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The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach's fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019]. [ABSTRACT FROM AUTHOR]

In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a function g in the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functions g and g ′ , where g ′ is a derived function of a 2π-periodic function g, in the generalized Zygmund class X z (η) , z ≥ 1 , using matrix-Cesàro (T C δ) means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1. [ABSTRACT FROM AUTHOR]

In this paper, we establish inequalities of Hermite–Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (İşcan in Hacet. J. Math. Stat. 43(6):935–942, 2014 and İşcan and Wu in Appl. Math. Comput. 238:237–244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of mild solutions for the abstract fractional nonlocal evolution equations with noncompact semigroup in partially ordered Banach spaces. Under some mixed conditions, a group of sufficient conditions for the existence of abstract fractional nonlocal evolution equations are obtained by using a Krasnoselskii type fixed point theorem. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to illustrate the applicability of abstract result. [ABSTRACT FROM AUTHOR]

In this paper, we mainly investigate upper semicontinuity and regularity of attractors for nonclassical diffusion equations with perturbed parameters ν and the nonlinear term f satisfying the polynomial growth of arbitrary order p − 1 (p ≥ 2 ). We extend the asymptotic a priori estimate method (see (Wang et al. in Appl. Math. Comput. 240:51–61, 2014)) to verify asymptotic compactness and upper semicontinuity of a family of semigroups for autonomous dynamical systems (see Theorems 2.2 and 2.3). By using the new operator decomposition method, we construct asymptotic contractive function and obtain the upper semicontinuity for our problem, which generalizes the results obtained in (Wang et al. in Appl. Math. Comput. 240:51–61, 2014). In particular, the regularity of global attractors is obtained, which extends and improves some results in (Xie et al. in J. Funct. Spaces 2016:5340489, 2016; Xie et al. in Nonlinear Anal. 31:23–37, 2016). [ABSTRACT FROM AUTHOR]

This paper is concerned with the number of limit cycles bifurcating from a period annulus for some planar piecewise smooth non-Hamiltonian systems. We construct a planar piecewise quadratic system with multiple parameters, obtain its lower bound for the maximum number of limit cycles by using Melnikov function method, and find more limit cycles than (Li and Liu in J. Math. Anal. Appl. 428:1354–1367, 2015). [ABSTRACT FROM AUTHOR]

Recently, Wu (Mathematics 6(11):219, 2018; Mathematics 6(6):90, 2018) introduced the concept of a near-fixed point and established some results on near fixed points in a metric interval space and hyperspace. Motivated by these papers, we studied the near-coincidence point theorem in these spaces via a simulation function. To show the authenticity of the established results and definitions, we also provide some examples. [ABSTRACT FROM AUTHOR]

In this paper, we use a new powerful technique of arbitrary-order fractional (AOF) characteristic method (CM) to solve the AOF hyperbolic nonlinear scalar conservation law (HNSCL) of time and space. We present the existence and uniqueness of this class of equations in time and one-dimensional space of fractional arbitrary order. We extend Jumarie's modification of Riemann–Liouville and Caputo's definition of the fractional arbitrary order to introduce some formulae (Appl. Math. Lett. 22:378–385, 2009; Appl. Math. Lett. 18:739–748, 2005). Then, we use these formulae to prove the main theorem. In the application section, we use the analytical technique that is presented in the theorem to solve examples that are given. [ABSTRACT FROM AUTHOR]

Işık et al. (Mathematics 7:862, 2019) presented an interesting generalization of the Banach contraction principle. In this paper, motivated by Işık et al., we give a new extension of the well-known Darbo inequality in a Banach space. Our results provide several generalizations of the Darbo inequality. As an application, we study the existence of solutions for a system of functional integral equations in C [ 0 , T ] . Finally, we expose a genuine example to support the effectiveness of our results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the uniqueness of q-difference-differential polynomials of different types. Using the idea of common zeros and common poles (Chin. Ann. Math., Ser. A 35:675–684, 2014), we improve the conditions of the former theorems and obtain some new results on the uniqueness of q-difference-differential polynomials of meromorphic functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015). [ABSTRACT FROM AUTHOR]

In this paper we study the powers under umbral composition and degeneration for Sheffer sequences, where we presented several applications related to Bernoulli polynomials, Frobenius-Euler polynomials, falling factorial polynomials and Bell polynomials and their degeneration cases. [ABSTRACT FROM AUTHOR]

In this paper, the problem of exponential lag synchronization for a class of neural networks with mixed delays including discrete and distributed delays is investigated via adaptive intermittent control. Based on piecewise analytic method, some sufficient conditions for globally exponential lag synchronization are established through constructing a piecewise continuous auxiliary function. It is noted that both the control periods and the control widths in our adaptive intermittent control strategy are allowed to be nonidentical, which extends the scope of application of periodically intermittent control with fixed both control period and control width employed widely in previous works. Moreover, it is shown that the derived globally exponential lag synchronization criteria are related to the control rates rather than the control periods, which facilitates the choice of the control periods for practical problems. Finally, a numerical example is given to illustrate the correctness of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Lazer-Leach-type condition depending on the delay for the existence of positive periodic solutions for singular differential equations with a deviating argument at resonance. The proof of the main result is based on the phase-plane analysis and topological degree methods. [ABSTRACT FROM AUTHOR]

The accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analytical and numerical techniques. Adomian decomposition (AD), El Kalla (EK), cubic B-spline (CBS), extended cubic B-spline (ECBS), and exponential cubic B-spline (ExCBS) schemes are used to explain the matching between analytical and approximate solutions, which shows the accuracy of constructed traveling wave solutions. In 1880, Oliver Heaviside derived the considered model to describe the cutting-edge or voltage of an electrified transmission. The matching between solutions has been explained by plotting them in some different sketches. [ABSTRACT FROM AUTHOR]

We consider degenerate identification problems with smoothing overdetermination in abstract spaces. We establish an identifiability result using a projection method and suitable hypotheses on the operators involved and develop an identification method by reformulating the problem into a nondegenerate problem. Then we use perturbation results for linear operators to solve the regular problem. The introduced identification method permits one to solve the problems under the minimum restrictions on the input data. Finally, we provide applications to degenerate differential equations that appear in mathematical physics to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present certain composition formulae of the pathway fractional integral operators associated with two extended Mittag-Leffler functions. Here, we find out the relevant connections of some particular cases of the main results with those earlier ones. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to extend the Hurwitz–Lerch zeta function Φ δ , ς ; γ (ξ , s , υ ; p) involving the extension of the beta function (Choi et al. in Honam Math. J. 36(2):357–385, 2014). We also study the basic properties of this extended Hurwitz–Lerch zeta function which comprises various integral formulas, a derivative formula, the Mellin transform, and the generating relation. The fractional kinetic equation for an extended Hurwitz–Lerch zeta function is also obtained from an application point of view. Furthermore, we obtain certain interesting relations in the form of particular cases. [ABSTRACT FROM AUTHOR]

Recently, Hoffman (Commun. Number Theory Phys. 13:529–567, 2019), Kaneko and Tsumura (Tsukuba J. Math. (in press), 2020) introduced and systematically studied two variants of multiple zeta values of level two, i.e., multiple t-values and multiple T-values, respectively. In this paper, by the contour integration and residue theorem, we establish two general identities, which further reduce to the expressions of the alternating double t-values and T-values. Some examples are also provided. [ABSTRACT FROM AUTHOR]

Let f be a transcendental meromorphic function of finite order and c be a nonzero complex number. Define Δ c f = f (z + c) − f (z) . The authors investigate the existence on the fixed points of Δ c f . The results obtained in this paper may be viewed as discrete analogues on the existing theorem on the fixed points of f ′ . The existing theorem on the fixed points of Δ c f generalizes the relevant results obtained by (Chen in Ann. Pol. Math. 109(2):153–163, 2013; Zhang and Chen in Acta Math. Sin. New Ser. 32(10):1189–1202, 2016; Cui and Yang in Acta Math. Sci. 33B(3):773–780, 2013) et al. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point and multi-strip boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We demonstrate the application of the obtained results with the aid of examples. Some new results are also deduced by fixing the parameters involved in the problem at hand. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

Let f be an entire function of finite order, let n ≥ 1 , m ≥ 1 , L (z , f) ≢ 0 be a linear difference polynomial of f with small meromorphic coefficients, and P d (z , f) ≢ 0 be a difference polynomial in f of degree d ≤ n − 1 with small meromorphic coefficients. We consider the growth and zeros of f n (z) L m (z , f) + P d (z , f) . And some counterexamples are given to show that Theorem 3.1 proved by I. Laine (J. Math. Anal. Appl. 469:808–826, 2019) is not valid. In addition, we study meromorphic solutions to the difference equation of type f n (z) + P d (z , f) = p 1 e α 1 z + p 2 e α 2 z , where n ≥ 2 , P d (z , f) ≢ 0 is a difference polynomial in f of degree d ≤ n − 2 with small mromorphic coefficients, p i , α i (i = 1 , 2 ) are nonzero constants such that α 1 ≠ α 2 . Our results are improvements and complements of Laine 2019, Latreuch 2017, Liu and Mao 2018. [ABSTRACT FROM AUTHOR]

In this work, we provide upper and lower bounds for the numerical radius of an n × n off-diagonal operator matrix, which extends some results by Abu-Omar and Kittaneh (Stud. Math. 216:69–75, 2013; Linear Algebra Appl. 468:18–26, 2015; Rocky Mt. J. Math. 45(4):1055–1065, 2015), and Paul and Bag (Appl. Math. Comput. 222:2731–2943, 2013). [ABSTRACT FROM AUTHOR]

We study the weighted pseudo almost periodic solutions of a Lasota–Wazewska system. With the aid of fixed point theory and differential inequality strategies, we give a set of new sufficient criteria that guarantee the existence and global exponential stability of weighted pseudo almost periodic solutions to a Lasota–Wazewska system. The obtained results of this manuscript are completely innovative and complement the work of Shao (Appl. Math. Lett. 43:90–95, 2015) to some degree. So far, no scholars have investigated this aspect. [ABSTRACT FROM AUTHOR]

Multiplicity results are obtained for systems of first order dynamic inclusions on time scales with the periodic boundary condition. To this aim, we introduce the notion of a strict solution-tube. We consider the case where the nonlinearity satisfies an upper semi-continuity condition. Our results are new even in the particular cases where the nonlinearity is single-valued or it has real values. [ABSTRACT FROM AUTHOR]

We examine the various types of stability for the solutions of linear dynamic systems on time scales and give two examples. [ABSTRACT FROM AUTHOR]

We show that every solution of the following system of difference equations x(1)n+1 = x(2)n/(x(2)n - 1), x(2)n+1 = x(3)n/(x(3)n -1 ), ...,x (k)n+1 = x(1)n/x(1)n -1) as well as of the system x(1)n+1 = x(k)n/ (x(k)n-1), x(2)n+1 = x(1)n/ x(1)n- 1),..., x(k)n+1 = x(k-1)n/ (x(k-1)n-1) is periodic with period 2k if k ≠ 0 (mod2), and with period k if k = 0 (mod 2) where the initial values are nonzero real numbers for x(1)0, x(2)0,...,x(k)0 ≠ 1. [ABSTRACT FROM AUTHOR]

We consider second-order difference expressions, with complex coefficients, of the form wn-1[-Δ(pn-1 Δxn-1) + qnxn] acting on infinite sequences. The discrete analog of some known relationships in the theory of differential operators such as Dirichlet, conditional Dirichlet, weak Dirichlet, and strong limit-point is considered. Also, connections and some relationships between these properties have been established. [ABSTRACT FROM AUTHOR]

Some existence conditions of periodic solutions are obtained for a class of nonautonomous subquadratic first-order discrete Hamiltonian systems by the minimax methods in the critical point theory. [ABSTRACT FROM AUTHOR]

For periodic and almost periodic functional difference equations with finite delay, the existence of periodic and almost periodic solutions is obtained by using stability properties of a bounded solution. [ABSTRACT FROM AUTHOR]

We consider the following system of Lyness-type difference equations: x1(n + 1) = (akxk(n) + bk)/ xk-1(n - 1), x2(n + 1) = (a1x1(n) + b1)/xk(n - 1), xi(n + 1) = (ai-1xi-1(n) + bi-1)/xi-2(n-1), i = 3, 4,...,k, where ai, bi, i = 1, 2,...,k, are positive constants, k ≥ 3 is an integer, and the initial values are positive real numbers. We study the existence of invariants, the boundedness, the persistence, and the periodicity of the positive solutions of this system. [ABSTRACT FROM AUTHOR]

In this note, we will show that an entire function is equal to its difference operator if it has a growth property and shares a set, where the set consists of two entire functions of smaller orders. This result generalizes a result of Li (Comput. Methods Funct. Theory 12:307-328, 2012 and partially answers Liu's (J. Math. Anal. Appl. 359:384-393, 2009) question. [ABSTRACT FROM AUTHOR]

In this article, we formulate nabla fractional sums and differences of order 0<α≤1 on the time scale hZ, where 0. Then, we prove that if the nabla h-Riemann-Liouville (RL) fractional difference operator (a∇hαy)(t)>0, then y(t) is α-increasing. Conversely, if y(t) is α-increasing and y(a)>0, then (a∇hαy)(t)>0. The monotonicity results for the nabla h-Caputo fractional difference operator are also concluded by using the relation between h-nabla RL and Caputo fractional difference operators. It is observed that the reported monotonicity coefficient is not affected by the step h. We formulate a nabla h-fractional difference initial value problem as well. Finally, we furniture our results by proving a fractional difference version of the Mean Value Theorem (MVT) on hZ. [ABSTRACT FROM AUTHOR]