On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
-
Kshitij Kumar Pandey
Abstract
Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.
1 Introduction
Over the past three decades, the subject of fractal interpolation has been one of the major research themes among the fractals community. The concept of fractal interpolation function (FIF) was introduced by Barnsley in his fundamental and pioneering work [1]. He proved the existence of a univariate continuous function interpolating a given finite data set, with its novelty lying in the fact that the graph of the constructed interpolant is a fractal, that is, the attractor of a suitable iterated function system (IFS) [2,3]. We do not attempt here to provide a complete list of references on fractal interpolation, as the field is quite large. Instead, we refer the reader to a few recent research on fractal interpolation [4–11] and the references therein. For a compendium of fractal interpolation and related topics, we refer the reader to the monograph [12].
As is well known, interpolation and approximation are two intimately related concepts. However, the interplay between these two theories is more subtle in the fractal setting. It is our opinion that the notion of
As with any such salient idea, numerous questions and results based on fractal interpolation were spawned. One such question is related to its extension to the multivariate case. In this direction, several works have been done to interpolate a given bivariate data using bivariate FIFs (fractal surfaces); see, for instance, [20–28] and a few research works deal with the multivariate FIFs [29,30]. Among various constructions available in the literature, the general framework to construct fractal surfaces given in [31] aroused our interest, partially due to the fact that the construction thereat is amenable to obtain a bivariate analogue of the
Much of the existing literature dealing with univariate and multivariate FIFs concentrates primarily on finite data sets. This is no happenstance, as fractal interpolation hinges on the Hutchinson’s fundamental result about the existence of an invariant set (attractor) for a set-valued map induced by a finite number of contractions [2]. Attempts were made in literature to define these concepts in the setting of countably infinite number of maps. For instance, Secelean adapted the Hutchinson approach so as to handle a countable number of contractions [34,35] and deduced the existence of a univariate fractal function interpolating a data set consisting of countably infinite points [35,36]. Bivariate FIF for an infinite sequence of data is not studied hitherto, and naturally, we want to generalize the construction of fractal surfaces so as to accommodate countably infinite data sets. Let us note that this is practically important. For instance, in the theory of sampling and reconstruction, often one works with infinite sequence of data points, and a general approach focuses to seek an approximate rather than the perfect reconstruction on some restricted class of signals.
One of the objectives of this article is to unveil a method for constructing fractal interpolation surface (FIS) for a countably infinite data set over rectangular grids. We are inspired by the finite case treated in [31], and we will adapt this construction to the setting of countable data. This part of the current findings may be viewed also as a sequel to [36], where the univariate fractal interpolation for a countable data set has been studied. The countable bivariate scenario explored in this study necessitates certain extra considerations and assumptions, making the analysis more difficult. As hinted earlier, among various constructions of FISs, our choice of [31] is guided by the fact that it offers an efficient platform to obtain a parameterized family of fractal functions corresponding to a prescribed bivariate continuous function. Crucial to further development will be the aforementioned parameterized family consisting of approximate fractal reconstructions of the original function (seed function) defined on a rectangular domain. This family of fractal functions is obtained by sampling the seed function at infinite number of grid points in the domain and applying the countable FIS scheme developed in the first part of this article.
We define a fractal operator that sends each (non-fractal) seed function to its fractal approximant, and study some analytical properties of this operator, which, in general, is nonlinear. Though the bounded linear fractal operator has its origin in the theory of univariate
2 Preliminaries
We list pertinent definitions and notation from the theory of countable IFS [34,36] and nonlinear operators for use in the subsequent parts of this article. The terminologies on perturbation theory of nonlinear operators assembled here can be found in [38], and many of these are obvious modifications to the notions present in the well known treatise for perturbation theory of linear operators by Kato [39].
2.1 Countable IFS
Definition 2.1
Let
Definition 2.2
A non-empty set
where the notation bar denotes the closure of the respective set. A non-empty set
Definition 2.3
Let
is said to be the Hutchinson operator associated with the CIFS
Theorem 2.4
[34] Suppose that the CIFS
Moreover,
for every nonempty closed set
the attractor A can be approximated with respect to the Hausdorff-Pompeiu metric by the attractors of the partial IFSs
2.2 A few basics of linear and nonlinear operator theory
In this subsection, let
Definition 2.5
An operator
Definition 2.6
An operator
Definition 2.7
Let
Definition 2.8
Let
then
Definition 2.9
Let
Definition 2.10
If
Definition 2.11
An operator
Remark 2.12
Let
Definition 2.13
Let
The infimum
Remark 2.14
If
Definition 2.15
An operator
For a Lipschitz operator
Definition 2.16
Let
The infimum of all such values of
Remark 2.17
Similar to Remark 2.14, let us note that the inequality (2.3) may not hold when
Definition 2.18
Let
Definition 2.19
Given a Banach space
Further, given a Banach space
Lemma 2.20
Let X be a non-separable Banach space and
Lemma 2.21
Let X be a Banach space and
3 Construction of countable bivariate FIS
Denote
Further, it is worth to recall that the existence of
iterated limits:
Definition 3.1
A set
the sequences
the double sequence
Notation 3.2
For a bivariate CSD, let
3.1 Construction of FIS for countable data
Extensions of Barnsley’s framework [1] in two ways – by considering a countably infinite one-dimensional data set [36] and a bivariate finite data set [31] – have been one of the directions of further studies in fractal interpolation. With an intent to generalize these two approaches and inspired by the two publications [31,36], we ask:
Question 3.3
Can we construct a continuous function
the graph of
In what follows, we answer the aforementioned question in the affirmative.
For
(i.e.,
For
where
Clearly
For every
For
Remark 3.4
From (3.10), (3.11) and the following inequality
we deduce that, for each
We note that with some mild conditions on the constants involved thereat, (3.7)–(3.11) imply that the maps
Proposition 3.5
Suppose that
Then
and
where
Proof
We use the elementary inequality
and (3.10) to obtain
The inequality
and (3.11) provide
completing the proof.□
Recall that
Then
For
It can be easily verified that the metric
Proposition 3.6
The countable IFS
where
Proof
For
By the choice of
Assume that for each
for all
(3.16)for all
(3.17)
Let us denote by
endowed with the uniform metric. It is plain to see that:
Lemma 3.7
The metric subspace
For each
In what follows, we intend to show that
Remark 3.8
The existence of limits in the definition of
However, condition (3.16) enables to determine
Next we state an elementary result from analysis [40], crucial for the continuity arguments in the upcoming lemma, but omit the proof.
Lemma 3.9
Let
Lemma 3.10
For each
Proof
It follows from (3.16) and (3.17) that
On the other hand, treating
From (3.6), one has
It is obvious that
Case 1
Let
For each
Subcase 1.1
Assume that
On the other hand,
In view of (3.18), equations (3.22) and (3.23) with some basic algebra of limits provide
where in the second step, we used the triangle inequality, the third step is consequent upon (3.10) and (3.11) and the last step is plain. Now using (3.7), (3.8),
Consequently, (3.24) yields
Subcase 1.2
Let
Similarly, if
By combining the two aforementioned subcases, we conclude that
Case 2
Let
Similarly, we can prove that
The continuity of
Theorem 3.11
Let
is the attractor of the CIFS defined earlier.
Proof
By the aforementioned lemma, we have
By the continuity of
That is,
Now, let
Hence,
Conversely, for
Since
and so,
This concludes the proof.□
Remark 3.12
As mentioned in Section 1, the motivation for the aforementioned construction of FIS corresponding to a countably infinite data set comes from the construction of FIS in the case of finite data set given by Ruan and Xu [31]. The interested reader is encouraged to compare the above construction with that in [31, Theorem 3.1], in particular, the conditions imposed on the maps
3.2 Approximation of the attractor of countable IFS
Let
Let us begin with the following theorem:
Theorem 3.13
Let
Assume that
If
Proof
It can be proved similar to [34, Lemma 2.4].□
Lemma 3.14
For every
Proof
Choose
Obviously,
The proof follows immediately from the aforementioned observation.□
Lemma 3.15
[34, Lemma 2.2] If
The following theorem is analogous to [34, Prop. 3.6].
Theorem 3.16
The set
Proof
For every
By Lemma 3.15, we have
By Lemma 3.14, we obtain
By combining (3.27) and (3.28), we obtain
Now, using the continuity of
Conversely, by using (3.29), we have
Equations (3.30) and (3.31) together yield the following:
Since
The aforementioned theorem in conjunction with Theorem 3.13 and Lemma 3.14 provides the promised approximation of the attractor of CIFS by the attractors of the partial IFSs. To be more precise, we have the following result.
Theorem 3.17
Let A be the attractor of the CIFS
where the limit is taken with respect to the Hausdorff-Pompeiu metric.
As a special case of the previous theorem, we have the following.
Corollary 3.18
Let us consider the CIFS
where the limit is taken with respect to the Hausdorff-Pompeiu metric.
4 A parameterized family of bivariate fractal functions and associated fractal operator
As mentioned in the introduction, in order to explore some approximation theoretic aspects, we consider here a special case of the countable bivariate FIF constructed in the previous section. This is influenced by the notion of
Definition 4.1
Let
the sequences
Let
Definition 4.2
Let
We say that
Remark 4.3
The aforementioned conditions imposed on the scaling function
Fix
where
For
Consider the affine function
For
Assume that
(4.1)
For
Let us consider the CIFS
as in the previous section. In what follows, we will denote by
Theorem 4.4
Assume that the partition
the graph of
Proof
Consider the countable data set
We shall first prove that the functions
Thus, the map
condition (3.10) with
conditions required in (3.7), since by the definition of scaling function
and since
interpolation condition at the four vertices of the rectangle
the existence of limits in (3.18).
Definition 4.5
Emphasizing its dependence on the function
Remark 4.6
In fact, we obtain a family of fractal functions
Remark 4.7
The fundamental impetus for the definition of parameterized family of fractal functions reported above was Navascues’s construction of the
As in the context of univariate
Definition 4.8
Let
which assigns to each
The following results, which point to the error committed in “fractalizing” the germ function
Proposition 4.9
Let
Proof
Choose
If
Similarly, if
Also,
completing the proof.□
Corollary 4.10
Let
In particular, if
Proof
We have
Therefore,
completing the proof.□
Corollary 4.11
Let
Proof
From Proposition 4.9 and the triangle inequality
proving the claim.□
As an immediate consequence of the previous corollary, we obtain sequences of fractal functions converging uniformly to a prescribed bivariate Lipschitz continuous function, as specified in the upcoming result.
Corollary 4.12
Let
Assume that the partition
Assume that the partition
The proof of the following proposition is similar to its univariate counterpart in [13], but included here at the referee’s behest.
Proposition 4.13
If
Proof
Let
for all
for all
as desired.□
Remark 4.14
As noted earlier, when the germ function is univariate and
Proposition 4.15
Let
Proof
Let
The aforementioned inequality in conjunction with the convergence of
Proposition 4.16
The
Proof
According to Corollary 4.11, we have
Thus,
proving the claim.□
Corollary 4.17
If L is a norm-bounded nonlinear operator, that is,
Proof
From the previous proposition, we have
Since
Consequently,
Hence, we have
as required.□
Similarly, one can prove the following.
Corollary 4.18
If L is a topologically bounded operator, then so is the fractal operator
Proposition 4.19
The operator
Proof
By using similar computations as in Proposition 4.15, one obtains
proving the claim.□
Corollary 4.20
If
Proof
The assertion follows clearly from the aforementioned computations.□
Proposition 4.21
If
Proof
Let us assume on the contrary that the fractal operator
Proposition 4.22
If
Proof
Let
Consequently,
Similarly, one can prove the following result.
Proposition 4.23
If
5 Extension of fractal operator and some properties
In this short section, we extend the fractal operator
Lemma 5.1
Let
The operators
Proposition 5.2
If
Definition 5.3
For a prescribed germ function
Remark 5.4
If
The notion of invariant subspace is fundamental in operator theory. We next provide a class of proper closed invariant subspaces for the bounded linear fractal operator
Theorem 5.5
The subspace
Proof
For any partition
Consequently,
and hence, by Lemma 2.20,
is a non-trivial closed invariant subspace of
is a non-trivial closed invariant subspace for
Remark 5.6
Along with the proof of the previous theorem, it is perhaps worth recalling that the closed invariant subspace
Theorem 5.7
Let
is a non-trivial closed invariant subspace for the
Proof
By the above theorem, we have
6 Concluding remarks
Fractal interpolation is one of the few methods of interpolation that can produce both smooth and nonsmooth functions interpolating a prescribed finite set of univariate data. The apparent intricacy of the definition of fractal interpolation belies the remarkable richness of its structure so much so that the theory of fractal interpolation has developed at a rapid pace and continues to flourish. Extensions of Barnsley’s original framework of FIF in two ways – considering a countably infinite one-dimensional data set and considering a finite bivariate data set – have been one of the topics of further research studies in fractal interpolation. On the other hand, another area of study in the theory of fractal interpolation is the concept of
Acknowledgements
We would like to thank the anonymous reviewers for several constructive and competent criticism on the earlier version of this article. As always, any remaining errors are our own.
-
Funding information: The second author is financially supported by Lucian Blaga University of Sibiu & Hasso Plattner Foundation research grants LBUS-IRG-2021-07. The third author expresses gratitude for the support provided by project CRG/2020/002309 from the Science and Engineering Research Board (SERB), Government of India.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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- Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
- Multiplicity of k-convex solutions for a singular k-Hessian system
- Poisson C*-algebra derivations in Poisson C*-algebras
- Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
- Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
- Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
- Pseudo compact almost automorphic solutions to a family of delay differential equations
- Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
- Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
- Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
- Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
- Review Articles
- Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
- Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
- Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
- Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
- Binary relations applied to the fuzzy substructures of quantales under rough environment
- Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
- A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
- Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
- Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
- A study on a type of degenerate poly-Dedekind sums
- Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
- Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
- Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
- The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
- Neural network quaternion-based controller for port-Hamiltonian system
- Matching ontologies with kernel principle component analysis and evolutionary algorithm
- Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
- Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
- Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
- Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
- Some fixed point results on ultrametric spaces endowed with a graph
- On the generalized Mellin integral operators
- On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
- Approximation process of a positive linear operator of hypergeometric type
- On Kantorovich variant of Brass-Stancu operators
- A higher-dimensional categorical perspective on 2-crossed modules
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
- On parameterized inequalities for fractional multiplicative integrals
- On inverse source term for heat equation with memory term
- On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
- New extensions related to Fejér-type inequalities for GA-convex functions
- Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
- Some Hardy's inequalities on conformable fractional calculus
- The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
- A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
- Some new fixed point theorems of α-partially nonexpansive mappings
- Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
- Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
- Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
- Topological structure of the solution sets to neutral evolution inclusions driven by measures
- (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
-
Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
- Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
- Klein-Gordon potential in characteristic coordinates
- Asymptotic analysis of Leray solution for the incompressible NSE with damping
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
- Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
- Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
- Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
- Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
- A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
- The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
- Absence of global solutions to wave equations with structural damping and nonlinear memory
- Special Issue on Differential Equations and Numerical Analysis - Part I
- Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
- Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
- A comparison of two nonconforming finite element methods for linear three-field poroelasticity
Articles in the same Issue
- Regular Articles
- On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
- L-Fuzzy fixed point results in ℱ -metric spaces with applications
- Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
- Nonparametric methods of statistical inference for double-censored data with applications
- LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
- Existence of projected solutions for quasi-variational hemivariational inequality
- Spectral collocation method for convection-diffusion equation
- New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
- On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
- On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
- The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
- Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
- Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
- An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
- An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
- On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
- Some properties of a class of holomorphic functions associated with tangent function
- The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
- On the continuity in q of the family of the limit q-Durrmeyer operators
- Results on solutions of several systems of the product type complex partial differential difference equations
- On Berezin norm and Berezin number inequalities for sum of operators
- Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
- On a generalization of the Opial inequality
- A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
- Holomorphic curves into projective spaces with some special hypersurfaces
- On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
- Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
- Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
- Jordan left derivations in infinite matrix rings
- Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
- Invariant means and lacunary sequence spaces of order (α, β)
- Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
- Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
- Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
- Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
- On three-dimensional q-Riordan arrays
- Some aspects of normal curve on smooth surface under isometry
- Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
- Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
- Remark on the Daugavet property for complex Banach spaces
- Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
- Uniqueness of meromorphic functions concerning small functions and derivatives-differences
- Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
- Hyers-Ulam stability of Davison functional equation on restricted domains
- Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
- Composition of some positive linear integral operators
- On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
- Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
- Online makespan minimization for MapReduce scheduling on multiple parallel machines
- The sequential Henstock-Kurzweil delta integral on time scales
- On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
- Existence of three solutions for two quasilinear Laplacian systems on graphs
- Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
- Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
- Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
- Fractional Sturm-Liouville operators on compact star graphs
- Exact controllability for nonlinear thermoviscoelastic plate problem
- Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
- Superposition operator problems of Hölder-Lipschitz spaces
- A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
- Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
- A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
- Fixed point results for generalized convex orbital Lipschitz operators
- Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
- Multiplicity of k-convex solutions for a singular k-Hessian system
- Poisson C*-algebra derivations in Poisson C*-algebras
- Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
- Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
- Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
- Pseudo compact almost automorphic solutions to a family of delay differential equations
- Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
- Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
- Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
- Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
- Review Articles
- Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
- Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
- Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
- Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
- Binary relations applied to the fuzzy substructures of quantales under rough environment
- Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
- A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
- Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
- Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
- A study on a type of degenerate poly-Dedekind sums
- Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
- Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
- Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
- The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
- Neural network quaternion-based controller for port-Hamiltonian system
- Matching ontologies with kernel principle component analysis and evolutionary algorithm
- Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
- Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
- Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
- Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
- Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
- Some fixed point results on ultrametric spaces endowed with a graph
- On the generalized Mellin integral operators
- On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
- Approximation process of a positive linear operator of hypergeometric type
- On Kantorovich variant of Brass-Stancu operators
- A higher-dimensional categorical perspective on 2-crossed modules
- Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
- On parameterized inequalities for fractional multiplicative integrals
- On inverse source term for heat equation with memory term
- On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
- New extensions related to Fejér-type inequalities for GA-convex functions
- Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
- Some Hardy's inequalities on conformable fractional calculus
- The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
- Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
- A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
- Some new fixed point theorems of α-partially nonexpansive mappings
- Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
- Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
- Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
- Topological structure of the solution sets to neutral evolution inclusions driven by measures
- (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
-
Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces
- Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
- Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
- Klein-Gordon potential in characteristic coordinates
- Asymptotic analysis of Leray solution for the incompressible NSE with damping
- Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
- Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
- Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
- Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
- Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
- A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
- The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
- Absence of global solutions to wave equations with structural damping and nonlinear memory
- Special Issue on Differential Equations and Numerical Analysis - Part I
- Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
- Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
- A comparison of two nonconforming finite element methods for linear three-field poroelasticity
