## Asymptotic behavior of solutions of nonlinear Volterra equations and mean points.(English)Zbl 0985.45007

The author studies an asymptotic behavior at infinity of the solutions of the nonlinear Volterra equation $(V_{b,g,f}) u(t)+\int_0^t b(t-s)(Au(s)+g(s)u(s)) ds \ni f(t), \quad t\geq 0$ where $$b\in AC_{\text{loc}}(R^+;R)$$, $$b(0)=1$$; $$b^\prime\in BV_{\text{loc}}(R^+;R)$$; $$g\in C(R^+;R^+)$$; $$f\in W^{1,1}_{\text{loc}}(R^+;X)$$, $$f(0)\in \overline{D(A)}$$ and $$R^+=[0,\infty).$$ Here $$A$$ is an accretive operator in real reflexive Banach space $$X$$. Basing on the mean point, the weak and strong convergences for the “unbounded behavior” of solutions are given. The case $$V_{1,0,0}$$ was earlier considered from this point of view by W. Takahashi [J. Math. Anal. Appl. 109, 130-139 (1985; Zbl 0593.47057)].

### MSC:

 45M05 Asymptotics of solutions to integral equations 45G10 Other nonlinear integral equations

Zbl 0593.47057
Full Text:

### References:

 [1] Aizicovici, S.; Londen, S.O.; Reich, S., Asymptotic behavior of solutions to a class of nonlinear Volterra equations, Differential integral equations, 3, 813-825, (1990) · Zbl 0724.45017 [2] Baillon, J.B.; Clément, P., Ergodic theorems for nonlinear Volterra equations in Hilbert space, Nonlinear anal., 5, 789-801, (1981) · Zbl 0541.45009 [3] Clément, P., On abstract Volterra equations with kernels having a positive resolvent, Israel J. math., 36, 193-200, (1980) · Zbl 0451.45012 [4] Clément, P.; Nohel, J.A., Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. math. anal., 12, 514-535, (1981) · Zbl 0462.45025 [5] Crandall, M.G.; Nohel, J.A., An abstract functional differential equation and a related nonlinear Volterra equation, Israel J. math., 29, 313-328, (1978) · Zbl 0373.34035 [6] Crandall, M.G., Nonlinear semigroups and evolution governed by accretive operators, (), 305-338 · Zbl 0637.47039 [7] Day, M.M., Amenable semigroups, Illinois J. math., 1, 509-544, (1957) · Zbl 0078.29402 [8] Fan, K.; Glicksberg, I., Some geometric properties of the sphere in a normed linear space, Duke math. J., 25, 553-568, (1958) · Zbl 0084.33101 [9] Gripenberg, G., Volterra integro-differential equations with accretive nonlinearity, J. differential equations, 60, 57-79, (1985) · Zbl 0575.45013 [10] Hirano, N., Asymptotic behavior of solutions of nonlinear Volterra equations, J. differential equations, 47, 163-179, (1983) · Zbl 0462.45026 [11] Hulbert, D.S.; Reich, S., Asymptotic behavior of solutions to nonlinear Volterra integral equations, J. math. anal. appl., 104, 155-172, (1984) · Zbl 0589.45006 [12] Israel, M.M.; Reich, S., Asymptotic behavior of solutions of a nonlinear evolution equation, J. math. anal. appl., 83, 43-53, (1981) · Zbl 0508.47060 [13] Kato, N.; Kobayashi, K.; Miyadera, I., On the asymptotic behavior of solutions of evolution equations associated with nonlinear Volterra equations, Nonlinear anal., 9, 419-430, (1985) · Zbl 0581.47053 [14] Kato, N., On the asymptotic behavior of solutions of nonlinear Volterra equations, J. math. anal. appl., 120, 419-430, (1986) · Zbl 0581.47053 [15] Kato, N., Unbounded behavior and convergence of solutions of nonlinear Volterra equations in Banach spaces, Nonlinear anal., 12, 1193-1201, (1988) · Zbl 0722.47054 [16] Kobayasi, K., On the asymptotic behavior for a certain nonlinear evolution equation, J. math. anal. appl., 101, 555-561, (1984) · Zbl 0556.47033 [17] Miller, R.K., Nonlinear Volterra integral equations, Mathematics lecture notes series, (1971), Benjamin New York [18] Lakshmikantham, V.; Leela, S., Nonlinear differential equations in abstract spaces, International series in nonlinear mathematics, 2, (1981), Pergamon New York · Zbl 0456.34002 [19] Nohel, J.A., Nonlinear Volterra equations for heat flow in materials with memory, Integral and functional differential equations, Lecture notes in pure and applied mathematics, 67, (1981), Dekker New York/Basel, p. 3-82 [20] Reich, S., On the asymptotic behavior of nonlinear semigroups and the range of accretive operators, J. math. anal. appl., 79, 113-126, (1981) · Zbl 0457.47053 [21] S. Reich, Admissible pairs and integral equations, Preprint MT-680, Technion, Preprint Series, 1985. · Zbl 0615.45009 [22] Takahashi, W., The asymptotic behavior of nonlinear semigroups and invariant means, J. math. anal. appl., 109, 130-139, (1985) · Zbl 0593.47057
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.