Avoid common mistakes on your manuscript.
Dajun Guo was born in Sichuan province of China on January 23, 1934. He was an undergraduate student in Department of Mathematics of Sichuan University (1951–1955) and worked in 1956 with the position of assistant in Department of Mathematics of Shandong University. And then, he was promoted there to a lecturer (1960), an associate professor (1978) and a professor (1980). He was appointed as the Chairman of Department of Mathematics in Shandong University (1984–1989). As a visiting professor, D. Guo visited the University of Texas at Arlington (USA) (1985–1986), the University of Alberta (Canada) (1989–1990), and the University of Waterloo (Canada) (1994–1995). He has been the Editor-in-Chief of Journal of Shandong University (Natural Science Edition) (China), and has been a member of the editorial boards of Advances in Mathematics (China), Northeastern Mathematical Journal (China), Abstract and Applied Analysis (USA) and Dynamics of Continuous, Discrete and Impulsive Systems (Canada). He was the reviewer of Mathematical Reviews (USA). He was also the supervisor of twenty-six Ph. D. students of Department of Mathematics in Shandong University. His research area included nonlinear functional analysis, nonlinear integral equations and ordinary differential equations in Banach spaces.
Professor D. Guo was one of the three leaders of nonlinear functional analysis in China and they organized together the first, second and third congresses of nonlinear functional analysis in China (Beijing 1980, Chengdu 1981 and Xiamen 1982), and he gave main reports in these congresses. He attended the “International Congress of Nonlinear Analysis and Applications” (Arlington, Texas, USA, 1986), and “International Congress of Differential Equations and Applications” (Columbus, Ohio, USA, 1988), and gave a twenty minutes talk in each congress. He also attended the “Third World Congress of Nonlinear Analysis” (Catania, Italy, 2000) and gave an one hour report in the congress.
His research project “The Solutions of Nonlinear Integral Equations and Applications” received the second prize of science and technology of Shandong Province of China (1981), and his monograph “Nonlinear Functional Analysis” (Shandong Science and Technology Press, Jinan, China, 1985, in Chinese) received the second prize of science and technology progress of the Ministry of Education of China (1998). The research project “Nonlinear Analysis and Applications” (D. Guo, J. Sun, D. Zhou, Z. Liu, P. Ji) received the first prize of science and technology progress of Shandong Province of China (1998). The research project “Nonlinear Integral Equations and Applications” (D. Guo, J. Sun, C. Huang) and the research project “Partial Ordering Method in Nonlinear Analysis” (D. Guo, J. Sun, Y. Du, D. Zhou) both received the second prize of science and technology progress of the National Education Committee of China (1988 and 1993). The research project “Nonlinear Functional Analysis and Applications” (D. Guo, J. Sun, C. Huang, G. Qi, Z. Liu) received the second prize of science and technology progress of Shandong Province of China (1993), and the research project “Nonlinear Functional Analysis and Its Applications” (D. Guo, Z. Liu, F. Liang, B. Lou) received the second prize of natural science of Shandong Province of China (2002). The teaching project “Writing Textbooks and Teaching Courses on Functional Analysis” (D. Guo, C. Huang, F. Liang) received the excellent teaching prize of the National Education Committee of China (1989).
In 1980–2000, Professor D. Guo was interested to establish new fixed point theorems for nonlinear operators. In 1981, by using the Leray-Schauder degree theory, he established in [31] (see also [51]) a new fixed point theorem: fixed point theorem of domain expansion and compression. This theorem was useful for discussing the existence of non-zero solutions and multiple solutions for nonlinear operator equations and global structure of eigenvalues and eigenvectors for nonlinear operators. In particular, he has used it to make an improvement of a theorem about the eigenvalues of nonlinear operators due to J. Cronin in 1972 (see [32, 51]). Several authors called it “Guo’s theorem” in their published papers (see, for example, J. Sun, A generalization of Guo’s theorem and applications, J. Math. Anal. Appl., 126 (2) (1987), 566–573; D. Caponetti, A. Trombetta, G. Trombetta, An extension of Guo’s theorem via k-\(\psi \)-contractive retractions, Nonlinear Anal., 64 (9) (2006), 1897–1907.) and it has been selected into three monographs (W. Chen, Nonlinear Functional Analysis, Gansu People’s Press, Lanzhou (China), 1982 (in Chinese), p.287, Theorem 18.9; [57] p.133, Theorem 2.5.1; [140] p.128, Theorem 3.6). Moreover, D. Guo collaborated with L.H. Erbe and W. Krawcewicz on extending this theorem to multivalued maps with an application to second order differential inclusions (see [73]). For decreasing and completely continuous operators, he obtained in [51] the existence and uniqueness of a positive fixed point and the convergence of the iterative sequence to the unique positive fixed point. As an application to a nonlinear integral equation interesting in nuclear physics, he improved a result due to C.A. Stuart in 1971 (Stuart’s result did not contain the conclusion of convergence of the iterative sequence). For decreasing operators without complete continuity, he obtained the same result (see [76, 110]) by strengthening an inequality condition and using a quite different method to drop the condition of complete continuity. In 1987, D. Guo and V. Lakshmikantham proposed the concept of a mixed monotone operator and its coupled fixed points and obtained the existence of a coupled fixed point with minimax property and some iterative sequence converging to this coupled fixed point (see [54]). Many authors were interested in this concept and published many papers on this subject. Later, D. Guo obtained in papers [62] and [74] the existence and uniqueness of a positive fixed point and the convergence of the iterative sequence to the unique fixed point for a class of mixed monotone operators without the hypothesis on the continuity and compactness of the operators.
Professor D. Guo investigated nonlinear integral equations (especially, Hammerstein integral equations) for a long time (1960–2000). He used the topological degree theory, variational method and partial ordering method to discuss the existence of solutions, multiple solutions, existence of infinitely many solutions and applications to differential equations, and obtained a lot of new results (see [19, 20, 23,24,25,26,27,28,29,30, 36,37,38,39, 42,43,45, 49, 50, 52, 53, 56, 59,60,61, 66, 68, 70, 72, 75, 77, 79, 83, 86, 88, 90, 92, 138]). In paper [50], he used the critical point theory combining with the theory of linear integral equations to obtain the existence of infinitely many solutions of Hammerstein integral equations with applications to two-point boundary value problems for nonlinear ordinary differential equations. This result improved a result due to A. Ambrosetti and P.H. Rabinowitz in 1973 (that result could not apply to the two-point boundary value problems for nonlinear ordinary differential equations because the Green function did not satisfy the related conditions). By using the topological degree theory and partial ordering method, he investigated in papers [26, 29, 38] and [39] the sublinear, superlinear and mixed Hammerstein integral equations and obtained the existence of one nontrivial nonnegative solution and the existence of two nontrivial nonnegative solutions, respectively, and applied these results to the Dirichlet problems of elliptic partial differential equations. He also used the fixed point index theory to obtain the existence of two nontrivial nonnegative solutions for impulsive nonlinear Fredholm integral equations with applications to two-point boundary value problems for impulsive nonlinear ordinary differential equations (see [75]). In an early paper [23], he improved a result due to M.A. Krasnoselskii in 1956 on the existence of solutions to a class of Hammerstein integral equations by using the variational method and applied to a boundary value problem of elliptic partial differential equations, and in a recent paper [138], he obtained the existence of infinitely many solutions for a class of impulsive differential equations by using critical point theory to the equivalent Hammerstein integral equations. On the other hand, he collaborated with V. Lakshmikantham on investigation of a nonlinear integral equation arising in infectious diseases and obtained the existence of one nontrivial nonnegative periodic solution and the existence of two nontrivial nonnegative periodic solutions (see [53, 61]), which improved a result due to L.R. Williams and R.W. Leggett in 1982 on the existence of one nontrivial nonnegative periodic solution. In another paper [28], he discussed a Hammerstein integral equation in neutron transport theory, which was investigated by A. Pazy and P.H. Rabinowitz in 1969 This equation contained several physical parameters. A. Pazy and P.H. Rabinowitz proved that this equation has an unique nontrivial solution. D. Guo proved that this unique nontrivial solution continuously depended on these physical parameters, so it was a complementary of the result due to A. Pazy and P.H. Rabinowitz.
In recent years, Professor D. Guo was interested in impulsive nonlinear integro-differential equations in Banach spaces. This was a new area of investigation and there were only a few results. He used method of nonlinear functional analysis (topological degree theory, partial ordering method, etc.) combined with technique of differential equations (comparison results, upper and lower solutions, etc.) to investigate the solutions of first-order, second order, third-order and in general, nth-order impulsive nonlinear integro-differential equations in Banach spaces, and obtained a series of basic and new results (see [78, 87, 89, 91, 94,95,96,97,98,99,100,101,102,103,104,105,106, 115,116,117,118,119, 123,124,125, 128, 129, 131,132,134, 136, 141]). These results made great contributions to the establishment of the general theory in this area. In paper [115], he established the general idea and some fundamental tools for discussing the existence of solutions of nth-order nonlinear impulsive integro-differential equations in Banach spaces. First, he selected the space of piecewise differentiable abstract functions as the basic space for discussion, and then, established the Taylor formula in this space ( [115] Lemma 2), which enabled us to reduce the initial value problem of an nth-order impulsive integro-differential equation in a Banach space to an impulsive integral equation in the basic space, and so, reduce to find a fixed point of the corresponding operator in the basic space. Next, he established a formula about the Kuratowski measure of non-compactness of bounded sets in the basic space ( [115], Lemma 7), which enabled us to prove the compactness of the corresponding operator in the basic space. Finally, the Schauder fixed point theorem implied that the corresponding operator has a fixed point in the basic space, and therefore, the original initial value problem had an solution. In addition, the construction of an example ( [115], Example 1) as an application of the obtained result to an infinite system of scalar impulsive integro-differential equations is difficult and interesting. In three papers [125, 128] and [129], he discussed three boundary value problems for nth-order impulsive nonlinear integro-differential equations in Banach spaces. By using the basic space and the Taylor formula and the formula about the measure of non-compactness in the basic space, which were established in [115], and by applying the fixed point index theory, he obtained the existence of two positive solutions for each of the three boundary value problems. Of course, the three equivalent impulsive integral equations, and so, the three corresponding operators were different from each other, which were also different from that in [115]. In paper [119], he used a completely different method to deal with an initial value problem of an nth-order impulsive nonlinear integro-differential equation in a Banach space. The key was to establish a comparison result in the basic space, which enabled us to construct two monotone iterative convergent sequences of abstract functions such that the two limit abstract functions were the minimal and maximal solutions of the original initial value problems between a lower solution and an upper solution. In papers [133] and [134], he investigated two boundary value problems for nth-order impulsive nonlinear singular integro-differential equations in a Banach space. In order to overcome the difficulty arising from the singularities, he constructed a bounded closed convex set in the basic space for each boundary value problem apart from the singularities such that the corresponding operator mapped this bounded closed convex set into itself, and then, used the Schauder fixed point theorem. In another paper [141], he discussed the multiple solutions of a boundary value problem for a third-order impulsive nonlinear singular integro-differential equation in a Banach space. Instead of using the Schauder fixed point theorem, he used the fixed point theorem of cone expansion and compression with norm type, which was established by himself in [40] (see also [55]), and obtained the existence of two positive solutions for the boundary value problem. The key was to construct a completely new cone such that we could apply the fixed point theorem of cone expansion and compression with norm type twice apart from the singularities.
References
Guo, D.: Properties of Stieltjes integral of order 2. J. Sichuan Univ. (Natural Science Edition). (1), 21–32 (1955) (In Chinese)
Guo, D.: Uniformly complete continuity and norml approximation J. Sichuan Univ. (Natural Science Edition). (1), 79–96 (1957) (In Chinese)
Guo, D.: Sufficient conditions of uniformly complete continuity. J. Sichuan Univ. (Natural Science Edition). (2), 197–211 (1957) (In Chinese)
Guo, D.: On the existence and uniqueness and convergence of solutions for infinitely many linear operator equations. J. Shandong Univ. (Natural Science Edition). (1), 129–140 (1957) (In Chinese)
Guo, D.: On dimensions of ranges of linear operators. J. Shandong Univ. (Natural Science Edition). (1), 141–143 (1957) (In Chinese)
Guo, D.: On the factor splitting of integral kernels. J. Shandong Univ. (Natural Science Edition). (1), 69–74 (1958) (In Chinese)
Guo, D.: Some theorems of uniformly complete continuity. J. Sichuan Univ. (Natural Science Edition). (1), 1–6 (1958) (In Chinese)
Guo, D.: On \(B^*\)-compactness and \(B\)-compactness of sets in Orlicz spaces. J. Sichuan Univ. (Natural Science Edition). (1), 37–62 (1958) (In Chinese)
Guo, D.: On stability of solutions of linear integral equations. Scientia Sinica 8(4), 331–356 (1959). (In Russian)
Guo, D.: Stability of solutions for Volterra integral equations. J. Shandong Univ. (Natural Science Edition). (1), 107–112 (1959) (In Chinese)
Guo, D.: Covexity properties of space \(H^p\). J. Shandong Univ. (Natural Science Edition). (3), 16–23 (1959) (In Chinese)
Guo, D.: Approximate solution of functional equation (I). J. Shandong Univ. (Natural Science Edition). (1), 67–80 (1960) (In Chinese)
Guo, D.: Approximate solution of functional equation (II). J. Shandong Univ. (Natural Science Edition). (3), 34–51 (1960) (In Chinese)
Guo, D.: Compactness of sets in Orlicz spaces. J. Shandong Univ. (Natural Science Edition). (3), 1–24 (1961). (In Chinese)
Guo, D.: Some properties of the Sobolev space \(W^{(l)}_p\). J. Shandong Univ. (Natural Science Edition). (4), 1–20 (1961) (In Chinese)
Guo, D.: On the uniformly complete continuity of sets of operators. J. Shandong Univ. (Natural Science Edition). (2), 1–25 (1962) (In Chinese)
Guo, D.: Complete continuity of Urison operator. Scientia Sinica 11(4), 437–452 (1962) (In Russian)
Guo, D.: Further discussion for complete continuity of Urison operator. J. Shandong Univ. (Natural Science Edition). (2), 26–39 (1963) (In Chinese)
Guo, D.: Properties of Nemiskii operator and applications. Adv. Math. (China) 6(1), 70–91 (1963) (In Chinese)
Guo, D.: Some results of nonlinear integral equations and integral operators. J. Shandong Univ. (Natural Science Edition). (3), 8–21 (1964) (In Chinese)
Guo, D.: On tangent hyperbola process and Chebisev process. J. Shandong Univ. (Natural Science Edition). (4), 12–28 (1964) (In Chinese)
Guo, D.: Differentiability of Urison operator. J. Shandong Univ. (Natural Science Edition) (1), 1–6 (1965) (In Chinese)
Guo, D.: Solvability of Hammerstein nonlinear integral equations. Acta Math. Sinica 16(2), 137–149 (1966). (In Chinese)
Guo, D.: Proper values of Hammerstein nonlinear integral equations. Acta Math. Sinica 20(2), 99–108 (1977). (In Chinese)
Guo, D.: The solution of a nonlinear integral equation in nuclear physics. Kexue Tongbao 23(1), 27–31 (1978).(In Chinese)
Guo, D.: Nontrivial solutions of Hammerstein nonlinear integral equations. Kexue Tongbao 24(5), 193–197 (1979). (In Chinese)
Guo, D., Zhang, Q.: On the uniqueness of the solution of a nonlinear integral equation in nuclear physics. Kexue Tongbao 24(15), 678–681 (1979). (In Chinese)
Guo, D.: The solution for a nonlinear integral equation in the neutron transport theory. Acta Math. Sinica 22(2), 231–236 (1979). (In Chinese)
Guo, D.: The number of positive solutions of Hammerstein nonlinear integral equations. Acta Math. Sinica 22(5), 584–595 (1979). (In Chinese)
Guo, D.: Eigenvalues and eigenvectors of Hammerstein integral equations with degenerate kernels. J. Shandong Univ. (Natural Science Edition) 2, 20–28 (1979). (In Chinese)
Guo, D.: A new fixed-point theorem. Acta Math. Sinica 24(3), 444–450 (1981). (In Chinese)
Guo, D.: Eigenvalues and eigenfunctions of nonlinear operators. Chin. Ann. Math. (English Edition) 2(4), 65–80 (1981)
Guo, D.: Eigenvalues and eigenvectors of nonlinear completely continuous operators. J. Shandong Univ. (Natural Science Edition). (1), 7–15 (1981). (In Chinese)
Guo, D.: The positive spectrum of the limit operator of some nonlinear operators. J. Shandong Univ. (Natural Science Edition). (3), 1–19 (1981). (In Chinese)
Guo, D.: On the fixed point theorem of cone expansion and compression. Kexue Tongbao (English Edition) 27(6), 685 (1982)
Guo, D.: Positive solutions of Lyapunov-Lichtenstein nonlinear integral equation. J. Shandong Univ. (Natural Science Edition). (3), 1–11 (1982). (In Chinese)
Guo, D.: Eigenvalues and eigenfunctions of Hammerstein integral equations. Acta Math. Sinica 25(4), 419–426 (1982). (In Chinese)
Guo, D.: The number of nontrivial solutions of Hammerstein integral equations and their applications. Kexue Tongbao (English Edition) 27(7), 694–698 (1982)
Guo, D.: Positive solutions of Hammerstein integral equations of polynomial type with applications. Chin. Ann. Math. Ser. A 4(5), 645–656 (1983). (In Chinese. An English summary appears in: Chinese Ann. Math. Ser. B, 4(4) (1983), 529–530.)
Guo, D.: On some fixed point theorems for cone mappings. Kexue Tongbao (Chinese Edition) 28(20), 1217–1219 (1983). (English Edition, 29(5) (1984), 575–578)
Guo, D.: A fixed point theorem for decreasing operators with applications. Kexue Tongbao (Chinese Edition) 29(3), 189 (1984)
Guo, D.: The number of nontrivial solutions of nonlinear two-point boundary value problems. J. Math. Res. Exposit. 4(1), 55–60 (1984)
Guo, D.: The number of positive solutions of a class of two-point boundary value problems. J. Shandong Univ. (Natural Science Edition). (Special issue), 53–60 (1984) (In Chinese)
Guo, D.: On the solutions of a nonliear integral equation in neutron transport theory. J. Shandong Univ. (Natural Science Edition). (Special issue), 61–75 (1984) (In Chinese)
Guo, D.: Positive solutions to nonlinear operator equations with applications to nonlinear integral equations. Adv. Math. (China) 13(4), 294–310 (1984). (In Chinese)
Guo, D.: On the existence and uniqueness of positive fixed points of decreasing operaors. J. Shandong Univ. (Natural Science Edition). (1), 1–8 (1985) (In Chinese)
Guo, D.: Positive eigenvectors of decreasing operators and applications. Northeast. Math. J. 1(1), 101–109 (1985)
Guo, D.: Fixed points and eigenvectors of concave and convex operators. Kexue Tongbao 30(15), 1132–1135 (1985). (In Chinese)
Guo, D.: Solutions of Hammerstein nonlinear integral equations and applications. J. Qufu Normal Univ. (4), 1–6 (1985) (In Chinese)
Guo, D.: The number of nontrivial solutions to Hammerstein nonlinear integral equations. Chin. Ann. Math. Ser. B 7(2), 191–204 (1986). (A Chinese summary appears in: Chin. Ann. Math. Ser. A., 7(2) (1986), 239.)
Guo, D.: Some fixed point theorems and applications. Nonlinear Anal. 10(11), 1293–1302 (1986)
Guo, D.: Multiple positive solutions of nonlinear integral equations and applications. Appl. Anal. 23(1–2), 77–84 (1986)
Guo, D.: The number of positive solutions of a nonlinear integral equation modeling infectious disease. Chin. Ann. Math. Ser. A 8(1), 27–31 (1987). (In Chinese. An English summary appears in: Chin. Ann. Math. Ser. B, 8)
Guo, D., Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11(5), 623–632 (1987)
Guo, D.: Some fixed point theorems of expansion and compression type with applications, Nonlinear Analysis and Applications (Arlington, Tex., : edited by V. Lakshmikantham, Lecture Notes in Pure and Appl. Math., 109, Marcel Dekker Inc., New York 1987, 213–221 (1986)
Guo, D., Sun, J.: Nonlinear Integral Equations. Shandong Sci. Tech. Press, Jinan (China) (1987). (In Chinese)
Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering, 5. Academic Press Inc, Boston (1988)
Guo, D., Sun, J.: Some global generalizations of the Birkhoff-Kellogg theorem and applications. J. Math. Anal. Appl. 129(1), 231–242 (1988)
Guo, D.: Multiple solutions of nonlinear Fredholm integral equations in Banach spaces, Differential Equations and Applications (Columbus, OH, : edited by A. R. Aftabizadeh, Ohio Univ. Press, Athens. OH (USA) 1989, 371–375 (1988)
Guo, D., Lakshmikantham, V.: Multiple solutions of two-point boundary value problems of ordinary differential equations in Banach spaces. J. Math. Anal. Appl. 129(1), 211–222 (1988)
Guo, D., Lakshmikantham, V.: Positive solutions of nonlinear integral equations arising in infectious diseases. J. Math. Anal. Appl. 134(1), 1–8 (1988)
Guo, D.: Fixed points of mixed monotone operators with applications. Appl. Anal. 31(3), 215–224 (1988)
Guo, D., Sun, J., Qi, G.: Some extensions of the mountain pass lemma. Differ. Integral Equ. 1(3), 351–358 (1988)
Guo, D., Sun, J.: Evaluation of topological degree and applications. J. Math. Res. Exposit. 8(3), 469–480 (1988). (In Chinese)
Guo, D.: Solutions of nonlinear integro-differential equations of mixed type in Banach spaces. J. Appl. Math. Simul. 2(1), 1–11 (1989)
Guo, D.: Multiple solutions of nonlinear Fredholm integral equations in Banach spaces. Chin. Ann. Math. Ser. B 10(4), 513–519 (1989). (A Chinese summary appears in: Chin. Ann. Math. Ser. A, 10 (5) (1989), 641)
Guo, D.: Fixed point theorem for domain expansion and compression. J. Qufu Normal Univ. 15(4), 1–7 (1989). (In Chinese)
Guo, D.: Extremal solutions of nonlinear Fredholm integral equations in ordered Banach spaces. Northeast. Math. J. 7(4), 416–423 (1991)
Guo, D.: Non-negative solutions of two-point boundary value problems for nonlinear second order integro-differential equations in Banach spaces. J. Appl. Math. Stochastic Anal. 4(1), 47–69 (1991)
Guo, D.: Impulsive integral equations in Banach spaces and applications. J. Appl. Math. Stochastic Anal. 5(2), 111–122 (1992)
Erbe, L.H., Guo, D.: Periodic boundary value problems for second order integro-differential equations of mixed type. Appl. Anal. 46(3–4), 249–258 (1992)
Erbe, L.H., Guo, D., Liu, X.: Positive solutions of a class of nonlinear integral equations and applications. J. Integral Equ. Appl. 4(2), 179–195 (1992)
Erbe, L.H., Krawcewicz, W., Guo, D.: Some fixed point theorems for multivalued maps and applications. Results Math. 21(1–2), 42–64 (1992)
Guo, D.: Existence and uniqueness of positive fixed points for mixed monotone operators and applications. Appl. Anal. 46(1–2), 91–100 (1992)
Guo, D.: Multiple positive solutions of impulsive nonlinear Fredholm integral equations and applications. J. Math. Anal. Appl. 173(1), 318–324 (1993)
Guo, D.: Positive fixed points and eigenvectors of noncompact decreasing operators with applications to nonlinear integral equations. Chin. Ann. Math. Ser. B 14(4), 419–426 (1993). (A Chinese summary appears in: Chin. Ann. Math. Ser. A, 14(6) (1993), 739)
Guo, D.: Nonlinear impulsive Volterra integral equations in Banach spaces and applications. J. Appl. Math. Stochastic Anal. 6(1), 35–48 (1993)
Guo, D., Liu, X.: Extremal solutions of nonlinear impulsive integro-differential equations in Banach spaces. J. Math. Anal. Appl. 177(2), 538–552 (1993)
Guo, D.: Existence of solutions of boundary value problems for nonlinear second order impulsive differential equations in Banach spaces. J. Math. Anal. Appl. 181(2), 407–421 (1994)
Erbe, L.H., Guo, D.: Method of upper and lower solutions for nonlinear integro-differential equations of mixed type in Banach spaces. Appl. Anal. 52(1–4), 143–154 (1994)
Guo, D.: Initial value problems for integro-differential equations of Volterra type in Banach spaces. J. Appl. Math. Stochastic Anal. 7(1), 13–23 (1994)
Guo, D., Sun, J.: Some questions in the theory of ordinary differential equations in Banach spaces. Adv. Math. (China) 23(6), 492–504 (1994). (In Chinese)
Guo, D.: Proper functions of nonlinear Fredholm integral equations in Banach spaces. In: Guo, D. (ed.) Nonlinear Analysis and Applications, pp. 1–11. Beijing Sci. Tech. Press, Beijing (China) (1994). (In Chinese)
Guo, D., Erbe, L.H., Krawcewicz, W. : Positive solutions of two-point boundary value problems for nonlinear integro-differential equations in Banach spaces. Differ. Equ. Dyn. Syst. 2(3), 161–171 (1994)
Guo, D.: Extreme solutions of nonlinear second order integro-differential equations in Banach spaces. J. Appl. Math. Stochastic Anal. 8(3), 319–329 (1995)
Guo, D., Liu, X.: Multiple positive solutions of boundary-value problems for impulsive differential equations. Nonlinear Anal. 25(4), 327–337 (1995)
Liu, X., Guo, D.: Initial value problems for first order impulsive integro-differential equations in Banach spaces. Commun. Appl. Nonlinear Anal. 2(1), 65–83 (1995)
Liu, X., Guo, D.: Extremal solutions of nonlinear impulsive Fredholm integral equations in Banach spaces and applications. Differ. Equ. Dyn. Syst. 3(1), 77–88 (1995)
Guo, D.: Initial value problems for nonlinear second order impulsive integro-differential equations in Banach spaces. J. Math. Anal. Appl. 200(1), 1–13 (1996)
Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications, 373. Kluwer Academic Publishers Group, Dordrecht (1996)
Guo, D., Liu, X.: Impulsive integro-differential equations on unbounded domain in a Banach space. Nonlinear Stud. 3(1), 49–57 (1996)
Guo, D.: On global solutions of nonlinear integral equations in Banach spaces. Indian J. Pure Appl. Math. 27(1), 57–71 (1996)
Guo, D.: Some results on nonlinear functional analysis. In: Li, B., et al. (eds.) Functional Analysis in China, Math. Appl., 356, pp. 53–70. Kluwer Academic Publishers Group, Dordrecht (1996)
Guo, D., Liu, X.: First order impulsive integro-differential equations on unbounded domain in a Banach space. Dyn. Contin. Discrete Impuls. Syst. 2(3), 381–394 (1996)
Guo, D., Liu, X.: Periodic boundary value problems for impulsive integro-differential equations in Banach spaces. Nonlinear World 3(3), 427–441 (1996)
Guo, D.: Periodic boundary value problems for second order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 28(6), 983–997 (1997)
Guo, D.: Initial value problems for second order impulsive integro-differential equations in Banach spaces. Chin. Ann. Math. Ser. B 18(4), 439–448 (1997). (A Chinese summary appears in: Chin. Ann. Math. Ser. A, 18(5) (1997), 665)
Guo, D.: Terminal value problems of impulsive integro-differential equations in Banach spaces. J. Appl. Math. Stochastic Anal. 10(1), 71–78 (1997)
Guo, D.: Existence and Uniqueness of Solutions of Impulsive Integro-Differential Equations in Banach Spaces, Advances in Nonlinear Dynamics, 5, pp. 245–251. Gordon and Breach, Amsterdam (1997)
Liu, X., Guo, D.: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces. J. Math. Anal. Appl. 216(1), 284–302 (1997)
Guo, D.: Minimal nonnegative solutions for impulsive integro-differential equations on unbounded domains in Banach spaces. Indian J. Pure Appl. Math. 29(1), 1–18 (1998)
Liu, X., Guo, D.: Periodic boundary value problems for impulsive integro-differential equations of mixed type in Banach spaces. Chin. Ann. Math. Ser. B 19(4), 517–528 (1998). (A Chinese summary appears in: Chin. Ann. Math. Ser. A, 19(5) (1998), 658)
Guo, D.: Boundary value problems for impulsive integro-differential equations on unbounded domains in a Banach space. Appl. Math. Comput. 99(1), 1–15 (1999)
Guo, D.: Second order impulsive integro-differential equations on unbounded domains in Banach spaces. Nonlinear Anal. Ser. A: Theory Methods 35(4), 413–423 (1999)
Guo, D.: Initial value problems for second-order integro-differential equations in Banach spaces. Nonlinear Anal. Ser. A: Theory Methods 37(3), 289–300 (1999)
Liu, X., Guo, D.: Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space. Comput. Math. Appl. 38(3–4), 213–223 (1999)
Guo, D.: Integro-differential equations on unbounded domains in Banach spaces. Chin. Ann. Math. Ser. B 20(4), 435–446 (1999)
Guo, D.: The method of partial ordering in nonlinear analysis. J. Ningxia Univ. Nat. Sci. Ed. 20(1), 5–11 (1999). (In Chinese)
Guo, D.: Second order integro-differential equations of Volterra type on unbounded domains in Banach spaces. Nonlinear Anal. Ser. A: Theory Methods 41(3–4), 465–476 (2000)
Guo, D.: Existence and uniqueness of positive fixed points for noncompact decreasing operators. Indian J. Pure Appl. Math. 31(5), 551–562 (2000)
Guo, D.: Minimal nonnegative solutions for nth order integro-differential equations in Banach spaces. Appl. Math. Comput. 113(1), 55–65 (2000)
Guo, D.: Nonlinear functional analysis and applications. Sci. Found. China 8(1), 50–53 (2000)
Guo, D.: Integro-differential equations on unbounded domains in Banach spaces, Integral and integrodifferential equations, Ser. Math. Anal. Appl., 2, Gordon and Breach, Amsterdam, 117–124 (2000)
Guo, D.: Existence of solutions for nth-order integro-differential equations in Banach spaces. Comput. Math. Appl. 41(5–6), 597–606 (2001)
Guo, D.: Existence of solutions for nth order impulsive integro-differential equations in a Banach space. Proceedings of the Third World Congress of Nonlinear Analysis, Part 2 (Catania, 2000). Nonlinear Anal., 47(2), 741–752 (2001)
Guo, D.: Initial value problems of first order impulsive integro-differential equations on unbounded domain in a Banach space. Commun. Appl. Anal. 5(4), 523–533 (2001)
Guo, D.: A class of second-order impulsive integro-differential equations on unbounded domain in a Banach space. Appl. Math. Comput. 125(1), 59–77 (2002)
Guo, D., Liu, X.: A boundary value problem for first order impulsive integro-differential equations in Banach spaces. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 9(3), 449–462 (2002)
Guo, D.: A class of nth-order impulsive integrodifferential equations in Banach spaces. Comput. Math. Appl. 44(10–11), 1339–1356 (2002)
Guo, D., Sun, J.: Ordinary Differential Equations in Abstract Spaces, 2nd edn. Shandong Sci. Tech Press, Jinan (2002). (In Chinese)
Guo, D.: Multiple positive solutions for first order nonlinear integro-differential equations in Banach spaces. Nonlinear Anal. 53(2), 183–195 (2003)
Guo, D.: A boundary value problem for nth order integro-differential equations in a Banach space. Appl. Math. Comput. 136(2–3), 571–592 (2003)
Guo, D.: Multiple positive solutions for first order nonlinear impulsive integro-differential equations in a Banach space. Appl. Math. Comput. 143(2–3), 233–249 (2003)
Guo, D., Liu, X.: A boundary value problem for second order impulsive integro-differential equations in a Banach space. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10(6), 1001–1016 (2003)
Guo, D.: Multiple positive solutions of a boundary value problem for n th-order impulsive integro-differential equations in a Banach space. Nonlinear Anal. 56(7), 985–1006 (2004)
Guo, D.: Multiple positive solutions for a class of nonlinear integro-differential equations in Banach spaces. Appl. Math. Comput. 154(2), 469–485 (2004)
Guo, D., Cho, Y.J., Zhu, J.: Partial Ordering Methods in Nonlinear Problems. Nova Science Publishers Inc, New York (2004)
Guo, D.: Multiple positive solutions for nth-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 60(5), 955–976 (2005)
Guo, D.: Multiple positive solutions of a boundary value problem for n th-order impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 63(4), 618–641 (2005)
Guo, D., Sun, J., Liu, Z.: Functional Analysis Methods in Nonlinear Ordinary Differential Equations, 2nd edn. Shandong Sci. Tech Press, Jinan (2005). (In Chinese)
Guo, D.: Extremal solutions for nth-order impulsive integro-differential equations on the half-line in Banach spaces. Nonlinear Anal. 65(3), 677–696 (2006)
Guo, D., Liu, X.: Extremal solutions for a boundary value problem of nth-order impulsive integro-differential equations in a Banach space. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13(5), 599–619 (2006)
Guo, D.: Existence of positive solutions for n th-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal. 68(9), 2727–2740 (2008)
Guo, D.: Positive solutions of an infinite boundary value problem for n th-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Anal. 70(5), 2078–2090 (2009)
Guo, D.: Multiple positive solutions for first order impulsive superlinear integro-differential equations on the half line. Acta Math. Sci. Ser. B (English Edition) 31(3), 1167–1178 (2011)
Guo, D.: Multiple positive solutions for first-order impulsive integrodifferential equations on the half line in Banach spaces, Inter. J. Differ. Equ., 2011, Art. ID 634612, 22 pp
Guo, D.: Multiple positive solutions for first order impulsive singular integro-differential equations on the half line. Acta Math. Sci. Ser. B (English Edition) 32(6), 2176–2190 (2012)
Guo, D.: Variational approach to a class of impulsive differential equations. Bound. Value Probl. 2014, 37 (2014)
Guo, D.: Existence of two positive solutions for a class of second order impulsive singular integro-differential equations on the half line. Bound. Value Probl. 2015, 76 (2015)
Guo, D.: Nonlinear Functional Analysis, 3rd edn. Higher Edu Press, Beijing (2015). (In Chinese)
Guo, D.: Existence of two positive solutions for a class of third-order impulsive singular integro-differential equations on the half-line in Banach spaces. Bound. Value Probl. 2016, 70 (2016)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
This article is part of the topical collection dedicated to Prof. Dajun Guo for his 85th birthday, edited by Yihong Du, Zhaoli Liu, Xingbin Pan, and Zhitao Zhang.
Rights and permissions
About this article
Cite this article
Sun, J., Pan, XB., Du, Y. et al. Professor Dajun Guo: a true mathematician and educator. SN Partial Differ. Equ. Appl. 1, 37 (2020). https://doi.org/10.1007/s42985-020-00037-9
Published:
DOI: https://doi.org/10.1007/s42985-020-00037-9