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In this paper, we establish a relation-theoretic set-valued version of the fixed point result of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007) on metric spaces endowed with an arbitrary binary relation. The results of this paper, generalize and unify the fixed point results of Ćirić and Prešić (Acta Math. Univ. Comen. LXXVI(2):143–147, 2007), Shukla and López (Quaest. Math. 45(3):1–16, 2019), and Shukla and Radenović (An. Ştiinţ. Univ. 'Al.I. Cuza' Iaşi, Mat. 63(2):339–350, 2017) in product spaces. Some examples are provided that justify and establish the importance of our results. As applications of our main result, we have established the existence of solutions to differential inclusion problems and the weak asymptotical stability and a global attractivity of the equilibrium point of a difference inclusion problem. The use of arbitrary binary relations in our results permits us to apply the results to the differential inclusion problems and difference inclusion problems with weaker assumptions than those used in the papers mentioned above. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D incompressible Hall-magnetohydrodynamics with partial dissipation. Based on the results in (Du in Bound. Value Probl. 2022:6, 2022; Du and Liu in Acta Math. Sci. 42A:5, 2022; Fei and Xiang in J. Math. Phys. 56:051504, 2015), we establish an improved blow-up criterion for classical solutions. Furthermore, using the blow-up criterion, we also obtain the existence of the classical solutions only under the condition that the initial data ∥ V 0 ∥ H 1 + ∥ B 0 ∥ H 2 are sufficiently small. [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]

In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up of the solution to a non-local reaction diffusion problem in ℝN for N = 3 under nonlinear boundary conditions. Utilizing the technique of a differential inequality, lower bounds for the blow-up time are derived when the blow-up does occur under some suitable assumptions. [ABSTRACT FROM AUTHOR]

The paper deals with the study of the existence result for a Kirchhoff elliptic system with additive right-hand side and variable parameters by using the sub-/supersolution method. Our study is a natural extension result of our previous one in (Boulaaras and Guefaifia in Math. Methods Appl. Sci. 41:5203–5210, 2018), where we discussed only the simple case when the parameters are constant. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Choquard type quasilinear Schrödinger equation: − Δ u + u − Δ (u 2) u = (I α ∗ G (u)) g (u) , x ∈ R N , where N ≥ 3 , 0 < α < N , and I α is a Riesz potential. By using the minimization method developed by (Tang and Chen in Adv. Nonlinear Anal. 9:413–437, 2020; Willem in Minimax Theorems, 1996), we establish the existence of ground state solutions with general Choquard type nonlinearity. Our results extend the results obtained by (Chen et al. in Appl. Math. Lett. 102:106141, 2020). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability of the solutions of a viscoelastic plate equation with a logarithmic nonlinearity. We assume that the relaxation function g satisfies the minimal condition g ′ (t) ≤ − ξ (t) G (g (t)) , where ξ and G satisfy some properties. With this very general assumption on the behavior of g, we establish explicit and general energy decay results from which we can recover the exponential and polynomial rates when G (s) = s p and p covers the full admissible range [ 1 , 2) . Our new results substantially improve and generalize several earlier related results in the literature such as Gorka (Acta Phys. Pol. 40:59–66, 2009), Hiramatsu et al. (J. Cosmol. Astropart. Phys. 2010(06):008, 2010), Han and Wang (Acta Appl. Math. 110(1):195–207, 2010), Messaoudi and Al-Khulaifi (Appl. Math. Lett. 66:16–22, 2017), Mustafa (Math. Methods Appl. Sci. 41(1):192–204, 2018), and Al-Gharabli et al. (Commun. Pure Appl. Anal. 18(1):159–180, 2019). [ABSTRACT FROM AUTHOR]

The problem in this paper is motivated by physical problems concerned with the case when a class of continuous and equimeasurable densities of a string is given then how to find minimal frequencies among these given densities, that is, what kind of densities minimize the frequencies. By taking Dirichlet eigenvalues into account, given a certain weight function ω, we will show the minimizer of the mth eigenvalue is the m-degree continuous symmetrical decreasing rearrangement of ω. The main result of this paper can be viewed as complementary to Schwarz’s work (Schwarz in J. Math. Mech. 10:401-422, 1961). [ABSTRACT FROM AUTHOR]

In this paper, we study the symmetric solutions of second-order BVP with integral boundary conditions. By using a generalized Leggett-Williams fixed point theorem and some other techniques, we obtain sufficient conditions for the existence of symmetric positive solutions for the system. Meanwhile, an example is devoted to demonstrate our results in the end. [ABSTRACT FROM AUTHOR]

In this paper, we derive the optimal boundary conditions of the Navier-Stokes fluid in a bounded domain with a thin layer. We get the effects of the layer by investigating the limiting δ, the thickness of the layer shrinking to zero. In the two-dimensional case, we derive effective boundary conditions by the limit of u and p on the boundary of the uncoated body. [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]

This paper is concerned with the solvability for fractional Sturm-Liouville boundary value problems with $p(t)$ -Laplacian operator at resonance using Mawhin's continuation theorem. Sufficient conditions for the existence of solutions have been acquired, and they would extend the existing results. Furthermore, an example is provided to illustrate the main result. [ABSTRACT FROM AUTHOR]

We investigate a generalized poly-Laplacian system with a parameter on weighted finite graphs, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a (p , q) -Laplacian system with a parameter on weighted locally finite graphs. We utilize a critical points theorem built by Bonanno and Bisci [Bonanno, Bisci, and Regan, Math. Comput. Model. 52(1-2):152–160, 2010], which is an abstract critical points theorem without compactness condition, to obtain that these systems have infinitely many nontrivial solutions with unbounded norm when the parameters locate some well-determined range. [ABSTRACT FROM AUTHOR]

In mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of triple positive solutions for a type of second-order singular boundary problems with general differential operators. By using the Leggett-Williams fixed point theorem, we establish an existence criterion for at least three positive solutions with suitable growth conditions imposed on the nonlinear term. [ABSTRACT FROM AUTHOR]

In this paper we consider a quasilinear viscoelastic wave equation with initial-boundary conditions, strong damping and source term. Under suitable assumptions on the initial data and the relaxation function, we establish a blow-up result of a solution for negative initial energy and some positive initial energy if the influence of the source term is greater than the dissipation. We show that the solution exists globally for any initial data if the influence of dissipation is greater than the source term. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of solutions for fractional differential equations of arbitrary order with nonlocal integral boundary conditions. The existence results are obtained by applying Krasnoselskii's fixed point theorem and Leray-Schauder degree theory, while the uniqueness of the solutions is established by means of Banach's contraction mapping principle. The paper concludes with illustrative examples. [ABSTRACT FROM AUTHOR]

Motivated by (Xu et al. in Bound. Value Probl. 2013:262, 2013) and (Yang and Ren in Proc. Indian Acad. Sci. Math. Sci. 124(2):175-178, 2014), in this paper we aim to construct a modified Green function in the upper-half space of the n-dimensional Euclidean space, which generalizes the boundary property of general Green potential. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a class of cellular neural networks with time-varying delays. Applying differential inequality strategies without assuming the boundedness conditions on the activation functions, we obtain a new sufficient condition that ensures that all solutions of the considered neural networks converge exponentially to the zero equilibrium point. We give an example to illustrate the effectiveness of the theoretical results. The results obtained in this paper are completely new and complement the previously known studies of Tang (Appl. Math. Lett. 21:872–876, 2008). [ABSTRACT FROM AUTHOR]

This paper is concerned with semilinear wave equations with nonlinear interior and boundary sources and subject to a nonlinear dynamical boundary condition. By using the potential well method combined with a standard continuous argument, under appropriate assumptions imposed on the source term, we establish global existence of solutions. Moreover, for certain initial data in the unstable set, the finite time blow-up phenomenon is exhibited. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the nonlinear boundary problem for first-order impulsive semilinear differential inclusions. We establish existence results by using Martelli’s fixed point theorem with upper and lower solutions method. We find that by giving different definitions of lower and upper solutions we can get all existence results. We also present an example. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the existence of periodic solutions of the second-order equation x" = f (t, x), where f is a Carathéodory function, by combining a new expression of Green's function together with Dancer's global bifurcation theorem. Our main results are sharp and improve the main results by Torres (2003). [ABSTRACT FROM AUTHOR]

Let L be a divergence form operator with Lipschitz continuous coefficients in a domain Ω, and let u be a continuous weak solution of Lu = 0 in {u ≠ 0}. In this paper, we show that if ɸ satisfies a suitable differential inequality, then vɸ(x) = supBɸ(x)(x)u is a subsolution of Lu = 0 away from its zero set. We apply this result to prove C1, regularity of Lipschitz free boundaries in two-phase problems. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the approximate controllability of the coupled system with boundary degeneracy. The control functions act on the degenerate boundary. We prove the Carleman estimate and the unique continuation of the adjoint system. Then we get the approximate controllability by constructing the control functions. [ABSTRACT FROM AUTHOR]

In this paper, by using Karamata regular variation theory and the method of upper and lower solutions, we mainly study the second order expansion of solutions to the following p-Laplacian problems: Δ p u = b (x) f (u) , u > 0 , x ∈ Ω , u | ∂ Ω = ∞ , where Ω is a bounded domain with smooth boundary in R N (N ≥ 2) , p > 1 , b ∈ C α (Ω ¯) which is positive in Ω and may be vanishing on the boundary. The absorption term f is normalized regularly varying at infinity with index σ > p − 1 . The results extend some previous findings of D. Repovš (J. Math. Anal. Appl. 395:78-85, 2012) in a certain sense. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Sturm-Liouville problem with finite discontinuous points inside an interval and with abstract linear functionals in the boundary and transmission conditions. For such a problem, the properties such as isomorphism, Fredholmness, and coerciveness with respect to the spectral parameter are investigated. [ABSTRACT FROM AUTHOR]

Department of Differential Equations and Statistics, Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, Rzeszów, 35-959, Poland [ABSTRACT FROM AUTHOR]

Recently, Hattori–Lagha established the global existence and asymptotic behavior of the solutions for a three-dimensional compressible chemotaxis system with chemoattractant and repellent (Hattori and Lagha in Discrete Contin. Dyn. Syst. 41(11):5141–5164, 2021). Motivated by Hattori–Lagha's work, we further investigated the optimal time-decay rates of strong solutions with small perturbation to the three-dimensional Keller–Segel system coupled to the compressible Navier–Stokes equations, which models for the motion of swimming bacteria in a compressible viscous fluid. First, we reformulate the system into a perturbation form. Then we establish a prior estimates of solutions and prove the existence of the global-in-time solutions based on the local existence of unique solutions. Finally, we will establish the optimal time-decay rates of the nonhomogeneous system by the decomposition technique of both low and high frequencies of solutions as in (Wang and Wen in Sci. China Math., 2020, https://doi.org/10.1007/s11425-020-1779-7). Moreover, the decay rate is optimal since it agrees with the solutions of the linearized system. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results. [ABSTRACT FROM AUTHOR]

We present some sufficient conditions of blowup of the solutions to Laplace equations with semilinear dynamical boundary conditions of hyperbolic type. [ABSTRACT FROM AUTHOR]

We report on some Liouville-type theorems for a class of linear second-order partial differential equation with nonnegative characteristic form. The theorems we show improve our previous results. [ABSTRACT FROM AUTHOR]

We study positive solutions to steady-state reaction–diffusion models of the form { − Δ u = λ f (v) ; Ω , − Δ v = λ g (u) ; Ω , ∂ u ∂ η + λ u = 0 ; ∂ Ω , ∂ v ∂ η + λ v = 0 ; ∂ Ω , where λ > 0 is a positive parameter, Ω is a bounded domain in R N (N > 1) with smooth boundary ∂Ω, or Ω = (0 , 1) , ∂ z ∂ η is the outward normal derivative of z. We assume that f and g are continuous increasing functions such that f (0) = 0 = g (0) and lim s → ∞ f (M g (s)) s = 0 for all M > 0 . In particular, we extend the results for the single equation case discussed in (Fonseka et al. in J. Math. Anal. Appl. 476(2):480-494, 2019) to the above system. [ABSTRACT FROM AUTHOR]

The existence of weak solutions for a class of $(p,q)$ -Laplacian systems is obtained by using a new linking theorem on the product space $W^{1,p}_{0}(\Omega)\times W^{1,q}_{0}(\Omega)$ . [ABSTRACT FROM AUTHOR]