Home Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
Article Open Access

Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation

  • Tingting Guan , Guotao Wang and Serkan Araci EMAIL logo
Published/Copyright: August 12, 2024
Download Article (PDF)
Become an author with De Gruyter Brill
Author Information Explore this Subject
Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

This study first establishes several maximum and minimum principles involving the nonlocal Monge-Ampère operator and the multi-term time-space fractional Caputo-Fabrizio derivative. Based on the maximum principle established above, on the one hand, we show that a family of multi-term time-space fractional parabolic Monge-Ampère equations has at most one solution; on the other hand, we establish some comparison principles of linear and nonlinear multi-term time-space fractional parabolic Monge-Ampère equations.

Keywords: nonlocal Monge-Ampère operator; multi-term time-space parabolic equations; fractional Caputo-Fabrizio derivative; maximum and minimum principles
MSC 2010: 35B50; 26A33

1 Introduction

As we all know, the famous Monge-Ampère operator is a fully nonlinear, degenerate operator. It is derived from several areas of analysis and geometry, in which it is often used to describe the prescribed Gaussian curvature, optimal transportation, affine geometry, etc. The original representation of this operator consists of prescribing the determinant of the Hessian of a convex function ω , denoted as det ( D 2 ω ) . For its recent research progress, see [1–5] (including its generalized k-Hessian problem).

Very recently, Caffarelli and Charro [6] successfully modified the classic Monge-Ampère operator to the nonlocal case, which is named as the nonlocal Monge-Ampère operator defined by

D s ω ( ξ ) = inf P.V. ∫ R N ω ( y ) − ω ( ξ ) A − 1 ∣ ξ − y ∣ N + 2 s d y ,

where A satisfies A > 0 and det A = 1 , here A > 0 implies A is positive definite square matrix.

In 2023, Wang et al. [7] studied the asymptotic radial solution of a class of parabolic tempered fractional Laplacian problems with logarithmic nonlinearity

∂ z ( x , t ) ∂ t − ( Δ + λ ) β 2 z ( x , t ) = a z ( x , t ) ln ∣ z ( x , t ) + 1 ∣ p ,

where ( Δ + λ ) β 2 is the tempered fractional Laplacian operator. Based on some new asymptotic maximum principles and asymptotic narrow region principles for antisymmetric functions, authors derived some properties of asymptotic radial solution to the above parabolic tempered fractional Laplacian problem in a unit ball. Inspired by the above two papers, in this work, we study a family of multi-term time-space fractional parabolic Monge-Ampère equations:

(1.1) ∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) = G ( ξ , τ , ω ) , ( ξ , τ ) ∈ Σ × [ 0 , + ∞ ) , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , + ∞ ) , ω ( ξ , 0 ) = g ( ξ ) , ξ ∈ Σ ,

where Σ ⊂ R N ( N ≥ 1 ) is a bounded domain in R N . D s is a nonlocal Monge-Ampère operator defined by

D s ω ( ξ , τ ) = inf P.V. ∫ R N ω ( y , τ ) − ω ( ξ , τ ) A − 1 ∣ ξ − y ∣ N + 2 s d y ,

where A satisfies A > 0 and det A = 1 . D τ α i CFF is the fractional Caputo-Fabrizio derivative [8,9] of order 0 < α i < 1 ( i = 1 , 2 , … , n ) defined by

D τ α i CFF f ( ξ , τ ) = 1 1 − α i ∫ 0 τ exp − α i 1 − α i ( τ − s ) ∂ f ∂ s ( ξ , s ) d s , t ≥ 0 ,

and μ i ≥ 0 , ∑ i = 1 n μ i > 0 ( i = 1 , 2 , … , n ) .

Fractional calculus, as a powerful supplement to classical calculus, has received great attention and carried forward. From the germination of fractional calculus to its vigorous growth, there have been many variants, which serve different theoretical and practical needs. The fractional Caputo-Fabrizio derivative is one of these brilliant achievements. In 2018, Atanacković et al. [10] analyzed the Caputo-Fabrizio fractional derivative in classical and distributional settings. In 2020, Baleanu et al. [11] proposed a new fractional model for human liver involving fractional Caputo-Fabrizio derivative with the exponential kernel. For some developments of fractional calculus, please refer to the literature [12–16].

Maximum principle is a useful tool to study the fractional partial differential equations without certain knowledge of solutions. In 2020, Kirane and Torebek [17] applied the maximum principles to obtain the properties of solutions for the space-time fractional diffusion and pseudo-parabolic equations with Caputo and Riemann-Liouville time-fractional derivatives. In 2020, Zeng et al. [18] established the maximum principles for multi-term space-time fractional diffusion equations with generalized variable-order fractional Caputo derivatives and variable-order fractional Riesz-Caputo derivatives. Other investigations of the maximum principle can be found in [19,20].

In this paper, we focus our attention on the maximum principle for equation (1.1). We emphasize that the introduction of the nonlocal Monge-Ampère operator and the multi-term time-space fractional Caputo-Fabrizio derivative bring the main difficulties to prove our main result (Theorem 2.2). To handle these difficulties, we first obtain inequality (2.3) by using Theorem 2.1. Then, we prove inequality (2.4) for nonlocal Monge-Ampère operator. Finally, we obtain our main result (Theorem 2.2). Our goal is to give several maximum and minimum principles involving the nonlocal Monge-Ampère operator and the multi-term time-space fractional Caputo-Fabrizio derivative, and to show some selected applications.

2 Maximum and minimum principles

In this section, we will give several maximum and minimum principles of multi-term time-space fractional parabolic Monge-Ampère equations (1.1). To successfully accomplish this task, we need an important theorem from [21], in which we take ( 2 − α ) M ( α ) = 2 .

Theorem 2.1

Let a function f ∈ C 1 ( [ 0 , D ] ) attain its maximum over the interval [ 0 , D ] at the point τ 0 ∈ ( 0 , D ] . Then, for any 0 < α < 1 , the fractional Caputo-Fabrizio derivative has the following property:

(2.1) D α CFF f ( τ 0 ) ≥ 1 1 − α exp − α τ 0 1 − α [ f ( τ 0 ) − f ( 0 ) ] .

Next we will apply inequality (2.1) to deduce several maximum principles.

Theorem 2.2

Let ω ∈ C 2 , 1 ( Σ × ( 0 , D ) ) and ω ∈ C ( Σ ¯ × [ 0 , D ] ) . If ω satisfies

(2.2) ∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) + h ( ξ , τ ) ω ( ξ , τ ) ≥ 0 , ( ξ , τ ) ∈ Σ × ( 0 , D ] , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , D ] , ω ( ξ , 0 ) ≥ 0 , ξ ∈ Σ ,

with h ( ξ , τ ) ≥ 0 , then ω ( ξ , τ ) ≥ 0 , ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] .

Proof

Assume that the conclusion is not true. Since the condition ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , D ] , and ω ( ξ , 0 ) ≥ 0 , ξ ∈ Σ , there exists a point ( ξ 0 , τ 0 ) ∈ Σ × ( 0 , D ] , such that

ω ( ξ 0 , τ 0 ) = min ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] ω ( ξ , τ ) < 0 .

By Theorem 2.1, one obtains

(2.3) ∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) ≤ ∑ i = 1 n μ i 1 − α i exp − α i τ 0 1 − α i [ ω ( ξ 0 , τ 0 ) − ω ( ξ 0 , 0 ) ] ≤ 0 .

Since ω ( ξ , τ 0 ) ∈ C 2 ( Σ ) ∩ C ( Σ ¯ ) and ω ( ξ , τ 0 ) = 0 , ξ ∈ R N \ Σ , fix ε > 0 , arbitrarily, we have

(2.4) D s ω ( ξ 0 , τ 0 ) = inf P.V. ∫ R N ω ( y , τ 0 ) − ω ( ξ 0 , τ 0 ) A − 1 ∣ y − ξ 0 ∣ N + 2 s d y = inf ∫ R N \ Σ ω ( y , τ 0 ) − ω ( ξ 0 , τ 0 ) A − 1 ∣ y − ξ 0 ∣ N + 2 s d y + P.V. ∫ Σ ω ( y , τ 0 ) − ω ( ξ 0 , τ 0 ) A − 1 ∣ y − ξ 0 ∣ N + 2 s d y ≥ ∫ R N \ Σ ω ( y , τ 0 ) − ω ( ξ 0 , τ 0 ) A − 1 ∣ y − ξ 0 ∣ N + 2 s d y − ε .

Let ε → 0 , by ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , + ∞ ) , and ω ( ξ 0 , τ 0 ) < 0 , we have

(2.5) D s ω ( ξ 0 , τ 0 ) > 0 .

In fact, if D s ω ( ξ 0 , τ 0 ) = 0 , then ω ( ξ , τ 0 ) = 0 , which contradicts with ω ( ξ 0 , τ 0 ) < 0 . So, (2.5) holds.

Applying inequalities (2.3), (2.5), and h ( ξ , τ ) ≥ 0 , we obtain

∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ 0 , τ 0 ) + h ( ξ 0 , τ 0 ) ω ( ξ 0 , τ 0 ) < 0 ,

which contradicts with (2.2). Therefore, it holds ω ( ξ , τ ) ≥ 0 , ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] .

The proof is complete.□

By a similar process, we can obtain

Theorem 2.3

Let ω ∈ C 2 , 1 ( Σ × ( 0 , D ) ) and ω ∈ C ( Σ ¯ × [ 0 , D ] ) . If ω satisfies

(2.6) ∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) + h ( ξ , τ ) ω ( ξ , τ ) ≤ 0 , ( ξ , τ ) ∈ Σ × ( 0 , D ] , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , D ] , ω ( ξ , 0 ) ≤ 0 , ξ ∈ Σ ,

with h ( ξ , τ ) ≤ 0 , then ω ( ξ , τ ) ≤ 0 , ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] .

Based on Theorems 2.2 and 2.3, the following two theorems are available immediately, respectively.

Theorem 2.4

Let ω ∈ C 2 , 1 ( Σ × ( 0 , D ) ) ∩ C ( Σ ¯ × [ 0 , D ] ) , if ω satisfies (2.2) with h ( ξ , τ ) ≥ 0 , then

min ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] ω ( ξ , τ ) = min ( ξ , τ ) ∈ ( Σ × { 0 } ) ∪ ( ∂ Σ × [ 0 , D ] ) ω ( ξ , τ ) .

Theorem 2.5

Let ω ∈ C 2 , 1 ( Σ × ( 0 , D ) ) ∩ C ( Σ ¯ × [ 0 , D ] ) , if ω satisfies (2.6) with h ( ξ , τ ) ≤ 0 , then

max ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] ω ( ξ , τ ) = max ( ξ , τ ) ∈ ( Σ × { 0 } ) ∪ ( ∂ Σ × [ 0 , D ] ) ω ( ξ , τ ) .

3 Some applications of maximum principles

In this section, some applications of maximum and minimum principles are established.

Consider a family of linear multi-term time-space fractional parabolic Monge-Ampère equations:

(3.1) ∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) + h ( ξ , τ ) ω ( ξ , τ ) = f ( ξ , τ ) , ( ξ , τ ) ∈ Σ × ( 0 , + ∞ ) , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , + ∞ ) , ω ( ξ , 0 ) = g ( ξ ) , ξ ∈ Σ ,

where functions h ( ξ , τ ) and f ( ξ , τ ) are bounded on Σ × [ 0 , D ] .

Theorem 3.1

Let ω ∈ C 2 , 1 ( Σ × ( 0 , D ) ) ∩ C ( Σ ¯ × [ 0 , D ] ) , if h ( ξ , τ ) ≥ 0 , ( ξ , τ ) ∈ Σ × [ 0 , D ] , the linear multi-term time-space fractional parabolic Monge-Ampère equation (3.1) have at most one solution on Σ ¯ × [ 0 , D ] .

Proof

We will apply a proof by contradiction. Assume ω 1 , ω 2 ∈ C 2 , 1 ( Σ × ( 0 , D ) ) ∩ C ( Σ ¯ × [ 0 , D ] ) are two solutions of the linear multi-term time-space fractional parabolic Monge-Ampère equations (3.1).

Put ω = ω 1 − ω 2 , then we have

∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) + h ( ξ , τ ) ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ Σ × ( 0 , D ] , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , D ] , ω ( ξ , 0 ) = 0 , ξ ∈ Σ .

Theorems 2.2 and 2.3 lead to ω ≥ 0 and ω ≤ 0 , respectively. So ω = 0 , which implies ω 1 = ω 2 . Thus, the conclusion of Theorem 3.1 is proved.□

Next we consider a family of multi-term time-space nonlinear fractional parabolic Monge-Ampère equations

(3.2) ∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) + h ( ξ , τ ) ω ( ξ , τ ) = G ( ξ , τ , ω ) , ( ξ , τ ) ∈ Σ × ( 0 , + ∞ ) , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , + ∞ ) , ω ( ξ , 0 ) = g ( ξ ) , ξ ∈ Σ .

Theorem 3.2

Let G ( ξ , τ , ω ) be a smooth function. If ∂ G ∂ ω − h ( ξ , τ ) ≤ 0 , ( ξ , τ ) ∈ Σ × [ 0 , D ] , then nonlinear multi-term time-space fractional parabolic Monge-Ampère equations (3.2) have at most one solution on Σ ¯ × [ 0 , D ] .

Proof

We shall employ the same method that was used in the proof of Theorem 3.1.

Let ω 1 , ω 2 ∈ C 2 , 1 ( Σ × ( 0 , D ) ) ∩ C ( Σ ¯ × [ 0 , D ] ) be the two solutions of nonlinear multi-term time-space fractional parabolic Monge-Ampère equations (3.2) on Σ ¯ × [ 0 , D ] . By using the auxiliary function ω = ω 1 − ω 2 , we have

∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) + h ( ξ , τ ) ω ( ξ , τ ) = G ( ξ , τ , ω 1 ) − G ( ξ , τ , ω 2 ) , ( ξ , τ ) ∈ Σ × ( 0 , + ∞ ) , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , + ∞ ) , ω ( ξ , 0 ) = g ( ξ ) , ξ ∈ Σ .

Because G is a smooth function, it holds that

G ( ξ , τ , ω 1 ) − G ( ξ , τ , ω 2 ) = ∂ G ∂ ω ( ζ ) ( ω 1 − ω 2 ) = ∂ G ∂ ω ( ξ , τ , ζ ) ω ,

where ζ = ( 1 − λ ) ω 1 + λ ω 2 , for λ ∈ [ 0 , 1 ] . It is obvious that ζ and ∂ G ∂ ω ( ξ , τ , ζ ) depend on the variables ξ and τ .

Since ∂ G ∂ ω − h ( ξ , τ ) ≤ 0 , ( ξ , τ ) ∈ Σ × [ 0 , D ] , the rest of the proof is similar to that of Theorem 3.1.□

Example 3.1

We consider the following multi-term time-space nonlinear fractional parabolic Monge-Ampère equation:

(3.3) ∑ i = 0 n μ i D τ α i CFF ω ( ξ , τ ) − D s ω ( ξ , τ ) + ξ 2 ω ( ξ , τ ) = ξ 2 τ 2 exp − ω , ( ξ , τ ) ∈ Σ × ( 0 , + ∞ ) , ω ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , + ∞ ) , ω ( ξ , 0 ) = ξ 3 , ξ ∈ Σ .

Owing to h ( ξ , τ ) = ξ 2 , G ( ξ , τ , ω ) = ξ 2 τ 2 exp − ω , ∂ G ∂ ω − h ( ξ , τ ) = − ξ 2 τ 2 exp − ζ − ξ 2 ≤ 0 , where ζ = ( 1 − λ ) ω 1 + λ ω 2 , for λ ∈ [ 0 , 1 ] . According to Theorem 3.2, the multi-term time-space nonlinear fractional parabolic Monge-Ampère equation (3.3) has at most one solution on Σ ¯ × [ 0 , D ] .

Corollary 3.1

In Theorem 3.2, let h ( ξ , τ ) ≥ 0 . If F ( ξ , τ , ω ) is non-increasing with respect to ω , then the nonlinear multi-term time-space fractional parabolic Monge-Ampère equations (3.2) have at most one solution.

An important application of maximum and minimum principles is that derive some comparison principles of linear and nonlinear multi-term time-space fractional parabolic Monge-Ampère equations.

Theorem 3.3

Let ω , v ∈ C 2 , 1 ( Σ × ( 0 , D ) ) ∩ C ( Σ ¯ × [ 0 , D ] ) satisfy

(3.4) ∑ i = 0 n μ i D τ α i CFF ω − D s ω + h ( ξ , τ ) ω − G ( ξ , τ , ω ) ≥ ∑ i = 0 n μ i D τ α i CFF v − D s v + h ( ξ , τ ) v − G ( ξ , τ , v ) , ( ξ , τ ) ∈ Σ × [ 0 , D ] , ω ( ξ , τ ) = v ( ξ , τ ) , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , D ] , ω ( ξ , 0 ) ≥ v ( ξ , 0 ) , ξ ∈ Σ .

If there exists a bounded function g ∈ Σ ¯ × [ 0 , D ] with g + h ≥ 0 , such that

G ( ξ , τ , ω ) − G ( ξ , τ , v ) ≥ − g ( ξ , τ ) ( ω − v ) , ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] ,

then the inequality ω ( ξ , τ ) ≥ v ( ξ , τ ) , ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] holds.

Proof

Let us introduce an auxiliary function σ = ω − v . Thus, we have

∑ i = 0 n μ i D τ α i CFF σ − D s σ + ( h + g ) ( ξ , τ ) σ ≥ 0 , ( ξ , τ ) ∈ Σ × [ 0 , D ] , σ ( ξ , τ ) = 0 , ( ξ , τ ) ∈ ( R N \ Σ ) × [ 0 , D ] , σ ( ξ , 0 ) ≥ 0 , ξ ∈ Σ .

It follows from Theorem 2.2 that σ ( ξ , τ ) ≥ 0 , i.e., ω ( ξ , τ ) ≥ v ( ξ , τ ) , ( ξ , τ ) ∈ Σ ¯ × [ 0 , D ] .□

4 Conclusion

The multi-term time-space fractional parabolic Monge-Ampère equations with the fractional Caputo-Fabrizio derivative are considered in this study. New maximum principles of the nonlocal Monge-Ampère operator and the multi-term time-space fractional Caputo-Fabrizio derivative are derived. Based on the maximum principle, some comparison principles and the properties of the solution of multi-term time-space fractional parabolic Monge-Ampère equations are proved.

Acknowledgement

We would like to express our gratitude to the editor for taking time to handle the manuscript.

  1. Funding information: This research received no funding.

  2. Author contributions: All authors equally contributed to this manuscript and approved the final version.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Ethical approval: Written informed consent for publication of this article was obtained from Shanxi Normal University, Hasan Kalyoncu University and all authors.

  5. Institutional review board statement: Not applicable.

  6. Informed consent statement: Not applicable.

  7. Data availability statement: Not applicable.

References

[1] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations I. Monge-Ampére equation, Comm. Pure Appl. Math. 37 (1984), 369–402. 10.1002/cpa.3160370306Search in Google Scholar

[2] Z. Zhang, Optimal global and boundary asymptotic behavior of large solutions to the Monge-Ampère equation, J. Funct. Anal. 278 (2020), 108512. 10.1016/j.jfa.2020.108512Search in Google Scholar

[3] L. Caffarelli and Y. Li, A Liouville theorem for solutions of the Monge-Ampère equation with periodic data, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), 97–120. 10.1016/j.anihpc.2003.01.005Search in Google Scholar

[4] B. Guan and H. Jian, The Monge-Ampère equation with infinite boundary value, Pacific J. Math. 216 (2004), 77–94, DOI: https://doi.org/10.2140/pjm.2004.216.77. 10.2140/pjm.2004.216.77Search in Google Scholar

[5] G. D. Philippis, A. Figalli, and O. Savin, A note on interior W2,1+ε estimates for the Monge-Ampère equation, Math. Ann. 357 (2013), 11–22. 10.1007/s00208-012-0895-9Search in Google Scholar

[6] L. Caffarelli and F. Charro, On a fractional Monge-Ampère operator, Ann. PDE 1 (2015), no. 1, 4–47. 10.1007/s40818-015-0005-xSearch in Google Scholar

[7] G. Wang, Y. Liu, J. Nieto, and L. Zhang, Asymptotic radial solution of parabolic tempered fractional Laplacian problem, Bull. Malays. Math. Sci. Soc. 46 (2023), no. 1, 1, DOI: https://doi.org/10.1007/s40840-022-01394-x. 10.1007/s40840-022-01394-xSearch in Google Scholar

[8] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), 73–78. Search in Google Scholar

[9] J. Juan, Solution of a fractional logistic ordinary differential equation, Appl. Math. Lett. 123 (2022), 107568. 10.1016/j.aml.2021.107568Search in Google Scholar

[10] T. M. Atanacković, S. Pilipović, and D. Zorica, Properties of the Caputo-Fabrizio fractional derivative and its distributional settings, Fract. Calc. Appl. Anal. 21 (2018), 29–44. 10.1515/fca-2018-0003Search in Google Scholar

[11] D. Baleanu, A. Jajarmi, H. Mohammadi, and S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Solitons Fractals 134 (2020), 109705, DOI: https://doi.org/10.1016/j.chaos.2020.109705. 10.1016/j.chaos.2020.109705Search in Google Scholar

[12] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, vol. 204, 2006. Search in Google Scholar

[13] G. Wang, K. Pei, R. P. Agarwal, L. Zhang, and B. Ahmad, Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math. 343 (2018), 230–239. 10.1016/j.cam.2018.04.062Search in Google Scholar

[14] D. Kumar, J. Singh, K. Tanwar, and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power exponential and Mittag-Leffler laws, Int. J. Heat Mass Transf. 138 (2019), 1222–1227. 10.1016/j.ijheatmasstransfer.2019.04.094Search in Google Scholar

[15] L. Zhang, N. Qin, and B. Ahmad, Explicit iterative solution of a Caputo-Hadamard-type fractional turbulent flow model, Math. Methods Appl. Sci. (2020), 1–11, DOI: https://doi.org/10.1002/mma.6277. 10.1002/mma.6277Search in Google Scholar

[16] D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simul. 59 (2018), 444–462. 10.1016/j.cnsns.2017.12.003Search in Google Scholar

[17] M. Kirane and B. Torebek, Maximum principle for space and time-space fractional partial differential equations, Mathematic 2020 (2020), 1–24. 10.1186/s13662-020-02744-4Search in Google Scholar

[18] S. Zeng, S. Miǵorski, V. Nguyen, and Y. Bai, Maximum principles for a class of generalized time-fractional diffusion equations, Fract. Calc. Appl. Anal. 23 (2020), 822–836. 10.1515/fca-2020-0041Search in Google Scholar

[19] Z. Liu, S. Zeng, and Y. Bai, Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications, Fract. Calc. Appl. Anal. 19 (2016), 188–211. 10.1515/fca-2016-0011Search in Google Scholar

[20] T. Guan and L. Zhang, Maximum principle for variable-order fractional conformable differential equation with a generalized tempered fractional Laplace operator, Fractal Fract. 7 (2023), 798, DOI: https://doi.org/10.3390/fractalfract7110798. 10.3390/fractalfract7110798Search in Google Scholar

[21] L. Zhang, B. Ahmad, and G. Wang, Analysis and application for a diffusion equations with a new fractional derivative without singular kernel, Electron. J. Differential Equations 289 (2017), 1–6. 10.1186/s13662-017-1356-2Search in Google Scholar
Received: 2023-03-31
Revised: 2023-12-28
Accepted: 2024-05-24
Published Online: 2024-08-12

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
https://doi.org/10.1515/dema-2024-0031
Keywords for this article
nonlocal Monge-Ampère operator; multi-term time-space parabolic equations; fractional Caputo-Fabrizio derivative; maximum and minimum principles
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 13.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/dema-2024-0031/html
Scroll to top button