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In this paper, the coupled system of sequential ψ-Hilfer fractional boundary value problems with non-instantaneous impulses is investigated. The existence results of the system are proved by means of topological degree theory. An example is constructed to demonstrate our results. Additionally, a graphical analysis is performed to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence and uniqueness of solutions for a system of the first-order q-difference equations with multi-point and q-integral boundary conditions using various fixed point (fp) theorems. Also, we give two examples to support our results. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. Nonlocal conditions are specified in the form of a connection between the values of the unknown function at different points of the boundary. In this case, the boundary operator is determined using matrices of involution-type mappings. Theorems on the existence and uniqueness of solutions to the studied problems are proved. Using Green's functions of the classical Dirichlet and Neumann boundary value problems, Green's functions of the studied problems are constructed and integral representations of solutions to these problems are obtained. [ABSTRACT FROM AUTHOR]

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 obtain conditions of the solvability of the Riemann boundary value problem for sectionally analytic functions in multiply connected domains in Besov spaces embedded into the class of continuous functions. We indicate a new class of Cauchy-type integrals, which are continuous on a closed domain with continuous (not Hölder) density in terms of Besov spaces, and for which the Sokhotski–Plemelj formulas are valid. [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]

In this paper, we study the existence, nonexistence, and multiplicity of positive solutions to a nonlinear impulsive Sturm–Liouville boundary value problem with a parameter. By using a variational method, we prove that the problem has at least two positive solutions for the parameter λ ∈ (0 , Λ) , one positive solution for λ = Λ , and no positive solution for λ > Λ , where Λ > 0 is a constant. [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]

In this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions { (C D 0 + α u) (t) + ∑ i = 1 m λ i (t) (C D 0 + α i u) (t) + ∑ j = 1 n μ j (t) (C D 0 + β j u) (t) + ∑ k = 1 p ξ k (t) (C D 0 + γ k u) (t) + ∑ l = 1 q ω l (t) (C D 0 + δ l u) (t) + σ (t) u (t) + f (t , u (t)) = 0 , t ∈ [ 0 , 1 ] , u ″ (0) = u ‴ (0) = 0 , u ′ (0) = η 1 ∫ 0 1 u (s) d s , u (1) = η 2 ∫ 0 1 u (s) d s , where 0 < δ 1 < δ 2 < ⋯ < δ q < 1 < γ 1 < γ 2 < ⋯ < γ p < 2 < β 1 < β 2 < ⋯ < β n < 3 < α 1 < α 2 < ⋯ < α m < α < 4 and η 1 + 2 (1 − η 2) ≠ 0 . Using a fixed point theorem and Banach contractive mapping principle, we obtain some existence and uniqueness results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel'skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively. [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 discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating the initial boundary value problem for a semilinear wave equation with strong damping and scattering damping on an exterior domain. By introducing suitable multipliers and applying the test-function technique together with an iteration method, we derive the blow-up dynamics and an upper-bound lifespan estimate of the solution to the problem with power-type nonlinearity | u | p , derivative-type nonlinearity | u t | p , and combined type nonlinearities | u t | p + | u | q in the scattering case, respectively. The novelty of the present paper is that we establish the upper-bound lifespan estimate of the solution to the problem with strong damping and scattering damping, which are associated with the well-known Strauss exponent and Glassey exponent. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies g ′ (t) ≤ − ξ (t) g r (t) , t ≥ 0 , 1 ≤ r < 3 2 . The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive solutions for a class of fractional boundary value problems involving an improper integral and the infinite-point on the half-line by making use of properties of the Green function and Avery–Peterson fixed point theorem. In addition, an example is presented to illustrate the applicability of our main result. [ABSTRACT FROM AUTHOR]

In this paper, the existence of the multiple positive solutions for a class of higher-order fractional differential equations on infinite intervals with infinite-point boundary value conditions is mainly studied. First, we construct the Green function and analyze its properties, and then by using the Leggett–Williams fixed point theorem, some new results on the existence of positive solutions for the boundary value problem are obtained. Finally, we illustrate the application of our conclusion by an example. [ABSTRACT FROM AUTHOR]

The novelty of this paper is that, based on Mawhin's continuation theorem, we present some sufficient conditions that ensure that there is at least one solution to a particular kind of a boundary value problem with the p-Laplacian and generalized fractional Caputo derivative. [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, we introduce a new coupled system of sequential fractional differential equations with coupled boundary conditions. We establish existence and uniqueness results using the Leray–Schauder alternative and Banach contraction principle. We examine the stability of the solutions involved in the Hyers–Ulam type. As an application, we present a few examples to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the blow-up of solutions to the initial boundary value problem for the wave equation with scale invariant damping term on the exterior domain, where the nonlinear terms are power nonlinearity | u | p , derivative nonlinearity | u t | p and combined nonlinearities | u t | p + | u | q , respectively. Upper bound lifespan estimates of solutions to the problem are obtained by constructing suitable test functions and utilizing the test function technique. The main novelty is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems 1.1–1.3 are new. [ABSTRACT FROM AUTHOR]

Let T ⊆ R be a time scale. The purpose of this paper is to present sufficient conditions for the existence of multiple positive solutions of the following Lidstone boundary value problem on time scales: (− 1) n y Δ (2 n) (t) = f (t , y (t)) , t ∈ [ a , b ] T , y Δ (2 i) (a) = y Δ (2 i) (σ 2 n − 2 i (b)) = 0 , i = 0 , 1 , ... , n − 1. Existence of multiple positive solutions is established using fixed point methods. At the end some examples are also given to illustrate our results. [ABSTRACT FROM AUTHOR]

The initial boundary value problem of the evolutionary p i -Laplacian equation u t = ∑ i = 1 N ∂ ∂ x i (a i (x) | u x i | p i − 2 u x i) is considered, where a i (x) is nonnegative but is with 0 measure degeneracy. The weak solutions do not belong to B V x (Q T) , how to define the trace in a reasonable way? This is the main topic of this paper. A suitable new boundary value condition is quoted and the stability of weak solutions follows naturally. [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]

In this paper, we consider fractional neutral differential equations with multipoint boundary value conditions involving Hadamard derivatives and integrals. We obtain the existence and uniqueness of the solution of the equation by using several fixed point theorems, and we also consider the Ulam–Hyers stability of the solution. In addition, we study the differential inclusion problem with multipoint boundary value conditions and prove the existence of the solution of the boundary value problem when the multivalued map has convex values. We also give several examples to illustrate the feasibility of the 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]

This paper presents a numerical algorithm for solving high-order BVPs. We introduce the construction method of multiscale orthonormal basis in W 2 m [ 0 , 1 ] . Based on the orthonormal basis, the numerical solution of the boundary value problem is obtained by finding the ε-approximate solution. In addition, the convergence order, stability, and time complexity of the method are discussed theoretically. At last, several numerical experiments show the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

The paper is devoted to finding a solution and restoring the right-hand side of the heat equation with reflection of the argument in the second derivative, with a complex-valued variable coefficient. We prove a theorem on the Riesz basis property for eigenfunctions of the second-order differential operator with involution in the second derivative. We establish the existence and uniqueness of the solution of the studied problems by the method of separation of variables [ABSTRACT FROM AUTHOR]

In this paper, we deal with the blow-up and global solutions of the following p-Laplacian parabolic problems with Neumann boundary conditions: where $p>2$ and Ω is a bounded domain in $\mathbb{R}^{n}$ ( $n\geq 2$ ) with smooth boundary ∂Ω. By introducing some appropriate auxiliary functions and technically using maximum principles, we establish conditions to guarantee that the solution blows up in some finite time or remains global. In addition, the upper estimates of blow-up rate and global solution are specified. We also obtain an upper bound of blow-up time. [ABSTRACT FROM AUTHOR]

Our main objectives in this paper are to investigate the existence of the solutions for an integro-differential inclusion of second order with hybrid nonlocal boundary value conditions. The sufficient condition for the uniqueness of the solution will be given and the continuous dependence of the solution on the set of selections and on other functions will be proved. As an application, the nonlocal problem of the Chandrasekhar hybrid second-order functional integrodifferential inclusion and some particular cases will be presented. Also, we provide some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Recently, it was demonstrated that a modification of the Kalman-filtering model with a pointwise delay of the signal noise could improve communication with considerably distanced spacecraft. However, a complete and correct derivation of the equations of the Kalman-type filter for this case has not yet been provided. In this paper, we close this gap. The method of derivation is based on a passage from the distributed delay to pointwise by means of a delta function. The derived equations constitute a system of first-order partial differential equations with the initial and boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of ρ-concave positive solutions for a p-Laplacian boundary value problem of two-sided fractional differential equations involving generalized-Caputo fractional derivatives. Using Guo–Krasnoselskii fixed point theorem and under some additional assumptions, we prove some important results and obtain the existence of at least three solutions. To establish the results, Green functions are used to transform the considered two-sided generalized Katugampola and Caputo fractional derivatives. Finally, applications with illustrative examples are presented to show the validity and correctness of the obtained results. [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]

This paper aims to consider the multiplicity of solutions for a kind of boundary value problem to a fractional quasilinear differential model with impulsive effects. By establishing a new variational structure and overcoming the difficulties brought by the influence of impulsive effects, some new results are acquired via the symmetry mountain-pass theorem, which extend and enrich some previous results. [ABSTRACT FROM AUTHOR]

The fourth-order discrete Dirichlet boundary value problem is also a discrete elastic beam problem. In this paper, the existence of infinitely many solutions to this problem is investigated through the critical point theory. By an important inequality we established and the oscillatory behavior of f either near the origin or at infinity, we obtain the existence of infinitely many solutions, which either converge to zero or unbounded. In the end, two examples are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of multiple solutions for discrete boundary value problems involving the mean curvature operator by means of Clark's Theorem, where the nonlinear terms do not need any asymptotic and superlinear conditions at 0 or at infinity. Further, the existence of a positive solution has been considered by the strong comparison principle. As an application, some examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper provides conditions for determining the sign of all the partial derivatives of the Green functions of n-th order boundary value problems subject to a wide set of homogeneous two-point boundary conditions, removing restrictions of previous results about the distance between the two extremes that define the problem. To do so, it analyzes the sign of the derivatives of the solutions of related two-point n-th order boundary value problems subject to n − 1 boundary conditions by introducing a new property denoted by 'hyperdisfocality'. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following singular third-order two-point boundary value problem on the half-line of the form { x ‴ + ϕ (t) f (t , x , x ′ , x ″) = 0 , 0 < t < + ∞ , x (0) = 0 , x ′ (0) = a 1 , x ′ (+ ∞) = b 1 , where ϕ ∈ C [ 0 , + ∞) , f ∈ C ([ 0 , + ∞) × (0 , + ∞) × R 2 , R) may be singular at x = 0 , and a 1 , b 1 are positive constants. Using the Leray–Schauder nonlinear alternative and the diagonalization method together with the truncation function technique, we obtain the existence and qualitative properties of positive solutions for the problem. As applications, an example is given to illustrate our result. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global existence of strong solutions for the 3D viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows with density-temperature-dependent viscosities in a bounded domain. We essentially show that for the initial boundary value problem with initial density allowed to vanish, the strong solution exists globally under some suitable small conditions. As a byproduct, we obtain the nonlinear exponential stability of the solution. [ABSTRACT FROM AUTHOR]

In this paper, in a cylindrical domain D = Ω × (0 , T) with Ω ⊂ R n , we consider a mixed Cauchy problem with a potential lateral boundary condition for the following noncharacteristic degenerated equation L u = u t t − k (t) Δ x u (x , t) = f (x , t) , where k (t) ≥ 0 . As in the case for strictly hyperbolic equations, we first establish that u ∈ W 2 1 (D) and u ∈ W 2 2 (D) under the assumptions ∥ f k ∥ L 2 (Ω) (t) < ∞ and ∥ grad x f k ∥ L 2 (Ω) (t) < ∞ for every t ∈ [ 0 , T ] , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of solutions for the nonlocal boundary value problem (BVP) using Caputo fractional derivative (CFD). We derive Green's function and give some estimation for it to derive our main results. The main principles applied to investigate our results are based on the Banach contraction fixed point theorem and Schauder fixed point approach. We dwell in detail on some results concerning the Hyers-Ulam (H-U) type and generalized H-U (g-H-U) type stability also for problem we are considering. We justify our results with an illustrative example. [ABSTRACT FROM AUTHOR]

The paper studies the equation with the boundary degeneracy coming from $a(x)\vert_{x\in \partial \Omega }=0$ . The paper introduces a new kind of weak solutions of the equation. One can study the stability of the new kind of weak solutions without any boundary value condition. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of three solutions to a partial difference equation with (p , q) -Laplacian operator by using critical point theory. Furthermore, based on the strong maximum principle, we prove that the three solutions are positive under appropriate nonlinearity assumptions. Finally, we also give an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

The article discusses various reports published within the issue including one by Moritz Kassmann on the main results and applications of the Harnack inequality and another by Ugur G. Abdulla on reaction-diffusion in nonsmooth and closed domains.

In this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz–Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the investigation of a parabolic equation with moving boundaries arising in ductal carcinoma in situ (DCIS) model. Approximation solution of this problem is implemented by Ritz-Galerkin, which is a first attempt at tackling such problem. In process of dealing with this moving boundary condition, we use a trick of introducing two transformations to convert moving boundary to nonclassical boundary that can be handled with Ritz-Galerkin method. Also, existence and uniqueness are proved. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: { u (4) (t) + β 1 u ″ (t) − α 1 u (t) = f 1 (t , u (t) , v (t)) , 0 < t < 1 ; v (4) (t) + β 2 v ″ (t) − α 2 v (t) = f 2 (t , u (t) , v (t)) , 0 < t < 1 ; u (0) = u (1) = u ″ (0) = u ″ (1) = 0 ; v (0) = v (1) = v ″ (0) = v ″ (1) = 0 , where f 1 , f 2 ∈ C [ (0 , 1) × R 0 + × R , R ] , R 0 + = (0 , + ∞) . By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results. [ABSTRACT FROM AUTHOR]