, by using a delayed feedback control term η g ( x , x ( t − τ ) ) of same regularity, where η is small and τ large so that ητ be bounded but non small. We prove that trajectories starting in a neighborhood (of size independent of the parameters η , τ ) of this original periodic orbit enter asymptotically a periodic regime, and that the number of such distinct periodic regimes increases (almost) linearly when ητ increases infinitely. Our result is based on the construction of an invariant manifold via a process inspired by the Lyapunov-Perron method for integral operators associated to ordinary differential equations, and on the persistence of normally hyperbolic invariant manifolds for semi-flows on Banach spaces. The statement we provide here complements already known results on periodic orbits of delay differential equations.

Linear first order Riemann-Liouville fractional differential equations are studied. These new equations unify and generalize the Riemann-Liouville, modified Caputo and Caputo fractional differential equations. The equivalences between the fractional differential equations and the corresponding perturbed Abel's integral equations are obtained. These results are useful not only to study the initial or boundary value problems for nonlinear first order Riemann-Liouville fractional differential equations but also to study the solutions of the perturbed Abel's integral equations arising in a problem of mechanics and many other physical problems. The well-known Tonelli's result on solvability of the Abel's integral equation is generalized. We exhibit that there are nonconstant equilibria for some first order Caputo fractional equations. This is different from nonlinear first order ordinary differential equations which have only constant equilibria. The equivalence results are applied to generalize the classical Mean Value Theorem to the first order Riemann-Liouville fractional derivatives.

We extend the theory of lower and upper solutions to planar systems of ordinary differential equations with separated boundary conditions, both in the well-ordered and in the non-well-ordered cases. We are able to deal with general Sturm–Liouville boundary conditions in the well-ordered case, and we analyze the Dirichlet problem in the non-well-ordered case. Our results apply in particular to scalar second order differential equations, including those driven by the mean curvature operator. Higher dimensional systems are also treated, with the same approach.

In this paper, the pantograph delay differential equation y′(t)=ay(t)+byct subject to the condition y(0)=λ is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for λ=1, is well-posed if c1. In addition, the solution is available in the form of a standard power series when λ=1. In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for c∈(−1,1) such that ba1 when aa. The current results are introduced for the first time and have not been reported in the relevant literature.

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

This research is conducted for studying some qualitative specifications of solution to a generalized fractional structure of the standard snap boundary problem. We first rewrite the mathematical model of the extended fractional snap problem by means of the $\mathbb{G}$ G -operators. After finding its equivalent solution as a form of the integral equation, we establish the existence criterion of this reformulated model with respect to some known fixed point techniques. Then we analyze its stability and further investigate the inclusion version of the problem with the help of some special contractions. We present numerical simulations for solutions of several examples regarding the fractional $\mathbb{G}$ G -snap system in different structures including the Caputo, Caputo–Hadamard, and Katugampola operators of different orders.

The purpose of this paper is to consider some F-contraction mappings in a dualistic partial metric space and to provide sufficient related conditions for the existence of a fixed point. The obtained results are extensions of several ones existing in the literature. Moreover, we present examples and an application to support our results.

The theory of fractional integral inequalities plays an intrinsic role in approximation theory also it has been a key in establishing the uniqueness of solutions for some fractional differential equations. Fractional calculus has been found to be the best for modeling physical and engineering processes. More precisely, the proportional fractional operators are one of the recent important notions of fractional calculus. Our aim in this research paper is developing some novel ways of fractional integral Hermite–Hadamard inequalities in the frame of a proportional fractional integral with respect to another strictly increasing continuous function. The considered fractional integral is applied to establish some new fractional integral Hermite–Hadamard-type inequalities. Moreover, we present some special cases throughout discussing this work.

We consider a family of variational problems on the equilibrium of a composite Kirchhoff–Love plate containing two flat rectilinear rigid inclusions, which are connected in a hinged manner. It is assumed that both inclusions are delaminated from an elastic matrix, thus forming an interfacial crack between the inclusions and the surrounding elastic media. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual nonpenetration of opposite crack faces. The problems of the family depend on a parameter specifying the coordinate of a connection point of the inclusions. For the considered family of problems, we formulate a new inverse problem of finding unknown coordinates of a hinge joint point. The continuity of solutions of the problems on this parameter is proved. The solvability of this inverse problem has been established. Using a passage to the limit, a qualitative connection between the problems for plates with flat and bulk hinged inclusions is shown.

We provide a unified analytic approach to study the asymptotic dynamics of Young differential equations, using the framework of random dynamical systems and random attractors. Our method helps to generalize recent results (Duc et al. in J Differ Equ 264:1119–1145, 2018, SIAM J Control Optim 57(4):3046–3071, 2019; Garrido-Atienza et al. in Int J Bifurc Chaos 20(9):2761–2782, 2010) on the existence of the global pullback attractors for the generated random dynamical systems. We also prove sufficient conditions for the attractor to be a singleton, thus the pathwise convergence is in both pullback and forward senses.

The main purpose of this paper is to prove the existence of positive solutions for a system of nonlinear Caputo-type fractional differential equations with two parameters. By using the Guo–Krasnosel’skii fixed point theorem, some existence theorems of positive solutions are obtained in terms of different values of parameters. Two examples are given to illustrate the main results.

In this article, we construct a family of iterative methods for finding a single root of nonlinear equation and then generalize this family of iterative methods for determining all roots of nonlinear equations simultaneously. Further we extend this family of root estimating methods for solving a system of nonlinear equations. Convergence analysis shows that the order of convergence is 3 in case of the single root finding method as well as for the system of nonlinear equations and is 5 for simultaneous determination of all distinct and multiple roots of a nonlinear equation. The computational cost, basin of attraction, efficiency, log of residual and numerical test examples show that the newly constructed methods are more efficient as compared to the existing methods in literature.

Convection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.

Two new iterative methods for the simultaneous determination of all multiple as well as distinct roots of nonlinear polynomial equation are established, using two suitable corrections to achieve a very high computational efficiency as compared to the existing methods in the literature. Convergence analysis shows that the orders of convergence of the newly constructed simultaneous methods are 10 and 12. At the end, numerical test examples are given to check the efficiency and numerical performance of these simultaneous methods.

The existence of nonmonotone traveling wave solutions of the three-species Lotka–Volterra competition diffusion system under strong competition is established. A traveling wave solution can be considered as a heteroclinic orbit of a vector field in $${\mathbb {R}}^6$$ . Under suitable assumptions on parameters of the equations, we apply a bifurcation theory of heteroclinic orbits to show that a three-species traveling wave can bifurcate from two two-species waves which connect to a common equilibrium. The three components of the three-species wave obtained are positive and have the profiles that one is a front, one is a back, and the third component is a pulse between the previous two with a long middle part close to a constant. As applications of our result, we find several explicit regions of parameters of the equations where the bifurcation of three-species traveling waves occur.

This paper is devoted the study of a generalized hybrid dynamical system, which consists of a history-dependent hemivariational inequality of parabolic type and a nonlinear evolution equation. The unique solvability for the system is established via applying surjectivity of multivalued pseudomonotone operators, fixed point theorem, and properties of the Clarke generalized gradient. As an illustrative application, a dynamic frictional contact problem for viscoelastic materials with history-dependent and adhesion is investigated, in which the friction condition is described by the Clarke generalized gradient of a nonconvex and nonsmooth function involving adhesion, and the normal damped response condition is expressed by a given nonnegative function depending on the normal velocity and adhesion.

We consider a $$(1+N)$$ -body problem in which one particle has mass $$m_0 \gg 1$$ and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form $$\begin{aligned} U \sim \frac{1}{r^\alpha }, \end{aligned}$$ where $$\alpha \in [1,2)$$ and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use $$\Gamma $$ -convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional $$\Gamma $$ -converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the $$\Gamma $$ -limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of $$m_0$$ . We discuss some examples, where the symmetry is defined by an action of the groups $${\mathbb {Z}}_4$$ , $${\mathbb {Z}}_2 \times {\mathbb {Z}}_2$$ and the rotation groups of Platonic polyhedra on the set of loops.

A computational method is developed for a family of functional differential equations in one independent variable with a single deviating argument, assumed to be a delay. These equations are mapped into a finite system of ODEs, a subsystem of which involves only the function controlling the delay. Key features include efficiency during method of steps, freedom from Jacobians and root finding techniques, and computing a continuous approximation. The nonlinear differential equations may be retarded, neutral or advanced. The method is established for state dependent delays, stiff equations, discontinuous initial history, a specific loss of monotonicity in the delay and extended naturally to distributed delays. When restricted to ODEs, the method extends naturally to PDEs; the hope is the delayed version here can eventually be extended to delayed PDEs. Conditions for convergence of the approximation are established, and results of numerical experiments are reported to indicate robustness of the implementation.

In this work, we establish some necessary results about existence theory to a class of boundary value problems (BVPs) of hybrid fractional differential equations (HFDEs) in the frame of Atangana–Baleanu–Caputo (ABC) fractional derivative. Making use of Krasnoselskii and Banach theorems, we obtain the required conditions. Some appropriate results of Hyers–Ulam (H–U) stability corresponding to the considered problem are also established. Also a pertinent example is given to demonstrate the results.

In this paper, the problem of finding the source function for the Rayleigh–Stokes equation is considered. According to Hadamard’s definition, the sought solution of this problem is both unstable and independent of continuous data. By using the fractional Tikhonov method, we give the regularized solutions and then deal with a priori error estimate between the exact solution and its regularized solutions. Finally, the proposed regularized methods have been verified by simple numerical experiments to check error estimate between the sought solution and the regularized solution.

[EN] The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short)., Direction Générale de la Recherche Scientifique et du Développement Technologique DGRSDT. MESRS Algeria. Projet PRFU: C00L03UN060120180009

A highly efficient new three-step derivative-free family of numerical iterative schemes for estimating all roots of polynomial equations is presented. Convergence analysis proved that the proposed simultaneous iterative method possesses 12th-order convergence locally. Numerical examples and computational cost are given to demonstrate the capability of the method presented.

In this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.

Simpson inequalities for differentiable convex functions and their fractional versions have been studied extensively. Simpson type inequalities for twice differentiable functions are also investigated. More precisely, Budak et al. established the first result on fractional Simpson inequality for twice differentiable functions. In the present article, we prove a new identity for twice differentiable functions. In addition to this, we establish several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. This paper is a new version of fractional Simpson type inequalities for twice differentiable functions.

The aim of this paper is to investigate the interpolative contractions involving rational forms in the framework of b-metric spaces. We prove the existence of a fixed point of such a mapping with different combinations of the rational forms. A certain example is considered to indicate the validity of the observed result.

In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entirek-convex radial solutions is established by the monotone iterative method. Moreover, a nonexistence result is also obtained.

In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized ODEs. Secondly, we use the well known correspondence between solutions of generalized ODEs and solutions of measure differential equations, obtaining analogues results for the last equations. Finally, we apply some of these results for dynamic equations on time scales.

The goal of this study is to propose the existence results for the Sobolev-type Hilfer fractional integro-differential systems with infinite delay. We intend to implement the outcomes and realities of fractional theory to obtain the main results by Monch’s fixed point technique. Moreover, we show the existence and controllability of the thought about the fractional system with the nonlocal condition. In addition, an application to illustrate the outcomes is also included.

In this article, we discuss about a series of infinite dimensional extensions of some theorems given in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018), (Fisher in Math. Mag. 48(4):223–225, 1975), and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). We also prove a similar Geraghty type construction for Fisher (Math. Mag. 48(4):223–225, 1975) in an infinite dimension using similar techniques as in (Shumrani et al. in SER Math. Inform. 33(2):197–202, 2018) and (Fogh, Behnamian and Pashaie in Int. J. Maps in Mathematics 2(41):1–13, 2019). As an application, we ensure the existence of solutions for infinite dimensional Fredholm integral equation and Uryshon type integral equation.

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping and general source terms. According to some suitable hypothesis, we study the blow-up of solutions. This is the general case of the recent results of Boulaaras’ works (Bull. Malays. Math. Sci. Soc. 43:725–755, 2020) and (Appl. Anal. 99:1724–1748, 2020).

A three-species non-autonomous stochastic Lotka–Volterra food web system in a polluted environment is proposed, and the existence of positive periodic solutions of this system is established by constructing a proper Lyapunov function. Then the extinction property and its threshold between persistence and extinction are discussed by using Itô’s formula and the strong law of large numbers of martingale, and the sufficient condition of a.s. exponential stability of equilibrium point is obtained. Finally, the conclusions are tested by several numerical simulations.

In this work, we consider a generalized quantum fractional Sturm–Liouville–Langevin difference problem with terminal boundary conditions. The relevant results rely on Mönch’s fixed point theorem along with a theoretical method by terms of Kuratowski measure of noncompactness (MNC) and the Banach contraction principle (BCP). Furthermore, two dynamical notions of Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability are addressed for solutions of the supposed Sturm–Liouville–Langevin quantum boundary value problem (q-FBVP). Two examples are presented to show the validity and also the effectiveness of theoretical results. In the last part of the paper, we conclude our exposition with some final remarks and observations.

The main target of this paper is to study a problem of recovering a spherically symmetric domain with fractional derivative from observed data of nonlocal type. This problem can be established as a new boundary value problem where a Cauchy condition is replaced with a prescribed time average of the solution. In this work, we set some of the results above existence and regularity of the mild solutions of the proposed problem in some suitable space. Next, we also show the ill-posedness of our problem in the sense of Hadamard. The regularized solution is given by the fractional Tikhonov method and convergence rate between the regularized solution and the exact solution under a priori parameter choice rule and under a posteriori parameter choice rule.

This paper proposes two numerical approaches for solving the coupled nonlinear time-fractional Burgers’ equations with initial or boundary conditions on the interval $[0, L]$ [ 0 , L ] . The first method is the non-polynomial B-spline method based on L1-approximation and the finite difference approximations for spatial derivatives. The method has been shown to be unconditionally stable by using the Von-Neumann technique. The second method is the shifted Jacobi spectral collocation method based on an operational matrix of fractional derivatives. The proposed algorithms’ main feature is that when solving the original problem it is converted into a nonlinear system of algebraic equations. The efficiency of these methods is demonstrated by applying several examples in time-fractional coupled Burgers equations. The error norms and figures show the effectiveness and reasonable accuracy of the proposed methods.

The central purpose of this effort is to investigate analytic and geometric properties of a class of normalized analytic functions in the open unit disk involving Bernoulli’s formula. As a consequence, some solutions are indicated by the well-known hypergeometric function. The class of starlike functions is investigated containing the suggested class.

We study the initial-boundary value problems of quadratic semilinear wave equations outside of nontrapping obstacles with Neumann boundary conditions in spatial dimensions $$n\ge 3$$ . We obtain some lower bounds of the lifespan of classical solutions, which coincide with the sharp results for the corresponding Cauchy problems. Particularly, our results generalize the works of Shibata and Tsutsumi (North-Holland Mathematics Studies, vol 128, pp 175–228, 1985) in $$n\ge 6$$ , of Hayashi (J. Funct. Anal. 131:302–344, 1995) in $$n=4,5$$ , and of Godin (Comm Partial Differ Equ 14:299–374, 1989) in $$n=3$$ .

In this paper, we consider a new coupled system of fractional boundary value problems based on the thermostat control model. With the help of fixed point theory, we investigate the existence criterion of the solution to the given coupled system. This property is proved by using the Krasnoselskii’s fixed point theorem and its uniqueness is proved via the Banach principle for contractions. Further, the Hyers–Ulam stability of solutions is investigated. Then, we find the approximate solution of the coupled fractional thermostat control system by using a numerical technique called the generalized differential transform method. To show the consistency and validity of our theoretical results, we provide two illustrative examples.

In this article, we consider the following quasilinear Schrödinger–Poisson system $$ \textstyle\begin{cases} -\Delta u+V(x)u-u\Delta (u^{2})+K(x)\phi (x)u=g(x,u), \quad x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, \quad x\in \mathbb{R}^{3}, \end{cases} $$ { − Δ u + V ( x ) u − u Δ ( u 2 ) + K ( x ) ϕ ( x ) u = g ( x , u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ V , K : R 3 → R and $g:\mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ g : R 3 × R → R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.

The Darboux problem and the definition of the Riemann–Hadamard function are presented for the Bianchi equation of the fourth order—a linear fourth-order equation with four independent variables that has a dominant derivative not containing multiple differentiation with respect to any of the independent variables. Sufficient conditions on the coefficients of this equation under which its Riemann–Hadamard function allows explicit construction in terms of hypergeometric functions are obtained.

In this paper, we consider a fractional-order model of a brushless DC motor. To develop a mathematical model, we use the concept of the Liouville–Caputo noninteger derivative with the Mittag-Lefler kernel. We find that the fractional-order brushless DC motor system exhibits the character of chaos. For the proposed system, we show the largest exponent to be 0.711625. We calculate the equilibrium points of the model and discuss their local stability. We apply an iterative scheme by using the Laplace transform to find a special solution in this case. By taking into account the rule of trapezoidal product integration we develop two iterative methods to find an approximate solution of the system. We also study the existence and uniqueness of solutions. We take into account the numerical solutions for Caputo Liouville product integration and Atangana–Baleanu Caputo product integration. This scheme has an implicit structure. The numerical simulations indicate that the obtained approximate solutions are in excellent agreement with the expected theoretical results.

We state and study the first and second boundary value problems and the transmission problem with complicated boundary conditions containing singular functions for three-dimensional filtration flows in an inhomogeneous medium. The flow sources are arbitrary discrete sources and are located both on the boundaries and outside the boundaries. When boundaries are modeled by canonical surfaces (a plane and a sphere), the solutions of the problem can be presented in closed form. In the case of arbitrary closed smooth boundary surfaces with a point sink (source) located on them, the generalized double layer potential is used; this permits one to reduce the transmission problem and the second boundary value problem to singular and hypersingular integral equations, respectively. The problems studied here are mathematical models of three-dimensional filtration processes in heterogeneous media, which are of interest, for example, for the practice of developing natural oil-bearing (water-bearing) soil layers.

We derive volume integral equations with time delay describing nonstationary problems of acoustic field scattering by transparent three-dimensional structures. An efficient method for the numerical solution of the resulting equations is proposed.

For a fractional diffusion equation with reaction coefficient depending only on the first two components of the spatial variable $$x=(x_1,x_2,x_3)\in \mathbb {R}^3 $$ and on time $$t\geq 0 $$ , we consider the inverse problem of determining this coefficient under the assumption that the initial value at $$t=0 $$ is known for the solution of the equation and the boundary value at $$ x_3=0$$ is given as an additional condition. This inverse problem is reduced to equivalent integral equations, and we apply the contraction mapping principle to prove the existence of solutions of these equations. Local existence and global uniqueness theorems are proved. We also obtain a stability estimate for the solution of the inverse problem.

We study a linear system of integral equations of the first kind arising in the problem of wave diffraction by a local permeable body. The equivalence of the system and the original boundary value diffraction problem is established. The existence and uniqueness of the solution of the system and the dissipativity of its matrix operator are proved. The questions of the existence of nonradiating currents and their relation to the properties of the matrix operator of the system are considered.

In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result.