• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.151-155
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• 1990

Recently, H.Z.Ming [7] obtained a necessary and sufficient condition for the existence of a solution to a general operator equation. In the present paper, we obtain such conditions in general forms and give some examples. We begin with the well-known Fan-Browder fixed point theorem, from which we deduce two general theorems on such necessary and sufficient conditiions. We give some examples of such conditions, which are improved versions of fixed point theorems of Halpern-Bergman [5], Ky Fan [3], [4], Kaczynski [6], Reich [9], Schauder [10], Tychonoff [11], and Ming [7]. In fact, we restate Ming's result in its correct form. The following is known as the Fan-Browder fixed point theorem [1], [2].

• Journal of the Chungcheong Mathematical Society
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• v.4 no.1
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• pp.39-44
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• 1991

In this note, we shall prove a new variational inequality in non-compact sets and as an application, we prove a generalization of the Schauder-Tychonoff fixed point theorem.

• Honam Mathematical Journal
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• v.26 no.4
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• pp.441-453
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• 2004

In this paper, sufficient conditions for the controllability of integrodifferential systems are established. The results are obtained by using the Schauder fixed point theorem and the resolvent operator. Example is provided to illustrate the theory.

• East Asian mathematical journal
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• v.32 no.5
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• pp.621-633
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• 2016

We establish the existence of positive solutions to nonlocal boundary value problems with integral boundary condition and non-negative real boundary parameter by mainly using the Schauder-Fixed point theorem.

• Korean Journal of Mathematics
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• v.16 no.1
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• pp.91-101
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• 2008

Let H be a Hilbert space. Assume that $0{\leq}{\alpha}$, ${\beta}{\leq}1$ and r(t) > 0 in I = [0, T]. By means of the fixed point theorem of Leray-Schauder type the existence principles of solutions for two point boundary value problems of the form $\array{(r(t)x^{\prime}(t))^{\prime}+f(t,x(t),r(t)x^{\prime}(t))=0,\;t{\in}I\\x(0)=x(T)=0}$ are established where f satisfies for positive constants a, b and c ${\mid}{f(t,x,y){\mid}{\leq}a{\mid}x{\mid}^{\alpha}+b{\mid}y{\mid}^{\beta}+c\;\;for\;all(t,x,y){\in}I{\times}H{\times}H$.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1361-1370
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• 2009

In this paper, we consider singular fourth-order two point boundary value problems $u^{(4)}$ (t) = f(t, u), 0 < t < 1, u(0) = u(l) = u'(0) = u'(l) = 0, where $f:(0,1){\times}(0,+{\infty}){\rightarrow}[0,+{\infty})$ may be singular at t = 0, 1 and u = 0. By using the upper and lower solution method, we obtained the existence of positive solutions to the above boundary value problems. An example is also given to illustrate the obtained theorems.

• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.83-98
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• 2024

In this paper, we investigate the existence and uniqueness of solutions to a new class of integro-differential equation boundary value problems (BVPs) with ㄒ-Hilfer operator. Our problem is converted into an equivalent fixed-point problem by introducing an operator whose fixed points coincide with the solutions to the given problem. Using Banach's and Schauder's fixed point techniques, the uniqueness and existence result for the given problem are demonstrated. The stability results for solutions of the given problem are also discussed. In the end. One example is provided to demonstrate the obtained results

• Journal of the Korean Mathematical Society
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• v.35 no.4
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• pp.803-829
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• 1998

We give general fixed point theorems for compact multimaps in the "better" admissible class $B^{K}$ defined on admissible convex subsets (in the sense of Klee) of a topological vector space not necessarily locally convex. Those theorems are used to obtain results for $\Phi$-condensing maps. Our new theorems subsume more than seventy known or possible particular forms, and generalize them in terms of the involving spaces and the multimaps as well. Further topics closely related to our new theorems are discussed and some related problems are given in the last section.n.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.281-292
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• 2005

In this paper a nonlinear alternative of Leray-Schauder type is proved in a Banach algebra involving three operators and it is further applied to a functional nonlinear integral equation of mixed type $$x(t)=k(t,x({\mu}(t)))+[f(t,x({\theta}(t)))]\(q(t)+{\int}_0{^{\sigma}^{(t)}}v(t,s)g(s,x({\eta}\(s)))ds\)$$ for proving the existence results in Banach algebras under generalized Lipschitz and $Carath{\acute{e}}odory$ conditions.

• Journal of the Korean Mathematical Society
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• v.36 no.2
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• pp.355-367
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• 1999

We prove the existence of 2N-1 distinct ordered positive solutions of a class of semilinear elliptic Dirichlet boundary value problems on Rn when the forcing term has N distinct positive stable zeros and the coefficient function decaying to the zero at infinity.