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For μ∈C1(I)\mu \in {C}^{1}\left(I), μ>0\mu \gt 0, and λ∈C(I)\lambda \in C\left(I), where II is an open interval of R{\mathbb{R}}, we consider the set of functions f∈C2(I)f\in {C}^{2}\left(I) satisfying the second-order differential inequality ddtμdfdt+λf≥0\frac{{\rm{d}}}{{\rm{d}}t}\left(\phantom{\rule[-0.75em]{}{0ex}},\mu \frac{{\rm{d}}f}{{\rm{d}}t}\right)+\lambda f\ge 0 in II. The considered set includes several classes of generalized convex functions from the literature. In particular, if μ≡1\mu \equiv 1 and λ=k2\lambda ={k}^{2}, k>0k\gt 0, we obtain the class of trigonometrically kk-convex functions, while if μ≡1\mu \equiv 1 and λ=−k2\lambda =-{k}^{2}, k>0k\gt 0, we obtain the class of hyperbolic kk-convex functions. In this article, we establish a Fejér-type inequality for the introduced set of functions without any symmetry condition imposed on the weight function and discuss some special cases of weight functions. Moreover, we provide characterizations of the classes of trigonometrically and hyperbolic kk-convex functions.

In this paper, we investigate the asymptotic stability of the trajectories governed by some delay differential inequalities on time scales. Based on time scale theory and the fixed-point theorem, some sufficient conditions are obtained for guaranteeing asymptotic stability. It is interesting that the inequalities studied in this paper include the generalized Halanay inequalities. Due to the fact that dynamic systems on a time scale unify discrete and continuous systems, the results of this paper have wider application value. Furthermore, some numerical examples verify the main results.

The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables \[u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},\] where \(\alpha\in\mathbb{R}\), \(p\gt 1\), and \({\mathtt a}(t)\) as well as \({\mathbf w}(x)\) are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example \(t^\sigma\,{\mathbf w}(x)\) as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit \(\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds\). The main novelty lies in our treatment of the nonstandard condition on the forcing term.

The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration. It is used to study the evolutionary pathway. In this article, we construct mathematical models for the morbidostat. First we consider the case of no mutations, we study limiting systems and obtain criteria for the large time behavior of the solutions. From the theoretical results and numerical simulations, we conclude that there are two competitive exclusion states of either wild type or mutant type as the threshold parameter $ U $ varies. There are three cases, wild type bacteria excludes all mutants; a mutant dominates in the competition; oscillation between the above two states. Next we study the systems of forward mutations and forward-backward mutations. Then we apply a result of pertubation for globally stable state.

In this paper, we focus on the asymptotic stability of the trajectories governed by the differential inequalities with mixed delays using the fixed-point theorem. It is interesting that the Halanay inequality is a special case of the differential inequality studied in this paper. Our results generalize and improve the existing results on Halanay inequality. Finally, three numerical examples are utilized to illustrate the effectiveness of the obtained results.

This paper deals with some differential inequalities for generalized fractional integro-differential equations by using the technique of upper and lower solutions. The fractional differential operator is taken in Caputo’s sense and the nonlinear term divided into two parts depends on the fractional integrals of an unknown function with two different fractional orders. The results are studied by employing a variety of coupled upper and lower solutions. These theorems have some potential for extending the iterative techniques to fractional order integro-differential equations and to coupled systems of integro-differential fractional equations to obtain the existence of solutions as well as approximate solutions for the considered problem.

Consideration is given to three different analytical methods for the computation of upper bounds for the rate of convergence to the limiting regime of one specific class of (in)homogeneous continuous-time Markov chains. This class is particularly well suited to describe evolutions of the total number of customers in (in)homogeneous M/M/S queueing systems with possibly state-dependent arrival and service intensities, batch arrivals and services. One of the methods is based on the logarithmic norm of a linear operator function; the other two rely on Lyapunov functions and differential inequalities, respectively. Less restrictive conditions (compared with those known from the literature) under which the methods are applicable are being formulated. Two numerical examples are given. It is also shown that, for homogeneous birth-death Markov processes defined on a finite state space with all transition rates being positive, all methods yield the same sharp upper bound.

Abstract In this paper, we develop the theory of differential equations with linear perturbations of second type on time scales. An existence theorem for differential equations with linear perturbations of second type on time scales is given under D $\mathscr{D}$ -Lipschitz conditions. Some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions, are also presented. The comparison principle on differential equations with linear perturbations of second type on time scales is established. Our results in this paper extend and improve some well-known results.

Abstract In this paper, we develop the theory of differential equations with mixed perturbations of the second type on time scales. We give an existence theorem for differential equations with mixed perturbations of the second type on time scales under Lipschitz condition. We also present some fundamental differential inequalities on time scales, which are utilized to investigate the existence of extremal solutions. We establish the comparison principle for differential equations with mixed perturbations of the second type on time scales. Our results in this paper extend and improve some well-known results.

We consider a higher-order evolution differential inequality in an exterior domain of ℝN{\mathbb{R}^{N}}, N≥3{N\geq 3}, with Dirichlet and Neumann boundary conditions. Using a unified approach, we obtain the critical exponents in the sense of Fujita for the considered problems. Moreover, the behavior of the solutions with respect to the initial data is discussed.

The second order nonlinear differential equation \begin{equation*} x''+h(t,x,x')x'+f(x)=0 \qquad \bigl(x\in\mathbb{R},\ t\in\mathbb{R}_+:=[0,\infty),\ ()':=\tfrac{\text{d}}{\text{d}t}()\bigr), \end{equation*} and a sequence $\{I_n\}_{n=1}^\infty$ of non-overlapping intervals are given, where the damping coefficient $h$ admits an estimate \begin{equation*} a(t)|y|^\alpha w(x,y)\le h(t,x,y)\le b(t) W(x,y)\qquad (t\in I:=\cup_{n=1}^\infty I_n; x,y\in \mathbb{R}). \end{equation*} It is known that if the equation is linear ($f(x)\equiv x$, $h(t,x,x')\equiv h(t)$, $a(t)\le h(t)\le b(t)$), $a(t)\ge \underline{a}=\text{const.}>0$ and $b(t)\le \overline{b}=\text{const.}0$ and/or $\overline{b}

In the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulation of the problem is investigated, with weaker sufficient conditions for the existence of a solution with an internal transition layer to be provided than in previous studies. For convenience, the known results are given, which ensure the fulfillment of the existence theorem of the contrast structure. The justification for the existence of a solution with an internal transition layer is based on the use of an asymptotic method of differential inequalities based on the modification of the terms of the constructed asymptotic expansion. Further, sufficient conditions are established to fulfill these requirements, and they have simple and concise formulations in the form of the algebraic equation w(x0,t) = 0 and the condition wx(x0,t) < 0, which is essentially a condition of simplicity of the root x0(t) and ensuring the stability of the solution found. The function w is a function of the known functions appearing in the reactive and advective terms of the original problem. The equation w(x0,t) = 0 is a problem for finding the zero approximation x0(t) to determine the localization region of the inner transition layer. In addition, the asymptotic Lyapunov stability of the found periodic solution is investigated, based on the application of the so-called compressible barrier method. The main result of the paper is formulated as a theorem.

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. Such an approach seems very general; the corresponding description and bounds were considered earlier for finite Markov chains with analytical in time intensity functions. Now we generalize this method to locally integrable intensity functions. Special attention is paid to the situation of a countable Markov chain. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences l1.

In this article we consider the problem of finding the solution of a system of differential inequalities. We reduce the original problem to the unconstrained minimization of a functional. We find the Gateaux gradient for this functional, and then obtain nucessary and sufficient conditions for the existence of a minimum. Based on these conditions we apply the steepest descent method, and present a numerical implementation of the method.

We consider a nonstationary process of spreading some substance in a one-dimensional spatially inhomogeneous system of cells. It is assumed that a change in the concentration of \(u_n (t)\) in a cell with the number \(n\) with time \(t\) is determined by the difference in concentration in this cell and in its two neighbors on the left and on the right, as well as the source density, which depends on \(n\) and depends on \(u_n (t)\). Such a model leads to the initial-boundary value problem for the differentialdifference equation (differentiation with respect to t variable, the difference expression with respect to \(n\)). With a sufficiently small difference in concentration in each pair of neighboring cells we can replace the difference expression by the second partial derivative with respect to the spatial coordinate, and describe the propagation by the reaction-diffusion equation. This equation belongs to the class of quasilinear parabolic equations. It is assumed that the density of the sources vanishes (with changing the sign) at three values of the concentration, two of which, lower and upper, are stable. There is also an intermediate unstable state with zero source density, in which the sign reversal also takes place. The peculiarity of our model is that we assume, that two extreme roots of the source density function are degenerate (with an integer or fractional exponent). We intend to show analytically and by the computer simulation, that this model leads to the fact, that the rate of asymptotic aspiration of concentration to equilibrium values for a moving front becomes power-law instead of exponential, which takes place for standard models. In the paper, we have constructed a formal asymptotics solution of the initialboundary value problem for the reaction-diffusion equation in a homogeneous medium with a power-law dependence of the source density on the temperature, an upper and lower solutions are constructed, a rigorous justification of the formal asymptotics is given. Precise solutions of the diffusion reaction equation are constructed for a wide class of source density functions.

An important problem in complex analysis is to determine properties of the image of an analytic function p defined on the unit disc U from an inclusion or containment relation involving several of the derivatives of p. Results dealing with differential inclusions have led to the development of the field of Differential Subordinations, while results dealing with differential containments have led to the development of the field of Differential Superordinations. In this article, the authors consider a mixed problem consisting of special differential inclusions implying a corresponding containment of the form D[p](U)⊂Ω⇒Δ⊂p(U), where Ω and Δ are sets in C, and D is a differential operator such that D[p] is an analytic function defined on U. We carry out this research by considering the more general case involving a system of two simultaneous differential operators in two unknown functions.

We consider two simple boundary value problems at resonance for an ordinary differential equation. Employing a shift argument, a regular fixed point operator is constructed. We employ the monotone method coupled with a method of upper and lower solutions and obtain sufficient conditions for the existence of solutions of boundary value problems at resonance for nonlinear boundary value problems. Three applications are presented in which explicit upper solutions and lower solutions are exhibited for the first boundary value problem. Two applications are presented for the second boundary value problem. Of interest, the upper and lower solutions are easily and explicitly constructed. Of primary interest, the upper and lower solutions are elements of the kernel of the linear problem at resonance.

Parabolic singularly perturbed problems have been actively studied in recent years in connection with a large number of practical applications: chemical kinetics, synergetics, astrophysics, biology, and so on. In this work a singularly perturbed periodic problem for a parabolic reaction-diffusion equation is studied in the two-dimensional case. The case when there is an internal transition layer under unbalanced nonlinearity is considered. The internal layer is localised near the so called transitional curve. An asymptotic expansion of the solution is constructed and an asymptotics for the transitional curve is determined. The asymptotical expansion consists of a regular part, an interior layer part and a boundary part. In this work we focus on the interior layer part. In order to describe it in the neighborhood of the transition curve the local coordinate system is introduced and the stretched variables are used. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The upper and lower solutions are constructed by sufficiently complicated modification of the asymptotic expansion of the solution. The Lyapunov asymptotical stability of the solution was proved by using the method of contracting barriers. This method is based on the asymptotic comparison principle and uses the upper and lower solutions which are exponentially tending to the solution to the problem. As a result, the solution is locally unique.The article is published in the authors’ wording.

This article furnishes qualitative properties for solutions to two-point boundary value problems (BVPs) which are systems of singular, second-order, nonlinear ordinary differential equations. The right-hand side of the differential equation is allowed to be unrestricted in the growth of its variables and may depend on the derivative of the solution, which incurs additional difficultly in the mathematical proofs. A new approach is introduced by using a singular differential inequality that ensures that all possible solutions satisfy certain a priori bounds, including their "derivatives", to the singular BVP under consideration. Topological methods, in particular Schauder's fixed point theorem are then applied to generate new existence results for solutions to the singular boundary value problems. Many of the results are novel for both the singular and the nonsingular cases.

We provide sufficient conditions for the blow-up of solutions to certain second-order differential inequalities and systems with advanced and delayed arguments. Our proofs are based on the test function method.

Using the test function technique, we obtain sufficient conditions for the blow-up of solutions to some differential equations and inequalities with advanced and delayed arguments, and for systems. Also we obtain upper estimates for the blow-up time.

In this article, we study a class of non-linear neutral delay differential equations of third order. We first prove criteria for non-existence of non-Kneser solutions, and criteria for non-existence of Kneser solutions. We then use these results to provide criteria for the under study differential equations to ensure that all its solutions are oscillatory. An example is given that illustrates our theory.

In this work, we present a new technique for the oscillatory properties of solutions of higher-order differential equations. We set new sufficient criteria for oscillation via comparison with higher-order differential inequalities. Moreover, we use the comparison with first-order differential equations. Finally, we provide an example to illustrate the importance of the results.

In this paper the author introduced a new generalized integral operator for meromorphic p-valent functions in U*={z:z in C and 0

In this article we apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a \(2n\)-th order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities that extend the notion of concavity to \(2n\)-th order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space.

In this paper, some fundamental differential inequalities for the implicit perturbations of nonlinear first order ordinary causal differential equations have been established.

Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear differential equations on a half-axis. In addition, we extend the methods and present new similar results for integral equations and Volterra-Stieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of Volterra-Stieltjes integral equations is also presented.

In the paper we present some new existence results for nonlinear second order generalized periodic boundary value problems of the form These results are based on the method of lower and upper functions defined as solutions of the system of differential inequalities associated with the problem and their relation to the Leray–Schauder topological degree of the corresponding operator. Our main goal consists in a fairly general definition of these functions as couples from . Some conditions ensuring their existence are indicated, as well.

The notions of lower and upper functions of the second order differential equations take their beginning from the classical work by C. Scorza-Dragoni and have been investigated till now because they play an important role in the theory of nonlinear boundary value problems. Most of them define lower and upper functions as solutions of the corresponding second order differential inequalities. The aim of this paper is to compare two more general approaches. One is due to Rachůnková and Tvrdý (Nonlinear systems of differential inequalities and solvability of certain boundary value problems (J. of Inequal. & Appl. (to appear))) who defined the lower and upper functions of the given equation as solutions of associated systems of two differential inequalities with solutions possibly not absolutely continuous. The second belongs to Fabry and Habets (Nonlinear Analysis, TMA 10 (1986), 985–1007) and requires the monotonicity of certain integro-differential expressions.

Four different well-posed regularizations of the improperly posed Cauchy problem for the backward heat equation are investigated in order to determine whether solutions of these problems depend continuously on a perturbation parameter. Using differential inequality techniques, we derive results implying continuous dependence in each case.