A Fixed-Point Theoretic Approach to the Unique Solvability of a High-Order Nonlinear Fractional Boundary Value Problem with Integral Conditions

Document Type : Research Article

Author

Department of Applied Mathematics, Imam Khomeini International University, Qazvin, 34148-96818, Iran

10.22128/ansne.2025.3027.1148

Abstract

This work is devoted to a rigorous analysis of the existence and uniqueness of solutions for a class of high-order nonlinear differential equations of fractional order. The considered problem is defined by a Caputo fractional derivative and is augmented by a set of nonlocal boundary constraints. A key feature of these constraints is an integral condition that couples the behavior of the solution across its entire spatial domain, reflecting a global dependency. Our primary analytical strategy is to recast the differential problem as a fixed-point equation for an equivalent integral operator. This is accomplished by first methodically constructing the Green's function associated with the corresponding linear problem. With the integral operator established, the existence of a unique solution for the full nonlinear problem is then proven by leveraging the power of the Banach contraction mapping principle. To demonstrate the practical relevance and applicability of our theoretical framework, a detailed illustrative example is presented and analyzed.

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[1] M. Derakhshan and Y. Ordokhani, Efficient numerical approximation of distributed-order fractional PDEs using gl1-2 time discretization and second-order Riesz operators, Analytical and Numerical Solutions for Nonlinear Equations, 9(2), 203–220, (2025).
[2] R. L. Magin, Fractional calculus in bioengineering, vol. 2. Begell House Redding, 2006.
[3] E. Shivanian, Z. Hajimohammadi, F. Baharifard, K. Parand, and R. Kazemi, A novel learning approach for different profile shapes of convecting–radiating fins based on shifted gegenbauer lssvm, New Mathematics and Natural Computation, 19(01), 195–215, (2023).
[4] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, vol. 204. elsevier, 2006.
[5] J. Sabatier, O. P. Agrawal, and J. T. Machado, Advances in fractional calculus, vol. 4. Springer, 2007.
[6] S. Mirzaei and A. Shokri, A pseudo-spectral approach to solving the fractional cable equation using lagrange polynomials, Analytical and Numerical Solutions for Nonlinear Equations, 9(1), 20–33, (2025).
[7] Z. Gilani, M. Alipour, and S. Rivaz, System of volterra fredholm integro-fractional differential equations: Application of fibonacci polynomials, Analytical and Numerical Solutions for Nonlinear Equations, 9(1), 89–101, (2025).
[8] N. Hosseinzadeh, E. Shivanian, M. Z. Fairooz, and T. G. Chegini, A robust rbf-fd technique combined with polynomial enhancements for valuing european options in jump-diffusion frameworks: N. hosseinzadeh et al., International Journal of Dynamics and Control, 13(6), 212, (2025).
[9] E. Shivanian, A. Jafarabadi, T. G. Chegini, and A. Dinmohammadi, Analysis of a time-dependent source function for the heat equation with nonlocal boundary conditions through a local meshless procedure, Computational and Applied Mathematics, 44(6), 282, (2025).
[10] A. Granas and J. Dugundji, Elementary fixed point theorems, in Fixed Point Theory, 9–84, Springer, (2003).
[11] A. Cabada and Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions, 228, 251–257, (2014).
[12] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, Journal of Mathematical Analysis and Applications, 389(1), 403–411, (2012).
[13] G. A. Anastassiou, Fractional differentiation inequalities, vol. 68. Springer, 2009. 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 9, Issue 2
September 2025
Pages 253-260

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History

  • Receive Date: 24 August 2025
  • Accept Date: 06 October 2025
  • Publish Date: 13 October 2025

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