On second-order linear Stieltjes differential equations with non-constant coefficients
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Francisco J. Fernández
Abstract
In this work, we define the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and study some of their properties in a similar manner to the context of time scales or the usual derivative. Later, we use these tools to investigate second-order linear differential equations with Stieltjes derivatives to find linearly independent solutions, as well as to derive the variation of parameters method for problems with
1 Introduction
The Wronskian has been a very useful tool in the theory of differential equations since its discovery [1]. This instrument allows us to check whether a given family of
These classical elements have also a role in newer theories of differential equations, such as the theory of Stieltjes differential equations, as this article will show. Unfortunately, the straightforward nature of the computations in the classical case (both regarding the Wronskian and the variation of parameters method) is not replicable in this setting due to the different nature of the product rule ([2, Proposition 3.9]) which, in this case, reads
where
In this article, we derive the definition of the Wronskian and the variation of parameters method in the context of Stieltjes calculus, taking into account the difficulties that arise from this theory and illustrating the applicability of the method with some examples. This endeavor will lead to the necessity of defining what we call a simplified Wronskian, as well as the study of a special family of functions which, despite having low regularity, preserve the smoothness of a function when multiplied by it – see Corollary 2.20.
Given the existing relations between Stieltjes differential equations and other differential problems ([4, Section 8]), our work draws from the classical theory as well as from the theory of dynamic equations in time scales, in which a notion of Wronskian also appears [5]. Remarkably, in this work, we will be able to apply our theory to the study of second-order differential equations with non-constant coefficients, which cannot be found in [5]. Also, we would like to acknowledge that the notion of Wronskian for the case of Stieltjes differential equations also appears in the Master Thesis [6], although in that work, it is not studied with an everywhere defined derivative, which is necessary for the finer points of the theory we develop below.
To reach our goal, in Section 2, we give a brief overview of Stieltjes calculus in order to set the basis for our research, as well as introduce some new tools that are important for the work ahead, such as a version of the integration by parts formula or some results concerning the regularity of maps. In Section 3, we introduce the Wronskian and its simplified counterpart, presenting some of their basic properties. We apply the Wronskian and the variation of parameters method to study second-order Stieltjes differential equations in Section 4, and, finally, in Section 5, we provide three examples to which we apply the theory developed. In particular, we study the one-dimensional linear Helmholtz equation with piecewise-constant coefficients and we show that the solution of the corresponding homogeneous equation arises naturally through the lense of Stieltjes differential problems, which is particularly remarkable as this function is not two times continuously differentiable in the usual setting, but it does present the corresponding regularity in the Stieltjes sense.
2 A brief overview of Stieltjes calculus
Let
see [7–9]. We will use the term g-measurable with respect to a set or function to refer to its
Similarly, we will talk about properties holding g-almost everywhere on a set
Define the sets
where
where
Before moving on to the study of the Stieltjes derivative, we present a result that contains some basic properties of the map
Proposition 2.1
[3, Proposition 3.1] For each
In particular,
After these considerations, we are finally in a position to define the Stieltjes derivative of a function on an interval as presented in [2], where the derivative is defined on the whole domain of the function, unlike in other papers such as [4,10,12], where they exclude the points of
In order to present the derivative, we recall the hypotheses required in the mentioned paper, which we will also assume throughout this work. Namely, we will consider some
Definition 2.2
[2, Definition 3.7] We define the Stieltjes derivative, or g-derivative, of a map
where
Remark 2.3
For
provided the corresponding limit exists. Similarly, as pointed out in [4],
Remark 2.4
It is possible to further simplify the definition of the Stieltjes derivative at a point
with
provided the corresponding limit exists. Note that the information in Remark 2.3 should still be taken into account. From now on, given a function
The following result, [2, Proposition 3.9], includes some basic properties of this derivative.
Proposition 2.5
Let
The function
(Product rule). The product
(Quotient rule). If
The following result presents some conditions ensuring that the map
Proposition 2.6
[3, Corollary 4.4] Consider the sets
and assume that
Then,
In the context of Stieltjes calculus, we also find a concept of continuity related to the map
Definition 2.7
[4, Definition 3.1] A function
If
Remark 2.8
Observe that we distinguish between
Remark 2.9
The set
Otherwise, we say that
Naturally, we have that the sum and product of
Proposition 2.10
If
if
if g is constant on some
Remark 2.11
As a direct consequence of Proposition 2.10, we see that if
In particular, this means that if
Next we introduce the concept of
Theorem 2.12
Let
The function F is g-absolutely continuous on
The function F satisfies the following conditions:
there exists
for each
(2.5)
We denote by
The following result is a version of the formula of integration by parts for
Lemma 2.13
(Integration by parts) Given
Proof
First, observe that [4, Proposition 5.4] ensures that
so, thanks to (2.3), we obtain
from which the integration by parts formula follows using (2.5).□
As pointed out in [4, Proposition 5.5],
Definition 2.14
Given
where
We also define
One of the most important examples of functions in the space
Definition 2.15
Given a function
Given a regressive function
where, denoting by
Remark 2.16
The
In particular, if
In order to make this work more self-contained, we highlight below some of the properties of the
Proposition 2.17
Let
For each
In particular,
For each
As a consequence,
One of the main problems that we will encounter when trying to apply the method of variation of parameters in the context of Stieltjes calculus is the fact that the product of two functions in the space
Therefore, if
with
Lemma 2.18
Let
is well-defined and belongs to
Proof
First, observe that
First, let
on the domain of the function, i.e.,
Thus, since
as we wanted.
Finally, if
Remark 2.19
Observe that in Lemma 2.18, even though we can only ensure that
As anticipated, we have the following corollary which, for a given function, provides an explicit expression for another function such that their product is an element of
Corollary 2.20
Let
is well-defined and bounded. Then, the map
is well-defined and belongs to
and, as a consequence,
Proof
Observe that it is enough to show that
Let
where we have used the fact that
Hence, we can apply Lemma 2.18 to see that
Finally, Proposition 2.5 ensures that, for each
which completes the proof.□
Remark 2.21
In Corollary 2.20, we are assuming that
which guarantees that
Corollary 2.22
Let
is well-defined and belongs to
Proof
Observe that, in order to obtain the result, it is enough to show that the map
satisfies the hypotheses of Lemma 2.18.
We start by proving that
Observe that, necessarily,
the set
Now, the proof of the continuity of
Therefore, we have that the hypotheses of Lemma 2.18 are satisfied, so
where the last equality is a consequence of the
Finally, we include a result that will be of interest for the work ahead and it can be directly obtained from Corollary 2.22.
Corollary 2.23
Let
Then, the map
Proof
First, observe that Corollary 2.22 and [2, Theorem 4.2] are enough to guarantee that
Let
which proves the case
which completes the proof.□
3 The
g
-Wronskian and second-order linear Stieltjes differential equations
In this section, we will define the concept of
Definition 3.1
(
Explicitly, with the notation
Similarly, given
Explicitly,
Remark 3.2
Observe that when condition (2.4) is satisfied, and noting that
We can further rewrite it as
or equivalently,
Remark 3.3
Observe that, given the order of derivation necessary in each of the definitions, we require different regularity on the functions for the definition of the
It is easy to see that both the
Bearing in mind Remark 2.3, we have that
so, since
which shows that
Lemma 3.4
Given
As a consequence,
Proof
Taking into account equation (3.3), it is enough to check that
This follows directly from [14, Corollary 4.1.9] since the maps
Remark 3.5
Lemma 3.4 ensures that
It is possible to check that
Therefore, we can obtain the explicit expression of
In this case, we have that
Similar to the usual setting, our definition of Wronskian functions allows us to obtain a sufficient condition for two maps to be linearly independent in the corresponding space of functions, as presented in the next result.
Lemma 3.6
Let
If
If
Proof
First, suppose
Now, the proof for the case where
The next result provides a partial converse to Lemma 3.6 for functions in
Lemma 3.7
Let
Proof
From the hypotheses, we know that there exists a
Define
Note that
which has a unique solution, cf. [12, Theorem 4.6]. Therefore,
Remark 3.8
A result analogous to Lemma 3.7 can be stated for the case where
which, in particular, yields that
so
So far, we have studied the properties of the
where
Definition 3.9
A solution of (3.4) is a function
If, in addition,
Remark 3.10
Observe that (3.4)–(3.5) is equivalent to the system
which we know to have a unique solution, cf. [13, Theorem 5.58]. Thus, (3.4)–(3.5) have a unique solution.
Remark 3.11
In general, we cannot ensure that a solution of (3.4)–(3.5) belongs to space
In order to study the application of the variation of parameters method to obtain the solution of (3.4)–(3.5), we consider the homogeneous problem
We have the following lemma whose proof is straightforward from the linearity of the
Lemma 3.12
Let
Then, the map
In the following Lemma – cf. [6, Theorem 5.2.3] – we will make the relationship between
Lemma 3.13
Let
Moreover, if
then,
In particular, if
which is a bounded function and continuous on
Proof
Given that
from which it is clear that (3.9) holds.
Assume that (3.10) holds. In order to prove (3.11), it is enough to show that
This is because
Clearly,
so
Finally, assume that (3.10) holds and
Hence, given (3.9) and (3.11), it follows that
which proves (3.12). Thus, all that is left to do is show that
As a direct consequence of Lemma 3.13, we have the following result ensuring the linear independence of two solutions of the homogeneous equation provided condition (3.8) is satisfied.
Corollary 3.14
Assume that (3.10) holds and let
Proof
Since
4 The variation of parameters method
One of the most important applications of the
4.1 Finding a solution of the homogeneous problem
The idea behind the classical variation of parameters method is to find a linearly independent solution of (3.6) with respect to a known solution of (3.6), say
To that end, suppose that
First, under suitable conditions, we can use Propositions 2.5 and 2.6 to compute the first and second
So, knowing that
Under suitable conditions, this is equivalent to requiring that
where we have used the fact that
On the other hand, since
Observe that once again, under sufficient conditions,
Thus, keeping in mind equation (4.2) and the definition of
which is a first-order linear equation. This means that, under suitable conditions, we can consider
where the last equality is a consequence of the product formula in Proposition 2.17 and (2.3). Furthermore, the quotient rule in Proposition 2.17 ensures that, under the corresponding conditions,
so, given the uniqueness of solution of first-order linear equations, it follows that
which means that
and, as a consequence,
Now, the expression of
We must note that in the formal deduction of formula (4.3) that we just derived, it has been necessary to obtain the
Theorem 4.1
Assume that (3.10) holds. If
is well-defined and bounded, and
belongs to
is well-defined, and the map
Proof
On the one hand, thanks to [3, Lemma 2.14], we have that the function
Now, on the other hand, thanks to Proposition 2.5,
Hence,
If we use the fact that
As a consequence,
Observe that
since
Finally, we study the linear independence by means of Lemma 3.6. Observe that
which is enough to guarantee that
Remark 4.2
Similar to Corollary 2.20, Theorem 4.1 presents some technical conditions on the map
4.2 Obtaining a particular solution of the nonhomogeneous problem
In a similar way to the classical case, to obtain the solution of the nonhomogeneous equation, we need to find a particular solution. In order to achieve this, we can use the method of variation of parameters, that is, we will look for a particular solution of the form
where
In order to avoid that second derivatives appear on the unknowns
Thus,
Substituting
Taking into account equations (4.5) and (4.6), we need
The determinant of the matrix associated with the previous system coincides with the
In the next result, we will see that the functions
are such that (4.4) is a particular solution of equation (3.4). We can now state and prove the following theorem for the solutions to the initial value problem (3.4)–(3.5) using the definition of
Theorem 4.3
Assume that (3.10) holds. Let
Then, the map
is a particular solution to the nonhomogeneous equation (3.4). Moreover,
is the solution of the initial value problem (3.6)–(3.7).
Proof
Thanks to Lemma 3.13,
is a bounded function and continuous on
satisfy the hypotheses of Lemma 2.18 and, therefore,
Let us now check that
Therefore,
for all
for all
Observe that it is immediate from equations (4.10) and (4.11) that
This proves that
5 Applications
In this final section, we illustrate the theory developed above with three examples: a second-order linear Stieltjes differential equation with constant coefficients, an example with variable coefficients, and the one-dimensional linear Helmholtz equation with piecewise-constant coefficients.
5.1 Second-order linear Stieltjes differential equations with constant coefficients
In [2], the authors obtained the solution of the second-order linear problem
where
they showed that a solution of the second-order problem,
where
By doing so, in [2, Proposition 4.12], they obtained the explicit expression of the solution, which is given by
In this section, we present an alternative method of obtaining the general solution given by (5.3) based on the variation of parameters method. This technique will be especially useful when equation
Corollary 5.1
Let
If
If
(5.4)are two linearly independent solutions of (3.6) which belong to
Proof
The first case is straightforward from Corollary 3.14 and Remark 2.16. Just observe that
Let us now consider the second case, namely,
First, observe that condition (3.10) is satisfied because
Furthermore, since
On the other hand,
is well-defined and bounded. Hence, the hypotheses of Theorem 4.1 are satisfied which implies that
is a solution of (3.6) and
First, since
so
Let us analyze next the existence of a solution of the initial value problem (3.6)–(3.7). We have the following corollary.
Corollary 5.2
Let
If
(5.5)is the solution of the initial value problem (3.6)–(3.7).
If
(5.6)is the solution of the initial value problem (3.6)–(3.7), where
(5.7)
Proof
The result is a direct consequence of Theorem 4.3. Indeed, let us consider the two cases separately.
If
Thus, Theorem 4.3 ensures that the solution is the map in (5.5).
On the other hand, if
Hence, Theorem 4.3 gives the solution in (5.6), which completes the result.□
Remark 5.3
Observe that the general solution in (5.3) coincides with the one in Corollary 5.2. Indeed, we show that this is the case going case by case.
If
Now, if in Lemma 2.13, we take the functions given by
we have, thanks to (2.3) and Proposition 2.17, that for each
Therefore, for each
On the other hand, if
Once again, applying Lemma 2.13 to the functions
we have, thanks to (2.3) and Proposition 2.17, that for each
Therefore, we have that, for each
Several numerical examples can be found in [2, Section 6]. In this section, the authors present an application to the Stieltjes harmonic oscillator, as well as an example, in which the resonance effect appears.
5.2 An example with variable coefficients
In this section, we will see how we can apply the theory we have developed to the following problem:
where
Thus, the solution of the initial value problem (5.8)–(5.9) can be obtained thanks to Theorem 4.3, which yields
In this case, we have that for
and as a consequence,
Therefore, for each
Using integration by parts (Lemma 2.13), we have that for each
Thus, for each
Note that the previous formula is the same as the one we obtain if we introduce the variable
Therefore, by [2, Proposition 4.12],
and, as a consequence, for each
5.3 One-dimensional linear Helmholtz equation with piecewise-constant coefficients
In the two previous examples, we have seen how the theory we have developed allows us to recover the same solution as the one obtained using order reduction methods. In the following example, we will see that it is possible to use the techniques that we have developed throughout this work to study the one-dimensional linear Helmholtz equation with piecewise-constant coefficients given by
where
for some
We have that
We have, thanks to Remark 3.10, that (5.10)–(5.11) has a unique solution
and denote by
Let us see that it is possible to obtain two linearly independent solutions of (5.13) of the form
for some
Therefore, if we want to ensure that the first
or, equivalently,
In the case in which (5.15) is fulfilled, we have that for
Observe that
In a similar fashion, for
Reasoning analogously to the previous case, for
Again, for
so, in the case in which (5.16) is fulfilled, we obtain, for
Observe that
Conditions (5.15) and (5.16) give us the following linear system of equations:
Note that, thanks to (5.12), the determinant of the matrix
Therefore,
We have the following result.
Corollary 5.4
Let
are two linearly independent solutions of the homogeneous equation (5.13). In particular,
is the solution of the initial value problem (5.13)–(5.14).
Proof
Given the reasoning above, it is clear that
Now, given that by definition
so Lemma 3.6 ensures that
We have the following result for the nonhomogeneous problem (5.10)–(5.11).
Corollary 5.5
Let
where
is a particular solution of (5.10) with
Proof
First, observe that the hypotheses of Theorem 4.3 are satisfied for
is a particular solution of (5.10). Now, observe that, for
Analogously, for
Therefore, for any
On the other hand, given
We have that
Analogously,
Therefore, for any
which completes the proof.□
In the following example, we obtain an explicit solution of problem (5.13)–(5.14) for a particular choice of a derivator
Example 5.6
Let
for some
In this case, the solution of system (5.17) is
Therefore, the solution of the corresponding problem (5.13)–(5.14) is given by the following expression:
for
is such that
where
Solution of the homogeneous problem (5.13)–(5.14) for different values of
First
Second
Remark 5.7
This last example suggests that there is some form of continuity of the solution of problem (5.13)–(5.14) with respect to
Finally, every function of bounded variation (for instance
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Funding information: Francisco J. Fernández and F. Adrián F. Tojo were supported by a research grant of the Agencia Estatal de Investigacion, Spain, Grant PID2020-113275GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, by the “European Union” and Xunta de Galicia, grant ED431C 2023/12 for Competitive Reference Research Groups (2023-2026). Ignacio Márquez Albés was funded by the Czech Academy of Sciences (RVO 67985840).
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, written and reviewed all the results and approved the final version of the manuscript.
-
Conflict of interest: The authors state no conflict of interest.
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- Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
- Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
- The curvature entropy inequalities of convex bodies
- Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
- High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
-
Riemannian invariants for warped product submanifolds in
- Generalized quadratic Gauss sums and their 2mth power mean
- Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
- Enochs conjecture for cotorsion pairs over recollements of exact categories
- Zeros distribution and interlacing property for certain polynomial sequences
- Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
- Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
- Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
- A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
- On the δ-chromatic numbers of the Cartesian products of graphs
- Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
- Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
- System of degenerate parabolic p-Laplacian
- Stochastic stability and instability of rumor model
- Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
- Stability of an additive-quadratic functional equation in modular spaces
- Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
- On k-prime graphs
- On the existence of tripartite graphs and n-partite graphs
- Classifying pentavalent symmetric graphs of order 12pq
- Almost periodic functions on time scales and their properties
- Some results on uniqueness and higher order difference equations
- Coding of hypersurfaces in Euclidean spaces by a constant vector
- Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
- Degrees of (L, M)-fuzzy bornologies
- A matrix approach to determine optimal predictors in a constrained linear mixed model
- On ideals of affine semigroups and affine semigroups with maximal embedding dimension
- Solutions of linear control systems on Lie groups
- A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
- On prime spaces of neutrosophic extended triplet groups
- On a generalized Krasnoselskii fixed point theorem
- On the relation between one-sided duoness and commutators
- Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
- Erratum
- Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
- Corrigendum
- Corrigendum to “Matrix stretching”
- Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
Articles in the same Issue
- Special Issue on Contemporary Developments in Graph Topological Indices
- On the maximum atom-bond sum-connectivity index of graphs
- Upper bounds for the global cyclicity index
- Zagreb connection indices on polyomino chains and random polyomino chains
- On the multiplicative sum Zagreb index of molecular graphs
- The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
- Special Issue on Convex Analysis and Applications - Part I
- Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
- Scattering threshold for the focusing energy-critical generalized Hartree equation
- (p, q)-Compactness in spaces of holomorphic mappings
- Characterizations of minimal elements of upper support with applications in minimizing DC functions
- Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
- Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
- Extension of Fejér's inequality to the class of sub-biharmonic functions
- On sup- and inf-attaining functionals
- Regularization method and a posteriori error estimates for the two membranes problem
- Rapid Communication
- Note on quasivarieties generated by finite pointed abelian groups
- Review Articles
- Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
- A comprehensive review of the recent numerical methods for solving FPDEs
- On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
- Pullback and uniform exponential attractors for non-autonomous Oregonator systems
- Regular Articles
- On certain functional equation related to derivations
- The product of a quartic and a sextic number cannot be octic
- Combined system of additive functional equations in Banach algebras
- Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
- Local and global solvability for the Boussinesq system in Besov spaces
- Construction of 4 x 4 symmetric stochastic matrices with given spectra
- A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
- The uniqueness of expression for generalized quadratic matrices
- On the generalized exponential sums and their fourth power mean
- Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
- Computing the determinant of a signed graph
- Two results on the value distribution of meromorphic functions
- Zariski topology on the secondary-like spectrum of a module
- On deferred f-statistical convergence for double sequences
-
About
- Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
- On the distribution of powered numbers
- Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
- A new distributionally robust reward-risk model for portfolio optimization
- Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
- Silting modules over a class of Morita rings
- Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
- Mutually unbiased bases via complex projective trigonometry
- Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
- On second-order linear Stieltjes differential equations with non-constant coefficients
- Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
- The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
- On discrete inequalities for some classes of sequences
- Boundary value problems for integro-differential and singular higher-order differential equations
- Existence and properties of soliton solution for the quasilinear Schrödinger system
- Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
- Endpoint boundedness of toroidal pseudo-differential operators
- Matrix stretching
- A singular perturbation result for a class of periodic-parabolic BVPs
- On Laguerre-Sobolev matrix orthogonal polynomials
- Pullback attractors for fractional lattice systems with delays in weighted space
- Singularities of spherical surface in R4
- Variational approach to Kirchhoff-type second-order impulsive differential systems
- Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
- On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
- The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
- Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
- Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
- Finite groups with gcd(χ(1), χc (1)) a prime
- Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
- The hull-kernel topology on prime ideals in ordered semigroups
- ℐ-sn-metrizable spaces and the images of semi-metric spaces
- Strong laws for weighted sums of widely orthant dependent random variables and applications
- An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
- Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
- Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
- Note on stability estimation of stochastic difference equations
- Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
- Purity and hybridness of two tensors on a real hypersurface in complex projective space
- Classification of positive solutions for a weighted integral system on the half-space
- A quasi-reversibility method for solving nonhomogeneous sideways heat equation
- Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
- Noetherian rings of composite generalized power series
- On generalized shifts of the Mellin transform of the Riemann zeta-function
- Further results on enumeration of perfect matchings of Cartesian product graphs
- A new extended Mulholland's inequality involving one partial sum
- Power vector inequalities for operator pairs in Hilbert spaces and their applications
- On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
- One special kind of Kloosterman sum and its fourth-power mean
- The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
- Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
- Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
- Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
- The curvature entropy inequalities of convex bodies
- Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
- High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
-
Riemannian invariants for warped product submanifolds in
- Generalized quadratic Gauss sums and their 2mth power mean
- Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
- Enochs conjecture for cotorsion pairs over recollements of exact categories
- Zeros distribution and interlacing property for certain polynomial sequences
- Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
- Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
- Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
- A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
- On the δ-chromatic numbers of the Cartesian products of graphs
- Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
- Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
- System of degenerate parabolic p-Laplacian
- Stochastic stability and instability of rumor model
- Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
- Stability of an additive-quadratic functional equation in modular spaces
- Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
- On k-prime graphs
- On the existence of tripartite graphs and n-partite graphs
- Classifying pentavalent symmetric graphs of order 12pq
- Almost periodic functions on time scales and their properties
- Some results on uniqueness and higher order difference equations
- Coding of hypersurfaces in Euclidean spaces by a constant vector
- Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
- Degrees of (L, M)-fuzzy bornologies
- A matrix approach to determine optimal predictors in a constrained linear mixed model
- On ideals of affine semigroups and affine semigroups with maximal embedding dimension
- Solutions of linear control systems on Lie groups
- A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
- On prime spaces of neutrosophic extended triplet groups
- On a generalized Krasnoselskii fixed point theorem
- On the relation between one-sided duoness and commutators
- Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
- Erratum
- Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
- Corrigendum
- Corrigendum to “Matrix stretching”
- Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
