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In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He's and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations. [ABSTRACT FROM AUTHOR]

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]

A hybrid system interacts with the discrete and continuous dynamics of a physical dynamical system. The notion of a hybrid system gives embedded control systems a great advantage. The Langevin differential equation can accurately depict many physical phenomena and help researchers effectively represent anomalous diffusion. This paper considers a fractional hybrid Langevin differential equation, including the ψ-Caputo fractional operator. Furthermore, some novel boundaries selected are considered to be a problem. We used the Schauder and Banach fixed-point theorems to prove the existence and uniqueness of solutions to the considered problem. Additionally, the Ulam-Hyer stability is evaluated. Finally, we present a representative example to verify the theoretical outcomes of our findings. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a mixed boundary value problem for a complete Sturm–Liouville equation, where the coefficients can also be negative. In particular, the existence of infinitely many distinct positive solutions to the given problem is obtained by using critical point theory. [ABSTRACT FROM AUTHOR]

This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result. [ABSTRACT FROM AUTHOR]

In this paper, we consider n-type Markov branching–immigration processes. The uniqueness criterion is first established. Then, we construct a related system of differential equations based on the branching property. Furthermore, the explicit expression of extinction probability and the mean extinction time are successfully obtained in the absorbing case by using the unique solution of the related system of differential equations and Kolmogorov forward equations. Finally, the recurrence and ergodicity criteria are given if the zero state 0 is not absorbing. [ABSTRACT FROM AUTHOR]

This paper is associated with Sturm–Liouville type boundary value problems and periodic boundary value problems for quaternion-valued differential equations (QDEs). Employing the theory of quaternionic matrices, we prove the conditions for the solvability of the linear boundary valued problem and find Green's function. [ABSTRACT FROM AUTHOR]

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch's fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given. [ABSTRACT FROM AUTHOR]

In this paper, three-point boundary value problems for second-order p-Laplacian differential equations with instantaneous and noninstantaneous impulses are studied. The existence of at least one classical solution and infinitely many classical solutions is obtained by using variational methods and critical point theory. In addition, some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

We prove the existence of the solutions for the new mixed differential equations, which is characteristic of the right-sided Caputo and the left-sided Riemann–Liouville fractional derivatives. There are four major ingredients. The first is composed of some basic definitions and lemmas. The second is the Green's function of the new mixed fractional differential equations. We calculate the corresponding Green's functions as well as their properties. The third, which is the main new ingredient of this paper, is demonstration of the existence of the solutions for fractional equations by the fixed-point theorem in cone expansion and compression of norm type. The fourth, as applications, is the example provided to illustrate our main results. [ABSTRACT FROM AUTHOR]

In the paper, we study the oscillatory and spectral properties of a fourth-order differential operator. These properties are established based on the validity of some weighted second-order differential inequality, where the inequality's weights are the coefficients of the operator. The inequality, in turn, is established for functions satisfying certain boundary conditions that depend on the boundary behavior of its weights at infinity and at zero. [ABSTRACT FROM AUTHOR]

In this paper, the existence and multiplicity of solutions for a coupled system of differential equations with instantaneous and noninstantaneous impulses are studied. By the virtue of variational methods, some new existence theorems of solutions are obtained. In addition, two examples are given to demonstrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we employ the well-known Krasnoselskii fixed point theorem to study the existence and n-multiplicity of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. The form including an impulsive term of the equations in this paper is rather general and incorporates as special cases various problems which have been studied extensively in the literature. Easily verifiable sufficient criteria are obtained for the existence and n-multiplicity of positive periodic solutions of the impulsive functional differential equations. [ABSTRACT FROM AUTHOR]

This paper focuses on second-order differential equations involving causal operators with nonlinear two-point boundary conditions. By applying the monotone iterative technique in the presence of upper and lower solutions, with a new comparison theorem, we obtain the existence of extremal solutions. This is an extension of classical theory of second-order differential equations. Finally, we present two examples to show the usefulness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the oscillatory behavior of solutions of a class of Super-linear fourth-order differential equations with several sub-linear neutral terms using the Riccati and generalized Riccati transformations. Some Kamenev–Philos-type oscillation criteria are established. New oscillation criteria are deduced in both canonical and non-canonical cases. An illustrative example is given. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has n − 1 zero in (0 , π) for n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem. [ABSTRACT FROM AUTHOR]

In the present paper, a new method is developed to study the existence of an almost periodic solution for the ordinary or functional differential equations. The approaches are based on topological degree and novel estimation techniques for the a priori bounds of unknown solutions for $Lx=\lambda Nx$. To investigate the existence of an almost periodic solution, a few good methods have been presented in the previous literature (such as using the Lyapunov function, averaging, exponential dichotomy, stability, separate conditions, and so on). But topological degree theory was never employed to study the almost periodic differential equations. Though Mawhin's coincidence degree is employed to study the existence of periodic differential equations extensively, it cannot be applied to study the almost periodic systems immediately. Some essentially new and interesting lemmas should be proved before applying topological degree theory to almost periodic systems. To the best knowledge of the authors', it is the first time that topological degree theory is employed to study the existence of almost periodic solution and this method can be seen as a good supplement to the known methods. Therefore, it will be of great significance to study the almost periodic systems by using this method. The approach followed in the paper could be further generalized to investigate the existence of almost periodic oscillatory in some other nonlinear dynamical systems. It is believed that it can be applied to image patterns, digital image processing, data processing, signal sparse decomposition and information technology, etc. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of positive classical solutions for a class of second-order differential equations with the nonlinearity dependent on the derivative. We also provide a range of the parameter in order to obtain the existence of multiple solutions. The approach is based on variational methods. An example illustrates the abstract results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we establish the new expression and properties of Green's function for an m-point boundary value problem with a delayed argument. Furthermore, using Hölder's inequality and a fixed point theorem due to Leggett and Williams, the existence of at least three positive solutions is also given. We discuss our problem with a delayed argument. In this case, our results cover m-point boundary value problems without delayed arguments and are compared with some recent results. An example is included to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper is devoted to prove the existence of positive solutions of a second order differential equation with a nonhomogeneous Dirichlet conditions given by a parameter dependence integral. The studied problem is a nonlocal perturbation of the Dirichlet conditions by considering a homogeneous Dirichlet-type condition at one extreme of the interval and an integral operator on the other one. We obtain the expression of the Green's function related to the linear part of the equation and characterize its constant sign. Such a property will be fundamental to deduce the existence of solutions of the nonlinear problem. The results hold from fixed point theory applied to related operators defined on suitable cones. [ABSTRACT FROM AUTHOR]

In this paper, the existence results of positive solutions for three-point Riemann-Stieltjes integral BVPs (boundary value problems) is considered. By applying shooting method and comparison principle, we obtain some new results which extend the known ones. At the same time, the theorems in one of our published articles are corrected by another theorem in this paper. [ABSTRACT FROM AUTHOR]

In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects. To the authors' knowledge, there is no paper about periodic solution of impulses delay differential systems via critical point theory. Our results are completely new. [ABSTRACT FROM AUTHOR]

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples. [ABSTRACT FROM AUTHOR]

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation $-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}$ with $f\in L^{m}(\Omega)$ ( $m\geqslant1$ ) and $\alpha(x)>0$ . Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent $\alpha(x)$ , some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of $\alpha(x)$ . [ABSTRACT FROM AUTHOR]

In this paper, we discuss a boundary value problem for an impulsive fractional differential equation. By transforming the boundary value problem into an equivalent integral equation, and employing the Banach fixed point theorem and the Schauder fixed point theorem, existence results for the solutions are obtained. For application, we provide some examples to illustrate our main results. [ABSTRACT FROM AUTHOR]

On the interval [ 0 , 1 ] we consider the nth order linear differential equation, the coefficient of the highest derivative of which is equivalent to the power function t μ when t → 0 . The main aim of the paper is to pose "generalized" Cauchy conditions for the given equation at the point of singularity t = 0 , which would be correct for any μ > 0 . [ABSTRACT FROM AUTHOR]

The 2D incompressible Boussinesq system with partial or fractional dissipation have recently attracted considerable attention. In this paper, we study the Cauchy problem for the 2D Boussinesq system in a periodic domain with fractional vertical dissipation in the subcritical case, and we prove the global well-posedness of strong solutions. Based on this, we also discuss the existence of the global attractor. [ABSTRACT FROM AUTHOR]

This paper is concerned with the quenching phenomenon for the one-dimensional non-Newtonian filtration equation with both source term and Neumann boundary condition. With two different kinds of initial data, we prove that the solution must quench in a finite time and the time derivative blows up at a quenching point. The corresponding quenching rate and a lower bound for the quenching time are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study persistence properties for a generalized Camassa-Holm equation with cubic nonlinearity, and we prove the persistence properties in weighted spaces of the solution to the equation, provided that the initial potential satisfies a certain sign condition. Our results extend the work of Brandolese (Int. Math. Res. Not. 22:5161-5181, ) on persistence properties to the Fokas-Olver-Rosenau-Qiao equation. In contrast to the Camassa-Holm equation with quadratic nonlinearity, the effect of cubic nonlinearity of the Fokas-Olver-Rosenau-Qiao equation on the persistence properties is rather delicate. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up for a class of nonlinear viscoelastic wave equation. Under certain conditions on the data, we construct a lower bound for the blow-up time when blow-up occurs. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of elliptic problems of p-Kirchhoff type with critical exponent in bounded domains and new results as regards the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial value problem for a semi-linear wave equation in n-dimensional space. Under a smallness condition on the initial value, the global existence and decay estimates of the solutions are established. Furthermore, time decay estimates for the spatial derivatives of the solution are provided. The proof is carried out by means of the decay property of the solution operator and a fixed point-contraction mapping argument. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of concave and monotone positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form $x^{(4)}(t)=f(t,x(t),x'(t),x''(t))$, $t\in[0,1]$, $x(0)=x'(1)=x'''(1)=0$, $x''(0)=\int_{0}^{1}g(s)x''(s) \, \mathrm{d}s$, where $f\in C([0,1]\times[0,+\infty)^{2}\times(-\infty,0],[0,+\infty))$, $g\in C([0,1],[0,+\infty))$. By using a fixed point theorem of cone expansion and compression of norm type, the existence and nonexistence of concave and monotone positive solutions for the above boundary value problems is obtained. Meanwhile, as applications of our results, some examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of nonlinear differential equations in a Banach space with boundary conditions on an infinite interval and provide sufficient conditions for the existence of solutions of the system. Our method relies upon the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem. An example is given to illustrate the main results. [ABSTRACT FROM AUTHOR]

The present research work investigates some new results for a fractional generalized Sturm–Liouville–Langevin (FGSLL) equation involving the Ψ-Caputo fractional derivative with a modified argument. We prove the uniqueness of the solution using the Banach contraction principle endowed with a norm of the Ψ-Bielecki-type. Meanwhile, the fixed-point theorems of the Leray–Schauder and Krasnoselskii type associated with the Ψ-Bielecki-type norm are used to derive the existence properties by removing some strong conditions. We use the generalized Gronwall-type inequality to discuss Ulam–Hyers (UH), generalized Ulam–Hyers (GUH), Ulam–Hyers–Rassias (UHR), and generalized Ulam–Hyers–Rassias (GUHR) stability of these solutions. Lastly, three examples are provided to show the effectiveness of our main results for different cases of (FGSLL)-problem such as Caputo-type Sturm–Liouville, Caputo-type Langevin, Caputo–Erdélyi–Kober-type Langevin problems. [ABSTRACT FROM AUTHOR]

This work is devoted to using topological degree theory to establish a mathematical analysis for a class of fractional-order evolution hybrid differential equations using a modified Mittag–Leffler-type derivative. In addition, two kinds of Ulam–Hyers (U–H) stability results are deduced for the mentioned problem. A pertinent example is given to verify the results. [ABSTRACT FROM AUTHOR]

This work contains sufficient conditions for the solvability of a third-order coupled system with two differential equations involving different Laplacians, fully discontinuous nonlinearities, two-point boundary conditions, and two sets of impulsive effects. The first existing result is obtained from Schauder's fixed point theorem, and the second one provides also the localization of a solution via the lower and upper solutions technique. We point out that it is the first time that impulsive coupled systems with strongly nonlinear fully differential equations and generalized impulse effects are considered simultaneously. Moreover, the singular case is applied to a special relativity model in classical electrodynamics. [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 $3

In this paper, the existence of Aubry-Mather sets and quasi-periodic solutions of the oscillator x″ + αx+3 - βx-3 + f(x)q(x′) + ψ(x) = p(t) are established, where f, q, and ψ belong to the class C¹(R), p is a continuous 2&#960-periodic function. Under some assumptions on the parities of f, ψ, and p, we prove that there are infinitely many generalized quasi-periodic solutions by a result of Shuinee Chow and Mingliang Pei from the Aubry-Mather theorem of reversible mappings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the asymptotic properties and oscillation of the nth order advanced differential equations (r(t)[x(n-1)(t)γ)]' + q(t)xγ [t(t)] = 0. The results obtained are based on the Riccati transformation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there exists a weak solution for Schrödingerean technique for order performance by similarity (TOPSIS) equations on cylinders. Meanwhile, the boundary behaviors of it are also obtained via the abstract theory of fuzzy multi-criterion decision making. As the main tools, we use Karamata regular variation theory and the method of upper and lower solutions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a class of new Picard-Mann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. We also give convergence and stability analysis of the new Picard-Mann iterative approximation and propose numerical examples to show that the new Picard-Mann iteration converges more effectively than the Picard iterative process, Mann iterative process, Picard-Mann iterative process due to Khan and other related iterative processes. Furthermore, as an application, we explore iterative approximation of solutions for an elliptic boundary value problem in Hilbert spaces by using the new Picard-Mann iterative methods with mixed errors for contraction operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solution (EUS) as well as Hyers-Ulam stability for a coupled system of FDEs in Caputo's sense with nonlinear p-Laplacian operator. For this purpose, the suggested coupled system is transferred to an integral system with the help of four Green functions $\mathcal{G}^{\alpha_{1}}(t,s)$ , $\mathcal{G}^{\beta_{1}}(t,s)$ , $\mathcal{G}^{\alpha_{2}}(t,s)$ , $\mathcal{G}^{\beta_{2}}(t,s)$ . Then using topological degree theory and Leray-Schauder's-type fixed point theorem, existence and uniqueness results are proved. An illustrative and expressive example is given as an application of the results. [ABSTRACT FROM AUTHOR]

In this paper, we study the following nonlinear problem of Kirchhoff type: where the parameter $\lambda > 0$ and $4 \leq p < 6$ , constants $a, b>0$ . By variational methods, the results of the existence of nontrivial solutions and the concentration phenomena of the solutions as $\lambda \to + \infty$ are obtained. It is worth pointing out that, for the case $p\in(4,6)$ , the potential V is permitted to be sign-changing. [ABSTRACT FROM AUTHOR]

The oscillatory behavior of solutions of an even-order differential equation with a superlinear neutral term is considered using Riccati and generalized Riccati transformations, the integral averaging technique, and the theory of comparison. New sufficient conditions are established in the noncanonical case. An example is given to support our results. [ABSTRACT FROM AUTHOR]

This paper is mainly devoted to studying one kind of the second order differential equation. Under periodic-integrable boundary value condition, the existence of the solutions of this equation is discussed by the method of the operator theory and the Schauder fixed point theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the positive periodic solutions of a kind of p-Laplacian neutral differential equation with a singularity. By applying the continuation theorem and some analytic techniques, we shall establish several new criteria for the existence of positive periodic solutions for the considered problem. Some recent results in the literature are generalized and improved. Three examples are given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following p-Kirchhoff equation: with Dirichlet boundary conditions, where Ω is a bounded domain in $\mathbb{R}^{N}$ . Under proper assumptions on M and f, we obtain three existence theorems of infinitely many solutions for problem (P) by the fountain theorem. Moreover, for a special nonlinearity $f(x,u)=\lambda |u|^{q-2}u+|u|^{r-2}u$ ( $1< q< p< r< p^{*}$ ), we prove that problem (P) has at least two nonnegative solutions via the Nehari manifold method and a sequence of solutions with negative energy by the dual fountain theorem. [ABSTRACT FROM AUTHOR]

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder's alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study. [ABSTRACT FROM AUTHOR]

The purpose of this study is to establish fixed-point results for new interpolative contraction mappings in the setting of Busemann space involving a convex hull. To illustrate our findings, we also offer helpful and compelling examples. Finally, the theoretical results are applied to study the existence of solutions to fractional Navier–Stokes and fractional-functional differential equations as applications. [ABSTRACT FROM AUTHOR]