Existence of Weak Solutions to a Kind of System of Fractional Semi-Linear Fredholm-Volterra Boundary Value Problem
Subject Areas : International Journal of Industrial Mathematics
1 - Department of Mathematics, Imam Khomeini International
University, Qazvin, Iran.
Keywords: Variational method, Critical point theory, System of fractional semi-linear Fredholm-Volterra integro-differential equations, Weak solution, Dirichlet condition,
Abstract :
This article is devoted to study the weak solutions of a class of nonlinear system of fractional boundary value problems including both Volterra and Fredholm linear integral terms. This system of fractional semi-linear Fredholm-Volterra integro-differential equations does have a gradient of a nonlinear source term as well. We apply the critical point theory and the variational structure to prove the existence of at least three distinct weak solutions to the system. Furthermore, it is presented an example to verify the legitimacy and applicability of the theory.
[1] S. Abbasbandy, E. Shivanian, Application of the variational iteration method for system of nonlinear volterra’s integro-differential equations, Mathematical and computational applications 14 (2009) 147-158.
[2] S. Abbasbandy, E. Shivanian, Application of variational iteration method for nth-order integro-differential equations, Zeitschrift f¨ur Naturforschung A 64 (2009) 439-444.
[3] S. Abbasbandy, E. Shivanian, Series solution of the system of integro-differential equations, Zeitschrift f¨ur Naturforschung A 64 (2009) 811-818.
[4] R. P. Agarwal, M. Benchohra, S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Applicandae Mathematicae 109 (2010) 973-1033.
[5] B. Ahmad, J. J. Nieto, Riemann-liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Boundary Value Problems 1 (2011) 36-46.
[6] M. Aslefallah, E. Shivanian. Nonlinear fractional integro-differential reaction-diffusion equation via radial basis functions, Eur Phys J Plus 130 (2015) 1-9.
[7] C. Bai, Infinitely many solutions for a perturbed nonlinear fractional boundary-value problem, Electronic Journal of Differential Equations 136 (2013) 1-12.
[8] K. Balachandran, J. Dauer, P. Balasubramaniam, Controllability of semilinear integrodi erential systems in banach spaces, Journal of Mathematical Systems, Estimation and Control 6 (1996) 1-10.
[9] K. Balachandran, J. J. Trujillo, The nonlocal cauchy problem for nonlinear fractional integrodifferential equations in banach spaces, Nonlinear Analysis: Theory, Methods & Applications 72 (2010) 4587-4593.
[10] D. Baleanu, R. Darzi, B. Agheli, New study of weakly singular kernel fractional fourthorder partial integro-differential equations based on the optimum q-homotopic analysis method, Journal of Computational and Applied Mathematics 320 (2017) 193-201.
[11] M. Benchohra, S. Hamani, S. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications 71 (2009) 2391-2396.
[12] G. Bonanno, S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Applicable Analysis 89 (2010) 1-10.
[13] J. Chen, X. Tang, Existence and multiplicity of solutions for some fractional boundary value problem via critical point theory, In Abstract and Applied Analysis, volume 2012, Hindawi Publishing Corporation, 2012.
[14] J. Chu, S. Heidarkhani, A. Salari, G. Caristi, Weak solutions and energy estimates for singular p-laplacian-type equations, Journal of Dynamical and Control Systems 11 (2017) 1-13.
[15] J. N. Corvellec, V. Motreanu, C. Saccon, Doubly resonant semilinear elliptic problems via nonsmooth critical point theory, Journal of Differential Equations 248 (2010) 2064-2091.
[16] A. Firouzjai, G. Afrouzi, S. Talebi, Existence results for kirchhoff type systems with singular nonlinearity, Opuscula Mathematica 38 (2018) 187-199.
[17] S. Heidarkhani, G. A. Afrouzi, S. Moradi, G. Caristi, Existence of multiple solutions for a perturbed discrete anisotropic equation, Journal of Difference Equations and Applications 22 (2017) 1-17.
[18] S. Heidarkhani, Y. Zhao, G. Caristi, G. A. Afrouzi, S. Moradi, Infinitely many solutions for perturbed impulsive fractional differential systems, Applicable Analysis 96 (2017) 1401-1424.
[19] S. Heidarkhani, Y. Zhou, G. Caristi, G. A. Afrouzi, S. Moradi. Existence results for fractional differential systems through a local minimization principle, Computers & Mathematics with Applications 2016.
[20] F. Jiao, Y. Zhou, Existence results for fractional boundary value problem via critical point theory, International Journal of Bifurcation and Chaos 22 (2012) 125-138.
[21] M. Kamrani, Convergence of galerkin method for the solution of stochastic fractional integro differential equations, OptikInternational Journal for Light and Electron Optics 127 (2016) 10049-10057.
[22] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, (2006).
[23] F. Li, Z. Liang, Q. Zhang. Existence of solutions to a class of nonlinear second order twopoint boundary value problems, Journal of mathematical analysis and applications 312 (2005) 357-373.
[24] Y. N. Li, H. R. Sun, Q. G. Zhang, Existence of solutions to fractional boundaryvalue problems with a parameter, Electronic Journal of Differential Equations 141 (2013) 1-12.
[25] X. Ma, C. Huang, Spectral collocation method for linear fractional integrodifferential equations, Applied Mathematical Modelling 38 (2014) 1434-1448.
[27] J. J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Analysis: Real World Applications 10 (2009) 680-690.
[28] S. Pashayi, M. Hashemi, S. Shahmorad, Analytical lie group approach for solving fractional integro-differential equations, Communications in Nonlinear Science and Numerical Simulation 51 (2017) 66-77.
[29] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, American Mathematical Soc. 65 (1986) 129-142.
[30] P. Rahimkhani, Y. Ordokhani, E. Babolian, Fractional-order bernoulli functions and their applications in solving fractional fredholemvolterra integro-differential equations, Applied Numerical Mathematics 122 (2017) 66-81.
[31] E. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Applied Mathematics and Computation 176 (2006) 1-6.
[32] E. Shivanian, Analysis of meshless local radial point interpolation (mlrpi) on a nonlinear partial integro-differential equation arising in population dynamics, Engineering Analysis with Boundary Elements 37 (2013) 1693-1702.
[33] H. R. Sun, Q. G. Zhang, Existence of solutions for a fractional boundary value problem via the mountain pass method and an iterative technique, Computers & Mathematics with Applications 64 (2012) 3436-3443.
[34] C. L. Tang, X. P. Wu, Some critical point theorems and their applications to periodic solution for second order hamiltonian systems, Journal of Differential Equations 248 (2010) 660-692.
[35] W. Xie, J. Xiao, Z. Luo, Existence of solutions for fractional boundary value problem with nonlinear derivative dependence, In Abstract and Applied Analysis, volume 2014, Hindawi Publishing Corporation, (2014).
[36] L. Zhang, B. Ahmad, G. Wang, R. P. Agarwal, M. Al-Yami, W. Shammakh, Nonlocal integrodifferential boundary value problem for nonlinear fractional differential equations on an unbounded domain, In Abstract and Applied Analysis, volume 2013, Hindawi Publishing Corporation, (2013).
[37] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Computers & Mathematics with Applications 59 (2010) 1300-1309.
[38] Y. Zhao, H. Chen, B. Qin, Multiple solutions for a coupled system of nonlinear fractional differential equations via variational methods, Applied Mathematics and Computation 257 (2015) 417-427.