On solutions of a system of hereditary and self-referred partial-differential equations. (English) Zbl 1202.35338
Summary: We present the local and global solutions of a system of hereditary and self-referred partial-differential equations. Namely, by the assumption on the Lipschitz continuity of the initial conditions \(u_0, v_0\), Theorem 1 states the existence of local solutions of the problem; furthermore, under the assumption that those initial conditions are non-negative, non-decreasing, bounded, and lower semi-continuous functions, Theorem 2 gives global solution which is also a non-negative, non-decreasing, bounded, and lower semi-continuous function (in variable \(x\) of even for any time \(t\)).
MSC:
| 35R10 | Partial functional-differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35B09 | Positive solutions to PDEs |
| 45K05 | Integro-partial differential equations |
Keywords:
evolution equations; hereditary and self-referred differential equations; nonlinear integral equations; recursive schemeReferences:
| [1] | Eder, E.: The functional-differential equation x ’(t)=x(x(t)). J. Differ. Equ. 54, 390–400 (1984) · doi:10.1016/0022-0396(84)90150-5 |
| [2] | Favini, A., Labbas, R., Maingot, S., Tanabe, H., Yagi, A.: A simplified approach in the study of elliptic differential equations in UMD spaces and new applications. Funkc. Ekvacioj 51, 165–187 (2008) · Zbl 1165.34035 · doi:10.1619/fesi.51.165 |
| [3] | Fichera, G.: Material with memory, cime, bressanone. Material with memory, CIME, Bressanone, pp. 110–169 (1977) |
| [4] | Fichera, G.: Analytic problems of hereditary phenomena. Arch. Ration. Mech. Anal. 70, 101–112 (1979) · Zbl 0425.73002 · doi:10.1007/BF00250347 |
| [5] | Fichera, G.: Sul principio della memoria evanescente. Rend. Semin. Mat. Univ. Padova 68, 245–259 (1983) · Zbl 0515.73003 |
| [6] | Fichera, G.: Analytic problems of hereditary phenomena. Material with memory, vol. 26, CIME, Bressanone (1997) · Zbl 0453.73012 |
| [7] | Israel, G.: On the contribution of Volterra and Lotka to the development of modern biomathematics. Hist. Philos. Life Sci. 10, 37–49 (1988) |
| [8] | Wang, X.P., Si, J.G.: Smooth solutions of a nonhomogeneous iterative functional differential equation with variable coefficients. J. Math. Anal. Appl. 226, 377–392 (1998) · Zbl 0917.34055 · doi:10.1006/jmaa.1998.6086 |
| [9] | Wang, X., Si, J.G., Cheng, S.S.: Analytic solutions of a functional differential equation with state derivative dependent delay. Aequ. Math. 1, 75–86 (1999) · Zbl 0959.34061 |
| [10] | Miranda, M., Pascali, E.: On a class of differential equations with self-reference. Rend. Mat., serie VII, vol. 25, pp. 155–164, Roma (2005) · Zbl 1117.47059 |
| [11] | Miranda, M., Pascali, E.: On a type of evolution of self-referred and hereditary phenomena. Aequ. Math. 71, 253–268 (2006) · Zbl 1100.47058 · doi:10.1007/s00010-005-2821-7 |
| [12] | Murray, J.D.: Mathematical Biology I: an Introduction. Springer, Berlin (2003) |
| [13] | Pascali, E.: Existence of solutions to a self-referred and hereditary system of differential equations. Electron. J. Diff. Eqs. 2006(7), 1–7 (2006) · Zbl 1095.47053 |
| [14] | Pelczar, A.: On a functional-differential equation (in a historical context). Opuscula Math. 19, 45–61 (1999) · Zbl 0996.35081 |
| [15] | Scudo, F.M.: Vito Volterra and theoretical ecology. Theor. Popul. Biol. 2, 1–23 (1971) · Zbl 0241.92001 · doi:10.1016/0040-5809(71)90002-5 |
| [16] | Si, J.G., Cheng, S.S.: Analytic solutions of a functional-differential equation with state dependent argument. Taiwan. J. Math. 4, 471–480 (1997) · Zbl 0892.30023 |
| [17] | Lê, U.V., Nguyen, L.T.T.: Existence of solutions for systems of self-referred and hereditary differential equations. Electron. J. Diff. Eqns. 2008(51), 1–7 (2008) |
| [18] | Lê, U.V., Pascali, E.: An existence theorem for self-referred and hereditary differential equations. Adv. Differ. Equ. Control Process 1, 25–32 (2008) · Zbl 1169.47060 |
| [19] | Volterra, V. : Opere Matematiche: Memorie e note, vol. V, 1926–1940. Accad. Naz. Lincei. Roma (1962) · Zbl 0099.00102 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
