About this Journal Submit a Manuscript Table of Contents
The Scientific World Journal
Volume 2014 (2014), Article ID 519814, 11 pages
http://dx.doi.org/10.1155/2014/519814
Research Article

Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

1Department of Mathematics, Fatih University, 34500 Istanbul, Turkey
2ITTU, 74400 Ashgabat, Turkmenistan
3Tejen High School, Tejen, Turkmenistan
4Department of Mathematics, Turkmen State University, 74400 Ashgabat, Turkmenistan

Received 21 August 2013; Accepted 2 October 2013; Published 12 January 2014

Academic Editors: G. Bonanno, F. Minhós, and G.-Q. Xu

Copyright © 2014 Allaberen Ashyralyev and Asker Hanalyev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The nonlocal boundary value problem for the parabolic differential equation in an arbitrary Banach space with the dependent linear positive operator is investigated. The well-posedness of this problem is established in Banach spaces of all -valued continuous functions on satisfying a Hölder condition with a weight . New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

1. Introduction: A Cauchy Problem

It is known that (see, e.g., [15] and the references given therein) several applied problems in fluid mechanics, physics, and mathematical biology were formulated into nonlocal mathematical models. In general, such nonlocal problems were not well studied.

Before going to discuss the well-posedness of nonlocal boundary value problem, we will give the definition of positive operators in a Banach space and introduce the fractional spaces generated by positive operators that will be needed in the sequel.

Let be a Banach space and let be a linear unbounded operator densely defined in . The operator is said to be positive in the Banach space if its spectrum lies in the interior of the sector of angle , , symmetric with respect to the real axis, and if, on the edges of this sector, and , and outside of the sector the resolvent is subject to the bound The infimum of all such angles is called the spectral angle of the positive operator and is denoted by . We call strongly positive in the Banach space if its spectral angle .

For positive operator in the Banach space , let us introduce the fractional spaces consisting of those for which the norm is finite.

In the paper [6], the well-posedness in spaces of smooth functions of the nonlocal boundary value problem for the differential equation in an arbitrary Banach space with the strongly positive operator was established. The importance of coercive inequalities (well-posedness) is well known [7, 8].

Noted that theory and methods for approximate solutions of local and nonlocal boundary value problems for evolution differential equations have been studied extensively by many researchers (see [939] and the references therein).

Before going to establish theorems on the well-posedness of nonlocal boundary value problem for parabolic equations in an arbitrary Banach space with the strongly positive dependent space operators, let us consider the abstract Cauchy problem for the differential equation in an arbitrary Banach space with the strongly positive operators in with domain , independent of and dense in .

A function is called a solution of the problem (4) if the following conditions are satisfied.(i) is continuously differentiable on the segment  . The derivative at the endpoints of the segment is understood as the appropriate unilateral derivatives.(ii)The element belongs to for all , and the function is continuous on .(iii) satisfies the equation and the initial condition (4).

A solution of problem (4) defined in this manner will from now on be referred to as a solution of problem (4) in the space . Here, stands for the Banach space of all continuous functions defined on with values in equipped with the norm From the existence of the such solutions, it evidently follows that and .

We say that the problem (4) is well posed in if the following conditions are satisfied.(1)Problem (4) is uniquely solvable for any and any . This means that an additive and homogeneous operator is defined which acts from to and gives the solution of problem (4) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs , , and , equipped with the norm By Banach’s theorem, in and these properties, one has coercive inequality where does not depend on  and .

Inequality (7) is called the coercivity inequality in for (4). If , then the coercivity inequality implies the analyticity of the semigroup , that is, the following estimates hold for some . Thus, the analyticity of the semigroup is  necessary for the well-posedness of problem (4) in . Unfortunately, the analyticity of the semigroup is not sufficient for the well-posedness of problem (4) in .

Suppose that, for each , the operator generates an analytic semigroup with exponentially decreasing norm, when , that is, the following estimates hold for some ,  . From this inequality, it follows that the operator exists and is bounded, and hence is closed in .

Suppose that the operator is Hölder continuous in in the uniform operator topology for each fixed , that is, where and are positive constants independent of , and for .

From (9) and (10), it follows that for all   and (see, [14]).

An operator-valued function , defined and strongly continuous jointly in and for , is called a fundamental solution of (4) ifthe operator is strongly continuous in and for ,the following identity holds: the operator maps the region into itself; the operator is bounded and strongly continuous in and for ,on the region , the operator is strongly differentiable relative to and , while is also called development family, evolution operator, Green’s function, and so forth [14].

If the function is not only continuous, but also continuously differentiable on , and , it is easy to show that the formula gives a solution of problem (4).

Now, we will give lemmas and estimates from [14] concerning the semigroup and the fundamental solution of (4) and theorem on well-posedness of (4) which will be useful in the sequel.

Lemma 1. For any ,  , and , one has the inequality where does not depend on , and .

Lemma 2. For any ,   ,   and , the following estimates hold: where and do not depend on , and .

Lemma 3. For any and , the following identities hold:

Lemma 4. For any , the following estimates hold: where does not depend on and .

A function is said to be a solution of problem (4) in if it is a solution of this problem in and the functions   and belong to .

As in the case of the space , we say that the problem (4) is well posed in if the following two conditions are satisfied.(1)For any and , there exists the unique solution in of problem (4). This means that an additive and homogeneous operator is defined which acts from to and gives the solution of (4) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs (,, and , equipped with the norm

We set equal to space, obtained by completion of the set of all smooth -valued functions on with respect to the norm Let us give the following theorems on well-posedness of (4) in from [14]. To these, there correspond the spaces of traces , which consist of elements for which the norm is finite.

Theorem 5. Suppose ,  . Suppose that the assumptions (9) and (10) hold and . Then, for the solution in of the Cauchy problem (4), the coercive inequality holds, where does not depend on , and .

Theorem 6. Suppose ,  . Suppose that the assumptions (9) and (10) hold and . Then, for the solution in of the Cauchy problem (4), the coercive inequality holds, where does not depend on , and .

Note that the spaces of smooth functions , in which coercive solvability has been established, depend on the parameters , and . However, the constants in the coercive inequalities depend only on . Hence, we can choose the parameters and freely, which increases the number of functional spaces in which problem (4)  is well posed. In particular, Theorems 5 and 6 imply theorems on well-posedness of the nonlocal boundary value problem (4) in .

Finally, in the paper [40], the initial-value problem for the fractional parabolic equation in an arbitrary Banach space with the strongly positive operators was investigated. Here, is standard Riemann-Liouville’s derivative of order . The well-posedness of problem (26) in spaces was established. New exact estimates in Hölder norms for the solution of initial-boundary value problems for fractional parabolic equations were obtained.

In the paper [41], the nonlocal boundary value problem for the parabolic differential equation in an arbitrary Banach space with the strongly positive operators in with domain , independent of and dense in , was investigated. The well-posedness of problem (27) in spaces was established. New exact estimates in Hölder norms for the solution of three nonlocal boundary value problems for parabolic equations were obtained.

In the present paper, the well-posedness of problem (27) in spaces is established. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

The paper is organized as follows. Section 1 is introduction. In Section 2, new theorems on well-posedness of problem (27) in spaces are established. In Section 3, theorems on the coercive stability estimates for the solution of two nonlocal boundary value parabolic problems are obtained. Finally, Section 4 is conclusion.

2. Well-Posedness of Nonlocal Boundary Value Problem (27)

Now, we will give lemmas on the fundamental solution of (4) from paper [41].

Lemma 7. Assume that for any . Then, for any and , the following identity holds

Lemma 8. Under the assumption of Lemma 7 there exists the inverse of the operator in and the following estimate holds:

A function is called a solution of the problem (27) if the following conditions are satisfied.(i) is continuously differentiable on the segment .(ii)The element belongs to for all , and the function is continuous on .(iii) satisfies the equation and the nonlocal boundary condition (27).

We say that the problem (27) is well posed in if the following conditions are satisfied.(1)Problem (27) is uniquely solvable for any and any . This means that an additive and homogeneous operator is does not depend on defined which acts from to and gives the solution of problem (4) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs ,  , and , equipped with the norm By Banach’s theorem in and these properties, one has coercive inequality where does not depend on   and .

Inequality (32) is called the coercivity inequality in for (27).  If , then the coercivity inequality implies the analyticity of the semigroup . Thus, the analyticity of the semigroup is necessary for the well-posedness of problem (27) in . Unfortunately, the analyticity of the semigroup is not sufficient for the well-posedness of problem (27) in .

A function is said to be a solution of problem (27) in if it is a solution of this problem in and the functions   and belong to .

As in the case of the space , we say that the problem (27) is well posed in if the following two conditions are satisfied.(1)For any and , there exists the unique solution in of problem (27). This means that an additive and homogeneous operator is defined which acts from to and gives the solution of (27) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs (,, and , equipped with the norm If is a solution in of problem (27), then it is a solution in of this problem. Hence, by (13), we get the following representation for the solution of problem (27): Using (27) and formula (34), we get where The main result of the present paper is the following theorem on well-posedness of (27) in the spaces .

Theorem 9. Suppose . Suppose that the assumptions (9) and (10) hold and . If , then for the solution in of the nonlocal boundary value problem (27) the coercive inequality holds, where does not depend on , and .

Proof. The proof of Theorem 9 is based on Theorem 5 and the following estimate: Let us estimate for any in , separately. We start with . Applying the inequality (30), we get Therefore, we will estimate We have that where By the definition of spaces and using estimates (9), we obtain for any . We will consider separately the cases and . If , then If , then Combining these estimates, we get for any . Therefore, By the definition of spaces and using estimates (9), (18), (19), and (28), we obtain for any . We will consider separately the cases and . If , then If , then Combining these estimates, we get for any . Therefore, Applying the triangle inequality and estimates (47) and (52), we obtain From this estimate and estimate (39), it follows the following estimate Now, we estimate . Applying inequality (30), we get Using estimates (9), (18), (19), and (28), we obtain for all . We will prove that for any . If , then If , then Combining these estimates, we get (57). Applying estimates (55), (56), and (57), we obtain Applying inequality (30), we get Finally, we estimate . Applying inequality (30), we get Using estimates (9), (18), (19), and (28), we obtain for all . Applying estimates (63) and (62), we get Combining the estimates (54), (60), (61), and (64), we get estimate (38). Theorem 9 is proved.

Furthermore, the method of proof of Theorem 6 and scheme of proof of Theorem 9 enable us to establish the following theorem on well-posedness of (27) in spaces .

Theorem 10. Suppose . Suppose that the assumptions (9) and (10) hold and . If , then for the solution in of the nonlocal boundary value problem (27) the coercive inequality holds, where does not depend on , and .

Note that the spaces of smooth functions , in which coercive solvability has been established, depend on the parameters , and . However, the constants in the coercive inequalities depend only on . Hence, we can choose the parameters and freely, which increases the number of functional spaces in which problem (27) is well posed. In particular, Theorems 9 and 10 imply theorems on well-posedness of the nonlocal boundary value problem (27) in which is established in the paper [41].

3. Applications

First, we consider the nonlocal boundary value problem for the parabolic equation where ,  , and    are given sufficiently smooth functions and and they satisfy every compatibility conditions which guarantee that the problem (66) has a smooth solution . Here,   is a sufficiently large number.

We introduce the Banach spaces of all continuous functions satisfying a Hölder condition for which the following norms are finite: where is the space of the all continuous functions defined on with the usual norm It is known that the differential expression defines a strongly positive operator acting in with the domain and satisfying nonlocal boundary conditions for any .

Therefore, we can replace the nonlocal boundary value problem (66) by the abstract nonlocal boundary value problem (27) in a Banach space with a strongly positive operator defined by formula (69).

Theorem 11. For the solution of nonlocal boundary value problem (66), the following coercive inequality: holds, where does not depend on , ,  ,  .

The proof of Theorem 11 is based on abstract Theorem 9 and on the following theorem on the structure of fractional spaces .

Theorem 12. for all and [42].

Second, on the range , we consider the nonlocal boundary value problem for the 2th order multidimensional parabolic equation where   and , and    are given sufficiently smooth functions. Here,   is a sufficiently large number.

Let us consider a differential operator with constant coefficients of the form acting on functions defined on the entire space  . Here, is a vector with nonnegative integer components,  . If is an infinitely differentiable function that decays at infinity together with all its derivatives, then, by means of the Fourier transformation one establishes the equality Here, the Fourier transform operator is defined by the rule The function is called the symbol of the operator and is given by We will assume that the symbol of the differential operator of the form acting on functions defined on the space , satisfies the inequalities for and for any . Assume that the all compatibility conditions hold which guarantees that the problem (71) has a unique smooth solution . This allows us to reduce the nonlocal boundary value problem (71) by the abstract nonlocal boundary value problem (27) in a Banach space of all continuous bounded functions defined on satisfying a Hölder condition with the indicator with a strongly positive operator defined by (77).

Theorem 13. For the solution of the nonlocal boundary value problem (71), the following coercivity inequality: holds, where does not depend on  ,  ,  ,   .

The proof of Theorem 13 is based on abstract Theorem 9, on the coercivity inequality for an elliptic operator in , and on the following theorem on the structure of fractional spaces .

Theorem 14. for all   and ,   [7].

4. Conclusion

In the present study, the well-posedness of the nonlocal boundary value parabolic problem with dependent coefficients in Hölder spaces with a weight is established. In practice, new Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are obtained. Moreover, applying the result of the monograph [14], the high order of accuracy single-step difference schemes for the numerical solution of the nonlocal boundary value parabolic problem with dependent coefficients can be presented. Of course, the coercive stability estimates for the solution of these difference schemes have been established without any assumptions about the grid steps.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors would like to thank Professor Pavel E. Sobolevskii, for his helpful suggestions. They  would like also to thank the referees for their efforts to improve the paper.

References

  1. M. Dehghan, “On the numerical solution of the diffusion equation with a nonlocal boundary condition,” Mathematical Problems in Engineering, vol. 2003, no. 1-2, pp. 81–92, 2003. View at Scopus
  2. J. R. Cannon, S. Perez Esteva, and J. van der Hoek, “Galerkin procedure for the diffiusion equation subject to the specifcation of mass,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 499–515, 1987. View at Scopus
  3. N. Gordeziani, P. Natalini, and P. E. Ricci, “Finite-difference methods for solution of nonlocal boundary value problems,” Computers and Mathematics with Applications, vol. 50, no. 8-9, pp. 1333–1344, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Dautray and J. L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, vol. 8, Masson, Paris, France, 1988.
  5. A. Ashyralyev and Y. Ozdemir, “Stability of difference schemes for hyperbolic-parabolic equations,” Computers and Mathematics with Applications, vol. 50, no. 8-9, pp. 1443–1476, 2005. View at Publisher · View at Google Scholar · View at Scopus
  6. A. Ashyralyev, A. Hanalyev, and P. E. Sobolevskii, “Coercive solvability of nonlocal boundary value problem for parabolic equations,” Abstract and Applied Analysis, vol. 6, no. 1, pp. 53–61, 2001.
  7. A. Ashyralyev and P. E. Sobolevskiĭ, Well-Posedness of Parabolic Difference Equations, vol. 69 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  8. R. Rautmann, “H2,r-convergent approximation schemes to the Navier-Stokes equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 30, no. 4, pp. 1915–1926, 1997.
  9. A. O. Ashyralyev and P. E. Sobolevskiĭ, “The theory of interpolation of linear operators and the stability of difference schemes,” Doklady Akademii Nauk SSSR, vol. 275, no. 6, pp. 1289–1291, 1984 (Russian). View at MathSciNet
  10. A. Ashyralyev, I. Karatay, and P. E. Sobolevskii, “On well-posedness of the nonlocal boundary value problem for parabolic difference equations,” Discrete Dynamics in Nature and Society, no. 2, pp. 273–286, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Ashyralyev, “Nonlocal boundary-value problems for abstract parabolic equations: well-posedness in Bochner spaces,” Journal of Evolution Equations, vol. 6, no. 1, pp. 1–28, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. A. Ashyralyev and P. E. Sobolevskiĭ, “Difference schemes of a high order of accuracy for parabolic equations with variable coefficients,” Akademiya Nauk Ukrainskoĭ SSR, no. 6, pp. 3–7, 1988 (Russian). View at MathSciNet
  13. A. Ashyralyev and P. E. Sobolevskii, “Well-posed solvability of the Cauchy problem for difference equations of parabolic type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24, no. 2, pp. 257–264, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. Ashyralyev, S. Piskarev, and L. Weis, “On well-posedness of difference schemes for abstract parabolic equations in Lp([0,1],E) spaces,” Numerical Functional Analysis and Optimization, vol. 23, no. 7-8, pp. 669–693, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. I. P. Gavrilyuk and V. L. Makarov, “Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces,” SIAM Journal on Numerical Analysis, vol. 43, no. 5, pp. 2144–2171, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. D. Gordeziani, H. Meladze, and G. Avalishvili, “On one class of nonlocal in time problems for first-order evolution equations,” Zhurnal Obchyslyuval'no Ta Prykladno I Matematyky, vol. 88, no. 1, pp. 66–78, 2003.
  18. A. Boucherif, “Second-order boundary value problems with integral boundary conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 364–371, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. A. Boucherif and H. Akca, “Nonlocal Cauchy problems for semilinear evolution equations,” Dynamic Systems and Applications, vol. 11, no. 3, pp. 415–420, 2002. View at Zentralblatt MATH · View at MathSciNet
  20. L. Byszewski and H. Akca, “Existence of solutions of a semilinear functional-differential evolution nonlocal problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 1, pp. 65–72, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. Boucherif and R. Precup, “Semilinear evolution equations with nonlocal initial conditions,” Dynamic Systems and Applications, vol. 16, no. 3, pp. 507–516, 2007. View at MathSciNet
  22. A. Ashyralyev and D. Agirseven, “On convergence of difference schemes for delay parabolic equations,” Computers & Mathematics with Applications, vol. 66, no. 7, pp. 1232–1244, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  23. A. Sahmurova and V. B. Shakhmurov, “Parabolic problems with parameter occuring in environmental engineering,” AIP Conference Proceedings, vol. 1470, no. 1, pp. 39–41, 2012.
  24. M. Ashyraliyev, “A note on the stability of the integral-differential equation of the parabolic type in a Banach space,” Abstract and Applied Analysis, vol. 2012, Article ID 178084, 18 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  25. M. Dehghan, “A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications,” Numerical Methods for Partial Differential Equations, vol. 22, no. 1, pp. 220–257, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. Y. Wang and S. Zheng, “The existence and behavior of solutions for nonlocal boundary problems,” Boundary Value Problems, vol. 2009, Article ID 484879, 17 pages, 2009. View at Zentralblatt MATH · View at MathSciNet
  27. R. P. Agarwal and V. B. Shakhmurov, “Multipoint problems for degenerate abstract differential equations,” Acta Mathematica Hungarica, vol. 123, no. 1-2, pp. 65–89, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. D. Agirseven, “Approximate solutions of delay parabolic equations with the Dirichlet condition,” Abstract and Applied Analysis, vol. 2012, Article ID 682752, 31 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  29. A. S. Erdogan and H. Uygun, “A note on the inverse problem for a fractional parabolic equation,” Abstract and Applied Analysis, vol. 2012, Article ID 276080, 26 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  30. A. S. Erdogan, “A note on the right-hand side identification problem arising in biofluid mechanics,” Abstract and Applied Analysis, vol. 2012, Article ID 548508, 25 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  31. R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Maximal regular boundary value problems in Banach-valued weighted space,” Boundary Value Problems, vol. 1, pp. 9–42, 2005. View at Zentralblatt MATH · View at MathSciNet
  32. J. I. Ramos, “Linearly-implicit, approximate factorization, exponential methods for multi-dimensional reaction-diffusion equations,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 1609–1633, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  33. A. V. Gulin and V. A. Morozova, “On the stability of a nonlocal difference boundary value problem,” Differential Equation, vol. 39, no. 7, pp. 912–917, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  35. Y. G. Wang and M. Oberguggenberger, “Nonlinear parabolic equations with regularized derivatives,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 644–658, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. W.-J. Beyn and B. M. Garay, “Estimates of variable stepsize Runge-Kutta methods for sectorial evolution equations with nonsmooth data,” Applied Numerical Mathematics, vol. 41, no. 3, pp. 369–400, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. A. M. Selitskii, “The space of initial data for the second boundary-value problem for parabolic differential-difference equation,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 1, pp. 34–41, 2013.
  38. A. M. Selitskii, “The space of initial data for the Robin boundary-value problem for parabolic differential-difference equation,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 2, pp. 91–97, 2013.
  39. J. M. Rassias and E. T. Karimov, “Boundary-value problems with non-local condition for degenerate parabolic equations,” Contemporary Analysis and Applied Mathematics, vol. 1, no. 1, pp. 42–48, 2013.
  40. A. Ashyralyev, “Well-posedness of fractional parabolic equations,” Boundary Value Problems, Article ID 31, pp. 1–18, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  41. A. Ashyralyev and A. Hanalyev, “Coercive solvability of parabolic differential equations with dependent operators,” TWMS Journal of Applied and Engineering Mathematics, vol. 2, no. 1, pp. 75–93, 2012. View at Zentralblatt MATH · View at MathSciNet
  42. A. Ashyralyev, “Fractional spaces generated by the positivite differential and difference operator in a Banach space,” in Proceedings of the Conference Mathematical Methods and Engineering, K. Taş, J. A. Tenreiro Machado, and D. Baleanu, Eds., pp. 13–22, Springer, 2007.