Existence of weak solutions for non-local fractional problems via Morse theory

Previous Article
Two-stage model of carcinogenic mutations with the influence of delays
DCDS-B Home
This Issue
Next Article
Modeling DNA thermal denaturation at the mesoscopic level

October 2014, 19(8): 2483-2499. doi: 10.3934/dcdsb.2014.19.2483

Existence of weak solutions for non-local fractional problems via Morse theory

Massimiliano Ferrara ^1, , Giovanni Molica Bisci ^2, and Binlin Zhang ^3,

1.	University of Reggio Calabria and CRIOS University Bocconi of Milan, Via dei Bianchi presso Palazzo Zani, 89127 Reggio Calabria, Italy
2.	Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio Calabria
3.	Department of Mathematics, Heilongjiang Institute of Technology, 150050 Harbin, China

Received November 2013 Revised February 2014 Published August 2014

Abstract
Related Papers

In this paper, we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. Non-trivial solutions are obtained by computing the critical groups and Morse theory. Our results extend some classical theorems for semilinear elliptic equations to the non-local fractional setting.

Keywords: Fractional Laplacian, Morse theory., integro-differential operator, critical groups.

Mathematics Subject Classification: 35J60, 35J91, 58E0.

Citation: Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483

References:

[1]	C. O. Alves and S. B. Liu, On superlinear $p(x)$-Laplacian equations in $\mathbbR^N$, Nonlinear Anal., 73 (2010), 2566-2579. doi: 10.1016/j.na.2010.06.033.
[2]	B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.
[3]	T. Bartsch, and S. J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.
[4]	K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.
[5]	K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198. doi: 10.1007/s00220-006-0178-y.
[6]	D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal., (2013), Art. ID 240863, 10 pp.
[7]	D. Bors, Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian, Scientific World Journal, (2014), Art. ID 920537, 10 pp. doi: 10.1155/2014/920537.
[8]	C. Brändle, E. Colorado, A.de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[9]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[10]	L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.} doi: 10.4171/JEMS/226.
[11]	L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.
[12]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[13]	A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662. doi: 10.3934/cpaa.2011.10.1645.
[14]	K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.
[15]	S. Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.
[16]	M. del Mar González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differential Equations, 36 (2009), 173-210. doi: 10.1007/s00526-009-0225-6.
[17]	F. Fang, and S. B. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math. Anal. Appl., 351 (2009), 138-146. doi: 10.1016/j.jmaa.2008.09.064.
[18]	A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwendungen, 32 (2013), 411-431. doi: 10.4171/ZAA/1492.
[19]	D. Idczak and R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in $\mathbbR^n$, Fractional Calculus & Applied Analysis, 14 (2011), 538-553. doi: 10.2478/s13540-011-0033-5.
[20]	D. Idczak and R. Kamocki, Fractional differential repetitive process with Riemann-Liouville and Caputo derivatives, Multidim. Syst. Sign Process, (2013). doi: 10.1007/s11045-013-0249-0.
[21]	D. Idczak, R. Kamocki and M. Majewski, Fractional continuous Roesser model with Reimann-Liouville derivative, 8th Int. workshop on multidimensional Systems, (2013), 33-38.
[22]	D. Idczak, R. Kamocki and M. Majewski, On a fractional continuous counterpart of Fornasini-Marchesini model, 8th Int. Workshop on Multidimensional Systems, (2013), 45-49.
[23]	D. Idczak and M. Majewski, Fractional fundamental lemma of order $\alpha\in (n-\frac{1}{2}, n)$ with $n\in\mathbbN, n\geq2$, Dynamic System and Applications, 12 (2012), 251-268.
[24]	D. Idczak and M. Majewski, Compactness of fractional embeddings for boundary value problems, IEEE, 7 (2013), 599-603.
[25]	D. Idczak, A. Skowron and S. Walczak, Sensitivity of a fractional integrodifferential Cauchy problem of Volterra type, Abstract and Applied Analysis, 2013 (2013), 1-8.
[26]	D. Idczak, and S. Walczak, A fractional imbedding theorem, Fractional Calculus & Applied Analysis, 15 (2012), 418-425. doi: 10.2478/s13540-012-0030-3.
[27]	D. Idczak and S. Walczak, Compactness of fractional imbeddings, IEEE, 2 (2012), 585-588. doi: 10.1109/MMAR.2012.6347820.
[28]	D. Idczak and S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, Journal of Function Spaces and Applications, 2013 (2013), 1-15.
[29]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.
[30]	K. Kaleta and T. Kulczycki, Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians, Potential Anal., 33 (2010), 313-339. doi: 10.1007/s11118-010-9170-4.
[31]	R. Kamocki and M. Majewski, On a fractional Dirichlet problem, IEEE, 2 (2012), 60-63. doi: 10.1109/MMAR.2012.6347911.
[32]	A. Kristály, V. Rădulescu and Cs. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.
[33]	S. B. Liu, On ground states of superlinear $p$-Laplacian equations in $\mathbbR^N$, J. Math. Anal. Appl., 361 (2010), 48-58. doi: 10.1016/j.jmaa.2009.09.016.
[34]	S. B. Liu, On superlinear problems without Ambrosetti-Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. doi: 10.1016/j.na.2010.04.016.
[35]	S. B. Liu, Existence of solutions to a superlinear $p$-Laplacian equation, Electron J. Differential Equations, 66 (2001), 1-6.
[36]	J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.
[37]	Q. S. Liu and J. B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601.
[38]	G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett., 27 (2014), 53-58. doi: 10.1016/j.aml.2013.07.011.
[39]	G. Molica Bisci, Sequences of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 1-13.
[40]	G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14 (2014), 591-601.
[41]	G. Molica Bisci and D. Repovs, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176.
[42]	G. Molica Bisci and R. Servadei, A bifurcation result for non-local fractional equations,, to appear in Analysis and Applications., (). doi: 10.1142/S0219530514500067.
[43]	K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operatiors, Mathematical Surveys and Monographs, 161, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/161.
[44]	X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508. doi: 10.1016/j.crma.2012.05.011.
[45]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.
[46]	S. Serfaty and J. Luis Vazquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differential Equations, 49 (2014), 1091-1120. doi: 10.1007/s00526-013-0613-9.
[47]	R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.
[48]	R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.
[49]	R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

show all references

References:

[1]	C. O. Alves and S. B. Liu, On superlinear $p(x)$-Laplacian equations in $\mathbbR^N$, Nonlinear Anal., 73 (2010), 2566-2579. doi: 10.1016/j.na.2010.06.033.
[2]	B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.
[3]	T. Bartsch, and S. J. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441. doi: 10.1016/0362-546X(95)00167-T.
[4]	K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.
[5]	K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198. doi: 10.1007/s00220-006-0178-y.
[6]	D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal., (2013), Art. ID 240863, 10 pp.
[7]	D. Bors, Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian, Scientific World Journal, (2014), Art. ID 920537, 10 pp. doi: 10.1155/2014/920537.
[8]	C. Brändle, E. Colorado, A.de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[9]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.
[10]	L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.} doi: 10.4171/JEMS/226.
[11]	L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.
[12]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[13]	A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662. doi: 10.3934/cpaa.2011.10.1645.
[14]	K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.
[15]	S. Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.
[16]	M. del Mar González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differential Equations, 36 (2009), 173-210. doi: 10.1007/s00526-009-0225-6.
[17]	F. Fang, and S. B. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math. Anal. Appl., 351 (2009), 138-146. doi: 10.1016/j.jmaa.2008.09.064.
[18]	A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwendungen, 32 (2013), 411-431. doi: 10.4171/ZAA/1492.
[19]	D. Idczak and R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in $\mathbbR^n$, Fractional Calculus & Applied Analysis, 14 (2011), 538-553. doi: 10.2478/s13540-011-0033-5.
[20]	D. Idczak and R. Kamocki, Fractional differential repetitive process with Riemann-Liouville and Caputo derivatives, Multidim. Syst. Sign Process, (2013). doi: 10.1007/s11045-013-0249-0.
[21]	D. Idczak, R. Kamocki and M. Majewski, Fractional continuous Roesser model with Reimann-Liouville derivative, 8th Int. workshop on multidimensional Systems, (2013), 33-38.
[22]	D. Idczak, R. Kamocki and M. Majewski, On a fractional continuous counterpart of Fornasini-Marchesini model, 8th Int. Workshop on Multidimensional Systems, (2013), 45-49.
[23]	D. Idczak and M. Majewski, Fractional fundamental lemma of order $\alpha\in (n-\frac{1}{2}, n)$ with $n\in\mathbbN, n\geq2$, Dynamic System and Applications, 12 (2012), 251-268.
[24]	D. Idczak and M. Majewski, Compactness of fractional embeddings for boundary value problems, IEEE, 7 (2013), 599-603.
[25]	D. Idczak, A. Skowron and S. Walczak, Sensitivity of a fractional integrodifferential Cauchy problem of Volterra type, Abstract and Applied Analysis, 2013 (2013), 1-8.
[26]	D. Idczak, and S. Walczak, A fractional imbedding theorem, Fractional Calculus & Applied Analysis, 15 (2012), 418-425. doi: 10.2478/s13540-012-0030-3.
[27]	D. Idczak and S. Walczak, Compactness of fractional imbeddings, IEEE, 2 (2012), 585-588. doi: 10.1109/MMAR.2012.6347820.
[28]	D. Idczak and S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, Journal of Function Spaces and Applications, 2013 (2013), 1-15.
[29]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.
[30]	K. Kaleta and T. Kulczycki, Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians, Potential Anal., 33 (2010), 313-339. doi: 10.1007/s11118-010-9170-4.
[31]	R. Kamocki and M. Majewski, On a fractional Dirichlet problem, IEEE, 2 (2012), 60-63. doi: 10.1109/MMAR.2012.6347911.
[32]	A. Kristály, V. Rădulescu and Cs. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.
[33]	S. B. Liu, On ground states of superlinear $p$-Laplacian equations in $\mathbbR^N$, J. Math. Anal. Appl., 361 (2010), 48-58. doi: 10.1016/j.jmaa.2009.09.016.
[34]	S. B. Liu, On superlinear problems without Ambrosetti-Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. doi: 10.1016/j.na.2010.04.016.
[35]	S. B. Liu, Existence of solutions to a superlinear $p$-Laplacian equation, Electron J. Differential Equations, 66 (2001), 1-6.
[36]	J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.
[37]	Q. S. Liu and J. B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601.
[38]	G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett., 27 (2014), 53-58. doi: 10.1016/j.aml.2013.07.011.
[39]	G. Molica Bisci, Sequences of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 1-13.
[40]	G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14 (2014), 591-601.
[41]	G. Molica Bisci and D. Repovs, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176.
[42]	G. Molica Bisci and R. Servadei, A bifurcation result for non-local fractional equations,, to appear in Analysis and Applications., (). doi: 10.1142/S0219530514500067.
[43]	K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operatiors, Mathematical Surveys and Monographs, 161, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/161.
[44]	X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508. doi: 10.1016/j.crma.2012.05.011.
[45]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.
[46]	S. Serfaty and J. Luis Vazquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differential Equations, 49 (2014), 1091-1120. doi: 10.1007/s00526-013-0613-9.
[47]	R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.
[48]	R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.
[49]	R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[1]	Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053
[2]	Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[3]	Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044
[4]	Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249
[5]	Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021021
[6]	Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277
[7]	Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064
[8]	Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053
[9]	Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021225
[10]	Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030
[11]	Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100
[12]	Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations and Control Theory, 2022, 11 (2) : 605-619. doi: 10.3934/eect.2021016
[13]	Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[14]	Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
[15]	Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026
[16]	Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021051
[17]	Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022025
[18]	Ichrak Bouacida, Mourad Kerboua, Sami Segni. Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022028
[19]	Jean-Michel Roquejoffre, Juan-Luis Vázquez. Ignition and propagation in an integro-differential model for spherical flames. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 379-387. doi: 10.3934/dcdsb.2002.2.379
[20]	Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Massimiliano Ferrara Giovanni Molica Bisci Binlin Zhang
on Google Scholar
Massimiliano Ferrara Giovanni Molica Bisci Binlin Zhang

[Back to Top]