Existence and multiplicity of solutions to an ab- stract Cauchy problem for an evolution equation involving fractional dissipation term of Caputo type
DOI:
https://doi.org/10.24193/subbmath.2025.4.09Keywords:
Existence, multiplicity, evolution equation, fractional derivative, fixed point theoremAbstract
The aim of this work is to investigate the existence and multiplicity of solutions to a nonhomogenious quasi-linear second order evolution equation involving a fractional dissipation term of Caputo type in abstract framework. Some criteria on the existence of at least one or two solutions are obtained by using some well-known fixed-point theorems for the sum of two operators. An example is presented to validate our analysis.
Mathematics Subject Classification (2010): 35J15, 35J25, 93B24, 26A33.
Received 05 May 2025; Accepted 21 July 2025.
References
Afrouzi, G. A., Chung, N. T., Shakeri, S., Existence of positive solutions for Kirchhoff type equations, Electron. J. Differential Equations., 180 (2013), no. 8.
Ames, W. F., Nonlinear partial differential equations in engineering, Academic press, 1965.
Blanc, E., Chiavassa, G., Lombard, B., Biot-JKD model: Simulation of 1D transient poroelastic waves with fractional derivatives, J. Comput. Phys., 237 (2013), 1-20.
Boudaoud, A., Benaissa, A., Stabilization of a Wave Equation with a General Internal Control of Diffusive Type, Discontin, Nonlinearity Complex, 12 (2023), no. 4, 879-891.
Calamai, A., Infante, G., On the solvability of parameter-dependent elliptic functional BVPs on annular-like domains, Discrete Contin. Dyn. Syst. Ser. B., 30 (2025), no. 11, 4287-4295.
Choi, J. U., Maccamy, R. C., Fractional order Volterra equations with applications to elasticity, J. Math. Anal. Appl., 139 (1989), 448-464.
De Brito, E. H., Hale, J., The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability, Appl. Anal., 13 (1982), no. 3, 219-33.
Djebali, S., Mebarki, K., Fixed point index theory for perturbation of expansive mappings by k-set contractions, Topol. Methods Nonlinear Anal., 54 (2019), no. 2, 613-640.
Figueiredo, G. M., Rodrigo, C. M., Santos Júnior, J. R., Study of a nonlinear Kirchhoff equation with non-homogeneous material, J. Math. Anal. Appl., 416 (2014), 597-608.
Georgiev, S., Kheloufi, A., Mebarki, K., Existence of classical solutions for a class of impulsive Hamilton-Jacobi equations, Palest. J. Math., 13 (2024), no. 4, 1084-1097.
Ghisi, M., Global solutions for dissipative Kirchhoff strings with m(r) = r^p, (p < 1), J. Math. Anal. Appl., 250 (2000), no. 1, 86-97.
Ghisi, M., Some remarks on global solutions to nonlinear dissipative mildly degenerate Kirchhoff strings, Rend. Semin. Mat. Univ. Padova., 106 (2001), 185-205.
Hallouz, A., The Klein-Gordon Equation with Dynamic Boundaries Dissipation of Fractional Derivative Type, An. Univ. Oradea Fasc. Mat., 21 (2024), No. 2, 35-53.
Hataff, K., Yousfi, N., A generalized HBV model with diffusion and two delays, Comput. Math. Appl., 69 (2015), no. 2, 31-40.
Houari, N., Haddouchi, F., Existence and nonexistence results for fifth-order multi-point boundary value problems involving integral boundary condition, Filomat, 37 (2023), no. 19, 6463-6486.
Ikehata, R., Okazawa, N., Yosida approximation and nonlinear hyperbolic equation, Nonlinear Anal., 15 (1990), no. 5, 479-495.
Ikehata, R., On solutions to some quasilinear hyperbolic equations with nonlinear inhomogeneous terms, Nonlinear Anal., 17 (1991), no. 2, 181-203.
Kato, T., Linear evolution equations of hyperbolic type, J. Fac. Sci. Univ. Tokyo Sect. I., 17 (1970), no. 6, 241-258.
Kato, T., Linear evolution equations of hyperbolic type II, J. math. Soc. Japan., 25 (1973), no. 4, 648-666.
Kato, T., Quasi-linear equations of evolution, with applications to partial differential equations. Spectral Theory and Differential Equations, Lecture Notes in Mathematics, 448, Springer, Berlin, Heidelberg, 1975.
Khemmar, K., Mebarki, K., Georgiev, S. G., Existence of solutions for a class of boundary value problems for weighted p(t)-Laplacian impulsive systems, Filomat, 38 (2024), no. 21, 7563-7577.
Kim, D., Hong, K. S., Jung, I. H., Global Existence and Energy Decay Rates for a Kirchhoff-Type Wave Equation with Nonlinear Dissipation, The Scientific World Journal, 1 (2014), no. 716740, 10.
Mbodje, B., Wave energy decay under fractional derivative controls, IMA Journal of Mathematical Control and Information, 23 (2006), no. 2, 237-257.
Mbodje, B., Montseny, G., Boundary fractional derivative control of the wave equation, IEEE Trans. Automat. Control., 40 (1995), 368-382.
Meradjah, I., Louhibi, N., Benaissa, A., Stability of a Schrödinger Equation with Internal Fractional Damping, An. Univ. Craiova Ser. Mat. Inform., 50 (2023), no 2, 427-441.
Precup, R., Rodríguez-López, J., Positive radial solutions for Dirichlet problems via a Harnack-type inequality, Math. Meth. Appl. Sci., 46 (2023), 2972-2985.
Precup, R., Stan, A., Stationnary Kirchhoff equations and systems with reaction terms, AIMS Mathematics, 7 (2022), no 8, 15258–15281.
Yan, B., O'Regan, D., Agarwal, R. P., Existence of solutions for Kirchhoff-type problems via the method of upper and lower solutions, Electron. J. Differential. Equations., 2019 (2019), no. 54, 1–19.
Zahar, S., Georgiev, S. G., Mebarki, K., Multiple solutions for a class of BVPs for second order ODEs via an extension of Leray-Schauder boundary condition, Nonlinear Studies., 30 (2023), no. 1, 113–125.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution 4.0 International License.
