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We analyzed the triple Higgs boson self-coupling at future $e^{+}e^{-}$colliders energies, with the reactions $e^{+}e^{-}\to b \bar b HH, t \bar tHH$. We evaluate the total cross-sections for both $b\bar bHH$ and $t\bar tHH$,and calculate the total number of events considering the complete set ofFeynman diagrams at tree-level. We vary the triple coupling$\kappa\lambda_{3H}$ within the range $\kappa=-1$ and +2. The numericalcomputation is done for the energies expected to be available at a possibleFuture Linear $e^{+}e^{-}$ Collider with a center-of-mass energy $800, 1000,1500$ $GeV$ and a luminosity 1000 $fb^{-1}$. Our analysis is also extended to acenter-of-mass energy 3 $TeV$ and luminosities of 1000 $fb^{-1}$ and 5000$fb^{-1}$. We found that for the process $e^{+}e^{-}\to b \bar b HH$, thecomplete calculation differs only by 3% from the approximate calculation$e^{+}e^{-}\to ZHH(Z\to b\bar b)$, while for the process $e^{+}e^{-}\to t \bartHH$, the expected number of events, considering the decay products of both $t$and $H$, is not enough to obtain an accurate determination of the triple Higgsboson self-coupling.

The Triple Higgs Boson Self-Coupling at Future Linear e+e- Colliders Energies: ILC and CLIC