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We prove a general existence theorem for non-self-adjoint Sturm-Liouville problems and we derive quite general asymptotic formulae for their eigenvalues and the corresponding eigenfunctions. The derived formulae are general, very accurate and remain valid for a large class of complex potentials with singularities. These results are obtained by using He's homotopy perturbation method (HPM) with an auxiliary parameter. It will be shown that this method is easy to use and makes the study of these problems more simple and more efficient. In order to illustrate the theory, interesting asymptotic and numerical results are discussed and presented from a wide range of examples. [ABSTRACT FROM AUTHOR]

Consider the number of limit cycles of a family of systems with homogeneous components: x ˙ = y , y ˙ = - x 3 + α x 2 y + y 3. We show that there is an α ∗ < 0 such that the system has exactly one limit cycle for α ∈ (α ∗ , 0) , while no limit cycle for the else region. This completes a previous result and also gives a positive answer to the second part of Gasull’s 3rd problem listed in the paper (SeMA J 78(3):233–269, 2021). To obtain this result, we mainly analyse the behavior of the heteroclinic separatrices at infinity. [ABSTRACT FROM AUTHOR]

This work seeks to establish new oscillation criteria to analyze the oscillatory characteristics of solutions to a specific class of neutral differential equations featuring distributed delay arguments r ψ x z ′ α ′ t + ∫ a b q t , s F x g t , s d s = 0 . These equations, which have numerous applications in the physical and engineering sciences, present intriguing analytical challenges. Utilizing novel monotonic properties, we apply the comparison principle and Riccati substitutions to derive effective oscillation criteria. This approach allows us to remove certain constraints previously imposed on the coefficients of neutral differential equations. Additionally, when utilized in the Euler-type differential equation, the new conditions provide a precise conclusion regarding oscillation. To highlight the novelty and significance of the results, we include examples, remarks, as well as tables and figures for comparison. [ABSTRACT FROM AUTHOR]

In this paper, the authors study a class of general mean-field BDSDEs whose coefficients satisfy some stochastic conditions. Specifically, the authors prove the existence and uniqueness theorem of solution under stochastic Lipschitz condition and obtain the related comparison theorem. Besides, the authors further relax the conditions and deduce the existence theorem of solutions under stochastic linear growth and continuous conditions, and the authors also prove the associated comparison theorem. Finally, an asset pricing problem is discussed, which demonstrates the application of the general mean-field BDSDEs in finance. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving a reflected backward stochastic differential equation (BSDE in short) with one continuous barrier and a quasi-linear growth generator g, which has a linear growth in (y, z), except the upper direction in case of y < 0, and is more general than the usual linear growth generator. By showing the convergence of a penalization scheme we prove existence and comparison theorem of the minimal Lp (p > 1) solutions for the reflected BSDEs. We also prove that the minimal Lp solution can be approximated by a sequence of Lp solutions of certain reflected BSDEs with Lipschitz generators. [ABSTRACT FROM AUTHOR]

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method's effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results. [ABSTRACT FROM AUTHOR]

In this article, we study dynamic risk measures by means of backward doubly stochastic Volterra integral equations (BDSVIEs, for short) with jumps. We establish the well-posedness of BDSVIEs with jumps in the sense of M-solution and prove a comparison theorem of BDSVIEs with jumps. Finally, we study properties of dynamic risk measures induced by BDSVIEs with jumps. Our results extend the well-posedness and the comparison theorem of BDSVIEs without jumps to the setting with jumps, and extend dynamic risk measures induced by BSDEs, BDSDEs, and BSVIEs to the case of BDSVIEs with jumps. [ABSTRACT FROM AUTHOR]

This paper focuses on discussing the stability of time-varying switched impulsive nonlinear systems (TVSINSs). To conquer the difficulties induced by time-varying parameters, we firstly put forward comparison theorem associated with a method that doesn't involve any Lyapunov function and is commonly utilized in positive systems to set up new stability criteria of TVSINSs under the average dwell time (ADT) switching. In comparison with recent results, the derived ones are not only in connection with state delay and impulse delay, but also possess less conservative conditions from the simulation analyses. In conclusion, the feasibility and superiority of the adopted approaches are testified by two appropriate examples. [ABSTRACT FROM AUTHOR]

A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod's representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved. [ABSTRACT FROM AUTHOR]

The authors discuss the comparison theorem for solutions of stochastic functional differential equations subject to external disturbances and its application to a stochastic control problem. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in solving backward doubly stochastic differential equations in Lp, for any p ∈ (1 , 2) , when the coefficients are continuous with stochastic linear growth. Via approximation and comparison theorem, the existence of L p − minimal and L p − maximal solutions are obtained. [ABSTRACT FROM AUTHOR]

The article discusses the comparison theorem for solutions of stochastic functional differential equations under the influence of external disturbances and its application to one stochastic control problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the reflected backward stochastic differential equations driven by G -Brownian motion (reflected G -BSDEs) with time-varying Lipschitz coefficients. We obtain the uniqueness result by establishing a priori estimates. For the existence, the solution can be approximated by a family of reflected G -BSDEs with Lipschitz conditions and by penalized G -BSDEs with time-varying coefficients. The latter approximation is useful to get the comparison theorem. Finally, we study the reflected G -BSDEs with infinite time horizon. [ABSTRACT FROM AUTHOR]

In this article, we study the viability for the multi-dimensional stochastic system described by Itô stochastic differential equations under the non-Lipschitzian assumption on the coefficients. We establish a necessary and sufficient condition for viability property with respect to a given nonempty closed convex set. And with this result, we explore the multi-dimensional comparison theorems about stochastic differential equations. [ABSTRACT FROM AUTHOR]

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions on the drift and reward coefficients, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland's variational principle on the Wasserstein space. [ABSTRACT FROM AUTHOR]

Preconditioning techniques have been focused on multi-linear systems or tensor equations. However, to our knowledge, there is no relevant research on tensor complementarity problems. In this paper, we present a general preconditioner P for solving the tensor complementarity problem (TCP) involved with an L -tensor A and a vector q, and we prove that the TCP (A , q) is equivalent to the TCP (P A , P q) under the assumption that P A is an L -tensor. Based on this equivalence, a preconditioned fixed point iteration method is proposed for solving the tensor complementarity problem and its convergence analysis is given. For actual computations, we provide a concrete choice for the preconditioner P associated with a parameter satisfying above-mentioned hypothesis. In addition, it is proved theoretically that the convergence rate of the fixed point method with the chosen preconditioner P is at least as fast as that of the corresponding method without preprocessing. Meanwhile, we also obtain the monotony of the parameter on the performance of the preconditioned iterative method. Lastly, numerical examples are used to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

This article describes a new form of an anticipated backward stochastic differential equation (BSDE) with a time-delayed generator driven by fractional Brownian motion, further known as fractional BSDE, with a Hurst parameter H ∈ (1 / 2 , 1) . This study expands upon the findings of the anticipated BSDE by considering the scenario when the driver is fractional Brownian motion rather instead of standard Brownian motion. Additionally, the generator incorporates not only the present and future but also the past. We will demonstrate the existence and uniqueness of the solutions to these equations by employing the fixed point theorem. Furthermore, an equivalent comparison theorem is derived. [ABSTRACT FROM AUTHOR]

We have investigated the generalized quasilinearization method under some convenient conditions for nonlinear initial value problem (IVP) of dynamic equation on time scale and constructed by monotone sequences of function by using comparison theorem. The solutions of linear IVPs of dynamic equation on time scale converge uniformly and monotonically to the unique solution of the original problem and the convergence is quadratic. [ABSTRACT FROM AUTHOR]

This paper deals with generalized backward doubly stochastic differential equations driven by a Lévy process (GBDSDEL, in short). Under left or right continuous and linear growth conditions, we prove the existence of minimal (resp. maximal) solutions. [ABSTRACT FROM AUTHOR]

We deal with fractional mean field backwardWe deal with fractional mean field backward stochastic differential equations with hurst parameter $H\in (\frac{1}{2},1)$ when the coefficient $f$ satisfy a stochastic Lipschitz conditions, we prove the existence and uniqueness of solution and provide a comparison theorem. Via an approximation and comparison theorem, we show the existence of a minimal solution when the drift satisfies a stochastic growth condition.

In this paper, we study spectral properties of a fourth‐order differential operator on a network, which is a model of the Euler‐Bernoulli beam system. We propose a new approach to the development of oscillatory spectral theory of a fourth‐order differential operator on a network. This approach is based on the concept of a sign‐constant zone for a continuous function on a graph. We show that eigenvalues and eigenfunctions of the corresponding operator on a network have oscillatory properties. We establish a condition of simplicity of eigenvalues. Finally, we study the distribution of the zeros of the eigenfunctions. To that end, we introduce the Weyl solutions and study their dependence on the spectral parameter. We show that the k$$ k $$th eigenfunction has exactly k$$ k $$ zeros in the graph. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an existence of minimal (resp. maximal) solutions for backward doubly stochastic differential equations (BDSDEs, in short) when the coefficient with respect to the forward integral is left (resp. right)-continuous in y and continuous in z with stochastic linear growth in (y, z). Also, the associated comparison theorem is obtained. [ABSTRACT FROM AUTHOR]

We prove the existence of maximal (and minimal) solution for one-dimensional generalized doubly reflected backward stochastic differential equation (RBSDE for short) with irregular barriers and stochastic quadratic growth, for which the solution Y has to remain between two rcll barriers L and U on [ 0 , T [ , and its left limit Y − has to stay respectively above and below two predictable barriers l and u on ] 0 , T ]. This is done without assuming any P − integrability conditions and under weaker assumptions on the input data. In particular, we construct a maximal solution for such a RBSDE when the terminal condition ξ is only F T − measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. Our result is based on a (generalized) penalization method. This method allows us find an equivalent form to our original RBSDE where its solution has to remain between two new rcll reflecting barriers Y ¯ and Y ̲ which are, roughly speaking, the limit of the penalizing equations driven by the dominating conditions assumed on the coefficients. A standard and equivalent form to our initial RBSDE as well as a characterization of the solution Y as a generalized Snell envelope of some given predictable process l are also given. [ABSTRACT FROM AUTHOR]

In this paper, we study multi-dimensional reflected backward stochastic differential equations (BSDEs) with diagonally quadratic generators. Using the comparison theorem for diagonally quadratic BSDEs established recently in Luo (Disc Contin Dyn Syst 41(6):2543–2557, 2021), we obtain the existence and uniqueness of a solution by a penalization method. Moreover, we provide a comparison theorem. [ABSTRACT FROM AUTHOR]

This article is interested in Lp ðp > 1Þ solutions of general time interval backward stochastic differential equations (BSDEs) with generator g satisfying a p-order weak stochastic-monotonicity condition and a general growth condition in y. An existence and uniqueness result of the Lp solution is proved for the multidimensional BSDEs where the generator g is stochastic-Lipschitz in z. And, a general comparison theorem and an existence and uniqueness theorem of the Lp solution is established for the one-dimensional BSDEs under a stochastic uniform-continuity condition of the generator g in z. These results strengthen some known works. The stochastic Gronwall-type inequality and the stochastic Bihari-type inequality play an important role in the whole article. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter 1 / 2 < H < 1 under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions. [ABSTRACT FROM AUTHOR]

For solving the linear complementarity problem (LCP), we propose a preconditioned new modulus-based matrix splitting (PNMMS) iteration method by extending the state-of-the-art new modulus-based matrix splitting (NMMS) iteration method to a more general framework with a customized preconditioner. We devise a generalized preconditioner that is associated with both H+-matrix A and vector q of the LCP. The convergence analysis is conducted under some mild conditions. In particular, we provide a comparison theorem to theoretically show the PNMMS method accelerates the convergence rate. Numerical experiments further illustrate that the PNMMS method is efficient and has better performance for solving the large and sparse LCP.

In this paper, a general class of time-symmetric mean-field stochastic systems, namely the so-called mean-field forward-backward doubly stochastic differential equations (mean-field FBDSDEs, in short) are studied, where coefficients depend not only on the solution processes but also on their law. We first verify the existence and uniqueness of solutions for the forward equation of general mean-field FBDSDEs under Lipschitz conditions, and we obtain the associated comparison theorem; similarly, we also verify those results about the backward equation. As the above two comparison theorems' application, we prove the existence of the maximal solution for general mean-field FBDSDEs under some much weaker monotone continuity conditions. Furthermore, under appropriate assumptions we prove the uniqueness of the solution for the equations. Finally, we also obtain a comparison theorem for coupled general mean-field FBDSDEs. [ABSTRACT FROM AUTHOR]

We establish well-posedness for a class of systems of SDEs with non-Lipschitz coefficients in the diffusion and jump terms and with two sources of interdependence: a monotone function of all the components in the drift of each SDE and the correlation between the driving Brownian motions and jump random measures. Pathwise uniqueness is derived by employing some standard techniques. Then, we use a comparison theorem along with our uniqueness result to construct non-negative, L 1 -integrable càdlàg solutions as monotone limits of solutions to approximating SDEs, allowing for time-inhomogeneous drift terms to be included. Our approach allows also for a comparison property to be established for the solutions to the systems we investigate. The applicability of certain systems in financial modeling is also discussed. [ABSTRACT FROM AUTHOR]

We study the solvability of the initial value problem for the semilinear heat equation u t - Δ u = u p in a Riemannian manifold M with a nonnegative Radon measure μ on M as initial data. We give sharp conditions on the local-in-time solvability of the problem for complete and connected M with positive injectivity radius and bounded sectional curvature. [ABSTRACT FROM AUTHOR]

This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in Haussmann and Suo [SIAM J. Control Optim., 33 (1995), pp. 916-936] and [SIAM J. Control Optim., 33 (1995), pp. 937-959]). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller [SIAM J. Control Optim., 43 (2004), pp. 708-730]. [ABSTRACT FROM AUTHOR]

This paper presents a full classification of the short-time behavior of the solution and the interfaces in the Cauchy problem for the nonlinear second order singular parabolic PDE u t - Δ u m + b u β = 0 , x ∈ R N , 0 < t < T with nonnegative initial function u 0 such that s u p p u 0 = { | x | < R } , u 0 ∼ C (R - | x |) α , as | x | → R - 0 , where 0 < m < 1 , b , β , C , α > 0 . Depending on the relative strength of the fast diffusion and absorption terms the problem may have infinite ( β ≥ m ) or finite ( β < m ) speed of propagation. In the latter case, the interface surface t = η (x) may shrink, expand or remain stationary depending on the relative strength of the fast diffusion and strong absorption terms near the boundary of support, expressed in terms of the parameters m , β , α , and C. In all cases we prove the existence or non-existence of the interfaces, explicit formula for the interface asymptotics, and local solution near the interface or at infinity. [ABSTRACT FROM AUTHOR]

We establish the existence and uniqueness of viscosity solutions to the weakly coupled second-order parabolic Hamilton–Jacobi–Bellman equations under Hölder continuous condition, for which the standard quasi-monotone condition does not hold. The existence theorem is established by solving the finite horizon optimal control problem for regime-switching stochastic processes with Hölder continuous coefficients such as the Cox–Ingersoll–Ross process. A comparison principle for this weakly coupled system without state constraint is established. [ABSTRACT FROM AUTHOR]

We consider a nonstationary initial‐boundary value problem governing a radiative–conductive heat transfer in an absolutely black body with semitransparent inclusions. To describe the radiative transfer, the integro‐differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. In addition, we proved the stability of solutions with respect to data, established a comparison theorem and results on improving the properties of solutions with an increasing on the summability of the data. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables y and z) in a super-linear way like | y | | ln | y | | δ + | z | | ln | z | | λ for some δ ∈ [ 0 , 1 ] and λ ≥ 0. Let k be the maximum of δ , λ + 1 / 2 and 2 λ. For the following four different ranges of the growth power parameter k : k = 1 / 2 , k ∈ (1 / 2 , 1) , k = 1 and k > 1 , we give reasonably weakest possible different integrability conditions on the terminal value for ensuring existence and uniqueness of the unbounded solution to the BSDE. In the first two cases, they are stronger than the L ln L -integrability and weaker than any L p -integrability with p > 1 ; in the third case, the integrability condition is just some L p -integrability for p > 1 ; and in the last case, the integrability condition is stronger than any L p -integrability with p > 1 and weaker than any exp (L ɛ) -integrability with ɛ ∈ (0 , 1). We also establish three comparison theorems, which yield immediately the uniqueness, when either one of generators of both BSDEs is convex (or concave) in both unknown variables (y , z) , or satisfies a one-sided Osgood condition in the first unknown variable y and a uniform continuity condition in the second unknown variable z. Finally, we give an application of our results in mathematical finance. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a new type of mean-field anticipated backward stochastic differential equation with a time-delayed generator (MF-DABSDEs) which extends the results of the anticipated backward stochastic differential equation to the case of mean-field limits, and in which the generator considers not only the present and future times but also the past time. By using the fixed point theorem, we shall demonstrate the existence and uniqueness of the solutions to these equations. Finally, we shall establish a comparison theorem for the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a comparison theorem and an invariant representation theorem for backward stochastic differential equations (BSDEs) without any assumption on the second variable z. Using the two results, we further develop the theory of g -expectations. Filtration-consistent nonlinear expectation (F -expectation) provides an ideal characterization for the dynamical risk measures, asset pricing and utilities. We propose two new conditions: an absolutely continuous condition and a (locally Lipschitz) domination condition. Under the two conditions respectively, we prove that any F -expectation can be represented as a g -expectation. Our results contain a representation theorem for n -dimensional F -expectations in the Lipschitz case, and two representation theorems for 1-dimensional F -expectations in the locally Lipschitz case, which contain quadratic F -expectations. [ABSTRACT FROM AUTHOR]

We establish the Bonnet–Myers theorem, Laplacian comparison theorem, and Bishop–Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with ϵ-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie–Yeroshkin and Kuwae–Li.

In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions.

In this note, we present a new preconditioner for solving the multi-linear systems, which arise from many practical problems and are different from the traditional linear systems. Based on the analysis of the spectral radius, we give new comparison results between some preconditioned tensor splitting iterative methods. Numerical examples are given to demonstrate the efficiency of the proposed preconditioned method.

In this paper, we focus on the problem of viability property for the multi-dimensional stochastic differential equations, where the coefficients of the stochastic differential equations may be random and discontinuous in the time variable. First, we establish a necessary and sufficient condition for viability property with respect to a given non-empty closed convex set. And furthermore, with the help of this result, we also study the multi-dimensional comparison theorems about stochastic differential equations. [ABSTRACT FROM AUTHOR]

A complex matrix is associated with a sequence of real numbers. The present study demonstrates the methodology for computing the actual perturbation values and highlights their resemblance to the singular values in terms of various properties. Real perturbation values can be utilized to compute the real pseudospectra and real stability radii. The primary outcome pertains to the signature exhibited by real quadratic forms within complex vector spaces. In this work I showed a result of the contraction semigroup LP norm for vector-valued functions, the Markovian semigroup which is symmetric and not the only one on L² contraction semigroup but also a LP contraction of semigroup for any P ∈ [1,∝) and I prove that the sharp estimate of the defective logarithmic Sobolev inequality holds for t > 0 and 1 < p ≤ 2. [ABSTRACT FROM AUTHOR]

For solving the linear complementarity problem (LCP), we propose a preconditioned new modulus-based matrix splitting (PNMMS) iteration method by extending the state-of-the-art new modulus-based matrix splitting (NMMS) iteration method to a more general framework with a customized preconditioner. We devise a generalized preconditioner that is associated with both H + -matrix A and vector q of the LCP. The convergence analysis is conducted under some mild conditions. In particular, we provide a comparison theorem to theoretically show the PNMMS method accelerates the convergence rate. Numerical experiments further illustrate that the PNMMS method is efficient and has better performance for solving the large and sparse LCP. [ABSTRACT FROM AUTHOR]

A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get an upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes. [ABSTRACT FROM AUTHOR]

In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author's previous results under integral Bakry-Émery Ricci tensor bounds, but also are generalizations of the Wei-Wylie's pointwise results. [ABSTRACT FROM AUTHOR]