Multi-term fractional differential equations with nonlocal boundary conditions
-
Bashir Ahmad
,
Najla Alghamdi
Abstract
We introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.
1 Introduction
Non-integer (arbitrary) order calculus has been extensively studied by many researchers in the recent years. The literature on the topic is now much enriched and contains a variety of results. The overwhelming interest in this branch of mathematical analysis results from its extensive applications in modeling several real world problems occurring in natural and social sciences. The mathematical models based on the tools of fractional calculus provide more insight into the characteristics of the associated phenomena in view of the nonlocal nature of fractional order operators in contrast to integer order operators. Examples include bioengineering [14], physics [9], thermoelasticity [15], etc. Boundary value problems of fractional order differential equations and inclusions have also attracted a significant attention and one can find a great deal of work on the topic involving different kinds of boundary conditions, for instance, see [1, 2, 3, 4, 5, 6, 7,] and the references cited therein.
Besides the equations involving only one differential operator, there are certain equations containing more than one differential operators. Such equations are called multi-term differential equations, see [5, 12, 13, 16].
In this paper we investigate a new type of boundary value problems of multi-term fractional differential equations and nonlocal three-point boundary conditions. Precisely, we consider the following problem:
where cDq denotes the Caputo fractional derivative of order q, f : [0, 1]×ℝ ➜ ℝ is a given continuous function and ai (i = 0, 1, 2) are real positive constants.
We prove the existence of solutions for the problem (1)-(2) by means of Krasnoselskii’s fixed point theorem and Leray-Schauder nonlinear alternative, while the uniqueness of solutions is established by Banach fixed point theorem. These results are presented in Section 3. An auxiliary lemma concerning the linear variant of (1)-(2) and some definitions are given in Section 2. Section 4 contains illustrative examples for the main results.
2 Basic results
We begin this Section with some definitions [10].
Definition 2.1
The Riemann-Liouville fractional integral of order τ > 0 of a function h : (0, ∞) ➝ ℝ is defined by
provided the right-hand side is point-wise defined on (0,∞), where Г is the Gamma function.
Definition 2.2
The Caputo derivative of order τ for a function h : [0, ∞) → R with h(x) 2 Cn[0, ∞) is defined by
Property 2.1
With the given notations, the following equality holds:
where ci (i = 1, ..., n − 1) are arbitrary constants.
The following lemma facilitates the transformation of the problem (1)-(2) into a fixed point problem.
Lemma 2.1
For any y ∈ C([0, 1], ℝ), the solution of linear multi-term fractional differential equation
supplemented with the boundary conditions (2) is given by
where
Proof. Case (i):
Applying the operator Iq on (4) and using (3), we get
where c1 is an arbitrary constant. By the method of variation of parameters, the solution of (11) can be written as
where m1 and m2 are given by (8). Using x(0) = 0 in (12), we get
which together with the conditions x(1) = 0 and x(η) = 0 yields the following system of equations in the unknown constants c1 and c2:
Solving the above system together with the notations (8), we find that
and
Substituting the value of c1 and c2 in (13), we obtain the solution (5). The converse of the lemma follows by direct computation.
The other two cases can be treated in a similar manner. This completes the proof. □
3 Existence and Uniqueness Results
Let 𝒞 = C([0, 1], ℝ) denote the Banach space of all continuous functions from [0, 1] to ℝ equipped with the norm defined by ∥u∥ = sup {|u(t)| : t ∈ [0, 1]}.
By Lemma 2.1, we transform the problem (1)-(2) into equivalent fixed point problems according to the cases (i) − (iii) as follows.
(i) For a12 − 4a0a2 > 0, we define
where the operator 𝒥 : 𝒞 ➝ 𝒞 is given by
where Φ(·), σ1(t) and σ2(t) are defined by (8).
(ii) In case a12 − 4a0a2 = 0, we have
where the operator 𝓗 : 𝒞 ➝ 𝒞 is given by
where Ѱ(·), ѱ1(t) and ѱ2(t) are given by (9).
(iii)When a12 − 4a0a2 < 0, let us define
where the operator 𝒦 : 𝒞 ➝ 𝒞 is given by
where Ω(·), φ1(t) and φ2(t) are defined by (10).
In the sequel, for the sake of computational convenience, we set
Before presenting our first existence result for the problem (1)-(2), let us state Krasnoselskii’s fixed point theorem [11] that plays a key role in its proof.
Theorem 3.1
(Krasnoselskii’s fixed point theorem). Let Y be a bounded, closed, convex, and nonempty subset of a Banach space X. Let F1and F2be the operators satisfying the conditions: (i) F1y1 + F2y2ϵ Y whenever y1, y2ϵ Y; (ii) F1is compact and continuous; (iii) F2is a contraction mapping. Then there exists y ∈ Y such that y = F1y + F2y.
In the forthcoming analysis, we need the following assumptions:
(A1) |f (t, x) − f (t, y)| ≤ ℓ|x − y|, for all t ϵ [0, 1], x, y ϵ ℝ, ℓ > 0.
(A2) |f (t, x)| ≤ ϑ(t), for all (t, x) ϵ [0, 1] × ℝ and ϑ ϵ C([0, 1], ℝ+).
Theorem 3.2
Let f : [0, 1] ×ℝ ➝ ℝ be a continuous function satisfying the conditions (A1) and (A2). Then the problem (1)-(2) has at least one solution on [0, 1] provided that
(i) ℓλ1 < 1 for a12 − 4a0a2 > 0, where λ1is given by (20);
(ii) ℓμ1 < 1 for a12 − 4a0a2 = 0, where μ1is given by (21);
(iii) ℓρ1 < 1 for If a12 − 4a0a2 < 0, where ρ1is given by (22).
Proof. (i) Setting supt∈[0,1]|ϑ(t)| = ∥ϑ∥ and choosing
we consider a closed ball Br1 = {x ϵ 𝒞 : ∥x∥ ≤ r1g. Introduce the operators 𝒥1 and 𝒥2 on Br1 as follows:
Observe that 𝒥 = 𝒥1 + 𝒥2. For x, y ϵ Br1 , we have
where we used (23). Thus 𝒥 1x + 𝒥 2y ϵ Br1 . Using the assumption (A1) together with the condition ℓλ1 < 1, we can show that 𝒥2 is a contraction as follows:
Note that continuity of f implies that the operator 𝒥1 is continuous. Also, 𝒥1 is uniformly bounded on Br1 as
Now we prove the compactness of operator 𝒥1.We define
independent of x. Thus 𝒥1 is relatively compact on Br1 .Hence, by the Arzelá-Ascoli Theorem, 𝒥1 is compact on Br1 . Thus all the assumptions of Theorem 3.1 are satisfied. So, by the conclusion of Theorem 3.1, the problem (1)-(2) has at least one solution on [0, 1].
(ii) Let us consider Br2 = {x ϵ 𝒞 : ∥x∥ ≤ r2}, where supt2[0,1]|ϑ(t)| = ∥ϑ∥ and
Introduce the operators 𝓗1 and 𝓗2 on Br2 as follows:
Observe that 𝓗 = 𝓗1 + 𝓗2. For x, y ϵ Br2 , we have
where we used (26). Thus 𝓗1x + 𝓗2y ϵ Br2 . Using the assumption (A1) together with ℓμ1 < 1, it is easy to show that 𝓗2 is a contraction. Note that continuity of f implies that the operator 𝓗1 is continuous. Also, 𝓗1 is uniformly bounded on Br2 as
Now we prove the compactness of operator 𝓗1. For that, let
independent of x. Thus, 𝓗1 is relatively compact on Br2 . Hence, by the Arzelá-Ascoli Theorem, 𝓗1 is compact on Br2 . Thus all the assumption of Theorem 3.1 are satisfied. So the conclusion of Theorem 3.1 applies and hence the problem (1)-(2) has at least one solution on [0, 1].
(iii) As before, letting supt2[0,1]|ϑ(t)| = ∥ϑ∥ and
we consider Br3 = {x ϵ 𝒞 : ∥x∥ ≤ r3}. Define the operators 𝒦1 and 𝒦2 on Br3 as follows:
Observe that 𝒦 = 𝒦1 + 𝒦2. For x, y ϵ Br3 , as before, it can be shown that 𝒦1x + 𝒦2y ϵ Br3 . Using the assumption (A1) together with ℓρ1 < 1, we can show that 𝒦2 is a contraction. Also, 𝒦1 is uniformly bounded on Br3 as
Fixing sup(t,x)ϵ[0,1]×Br3|f (t, x)| = f̄ , we have
which is independent of x and tends to zero as t2 − t1 ➝ 0 (0 < t1 < t2 < 1). Thus, employing the earlier arguments, 𝒦1 is compact on Br3 . In view of the foregoing arguments, it follows that the problem (1)-(2) has at least one solution on [0, 1]. The proof is completed. □
Remark 3.1
In the above theorem, we can interchange the roles of the operators
(1) 𝒥1and 𝒥2to obtain a second result by replacing ℓλ1 < 1 with the condition:
(2) 𝓗1and 𝓗2to obtain a second result by replacing ℓμ1 < 1 with the condition:
(3) 𝒦1and 𝒦2to obtain a second result by replacing ℓρ1 < 1 with the condition:
Now we establish the uniqueness of solutions for the problem (1)-(2) by means of Banach’s contraction mapping principle.
Theorem 3.3
Assume that f : [0, 1] × ℝ ➝ ℝ is a continuous function such that (A1) is satisfied. Then the problem (1)-(2) has a unique solution on [0, 1] if
(i) ℓλ < 1 for a12 − 4a0a2 > 0, where λ is given by (20);
(ii) ℓμ < 1 for a12 − 4a0a2 = 0, where μ is given by (21);
(iii) ℓρ < 1 for If a12 − 4a0a2 < 0, where ρ is given by (22).
Proof. (i) Let us define supt2[0,1]|f (t, 0)| = M and select
Then, for x ϵ Bk1 , we obtain
which clearly shows that 𝒥x ϵ Bk1 for any x ϵ Bk1 . Thus 𝒥Bk1⊂ Bk1 . Now, for x, y ϵ 𝒞 and for each t ϵ [0, 1], we have
where λ is given by (20) and depends only on the parameters involved in the problem. In view of the condition ℓ < 1/λ, it follows that J is a contraction. Thus, by the contraction mapping principle (Banach fixed point theorem), the problem (1)-(2) with a12 − 4a0a2 > 0 has a unique solution on [0, 1].
(ii) Let us define suptϵ[0,1]|f (t, 0)| = M and select
where μ is given by (21). By the condition ℓ < 1/μ, we deduce that the operator 𝓗 is a contraction. Thus, by the contraction mapping principle, the problem (1)-(2) with a12 − 4a0a2 = 0 has a unique solution on [0, 1].
(iii) Letting suptϵ[0,1]ϵf (t, 0)ϵ = M and selecting
where ρ is given by (22). Evidently, it follows by the condition ℓ < 1/ρ that K is a contraction. Thus, by the contraction mapping principle, the problem (1)-(2) with a12 − 4a0a2 < 0 has a unique solution on [0, 1]. This completes the proof. □
The next existence result is based on Leray-Schauder nonlinear alternative [8], which is stated below.
Theorem 3.4
(Nonlinear alternative for single valued maps). Let Y be a closed, convex subset of a Banach space X and V be an open subset of Y with 0 ϵ V. Let G : V ➝ Y be a continuous and compact (that is, G(V) is a relatively compact subset of Y) map. Then either G has a fixed point in V or there is a v ϵ ∂V (the boundary of V in Y) and ε ϵ (0, 1) with u = εG(u).
In order to establish our last result, we need the following conditions.
(H1)There exist a function g ϵ C([0, 1], ℝ+), and a nondecreasing function Q : ℝ+ ➝ ℝ+ such that |f (t, y)| ≤ g(t)Q(∥y∥), ∀(t, y) ϵ [0, 1] × ℝ.
(H2)−(i, P) There exists a constant 𝒦i > 0 such that
Theorem 3.5
Let f : [0, 1]×ℝ ➝ ℝ be a continuous function. Then the problem (1)-(2) has at least one solution on [0, 1] if
(a) (H1) and (H2)− (1, λ) are satisfied for
(b) (H1) and (H2)− (2, μ) are satisfied for
(c) (H1) and (H2)− (3, ρ) are satisfied for
Proof. (a) Let us first show that the operator 𝒥 : 𝒞 ➝ 𝒞 defined by (15) maps bounded sets into bounded sets in 𝒥 = C([0, 1], ℝ). For a positive number ζ1, let 𝓑ζ1 = {x ϵ 𝒞 : ∥x∥ ≤ ζ1} be a bounded set in 𝒥. Then we have
which yields
Next we show that 𝒥 maps bounded sets into equicontinuous sets of 𝒞. Let t1, t2ϵ [0, 1] with t1 < t2 and y ϵ 𝓑ζ1 , where 𝓑ζ1 is a bounded set of 𝒞. Then we obtain
which tends to zero as t2 − t1 ➝ 0 independently of x ϵ 𝓑ζ1 . From the foregoing arguments, it follows by the Arzelá-Ascoli theorem that 𝒥 : 𝒞 ➝ 𝒞 is completely continuous.
The proof will be complete by virtue of Theorem 3.4 once we establish that the set of all solutions to the equation x = θ𝒥x is bounded for θ ϵ [0, 1]. To do so, let x be a solution of x = θ𝒥x for θ ϵ [0, 1]. Then, for t ϵ [0, 1], we get
which on taking the norm for t ϵ [0, 1], yields
In view of (H2)− (1, λ), there is no solution x such that ∥x∥ 6= K1. Let us set
As the operator 𝒥 :
(b) As in part (a), it can be shown that the operator H : 𝒞 ➝ 𝒞 defined by (17) maps bounded sets into bounded sets in 𝒞 = C([0, 1], ℝ). For that, let ζ2 be a positive number and let 𝓑ζ2 = {x ϵ 𝒞 : ∥x∥ ≤ ζ2} be a bounded set in 𝒞. Then we have
In order to show that 𝓗 maps bounded sets into equicontinuous sets of 𝒞, let t1, t2ϵ [0, 1] with t1 < t2 and ϵ 2 𝓑ζ2 , where 𝓑ζ2 is a bounded set of 𝒞. Then we get
which tends to zero as t2 − t1 ➝ 0 independently of x ϵ 𝓑ζ2 . As argued before, 𝓗 : 𝒞 ➝ 𝒞 is completely continuous. To show that the set of all solutions to the equation x = θ𝓗x is bounded for θ ϵ [0, 1], let x be a solution of x = θ𝓗x for θ ϵ [0, 1]. Then, for t ϵ [0, 1], we find that
Thus
In view of (H2)− (2, μ), there is no solution x such that ∥x∥ 6= K2. Let us define
Since the operator
(c) As in the preceding cases, one can show that the operator 𝒦 defined by (19) is continuous and completely continuous. We only provide the outline for the last part (a-priori bounds) of the proof. Let x be a solution of x = θKx for θ ϵ [0, 1], where K is defined by (19). Then, for t ϵ [0, 1], we have
which yields
In view of (H2)− (3, ρ), there is no solution x such that ∥x∥ 6= K3. Let us set
As before, one can show that the operator K has a fixed point u ϵ Ū3, which is a solution of the problem (1)-(2). This completes the proof. □
4 Examples
Example 4.1
Consider the following boundary value problem
Here,
Clearly
with ℓ = 0.04 A. Using the given values, we find that
As all the conditions of Theorem 3.2 (i) are satisfied, the problem (33)-(34) has at least one solution on [0, 1]. On the other hand, ℓλ < 1 whenever A < 37.184674 and thus there exists a unique solution for the problem (33)-(34) on [0, 1] by Theorem 3.3 (i).
Example 4.2
Consider the multi-term fractional differential equation
supplemented with the boundary conditions
Here,
Clearly
where ℓ = 0.2 B. Using the given values, it is found that
Example 4.3
consider the multi-term fractional boundary value problem given by
where,
Clearly
with ℓ = C /6. Using the given values, we have
and
Example 4.4
Consider the following nonlocal boundary value problem of multi-term fractional differential equation
where,
Clearly
with
Example 4.5
Consider the following boundary value problem
where,
Clearly
with
Example 4.6
consider the following problem
Here,
Clearly
with
Acknowledgement
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-11-130-39). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs
Articles in the same Issue
- Regular Articles
- Algebraic proofs for shallow water bi–Hamiltonian systems for three cocycle of the semi-direct product of Kac–Moody and Virasoro Lie algebras
- On a viscous two-fluid channel flow including evaporation
- Generation of pseudo-random numbers with the use of inverse chaotic transformation
- Singular Cauchy problem for the general Euler-Poisson-Darboux equation
- Ternary and n-ary f-distributive structures
- On the fine Simpson moduli spaces of 1-dimensional sheaves supported on plane quartics
- Evaluation of integrals with hypergeometric and logarithmic functions
- Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients
- Oscillation of first order linear differential equations with several non-monotone delays
- Existence and regularity of mild solutions in some interpolation spaces for functional partial differential equations with nonlocal initial conditions
- The log-concavity of the q-derangement numbers of type B
- Generalized state maps and states on pseudo equality algebras
- Monotone subsequence via ultrapower
- Note on group irregularity strength of disconnected graphs
- On the security of the Courtois-Finiasz-Sendrier signature
- A further study on ordered regular equivalence relations in ordered semihypergroups
- On the structure vector field of a real hypersurface in complex quadric
- Rank relations between a {0, 1}-matrix and its complement
- Lie n superderivations and generalized Lie n superderivations of superalgebras
- Time parallelization scheme with an adaptive time step size for solving stiff initial value problems
- Stability problems and numerical integration on the Lie group SO(3) × R3 × R3
- On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations
- On algebraic characterization of SSC of the Jahangir’s graph 𝓙n,m
- A greedy algorithm for interval greedoids
- On nonlinear evolution equation of second order in Banach spaces
- A primal-dual approach of weak vector equilibrium problems
- On new strong versions of Browder type theorems
- A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
- Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
- Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
- Introduction to disoriented knot theory
- Restricted triangulation on circulant graphs
- Boundedness control sets for linear systems on Lie groups
- Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
- Disjointed sum of products by a novel technique of orthogonalizing ORing
- A parametric linearizing approach for quadratically inequality constrained quadratic programs
- Generalizations of Steffensen’s inequality via the extension of Montgomery identity
- Vector fields satisfying the barycenter property
- On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
- Biderivations of the higher rank Witt algebra without anti-symmetric condition
- Some remarks on spectra of nuclear operators
- Recursive interpolating sequences
- Involutory biquandles and singular knots and links
- Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
- Topological entropy for positively weak measure expansive shadowable maps
- Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
- On 𝓠-regular semigroups
- One kind power mean of the hybrid Gauss sums
- A reduced space branch and bound algorithm for a class of sum of ratios problems
- Some recurrence formulas for the Hermite polynomials and their squares
- A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
- On f - prime radical in ordered semigroups
- Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
- Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
- A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
- Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
- The points and diameters of quantales
- Directed colimits of some flatness properties and purity of epimorphisms in S-posets
- Super (a, d)-H-antimagic labeling of subdivided graphs
- On the power sum problem of Lucas polynomials and its divisible property
- Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
- On generalized P-reducible Finsler manifolds
- On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
- On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
- On the different kinds of separability of the space of Borel functions
- Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
- Functional analysis method for the M/G/1 queueing model with single working vacation
- Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
- The existence of solutions to certain type of nonlinear difference-differential equations
- Domination in 4-regular Knödel graphs
- Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
- Algebras of right ample semigroups
- Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
- Nontrivial periodic solutions to delay difference equations via Morse theory
- A note on the three-way generalization of the Jordan canonical form
- On some varieties of ai-semirings satisfying xp+1 ≈ x
- Abstract-valued Orlicz spaces of range-varying type
- On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
- Arithmetic of generalized Dedekind sums and their modularity
- Multipreconditioned GMRES for simulating stochastic automata networks
- Regularization and error estimates for an inverse heat problem under the conformable derivative
- Transitivity of the εm-relation on (m-idempotent) hyperrings
- Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
- Simultaneous prediction in the generalized linear model
- Two asymptotic expansions for gamma function developed by Windschitl’s formula
- State maps on semihoops
- 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
- Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
- New topology in residuated lattices
- Optimality and duality in set-valued optimization utilizing limit sets
- An improved Schwarz Lemma at the boundary
- Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
- Toeplitz matrices whose elements are coefficients of Bazilevič functions
- Epi-mild normality
- Nonlinear elastic beam problems with the parameter near resonance
- Orlicz difference bodies
- The Picard group of Brauer-Severi varieties
- Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
- Weak group inverse
- Infinite growth of solutions of second order complex differential equation
- Semi-Hurewicz-Type properties in ditopological texture spaces
- Chaos and bifurcation in the controlled chaotic system
- Translatability and translatable semigroups
- Sharp bounds for partition dimension of generalized Möbius ladders
- Uniqueness theorems for L-functions in the extended Selberg class
- An effective algorithm for globally solving quadratic programs using parametric linearization technique
- Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
- On categorical aspects of S -quantales
- On the algebraicity of coefficients of half-integral weight mock modular forms
- Dunkl analogue of Szász-mirakjan operators of blending type
- Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
- Global stability of a distributed delayed viral model with general incidence rate
- Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
- Boundary value problems of a discrete generalized beam equation via variational methods
- Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
- Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
- Spectrum of free-form Sudoku graphs
- Regularity of fuzzy convergence spaces
- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs
