The role of w-tilting modules in relative Gorenstein (co)homology
-
Driss Bennis
and
Luis Oyonarte
Abstract
Let
1 Introduction
Foxby classes (Auslander and Bass classes) have proven to be very useful when studying Gorenstein injective and projective dimensions: over Gorenstein rings, every module has finite Gorenstein projective and finite Gorenstein injective dimensions [1]. However, when the ring is no longer Gorenstein, modules of finite Gorenstein injective or projective dimensions constitute two new full subcategories. These were identified in the case of a commutative local Cohen-Macaulay ring with a dualizing module, at least for finitely generated modules. Foxby proved that the Auslander and Bass classes behave in a duality and that finitely generated modules in the Auslander class are precisely those of finite Gorenstein projective dimension, while finitely generated modules in the Bass class are those of finite Gorenstein injective dimension [2].
This result has been much appreciated and later on many authors have studied and extended the situation given by Foxby (see [3,4,5] or [6] for instance).
On the other hand, the existence of Gorenstein projective precovers and Gorenstein injective preenvelopes is an active subject of study nowadays, not only in the category of modules but also in any category where the knowledge of homological properties is interesting, as for example the categories of sheaves over a topological space, the ones of complexes of modules, those of representations of quivers by modules, etc. (read for instance Enochs et al. in [7], Enochs et al. in [8], Gillespie in [9] or Krause in [10]). At the same time, generalizations of these concepts by means of a semidualizing or even a weakly Wakamatsu tilting (cotilting), module (see Definition 2.1) have been given (
In this paper, we will address that problem, and we will do so by a systematic use of the duality between the Auslander and Bass classes. We will study when the pairs of classes
Moreover, we will see (Theorem 3.10 and Corollary 4.8) that, with the extra assumption that
And finally we will be able to prove the last two main results of the paper: Corollary 5.7, which says that, under certain circumstances, the class of all
The paper is organized as follows. Section 2 is devoted to setting the conditions assumed in all this work, to introducing the definitions of all the concepts related to this problem, and to stating and, sometimes proving, the basic results needed throughout the paper.
In Section 3, we study the problems addressed above regarding the pairs of classes
In Section 4, we study when the class
Finally, in Section 5 we study the existence of
2 Preliminaries
Throughout this paper
By
From now on, and unless otherwise specified,
The character module of any module
If
such that
Given two classes of modules
with
A degreewise finite projective resolution is a projective resolution (that is, an exact left
For any class of left
Similarly,
Recall that if
A class of modules
A cotorsion theory is defined as a pair of classes
Cogenerated by a set if there is a set
Complete if any left
Perfect if any left
Hereditary if
Associated with every bimodule
The canonical map
The canonical map
So we see that the functors
Of course the classes
is an isomorphism of
is an isomorphism of
We now recall some concepts that will be needed throughout the paper.
Definition 2.1
[11] A left
Dually, a left
There is an injective cogenerator
We will say that a bimodule
Definition 2.2
[3, Definition 2.1] Let
When
On the other hand,
Again,
A right
A module
In what follows,
Proposition 2.3
For any bimodule
Proof
Take any
But we know that
On the other hand, suppose that
and since
3 Cotorsion theories via Foxby classes
In this section, we study the properties of the different cotorsion theories arising from the Auslander and Bass classes. Namely, we will see under what conditions the pairs of classes
Moreover, we will prove that when
We start by showing that the classes
It is known (see [17]) that when
Proposition 3.1
If
Proof
The class
Let then
This implies both that
so we have two commutative diagrams 
and 
which show that
As for the classes
Proposition 3.2
Let
Proof
The inclusion
On the other hand,
If we now let
(
There exist in the literature a number of results providing information about when the image of a (pre)cover or a (pre)envelope by means of a functor in an adjoint situation is again a (pre)cover or a (pre)envelope. And surely there are solutions to the question of whether or not the “image” of a (pre)covering or (pre)enveloping class is (pre)covering or (pre)enveloping. However, we have not found them and so we give below two results that may shed some light on this problem and that will be useful later in the paper.
Theorem 3.3
Let
If
If
Proof
We only prove 1. since 2. is just a dualization.
Assume that
Take any morphism
where
Now, using once more the properties of the unit and the co-unit of the adjunction we have
and since
so we see that
If
can be completed commutatively by some
On the other hand, since
But
We know (see [18]) that the classes
Corollary 3.4
Let
Proof
By Proposition 3.2, we know that
Following the same type of arguments we also get the following.
Corollary 3.5
Assume that
We now turn to study conditions which guarantee when a module is w-(co)tilting.
Recall that given a subcategory
The next result follows somewhat similar arguments to those of [19, Proposition 3.6].
Proposition 3.6
Let
Proof
Since
Start by choosing a
can be completed commutatively. This shows that every
Now,
with
But
Thus, the exact sequence
is also
Dually, the following result can be easily proved.
Proposition 3.7
Let
We now give results which find sufficient conditions on
Proposition 3.8
Let
Proof
By [11, Proposition 5.4] we have that
Now,
we immediately get that
Similarly, the fact that
we get that
Finally, the commutative diagram with exact rows 
shows that
Proposition 3.9
Suppose
Proof
By Proposition 3.2 we have that
Now, applying
we immediately get that
and then
But then the sequence
is also exact, and applying
which shows that
We will now see that if
Theorem 3.10
Let
be an exact right
Proof
Call
is
Applying both
we get a commutative diagram with exact rows 
with
Now, the commutativity of the diagram shows that
Now, for any
shows that
On the other hand, applying
and noting that
Therefore ,we get
But we know that
Conversely,
Let
Since
Now, applying
we get that
after applying
Now let us show that when
Definition 3.11
A class of modules
is a short exact sequence and
Dually,
Corollary 3.12
Let
Proof
Call
call
For any left
and of course
Therefore, we see that
Now, it is well known that when a cotorsion theory is cogenerated by a set, then it is complete (see for instance [1, Theorem 7.4.1]), and
We now recall the definition of duality pairs.
Definition 3.13
[21, Definition 2.1] A left duality pair over
For a left
Analogously, right duality pairs with
A left duality pair
A left duality pair
We now want to check when
We start our way with the next result, whose proof follows the arguments of [6, Proposition 7.2].
Proposition 3.14
Assume that
Corollary 3.15
Let
If, in addition,
On the other hand, if
If, in addition,
Proof
Items (a) and (b): by Proposition 3.14 we see that the two pairs are left duality pairs. Of course both are coproduct-closed as well (and product closed when
Items (c) and (d) are proved dually.□
With the tools developed so far we can answer several of the questions raised in Section 1. This answer generalizes [21, Theorem 3.2] and improves [3, Theorem 4.4].
Theorem 3.16
Let
The classes
If, furthermore,
Assume now that
The classes
If, in addition,
If both
Proof
By Corollary 3.15
Now, if
Dual to (i).
If both
Corollary 3.17
Suppose that
Proof
If
Conversely, if
is exact with
shows, using induction, that
Using Theorem 3.16 and Proposition 3.6 we can see what conditions we need on
Corollary 3.18
Suppose that
Proof
The class
Now, any
Finally, applying Proposition 3.6 we get that
In a similar way we have the following result.
Corollary 3.19
Suppose that
Proof
Theorem 3.16 shows that
4 Gorenstein
C
-injective preenvelopes and Gorenstein
C
-projective precovers. Applications
In this section, we address another interesting question: when are the classes of syzygies of complete
As applications we will see that, dually to Theorem 3.10, the Auslander class can also be recovered from a right
We start by defining all these
Definition 4.1
A left
A module
with
A right
The concepts of
We denote by
It is worth noting that when
with all
Similarly, if
We start with the case of complete
Lemma 4.2
Let
Proof
We know that
Thus,
Now, the naturality of
is commutative, that is,
Proposition 4.3
Let
Proof
We start by noting that by [11, Theorem 5.9] we have that
Call
Applying
since
Now, for every
is exact since
We then see that to have
Choose any injective left
If we apply
since
But
Conversely, let
and call
It only remains to prove that
Let then
can be completed commutatively.
Applying
since all
is the natural homomorphism given in the definition of the Bass class, that is, the co-unit of the adjunction
But
and
The
Corollary 4.4
Suppose that
If, in addition,
Proof
By [11, Theorem 5.9] we have
Now, if
We now give the dual results concerning complete
Proposition 4.5
Let
Corollary 4.6
Let
As a consequence of [17, Proposition 2.12], we have that the class
Proposition 4.7
Suppose that both
Proof
The class
Let then
In Theorem 3.10 we showed how, under certain circumstances, the Bass class
Corollary 4.8
Suppose that
be an exact right
Proof
Using Proposition 3.14 we get that
But
But at the same time, we have
and we finally get
We finish this section by proving that, in some circumstances, the common part of the two components of the pairs
Proposition 4.9
The following statements hold.
Suppose that
Proof
(1)
so
We then have
Conversely, choose any
(
(2)
On the other hand, for any injective
by [1, Theorem 3.2.1]. But
Conversely, if
splits, and since
(
5
G
C
-flat and
G
C
-projective covers and dimensions
The information existing in the literature concerning the existence of relative Gorenstein flat precovers is very limited. In this section, we want to address two natural problems which arise when studying this class of modules: when do relative Gorenstein flat (pre)covers exist? and, when is the class of relative Gorenstein projective modules included in the class of relative Gorenstein flats? And, related to this last problem, can we compare relative Gorenstein projective and flat dimensions? Answers to the first two questions will be given in Corollary 5.7 and Theorem 5.9, respectively, and a partial answer to the third question will be given in Theorem 5.13.
To better know how the class of relative Gorenstein flat modules look like, we will study its relations with the class of relative Gorenstein injective modules in Proposition 5.6.
As a consequence of the results to be obtained in this section, we will also give sufficient conditions for the class of relative projective modules to be special precovering in
We start by recalling the concept of relative Gorenstein flat module.
Definition 5.1
A left
where the
The class of all
It is immediate to see that the class
Proposition 5.2
Suppose that
Proof
By [11, Proposition 5.4] we know that
so
From Proposition 5.2, we can give conditions for the class
Proposition 5.3
Let
If
As a consequence, when all conditions above are satisfied,
Proof
To prove that
The statement about
Finally, we know that if a Kaplansky class is closed under direct products, then it is preenveloping [1, Theorem 11.9.4] so we only have to prove that
For let
Now,
is exact, so since
It is a very interesting question to know whether or not the class of
Proposition 5.4
Suppose that
Proof
For any injective right
be an injective resolution of the left
Call
We get that
But
by [1, Theorem 3.2.1], and
by [1, Theorem 3.2.13], so we get
Now,
Now we give a slight modification of the dual of Holm’s Corollary 2.11 in [22] for Gorenstein projective modules, extended by White in [14, Corollary 3.8] to the case of
Proposition 5.5
Suppose that
Proof
Since
for some injective right
Of course,
be an injective resolution of
Call then
Applying
we get that
so we actually have
But, as in the proof of Proposition 5.4,
we see that
As for
Therefore,
Now, constructing the pullback of
Since
Finally,
Therefore,
The following result has been proved when
Proposition 5.6
Let
If
Suppose now that
Corollary 5.7
Suppose that
Proof
Since
Lemma 5.8
Let
is an exact complex and all
is also exact.
Proof
We have to prove that for any integer
is exact.
If
On the other hand, if
But
The same argument shows that
we see that
An inductive process gives now that indeed
are all exact.□
Theorem 5.9
Suppose that
Proof
Let
If we prove that the complex
Of course
Now,
with all
Then, by Lemma 5.8 we get that the sequence of complexes
is exact. Since each
Our next goal will be to find conditions for every
Recall that a module
with all
The following result gives an interesting property of flat modules over
Proposition 5.10
The flat cotorsion dimension of any flat left
Proof
Let
be a cotorsion resolution of
Now, since
and
Lemma 5.11
Suppose that
Proof
In first place, we note that the natural morphism of left
Now,
Thus, since
Now, since
in which every
is exact and, if we prove that
is exact, we will have that the complex
But each
We now give an extension of the concept of
Definition 5.12
Given a nonnegative integer number
Theorem 5.13
Suppose that
Proof
Let
in which
Therefore,
Now, for any
be a short exact sequence in
in
are also exact.
For any
with exact rows and columns. Similarly, if
and if
Therefore, if for any module
with
in which every
For
For
But
Moreover, we see from the pullbacks above that
Now, for any
Since
so we see that
Therefore, the complex
Moreover, for every
we have
(unless
Hence, by [11, Proposition 3.11] we get that
This means that we can apply the same argument to the complex
with
Now,
shows that
We now give conditions for the class
Corollary 5.14
Let
Proof
By Theorems 5.13 and 5.7, we have the inclusions
where
Now, given any left
Since
we see that
-
Funding information: Juan R. García Rozas and Luis Oyonarte were partially supported by Ministerio de Economía y Competitividad, grant reference 2017MTM2017-86987-P.
-
Conflict of interest: Authors state no conflict of interest.
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© 2021 Driss Bennis et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
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- Almost factorizable weakly type B semigroups
- The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
- Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
- Epi-quasi normality
- Derivative and higher-order Cauchy integral formula of matrix functions
- Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
- Solutions to a multi-phase model of sea ice growth
- Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
- Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
- Review Article
- Semiprimeness of semigroup algebras
- Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
- Third-order differential equations with three-point boundary conditions
- Fractional calculus, zeta functions and Shannon entropy
- Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
- Synchronization of Caputo fractional neural networks with bounded time variable delays
- On quasilinear elliptic problems with finite or infinite potential wells
- Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
- On a fractional Schrödinger-Poisson system with strong singularity
- Parabolic inequalities in Orlicz spaces with data in L1
- Special Issue on Evolution Equations, Theory and Applications (Part II)
- Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
- Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
- On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
- Blow-up results of the positive solution for a class of degenerate parabolic equations
- Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
- On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
- General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
- On hyponormality on a weighted annulus
- Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
- Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
- Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
- Marangoni convection in layers of water-based nanofluids under the effect of rotation
- A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
- Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
- Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
- Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
- On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
- The Lp dual Minkowski problem about 0 < p < 1 and q > 0
Articles in the same Issue
- Regular Articles
- Sharp conditions for the convergence of greedy expansions with prescribed coefficients
- Range-kernel weak orthogonality of some elementary operators
- Stability analysis for Selkov-Schnakenberg reaction-diffusion system
- On non-normal cyclic subgroups of prime order or order 4 of finite groups
- Some results on semigroups of transformations with restricted range
- Quasi-ideal Ehresmann transversals: The spined product structure
- On the regulator problem for linear systems over rings and algebras
- Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
- Resolving resolution dimensions in triangulated categories
- Entire functions that share two pairs of small functions
- On stochastic inverse problem of construction of stable program motion
- Pentagonal quasigroups, their translatability and parastrophes
- Counting certain quadratic partitions of zero modulo a prime number
- Global attractors for a class of semilinear degenerate parabolic equations
- A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
- On sub-class sizes of mutually permutable products
- Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
- Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
- On kernels by rainbow paths in arc-coloured digraphs
- Fully degenerate Bell polynomials associated with degenerate Poisson random variables
- Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
- A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
- Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
- Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
- Hodge-Deligne polynomials of character varieties of free abelian groups
- Diophantine approximation with one prime, two squares of primes and one kth power of a prime
- The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
- Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
- On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
- Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
- Asymptotic measure-expansiveness for generic diffeomorphisms
- Infinitesimals via Cauchy sequences: Refining the classical equivalence
- The (1, 2)-step competition graph of a hypertournament
- Properties of multiplication operators on the space of functions of bounded φ-variation
- Disproving a conjecture of Thornton on Bohemian matrices
- Some estimates for the commutators of multilinear maximal function on Morrey-type space
- Inviscid, zero Froude number limit of the viscous shallow water system
- Inequalities between height and deviation of polynomials
- New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
- Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
- On a new generalization of some Hilbert-type inequalities
- On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
- On split regular BiHom-Poisson color algebras
- Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
- The mixed metric dimension of flower snarks and wheels
- Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
- The B-topology on S∗-doubly quasicontinuous posets
- Hyers-Ulam stability of isometries on bounded domains
- Inhomogeneous conformable abstract Cauchy problem
- Path homology theory of edge-colored graphs
- Refinements of quantum Hermite-Hadamard-type inequalities
- Symmetric graphs of valency seven and their basic normal quotient graphs
- Mean oscillation and boundedness of multilinear operator related to multiplier operator
- Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
- Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
- Finite groups whose intersection power graphs are toroidal and projective-planar
- On primitive solutions of the Diophantine equation x2 + y2 = M
- A note on polyexponential and unipoly Bernoulli polynomials of the second kind
- On the type 2 poly-Bernoulli polynomials associated with umbral calculus
- Some estimates for commutators of Littlewood-Paley g-functions
- Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
- On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
- On intersections of two non-incident subgroups of finite p-groups
- Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
- Finite groups with 4p2q elements of maximal order
- Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
- Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
- Entire solutions for several general quadratic trinomial differential difference equations
- Strong consistency of regression function estimator with martingale difference errors
- Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
- Montgomery identity and Ostrowski-type inequalities via quantum calculus
- Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
- On reducible non-Weierstrass semigroups
- so-metrizable spaces and images of metric spaces
- Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
- The concept of cone b-Banach space and fixed point theorems
- Complete consistency for the estimator of nonparametric regression model based on m-END errors
- A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
- Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
- Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
- A new characterization of the automorphism groups of Mathieu groups
- The role of w-tilting modules in relative Gorenstein (co)homology
- Primitive and decomposable elements in homology of ΩΣℂP∞
- The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
- Classification of f-biharmonic submanifolds in Lorentz space forms
- Some new results on the weaving of K-g-frames in Hilbert spaces
- Matrix representation of a cross product and related curl-based differential operators in all space dimensions
- Global optimization and applications to a variational inequality problem
- Functional equations related to higher derivations in semiprime rings
- A partial order on transformation semigroups with restricted range that preserve double direction equivalence
- On multi-step methods for singular fractional q-integro-differential equations
- Compact perturbations of operators with property (t)
- Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
- Random attractors for stochastic plate equations with memory in unbounded domains
- On the convergence of two-step modulus-based matrix splitting iteration method
- On the separation method in stochastic reconstruction problem
- Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
- Structure of coincidence isometry groups
- Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
- Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
- Euler-type sums involving multiple harmonic sums and binomial coefficients
- Poly-falling factorial sequences and poly-rising factorial sequences
- Geometric approximations to transition densities of Jump-type Markov processes
- Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
- Bifurcations and exact traveling wave solutions for the regularized Schamel equation
- Almost factorizable weakly type B semigroups
- The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
- Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
- Epi-quasi normality
- Derivative and higher-order Cauchy integral formula of matrix functions
- Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
- Solutions to a multi-phase model of sea ice growth
- Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
- Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
- Review Article
- Semiprimeness of semigroup algebras
- Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
- Third-order differential equations with three-point boundary conditions
- Fractional calculus, zeta functions and Shannon entropy
- Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
- Synchronization of Caputo fractional neural networks with bounded time variable delays
- On quasilinear elliptic problems with finite or infinite potential wells
- Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
- On a fractional Schrödinger-Poisson system with strong singularity
- Parabolic inequalities in Orlicz spaces with data in L1
- Special Issue on Evolution Equations, Theory and Applications (Part II)
- Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
- Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
- On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
- Blow-up results of the positive solution for a class of degenerate parabolic equations
- Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
- On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
- General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
- On hyponormality on a weighted annulus
- Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
- Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
- Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
- Marangoni convection in layers of water-based nanofluids under the effect of rotation
- A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
- Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
- Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
- Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
- On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
- The Lp dual Minkowski problem about 0 < p < 1 and q > 0
